Theorem grpoidinv 28870

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 483	. . . . . . . 8 ⊢ (((𝑢𝐺𝑧) = 𝑧 ∧ ∃𝑤 ∈ 𝑋 (𝑤𝐺𝑧) = 𝑢) → (𝑢𝐺𝑧) = 𝑧)
2	1	ralimi 3087	. . . . . . 7 ⊢ (∀𝑧 ∈ 𝑋 ((𝑢𝐺𝑧) = 𝑧 ∧ ∃𝑤 ∈ 𝑋 (𝑤𝐺𝑧) = 𝑢) → ∀𝑧 ∈ 𝑋 (𝑢𝐺𝑧) = 𝑧)
3		oveq2 7283	. . . . . . . . 9 ⊢ (𝑧 = 𝑥 → (𝑢𝐺𝑧) = (𝑢𝐺𝑥))
4		id 22	. . . . . . . . 9 ⊢ (𝑧 = 𝑥 → 𝑧 = 𝑥)
5	3, 4	eqeq12d 2754	. . . . . . . 8 ⊢ (𝑧 = 𝑥 → ((𝑢𝐺𝑧) = 𝑧 ↔ (𝑢𝐺𝑥) = 𝑥))
6	5	rspccva 3560	. . . . . . 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 711	. . . . 5 ⊢ (((𝑢 ∈ 𝑋 ∧ ∀𝑧 ∈ 𝑋 ((𝑢𝐺𝑧) = 𝑧 ∧ ∃𝑤 ∈ 𝑋 (𝑤𝐺𝑧) = 𝑢)) ∧ 𝑥 ∈ 𝑋) → (𝑢𝐺𝑥) = 𝑥)
9	8	adantll 711	. . . 4 ⊢ (((𝐺 ∈ GrpOp ∧ (𝑢 ∈ 𝑋 ∧ ∀𝑧 ∈ 𝑋 ((𝑢𝐺𝑧) = 𝑧 ∧ ∃𝑤 ∈ 𝑋 (𝑤𝐺𝑧) = 𝑢))) ∧ 𝑥 ∈ 𝑋) → (𝑢𝐺𝑥) = 𝑥)
10		simpl 483	. . . . . . 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 714	. . . . . . . . 9 ⊢ ((𝐺 ∈ GrpOp ∧ (𝑢 ∈ 𝑋 ∧ ∀𝑧 ∈ 𝑋 ((𝑢𝐺𝑧) = 𝑧 ∧ ∃𝑤 ∈ 𝑋 (𝑤𝐺𝑧) = 𝑢))) → (𝐺 ∈ GrpOp ∧ 𝑢 ∈ 𝑋))
14	13	adantr 481	. . . . . . . 8 ⊢ (((𝐺 ∈ GrpOp ∧ (𝑢 ∈ 𝑋 ∧ ∀𝑧 ∈ 𝑋 ((𝑢𝐺𝑧) = 𝑧 ∧ ∃𝑤 ∈ 𝑋 (𝑤𝐺𝑧) = 𝑢))) ∧ 𝑥 ∈ 𝑋) → (𝐺 ∈ GrpOp ∧ 𝑢 ∈ 𝑋))
15	2	adantl 482	. . . . . . . . 9 ⊢ ((𝑢 ∈ 𝑋 ∧ ∀𝑧 ∈ 𝑋 ((𝑢𝐺𝑧) = 𝑧 ∧ ∃𝑤 ∈ 𝑋 (𝑤𝐺𝑧) = 𝑢)) → ∀𝑧 ∈ 𝑋 (𝑢𝐺𝑧) = 𝑧)
16	15	ad2antlr 724	. . . . . . . 8 ⊢ (((𝐺 ∈ GrpOp ∧ (𝑢 ∈ 𝑋 ∧ ∀𝑧 ∈ 𝑋 ((𝑢𝐺𝑧) = 𝑧 ∧ ∃𝑤 ∈ 𝑋 (𝑤𝐺𝑧) = 𝑢))) ∧ 𝑥 ∈ 𝑋) → ∀𝑧 ∈ 𝑋 (𝑢𝐺𝑧) = 𝑧)
17		simpr 485	. . . . . . . . . . 11 ⊢ (((𝑢𝐺𝑧) = 𝑧 ∧ ∃𝑤 ∈ 𝑋 (𝑤𝐺𝑧) = 𝑢) → ∃𝑤 ∈ 𝑋 (𝑤𝐺𝑧) = 𝑢)
18	17	ralimi 3087	. . . . . . . . . 10 ⊢ (∀𝑧 ∈ 𝑋 ((𝑢𝐺𝑧) = 𝑧 ∧ ∃𝑤 ∈ 𝑋 (𝑤𝐺𝑧) = 𝑢) → ∀𝑧 ∈ 𝑋 ∃𝑤 ∈ 𝑋 (𝑤𝐺𝑧) = 𝑢)
19	18	adantl 482	. . . . . . . . 9 ⊢ ((𝑢 ∈ 𝑋 ∧ ∀𝑧 ∈ 𝑋 ((𝑢𝐺𝑧) = 𝑧 ∧ ∃𝑤 ∈ 𝑋 (𝑤𝐺𝑧) = 𝑢)) → ∀𝑧 ∈ 𝑋 ∃𝑤 ∈ 𝑋 (𝑤𝐺𝑧) = 𝑢)
