Theorem grprinvd 17878

Description: Deduce right inverse 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
grprinvlem.c	⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) → (𝑥 + 𝑦) ∈ 𝐵)
grprinvlem.o	⊢ (𝜑 → 𝑂 ∈ 𝐵)
grprinvlem.i	⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → (𝑂 + 𝑥) = 𝑥)
grprinvlem.a	⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵)) → ((𝑥 + 𝑦) + 𝑧) = (𝑥 + (𝑦 + 𝑧)))
grprinvlem.n	⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → ∃𝑦 ∈ 𝐵 (𝑦 + 𝑥) = 𝑂)
grprinvd.x	⊢ ((𝜑 ∧ 𝜓) → 𝑋 ∈ 𝐵)
grprinvd.n	⊢ ((𝜑 ∧ 𝜓) → 𝑁 ∈ 𝐵)
grprinvd.e	⊢ ((𝜑 ∧ 𝜓) → (𝑁 + 𝑋) = 𝑂)

Assertion

Ref	Expression
grprinvd	⊢ ((𝜑 ∧ 𝜓) → (𝑋 + 𝑁) = 𝑂)

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

Allowed substitution hints:   𝜓(𝑥,𝑧)   𝑁(𝑥)   𝑋(𝑥)

Proof of Theorem grprinvd

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

Step	Hyp	Ref	Expression
1		grprinvlem.c	. 2 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) → (𝑥 + 𝑦) ∈ 𝐵)
2		grprinvlem.o	. 2 ⊢ (𝜑 → 𝑂 ∈ 𝐵)
3		grprinvlem.i	. 2 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → (𝑂 + 𝑥) = 𝑥)
4		grprinvlem.a	. 2 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵)) → ((𝑥 + 𝑦) + 𝑧) = (𝑥 + (𝑦 + 𝑧)))
5		grprinvlem.n	. 2 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → ∃𝑦 ∈ 𝐵 (𝑦 + 𝑥) = 𝑂)
6	1	3expb 1116	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑥 + 𝑦) ∈ 𝐵)
7	6	caovclg 7334	. . . 4 ⊢ ((𝜑 ∧ (𝑢 ∈ 𝐵 ∧ 𝑣 ∈ 𝐵)) → (𝑢 + 𝑣) ∈ 𝐵)
8	7	adantlr 713	. . 3 ⊢ (((𝜑 ∧ 𝜓) ∧ (𝑢 ∈ 𝐵 ∧ 𝑣 ∈ 𝐵)) → (𝑢 + 𝑣) ∈ 𝐵)
9		grprinvd.x	. . 3 ⊢ ((𝜑 ∧ 𝜓) → 𝑋 ∈ 𝐵)
10		grprinvd.n	. . 3 ⊢ ((𝜑 ∧ 𝜓) → 𝑁 ∈ 𝐵)
11	8, 9, 10	caovcld 7335	. 2 ⊢ ((𝜑 ∧ 𝜓) → (𝑋 + 𝑁) ∈ 𝐵)
12	4	caovassg 7340	. . . . 5 ⊢ ((𝜑 ∧ (𝑢 ∈ 𝐵 ∧ 𝑣 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) → ((𝑢 + 𝑣) + 𝑤) = (𝑢 + (𝑣 + 𝑤)))
13	12	adantlr 713	. . . 4 ⊢ (((𝜑 ∧ 𝜓) ∧ (𝑢 ∈ 𝐵 ∧ 𝑣 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) → ((𝑢 + 𝑣) + 𝑤) = (𝑢 + (𝑣 + 𝑤)))
