Theorem grpoidinv 27952

Description: A group has a left and right identity element, and every member has a left and right inverse. (Contributed by NM, 14-Oct-2006.) (New usage is discouraged.)

Hypothesis

Ref	Expression
grpfo.1	⊢ 𝑋 = ran 𝐺

Assertion

Ref	Expression
grpoidinv	⊢ (𝐺 ∈ GrpOp → ∃𝑢 ∈ 𝑋 ∀𝑥 ∈ 𝑋 (((𝑢𝐺𝑥) = 𝑥 ∧ (𝑥𝐺𝑢) = 𝑥) ∧ ∃𝑦 ∈ 𝑋 ((𝑦𝐺𝑥) = 𝑢 ∧ (𝑥𝐺𝑦) = 𝑢)))

Distinct variable groups:   𝑥,𝑦,𝑢,𝐺   𝑢,𝑋,𝑥,𝑦

Proof of Theorem grpoidinv

Dummy variables 𝑤 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		simpl 476	. . . . . . . 8 ⊢ (((𝑢𝐺𝑧) = 𝑧 ∧ ∃𝑤 ∈ 𝑋 (𝑤𝐺𝑧) = 𝑢) → (𝑢𝐺𝑧) = 𝑧)
2	1	ralimi 3134	. . . . . . 7 ⊢ (∀𝑧 ∈ 𝑋 ((𝑢𝐺𝑧) = 𝑧 ∧ ∃𝑤 ∈ 𝑋 (𝑤𝐺𝑧) = 𝑢) → ∀𝑧 ∈ 𝑋 (𝑢𝐺𝑧) = 𝑧)
3		oveq2 6932	. . . . . . . . 9 ⊢ (𝑧 = 𝑥 → (𝑢𝐺𝑧) = (𝑢𝐺𝑥))
4		id 22	. . . . . . . . 9 ⊢ (𝑧 = 𝑥 → 𝑧 = 𝑥)
5	3, 4	eqeq12d 2793	. . . . . . . 8 ⊢ (𝑧 = 𝑥 → ((𝑢𝐺𝑧) = 𝑧 ↔ (𝑢𝐺𝑥) = 𝑥))
6	5	rspccva 3510	. . . . . . 7 ⊢ ((∀𝑧 ∈ 𝑋 (𝑢𝐺𝑧) = 𝑧 ∧ 𝑥 ∈ 𝑋) → (𝑢𝐺𝑥) = 𝑥)
7	2, 6	sylan 575	. . . . . 6 ⊢ ((∀𝑧 ∈ 𝑋 ((𝑢𝐺𝑧) = 𝑧 ∧ ∃𝑤 ∈ 𝑋 (𝑤𝐺𝑧) = 𝑢) ∧ 𝑥 ∈ 𝑋) → (𝑢𝐺𝑥) = 𝑥)
8	7	adantll 704	. . . . 5 ⊢ (((𝑢 ∈ 𝑋 ∧ ∀𝑧 ∈ 𝑋 ((𝑢𝐺𝑧) = 𝑧 ∧ ∃𝑤 ∈ 𝑋 (𝑤𝐺𝑧) = 𝑢)) ∧ 𝑥 ∈ 𝑋) → (𝑢𝐺𝑥) = 𝑥)
9	8	adantll 704	. . . 4 ⊢ (((𝐺 ∈ GrpOp ∧ (𝑢 ∈ 𝑋 ∧ ∀𝑧 ∈ 𝑋 ((𝑢𝐺𝑧) = 𝑧 ∧ ∃𝑤 ∈ 𝑋 (𝑤𝐺𝑧) = 𝑢))) ∧ 𝑥 ∈ 𝑋) → (𝑢𝐺𝑥) = 𝑥)
10		simpl 476	. . . . . . 7 ⊢ ((𝐺 ∈ GrpOp ∧ (𝑢 ∈ 𝑋 ∧ ∀𝑧 ∈ 𝑋 ((𝑢𝐺𝑧) = 𝑧 ∧ ∃𝑤 ∈ 𝑋 (𝑤𝐺𝑧) = 𝑢))) → 𝐺 ∈ GrpOp)
