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Theorem cdlemk41 40433
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 2900 . 2 (𝐺𝑇𝑔((𝑃 (𝑅𝐺)) (𝑍 (𝑅‘(𝐺𝑏)))))
2 cdlemk41.y . . 3 𝑌 = ((𝑃 (𝑅𝑔)) (𝑍 (𝑅‘(𝑔𝑏))))
3 fveq2 6902 . . . . 5 (𝑔 = 𝐺 → (𝑅𝑔) = (𝑅𝐺))
43oveq2d 7442 . . . 4 (𝑔 = 𝐺 → (𝑃 (𝑅𝑔)) = (𝑃 (𝑅𝐺)))
5 coeq1 5864 . . . . . 6 (𝑔 = 𝐺 → (𝑔𝑏) = (𝐺𝑏))
65fveq2d 6906 . . . . 5 (𝑔 = 𝐺 → (𝑅‘(𝑔𝑏)) = (𝑅‘(𝐺𝑏)))
76oveq2d 7442 . . . 4 (𝑔 = 𝐺 → (𝑍 (𝑅‘(𝑔𝑏))) = (𝑍 (𝑅‘(𝐺𝑏))))
84, 7oveq12d 7444 . . 3 (𝑔 = 𝐺 → ((𝑃 (𝑅𝑔)) (𝑍 (𝑅‘(𝑔𝑏)))) = ((𝑃 (𝑅𝐺)) (𝑍 (𝑅‘(𝐺𝑏)))))
92, 8eqtrid 2780 . 2 (𝑔 = 𝐺𝑌 = ((𝑃 (𝑅𝐺)) (𝑍 (𝑅‘(𝐺𝑏)))))
101, 9csbiegf 3928 1 (𝐺𝑇𝐺 / 𝑔𝑌 = ((𝑃 (𝑅𝐺)) (𝑍 (𝑅‘(𝐺𝑏)))))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1533  wcel 2098  csb 3894  ccnv 5681  ccom 5686  cfv 6553  (class class class)co 7426
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2699
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-rab 3431  df-v 3475  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4327  df-if 4533  df-sn 4633  df-pr 4635  df-op 4639  df-uni 4913  df-br 5153  df-opab 5215  df-co 5691  df-iota 6505  df-fv 6561  df-ov 7429
This theorem is referenced by:  cdlemkid2  40437  cdlemkfid3N  40438  cdlemky  40439  cdlemk42yN  40457
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