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Metamath Proof Explorer

Theorem cdleme4gfv

Description: Part of proof of Lemma D in Crawley p. 113. TODO: Can this replace uses of cdleme32a ? TODO: Can this be used to help prove the R or S case where X is an atom? TODO: Would an antecedent transformer like cdleme46f2g2 help? (Contributed by NM, 8-Apr-2013)

Ref Expression
Hypotheses cdlemef47.b ⊢ 𝐵 = ( Base ‘ 𝐾 )
cdlemef47.l ⊢ ≤ = ( le ‘ 𝐾 )
cdlemef47.j ⊢ ∨ = ( join ‘ 𝐾 )
cdlemef47.m ⊢ ∧ = ( meet ‘ 𝐾 )
cdlemef47.a ⊢ 𝐴 = ( Atoms ‘ 𝐾 )
cdlemef47.h ⊢ 𝐻 = ( LHyp ‘ 𝐾 )
cdlemef47.v ⊢ 𝑉 = ( ( 𝑄 ∨ 𝑃 ) ∧ 𝑊 )
cdlemef47.n ⊢ 𝑁 = ( ( 𝑣 ∨ 𝑉 ) ∧ ( 𝑃 ∨ ( ( 𝑄 ∨ 𝑣 ) ∧ 𝑊 ) ) )
cdlemefs47.o ⊢ 𝑂 = ( ( 𝑄 ∨ 𝑃 ) ∧ ( 𝑁 ∨ ( ( 𝑢 ∨ 𝑣 ) ∧ 𝑊 ) ) )
cdlemef47.g ⊢ 𝐺 = ( 𝑎 ∈ 𝐵 ↦ if ( ( 𝑄 ≠ 𝑃 ∧ ¬ 𝑎 ≤ 𝑊 ) , ( ℩ 𝑐 ∈ 𝐵 ∀ 𝑢 ∈ 𝐴 ( ( ¬ 𝑢 ≤ 𝑊 ∧ ( 𝑢 ∨ ( 𝑎 ∧ 𝑊 ) ) = 𝑎 ) → 𝑐 = ( if ( 𝑢 ≤ ( 𝑄 ∨ 𝑃 ) , ( ℩ 𝑏 ∈ 𝐵 ∀ 𝑣 ∈ 𝐴 ( ( ¬ 𝑣 ≤ 𝑊 ∧ ¬ 𝑣 ≤ ( 𝑄 ∨ 𝑃 ) ) → 𝑏 = 𝑂 ) ) , ⦋ 𝑢 / 𝑣 ⦌ 𝑁 ) ∨ ( 𝑎 ∧ 𝑊 ) ) ) ) , 𝑎 ) )
Assertion cdleme4gfv ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( 𝑃 ≠ 𝑄 ∧ ( 𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊 ) ) ∧ ( ( 𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊 ) ∧ ( 𝑆 ∨ ( 𝑋 ∧ 𝑊 ) ) = 𝑋 ) ) → ( 𝐺 ‘ 𝑋 ) = ( ( 𝐺 ‘ 𝑆 ) ∨ ( 𝑋 ∧ 𝑊 ) ) )

