Theorem grpoidinv 28291

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 486	. . . . . . . 8 ⊢ (((𝑢𝐺𝑧) = 𝑧 ∧ ∃𝑤 ∈ 𝑋 (𝑤𝐺𝑧) = 𝑢) → (𝑢𝐺𝑧) = 𝑧)
2	1	ralimi 3128	. . . . . . 7 ⊢ (∀𝑧 ∈ 𝑋 ((𝑢𝐺𝑧) = 𝑧 ∧ ∃𝑤 ∈ 𝑋 (𝑤𝐺𝑧) = 𝑢) → ∀𝑧 ∈ 𝑋 (𝑢𝐺𝑧) = 𝑧)
3		oveq2 7143	. . . . . . . . 9 ⊢ (𝑧 = 𝑥 → (𝑢𝐺𝑧) = (𝑢𝐺𝑥))
4		id 22	. . . . . . . . 9 ⊢ (𝑧 = 𝑥 → 𝑧 = 𝑥)
5	3, 4	eqeq12d 2814	. . . . . . . 8 ⊢ (𝑧 = 𝑥 → ((𝑢𝐺𝑧) = 𝑧 ↔ (𝑢𝐺𝑥) = 𝑥))
6	5	rspccva 3570	. . . . . . 7 ⊢ ((∀𝑧 ∈ 𝑋 (𝑢𝐺𝑧) = 𝑧 ∧ 𝑥 ∈ 𝑋) → (𝑢𝐺𝑥) = 𝑥)
7	2, 6	sylan 583	. . . . . 6 ⊢ ((∀𝑧 ∈ 𝑋 ((𝑢𝐺𝑧) = 𝑧 ∧ ∃𝑤 ∈ 𝑋 (𝑤𝐺𝑧) = 𝑢) ∧ 𝑥 ∈ 𝑋) → (𝑢𝐺𝑥) = 𝑥)
8	7	adantll 713	. . . . 5 ⊢ (((𝑢 ∈ 𝑋 ∧ ∀𝑧 ∈ 𝑋 ((𝑢𝐺𝑧) = 𝑧 ∧ ∃𝑤 ∈ 𝑋 (𝑤𝐺𝑧) = 𝑢)) ∧ 𝑥 ∈ 𝑋) → (𝑢𝐺𝑥) = 𝑥)
9	8	adantll 713	. . . 4 ⊢ (((𝐺 ∈ GrpOp ∧ (𝑢 ∈ 𝑋 ∧ ∀𝑧 ∈ 𝑋 ((𝑢𝐺𝑧) = 𝑧 ∧ ∃𝑤 ∈ 𝑋 (𝑤𝐺𝑧) = 𝑢))) ∧ 𝑥 ∈ 𝑋) → (𝑢𝐺𝑥) = 𝑥)
10		simpl 486	. . . . . . 7 ⊢ ((𝐺 ∈ GrpOp ∧ (𝑢 ∈ 𝑋 ∧ ∀𝑧 ∈ 𝑋 ((𝑢𝐺𝑧) = 𝑧 ∧ ∃𝑤 ∈ 𝑋 (𝑤𝐺𝑧) = 𝑢))) → 𝐺 ∈ GrpOp)
11	10	anim1i 617	. . . . . 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 716	. . . . . . . . 9 ⊢ ((𝐺 ∈ GrpOp ∧ (𝑢 ∈ 𝑋 ∧ ∀𝑧 ∈ 𝑋 ((𝑢𝐺𝑧) = 𝑧 ∧ ∃𝑤 ∈ 𝑋 (𝑤𝐺𝑧) = 𝑢))) → (𝐺 ∈ GrpOp ∧ 𝑢 ∈ 𝑋))
14	13	adantr 484	. . . . . . . 8 ⊢ (((𝐺 ∈ GrpOp ∧ (𝑢 ∈ 𝑋 ∧ ∀𝑧 ∈ 𝑋 ((𝑢𝐺𝑧) = 𝑧 ∧ ∃𝑤 ∈ 𝑋 (𝑤𝐺𝑧) = 𝑢))) ∧ 𝑥 ∈ 𝑋) → (𝐺 ∈ GrpOp ∧ 𝑢 ∈ 𝑋))
15	2	adantl 485	. . . . . . . . 9 ⊢ ((𝑢 ∈ 𝑋 ∧ ∀𝑧 ∈ 𝑋 ((𝑢𝐺𝑧) = 𝑧 ∧ ∃𝑤 ∈ 𝑋 (𝑤𝐺𝑧) = 𝑢)) → ∀𝑧 ∈ 𝑋 (𝑢𝐺𝑧) = 𝑧)
16	15	ad2antlr 726	. . . . . . . 8 ⊢ (((𝐺 ∈ GrpOp ∧ (𝑢 ∈ 𝑋 ∧ ∀𝑧 ∈ 𝑋 ((𝑢𝐺𝑧) = 𝑧 ∧ ∃𝑤 ∈ 𝑋 (𝑤𝐺𝑧) = 𝑢))) ∧ 𝑥 ∈ 𝑋) → ∀𝑧 ∈ 𝑋 (𝑢𝐺𝑧) = 𝑧)
17		simpr 488	. . . . . . . . . . 11 ⊢ (((𝑢𝐺𝑧) = 𝑧 ∧ ∃𝑤 ∈ 𝑋 (𝑤𝐺𝑧) = 𝑢) → ∃𝑤 ∈ 𝑋 (𝑤𝐺𝑧) = 𝑢)
18	17	ralimi 3128	. . . . . . . . . 10 ⊢ (∀𝑧 ∈ 𝑋 ((𝑢𝐺𝑧) = 𝑧 ∧ ∃𝑤 ∈ 𝑋 (𝑤𝐺𝑧) = 𝑢) → ∀𝑧 ∈ 𝑋 ∃𝑤 ∈ 𝑋 (𝑤𝐺𝑧) = 𝑢)
19	18	adantl 485	. . . . . . . . 9 ⊢ ((𝑢 ∈ 𝑋 ∧ ∀𝑧 ∈ 𝑋 ((𝑢𝐺𝑧) = 𝑧 ∧ ∃𝑤 ∈ 𝑋 (𝑤𝐺𝑧) = 𝑢)) → ∀𝑧 ∈ 𝑋 ∃𝑤 ∈ 𝑋 (𝑤𝐺𝑧) = 𝑢)
20	19	ad2antlr 726	. . . . . . . 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 519	. . . . . . 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 264	. . . . . . . 8 ⊢ (∀𝑧 ∈ 𝑋 (𝑢𝐺𝑧) = 𝑧 ↔ ∀𝑧 ∈ 𝑋 (𝑢𝐺𝑧) = 𝑧)
