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Metamath Proof Explorer

Theorem cdlemg2fvlem

Description: Lemma for cdlemg2fv . (Contributed by NM, 23-Apr-2013)

Ref Expression
Hypotheses cdlemg2.b ⊢ 𝐵 = ( Base ‘ 𝐾 )
cdlemg2.l ⊢ ≤ = ( le ‘ 𝐾 )
cdlemg2.j ⊢ ∨ = ( join ‘ 𝐾 )
cdlemg2.m ⊢ ∧ = ( meet ‘ 𝐾 )
cdlemg2.a ⊢ 𝐴 = ( Atoms ‘ 𝐾 )
cdlemg2.h ⊢ 𝐻 = ( LHyp ‘ 𝐾 )
cdlemg2.t ⊢ 𝑇 = ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 )
cdlemg2ex.u ⊢ 𝑈 = ( ( 𝑝 ∨ 𝑞 ) ∧ 𝑊 )
cdlemg2ex.d ⊢ 𝐷 = ( ( 𝑡 ∨ 𝑈 ) ∧ ( 𝑞 ∨ ( ( 𝑝 ∨ 𝑡 ) ∧ 𝑊 ) ) )
cdlemg2ex.e ⊢ 𝐸 = ( ( 𝑝 ∨ 𝑞 ) ∧ ( 𝐷 ∨ ( ( 𝑠 ∨ 𝑡 ) ∧ 𝑊 ) ) )
cdlemg2ex.g ⊢ 𝐺 = ( 𝑥 ∈ 𝐵 ↦ if ( ( 𝑝 ≠ 𝑞 ∧ ¬ 𝑥 ≤ 𝑊 ) , ( ℩ 𝑧 ∈ 𝐵 ∀ 𝑠 ∈ 𝐴 ( ( ¬ 𝑠 ≤ 𝑊 ∧ ( 𝑠 ∨ ( 𝑥 ∧ 𝑊 ) ) = 𝑥 ) → 𝑧 = ( if ( 𝑠 ≤ ( 𝑝 ∨ 𝑞 ) , ( ℩ 𝑦 ∈ 𝐵 ∀ 𝑡 ∈ 𝐴 ( ( ¬ 𝑡 ≤ 𝑊 ∧ ¬ 𝑡 ≤ ( 𝑝 ∨ 𝑞 ) ) → 𝑦 = 𝐸 ) ) , ⦋ 𝑠 / 𝑡 ⦌ 𝐷 ) ∨ ( 𝑥 ∧ 𝑊 ) ) ) ) , 𝑥 ) )
Assertion cdlemg2fvlem ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊 ) ) ∧ ( 𝐹 ∈ 𝑇 ∧ ( 𝑃 ∨ ( 𝑋 ∧ 𝑊 ) ) = 𝑋 ) ) → ( 𝐹 ‘ 𝑋 ) = ( ( 𝐹 ‘ 𝑃 ) ∨ ( 𝑋 ∧ 𝑊 ) ) )

