Theorem grpoidinv 30489

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 482	. . . . . . . 8 ⊢ (((𝑢𝐺𝑧) = 𝑧 ∧ ∃𝑤 ∈ 𝑋 (𝑤𝐺𝑧) = 𝑢) → (𝑢𝐺𝑧) = 𝑧)
2	1	ralimi 3073	. . . . . . 7 ⊢ (∀𝑧 ∈ 𝑋 ((𝑢𝐺𝑧) = 𝑧 ∧ ∃𝑤 ∈ 𝑋 (𝑤𝐺𝑧) = 𝑢) → ∀𝑧 ∈ 𝑋 (𝑢𝐺𝑧) = 𝑧)
3		oveq2 7413	. . . . . . . . 9 ⊢ (𝑧 = 𝑥 → (𝑢𝐺𝑧) = (𝑢𝐺𝑥))
4		id 22	. . . . . . . . 9 ⊢ (𝑧 = 𝑥 → 𝑧 = 𝑥)
5	3, 4	eqeq12d 2751	. . . . . . . 8 ⊢ (𝑧 = 𝑥 → ((𝑢𝐺𝑧) = 𝑧 ↔ (𝑢𝐺𝑥) = 𝑥))
6	5	rspccva 3600	. . . . . . 7 ⊢ ((∀𝑧 ∈ 𝑋 (𝑢𝐺𝑧) = 𝑧 ∧ 𝑥 ∈ 𝑋) → (𝑢𝐺𝑥) = 𝑥)
7	2, 6	sylan 580	. . . . . 6 ⊢ ((∀𝑧 ∈ 𝑋 ((𝑢𝐺𝑧) = 𝑧 ∧ ∃𝑤 ∈ 𝑋 (𝑤𝐺𝑧) = 𝑢) ∧ 𝑥 ∈ 𝑋) → (𝑢𝐺𝑥) = 𝑥)
8	7	adantll 714	. . . . 5 ⊢ (((𝑢 ∈ 𝑋 ∧ ∀𝑧 ∈ 𝑋 ((𝑢𝐺𝑧) = 𝑧 ∧ ∃𝑤 ∈ 𝑋 (𝑤𝐺𝑧) = 𝑢)) ∧ 𝑥 ∈ 𝑋) → (𝑢𝐺𝑥) = 𝑥)
9	8	adantll 714	. . . 4 ⊢ (((𝐺 ∈ GrpOp ∧ (𝑢 ∈ 𝑋 ∧ ∀𝑧 ∈ 𝑋 ((𝑢𝐺𝑧) = 𝑧 ∧ ∃𝑤 ∈ 𝑋 (𝑤𝐺𝑧) = 𝑢))) ∧ 𝑥 ∈ 𝑋) → (𝑢𝐺𝑥) = 𝑥)
10		simpl 482	. . . . . . 7 ⊢ ((𝐺 ∈ GrpOp ∧ (𝑢 ∈ 𝑋 ∧ ∀𝑧 ∈ 𝑋 ((𝑢𝐺𝑧) = 𝑧 ∧ ∃𝑤 ∈ 𝑋 (𝑤𝐺𝑧) = 𝑢))) → 𝐺 ∈ GrpOp)
11	10	anim1i 615	. . . . . 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 717	. . . . . . . . 9 ⊢ ((𝐺 ∈ GrpOp ∧ (𝑢 ∈ 𝑋 ∧ ∀𝑧 ∈ 𝑋 ((𝑢𝐺𝑧) = 𝑧 ∧ ∃𝑤 ∈ 𝑋 (𝑤𝐺𝑧) = 𝑢))) → (𝐺 ∈ GrpOp ∧ 𝑢 ∈ 𝑋))
14	13	adantr 480	. . . . . . . 8 ⊢ (((𝐺 ∈ GrpOp ∧ (𝑢 ∈ 𝑋 ∧ ∀𝑧 ∈ 𝑋 ((𝑢𝐺𝑧) = 𝑧 ∧ ∃𝑤 ∈ 𝑋 (𝑤𝐺𝑧) = 𝑢))) ∧ 𝑥 ∈ 𝑋) → (𝐺 ∈ GrpOp ∧ 𝑢 ∈ 𝑋))
15	2	adantl 481	. . . . . . . . 9 ⊢ ((𝑢 ∈ 𝑋 ∧ ∀𝑧 ∈ 𝑋 ((𝑢𝐺𝑧) = 𝑧 ∧ ∃𝑤 ∈ 𝑋 (𝑤𝐺𝑧) = 𝑢)) → ∀𝑧 ∈ 𝑋 (𝑢𝐺𝑧) = 𝑧)
16	15	ad2antlr 727	. . . . . . . 8 ⊢ (((𝐺 ∈ GrpOp ∧ (𝑢 ∈ 𝑋 ∧ ∀𝑧 ∈ 𝑋 ((𝑢𝐺𝑧) = 𝑧 ∧ ∃𝑤 ∈ 𝑋 (𝑤𝐺𝑧) = 𝑢))) ∧ 𝑥 ∈ 𝑋) → ∀𝑧 ∈ 𝑋 (𝑢𝐺𝑧) = 𝑧)
17		simpr 484	. . . . . . . . . . 11 ⊢ (((𝑢𝐺𝑧) = 𝑧 ∧ ∃𝑤 ∈ 𝑋 (𝑤𝐺𝑧) = 𝑢) → ∃𝑤 ∈ 𝑋 (𝑤𝐺𝑧) = 𝑢)
18	17	ralimi 3073	. . . . . . . . . 10 ⊢ (∀𝑧 ∈ 𝑋 ((𝑢𝐺𝑧) = 𝑧 ∧ ∃𝑤 ∈ 𝑋 (𝑤𝐺𝑧) = 𝑢) → ∀𝑧 ∈ 𝑋 ∃𝑤 ∈ 𝑋 (𝑤𝐺𝑧) = 𝑢)
19	18	adantl 481	. . . . . . . . 9 ⊢ ((𝑢 ∈ 𝑋 ∧ ∀𝑧 ∈ 𝑋 ((𝑢𝐺𝑧) = 𝑧 ∧ ∃𝑤 ∈ 𝑋 (𝑤𝐺𝑧) = 𝑢)) → ∀𝑧 ∈ 𝑋 ∃𝑤 ∈ 𝑋 (𝑤𝐺𝑧) = 𝑢)
20	19	ad2antlr 727	. . . . . . . 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 515	. . . . . . 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 261	. . . . . . . 8 ⊢ (∀𝑧 ∈ 𝑋 (𝑢𝐺𝑧) = 𝑧 ↔ ∀𝑧 ∈ 𝑋 (𝑢𝐺𝑧) = 𝑧)
