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Theorem cdleme31sn2 38048
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 2738 . . . . 5 if(𝑅 (𝑃 𝑄), 𝑅 / 𝑠𝐼, 𝑅 / 𝑠𝐷) = if(𝑅 (𝑃 𝑄), 𝑅 / 𝑠𝐼, 𝑅 / 𝑠𝐷)
31, 2cdleme31sn 38039 . . . 4 (𝑅𝐴𝑅 / 𝑠𝑁 = if(𝑅 (𝑃 𝑄), 𝑅 / 𝑠𝐼, 𝑅 / 𝑠𝐷))
43adantr 484 . . 3 ((𝑅𝐴 ∧ ¬ 𝑅 (𝑃 𝑄)) → 𝑅 / 𝑠𝑁 = if(𝑅 (𝑃 𝑄), 𝑅 / 𝑠𝐼, 𝑅 / 𝑠𝐷))
5 iffalse 4423 . . . . 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 3798 . . . . 5 𝑅 / 𝑠𝐷 = 𝑅 / 𝑠((𝑠 𝑈) (𝑄 ((𝑃 𝑠) 𝑊)))
85, 7eqtrdi 2789 . . . 4 𝑅 (𝑃 𝑄) → if(𝑅 (𝑃 𝑄), 𝑅 / 𝑠𝐼, 𝑅 / 𝑠𝐷) = 𝑅 / 𝑠((𝑠 𝑈) (𝑄 ((𝑃 𝑠) 𝑊))))
9 nfcvd 2900 . . . . 5 (𝑅𝐴𝑠((𝑅 𝑈) (𝑄 ((𝑃 𝑅) 𝑊))))
10 oveq1 7179 . . . . . 6 (𝑠 = 𝑅 → (𝑠 𝑈) = (𝑅 𝑈))
11 oveq2 7180 . . . . . . . 8 (𝑠 = 𝑅 → (𝑃 𝑠) = (𝑃 𝑅))
1211oveq1d 7187 . . . . . . 7 (𝑠 = 𝑅 → ((𝑃 𝑠) 𝑊) = ((𝑃 𝑅) 𝑊))
1312oveq2d 7188 . . . . . 6 (𝑠 = 𝑅 → (𝑄 ((𝑃 𝑠) 𝑊)) = (𝑄 ((𝑃 𝑅) 𝑊)))
1410, 13oveq12d 7190 . . . . 5 (𝑠 = 𝑅 → ((𝑠 𝑈) (𝑄 ((𝑃 𝑠) 𝑊))) = ((𝑅 𝑈) (𝑄 ((𝑃 𝑅) 𝑊))))
159, 14csbiegf 3823 . . . 4 (𝑅𝐴𝑅 / 𝑠((𝑠 𝑈) (𝑄 ((𝑃 𝑠) 𝑊))) = ((𝑅 𝑈) (𝑄 ((𝑃 𝑅) 𝑊))))
168, 15sylan9eqr 2795 . . 3 ((𝑅𝐴 ∧ ¬ 𝑅 (𝑃 𝑄)) → if(𝑅 (𝑃 𝑄), 𝑅 / 𝑠𝐼, 𝑅 / 𝑠𝐷) = ((𝑅 𝑈) (𝑄 ((𝑃 𝑅) 𝑊))))
174, 16eqtrd 2773 . 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 2791 1 ((𝑅𝐴 ∧ ¬ 𝑅 (𝑃 𝑄)) → 𝑅 / 𝑠𝑁 = 𝐶)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 399   = wceq 1542  wcel 2114  csb 3790  ifcif 4414   class class class wbr 5030  (class class class)co 7172
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2020  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2162  ax-12 2179  ax-ext 2710
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 847  df-3an 1090  df-tru 1545  df-ex 1787  df-nf 1791  df-sb 2075  df-clab 2717  df-cleq 2730  df-clel 2811  df-nfc 2881  df-rab 3062  df-v 3400  df-sbc 3681  df-csb 3791  df-un 3848  df-in 3850  df-ss 3860  df-if 4415  df-sn 4517  df-pr 4519  df-op 4523  df-uni 4797  df-br 5031  df-iota 6297  df-fv 6347  df-ov 7175
This theorem is referenced by:  cdlemefr32sn2aw  38063  cdleme43frv1snN  38067  cdlemefr31fv1  38070  cdleme35sn2aw  38117  cdleme35sn3a  38118
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