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Theorem cdleme43frv1snN 40410
Description: Value of 𝑅 / 𝑠𝑁 when ¬ 𝑅 (𝑃 𝑄). (Contributed by NM, 30-Mar-2013.) (New usage is discouraged.)
Hypotheses
Ref Expression
cdlemefr27.b 𝐵 = (Base‘𝐾)
cdlemefr27.l = (le‘𝐾)
cdlemefr27.j = (join‘𝐾)
cdlemefr27.m = (meet‘𝐾)
cdlemefr27.a 𝐴 = (Atoms‘𝐾)
cdlemefr27.h 𝐻 = (LHyp‘𝐾)
cdlemefr27.u 𝑈 = ((𝑃 𝑄) 𝑊)
cdlemefr27.c 𝐶 = ((𝑠 𝑈) (𝑄 ((𝑃 𝑠) 𝑊)))
cdlemefr27.n 𝑁 = if(𝑠 (𝑃 𝑄), 𝐼, 𝐶)
cdleme43fr.x 𝑋 = ((𝑅 𝑈) (𝑄 ((𝑃 𝑅) 𝑊)))
Assertion
Ref Expression
cdleme43frv1snN ((𝑅𝐴 ∧ ¬ 𝑅 (𝑃 𝑄)) → 𝑅 / 𝑠𝑁 = 𝑋)
Distinct variable groups:   𝐴,𝑠   ,𝑠   ,𝑠   ,𝑠   𝑃,𝑠   𝑄,𝑠   𝑅,𝑠   𝑈,𝑠   𝑊,𝑠   𝐻,𝑠   𝐾,𝑠   𝐵,𝑠
Allowed substitution hints:   𝐶(𝑠)   𝐼(𝑠)   𝑁(𝑠)   𝑋(𝑠)

Proof of Theorem cdleme43frv1snN
StepHypRef Expression
1 cdlemefr27.c . 2 𝐶 = ((𝑠 𝑈) (𝑄 ((𝑃 𝑠) 𝑊)))
2 cdlemefr27.n . 2 𝑁 = if(𝑠 (𝑃 𝑄), 𝐼, 𝐶)
3 cdleme43fr.x . 2 𝑋 = ((𝑅 𝑈) (𝑄 ((𝑃 𝑅) 𝑊)))
41, 2, 3cdleme31sn2 40391 1 ((𝑅𝐴 ∧ ¬ 𝑅 (𝑃 𝑄)) → 𝑅 / 𝑠𝑁 = 𝑋)
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  cfv 6561  (class class class)co 7431  Basecbs 17247  lecple 17304  joincjn 18357  meetcmee 18358  Atomscatm 39264  LHypclh 39986
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: (None)
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