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Theorem cdleme31sn1 40383
Description: Part of proof of Lemma E in [Crawley] p. 113. (Contributed by NM, 26-Feb-2013.)
Hypotheses
Ref Expression
cdleme31sn1.i 𝐼 = (𝑦𝐵𝑡𝐴 ((¬ 𝑡 𝑊 ∧ ¬ 𝑡 (𝑃 𝑄)) → 𝑦 = 𝐺))
cdleme31sn1.n 𝑁 = if(𝑠 (𝑃 𝑄), 𝐼, 𝐷)
cdleme31sn1.c 𝐶 = (𝑦𝐵𝑡𝐴 ((¬ 𝑡 𝑊 ∧ ¬ 𝑡 (𝑃 𝑄)) → 𝑦 = 𝑅 / 𝑠𝐺))
Assertion
Ref Expression
cdleme31sn1 ((𝑅𝐴𝑅 (𝑃 𝑄)) → 𝑅 / 𝑠𝑁 = 𝐶)
Distinct variable groups:   𝑡,𝑠,𝑦,𝐴   𝐵,𝑠   ,𝑠   ,𝑠   𝑃,𝑠   𝑄,𝑠   𝑅,𝑠,𝑡,𝑦   𝑊,𝑠
Allowed substitution hints:   𝐵(𝑦,𝑡)   𝐶(𝑦,𝑡,𝑠)   𝐷(𝑦,𝑡,𝑠)   𝑃(𝑦,𝑡)   𝑄(𝑦,𝑡)   𝐺(𝑦,𝑡,𝑠)   𝐼(𝑦,𝑡,𝑠)   (𝑦,𝑡)   (𝑦,𝑡)   𝑁(𝑦,𝑡,𝑠)   𝑊(𝑦,𝑡)

Proof of Theorem cdleme31sn1
StepHypRef Expression
1 cdleme31sn1.n . . . 4 𝑁 = if(𝑠 (𝑃 𝑄), 𝐼, 𝐷)
2 eqid 2737 . . . 4 if(𝑅 (𝑃 𝑄), 𝑅 / 𝑠𝐼, 𝑅 / 𝑠𝐷) = if(𝑅 (𝑃 𝑄), 𝑅 / 𝑠𝐼, 𝑅 / 𝑠𝐷)
31, 2cdleme31sn 40382 . . 3 (𝑅𝐴𝑅 / 𝑠𝑁 = if(𝑅 (𝑃 𝑄), 𝑅 / 𝑠𝐼, 𝑅 / 𝑠𝐷))
43adantr 480 . 2 ((𝑅𝐴𝑅 (𝑃 𝑄)) → 𝑅 / 𝑠𝑁 = if(𝑅 (𝑃 𝑄), 𝑅 / 𝑠𝐼, 𝑅 / 𝑠𝐷))
5 iftrue 4531 . . . . 5 (𝑅 (𝑃 𝑄) → if(𝑅 (𝑃 𝑄), 𝑅 / 𝑠𝐼, 𝑅 / 𝑠𝐷) = 𝑅 / 𝑠𝐼)
6 cdleme31sn1.i . . . . . 6 𝐼 = (𝑦𝐵𝑡𝐴 ((¬ 𝑡 𝑊 ∧ ¬ 𝑡 (𝑃 𝑄)) → 𝑦 = 𝐺))
76csbeq2i 3907 . . . . 5 𝑅 / 𝑠𝐼 = 𝑅 / 𝑠(𝑦𝐵𝑡𝐴 ((¬ 𝑡 𝑊 ∧ ¬ 𝑡 (𝑃 𝑄)) → 𝑦 = 𝐺))
85, 7eqtrdi 2793 . . . 4 (𝑅 (𝑃 𝑄) → if(𝑅 (𝑃 𝑄), 𝑅 / 𝑠𝐼, 𝑅 / 𝑠𝐷) = 𝑅 / 𝑠(𝑦𝐵𝑡𝐴 ((¬ 𝑡 𝑊 ∧ ¬ 𝑡 (𝑃 𝑄)) → 𝑦 = 𝐺)))
9 nfcv 2905 . . . . . . . 8 𝑠𝐴
10 nfv 1914 . . . . . . . . 9 𝑠𝑡 𝑊 ∧ ¬ 𝑡 (𝑃 𝑄))
11 nfcsb1v 3923 . . . . . . . . . 10 𝑠𝑅 / 𝑠𝐺
1211nfeq2 2923 . . . . . . . . 9 𝑠 𝑦 = 𝑅 / 𝑠𝐺
1310, 12nfim 1896 . . . . . . . 8 𝑠((¬ 𝑡 𝑊 ∧ ¬ 𝑡 (𝑃 𝑄)) → 𝑦 = 𝑅 / 𝑠𝐺)
