Users' Mathboxes Mathbox for Norm Megill < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  cdlemk41 Structured version   Visualization version   GIF version

Theorem cdlemk41 36995
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 2970 . 2 (𝐺𝑇𝑔((𝑃 (𝑅𝐺)) (𝑍 (𝑅‘(𝐺𝑏)))))
2 cdlemk41.y . . 3 𝑌 = ((𝑃 (𝑅𝑔)) (𝑍 (𝑅‘(𝑔𝑏))))
3 fveq2 6433 . . . . 5 (𝑔 = 𝐺 → (𝑅𝑔) = (𝑅𝐺))
43oveq2d 6921 . . . 4 (𝑔 = 𝐺 → (𝑃 (𝑅𝑔)) = (𝑃 (𝑅𝐺)))
5 coeq1 5512 . . . . . 6 (𝑔 = 𝐺 → (𝑔𝑏) = (𝐺𝑏))
65fveq2d 6437 . . . . 5 (𝑔 = 𝐺 → (𝑅‘(𝑔𝑏)) = (𝑅‘(𝐺𝑏)))
76oveq2d 6921 . . . 4 (𝑔 = 𝐺 → (𝑍 (𝑅‘(𝑔𝑏))) = (𝑍 (𝑅‘(𝐺𝑏))))
84, 7oveq12d 6923 . . 3 (𝑔 = 𝐺 → ((𝑃 (𝑅𝑔)) (𝑍 (𝑅‘(𝑔𝑏)))) = ((𝑃 (𝑅𝐺)) (𝑍 (𝑅‘(𝐺𝑏)))))
92, 8syl5eq 2873 . 2 (𝑔 = 𝐺𝑌 = ((𝑃 (𝑅𝐺)) (𝑍 (𝑅‘(𝐺𝑏)))))
101, 9csbiegf 3781 1 (𝐺𝑇𝐺 / 𝑔𝑌 = ((𝑃 (𝑅𝐺)) (𝑍 (𝑅‘(𝐺𝑏)))))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1658  wcel 2166  csb 3757  ccnv 5341  ccom 5346  cfv 6123  (class class class)co 6905
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1896  ax-4 1910  ax-5 2011  ax-6 2077  ax-7 2114  ax-9 2175  ax-10 2194  ax-11 2209  ax-12 2222  ax-13 2391  ax-ext 2803
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 881  df-3an 1115  df-tru 1662  df-ex 1881  df-nf 1885  df-sb 2070  df-clab 2812  df-cleq 2818  df-clel 2821  df-nfc 2958  df-rex 3123  df-rab 3126  df-v 3416  df-sbc 3663  df-csb 3758  df-dif 3801  df-un 3803  df-in 3805  df-ss 3812  df-nul 4145  df-if 4307  df-sn 4398  df-pr 4400  df-op 4404  df-uni 4659  df-br 4874  df-opab 4936  df-co 5351  df-iota 6086  df-fv 6131  df-ov 6908
This theorem is referenced by:  cdlemkid2  36999  cdlemkfid3N  37000  cdlemky  37001  cdlemk42yN  37019
  Copyright terms: Public domain W3C validator