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Theorem grpoidinvlem2 28286
Description: Lemma for grpoidinv 28289. (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 772 . . . . 5 ((𝐺 ∈ GrpOp ∧ (𝑌𝑋𝐴𝑋)) → 𝐴𝑋)
2 simprl 770 . . . . 5 ((𝐺 ∈ GrpOp ∧ (𝑌𝑋𝐴𝑋)) → 𝑌𝑋)
3 grpfo.1 . . . . . . . 8 𝑋 = ran 𝐺
43grpocl 28281 . . . . . . 7 ((𝐺 ∈ GrpOp ∧ 𝐴𝑋𝑌𝑋) → (𝐴𝐺𝑌) ∈ 𝑋)
543com23 1123 . . . . . 6 ((𝐺 ∈ GrpOp ∧ 𝑌𝑋𝐴𝑋) → (𝐴𝐺𝑌) ∈ 𝑋)
653expb 1117 . . . . 5 ((𝐺 ∈ GrpOp ∧ (𝑌𝑋𝐴𝑋)) → (𝐴𝐺𝑌) ∈ 𝑋)
71, 2, 63jca 1125 . . . 4 ((𝐺 ∈ GrpOp ∧ (𝑌𝑋𝐴𝑋)) → (𝐴𝑋𝑌𝑋 ∧ (𝐴𝐺𝑌) ∈ 𝑋))
83grpoass 28284 . . . 4 ((𝐺 ∈ GrpOp ∧ (𝐴𝑋𝑌𝑋 ∧ (𝐴𝐺𝑌) ∈ 𝑋)) → ((𝐴𝐺𝑌)𝐺(𝐴𝐺𝑌)) = (𝐴𝐺(𝑌𝐺(𝐴𝐺𝑌))))
97, 8syldan 594 . . 3 ((𝐺 ∈ GrpOp ∧ (𝑌𝑋𝐴𝑋)) → ((𝐴𝐺𝑌)𝐺(𝐴𝐺𝑌)) = (𝐴𝐺(𝑌𝐺(𝐴𝐺𝑌))))
109adantr 484 . 2 (((𝐺 ∈ GrpOp ∧ (𝑌𝑋𝐴𝑋)) ∧ ((𝑈𝐺𝑌) = 𝑌 ∧ (𝑌𝐺𝐴) = 𝑈)) → ((𝐴𝐺𝑌)𝐺(𝐴𝐺𝑌)) = (𝐴𝐺(𝑌𝐺(𝐴𝐺𝑌))))
11 oveq1 7147 . . . . . . 7 ((𝑌𝐺𝐴) = 𝑈 → ((𝑌𝐺𝐴)𝐺𝑌) = (𝑈𝐺𝑌))
1211adantl 485 . . . . . 6 (((𝑈𝐺𝑌) = 𝑌 ∧ (𝑌𝐺𝐴) = 𝑈) → ((𝑌𝐺𝐴)𝐺𝑌) = (𝑈𝐺𝑌))
13 simpl 486 . . . . . 6 (((𝑈𝐺𝑌) = 𝑌 ∧ (𝑌𝐺𝐴) = 𝑈) → (𝑈𝐺𝑌) = 𝑌)
1412, 13eqtr2d 2858 . . . . 5 (((𝑈𝐺𝑌) = 𝑌 ∧ (𝑌𝐺𝐴) = 𝑈) → 𝑌 = ((𝑌𝐺𝐴)𝐺𝑌))
15 id 22 . . . . . . 7 ((𝑌𝑋𝐴𝑋𝑌𝑋) → (𝑌𝑋𝐴𝑋𝑌𝑋))
16153anidm13 1417 . . . . . 6 ((𝑌𝑋𝐴𝑋) → (𝑌𝑋𝐴𝑋𝑌𝑋))
173grpoass 28284 . . . . . 6 ((𝐺 ∈ GrpOp ∧ (𝑌𝑋𝐴𝑋𝑌𝑋)) → ((𝑌𝐺𝐴)𝐺𝑌) = (𝑌𝐺(𝐴𝐺𝑌)))
1816, 17sylan2 595 . . . . 5 ((𝐺 ∈ GrpOp ∧ (𝑌𝑋𝐴𝑋)) → ((𝑌𝐺𝐴)𝐺𝑌) = (𝑌𝐺(𝐴𝐺𝑌)))
1914, 18sylan9eqr 2879 . . . 4 (((𝐺 ∈ GrpOp ∧ (𝑌𝑋𝐴𝑋)) ∧ ((𝑈𝐺𝑌) = 𝑌 ∧ (𝑌𝐺𝐴) = 𝑈)) → 𝑌 = (𝑌𝐺(𝐴𝐺𝑌)))
2019eqcomd 2828 . . 3 (((𝐺 ∈ GrpOp ∧ (𝑌𝑋𝐴𝑋)) ∧ ((𝑈𝐺𝑌) = 𝑌 ∧ (𝑌𝐺𝐴) = 𝑈)) → (𝑌𝐺(𝐴𝐺𝑌)) = 𝑌)
2120oveq2d 7156 . 2 (((𝐺 ∈ GrpOp ∧ (𝑌𝑋𝐴𝑋)) ∧ ((𝑈𝐺𝑌) = 𝑌 ∧ (𝑌𝐺𝐴) = 𝑈)) → (𝐴𝐺(𝑌𝐺(𝐴𝐺𝑌))) = (𝐴𝐺𝑌))
2210, 21eqtrd 2857 1 (((𝐺 ∈ GrpOp ∧ (𝑌𝑋𝐴𝑋)) ∧ ((𝑈𝐺𝑌) = 𝑌 ∧ (𝑌𝐺𝐴) = 𝑈)) → ((𝐴𝐺𝑌)𝐺(𝐴𝐺𝑌)) = (𝐴𝐺𝑌))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 399  w3a 1084   = wceq 1538  wcel 2114  ran crn 5533  (class class class)co 7140  GrpOpcgr 28270
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2178  ax-ext 2794  ax-sep 5179  ax-nul 5186  ax-pr 5307  ax-un 7446
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2622  df-eu 2653  df-clab 2801  df-cleq 2815  df-clel 2894  df-nfc 2962  df-ral 3135  df-rex 3136  df-rab 3139  df-v 3471  df-sbc 3748  df-csb 3856  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-nul 4266  df-if 4440  df-sn 4540  df-pr 4542  df-op 4546  df-uni 4814  df-iun 4896  df-br 5043  df-opab 5105  df-mpt 5123  df-id 5437  df-xp 5538  df-rel 5539  df-cnv 5540  df-co 5541  df-dm 5542  df-rn 5543  df-iota 6293  df-fun 6336  df-fn 6337  df-f 6338  df-fo 6340  df-fv 6342  df-ov 7143  df-grpo 28274
This theorem is referenced by:  grpoidinvlem3  28287
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