11	10	anim1i 608	. . . . . 6 ⊢ (((𝐺 ∈ GrpOp ∧ (𝑢 ∈ 𝑋 ∧ ∀𝑧 ∈ 𝑋 ((𝑢𝐺𝑧) = 𝑧 ∧ ∃𝑤 ∈ 𝑋 (𝑤𝐺𝑧) = 𝑢))) ∧ 𝑥 ∈ 𝑋) → (𝐺 ∈ GrpOp ∧ 𝑥 ∈ 𝑋))
12		id 22	. . . . . . . . . 10 ⊢ ((𝐺 ∈ GrpOp ∧ 𝑢 ∈ 𝑋) → (𝐺 ∈ GrpOp ∧ 𝑢 ∈ 𝑋))
13	12	adantrr 707	. . . . . . . . 9 ⊢ ((𝐺 ∈ GrpOp ∧ (𝑢 ∈ 𝑋 ∧ ∀𝑧 ∈ 𝑋 ((𝑢𝐺𝑧) = 𝑧 ∧ ∃𝑤 ∈ 𝑋 (𝑤𝐺𝑧) = 𝑢))) → (𝐺 ∈ GrpOp ∧ 𝑢 ∈ 𝑋))
14	13	adantr 474	. . . . . . . 8 ⊢ (((𝐺 ∈ GrpOp ∧ (𝑢 ∈ 𝑋 ∧ ∀𝑧 ∈ 𝑋 ((𝑢𝐺𝑧) = 𝑧 ∧ ∃𝑤 ∈ 𝑋 (𝑤𝐺𝑧) = 𝑢))) ∧ 𝑥 ∈ 𝑋) → (𝐺 ∈ GrpOp ∧ 𝑢 ∈ 𝑋))
15	2	adantl 475	. . . . . . . . 9 ⊢ ((𝑢 ∈ 𝑋 ∧ ∀𝑧 ∈ 𝑋 ((𝑢𝐺𝑧) = 𝑧 ∧ ∃𝑤 ∈ 𝑋 (𝑤𝐺𝑧) = 𝑢)) → ∀𝑧 ∈ 𝑋 (𝑢𝐺𝑧) = 𝑧)
16	15	ad2antlr 717	. . . . . . . 8 ⊢ (((𝐺 ∈ GrpOp ∧ (𝑢 ∈ 𝑋 ∧ ∀𝑧 ∈ 𝑋 ((𝑢𝐺𝑧) = 𝑧 ∧ ∃𝑤 ∈ 𝑋 (𝑤𝐺𝑧) = 𝑢))) ∧ 𝑥 ∈ 𝑋) → ∀𝑧 ∈ 𝑋 (𝑢𝐺𝑧) = 𝑧)
17		simpr 479	. . . . . . . . . . 11 ⊢ (((𝑢𝐺𝑧) = 𝑧 ∧ ∃𝑤 ∈ 𝑋 (𝑤𝐺𝑧) = 𝑢) → ∃𝑤 ∈ 𝑋 (𝑤𝐺𝑧) = 𝑢)
18	17	ralimi 3134	. . . . . . . . . 10 ⊢ (∀𝑧 ∈ 𝑋 ((𝑢𝐺𝑧) = 𝑧 ∧ ∃𝑤 ∈ 𝑋 (𝑤𝐺𝑧) = 𝑢) → ∀𝑧 ∈ 𝑋 ∃𝑤 ∈ 𝑋 (𝑤𝐺𝑧) = 𝑢)
19	18	adantl 475	. . . . . . . . 9 ⊢ ((𝑢 ∈ 𝑋 ∧ ∀𝑧 ∈ 𝑋 ((𝑢𝐺𝑧) = 𝑧 ∧ ∃𝑤 ∈ 𝑋 (𝑤𝐺𝑧) = 𝑢)) → ∀𝑧 ∈ 𝑋 ∃𝑤 ∈ 𝑋 (𝑤𝐺𝑧) = 𝑢)
20	19	ad2antlr 717	. . . . . . . 8 ⊢ (((𝐺 ∈ GrpOp ∧ (𝑢 ∈ 𝑋 ∧ ∀𝑧 ∈ 𝑋 ((𝑢𝐺𝑧) = 𝑧 ∧ ∃𝑤 ∈ 𝑋 (𝑤𝐺𝑧) = 𝑢))) ∧ 𝑥 ∈ 𝑋) → ∀𝑧 ∈ 𝑋 ∃𝑤 ∈ 𝑋 (𝑤𝐺𝑧) = 𝑢)
21	14, 16, 20	jca32 511	. . . . . . 7 ⊢ (((𝐺 ∈ GrpOp ∧ (𝑢 ∈ 𝑋 ∧ ∀𝑧 ∈ 𝑋 ((𝑢𝐺𝑧) = 𝑧 ∧ ∃𝑤 ∈ 𝑋 (𝑤𝐺𝑧) = 𝑢))) ∧ 𝑥 ∈ 𝑋) → ((𝐺 ∈ GrpOp ∧ 𝑢 ∈ 𝑋) ∧ (∀𝑧 ∈ 𝑋 (𝑢𝐺𝑧) = 𝑧 ∧ ∀𝑧 ∈ 𝑋 ∃𝑤 ∈ 𝑋 (𝑤𝐺𝑧) = 𝑢)))
22		grpfo.1	. . . . . . . 8 ⊢ 𝑋 = ran 𝐺
