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Theorem grpoidinvlem2 27961
Description: Lemma for grpoidinv 27964. (Contributed by NM, 10-Oct-2006.) (New usage is discouraged.)
Hypothesis
Ref Expression
grpfo.1 𝑋 = ran 𝐺
Assertion
Ref Expression
grpoidinvlem2 (((𝐺 ∈ GrpOp ∧ (𝑌𝑋𝐴𝑋)) ∧ ((𝑈𝐺𝑌) = 𝑌 ∧ (𝑌𝐺𝐴) = 𝑈)) → ((𝐴𝐺𝑌)𝐺(𝐴𝐺𝑌)) = (𝐴𝐺𝑌))

Proof of Theorem grpoidinvlem2
StepHypRef Expression
1 simprr 769 . . . . 5 ((𝐺 ∈ GrpOp ∧ (𝑌𝑋𝐴𝑋)) → 𝐴𝑋)
2 simprl 767 . . . . 5 ((𝐺 ∈ GrpOp ∧ (𝑌𝑋𝐴𝑋)) → 𝑌𝑋)
3 grpfo.1 . . . . . . . 8 𝑋 = ran 𝐺
43grpocl 27956 . . . . . . 7 ((𝐺 ∈ GrpOp ∧ 𝐴𝑋𝑌𝑋) → (𝐴𝐺𝑌) ∈ 𝑋)
543com23 1117 . . . . . 6 ((𝐺 ∈ GrpOp ∧ 𝑌𝑋𝐴𝑋) → (𝐴𝐺𝑌) ∈ 𝑋)
653expb 1111 . . . . 5 ((𝐺 ∈ GrpOp ∧ (𝑌𝑋𝐴𝑋)) → (𝐴𝐺𝑌) ∈ 𝑋)
71, 2, 63jca 1119 . . . 4 ((𝐺 ∈ GrpOp ∧ (𝑌𝑋𝐴𝑋)) → (𝐴𝑋𝑌𝑋 ∧ (𝐴𝐺𝑌) ∈ 𝑋))
83grpoass 27959 . . . 4 ((𝐺 ∈ GrpOp ∧ (𝐴𝑋𝑌𝑋 ∧ (𝐴𝐺𝑌) ∈ 𝑋)) → ((𝐴𝐺𝑌)𝐺(𝐴𝐺𝑌)) = (𝐴𝐺(𝑌𝐺(𝐴𝐺𝑌))))
97, 8syldan 591 . . 3 ((𝐺 ∈ GrpOp ∧ (𝑌𝑋𝐴𝑋)) → ((𝐴𝐺𝑌)𝐺(𝐴𝐺𝑌)) = (𝐴𝐺(𝑌𝐺(𝐴𝐺𝑌))))
109adantr 481 . 2 (((𝐺 ∈ GrpOp ∧ (𝑌𝑋𝐴𝑋)) ∧ ((𝑈𝐺𝑌) = 𝑌 ∧ (𝑌𝐺𝐴) = 𝑈)) → ((𝐴𝐺𝑌)𝐺(𝐴𝐺𝑌)) = (𝐴𝐺(𝑌𝐺(𝐴𝐺𝑌))))
11 oveq1 7014 . . . . . . 7 ((𝑌𝐺𝐴) = 𝑈 → ((𝑌𝐺𝐴)𝐺𝑌) = (𝑈𝐺𝑌))
1211adantl 482 . . . . . 6 (((𝑈𝐺𝑌) = 𝑌 ∧ (𝑌𝐺𝐴) = 𝑈) → ((𝑌𝐺𝐴)𝐺𝑌) = (𝑈𝐺𝑌))
13 simpl 483 . . . . . 6 (((𝑈𝐺𝑌) = 𝑌 ∧ (𝑌𝐺𝐴) = 𝑈) → (𝑈𝐺𝑌) = 𝑌)
1412, 13eqtr2d 2830 . . . . 5 (((𝑈𝐺𝑌) = 𝑌 ∧ (𝑌𝐺𝐴) = 𝑈) → 𝑌 = ((𝑌𝐺𝐴)𝐺𝑌))
15 id 22 . . . . . . 7 ((𝑌𝑋𝐴𝑋𝑌𝑋) → (𝑌𝑋𝐴𝑋𝑌𝑋))
16153anidm13 1411 . . . . . 6 ((𝑌𝑋𝐴𝑋) → (𝑌𝑋𝐴𝑋𝑌𝑋))
173grpoass 27959 . . . . . 6 ((𝐺 ∈ GrpOp ∧ (𝑌𝑋𝐴𝑋𝑌𝑋)) → ((𝑌𝐺𝐴)𝐺𝑌) = (𝑌𝐺(𝐴𝐺𝑌)))
1816, 17sylan2 592 . . . . 5 ((𝐺 ∈ GrpOp ∧ (𝑌𝑋𝐴𝑋)) → ((𝑌𝐺𝐴)𝐺𝑌) = (𝑌𝐺(𝐴𝐺𝑌)))