Proof

Step Hyp Ref Expression
1 cdlemg2.b ⊢ 𝐵 = ( Base ‘ 𝐾 )
2 cdlemg2.l ⊢ ≤ = ( le ‘ 𝐾 )
3 cdlemg2.j ⊢ ∨ = ( join ‘ 𝐾 )
4 cdlemg2.m ⊢ ∧ = ( meet ‘ 𝐾 )
5 cdlemg2.a ⊢ 𝐴 = ( Atoms ‘ 𝐾 )
6 cdlemg2.h ⊢ 𝐻 = ( LHyp ‘ 𝐾 )
7 cdlemg2.t ⊢ 𝑇 = ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 )
8 cdlemg2ex.u ⊢ 𝑈 = ( ( 𝑝 ∨ 𝑞 ) ∧ 𝑊 )
9 cdlemg2ex.d ⊢ 𝐷 = ( ( 𝑡 ∨ 𝑈 ) ∧ ( 𝑞 ∨ ( ( 𝑝 ∨ 𝑡 ) ∧ 𝑊 ) ) )
10 cdlemg2ex.e ⊢ 𝐸 = ( ( 𝑝 ∨ 𝑞 ) ∧ ( 𝐷 ∨ ( ( 𝑠 ∨ 𝑡 ) ∧ 𝑊 ) ) )
11 cdlemg2ex.g ⊢ 𝐺 = ( 𝑥 ∈ 𝐵 ↦ if ( ( 𝑝 ≠ 𝑞 ∧ ¬ 𝑥 ≤ 𝑊 ) , ( ℩ 𝑧 ∈ 𝐵 ∀ 𝑠 ∈ 𝐴 ( ( ¬ 𝑠 ≤ 𝑊 ∧ ( 𝑠 ∨ ( 𝑥 ∧ 𝑊 ) ) = 𝑥 ) → 𝑧 = ( if ( 𝑠 ≤ ( 𝑝 ∨ 𝑞 ) , ( ℩ 𝑦 ∈ 𝐵 ∀ 𝑡 ∈ 𝐴 ( ( ¬ 𝑡 ≤ 𝑊 ∧ ¬ 𝑡 ≤ ( 𝑝 ∨ 𝑞 ) ) → 𝑦 = 𝐸 ) ) , ⦋ 𝑠 / 𝑡 ⦌ 𝐷 ) ∨ ( 𝑥 ∧ 𝑊 ) ) ) ) , 𝑥 ) )
12 simp1 ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊 ) ) ∧ ( 𝐹 ∈ 𝑇 ∧ ( 𝑃 ∨ ( 𝑋 ∧ 𝑊 ) ) = 𝑋 ) ) → ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) )
13 simp3l ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊 ) ) ∧ ( 𝐹 ∈ 𝑇 ∧ ( 𝑃 ∨ ( 𝑋 ∧ 𝑊 ) ) = 𝑋 ) ) → 𝐹 ∈ 𝑇 )
14 simp2r ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊 ) ) ∧ ( 𝐹 ∈ 𝑇 ∧ ( 𝑃 ∨ ( 𝑋 ∧ 𝑊 ) ) = 𝑋 ) ) → ( 𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊 ) )
15 simp2l ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊 ) ) ∧ ( 𝐹 ∈ 𝑇 ∧ ( 𝑃 ∨ ( 𝑋 ∧ 𝑊 ) ) = 𝑋 ) ) → ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) )
16 simp3r ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊 ) ) ∧ ( 𝐹 ∈ 𝑇 ∧ ( 𝑃 ∨ ( 𝑋 ∧ 𝑊 ) ) = 𝑋 ) ) → ( 𝑃 ∨ ( 𝑋 ∧ 𝑊 ) ) = 𝑋 )
17 15 16 jca ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊 ) ) ∧ ( 𝐹 ∈ 𝑇 ∧ ( 𝑃 ∨ ( 𝑋 ∧ 𝑊 ) ) = 𝑋 ) ) → ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑃 ∨ ( 𝑋 ∧ 𝑊 ) ) = 𝑋 ) )
18 fveq1 ⊢ ( 𝐹 = 𝐺 → ( 𝐹 ‘ 𝑋 ) = ( 𝐺 ‘ 𝑋 ) )
19 fveq1 ⊢ ( 𝐹 = 𝐺 → ( 𝐹 ‘ 𝑃 ) = ( 𝐺 ‘ 𝑃 ) )
20 19 oveq1d ⊢ ( 𝐹 = 𝐺 → ( ( 𝐹 ‘ 𝑃 ) ∨ ( 𝑋 ∧ 𝑊 ) ) = ( ( 𝐺 ‘ 𝑃 ) ∨ ( 𝑋 ∧ 𝑊 ) ) )
21 18 20 eqeq12d ⊢ ( 𝐹 = 𝐺 → ( ( 𝐹 ‘ 𝑋 ) = ( ( 𝐹 ‘ 𝑃 ) ∨ ( 𝑋 ∧ 𝑊 ) ) ↔ ( 𝐺 ‘ 𝑋 ) = ( ( 𝐺 ‘ 𝑃 ) ∨ ( 𝑋 ∧ 𝑊 ) ) ) )
22 1 2 3 4 5 6 8 9 10 11 cdleme48fvg ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑝 ∈ 𝐴 ∧ ¬ 𝑝 ≤ 𝑊 ) ∧ ( 𝑞 ∈ 𝐴 ∧ ¬ 𝑞 ≤ 𝑊 ) ) ∧ ( 𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊 ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑃 ∨ ( 𝑋 ∧ 𝑊 ) ) = 𝑋 ) ) → ( 𝐺 ‘ 𝑋 ) = ( ( 𝐺 ‘ 𝑃 ) ∨ ( 𝑋 ∧ 𝑊 ) ) )
23 22 3expb ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑝 ∈ 𝐴 ∧ ¬ 𝑝 ≤ 𝑊 ) ∧ ( 𝑞 ∈ 𝐴 ∧ ¬ 𝑞 ≤ 𝑊 ) ) ∧ ( ( 𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊 ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑃 ∨ ( 𝑋 ∧ 𝑊 ) ) = 𝑋 ) ) ) → ( 𝐺 ‘ 𝑋 ) = ( ( 𝐺 ‘ 𝑃 ) ∨ ( 𝑋 ∧ 𝑊 ) ) )
24 1 2 3 4 5 6 7 8 9 10 11 21 23 cdlemg2ce ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ∧ ( ( 𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊 ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑃 ∨ ( 𝑋 ∧ 𝑊 ) ) = 𝑋 ) ) ) → ( 𝐹 ‘ 𝑋 ) = ( ( 𝐹 ‘ 𝑃 ) ∨ ( 𝑋 ∧ 𝑊 ) ) )
25 12 13 14 17 24 syl112anc ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊 ) ) ∧ ( 𝐹 ∈ 𝑇 ∧ ( 𝑃 ∨ ( 𝑋 ∧ 𝑊 ) ) = 𝑋 ) ) → ( 𝐹 ‘ 𝑋 ) = ( ( 𝐹 ‘ 𝑃 ) ∨ ( 𝑋 ∧ 𝑊 ) ) )