23		biid 253	. . . . . . . 8 ⊢ (∀𝑧 ∈ 𝑋 (𝑢𝐺𝑧) = 𝑧 ↔ ∀𝑧 ∈ 𝑋 (𝑢𝐺𝑧) = 𝑧)
24		biid 253	. . . . . . . 8 ⊢ (∀𝑧 ∈ 𝑋 ∃𝑤 ∈ 𝑋 (𝑤𝐺𝑧) = 𝑢 ↔ ∀𝑧 ∈ 𝑋 ∃𝑤 ∈ 𝑋 (𝑤𝐺𝑧) = 𝑢)
25	22, 23, 24	grpoidinvlem3 27950	. . . . . . 7 ⊢ ((((𝐺 ∈ GrpOp ∧ 𝑢 ∈ 𝑋) ∧ (∀𝑧 ∈ 𝑋 (𝑢𝐺𝑧) = 𝑧 ∧ ∀𝑧 ∈ 𝑋 ∃𝑤 ∈ 𝑋 (𝑤𝐺𝑧) = 𝑢)) ∧ 𝑥 ∈ 𝑋) → ∃𝑦 ∈ 𝑋 ((𝑦𝐺𝑥) = 𝑢 ∧ (𝑥𝐺𝑦) = 𝑢))
26	21, 25	sylancom 582	. . . . . 6 ⊢ (((𝐺 ∈ GrpOp ∧ (𝑢 ∈ 𝑋 ∧ ∀𝑧 ∈ 𝑋 ((𝑢𝐺𝑧) = 𝑧 ∧ ∃𝑤 ∈ 𝑋 (𝑤𝐺𝑧) = 𝑢))) ∧ 𝑥 ∈ 𝑋) → ∃𝑦 ∈ 𝑋 ((𝑦𝐺𝑥) = 𝑢 ∧ (𝑥𝐺𝑦) = 𝑢))
27	22	grpoidinvlem4 27951	. . . . . 6 ⊢ (((𝐺 ∈ GrpOp ∧ 𝑥 ∈ 𝑋) ∧ ∃𝑦 ∈ 𝑋 ((𝑦𝐺𝑥) = 𝑢 ∧ (𝑥𝐺𝑦) = 𝑢)) → (𝑥𝐺𝑢) = (𝑢𝐺𝑥))
28	11, 26, 27	syl2anc 579	. . . . 5 ⊢ (((𝐺 ∈ GrpOp ∧ (𝑢 ∈ 𝑋 ∧ ∀𝑧 ∈ 𝑋 ((𝑢𝐺𝑧) = 𝑧 ∧ ∃𝑤 ∈ 𝑋 (𝑤𝐺𝑧) = 𝑢))) ∧ 𝑥 ∈ 𝑋) → (𝑥𝐺𝑢) = (𝑢𝐺𝑥))
29	28, 9	eqtrd 2814	. . . 4 ⊢ (((𝐺 ∈ GrpOp ∧ (𝑢 ∈ 𝑋 ∧ ∀𝑧 ∈ 𝑋 ((𝑢𝐺𝑧) = 𝑧 ∧ ∃𝑤 ∈ 𝑋 (𝑤𝐺𝑧) = 𝑢))) ∧ 𝑥 ∈ 𝑋) → (𝑥𝐺𝑢) = 𝑥)
30	9, 29, 26	jca31 510	. . 3 ⊢ (((𝐺 ∈ GrpOp ∧ (𝑢 ∈ 𝑋 ∧ ∀𝑧 ∈ 𝑋 ((𝑢𝐺𝑧) = 𝑧 ∧ ∃𝑤 ∈ 𝑋 (𝑤𝐺𝑧) = 𝑢))) ∧ 𝑥 ∈ 𝑋) → (((𝑢𝐺𝑥) = 𝑥 ∧ (𝑥𝐺𝑢) = 𝑥) ∧ ∃𝑦 ∈ 𝑋 ((𝑦𝐺𝑥) = 𝑢 ∧ (𝑥𝐺𝑦) = 𝑢)))
31	30	ralrimiva 3148	. 2 ⊢ ((𝐺 ∈ GrpOp ∧ (𝑢 ∈ 𝑋 ∧ ∀𝑧 ∈ 𝑋 ((𝑢𝐺𝑧) = 𝑧 ∧ ∃𝑤 ∈ 𝑋 (𝑤𝐺𝑧) = 𝑢))) → ∀𝑥 ∈ 𝑋 (((𝑢𝐺𝑥) = 𝑥 ∧ (𝑥𝐺𝑢) = 𝑥) ∧ ∃𝑦 ∈ 𝑋 ((𝑦𝐺𝑥) = 𝑢 ∧ (𝑥𝐺𝑦) = 𝑢)))
32	22	grpolidinv 27945	. 2 ⊢ (𝐺 ∈ GrpOp → ∃𝑢 ∈ 𝑋 ∀𝑧 ∈ 𝑋 ((𝑢𝐺𝑧) = 𝑧 ∧ ∃𝑤 ∈ 𝑋 (𝑤𝐺𝑧) = 𝑢))
33	31, 32	reximddv 3199	1 ⊢ (𝐺 ∈ GrpOp → ∃𝑢 ∈ 𝑋 ∀𝑥 ∈ 𝑋 (((𝑢𝐺𝑥) = 𝑥 ∧ (𝑥𝐺𝑢) = 𝑥) ∧ ∃𝑦 ∈ 𝑋 ((𝑦𝐺𝑥) = 𝑢 ∧ (𝑥𝐺𝑦) = 𝑢)))

