Theorem grpinva 13029

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
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)
grpinva.x	⊢ ((𝜑 ∧ 𝜓) → 𝑋 ∈ 𝐵)
grpinva.n	⊢ ((𝜑 ∧ 𝜓) → 𝑁 ∈ 𝐵)
grpinva.e	⊢ ((𝜑 ∧ 𝜓) → (𝑁 + 𝑋) = 𝑂)

Assertion

Ref	Expression
grpinva	⊢ ((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 grpinva

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

Step	Hyp	Ref	Expression
1		grpinva.c	. 2 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) → (𝑥 + 𝑦) ∈ 𝐵)
2		grpinva.o	. 2 ⊢ (𝜑 → 𝑂 ∈ 𝐵)
3		grpinva.i	. 2 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → (𝑂 + 𝑥) = 𝑥)
4		grpinva.a	. 2 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵)) → ((𝑥 + 𝑦) + 𝑧) = (𝑥 + (𝑦 + 𝑧)))
5		grpinva.r	. 2 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → ∃𝑦 ∈ 𝐵 (𝑦 + 𝑥) = 𝑂)
6	1	3expb 1206	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑥 + 𝑦) ∈ 𝐵)
7	6	caovclg 6076	. . . 4 ⊢ ((𝜑 ∧ (𝑢 ∈ 𝐵 ∧ 𝑣 ∈ 𝐵)) → (𝑢 + 𝑣) ∈ 𝐵)
8	7	adantlr 477	. . 3 ⊢ (((𝜑 ∧ 𝜓) ∧ (𝑢 ∈ 𝐵 ∧ 𝑣 ∈ 𝐵)) → (𝑢 + 𝑣) ∈ 𝐵)
9		grpinva.x	. . 3 ⊢ ((𝜑 ∧ 𝜓) → 𝑋 ∈ 𝐵)
10		grpinva.n	. . 3 ⊢ ((𝜑 ∧ 𝜓) → 𝑁 ∈ 𝐵)
11	8, 9, 10	caovcld 6077	. 2 ⊢ ((𝜑 ∧ 𝜓) → (𝑋 + 𝑁) ∈ 𝐵)
12	4	caovassg 6082	. . . . 5 ⊢ ((𝜑 ∧ (𝑢 ∈ 𝐵 ∧ 𝑣 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) → ((𝑢 + 𝑣) + 𝑤) = (𝑢 + (𝑣 + 𝑤)))
13	12	adantlr 477	. . . 4 ⊢ (((𝜑 ∧ 𝜓) ∧ (𝑢 ∈ 𝐵 ∧ 𝑣 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) → ((𝑢 + 𝑣) + 𝑤) = (𝑢 + (𝑣 + 𝑤)))
14	13, 9, 10, 11	caovassd 6083	. . 3 ⊢ ((𝜑 ∧ 𝜓) → ((𝑋 + 𝑁) + (𝑋 + 𝑁)) = (𝑋 + (𝑁 + (𝑋 + 𝑁))))
15		grpinva.e	. . . . . 6 ⊢ ((𝜑 ∧ 𝜓) → (𝑁 + 𝑋) = 𝑂)
16	15	oveq1d 5937	. . . . 5 ⊢ ((𝜑 ∧ 𝜓) → ((𝑁 + 𝑋) + 𝑁) = (𝑂 + 𝑁))
17	13, 10, 9, 10	caovassd 6083	. . . . 5 ⊢ ((𝜑 ∧ 𝜓) → ((𝑁 + 𝑋) + 𝑁) = (𝑁 + (𝑋 + 𝑁)))
18		oveq2 5930	. . . . . . 7 ⊢ (𝑦 = 𝑁 → (𝑂 + 𝑦) = (𝑂 + 𝑁))
19		id 19	. . . . . . 7 ⊢ (𝑦 = 𝑁 → 𝑦 = 𝑁)
20	18, 19	eqeq12d 2211	. . . . . 6 ⊢ (𝑦 = 𝑁 → ((𝑂 + 𝑦) = 𝑦 ↔ (𝑂 + 𝑁) = 𝑁))
21	3	ralrimiva 2570	. . . . . . . 8 ⊢ (𝜑 → ∀𝑥 ∈ 𝐵 (𝑂 + 𝑥) = 𝑥)
22		oveq2 5930	. . . . . . . . . 10 ⊢ (𝑥 = 𝑦 → (𝑂 + 𝑥) = (𝑂 + 𝑦))
23		id 19	. . . . . . . . . 10 ⊢ (𝑥 = 𝑦 → 𝑥 = 𝑦)
24	22, 23	eqeq12d 2211	. . . . . . . . 9 ⊢ (𝑥 = 𝑦 → ((𝑂 + 𝑥) = 𝑥 ↔ (𝑂 + 𝑦) = 𝑦))
25	24	cbvralvw 2733	. . . . . . . 8 ⊢ (∀𝑥 ∈ 𝐵 (𝑂 + 𝑥) = 𝑥 ↔ ∀𝑦 ∈ 𝐵 (𝑂 + 𝑦) = 𝑦)
26	21, 25	sylib 122	. . . . . . 7 ⊢ (𝜑 → ∀𝑦 ∈ 𝐵 (𝑂 + 𝑦) = 𝑦)
27	26	adantr 276	. . . . . 6 ⊢ ((𝜑 ∧ 𝜓) → ∀𝑦 ∈ 𝐵 (𝑂 + 𝑦) = 𝑦)
28	20, 27, 10	rspcdva 2873	. . . . 5 ⊢ ((𝜑 ∧ 𝜓) → (𝑂 + 𝑁) = 𝑁)
29	16, 17, 28	3eqtr3d 2237	. . . 4 ⊢ ((𝜑 ∧ 𝜓) → (𝑁 + (𝑋 + 𝑁)) = 𝑁)
30	29	oveq2d 5938	. . 3 ⊢ ((𝜑 ∧ 𝜓) → (𝑋 + (𝑁 + (𝑋 + 𝑁))) = (𝑋 + 𝑁))
31	14, 30	eqtrd 2229	. 2 ⊢ ((𝜑 ∧ 𝜓) → ((𝑋 + 𝑁) + (𝑋 + 𝑁)) = (𝑋 + 𝑁))
32	1, 2, 3, 4, 5, 11, 31	grpinvalem 13028	1 ⊢ ((𝜑 ∧ 𝜓) → (𝑋 + 𝑁) = 𝑂)

Syntax hints:   → wi 4   ∧ wa 104   ∧ w3a 980   = wceq 1364   ∈ wcel 2167  ∀wral 2475  ∃wrex 2476  (class class class)co 5922

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-ext 2178

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1475  df-sb 1777  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ral 2480  df-rex 2481  df-v 2765  df-un 3161  df-sn 3628  df-pr 3629  df-op 3631  df-uni 3840  df-br 4034  df-iota 5219  df-fv 5266  df-ov 5925

This theorem is referenced by:  grprida  13030  grprcan  13169  grprinv  13183