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Theorem cdlemk41 38942
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 2908 . 2 (𝐺𝑇𝑔((𝑃 (𝑅𝐺)) (𝑍 (𝑅‘(𝐺𝑏)))))
2 cdlemk41.y . . 3 𝑌 = ((𝑃 (𝑅𝑔)) (𝑍 (𝑅‘(𝑔𝑏))))
3 fveq2 6766 . . . . 5 (𝑔 = 𝐺 → (𝑅𝑔) = (𝑅𝐺))
43oveq2d 7283 . . . 4 (𝑔 = 𝐺 → (𝑃 (𝑅𝑔)) = (𝑃 (𝑅𝐺)))
5 coeq1 5759 . . . . . 6 (𝑔 = 𝐺 → (𝑔𝑏) = (𝐺𝑏))
65fveq2d 6770 . . . . 5 (𝑔 = 𝐺 → (𝑅‘(𝑔𝑏)) = (𝑅‘(𝐺𝑏)))
76oveq2d 7283 . . . 4 (𝑔 = 𝐺 → (𝑍 (𝑅‘(𝑔𝑏))) = (𝑍 (𝑅‘(𝐺𝑏))))
84, 7oveq12d 7285 . . 3 (𝑔 = 𝐺 → ((𝑃 (𝑅𝑔)) (𝑍 (𝑅‘(𝑔𝑏)))) = ((𝑃 (𝑅𝐺)) (𝑍 (𝑅‘(𝐺𝑏)))))
92, 8eqtrid 2790 . 2 (𝑔 = 𝐺𝑌 = ((𝑃 (𝑅𝐺)) (𝑍 (𝑅‘(𝐺𝑏)))))
101, 9csbiegf 3865 1 (𝐺𝑇𝐺 / 𝑔𝑌 = ((𝑃 (𝑅𝐺)) (𝑍 (𝑅‘(𝐺𝑏)))))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1539  wcel 2106  csb 3831  ccnv 5583  ccom 5588  cfv 6426  (class class class)co 7267
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-rab 3073  df-v 3431  df-sbc 3716  df-csb 3832  df-dif 3889  df-un 3891  df-in 3893  df-ss 3903  df-nul 4257  df-if 4460  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-br 5074  df-opab 5136  df-co 5593  df-iota 6384  df-fv 6434  df-ov 7270
This theorem is referenced by:  cdlemkid2  38946  cdlemkfid3N  38947  cdlemky  38948  cdlemk42yN  38966
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