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Theorem grpoidinvlem1 30533
Description: Lemma for grpoidinv 30537. (Contributed by NM, 10-Oct-2006.) (New usage is discouraged.)
Hypothesis
Ref Expression
grpfo.1 𝑋 = ran 𝐺
Assertion
Ref Expression
grpoidinvlem1 (((𝐺 ∈ GrpOp ∧ (𝑌𝑋𝐴𝑋)) ∧ ((𝑌𝐺𝐴) = 𝑈 ∧ (𝐴𝐺𝐴) = 𝐴)) → (𝑈𝐺𝐴) = 𝑈)

Proof of Theorem grpoidinvlem1
StepHypRef Expression
1 id 22 . . . . 5 ((𝑌𝑋𝐴𝑋𝐴𝑋) → (𝑌𝑋𝐴𝑋𝐴𝑋))
213anidm23 1420 . . . 4 ((𝑌𝑋𝐴𝑋) → (𝑌𝑋𝐴𝑋𝐴𝑋))
3 grpfo.1 . . . . 5 𝑋 = ran 𝐺
43grpoass 30532 . . . 4 ((𝐺 ∈ GrpOp ∧ (𝑌𝑋𝐴𝑋𝐴𝑋)) → ((𝑌𝐺𝐴)𝐺𝐴) = (𝑌𝐺(𝐴𝐺𝐴)))
52, 4sylan2 593 . . 3 ((𝐺 ∈ GrpOp ∧ (𝑌𝑋𝐴𝑋)) → ((𝑌𝐺𝐴)𝐺𝐴) = (𝑌𝐺(𝐴𝐺𝐴)))
65adantr 480 . 2 (((𝐺 ∈ GrpOp ∧ (𝑌𝑋𝐴𝑋)) ∧ ((𝑌𝐺𝐴) = 𝑈 ∧ (𝐴𝐺𝐴) = 𝐴)) → ((𝑌𝐺𝐴)𝐺𝐴) = (𝑌𝐺(𝐴𝐺𝐴)))
7 oveq1 7438 . . 3 ((𝑌𝐺𝐴) = 𝑈 → ((𝑌𝐺𝐴)𝐺𝐴) = (𝑈𝐺𝐴))
87ad2antrl 728 . 2 (((𝐺 ∈ GrpOp ∧ (𝑌𝑋𝐴𝑋)) ∧ ((𝑌𝐺𝐴) = 𝑈 ∧ (𝐴𝐺𝐴) = 𝐴)) → ((𝑌𝐺𝐴)𝐺𝐴) = (𝑈𝐺𝐴))
9 oveq2 7439 . . . 4 ((𝐴𝐺𝐴) = 𝐴 → (𝑌𝐺(𝐴𝐺𝐴)) = (𝑌𝐺𝐴))
109ad2antll 729 . . 3 (((𝐺 ∈ GrpOp ∧ (𝑌𝑋𝐴𝑋)) ∧ ((𝑌𝐺𝐴) = 𝑈 ∧ (𝐴𝐺𝐴) = 𝐴)) → (𝑌𝐺(𝐴𝐺𝐴)) = (𝑌𝐺𝐴))
11 simprl 771 . . 3 (((𝐺 ∈ GrpOp ∧ (𝑌𝑋𝐴𝑋)) ∧ ((𝑌𝐺𝐴) = 𝑈 ∧ (𝐴𝐺𝐴) = 𝐴)) → (𝑌𝐺𝐴) = 𝑈)
1210, 11eqtrd 2775 . 2 (((𝐺 ∈ GrpOp ∧ (𝑌𝑋𝐴𝑋)) ∧ ((𝑌𝐺𝐴) = 𝑈 ∧ (𝐴𝐺𝐴) = 𝐴)) → (𝑌𝐺(𝐴𝐺𝐴)) = 𝑈)
136, 8, 123eqtr3d 2783 1 (((𝐺 ∈ GrpOp ∧ (𝑌𝑋𝐴𝑋)) ∧ ((𝑌𝐺𝐴) = 𝑈 ∧ (𝐴𝐺𝐴) = 𝐴)) → (𝑈𝐺𝐴) = 𝑈)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  w3a 1086   = wceq 1537  wcel 2106  ran crn 5690  (class class class)co 7431  GrpOpcgr 30518
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438  ax-un 7754
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-fo 6569  df-fv 6571  df-ov 7434  df-grpo 30522
This theorem is referenced by:  grpoidinvlem3  30535
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