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Theorem grpoidinvlem2 26743
Description: Lemma for grpoidinv 26746. (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 792 . . . . 5 ((𝐺 ∈ GrpOp ∧ (𝑌𝑋𝐴𝑋)) → 𝐴𝑋)
2 simprl 790 . . . . 5 ((𝐺 ∈ GrpOp ∧ (𝑌𝑋𝐴𝑋)) → 𝑌𝑋)
3 grpfo.1 . . . . . . . 8 𝑋 = ran 𝐺
43grpocl 26738 . . . . . . 7 ((𝐺 ∈ GrpOp ∧ 𝐴𝑋𝑌𝑋) → (𝐴𝐺𝑌) ∈ 𝑋)
543com23 1263 . . . . . 6 ((𝐺 ∈ GrpOp ∧ 𝑌𝑋𝐴𝑋) → (𝐴𝐺𝑌) ∈ 𝑋)
653expb 1258 . . . . 5 ((𝐺 ∈ GrpOp ∧ (𝑌𝑋𝐴𝑋)) → (𝐴𝐺𝑌) ∈ 𝑋)
71, 2, 63jca 1235 . . . 4 ((𝐺 ∈ GrpOp ∧ (𝑌𝑋𝐴𝑋)) → (𝐴𝑋𝑌𝑋 ∧ (𝐴𝐺𝑌) ∈ 𝑋))
83grpoass 26741 . . . 4 ((𝐺 ∈ GrpOp ∧ (𝐴𝑋𝑌𝑋 ∧ (𝐴𝐺𝑌) ∈ 𝑋)) → ((𝐴𝐺𝑌)𝐺(𝐴𝐺𝑌)) = (𝐴𝐺(𝑌𝐺(𝐴𝐺𝑌))))
97, 8syldan 486 . . 3 ((𝐺 ∈ GrpOp ∧ (𝑌𝑋𝐴𝑋)) → ((𝐴𝐺𝑌)𝐺(𝐴𝐺𝑌)) = (𝐴𝐺(𝑌𝐺(𝐴𝐺𝑌))))
109adantr 480 . 2 (((𝐺 ∈ GrpOp ∧ (𝑌𝑋𝐴𝑋)) ∧ ((𝑈𝐺𝑌) = 𝑌 ∧ (𝑌𝐺𝐴) = 𝑈)) → ((𝐴𝐺𝑌)𝐺(𝐴𝐺𝑌)) = (𝐴𝐺(𝑌𝐺(𝐴𝐺𝑌))))
11 oveq1 6556 . . . . . . 7 ((𝑌𝐺𝐴) = 𝑈 → ((𝑌𝐺𝐴)𝐺𝑌) = (𝑈𝐺𝑌))
1211adantl 481 . . . . . 6 (((𝑈𝐺𝑌) = 𝑌 ∧ (𝑌𝐺𝐴) = 𝑈) → ((𝑌𝐺𝐴)𝐺𝑌) = (𝑈𝐺𝑌))
13 simpl 472 . . . . . 6 (((𝑈𝐺𝑌) = 𝑌 ∧ (𝑌𝐺𝐴) = 𝑈) → (𝑈𝐺𝑌) = 𝑌)
1412, 13eqtr2d 2645 . . . . 5 (((𝑈𝐺𝑌) = 𝑌 ∧ (𝑌𝐺𝐴) = 𝑈) → 𝑌 = ((𝑌𝐺𝐴)𝐺𝑌))
15 id 22 . . . . . . 7 ((𝑌𝑋𝐴𝑋𝑌𝑋) → (𝑌𝑋𝐴𝑋𝑌𝑋))
16153anidm13 1376 . . . . . 6 ((𝑌𝑋𝐴𝑋) → (𝑌𝑋𝐴𝑋𝑌𝑋))
173grpoass 26741 . . . . . 6 ((𝐺 ∈ GrpOp ∧ (𝑌𝑋𝐴𝑋𝑌𝑋)) → ((𝑌𝐺𝐴)𝐺𝑌) = (𝑌𝐺(𝐴𝐺𝑌)))
1816, 17sylan2 490 . . . . 5 ((𝐺 ∈ GrpOp ∧ (𝑌𝑋𝐴𝑋)) → ((𝑌𝐺𝐴)𝐺𝑌) = (𝑌𝐺(𝐴𝐺𝑌)))
1914, 18sylan9eqr 2666 . . . 4 (((𝐺 ∈ GrpOp ∧ (𝑌𝑋𝐴𝑋)) ∧ ((𝑈𝐺𝑌) = 𝑌 ∧ (𝑌𝐺𝐴) = 𝑈)) → 𝑌 = (𝑌𝐺(𝐴𝐺𝑌)))
2019eqcomd 2616 . . 3 (((𝐺 ∈ GrpOp ∧ (𝑌𝑋𝐴𝑋)) ∧ ((𝑈𝐺𝑌) = 𝑌 ∧ (𝑌𝐺𝐴) = 𝑈)) → (𝑌𝐺(𝐴𝐺𝑌)) = 𝑌)
2120oveq2d 6565 . 2 (((𝐺 ∈ GrpOp ∧ (𝑌𝑋𝐴𝑋)) ∧ ((𝑈𝐺𝑌) = 𝑌 ∧ (𝑌𝐺𝐴) = 𝑈)) → (𝐴𝐺(𝑌𝐺(𝐴𝐺𝑌))) = (𝐴𝐺𝑌))
2210, 21eqtrd 2644 1 (((𝐺 ∈ GrpOp ∧ (𝑌𝑋𝐴𝑋)) ∧ ((𝑈𝐺𝑌) = 𝑌 ∧ (𝑌𝐺𝐴) = 𝑈)) → ((𝐴𝐺𝑌)𝐺(𝐴𝐺𝑌)) = (𝐴𝐺𝑌))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383  w3a 1031   = wceq 1475  wcel 1977  ran crn 5039  (class class class)co 6549  GrpOpcgr 26727
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pr 4833  ax-un 6847
This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-fo 5810  df-fv 5812  df-ov 6552  df-grpo 26731
This theorem is referenced by:  grpoidinvlem3  26744
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