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Theorem cdleme31sn2 40391
Description: Part of proof of Lemma E in [Crawley] p. 113. (Contributed by NM, 26-Feb-2013.)
Hypotheses
Ref Expression
cdleme32sn2.d 𝐷 = ((𝑠 𝑈) (𝑄 ((𝑃 𝑠) 𝑊)))
cdleme31sn2.n 𝑁 = if(𝑠 (𝑃 𝑄), 𝐼, 𝐷)
cdleme31sn2.c 𝐶 = ((𝑅 𝑈) (𝑄 ((𝑃 𝑅) 𝑊)))
Assertion
Ref Expression
cdleme31sn2 ((𝑅𝐴 ∧ ¬ 𝑅 (𝑃 𝑄)) → 𝑅 / 𝑠𝑁 = 𝐶)
Distinct variable groups:   𝐴,𝑠   ,𝑠   ,𝑠   ,𝑠   𝑃,𝑠   𝑄,𝑠   𝑅,𝑠   𝑈,𝑠   𝑊,𝑠
Allowed substitution hints:   𝐶(𝑠)   𝐷(𝑠)   𝐼(𝑠)   𝑁(𝑠)

Proof of Theorem cdleme31sn2
StepHypRef Expression
1 cdleme31sn2.n . . . . 5 𝑁 = if(𝑠 (𝑃 𝑄), 𝐼, 𝐷)
2 eqid 2737 . . . . 5 if(𝑅 (𝑃 𝑄), 𝑅 / 𝑠𝐼, 𝑅 / 𝑠𝐷) = if(𝑅 (𝑃 𝑄), 𝑅 / 𝑠𝐼, 𝑅 / 𝑠𝐷)
31, 2cdleme31sn 40382 . . . 4 (𝑅𝐴𝑅 / 𝑠𝑁 = if(𝑅 (𝑃 𝑄), 𝑅 / 𝑠𝐼, 𝑅 / 𝑠𝐷))
43adantr 480 . . 3 ((𝑅𝐴 ∧ ¬ 𝑅 (𝑃 𝑄)) → 𝑅 / 𝑠𝑁 = if(𝑅 (𝑃 𝑄), 𝑅 / 𝑠𝐼, 𝑅 / 𝑠𝐷))
5 iffalse 4534 . . . . 5 𝑅 (𝑃 𝑄) → if(𝑅 (𝑃 𝑄), 𝑅 / 𝑠𝐼, 𝑅 / 𝑠𝐷) = 𝑅 / 𝑠𝐷)
6 cdleme32sn2.d . . . . . 6 𝐷 = ((𝑠 𝑈) (𝑄 ((𝑃 𝑠) 𝑊)))
76csbeq2i 3907 . . . . 5 𝑅 / 𝑠𝐷 = 𝑅 / 𝑠((𝑠 𝑈) (𝑄 ((𝑃 𝑠) 𝑊)))
85, 7eqtrdi 2793 . . . 4 𝑅 (𝑃 𝑄) → if(𝑅 (𝑃 𝑄), 𝑅 / 𝑠𝐼, 𝑅 / 𝑠𝐷) = 𝑅 / 𝑠((𝑠 𝑈) (𝑄 ((𝑃 𝑠) 𝑊))))
9 nfcvd 2906 . . . . 5 (𝑅𝐴𝑠((𝑅 𝑈) (𝑄 ((𝑃 𝑅) 𝑊))))
10 oveq1 7438 . . . . . 6 (𝑠 = 𝑅 → (𝑠 𝑈) = (𝑅 𝑈))
11 oveq2 7439 . . . . . . . 8 (𝑠 = 𝑅 → (𝑃 𝑠) = (𝑃 𝑅))
1211oveq1d 7446 . . . . . . 7 (𝑠 = 𝑅 → ((𝑃 𝑠) 𝑊) = ((𝑃 𝑅) 𝑊))
1312oveq2d 7447 . . . . . 6 (𝑠 = 𝑅 → (𝑄 ((𝑃 𝑠) 𝑊)) = (𝑄 ((𝑃 𝑅) 𝑊)))
1410, 13oveq12d 7449 . . . . 5 (𝑠 = 𝑅 → ((𝑠 𝑈) (𝑄 ((𝑃 𝑠) 𝑊))) = ((𝑅 𝑈) (𝑄 ((𝑃 𝑅) 𝑊))))
159, 14csbiegf 3932 . . . 4 (𝑅𝐴𝑅 / 𝑠((𝑠 𝑈) (𝑄 ((𝑃 𝑠) 𝑊))) = ((𝑅 𝑈) (𝑄 ((𝑃 𝑅) 𝑊))))
168, 15sylan9eqr 2799 . . 3 ((𝑅𝐴 ∧ ¬ 𝑅 (𝑃 𝑄)) → if(𝑅 (𝑃 𝑄), 𝑅 / 𝑠𝐼, 𝑅 / 𝑠𝐷) = ((𝑅 𝑈) (𝑄 ((𝑃 𝑅) 𝑊))))
174, 16eqtrd 2777 . 2 ((𝑅𝐴 ∧ ¬ 𝑅 (𝑃 𝑄)) → 𝑅 / 𝑠𝑁 = ((𝑅 𝑈) (𝑄 ((𝑃 𝑅) 𝑊))))
18 cdleme31sn2.c . 2 𝐶 = ((𝑅 𝑈) (𝑄 ((𝑃 𝑅) 𝑊)))
1917, 18eqtr4di 2795 1 ((𝑅𝐴 ∧ ¬ 𝑅 (𝑃 𝑄)) → 𝑅 / 𝑠𝑁 = 𝐶)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 395   = wceq 1540  wcel 2108  csb 3899  ifcif 4525   class class class wbr 5143  (class class class)co 7431
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-iota 6514  df-fv 6569  df-ov 7434
This theorem is referenced by:  cdlemefr32sn2aw  40406  cdleme43frv1snN  40410  cdlemefr31fv1  40413  cdleme35sn2aw  40460  cdleme35sn3a  40461
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