14	13, 9, 10, 11	caovassd 7341	. . 3 ⊢ ((𝜑 ∧ 𝜓) → ((𝑋 + 𝑁) + (𝑋 + 𝑁)) = (𝑋 + (𝑁 + (𝑋 + 𝑁))))
15		grprinvd.e	. . . . . 6 ⊢ ((𝜑 ∧ 𝜓) → (𝑁 + 𝑋) = 𝑂)
16	15	oveq1d 7165	. . . . 5 ⊢ ((𝜑 ∧ 𝜓) → ((𝑁 + 𝑋) + 𝑁) = (𝑂 + 𝑁))
17	13, 10, 9, 10	caovassd 7341	. . . . 5 ⊢ ((𝜑 ∧ 𝜓) → ((𝑁 + 𝑋) + 𝑁) = (𝑁 + (𝑋 + 𝑁)))
18		oveq2 7158	. . . . . . 7 ⊢ (𝑦 = 𝑁 → (𝑂 + 𝑦) = (𝑂 + 𝑁))
19		id 22	. . . . . . 7 ⊢ (𝑦 = 𝑁 → 𝑦 = 𝑁)
20	18, 19	eqeq12d 2837	. . . . . 6 ⊢ (𝑦 = 𝑁 → ((𝑂 + 𝑦) = 𝑦 ↔ (𝑂 + 𝑁) = 𝑁))
21	3	ralrimiva 3182	. . . . . . . 8 ⊢ (𝜑 → ∀𝑥 ∈ 𝐵 (𝑂 + 𝑥) = 𝑥)
22		oveq2 7158	. . . . . . . . . 10 ⊢ (𝑥 = 𝑦 → (𝑂 + 𝑥) = (𝑂 + 𝑦))
23		id 22	. . . . . . . . . 10 ⊢ (𝑥 = 𝑦 → 𝑥 = 𝑦)
24	22, 23	eqeq12d 2837	. . . . . . . . 9 ⊢ (𝑥 = 𝑦 → ((𝑂 + 𝑥) = 𝑥 ↔ (𝑂 + 𝑦) = 𝑦))
25	24	cbvralvw 3450	. . . . . . . 8 ⊢ (∀𝑥 ∈ 𝐵 (𝑂 + 𝑥) = 𝑥 ↔ ∀𝑦 ∈ 𝐵 (𝑂 + 𝑦) = 𝑦)
26	21, 25	sylib 220	. . . . . . 7 ⊢ (𝜑 → ∀𝑦 ∈ 𝐵 (𝑂 + 𝑦) = 𝑦)
27	26	adantr 483	. . . . . 6 ⊢ ((𝜑 ∧ 𝜓) → ∀𝑦 ∈ 𝐵 (𝑂 + 𝑦) = 𝑦)
28	20, 27, 10	rspcdva 3625	. . . . 5 ⊢ ((𝜑 ∧ 𝜓) → (𝑂 + 𝑁) = 𝑁)
29	16, 17, 28	3eqtr3d 2864	. . . 4 ⊢ ((𝜑 ∧ 𝜓) → (𝑁 + (𝑋 + 𝑁)) = 𝑁)
30	29	oveq2d 7166	. . 3 ⊢ ((𝜑 ∧ 𝜓) → (𝑋 + (𝑁 + (𝑋 + 𝑁))) = (𝑋 + 𝑁))
31	14, 30	eqtrd 2856	. 2 ⊢ ((𝜑 ∧ 𝜓) → ((𝑋 + 𝑁) + (𝑋 + 𝑁)) = (𝑋 + 𝑁))
32	1, 2, 3, 4, 5, 11, 31	grprinvlem 17877	1 ⊢ ((𝜑 ∧ 𝜓) → (𝑋 + 𝑁) = 𝑂)

Syntax hints:   → wi 4   ∧ wa 398   ∧ w3a 1083   = wceq 1533   ∈ wcel 2110  ∀wral 3138  ∃wrex 3139  (class class class)co 7150

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2156  ax-12 2172  ax-ext 2793

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ral 3143  df-rex 3144  df-rab 3147  df-v 3497  df-dif 3939  df-un 3941  df-in 3943  df-ss 3952  df-nul 4292  df-if 4468  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4833  df-br 5060  df-iota 6309  df-fv 6358  df-ov 7153

This theorem is referenced by:  grpridd  17879  grprcan  18131  grprinv  18147