Theorem grprida 18723

Description: Deduce right identity from left inverse and left identity in an associative structure (such as a group). (Contributed by NM, 10-Aug-2013.) (Proof shortened by Mario Carneiro, 6-Jan-2015.)

Hypotheses

Ref	Expression
grpinva.c	⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) → (𝑥 + 𝑦) ∈ 𝐵)
grpinva.o	⊢ (𝜑 → 𝑂 ∈ 𝐵)
grpinva.i	⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → (𝑂 + 𝑥) = 𝑥)
grpinva.a	⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵)) → ((𝑥 + 𝑦) + 𝑧) = (𝑥 + (𝑦 + 𝑧)))
grpinva.r	⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → ∃𝑦 ∈ 𝐵 (𝑦 + 𝑥) = 𝑂)

Assertion

Ref	Expression
grprida	⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → (𝑥 + 𝑂) = 𝑥)

Distinct variable groups:   𝑥,𝑦,𝑧,𝐵   𝑥,𝑂,𝑦,𝑧   𝜑,𝑥,𝑦,𝑧   𝑥, + ,𝑦,𝑧

Proof of Theorem grprida

Dummy variables 𝑢 𝑛 𝑣 𝑤 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		grpinva.r	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → ∃𝑦 ∈ 𝐵 (𝑦 + 𝑥) = 𝑂)
2		oveq1 7407	. . . . . 6 ⊢ (𝑦 = 𝑛 → (𝑦 + 𝑥) = (𝑛 + 𝑥))
3	2	eqeq1d 2767	. . . . 5 ⊢ (𝑦 = 𝑛 → ((𝑦 + 𝑥) = 𝑂 ↔ (𝑛 + 𝑥) = 𝑂))
4	3	cbvrexvw 3244	. . . 4 ⊢ (∃𝑦 ∈ 𝐵 (𝑦 + 𝑥) = 𝑂 ↔ ∃𝑛 ∈ 𝐵 (𝑛 + 𝑥) = 𝑂)
5	1, 4	sylib 221	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → ∃𝑛 ∈ 𝐵 (𝑛 + 𝑥) = 𝑂)
6		grpinva.a	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵)) → ((𝑥 + 𝑦) + 𝑧) = (𝑥 + (𝑦 + 𝑧)))
7	6	caovassg 7598	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑢 ∈ 𝐵 ∧ 𝑣 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) → ((𝑢 + 𝑣) + 𝑤) = (𝑢 + (𝑣 + 𝑤)))
8	7	adantlr 727	. . . . . 6 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ (𝑛 ∈ 𝐵 ∧ (𝑛 + 𝑥) = 𝑂))) ∧ (𝑢 ∈ 𝐵 ∧ 𝑣 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) → ((𝑢 + 𝑣) + 𝑤) = (𝑢 + (𝑣 + 𝑤)))
9		simprl 782	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ (𝑛 ∈ 𝐵 ∧ (𝑛 + 𝑥) = 𝑂))) → 𝑥 ∈ 𝐵)
10		simprrl 792	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ (𝑛 ∈ 𝐵 ∧ (𝑛 + 𝑥) = 𝑂))) → 𝑛 ∈ 𝐵)
11	8, 9, 10, 9	caovassd 7599	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ (𝑛 ∈ 𝐵 ∧ (𝑛 + 𝑥) = 𝑂))) → ((𝑥 + 𝑛) + 𝑥) = (𝑥 + (𝑛 + 𝑥)))
12		grpinva.c	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) → (𝑥 + 𝑦) ∈ 𝐵)
13		grpinva.o	. . . . . . 7 ⊢ (𝜑 → 𝑂 ∈ 𝐵)
14		grpinva.i	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → (𝑂 + 𝑥) = 𝑥)
15		simprrr 793	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ (𝑛 ∈ 𝐵 ∧ (𝑛 + 𝑥) = 𝑂))) → (𝑛 + 𝑥) = 𝑂)
16	12, 13, 14, 6, 1, 9, 10, 15	grpinva 18722	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ (𝑛 ∈ 𝐵 ∧ (𝑛 + 𝑥) = 𝑂))) → (𝑥 + 𝑛) = 𝑂)
17	16	oveq1d 7415	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ (𝑛 ∈ 𝐵 ∧ (𝑛 + 𝑥) = 𝑂))) → ((𝑥 + 𝑛) + 𝑥) = (𝑂 + 𝑥))
18	15	oveq2d 7416	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ (𝑛 ∈ 𝐵 ∧ (𝑛 + 𝑥) = 𝑂))) → (𝑥 + (𝑛 + 𝑥)) = (𝑥 + 𝑂))
19	11, 17, 18	3eqtr3d 2808	. . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ (𝑛 ∈ 𝐵 ∧ (𝑛 + 𝑥) = 𝑂))) → (𝑂 + 𝑥) = (𝑥 + 𝑂))
20	19	anassrs 472	. . 3 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐵) ∧ (𝑛 ∈ 𝐵 ∧ (𝑛 + 𝑥) = 𝑂)) → (𝑂 + 𝑥) = (𝑥 + 𝑂))
21	5, 20	rexlimddv 3172	. 2 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → (𝑂 + 𝑥) = (𝑥 + 𝑂))
22	21, 14	eqtr3d 2802	1 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → (𝑥 + 𝑂) = 𝑥)

Syntax hints:   → wi 4   ∧ wa 400   ∧ w3a 1101   = wceq 1563   ∈ wcel 2145  ∃wrex 3089  (class class class)co 7400

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1818  ax-4 1832  ax-5 1933  ax-6 1990  ax-7 2031  ax-8 2147  ax-9 2155  ax-ext 2737

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1566  df-fal 1576  df-ex 1803  df-sb 2094  df-clab 2744  df-cleq 2757  df-clel 2840  df-ral 3080  df-rex 3090  df-rab 3418  df-v 3459  df-dif 3910  df-un 3912  df-ss 3924  df-nul 4289  df-if 4484  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4869  df-br 5106  df-iota 6481  df-fv 6533  df-ov 7403

This theorem is referenced by:  isgrpde  19014