20	19	ad2antlr 724	. . . . . . . 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 516	. . . . . . 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 260	. . . . . . . 8 ⊢ (∀𝑧 ∈ 𝑋 (𝑢𝐺𝑧) = 𝑧 ↔ ∀𝑧 ∈ 𝑋 (𝑢𝐺𝑧) = 𝑧)
24		biid 260	. . . . . . . 8 ⊢ (∀𝑧 ∈ 𝑋 ∃𝑤 ∈ 𝑋 (𝑤𝐺𝑧) = 𝑢 ↔ ∀𝑧 ∈ 𝑋 ∃𝑤 ∈ 𝑋 (𝑤𝐺𝑧) = 𝑢)
25	22, 23, 24	grpoidinvlem3 28868	. . . . . . 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 28869	. . . . . 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 2778	. . . 4 ⊢ (((𝐺 ∈ GrpOp ∧ (𝑢 ∈ 𝑋 ∧ ∀𝑧 ∈ 𝑋 ((𝑢𝐺𝑧) = 𝑧 ∧ ∃𝑤 ∈ 𝑋 (𝑤𝐺𝑧) = 𝑢))) ∧ 𝑥 ∈ 𝑋) → (𝑥𝐺𝑢) = 𝑥)
30	9, 29, 26	jca31 515	. . 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 3103	. 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 28863	. 2 ⊢ (𝐺 ∈ GrpOp → ∃𝑢 ∈ 𝑋 ∀𝑧 ∈ 𝑋 ((𝑢𝐺𝑧) = 𝑧 ∧ ∃𝑤 ∈ 𝑋 (𝑤𝐺𝑧) = 𝑢))
33	31, 32	reximddv 3204	1 ⊢ (𝐺 ∈ GrpOp → ∃𝑢 ∈ 𝑋 ∀𝑥 ∈ 𝑋 (((𝑢𝐺𝑥) = 𝑥 ∧ (𝑥𝐺𝑢) = 𝑥) ∧ ∃𝑦 ∈ 𝑋 ((𝑦𝐺𝑥) = 𝑢 ∧ (𝑥𝐺𝑦) = 𝑢)))

Syntax hints:   → wi 4   ∧ wa 396   = wceq 1539   ∈ wcel 2106  ∀wral 3064  ∃wrex 3065  ran crn 5590  (class class class)co 7275  GrpOpcgr 28851

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-sep 5223  ax-nul 5230  ax-pr 5352  ax-un 7588

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ral 3069  df-rex 3070  df-rab 3073  df-v 3434  df-sbc 3717  df-csb 3833  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-iun 4926  df-br 5075  df-opab 5137  df-mpt 5158  df-id 5489  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-iota 6391  df-fun 6435  df-fn 6436  df-f 6437  df-fo 6439  df-fv 6441  df-ov 7278  df-grpo 28855

This theorem is referenced by:  grpoideu  28871  grpoidval  28875  grpoidinv2  28877  grpomndo  36033