Theorem grpoidinv 31103

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 488	. . . . . . . 8 ⊢ (((𝑢𝐺𝑧) = 𝑧 ∧ ∃𝑤 ∈ 𝑋 (𝑤𝐺𝑧) = 𝑢) → (𝑢𝐺𝑧) = 𝑧)
2	1	ralimi 3100	. . . . . . 7 ⊢ (∀𝑧 ∈ 𝑋 ((𝑢𝐺𝑧) = 𝑧 ∧ ∃𝑤 ∈ 𝑋 (𝑤𝐺𝑧) = 𝑢) → ∀𝑧 ∈ 𝑋 (𝑢𝐺𝑧) = 𝑧)
3		oveq2 7426	. . . . . . . . 9 ⊢ (𝑧 = 𝑥 → (𝑢𝐺𝑧) = (𝑢𝐺𝑥))
4		id 23	. . . . . . . . 9 ⊢ (𝑧 = 𝑥 → 𝑧 = 𝑥)
5	3, 4	eqeq12d 2777	. . . . . . . 8 ⊢ (𝑧 = 𝑥 → ((𝑢𝐺𝑧) = 𝑧 ↔ (𝑢𝐺𝑥) = 𝑥))
6	5	rspccva 3576	. . . . . . 7 ⊢ ((∀𝑧 ∈ 𝑋 (𝑢𝐺𝑧) = 𝑧 ∧ 𝑥 ∈ 𝑋) → (𝑢𝐺𝑥) = 𝑥)
7	2, 6	sylan 592	. . . . . 6 ⊢ ((∀𝑧 ∈ 𝑋 ((𝑢𝐺𝑧) = 𝑧 ∧ ∃𝑤 ∈ 𝑋 (𝑤𝐺𝑧) = 𝑢) ∧ 𝑥 ∈ 𝑋) → (𝑢𝐺𝑥) = 𝑥)
8	7	adantll 727	. . . . 5 ⊢ (((𝑢 ∈ 𝑋 ∧ ∀𝑧 ∈ 𝑋 ((𝑢𝐺𝑧) = 𝑧 ∧ ∃𝑤 ∈ 𝑋 (𝑤𝐺𝑧) = 𝑢)) ∧ 𝑥 ∈ 𝑋) → (𝑢𝐺𝑥) = 𝑥)
9	8	adantll 727	. . . 4 ⊢ (((𝐺 ∈ GrpOp ∧ (𝑢 ∈ 𝑋 ∧ ∀𝑧 ∈ 𝑋 ((𝑢𝐺𝑧) = 𝑧 ∧ ∃𝑤 ∈ 𝑋 (𝑤𝐺𝑧) = 𝑢))) ∧ 𝑥 ∈ 𝑋) → (𝑢𝐺𝑥) = 𝑥)
10		simpl 488	. . . . . . 7 ⊢ ((𝐺 ∈ GrpOp ∧ (𝑢 ∈ 𝑋 ∧ ∀𝑧 ∈ 𝑋 ((𝑢𝐺𝑧) = 𝑧 ∧ ∃𝑤 ∈ 𝑋 (𝑤𝐺𝑧) = 𝑢))) → 𝐺 ∈ GrpOp)
11	10	anim1i 627	. . . . . 6 ⊢ (((𝐺 ∈ GrpOp ∧ (𝑢 ∈ 𝑋 ∧ ∀𝑧 ∈ 𝑋 ((𝑢𝐺𝑧) = 𝑧 ∧ ∃𝑤 ∈ 𝑋 (𝑤𝐺𝑧) = 𝑢))) ∧ 𝑥 ∈ 𝑋) → (𝐺 ∈ GrpOp ∧ 𝑥 ∈ 𝑋))
12		id 23	. . . . . . . . . 10 ⊢ ((𝐺 ∈ GrpOp ∧ 𝑢 ∈ 𝑋) → (𝐺 ∈ GrpOp ∧ 𝑢 ∈ 𝑋))
13	12	adantrr 730	. . . . . . . . 9 ⊢ ((𝐺 ∈ GrpOp ∧ (𝑢 ∈ 𝑋 ∧ ∀𝑧 ∈ 𝑋 ((𝑢𝐺𝑧) = 𝑧 ∧ ∃𝑤 ∈ 𝑋 (𝑤𝐺𝑧) = 𝑢))) → (𝐺 ∈ GrpOp ∧ 𝑢 ∈ 𝑋))
14	13	adantr 486	. . . . . . . 8 ⊢ (((𝐺 ∈ GrpOp ∧ (𝑢 ∈ 𝑋 ∧ ∀𝑧 ∈ 𝑋 ((𝑢𝐺𝑧) = 𝑧 ∧ ∃𝑤 ∈ 𝑋 (𝑤𝐺𝑧) = 𝑢))) ∧ 𝑥 ∈ 𝑋) → (𝐺 ∈ GrpOp ∧ 𝑢 ∈ 𝑋))
15	2	adantl 487	. . . . . . . . 9 ⊢ ((𝑢 ∈ 𝑋 ∧ ∀𝑧 ∈ 𝑋 ((𝑢𝐺𝑧) = 𝑧 ∧ ∃𝑤 ∈ 𝑋 (𝑤𝐺𝑧) = 𝑢)) → ∀𝑧 ∈ 𝑋 (𝑢𝐺𝑧) = 𝑧)
16	15	ad2antlr 740	. . . . . . . 8 ⊢ (((𝐺 ∈ GrpOp ∧ (𝑢 ∈ 𝑋 ∧ ∀𝑧 ∈ 𝑋 ((𝑢𝐺𝑧) = 𝑧 ∧ ∃𝑤 ∈ 𝑋 (𝑤𝐺𝑧) = 𝑢))) ∧ 𝑥 ∈ 𝑋) → ∀𝑧 ∈ 𝑋 (𝑢𝐺𝑧) = 𝑧)
17		simpr 490	. . . . . . . . . . 11 ⊢ (((𝑢𝐺𝑧) = 𝑧 ∧ ∃𝑤 ∈ 𝑋 (𝑤𝐺𝑧) = 𝑢) → ∃𝑤 ∈ 𝑋 (𝑤𝐺𝑧) = 𝑢)
18	17	ralimi 3100	. . . . . . . . . 10 ⊢ (∀𝑧 ∈ 𝑋 ((𝑢𝐺𝑧) = 𝑧 ∧ ∃𝑤 ∈ 𝑋 (𝑤𝐺𝑧) = 𝑢) → ∀𝑧 ∈ 𝑋 ∃𝑤 ∈ 𝑋 (𝑤𝐺𝑧) = 𝑢)
19	18	adantl 487	. . . . . . . . 9 ⊢ ((𝑢 ∈ 𝑋 ∧ ∀𝑧 ∈ 𝑋 ((𝑢𝐺𝑧) = 𝑧 ∧ ∃𝑤 ∈ 𝑋 (𝑤𝐺𝑧) = 𝑢)) → ∀𝑧 ∈ 𝑋 ∃𝑤 ∈ 𝑋 (𝑤𝐺𝑧) = 𝑢)
