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Metamath Proof Explorer

Theorem cdlemefr32snb

Description: Show closure of [_ R / s ]_ N . (Contributed by NM, 28-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 )
Assertion cdlemefr32snb ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( 𝑃 ≠ 𝑄 ∧ ( 𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊 ) ) ∧ ¬ 𝑅 ≤ ( 𝑃 ∨ 𝑄 ) ) → ⦋ 𝑅 / 𝑠 ⦌ 𝑁 ∈ 𝐵 )

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 1 2 3 4 5 6 7 8 9 cdlemefr32sn2aw ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( 𝑃 ≠ 𝑄 ∧ ( 𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊 ) ) ∧ ¬ 𝑅 ≤ ( 𝑃 ∨ 𝑄 ) ) → ( ⦋ 𝑅 / 𝑠 ⦌ 𝑁 ∈ 𝐴 ∧ ¬ ⦋ 𝑅 / 𝑠 ⦌ 𝑁 ≤ 𝑊 ) )
11 10 simpld ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( 𝑃 ≠ 𝑄 ∧ ( 𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊 ) ) ∧ ¬ 𝑅 ≤ ( 𝑃 ∨ 𝑄 ) ) → ⦋ 𝑅 / 𝑠 ⦌ 𝑁 ∈ 𝐴 )
12 1 5 atbase ⊢ ( ⦋ 𝑅 / 𝑠 ⦌ 𝑁 ∈ 𝐴 → ⦋ 𝑅 / 𝑠 ⦌ 𝑁 ∈ 𝐵 )
13 11 12 syl ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( 𝑃 ≠ 𝑄 ∧ ( 𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊 ) ) ∧ ¬ 𝑅 ≤ ( 𝑃 ∨ 𝑄 ) ) → ⦋ 𝑅 / 𝑠 ⦌ 𝑁 ∈ 𝐵 )