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Theorem cdlemk41 35727
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 2762 . 2 (𝐺𝑇𝑔((𝑃 (𝑅𝐺)) (𝑍 (𝑅‘(𝐺𝑏)))))
2 cdlemk41.y . . 3 𝑌 = ((𝑃 (𝑅𝑔)) (𝑍 (𝑅‘(𝑔𝑏))))
3 fveq2 6158 . . . . 5 (𝑔 = 𝐺 → (𝑅𝑔) = (𝑅𝐺))
43oveq2d 6631 . . . 4 (𝑔 = 𝐺 → (𝑃 (𝑅𝑔)) = (𝑃 (𝑅𝐺)))
5 coeq1 5249 . . . . . 6 (𝑔 = 𝐺 → (𝑔𝑏) = (𝐺𝑏))
65fveq2d 6162 . . . . 5 (𝑔 = 𝐺 → (𝑅‘(𝑔𝑏)) = (𝑅‘(𝐺𝑏)))
76oveq2d 6631 . . . 4 (𝑔 = 𝐺 → (𝑍 (𝑅‘(𝑔𝑏))) = (𝑍 (𝑅‘(𝐺𝑏))))
84, 7oveq12d 6633 . . 3 (𝑔 = 𝐺 → ((𝑃 (𝑅𝑔)) (𝑍 (𝑅‘(𝑔𝑏)))) = ((𝑃 (𝑅𝐺)) (𝑍 (𝑅‘(𝐺𝑏)))))
92, 8syl5eq 2667 . 2 (𝑔 = 𝐺𝑌 = ((𝑃 (𝑅𝐺)) (𝑍 (𝑅‘(𝐺𝑏)))))
101, 9csbiegf 3543 1 (𝐺𝑇𝐺 / 𝑔𝑌 = ((𝑃 (𝑅𝐺)) (𝑍 (𝑅‘(𝐺𝑏)))))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1480  wcel 1987  csb 3519  ccnv 5083  ccom 5088  cfv 5857  (class class class)co 6615
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-rex 2914  df-rab 2917  df-v 3192  df-sbc 3423  df-csb 3520  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-nul 3898  df-if 4065  df-sn 4156  df-pr 4158  df-op 4162  df-uni 4410  df-br 4624  df-opab 4684  df-co 5093  df-iota 5820  df-fv 5865  df-ov 6618
This theorem is referenced by:  cdlemkid2  35731  cdlemkfid3N  35732  cdlemky  35733  cdlemk42yN  35751
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