20	19	ad2antlr 740	. . . . . . . 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 525	. . . . . . 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 31101	. . . . . . 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 600	. . . . . 6 ⊢ (((𝐺 ∈ GrpOp ∧ (𝑢 ∈ 𝑋 ∧ ∀𝑧 ∈ 𝑋 ((𝑢𝐺𝑧) = 𝑧 ∧ ∃𝑤 ∈ 𝑋 (𝑤𝐺𝑧) = 𝑢))) ∧ 𝑥 ∈ 𝑋) → ∃𝑦 ∈ 𝑋 ((𝑦𝐺𝑥) = 𝑢 ∧ (𝑥𝐺𝑦) = 𝑢))
27	22	grpoidinvlem4 31102	. . . . . 6 ⊢ (((𝐺 ∈ GrpOp ∧ 𝑥 ∈ 𝑋) ∧ ∃𝑦 ∈ 𝑋 ((𝑦𝐺𝑥) = 𝑢 ∧ (𝑥𝐺𝑦) = 𝑢)) → (𝑥𝐺𝑢) = (𝑢𝐺𝑥))
28	11, 26, 27	syl2anc 596	. . . . 5 ⊢ (((𝐺 ∈ GrpOp ∧ (𝑢 ∈ 𝑋 ∧ ∀𝑧 ∈ 𝑋 ((𝑢𝐺𝑧) = 𝑧 ∧ ∃𝑤 ∈ 𝑋 (𝑤𝐺𝑧) = 𝑢))) ∧ 𝑥 ∈ 𝑋) → (𝑥𝐺𝑢) = (𝑢𝐺𝑥))
29	28, 9	eqtrd 2796	. . . 4 ⊢ (((𝐺 ∈ GrpOp ∧ (𝑢 ∈ 𝑋 ∧ ∀𝑧 ∈ 𝑋 ((𝑢𝐺𝑧) = 𝑧 ∧ ∃𝑤 ∈ 𝑋 (𝑤𝐺𝑧) = 𝑢))) ∧ 𝑥 ∈ 𝑋) → (𝑥𝐺𝑢) = 𝑥)
30	9, 29, 26	jca31 524	. . 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 3155	. 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 31096	. 2 ⊢ (𝐺 ∈ GrpOp → ∃𝑢 ∈ 𝑋 ∀𝑧 ∈ 𝑋 ((𝑢𝐺𝑧) = 𝑧 ∧ ∃𝑤 ∈ 𝑋 (𝑤𝐺𝑧) = 𝑢))
33	31, 32	reximddv 3179	1 ⊢ (𝐺 ∈ GrpOp → ∃𝑢 ∈ 𝑋 ∀𝑥 ∈ 𝑋 (((𝑢𝐺𝑥) = 𝑥 ∧ (𝑥𝐺𝑢) = 𝑥) ∧ ∃𝑦 ∈ 𝑋 ((𝑦𝐺𝑥) = 𝑢 ∧ (𝑥𝐺𝑦) = 𝑢)))

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1570   ∈ wcel 2145  ∀wral 3077  ∃wrex 3087  ran crn 5652  (class class class)co 7418  GrpOpcgr 31084

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2733  ax-sep 5249  ax-nul 5260  ax-pr 5391  ax-un 7749

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-ral 3078  df-rex 3088  df-rab 3414  df-v 3453  df-sbc 3740  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4280  df-if 4483  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-iun 4953  df-br 5104  df-opab 5168  df-mpt 5187  df-id 5546  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-rn 5662  df-iota 6493  df-fun 6539  df-fn 6540  df-f 6541  df-fo 6543  df-fv 6545  df-ov 7421  df-grpo 31088

This theorem is used by:  grpoideu  31104  grpoidval  31108  grpoidinv2  31110  grpomndo  38789