24		biid 261	. . . . . . . 8 ⊢ (∀𝑧 ∈ 𝑋 ∃𝑤 ∈ 𝑋 (𝑤𝐺𝑧) = 𝑢 ↔ ∀𝑧 ∈ 𝑋 ∃𝑤 ∈ 𝑋 (𝑤𝐺𝑧) = 𝑢)
25	22, 23, 24	grpoidinvlem3 30487	. . . . . . 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 588	. . . . . 6 ⊢ (((𝐺 ∈ GrpOp ∧ (𝑢 ∈ 𝑋 ∧ ∀𝑧 ∈ 𝑋 ((𝑢𝐺𝑧) = 𝑧 ∧ ∃𝑤 ∈ 𝑋 (𝑤𝐺𝑧) = 𝑢))) ∧ 𝑥 ∈ 𝑋) → ∃𝑦 ∈ 𝑋 ((𝑦𝐺𝑥) = 𝑢 ∧ (𝑥𝐺𝑦) = 𝑢))
27	22	grpoidinvlem4 30488	. . . . . 6 ⊢ (((𝐺 ∈ GrpOp ∧ 𝑥 ∈ 𝑋) ∧ ∃𝑦 ∈ 𝑋 ((𝑦𝐺𝑥) = 𝑢 ∧ (𝑥𝐺𝑦) = 𝑢)) → (𝑥𝐺𝑢) = (𝑢𝐺𝑥))
28	11, 26, 27	syl2anc 584	. . . . 5 ⊢ (((𝐺 ∈ GrpOp ∧ (𝑢 ∈ 𝑋 ∧ ∀𝑧 ∈ 𝑋 ((𝑢𝐺𝑧) = 𝑧 ∧ ∃𝑤 ∈ 𝑋 (𝑤𝐺𝑧) = 𝑢))) ∧ 𝑥 ∈ 𝑋) → (𝑥𝐺𝑢) = (𝑢𝐺𝑥))
29	28, 9	eqtrd 2770	. . . 4 ⊢ (((𝐺 ∈ GrpOp ∧ (𝑢 ∈ 𝑋 ∧ ∀𝑧 ∈ 𝑋 ((𝑢𝐺𝑧) = 𝑧 ∧ ∃𝑤 ∈ 𝑋 (𝑤𝐺𝑧) = 𝑢))) ∧ 𝑥 ∈ 𝑋) → (𝑥𝐺𝑢) = 𝑥)
30	9, 29, 26	jca31 514	. . 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 3132	. 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 30482	. 2 ⊢ (𝐺 ∈ GrpOp → ∃𝑢 ∈ 𝑋 ∀𝑧 ∈ 𝑋 ((𝑢𝐺𝑧) = 𝑧 ∧ ∃𝑤 ∈ 𝑋 (𝑤𝐺𝑧) = 𝑢))
33	31, 32	reximddv 3156	1 ⊢ (𝐺 ∈ GrpOp → ∃𝑢 ∈ 𝑋 ∀𝑥 ∈ 𝑋 (((𝑢𝐺𝑥) = 𝑥 ∧ (𝑥𝐺𝑢) = 𝑥) ∧ ∃𝑦 ∈ 𝑋 ((𝑦𝐺𝑥) = 𝑢 ∧ (𝑥𝐺𝑦) = 𝑢)))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1540   ∈ wcel 2108  ∀wral 3051  ∃wrex 3060  ran crn 5655  (class class class)co 7405  GrpOpcgr 30470

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2707  ax-sep 5266  ax-nul 5276  ax-pr 5402  ax-un 7729

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2727  df-clel 2809  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3061  df-rab 3416  df-v 3461  df-sbc 3766  df-csb 3875  df-dif 3929  df-un 3931  df-in 3933  df-ss 3943  df-nul 4309  df-if 4501  df-sn 4602  df-pr 4604  df-op 4608  df-uni 4884  df-iun 4969  df-br 5120  df-opab 5182  df-mpt 5202  df-id 5548  df-xp 5660  df-rel 5661  df-cnv 5662  df-co 5663  df-dm 5664  df-rn 5665  df-iota 6484  df-fun 6533  df-fn 6534  df-f 6535  df-fo 6537  df-fv 6539  df-ov 7408  df-grpo 30474

This theorem is referenced by:  grpoideu  30490  grpoidval  30494  grpoidinv2  30496  grpomndo  37899