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Theorem grpoidinvlem2 27687
Description: Lemma for grpoidinv 27690. (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 780 . . . . 5 ((𝐺 ∈ GrpOp ∧ (𝑌𝑋𝐴𝑋)) → 𝐴𝑋)
2 simprl 778 . . . . 5 ((𝐺 ∈ GrpOp ∧ (𝑌𝑋𝐴𝑋)) → 𝑌𝑋)
3 grpfo.1 . . . . . . . 8 𝑋 = ran 𝐺
43grpocl 27682 . . . . . . 7 ((𝐺 ∈ GrpOp ∧ 𝐴𝑋𝑌𝑋) → (𝐴𝐺𝑌) ∈ 𝑋)
543com23 1149 . . . . . 6 ((𝐺 ∈ GrpOp ∧ 𝑌𝑋𝐴𝑋) → (𝐴𝐺𝑌) ∈ 𝑋)
653expb 1142 . . . . 5 ((𝐺 ∈ GrpOp ∧ (𝑌𝑋𝐴𝑋)) → (𝐴𝐺𝑌) ∈ 𝑋)
71, 2, 63jca 1151 . . . 4 ((𝐺 ∈ GrpOp ∧ (𝑌𝑋𝐴𝑋)) → (𝐴𝑋𝑌𝑋 ∧ (𝐴𝐺𝑌) ∈ 𝑋))
83grpoass 27685 . . . 4 ((𝐺 ∈ GrpOp ∧ (𝐴𝑋𝑌𝑋 ∧ (𝐴𝐺𝑌) ∈ 𝑋)) → ((𝐴𝐺𝑌)𝐺(𝐴𝐺𝑌)) = (𝐴𝐺(𝑌𝐺(𝐴𝐺𝑌))))
97, 8syldan 581 . . 3 ((𝐺 ∈ GrpOp ∧ (𝑌𝑋𝐴𝑋)) → ((𝐴𝐺𝑌)𝐺(𝐴𝐺𝑌)) = (𝐴𝐺(𝑌𝐺(𝐴𝐺𝑌))))
109adantr 468 . 2 (((𝐺 ∈ GrpOp ∧ (𝑌𝑋𝐴𝑋)) ∧ ((𝑈𝐺𝑌) = 𝑌 ∧ (𝑌𝐺𝐴) = 𝑈)) → ((𝐴𝐺𝑌)𝐺(𝐴𝐺𝑌)) = (𝐴𝐺(𝑌𝐺(𝐴𝐺𝑌))))
11 oveq1 6877 . . . . . . 7 ((𝑌𝐺𝐴) = 𝑈 → ((𝑌𝐺𝐴)𝐺𝑌) = (𝑈𝐺𝑌))
1211adantl 469 . . . . . 6 (((𝑈𝐺𝑌) = 𝑌 ∧ (𝑌𝐺𝐴) = 𝑈) → ((𝑌𝐺𝐴)𝐺𝑌) = (𝑈𝐺𝑌))
13 simpl 470 . . . . . 6 (((𝑈𝐺𝑌) = 𝑌 ∧ (𝑌𝐺𝐴) = 𝑈) → (𝑈𝐺𝑌) = 𝑌)
1412, 13eqtr2d 2841 . . . . 5 (((𝑈𝐺𝑌) = 𝑌 ∧ (𝑌𝐺𝐴) = 𝑈) → 𝑌 = ((𝑌𝐺𝐴)𝐺𝑌))
15 id 22 . . . . . . 7 ((𝑌𝑋𝐴𝑋𝑌𝑋) → (𝑌𝑋𝐴𝑋𝑌𝑋))
16153anidm13 1536 . . . . . 6 ((𝑌𝑋𝐴𝑋) → (𝑌𝑋𝐴𝑋𝑌𝑋))
173grpoass 27685 . . . . . 6 ((𝐺 ∈ GrpOp ∧ (𝑌𝑋𝐴𝑋𝑌𝑋)) → ((𝑌𝐺𝐴)𝐺𝑌) = (𝑌𝐺(𝐴𝐺𝑌)))
1816, 17sylan2 582 . . . . 5 ((𝐺 ∈ GrpOp ∧ (𝑌𝑋𝐴𝑋)) → ((𝑌𝐺𝐴)𝐺𝑌) = (𝑌𝐺(𝐴𝐺𝑌)))
1914, 18sylan9eqr 2862 . . . 4 (((𝐺 ∈ GrpOp ∧ (𝑌𝑋𝐴𝑋)) ∧ ((𝑈𝐺𝑌) = 𝑌 ∧ (𝑌𝐺𝐴) = 𝑈)) → 𝑌 = (𝑌𝐺(𝐴𝐺𝑌)))
2019eqcomd 2812 . . 3 (((𝐺 ∈ GrpOp ∧ (𝑌𝑋𝐴𝑋)) ∧ ((𝑈𝐺𝑌) = 𝑌 ∧ (𝑌𝐺𝐴) = 𝑈)) → (𝑌𝐺(𝐴𝐺𝑌)) = 𝑌)
2120oveq2d 6886 . 2 (((𝐺 ∈ GrpOp ∧ (𝑌𝑋𝐴𝑋)) ∧ ((𝑈𝐺𝑌) = 𝑌 ∧ (𝑌𝐺𝐴) = 𝑈)) → (𝐴𝐺(𝑌𝐺(𝐴𝐺𝑌))) = (𝐴𝐺𝑌))
2210, 21eqtrd 2840 1 (((𝐺 ∈ GrpOp ∧ (𝑌𝑋𝐴𝑋)) ∧ ((𝑈𝐺𝑌) = 𝑌 ∧ (𝑌𝐺𝐴) = 𝑈)) → ((𝐴𝐺𝑌)𝐺(𝐴𝐺𝑌)) = (𝐴𝐺𝑌))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384  w3a 1100   = wceq 1637  wcel 2156  ran crn 5312  (class class class)co 6870  GrpOpcgr 27671
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1877  ax-4 1894  ax-5 2001  ax-6 2068  ax-7 2104  ax-8 2158  ax-9 2165  ax-10 2185  ax-11 2201  ax-12 2214  ax-13 2420  ax-ext 2784  ax-sep 4975  ax-nul 4983  ax-pr 5096  ax-un 7175
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 866  df-3an 1102  df-tru 1641  df-ex 1860  df-nf 1864  df-sb 2061  df-eu 2634  df-mo 2635  df-clab 2793  df-cleq 2799  df-clel 2802  df-nfc 2937  df-ral 3101  df-rex 3102  df-rab 3105  df-v 3393  df-sbc 3634  df-csb 3729  df-dif 3772  df-un 3774  df-in 3776  df-ss 3783  df-nul 4117  df-if 4280  df-sn 4371  df-pr 4373  df-op 4377  df-uni 4631  df-iun 4714  df-br 4845  df-opab 4907  df-mpt 4924  df-id 5219  df-xp 5317  df-rel 5318  df-cnv 5319  df-co 5320  df-dm 5321  df-rn 5322  df-iota 6060  df-fun 6099  df-fn 6100  df-f 6101  df-fo 6103  df-fv 6105  df-ov 6873  df-grpo 27675
This theorem is referenced by:  grpoidinvlem3  27688
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