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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 ∧ 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 807 . . . . . 6 (((𝐺 ∈ GrpOp ∧ 𝐴𝑋) ∧ 𝑦𝑋) → 𝐺 ∈ GrpOp)
2 simplr 809 . . . . . 6 (((𝐺 ∈ GrpOp ∧ 𝐴𝑋) ∧ 𝑦𝑋) → 𝐴𝑋)
3 simpr 479 . . . . . 6 (((𝐺 ∈ GrpOp ∧ 𝐴𝑋) ∧ 𝑦𝑋) → 𝑦𝑋)
4 grpfo.1 . . . . . . 7 𝑋 = ran 𝐺
54grpoass 27587 . . . . . 6 ((𝐺 ∈ GrpOp ∧ (𝐴𝑋𝑦𝑋𝐴𝑋)) → ((𝐴𝐺𝑦)𝐺𝐴) = (𝐴𝐺(𝑦𝐺𝐴)))
61, 2, 3, 2, 5syl13anc 1441 . . . . 5 (((𝐺 ∈ GrpOp ∧ 𝐴𝑋) ∧ 𝑦𝑋) → ((𝐴𝐺𝑦)𝐺𝐴) = (𝐴𝐺(𝑦𝐺𝐴)))
7 oveq2 6773 . . . . 5 ((𝑦𝐺𝐴) = 𝑈 → (𝐴𝐺(𝑦𝐺𝐴)) = (𝐴𝐺𝑈))
86, 7sylan9eq 2778 . . . 4 ((((𝐺 ∈ GrpOp ∧ 𝐴𝑋) ∧ 𝑦𝑋) ∧ (𝑦𝐺𝐴) = 𝑈) → ((𝐴𝐺𝑦)𝐺𝐴) = (𝐴𝐺𝑈))
9 oveq1 6772 . . . 4 ((𝐴𝐺𝑦) = 𝑈 → ((𝐴𝐺𝑦)𝐺𝐴) = (𝑈𝐺𝐴))
108, 9sylan9req 2779 . . 3 (((((𝐺 ∈ GrpOp ∧ 𝐴𝑋) ∧ 𝑦𝑋) ∧ (𝑦𝐺𝐴) = 𝑈) ∧ (𝐴𝐺𝑦) = 𝑈) → (𝐴𝐺𝑈) = (𝑈𝐺𝐴))
1110anasss 682 . 2 ((((𝐺 ∈ GrpOp ∧ 𝐴𝑋) ∧ 𝑦𝑋) ∧ ((𝑦𝐺𝐴) = 𝑈 ∧ (𝐴𝐺𝑦) = 𝑈)) → (𝐴𝐺𝑈) = (𝑈𝐺𝐴))
1211r19.29an 3179 1 (((𝐺 ∈ GrpOp ∧ 𝐴𝑋) ∧ ∃𝑦𝑋 ((𝑦𝐺𝐴) = 𝑈 ∧ (𝐴𝐺𝑦) = 𝑈)) → (𝐴𝐺𝑈) = (𝑈𝐺𝐴))
Colors of variables: wff setvar class
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
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