Proof

Step Hyp Ref Expression
1 cdlemef47.b ⊢ 𝐵 = ( Base ‘ 𝐾 )
2 cdlemef47.l ⊢ ≤ = ( le ‘ 𝐾 )
3 cdlemef47.j ⊢ ∨ = ( join ‘ 𝐾 )
4 cdlemef47.m ⊢ ∧ = ( meet ‘ 𝐾 )
5 cdlemef47.a ⊢ 𝐴 = ( Atoms ‘ 𝐾 )
6 cdlemef47.h ⊢ 𝐻 = ( LHyp ‘ 𝐾 )
7 cdlemef47.v ⊢ 𝑉 = ( ( 𝑄 ∨ 𝑃 ) ∧ 𝑊 )
8 cdlemef47.n ⊢ 𝑁 = ( ( 𝑣 ∨ 𝑉 ) ∧ ( 𝑃 ∨ ( ( 𝑄 ∨ 𝑣 ) ∧ 𝑊 ) ) )
9 cdlemefs47.o ⊢ 𝑂 = ( ( 𝑄 ∨ 𝑃 ) ∧ ( 𝑁 ∨ ( ( 𝑢 ∨ 𝑣 ) ∧ 𝑊 ) ) )
10 cdlemef47.g ⊢ 𝐺 = ( 𝑎 ∈ 𝐵 ↦ if ( ( 𝑄 ≠ 𝑃 ∧ ¬ 𝑎 ≤ 𝑊 ) , ( ℩ 𝑐 ∈ 𝐵 ∀ 𝑢 ∈ 𝐴 ( ( ¬ 𝑢 ≤ 𝑊 ∧ ( 𝑢 ∨ ( 𝑎 ∧ 𝑊 ) ) = 𝑎 ) → 𝑐 = ( if ( 𝑢 ≤ ( 𝑄 ∨ 𝑃 ) , ( ℩ 𝑏 ∈ 𝐵 ∀ 𝑣 ∈ 𝐴 ( ( ¬ 𝑣 ≤ 𝑊 ∧ ¬ 𝑣 ≤ ( 𝑄 ∨ 𝑃 ) ) → 𝑏 = 𝑂 ) ) , ⦋ 𝑢 / 𝑣 ⦌ 𝑁 ) ∨ ( 𝑎 ∧ 𝑊 ) ) ) ) , 𝑎 ) )
11 simp11 ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( 𝑃 ≠ 𝑄 ∧ ( 𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊 ) ) ∧ ( ( 𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊 ) ∧ ( 𝑆 ∨ ( 𝑋 ∧ 𝑊 ) ) = 𝑋 ) ) → ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) )
12 simp13 ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( 𝑃 ≠ 𝑄 ∧ ( 𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊 ) ) ∧ ( ( 𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊 ) ∧ ( 𝑆 ∨ ( 𝑋 ∧ 𝑊 ) ) = 𝑋 ) ) → ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) )
13 simp12 ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( 𝑃 ≠ 𝑄 ∧ ( 𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊 ) ) ∧ ( ( 𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊 ) ∧ ( 𝑆 ∨ ( 𝑋 ∧ 𝑊 ) ) = 𝑋 ) ) → ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) )
14 simp2l ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( 𝑃 ≠ 𝑄 ∧ ( 𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊 ) ) ∧ ( ( 𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊 ) ∧ ( 𝑆 ∨ ( 𝑋 ∧ 𝑊 ) ) = 𝑋 ) ) → 𝑃 ≠ 𝑄 )
15 14 necomd ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( 𝑃 ≠ 𝑄 ∧ ( 𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊 ) ) ∧ ( ( 𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊 ) ∧ ( 𝑆 ∨ ( 𝑋 ∧ 𝑊 ) ) = 𝑋 ) ) → 𝑄 ≠ 𝑃 )
16 simp2r ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( 𝑃 ≠ 𝑄 ∧ ( 𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊 ) ) ∧ ( ( 𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊 ) ∧ ( 𝑆 ∨ ( 𝑋 ∧ 𝑊 ) ) = 𝑋 ) ) → ( 𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊 ) )
17 simp3 ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( 𝑃 ≠ 𝑄 ∧ ( 𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊 ) ) ∧ ( ( 𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊 ) ∧ ( 𝑆 ∨ ( 𝑋 ∧ 𝑊 ) ) = 𝑋 ) ) → ( ( 𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊 ) ∧ ( 𝑆 ∨ ( 𝑋 ∧ 𝑊 ) ) = 𝑋 ) )
18 1 2 3 4 5 6 7 8 9 10 cdleme48fv ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ) ∧ ( 𝑄 ≠ 𝑃 ∧ ( 𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊 ) ) ∧ ( ( 𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊 ) ∧ ( 𝑆 ∨ ( 𝑋 ∧ 𝑊 ) ) = 𝑋 ) ) → ( 𝐺 ‘ 𝑋 ) = ( ( 𝐺 ‘ 𝑆 ) ∨ ( 𝑋 ∧ 𝑊 ) ) )
19 11 12 13 15 16 17 18 syl321anc ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( 𝑃 ≠ 𝑄 ∧ ( 𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊 ) ) ∧ ( ( 𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊 ) ∧ ( 𝑆 ∨ ( 𝑋 ∧ 𝑊 ) ) = 𝑋 ) ) → ( 𝐺 ‘ 𝑋 ) = ( ( 𝐺 ‘ 𝑆 ) ∨ ( 𝑋 ∧ 𝑊 ) ) )