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Theorem grpoidinvlem4 30304
Description: Lemma for grpoidinv 30305. (Contributed by NM, 14-Oct-2006.) (New usage is discouraged.)
Hypothesis
Ref Expression
grpfo.1 𝑋 = ran 𝐺
Assertion
Ref Expression
grpoidinvlem4 (((𝐺 ∈ GrpOp ∧ 𝐴𝑋) ∧ ∃𝑦𝑋 ((𝑦𝐺𝐴) = 𝑈 ∧ (𝐴𝐺𝑦) = 𝑈)) → (𝐴𝐺𝑈) = (𝑈𝐺𝐴))
Distinct variable groups:   𝑦,𝐴   𝑦,𝐺   𝑦,𝑋   𝑦,𝑈

Proof of Theorem grpoidinvlem4
StepHypRef Expression
1 simpll 766 . . . . . 6 (((𝐺 ∈ GrpOp ∧ 𝐴𝑋) ∧ 𝑦𝑋) → 𝐺 ∈ GrpOp)
2 simplr 768 . . . . . 6 (((𝐺 ∈ GrpOp ∧ 𝐴𝑋) ∧ 𝑦𝑋) → 𝐴𝑋)
3 simpr 484 . . . . . 6 (((𝐺 ∈ GrpOp ∧ 𝐴𝑋) ∧ 𝑦𝑋) → 𝑦𝑋)
4 grpfo.1 . . . . . . 7 𝑋 = ran 𝐺
54grpoass 30300 . . . . . 6 ((𝐺 ∈ GrpOp ∧ (𝐴𝑋𝑦𝑋𝐴𝑋)) → ((𝐴𝐺𝑦)𝐺𝐴) = (𝐴𝐺(𝑦𝐺𝐴)))
61, 2, 3, 2, 5syl13anc 1370 . . . . 5 (((𝐺 ∈ GrpOp ∧ 𝐴𝑋) ∧ 𝑦𝑋) → ((𝐴𝐺𝑦)𝐺𝐴) = (𝐴𝐺(𝑦𝐺𝐴)))
7 oveq2 7422 . . . . 5 ((𝑦𝐺𝐴) = 𝑈 → (𝐴𝐺(𝑦𝐺𝐴)) = (𝐴𝐺𝑈))
86, 7sylan9eq 2787 . . . 4 ((((𝐺 ∈ GrpOp ∧ 𝐴𝑋) ∧ 𝑦𝑋) ∧ (𝑦𝐺𝐴) = 𝑈) → ((𝐴𝐺𝑦)𝐺𝐴) = (𝐴𝐺𝑈))
9 oveq1 7421 . . . 4 ((𝐴𝐺𝑦) = 𝑈 → ((𝐴𝐺𝑦)𝐺𝐴) = (𝑈𝐺𝐴))
108, 9sylan9req 2788 . . 3 (((((𝐺 ∈ GrpOp ∧ 𝐴𝑋) ∧ 𝑦𝑋) ∧ (𝑦𝐺𝐴) = 𝑈) ∧ (𝐴𝐺𝑦) = 𝑈) → (𝐴𝐺𝑈) = (𝑈𝐺𝐴))
1110anasss 466 . 2 ((((𝐺 ∈ GrpOp ∧ 𝐴𝑋) ∧ 𝑦𝑋) ∧ ((𝑦𝐺𝐴) = 𝑈 ∧ (𝐴𝐺𝑦) = 𝑈)) → (𝐴𝐺𝑈) = (𝑈𝐺𝐴))
1211r19.29an 3153 1 (((𝐺 ∈ GrpOp ∧ 𝐴𝑋) ∧ ∃𝑦𝑋 ((𝑦𝐺𝐴) = 𝑈 ∧ (𝐴𝐺𝑦) = 𝑈)) → (𝐴𝐺𝑈) = (𝑈𝐺𝐴))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1534  wcel 2099  wrex 3065  ran crn 5673  (class class class)co 7414  GrpOpcgr 30286
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2164  ax-ext 2698  ax-sep 5293  ax-nul 5300  ax-pr 5423  ax-un 7734
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2705  df-cleq 2719  df-clel 2805  df-nfc 2880  df-ne 2936  df-ral 3057  df-rex 3066  df-rab 3428  df-v 3471  df-sbc 3775  df-csb 3890  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-nul 4319  df-if 4525  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4904  df-iun 4993  df-br 5143  df-opab 5205  df-mpt 5226  df-id 5570  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-rn 5683  df-iota 6494  df-fun 6544  df-fn 6545  df-f 6546  df-fo 6548  df-fv 6550  df-ov 7417  df-grpo 30290
This theorem is referenced by:  grpoidinv  30305  grpoideu  30306
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