Syntax hints:   → wi 4   ∧ wa 386   = wceq 1601   ∈ wcel 2107  ∀wral 3090  ∃wrex 3091  ran crn 5358  (class class class)co 6924  GrpOpcgr 27933

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2055  ax-8 2109  ax-9 2116  ax-10 2135  ax-11 2150  ax-12 2163  ax-13 2334  ax-ext 2754  ax-sep 5019  ax-nul 5027  ax-pr 5140  ax-un 7228

This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-3an 1073  df-tru 1605  df-ex 1824  df-nf 1828  df-sb 2012  df-mo 2551  df-eu 2587  df-clab 2764  df-cleq 2770  df-clel 2774  df-nfc 2921  df-ral 3095  df-rex 3096  df-rab 3099  df-v 3400  df-sbc 3653  df-csb 3752  df-dif 3795  df-un 3797  df-in 3799  df-ss 3806  df-nul 4142  df-if 4308  df-sn 4399  df-pr 4401  df-op 4405  df-uni 4674  df-iun 4757  df-br 4889  df-opab 4951  df-mpt 4968  df-id 5263  df-xp 5363  df-rel 5364  df-cnv 5365  df-co 5366  df-dm 5367  df-rn 5368  df-iota 6101  df-fun 6139  df-fn 6140  df-f 6141  df-fo 6143  df-fv 6145  df-ov 6927  df-grpo 27937

This theorem is referenced by:  grpoideu  27953  grpoidval  27957  grpoidinv2  27959  grpomndo  34307