24		biid 264	. . . . . . . 8 ⊢ (∀𝑧 ∈ 𝑋 ∃𝑤 ∈ 𝑋 (𝑤𝐺𝑧) = 𝑢 ↔ ∀𝑧 ∈ 𝑋 ∃𝑤 ∈ 𝑋 (𝑤𝐺𝑧) = 𝑢)
25	22, 23, 24	grpoidinvlem3 28289	. . . . . . 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 591	. . . . . 6 ⊢ (((𝐺 ∈ GrpOp ∧ (𝑢 ∈ 𝑋 ∧ ∀𝑧 ∈ 𝑋 ((𝑢𝐺𝑧) = 𝑧 ∧ ∃𝑤 ∈ 𝑋 (𝑤𝐺𝑧) = 𝑢))) ∧ 𝑥 ∈ 𝑋) → ∃𝑦 ∈ 𝑋 ((𝑦𝐺𝑥) = 𝑢 ∧ (𝑥𝐺𝑦) = 𝑢))
27	22	grpoidinvlem4 28290	. . . . . 6 ⊢ (((𝐺 ∈ GrpOp ∧ 𝑥 ∈ 𝑋) ∧ ∃𝑦 ∈ 𝑋 ((𝑦𝐺𝑥) = 𝑢 ∧ (𝑥𝐺𝑦) = 𝑢)) → (𝑥𝐺𝑢) = (𝑢𝐺𝑥))
28	11, 26, 27	syl2anc 587	. . . . 5 ⊢ (((𝐺 ∈ GrpOp ∧ (𝑢 ∈ 𝑋 ∧ ∀𝑧 ∈ 𝑋 ((𝑢𝐺𝑧) = 𝑧 ∧ ∃𝑤 ∈ 𝑋 (𝑤𝐺𝑧) = 𝑢))) ∧ 𝑥 ∈ 𝑋) → (𝑥𝐺𝑢) = (𝑢𝐺𝑥))
29	28, 9	eqtrd 2833	. . . 4 ⊢ (((𝐺 ∈ GrpOp ∧ (𝑢 ∈ 𝑋 ∧ ∀𝑧 ∈ 𝑋 ((𝑢𝐺𝑧) = 𝑧 ∧ ∃𝑤 ∈ 𝑋 (𝑤𝐺𝑧) = 𝑢))) ∧ 𝑥 ∈ 𝑋) → (𝑥𝐺𝑢) = 𝑥)
30	9, 29, 26	jca31 518	. . 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 3149	. 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 28284	. 2 ⊢ (𝐺 ∈ GrpOp → ∃𝑢 ∈ 𝑋 ∀𝑧 ∈ 𝑋 ((𝑢𝐺𝑧) = 𝑧 ∧ ∃𝑤 ∈ 𝑋 (𝑤𝐺𝑧) = 𝑢))
33	31, 32	reximddv 3234	1 ⊢ (𝐺 ∈ GrpOp → ∃𝑢 ∈ 𝑋 ∀𝑥 ∈ 𝑋 (((𝑢𝐺𝑥) = 𝑥 ∧ (𝑥𝐺𝑢) = 𝑥) ∧ ∃𝑦 ∈ 𝑋 ((𝑦𝐺𝑥) = 𝑢 ∧ (𝑥𝐺𝑦) = 𝑢)))

Syntax hints:   → wi 4   ∧ wa 399   = wceq 1538   ∈ wcel 2111  ∀wral 3106  ∃wrex 3107  ran crn 5520  (class class class)co 7135  GrpOpcgr 28272

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-sep 5167  ax-nul 5174  ax-pr 5295  ax-un 7441

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ral 3111  df-rex 3112  df-rab 3115  df-v 3443  df-sbc 3721  df-csb 3829  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-nul 4244  df-if 4426  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4801  df-iun 4883  df-br 5031  df-opab 5093  df-mpt 5111  df-id 5425  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-rn 5530  df-iota 6283  df-fun 6326  df-fn 6327  df-f 6328  df-fo 6330  df-fv 6332  df-ov 7138  df-grpo 28276

This theorem is referenced by:  grpoideu  28292  grpoidval  28296  grpoidinv2  28298  grpomndo  35313