Theorem grprinvd 6771

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 1258	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑥 + 𝑦) ∈ 𝐵)
7	6	caovclg 6724	. . . 4 ⊢ ((𝜑 ∧ (𝑢 ∈ 𝐵 ∧ 𝑣 ∈ 𝐵)) → (𝑢 + 𝑣) ∈ 𝐵)
8	7	adantlr 747	. . 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 6725	. 2 ⊢ ((𝜑 ∧ 𝜓) → (𝑋 + 𝑁) ∈ 𝐵)
12	4	caovassg 6730	. . . . 5 ⊢ ((𝜑 ∧ (𝑢 ∈ 𝐵 ∧ 𝑣 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) → ((𝑢 + 𝑣) + 𝑤) = (𝑢 + (𝑣 + 𝑤)))
13	12	adantlr 747	. . . 4 ⊢ (((𝜑 ∧ 𝜓) ∧ (𝑢 ∈ 𝐵 ∧ 𝑣 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) → ((𝑢 + 𝑣) + 𝑤) = (𝑢 + (𝑣 + 𝑤)))
14	13, 9, 10, 11	caovassd 6731	. . 3 ⊢ ((𝜑 ∧ 𝜓) → ((𝑋 + 𝑁) + (𝑋 + 𝑁)) = (𝑋 + (𝑁 + (𝑋 + 𝑁))))
15		grprinvd.e	. . . . . 6 ⊢ ((𝜑 ∧ 𝜓) → (𝑁 + 𝑋) = 𝑂)
16	15	oveq1d 6564	. . . . 5 ⊢ ((𝜑 ∧ 𝜓) → ((𝑁 + 𝑋) + 𝑁) = (𝑂 + 𝑁))
17	13, 10, 9, 10	caovassd 6731	. . . . 5 ⊢ ((𝜑 ∧ 𝜓) → ((𝑁 + 𝑋) + 𝑁) = (𝑁 + (𝑋 + 𝑁)))
18	3	ralrimiva 2949	. . . . . . . 8 ⊢ (𝜑 → ∀𝑥 ∈ 𝐵 (𝑂 + 𝑥) = 𝑥)
19		oveq2 6557	. . . . . . . . . 10 ⊢ (𝑥 = 𝑦 → (𝑂 + 𝑥) = (𝑂 + 𝑦))
20		id 22	. . . . . . . . . 10 ⊢ (𝑥 = 𝑦 → 𝑥 = 𝑦)
21	19, 20	eqeq12d 2625	. . . . . . . . 9 ⊢ (𝑥 = 𝑦 → ((𝑂 + 𝑥) = 𝑥 ↔ (𝑂 + 𝑦) = 𝑦))
22	21	cbvralv 3147	. . . . . . . 8 ⊢ (∀𝑥 ∈ 𝐵 (𝑂 + 𝑥) = 𝑥 ↔ ∀𝑦 ∈ 𝐵 (𝑂 + 𝑦) = 𝑦)
23	18, 22	sylib 207	. . . . . . 7 ⊢ (𝜑 → ∀𝑦 ∈ 𝐵 (𝑂 + 𝑦) = 𝑦)
24	23	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ 𝜓) → ∀𝑦 ∈ 𝐵 (𝑂 + 𝑦) = 𝑦)
25		oveq2 6557	. . . . . . . 8 ⊢ (𝑦 = 𝑁 → (𝑂 + 𝑦) = (𝑂 + 𝑁))
26		id 22	. . . . . . . 8 ⊢ (𝑦 = 𝑁 → 𝑦 = 𝑁)
27	25, 26	eqeq12d 2625	. . . . . . 7 ⊢ (𝑦 = 𝑁 → ((𝑂 + 𝑦) = 𝑦 ↔ (𝑂 + 𝑁) = 𝑁))
28	27	rspcv 3278	. . . . . 6 ⊢ (𝑁 ∈ 𝐵 → (∀𝑦 ∈ 𝐵 (𝑂 + 𝑦) = 𝑦 → (𝑂 + 𝑁) = 𝑁))
29	10, 24, 28	sylc 63	. . . . 5 ⊢ ((𝜑 ∧ 𝜓) → (𝑂 + 𝑁) = 𝑁)
30	16, 17, 29	3eqtr3d 2652	. . . 4 ⊢ ((𝜑 ∧ 𝜓) → (𝑁 + (𝑋 + 𝑁)) = 𝑁)
31	30	oveq2d 6565	. . 3 ⊢ ((𝜑 ∧ 𝜓) → (𝑋 + (𝑁 + (𝑋 + 𝑁))) = (𝑋 + 𝑁))
32	14, 31	eqtrd 2644	. 2 ⊢ ((𝜑 ∧ 𝜓) → ((𝑋 + 𝑁) + (𝑋 + 𝑁)) = (𝑋 + 𝑁))
33	1, 2, 3, 4, 5, 11, 32	grprinvlem 6770	1 ⊢ ((𝜑 ∧ 𝜓) → (𝑋 + 𝑁) = 𝑂)

Syntax hints:   → wi 4   ∧ wa 383   ∧ w3a 1031   = wceq 1475   ∈ wcel 1977  ∀wral 2896  ∃wrex 2897  (class class class)co 6549

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-iota 5768  df-fv 5812  df-ov 6552

This theorem is referenced by:  grpridd  6772  grprcan  17278  grprinv  17292