Metamath Proof Explorer

Theorem cdleme48gfv

Description: TODO: fix comment. (Contributed by NM, 9-Apr-2013)

Ref Expression
Hypotheses cdlemef46g.b ⊢ 𝐵 = ( Base ‘ 𝐾 )
cdlemef46g.l ⊢ ≤ = ( le ‘ 𝐾 )
cdlemef46g.j ⊢ ∨ = ( join ‘ 𝐾 )
cdlemef46g.m ⊢ ∧ = ( meet ‘ 𝐾 )
cdlemef46g.a ⊢ 𝐴 = ( Atoms ‘ 𝐾 )
cdlemef46g.h ⊢ 𝐻 = ( LHyp ‘ 𝐾 )
cdlemef46g.u ⊢ 𝑈 = ( ( 𝑃 ∨ 𝑄 ) ∧ 𝑊 )
cdlemef46g.d ⊢ 𝐷 = ( ( 𝑡 ∨ 𝑈 ) ∧ ( 𝑄 ∨ ( ( 𝑃 ∨ 𝑡 ) ∧ 𝑊 ) ) )
cdlemefs46g.e ⊢ 𝐸 = ( ( 𝑃 ∨ 𝑄 ) ∧ ( 𝐷 ∨ ( ( 𝑠 ∨ 𝑡 ) ∧ 𝑊 ) ) )
cdlemef46g.f ⊢ 𝐹 = ( 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 ) )
cdlemef46.v ⊢ 𝑉 = ( ( 𝑄 ∨ 𝑃 ) ∧ 𝑊 )
cdlemef46.n ⊢ 𝑁 = ( ( 𝑣 ∨ 𝑉 ) ∧ ( 𝑃 ∨ ( ( 𝑄 ∨ 𝑣 ) ∧ 𝑊 ) ) )
cdlemefs46.o ⊢ 𝑂 = ( ( 𝑄 ∨ 𝑃 ) ∧ ( 𝑁 ∨ ( ( 𝑢 ∨ 𝑣 ) ∧ 𝑊 ) ) )
cdlemef46.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 cdleme48gfv ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ 𝑋 ∈ 𝐵 ) → ( 𝐺 ‘ ( 𝐹 ‘ 𝑋 ) ) = 𝑋 )

Proof

Step Hyp Ref Expression
1 cdlemef46g.b ⊢ 𝐵 = ( Base ‘ 𝐾 )
2 cdlemef46g.l ⊢ ≤ = ( le ‘ 𝐾 )
3 cdlemef46g.j ⊢ ∨ = ( join ‘ 𝐾 )
4 cdlemef46g.m ⊢ ∧ = ( meet ‘ 𝐾 )
5 cdlemef46g.a ⊢ 𝐴 = ( Atoms ‘ 𝐾 )
6 cdlemef46g.h ⊢ 𝐻 = ( LHyp ‘ 𝐾 )
7 cdlemef46g.u ⊢ 𝑈 = ( ( 𝑃 ∨ 𝑄 ) ∧ 𝑊 )
8 cdlemef46g.d ⊢ 𝐷 = ( ( 𝑡 ∨ 𝑈 ) ∧ ( 𝑄 ∨ ( ( 𝑃 ∨ 𝑡 ) ∧ 𝑊 ) ) )
9 cdlemefs46g.e ