Theorem cdleme43frv1snN 40409

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 40390	1 ⊢ ((𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ (𝑃 ∨ 𝑄)) → ⦋𝑅 / 𝑠⦌𝑁 = 𝑋)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 395   = wceq 1540   ∈ wcel 2109  ⦋csb 3865  ifcif 4491   class class class wbr 5110  ‘cfv 6514  (class class class)co 7390  Basecbs 17186  lecple 17234  joincjn 18279  meetcmee 18280  Atomscatm 39263  LHypclh 39985

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 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2702

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-clab 2709  df-cleq 2722  df-clel 2804  df-nfc 2879  df-rab 3409  df-v 3452  df-sbc 3757  df-csb 3866  df-dif 3920  df-un 3922  df-ss 3934  df-nul 4300  df-if 4492  df-sn 4593  df-pr 4595  df-op 4599  df-uni 4875  df-br 5111  df-iota 6467  df-fv 6522  df-ov 7393

This theorem is referenced by: (None)