Theorem cdleme43frv1snN 41037

Description: Value of ⦋𝑅 / 𝑠⦌𝑁 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	⊢ ((𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ (𝑃 ∨ 𝑄)) → ⦋𝑅 / 𝑠⦌𝑁 = 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

Step	Hyp	Ref	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)))
4	1, 2, 3	cdleme31sn2 41018	1 ⊢ ((𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ (𝑃 ∨ 𝑄)) → ⦋𝑅 / 𝑠⦌𝑁 = 𝑋)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 399   = wceq 1562   ∈ wcel 2144  ⦋csb 3854  ifcif 4482   class class class wbr 5102  ‘cfv 6523  (class class class)co 7398  Basecbs 17247  lecple 17295  joincjn 18345  meetcmee 18346  Atomscatm 39892  LHypclh 40613

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1817  ax-4 1831  ax-5 1932  ax-6 1989  ax-7 2030  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2214  ax-ext 2736

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1101  df-tru 1565  df-fal 1575  df-ex 1802  df-nf 1806  df-sb 2093  df-clab 2743  df-cleq 2756  df-clel 2839  df-nfc 2913  df-rab 3417  df-v 3458  df-sbc 3747  df-csb 3855  df-dif 3909  df-un 3911  df-ss 3923  df-nul 4288  df-if 4483  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-br 5103  df-iota 6479  df-fv 6531  df-ov 7401

This theorem is referenced by: (None)