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Theorem cdlemk41 40630
Description: Part of proof of Lemma K of [Crawley] p. 118. TODO: fix comment. (Contributed by NM, 19-Jul-2013.)
Hypothesis
Ref Expression
cdlemk41.y 𝑌 = ((𝑃 (𝑅𝑔)) (𝑍 (𝑅‘(𝑔𝑏))))
Assertion
Ref Expression
cdlemk41 (𝐺𝑇𝐺 / 𝑔𝑌 = ((𝑃 (𝑅𝐺)) (𝑍 (𝑅‘(𝐺𝑏)))))
Distinct variable groups:   ,𝑔   ,𝑔   𝑔,𝐺   𝑃,𝑔   𝑅,𝑔   𝑇,𝑔   𝑔,𝑍   𝑔,𝑏
Allowed substitution hints:   𝑃(𝑏)   𝑅(𝑏)   𝑇(𝑏)   𝐺(𝑏)   (𝑏)   (𝑏)   𝑌(𝑔,𝑏)   𝑍(𝑏)

Proof of Theorem cdlemk41
StepHypRef Expression
1 nfcvd 2893 . 2 (𝐺𝑇𝑔((𝑃 (𝑅𝐺)) (𝑍 (𝑅‘(𝐺𝑏)))))
2 cdlemk41.y . . 3 𝑌 = ((𝑃 (𝑅𝑔)) (𝑍 (𝑅‘(𝑔𝑏))))
3 fveq2 6891 . . . . 5 (𝑔 = 𝐺 → (𝑅𝑔) = (𝑅𝐺))
43oveq2d 7430 . . . 4 (𝑔 = 𝐺 → (𝑃 (𝑅𝑔)) = (𝑃 (𝑅𝐺)))
5 coeq1 5855 . . . . . 6 (𝑔 = 𝐺 → (𝑔𝑏) = (𝐺𝑏))
65fveq2d 6895 . . . . 5 (𝑔 = 𝐺 → (𝑅‘(𝑔𝑏)) = (𝑅‘(𝐺𝑏)))
76oveq2d 7430 . . . 4 (𝑔 = 𝐺 → (𝑍 (𝑅‘(𝑔𝑏))) = (𝑍 (𝑅‘(𝐺𝑏))))
84, 7oveq12d 7432 . . 3 (𝑔 = 𝐺 → ((𝑃 (𝑅𝑔)) (𝑍 (𝑅‘(𝑔𝑏)))) = ((𝑃 (𝑅𝐺)) (𝑍 (𝑅‘(𝐺𝑏)))))
92, 8eqtrid 2778 . 2 (𝑔 = 𝐺𝑌 = ((𝑃 (𝑅𝐺)) (𝑍 (𝑅‘(𝐺𝑏)))))
101, 9csbiegf 3926 1 (𝐺𝑇𝐺 / 𝑔𝑌 = ((𝑃 (𝑅𝐺)) (𝑍 (𝑅‘(𝐺𝑏)))))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1534  wcel 2099  csb 3892  ccnv 5672  ccom 5677  cfv 6544  (class class class)co 7414
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2697
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-clab 2704  df-cleq 2718  df-clel 2803  df-nfc 2878  df-rab 3421  df-v 3465  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-ss 3964  df-nul 4324  df-if 4525  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4907  df-br 5145  df-opab 5207  df-co 5682  df-iota 6496  df-fv 6552  df-ov 7417
This theorem is referenced by:  cdlemkid2  40634  cdlemkfid3N  40635  cdlemky  40636  cdlemk42yN  40654
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