149, 13nfralw 3311 . . . . . . 7 𝑠𝑡𝐴 ((¬ 𝑡 𝑊 ∧ ¬ 𝑡 (𝑃 𝑄)) → 𝑦 = 𝑅 / 𝑠𝐺)
15 nfcv 2905 . . . . . . 7 𝑠𝐵
1614, 15nfriota 7400 . . . . . 6 𝑠(𝑦𝐵𝑡𝐴 ((¬ 𝑡 𝑊 ∧ ¬ 𝑡 (𝑃 𝑄)) → 𝑦 = 𝑅 / 𝑠𝐺))
1716a1i 11 . . . . 5 (𝑅𝐴𝑠(𝑦𝐵𝑡𝐴 ((¬ 𝑡 𝑊 ∧ ¬ 𝑡 (𝑃 𝑄)) → 𝑦 = 𝑅 / 𝑠𝐺)))
18 csbeq1a 3913 . . . . . . . . 9 (𝑠 = 𝑅𝐺 = 𝑅 / 𝑠𝐺)
1918eqeq2d 2748 . . . . . . . 8 (𝑠 = 𝑅 → (𝑦 = 𝐺𝑦 = 𝑅 / 𝑠𝐺))
2019imbi2d 340 . . . . . . 7 (𝑠 = 𝑅 → (((¬ 𝑡 𝑊 ∧ ¬ 𝑡 (𝑃 𝑄)) → 𝑦 = 𝐺) ↔ ((¬ 𝑡 𝑊 ∧ ¬ 𝑡 (𝑃 𝑄)) → 𝑦 = 𝑅 / 𝑠𝐺)))
2120ralbidv 3178 . . . . . 6 (𝑠 = 𝑅 → (∀𝑡𝐴 ((¬ 𝑡 𝑊 ∧ ¬ 𝑡 (𝑃 𝑄)) → 𝑦 = 𝐺) ↔ ∀𝑡𝐴 ((¬ 𝑡 𝑊 ∧ ¬ 𝑡 (𝑃 𝑄)) → 𝑦 = 𝑅 / 𝑠𝐺)))
2221riotabidv 7390 . . . . 5 (𝑠 = 𝑅 → (𝑦𝐵𝑡𝐴 ((¬ 𝑡 𝑊 ∧ ¬ 𝑡 (𝑃 𝑄)) → 𝑦 = 𝐺)) = (𝑦𝐵𝑡𝐴 ((¬ 𝑡 𝑊 ∧ ¬ 𝑡 (𝑃 𝑄)) → 𝑦 = 𝑅 / 𝑠𝐺)))
2317, 22csbiegf 3932 . . . 4 (𝑅𝐴𝑅 / 𝑠(𝑦𝐵𝑡𝐴 ((¬ 𝑡 𝑊 ∧ ¬ 𝑡 (𝑃 𝑄)) → 𝑦 = 𝐺)) = (𝑦𝐵𝑡𝐴 ((¬ 𝑡 𝑊 ∧ ¬ 𝑡 (𝑃 𝑄)) → 𝑦 = 𝑅 / 𝑠𝐺)))
248, 23sylan9eqr 2799 . . 3 ((𝑅𝐴𝑅 (𝑃 𝑄)) → if(𝑅 (𝑃 𝑄), 𝑅 / 𝑠𝐼, 𝑅 / 𝑠𝐷) = (𝑦𝐵𝑡𝐴 ((¬ 𝑡 𝑊 ∧ ¬ 𝑡 (𝑃 𝑄)) → 𝑦 = 𝑅 / 𝑠𝐺)))
25 cdleme31sn1.c . . 3 𝐶 = (𝑦𝐵𝑡𝐴 ((¬ 𝑡 𝑊 ∧ ¬ 𝑡 (𝑃 𝑄)) → 𝑦 = 𝑅 / 𝑠𝐺))
2624, 25eqtr4di 2795 . 2 ((𝑅𝐴𝑅 (𝑃 𝑄)) → if(𝑅 (𝑃 𝑄), 𝑅 / 𝑠𝐼, 𝑅 / 𝑠𝐷) = 𝐶)
274, 26eqtrd 2777 1 ((𝑅𝐴𝑅 (𝑃 𝑄)) → 𝑅 / 𝑠𝑁 = 𝐶)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 395   = wceq 1540  wcel 2108  wnfc 2890  wral 3061  csb 3899  ifcif 4525   class class class wbr 5143  crio 7387  (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-ral 3062  df-rex 3071  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-riota 7388
This theorem is referenced by:  cdleme31sn1c  40390
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