MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  grpoidinvlem2 Structured version   Visualization version   GIF version

Theorem grpoidinvlem2 27699
Description: Lemma for grpoidinv 27702. (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 756 . . . . 5 ((𝐺 ∈ GrpOp ∧ (𝑌𝑋𝐴𝑋)) → 𝐴𝑋)
2 simprl 754 . . . . 5 ((𝐺 ∈ GrpOp ∧ (𝑌𝑋𝐴𝑋)) → 𝑌𝑋)
3 grpfo.1 . . . . . . . 8 𝑋 = ran 𝐺
43grpocl 27694 . . . . . . 7 ((𝐺 ∈ GrpOp ∧ 𝐴𝑋𝑌𝑋) → (𝐴𝐺𝑌) ∈ 𝑋)
543com23 1120 . . . . . 6 ((𝐺 ∈ GrpOp ∧ 𝑌𝑋𝐴𝑋) → (𝐴𝐺𝑌) ∈ 𝑋)
653expb 1113 . . . . 5 ((𝐺 ∈ GrpOp ∧ (𝑌𝑋𝐴𝑋)) → (𝐴𝐺𝑌) ∈ 𝑋)
71, 2, 63jca 1122 . . . 4 ((𝐺 ∈ GrpOp ∧ (𝑌𝑋𝐴𝑋)) → (𝐴𝑋𝑌𝑋 ∧ (𝐴𝐺𝑌) ∈ 𝑋))
83grpoass 27697 . . . 4 ((𝐺 ∈ GrpOp ∧ (𝐴𝑋𝑌𝑋 ∧ (𝐴𝐺𝑌) ∈ 𝑋)) → ((𝐴𝐺𝑌)𝐺(𝐴𝐺𝑌)) = (𝐴𝐺(𝑌𝐺(𝐴𝐺𝑌))))
97, 8syldan 579 . . 3 ((𝐺 ∈ GrpOp ∧ (𝑌𝑋𝐴𝑋)) → ((𝐴𝐺𝑌)𝐺(𝐴𝐺𝑌)) = (𝐴𝐺(𝑌𝐺(𝐴𝐺𝑌))))
109adantr 466 . 2 (((𝐺 ∈ GrpOp ∧ (𝑌𝑋𝐴𝑋)) ∧ ((𝑈𝐺𝑌) = 𝑌 ∧ (𝑌𝐺𝐴) = 𝑈)) → ((𝐴𝐺𝑌)𝐺(𝐴𝐺𝑌)) = (𝐴𝐺(𝑌𝐺(𝐴𝐺𝑌))))
11 oveq1 6800 . . . . . . 7 ((𝑌𝐺𝐴) = 𝑈 → ((𝑌𝐺𝐴)𝐺𝑌) = (𝑈𝐺𝑌))
1211adantl 467 . . . . . 6 (((𝑈𝐺𝑌) = 𝑌 ∧ (𝑌𝐺𝐴) = 𝑈) → ((𝑌𝐺𝐴)𝐺𝑌) = (𝑈𝐺𝑌))
13 simpl 468 . . . . . 6 (((𝑈𝐺𝑌) = 𝑌 ∧ (𝑌𝐺𝐴) = 𝑈) → (𝑈𝐺𝑌) = 𝑌)
1412, 13eqtr2d 2806 . . . . 5 (((𝑈𝐺𝑌) = 𝑌 ∧ (𝑌𝐺𝐴) = 𝑈) → 𝑌 = ((𝑌𝐺𝐴)𝐺𝑌))
15 id 22 . . . . . . 7 ((𝑌𝑋𝐴𝑋𝑌𝑋) → (𝑌𝑋𝐴𝑋𝑌𝑋))
16153anidm13 1530 . . . . . 6 ((𝑌𝑋𝐴𝑋) → (𝑌𝑋𝐴𝑋𝑌𝑋))
173grpoass 27697 . . . . . 6 ((𝐺 ∈ GrpOp ∧ (𝑌𝑋𝐴𝑋𝑌𝑋)) → ((𝑌𝐺𝐴)𝐺𝑌) = (𝑌𝐺(𝐴𝐺𝑌)))
1816, 17sylan2 580 . . . . 5 ((𝐺 ∈ GrpOp ∧ (𝑌𝑋𝐴𝑋)) → ((𝑌𝐺𝐴)𝐺𝑌) = (𝑌𝐺(𝐴𝐺𝑌)))
1914, 18sylan9eqr 2827 . . . 4 (((𝐺 ∈ GrpOp ∧ (𝑌𝑋𝐴𝑋)) ∧ ((𝑈𝐺𝑌) = 𝑌 ∧ (𝑌𝐺𝐴) = 𝑈)) → 𝑌 = (𝑌𝐺(𝐴𝐺𝑌)))
2019eqcomd 2777 . . 3 (((𝐺 ∈ GrpOp ∧ (𝑌𝑋𝐴𝑋)) ∧ ((𝑈𝐺𝑌) = 𝑌 ∧ (𝑌𝐺𝐴) = 𝑈)) → (𝑌𝐺(𝐴𝐺𝑌)) = 𝑌)
2120oveq2d 6809 . 2 (((𝐺 ∈ GrpOp ∧ (𝑌𝑋𝐴𝑋)) ∧ ((𝑈𝐺𝑌) = 𝑌 ∧ (𝑌𝐺𝐴) = 𝑈)) → (𝐴𝐺(𝑌𝐺(𝐴𝐺𝑌))) = (𝐴𝐺𝑌))
2210, 21eqtrd 2805 1 (((𝐺 ∈ GrpOp ∧ (𝑌𝑋𝐴𝑋)) ∧ ((𝑈𝐺𝑌) = 𝑌 ∧ (𝑌𝐺𝐴) = 𝑈)) → ((𝐴𝐺𝑌)𝐺(𝐴𝐺𝑌)) = (𝐴𝐺𝑌))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 382  w3a 1071   = wceq 1631  wcel 2145  ran crn 5250  (class class class)co 6793  GrpOpcgr 27683
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-sep 4915  ax-nul 4923  ax-pr 5034  ax-un 7096
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 835  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3353  df-sbc 3588  df-csb 3683  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4226  df-sn 4317  df-pr 4319  df-op 4323  df-uni 4575  df-iun 4656  df-br 4787  df-opab 4847  df-mpt 4864  df-id 5157  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-iota 5994  df-fun 6033  df-fn 6034  df-f 6035  df-fo 6037  df-fv 6039  df-ov 6796  df-grpo 27687
This theorem is referenced by:  grpoidinvlem3  27700
  Copyright terms: Public domain W3C validator