Theorem grpoidinvlem4 28278

Description: Lemma for grpoidinv 28279. (Contributed by NM, 14-Oct-2006.) (New usage is discouraged.)

Hypothesis

Ref	Expression
grpfo.1	⊢ 𝑋 = ran 𝐺

Assertion

Ref	Expression
grpoidinvlem4	⊢ (((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋) ∧ ∃𝑦 ∈ 𝑋 ((𝑦𝐺𝐴) = 𝑈 ∧ (𝐴𝐺𝑦) = 𝑈)) → (𝐴𝐺𝑈) = (𝑈𝐺𝐴))

Distinct variable groups:   𝑦,𝐴   𝑦,𝐺   𝑦,𝑋   𝑦,𝑈

Proof of Theorem grpoidinvlem4

Step	Hyp	Ref	Expression
1		simpll 765	. . . . . 6 ⊢ (((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋) ∧ 𝑦 ∈ 𝑋) → 𝐺 ∈ GrpOp)
2		simplr 767	. . . . . 6 ⊢ (((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋) ∧ 𝑦 ∈ 𝑋) → 𝐴 ∈ 𝑋)
3		simpr 487	. . . . . 6 ⊢ (((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋) ∧ 𝑦 ∈ 𝑋) → 𝑦 ∈ 𝑋)
4		grpfo.1	. . . . . . 7 ⊢ 𝑋 = ran 𝐺
5	4	grpoass 28274	. . . . . 6 ⊢ ((𝐺 ∈ GrpOp ∧ (𝐴 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋 ∧ 𝐴 ∈ 𝑋)) → ((𝐴𝐺𝑦)𝐺𝐴) = (𝐴𝐺(𝑦𝐺𝐴)))
6	1, 2, 3, 2, 5	syl13anc 1368	. . . . 5 ⊢ (((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋) ∧ 𝑦 ∈ 𝑋) → ((𝐴𝐺𝑦)𝐺𝐴) = (𝐴𝐺(𝑦𝐺𝐴)))
7		oveq2 7158	. . . . 5 ⊢ ((𝑦𝐺𝐴) = 𝑈 → (𝐴𝐺(𝑦𝐺𝐴)) = (𝐴𝐺𝑈))
8	6, 7	sylan9eq 2876	. . . 4 ⊢ ((((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋) ∧ 𝑦 ∈ 𝑋) ∧ (𝑦𝐺𝐴) = 𝑈) → ((𝐴𝐺𝑦)𝐺𝐴) = (𝐴𝐺𝑈))
9		oveq1 7157	. . . 4 ⊢ ((𝐴𝐺𝑦) = 𝑈 → ((𝐴𝐺𝑦)𝐺𝐴) = (𝑈𝐺𝐴))
10	8, 9	sylan9req 2877	. . 3 ⊢ (((((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋) ∧ 𝑦 ∈ 𝑋) ∧ (𝑦𝐺𝐴) = 𝑈) ∧ (𝐴𝐺𝑦) = 𝑈) → (𝐴𝐺𝑈) = (𝑈𝐺𝐴))
11	10	anasss 469	. 2 ⊢ ((((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋) ∧ 𝑦 ∈ 𝑋) ∧ ((𝑦𝐺𝐴) = 𝑈 ∧ (𝐴𝐺𝑦) = 𝑈)) → (𝐴𝐺𝑈) = (𝑈𝐺𝐴))
12	11	r19.29an 3288	1 ⊢ (((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋) ∧ ∃𝑦 ∈ 𝑋 ((𝑦𝐺𝐴) = 𝑈 ∧ (𝐴𝐺𝑦) = 𝑈)) → (𝐴𝐺𝑈) = (𝑈𝐺𝐴))

Syntax hints:   → wi 4   ∧ wa 398   = wceq 1533   ∈ wcel 2110  ∃wrex 3139  ran crn 5550  (class class class)co 7150  GrpOpcgr 28260

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2157  ax-12 2173  ax-ext 2793  ax-sep 5195  ax-nul 5202  ax-pr 5321  ax-un 7455

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-mo 2618  df-eu 2650  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ral 3143  df-rex 3144  df-rab 3147  df-v 3496  df-sbc 3772  df-csb 3883  df-dif 3938  df-un 3940  df-in 3942  df-ss 3951  df-nul 4291  df-if 4467  df-sn 4561  df-pr 4563  df-op 4567  df-uni 4832  df-iun 4913  df-br 5059  df-opab 5121  df-mpt 5139  df-id 5454  df-xp 5555  df-rel 5556  df-cnv 5557  df-co 5558  df-dm 5559  df-rn 5560  df-iota 6308  df-fun 6351  df-fn 6352  df-f 6353  df-fo 6355  df-fv 6357  df-ov 7153  df-grpo 28264

This theorem is referenced by:  grpoidinv  28279  grpoideu  28280