Users' Mathboxes Mathbox for Norm Megill < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  cdleme31sn Structured version   Visualization version   GIF version

Theorem cdleme31sn 36087
Description: Part of proof of Lemma E in [Crawley] p. 113. (Contributed by NM, 26-Feb-2013.)
Hypotheses
Ref Expression
cdleme31sn.n 𝑁 = if(𝑠 (𝑃 𝑄), 𝐼, 𝐷)
cdleme31sn.c 𝐶 = if(𝑅 (𝑃 𝑄), 𝑅 / 𝑠𝐼, 𝑅 / 𝑠𝐷)
Assertion
Ref Expression
cdleme31sn (𝑅𝐴𝑅 / 𝑠𝑁 = 𝐶)
Distinct variable groups:   𝐴,𝑠   ,𝑠   ,𝑠   𝑃,𝑠   𝑄,𝑠   𝑅,𝑠
Allowed substitution hints:   𝐶(𝑠)   𝐷(𝑠)   𝐼(𝑠)   𝑁(𝑠)

Proof of Theorem cdleme31sn
StepHypRef Expression
1 nfv 1956 . . . . 5 𝑠 𝑅 (𝑃 𝑄)
2 nfcsb1v 3655 . . . . 5 𝑠𝑅 / 𝑠𝐼
3 nfcsb1v 3655 . . . . 5 𝑠𝑅 / 𝑠𝐷
41, 2, 3nfif 4223 . . . 4 𝑠if(𝑅 (𝑃 𝑄), 𝑅 / 𝑠𝐼, 𝑅 / 𝑠𝐷)
54a1i 11 . . 3 (𝑅𝐴𝑠if(𝑅 (𝑃 𝑄), 𝑅 / 𝑠𝐼, 𝑅 / 𝑠𝐷))
6 breq1 4763 . . . 4 (𝑠 = 𝑅 → (𝑠 (𝑃 𝑄) ↔ 𝑅 (𝑃 𝑄)))
7 csbeq1a 3648 . . . 4 (𝑠 = 𝑅𝐼 = 𝑅 / 𝑠𝐼)
8 csbeq1a 3648 . . . 4 (𝑠 = 𝑅𝐷 = 𝑅 / 𝑠𝐷)
96, 7, 8ifbieq12d 4221 . . 3 (𝑠 = 𝑅 → if(𝑠 (𝑃 𝑄), 𝐼, 𝐷) = if(𝑅 (𝑃 𝑄), 𝑅 / 𝑠𝐼, 𝑅 / 𝑠𝐷))
105, 9csbiegf 3663 . 2 (𝑅𝐴𝑅 / 𝑠if(𝑠 (𝑃 𝑄), 𝐼, 𝐷) = if(𝑅 (𝑃 𝑄), 𝑅 / 𝑠𝐼, 𝑅 / 𝑠𝐷))
11 cdleme31sn.n . . 3 𝑁 = if(𝑠 (𝑃 𝑄), 𝐼, 𝐷)
1211csbeq2i 4101 . 2 𝑅 / 𝑠𝑁 = 𝑅 / 𝑠if(𝑠 (𝑃 𝑄), 𝐼, 𝐷)
13 cdleme31sn.c . 2 𝐶 = if(𝑅 (𝑃 𝑄), 𝑅 / 𝑠𝐼, 𝑅 / 𝑠𝐷)
1410, 12, 133eqtr4g 2783 1 (𝑅𝐴𝑅 / 𝑠𝑁 = 𝐶)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1596  wcel 2103  wnfc 2853  csb 3639  ifcif 4194   class class class wbr 4760  (class class class)co 6765
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1835  ax-4 1850  ax-5 1952  ax-6 2018  ax-7 2054  ax-9 2112  ax-10 2132  ax-11 2147  ax-12 2160  ax-13 2355  ax-ext 2704
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1599  df-ex 1818  df-nf 1823  df-sb 2011  df-clab 2711  df-cleq 2717  df-clel 2720  df-nfc 2855  df-rab 3023  df-v 3306  df-sbc 3542  df-csb 3640  df-dif 3683  df-un 3685  df-in 3687  df-ss 3694  df-nul 4024  df-if 4195  df-sn 4286  df-pr 4288  df-op 4292  df-br 4761
This theorem is referenced by:  cdleme31sn1  36088  cdleme31sn2  36096
  Copyright terms: Public domain W3C validator