User:Kc kennylau/sandbox
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Theorem
Let $\struct {R, +, \circ}$ be a ring with unity $1_R$.
Let $\paren {-1}_R$ be the additive inverse of $1_R$.
Then:
- $\paren {-1}_R \circ \paren {-1}_R = 1_R$
Proof
Let $0_R$ be the ring zero of $\left({R, +, \circ}\right)$.
Then:
| \(\ds \paren {-1}_R \circ \paren {-1}_R\) | \(=\) | \(\ds 0_R + \paren {\paren {-1}_R \circ \paren {-1}_R}\) | Ring Axiom $\text A3$: Identity for Addition | |||||||||||
| \(\ds \) | \(=\) | \(\ds \paren {1_R + \paren {-1}_R} + \paren {\paren {-1}_R \circ \paren {-1}_R}\) | Ring Axiom $\text A4$: Inverses for Addition | |||||||||||
| \(\ds \) | \(=\) | \(\ds 1_R + \paren {\paren {-1}_R + \paren {\paren {-1}_R \circ \paren {-1}_R} }\) | Ring Axiom $\text A1$: Associativity of Addition | |||||||||||
| \(\ds \) | \(=\) | \(\ds 1_R + \paren {\paren {\paren {-1}_R \circ 1_R} + \paren {\paren {-1}_R \circ \paren {-1}_R} }\) | Ring Axiom $\text M2$: Identity Element for Ring Product | |||||||||||
| \(\ds \) | \(=\) | \(\ds 1_R + \paren {\paren {-1}_R \circ \paren {1_R + \paren {-1}_R} }\) | Ring Axiom $\text D$: Distributivity of Product over Addition | |||||||||||
| \(\ds \) | \(=\) | \(\ds 1_R + \paren {\paren {-1}_R \circ 0_R}\) | Ring Axiom $\text A4$: Inverses for Addition | |||||||||||
| \(\ds \) | \(=\) | \(\ds 1_R + \paren {\paren {\paren {-1}_R \circ 0_R} + 0_R}\) | Ring Axiom $\text A3$: Identity for Addition | |||||||||||
| \(\ds \) | \(=\) | \(\ds 1_R + \paren {\paren {\paren {-1}_R \circ 0_R} + \paren {\paren {-1}_R + 1_R} }\) | Ring Axiom $\text A4$: Inverses for Addition | |||||||||||
| \(\ds \) | \(=\) | \(\ds 1_R + \paren {\paren {\paren {\paren {-1}_R \circ 0_R} + \paren {-1}_R} + 1_R}\) | Ring Axiom $\text A1$: Associativity of Addition | |||||||||||
| \(\ds \) | \(=\) | \(\ds 1_R + \paren {\paren {\paren {\paren {-1}_R \circ 0_R} + \paren {\paren {-1}_R \circ 1_R} } + 1_R}\) | Ring Axiom $\text M2$: Identity Element for Ring Product | |||||||||||
| \(\ds \) | \(=\) | \(\ds 1_R + \paren {\paren {\paren {-1}_R \circ \paren {0_R + 1_R} } + 1_R}\) | Ring Axiom $\text D$: Distributivity of Product over Addition | |||||||||||
| \(\ds \) | \(=\) | \(\ds 1_R + \paren {\paren {\paren {-1}_R \circ 1_R} + 1_R}\) | Ring Axiom $\text A3$: Identity for Addition | |||||||||||
| \(\ds \) | \(=\) | \(\ds 1_R + \paren {\paren {-1}_R + 1_R}\) | Ring Axiom $\text M2$: Identity Element for Ring Product | |||||||||||
| \(\ds \) | \(=\) | \(\ds 1_R + 0_R\) | Ring Axiom $\text A4$: Inverses for Addition | |||||||||||
| \(\ds \) | \(=\) | \(\ds 1\) | Ring Axiom $\text A3$: Identity for Addition |
$\blacksquare$