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Metamath Proof Explorer

Theorem cdlemefr31fv1

Description: Value of ( FR ) when -. R .<_ ( P .\/ Q ) . TODO This may be useful for shortening others that now use riotasv 3d . TODO: FIX COMMENT. (Contributed by NM, 30-Mar-2013)

Ref Expression
Hypotheses 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 )
cdleme29fr.o ⊢ 𝑂 = ( ℩ 𝑧 ∈ 𝐵 ∀ 𝑠 ∈ 𝐴 ( ( ¬ 𝑠 ≤ 𝑊 ∧ ( 𝑠 ∨ ( 𝑥 ∧ 𝑊 ) ) = 𝑥 ) → 𝑧 = ( 𝑁 ∨ ( 𝑥 ∧ 𝑊 ) ) ) )
cdleme29fr.f ⊢ 𝐹 = ( 𝑥 ∈ 𝐵 ↦ if ( ( 𝑃 ≠ 𝑄 ∧ ¬ 𝑥 ≤ 𝑊 ) , 𝑂 , 𝑥 ) )
cdleme43frv.x ⊢ 𝑋 = ( ( 𝑅 ∨ 𝑈 ) ∧ ( 𝑄 ∨ ( ( 𝑃 ∨ 𝑅 ) ∧ 𝑊 ) ) )
Assertion cdlemefr31fv1 ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( 𝑃 ≠ 𝑄 ∧ ( 𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊 ) ) ∧ ¬ 𝑅 ≤ ( 𝑃 ∨ 𝑄 ) ) → ( 𝐹 ‘ 𝑅 ) = 𝑋 )

Proof

Step Hyp Ref Expression
1 cdlemefr27.b ⊢ 𝐵 = ( Base ‘ 𝐾 )
2 cdlemefr27.l ⊢ ≤ = ( le ‘ 𝐾 )
3 cdlemefr27.j ⊢ ∨ = ( join ‘ 𝐾 )
4 cdlemefr27.m ⊢ ∧ = ( meet ‘ 𝐾 )
5 cdlemefr27.a ⊢ 𝐴 = ( Atoms ‘ 𝐾 )
6 cdlemefr27.h ⊢ 𝐻 = ( LHyp ‘ 𝐾 )
7 cdlemefr27.u ⊢ 𝑈 = ( ( 𝑃 ∨ 𝑄 ) ∧ 𝑊 )
8 cdlemefr27.c ⊢ 𝐶 = ( ( 𝑠 ∨ 𝑈 ) ∧ ( 𝑄 ∨ ( ( 𝑃 ∨ 𝑠 ) ∧ 𝑊 ) ) )
9 cdlemefr27.n ⊢ 𝑁 = if ( 𝑠 ≤ ( 𝑃 ∨ 𝑄 ) , 𝐼 , 𝐶 )
10 cdleme29fr.o ⊢ 𝑂 = ( ℩ 𝑧 ∈ 𝐵 ∀ 𝑠 ∈ 𝐴 ( ( ¬ 𝑠 ≤ 𝑊 ∧ ( 𝑠 ∨ ( 𝑥 ∧ 𝑊 ) ) = 𝑥 ) → 𝑧 = ( 𝑁 ∨ ( 𝑥 ∧ 𝑊 ) ) ) )
11 cdleme29fr.f ⊢ 𝐹 = ( 𝑥 ∈ 𝐵 ↦ if ( ( 𝑃 ≠ 𝑄 ∧ ¬ 𝑥 ≤ 𝑊 ) , 𝑂 , 𝑥 ) )
12 cdleme43frv.x ⊢ 𝑋 = ( ( 𝑅 ∨ 𝑈 ) ∧ ( 𝑄 ∨ ( ( 𝑃 ∨ 𝑅 ) ∧ 𝑊 ) ) )
13 1 2 3 4 5 6 7 8 9 10 11 cdlemefr32fva1 ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( 𝑃 ≠ 𝑄 ∧ ( 𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊 ) ) ∧ ¬ 𝑅 ≤ ( 𝑃 ∨ 𝑄 ) ) → ( 𝐹 ‘ 𝑅 ) = ⦋ 𝑅 / 𝑠 ⦌ 𝑁 )
14 simp2rl ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( 𝑃 ≠ 𝑄 ∧ ( 𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊 ) ) ∧ ¬ 𝑅 ≤ ( 𝑃 ∨ 𝑄 ) ) → 𝑅 ∈ 𝐴 )
15 simp3 ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( 𝑃 ≠ 𝑄 ∧ ( 𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊 ) ) ∧ ¬ 𝑅 ≤ ( 𝑃 ∨ 𝑄 ) ) → ¬ 𝑅 ≤ ( 𝑃 ∨ 𝑄 ) )
16 8 9 12 cdleme31sn2 ⊢ ( ( 𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ ( 𝑃 ∨ 𝑄 ) ) → ⦋ 𝑅 / 𝑠 ⦌ 𝑁 = 𝑋 )
17 14 15 16 syl2anc ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( 𝑃 ≠ 𝑄 ∧ ( 𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊 ) ) ∧ ¬ 𝑅 ≤ ( 𝑃 ∨ 𝑄 ) ) → ⦋ 𝑅 / 𝑠 ⦌ 𝑁 = 𝑋 )
18 13 17 eqtrd ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( 𝑃 ≠ 𝑄 ∧ ( 𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊 ) ) ∧ ¬ 𝑅 ≤ ( 𝑃 ∨ 𝑄 ) ) → ( 𝐹 ‘ 𝑅 ) = 𝑋 )