1914, 18sylan9eqr 2851 . . . 4 (((𝐺 ∈ GrpOp ∧ (𝑌𝑋𝐴𝑋)) ∧ ((𝑈𝐺𝑌) = 𝑌 ∧ (𝑌𝐺𝐴) = 𝑈)) → 𝑌 = (𝑌𝐺(𝐴𝐺𝑌)))
2019eqcomd 2799 . . 3 (((𝐺 ∈ GrpOp ∧ (𝑌𝑋𝐴𝑋)) ∧ ((𝑈𝐺𝑌) = 𝑌 ∧ (𝑌𝐺𝐴) = 𝑈)) → (𝑌𝐺(𝐴𝐺𝑌)) = 𝑌)
2120oveq2d 7023 . 2 (((𝐺 ∈ GrpOp ∧ (𝑌𝑋𝐴𝑋)) ∧ ((𝑈𝐺𝑌) = 𝑌 ∧ (𝑌𝐺𝐴) = 𝑈)) → (𝐴𝐺(𝑌𝐺(𝐴𝐺𝑌))) = (𝐴𝐺𝑌))
2210, 21eqtrd 2829 1 (((𝐺 ∈ GrpOp ∧ (𝑌𝑋𝐴𝑋)) ∧ ((𝑈𝐺𝑌) = 𝑌 ∧ (𝑌𝐺𝐴) = 𝑈)) → ((𝐴𝐺𝑌)𝐺(𝐴𝐺𝑌)) = (𝐴𝐺𝑌))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396  w3a 1078   = wceq 1520  wcel 2079  ran crn 5436  (class class class)co 7007  GrpOpcgr 27945
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1775  ax-4 1789  ax-5 1886  ax-6 1945  ax-7 1990  ax-8 2081  ax-9 2089  ax-10 2110  ax-11 2124  ax-12 2139  ax-13 2342  ax-ext 2767  ax-sep 5088  ax-nul 5095  ax-pr 5214  ax-un 7310
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 843  df-3an 1080  df-tru 1523  df-ex 1760  df-nf 1764  df-sb 2041  df-mo 2574  df-eu 2610  df-clab 2774  df-cleq 2786  df-clel 2861  df-nfc 2933  df-ral 3108  df-rex 3109  df-rab 3112  df-v 3434  df-sbc 3702  df-csb 3807  df-dif 3857  df-un 3859  df-in 3861  df-ss 3869  df-nul 4207  df-if 4376  df-sn 4467  df-pr 4469  df-op 4473  df-uni 4740  df-iun 4821  df-br 4957  df-opab 5019  df-mpt 5036  df-id 5340  df-xp 5441  df-rel 5442  df-cnv 5443  df-co 5444  df-dm 5445  df-rn 5446  df-iota 6181  df-fun 6219  df-fn 6220  df-f 6221  df-fo 6223  df-fv 6225  df-ov 7010  df-grpo 27949
This theorem is referenced by:  grpoidinvlem3  27962
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