Users' Mathboxes Mathbox for Norm Megill < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  cdleme31sc Structured version   Visualization version   GIF version

Theorem cdleme31sc 37639
Description: Part of proof of Lemma E in [Crawley] p. 113. (Contributed by NM, 31-Mar-2013.)
Hypotheses
Ref Expression
cdleme31sc.c 𝐶 = ((𝑠 𝑈) (𝑄 ((𝑃 𝑠) 𝑊)))
cdleme31sc.x 𝑋 = ((𝑅 𝑈) (𝑄 ((𝑃 𝑅) 𝑊)))
Assertion
Ref Expression
cdleme31sc (𝑅𝐴𝑅 / 𝑠𝐶 = 𝑋)
Distinct variable groups:   𝐴,𝑠   ,𝑠   ,𝑠   𝑃,𝑠   𝑄,𝑠   𝑅,𝑠   𝑈,𝑠   𝑊,𝑠
Allowed substitution hints:   𝐶(𝑠)   𝑋(𝑠)

Proof of Theorem cdleme31sc
StepHypRef Expression
1 nfcvd 2980 . . 3 (𝑅𝐴𝑠((𝑅 𝑈) (𝑄 ((𝑃 𝑅) 𝑊))))
2 oveq1 7147 . . . 4 (𝑠 = 𝑅 → (𝑠 𝑈) = (𝑅 𝑈))
3 oveq2 7148 . . . . . 6 (𝑠 = 𝑅 → (𝑃 𝑠) = (𝑃 𝑅))
43oveq1d 7155 . . . . 5 (𝑠 = 𝑅 → ((𝑃 𝑠) 𝑊) = ((𝑃 𝑅) 𝑊))
54oveq2d 7156 . . . 4 (𝑠 = 𝑅 → (𝑄 ((𝑃 𝑠) 𝑊)) = (𝑄 ((𝑃 𝑅) 𝑊)))
62, 5oveq12d 7158 . . 3 (𝑠 = 𝑅 → ((𝑠 𝑈) (𝑄 ((𝑃 𝑠) 𝑊))) = ((𝑅 𝑈) (𝑄 ((𝑃 𝑅) 𝑊))))
71, 6csbiegf 3888 . 2 (𝑅𝐴𝑅 / 𝑠((𝑠 𝑈) (𝑄 ((𝑃 𝑠) 𝑊))) = ((𝑅 𝑈) (𝑄 ((𝑃 𝑅) 𝑊))))
8 cdleme31sc.c . . 3 𝐶 = ((𝑠 𝑈) (𝑄 ((𝑃 𝑠) 𝑊)))
98csbeq2i 3863 . 2 𝑅 / 𝑠𝐶 = 𝑅 / 𝑠((𝑠 𝑈) (𝑄 ((𝑃 𝑠) 𝑊)))
10 cdleme31sc.x . 2 𝑋 = ((𝑅 𝑈) (𝑄 ((𝑃 𝑅) 𝑊)))
117, 9, 103eqtr4g 2882 1 (𝑅𝐴𝑅 / 𝑠𝐶 = 𝑋)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1538  wcel 2114  csb 3855  (class class class)co 7140
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2178  ax-ext 2794
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-clab 2801  df-cleq 2815  df-clel 2894  df-nfc 2962  df-v 3471  df-sbc 3748  df-csb 3856  df-un 3913  df-in 3915  df-ss 3925  df-sn 4540  df-pr 4542  df-op 4546  df-uni 4814  df-br 5043  df-iota 6293  df-fv 6342  df-ov 7143
This theorem is referenced by:  cdleme31snd  37641  cdleme31sdnN  37642  cdlemefr44  37680  cdlemefr45e  37683  cdleme48fv  37754  cdleme46fvaw  37756  cdleme48bw  37757  cdleme46fsvlpq  37760  cdlemeg46fvcl  37761  cdlemeg49le  37766  cdlemeg46fjgN  37776  cdlemeg46rjgN  37777  cdlemeg46fjv  37778  cdleme48d  37790  cdlemeg49lebilem  37794  cdleme50eq  37796  cdleme50f  37797  cdlemg2jlemOLDN  37848  cdlemg2klem  37850
  Copyright terms: Public domain W3C validator