Theorem grpoidinvlem4 27591

Description: Lemma for grpoidinv 27592. (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 807	. . . . . 6 ⊢ (((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋) ∧ 𝑦 ∈ 𝑋) → 𝐺 ∈ GrpOp)
2		simplr 809	. . . . . 6 ⊢ (((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋) ∧ 𝑦 ∈ 𝑋) → 𝐴 ∈ 𝑋)
3		simpr 479	. . . . . 6 ⊢ (((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋) ∧ 𝑦 ∈ 𝑋) → 𝑦 ∈ 𝑋)
4		grpfo.1	. . . . . . 7 ⊢ 𝑋 = ran 𝐺
5	4	grpoass 27587	. . . . . 6 ⊢ ((𝐺 ∈ GrpOp ∧ (𝐴 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋 ∧ 𝐴 ∈ 𝑋)) → ((𝐴𝐺𝑦)𝐺𝐴) = (𝐴𝐺(𝑦𝐺𝐴)))
6	1, 2, 3, 2, 5	syl13anc 1441	. . . . 5 ⊢ (((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋) ∧ 𝑦 ∈ 𝑋) → ((𝐴𝐺𝑦)𝐺𝐴) = (𝐴𝐺(𝑦𝐺𝐴)))
7		oveq2 6773	. . . . 5 ⊢ ((𝑦𝐺𝐴) = 𝑈 → (𝐴𝐺(𝑦𝐺𝐴)) = (𝐴𝐺𝑈))
8	6, 7	sylan9eq 2778	. . . 4 ⊢ ((((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋) ∧ 𝑦 ∈ 𝑋) ∧ (𝑦𝐺𝐴) = 𝑈) → ((𝐴𝐺𝑦)𝐺𝐴) = (𝐴𝐺𝑈))
9		oveq1 6772	. . . 4 ⊢ ((𝐴𝐺𝑦) = 𝑈 → ((𝐴𝐺𝑦)𝐺𝐴) = (𝑈𝐺𝐴))
10	8, 9	sylan9req 2779	. . 3 ⊢ (((((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋) ∧ 𝑦 ∈ 𝑋) ∧ (𝑦𝐺𝐴) = 𝑈) ∧ (𝐴𝐺𝑦) = 𝑈) → (𝐴𝐺𝑈) = (𝑈𝐺𝐴))
11	10	anasss 682	. 2 ⊢ ((((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋) ∧ 𝑦 ∈ 𝑋) ∧ ((𝑦𝐺𝐴) = 𝑈 ∧ (𝐴𝐺𝑦) = 𝑈)) → (𝐴𝐺𝑈) = (𝑈𝐺𝐴))
12	11	r19.29an 3179	1 ⊢ (((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋) ∧ ∃𝑦 ∈ 𝑋 ((𝑦𝐺𝐴) = 𝑈 ∧ (𝐴𝐺𝑦) = 𝑈)) → (𝐴𝐺𝑈) = (𝑈𝐺𝐴))

Syntax hints:   → wi 4   ∧ wa 383   = wceq 1596   ∈ wcel 2103  ∃wrex 3015  ran crn 5219  (class class class)co 6765  GrpOpcgr 27573

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1835  ax-4 1850  ax-5 1952  ax-6 2018  ax-7 2054  ax-8 2105  ax-9 2112  ax-10 2132  ax-11 2147  ax-12 2160  ax-13 2355  ax-ext 2704  ax-sep 4889  ax-nul 4897  ax-pr 5011  ax-un 7066

This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1599  df-ex 1818  df-nf 1823  df-sb 2011  df-eu 2575  df-mo 2576  df-clab 2711  df-cleq 2717  df-clel 2720  df-nfc 2855  df-ral 3019  df-rex 3020  df-rab 3023  df-v 3306  df-sbc 3542  df-csb 3640  df-dif 3683  df-un 3685  df-in 3687  df-ss 3694  df-nul 4024  df-if 4195  df-sn 4286  df-pr 4288  df-op 4292  df-uni 4545  df-iun 4630  df-br 4761  df-opab 4821  df-mpt 4838  df-id 5128  df-xp 5224  df-rel 5225  df-cnv 5226  df-co 5227  df-dm 5228  df-rn 5229  df-iota 5964  df-fun 6003  df-fn 6004  df-f 6005  df-fo 6007  df-fv 6009  df-ov 6768  df-grpo 27577

This theorem is referenced by:  grpoidinv  27592  grpoideu  27593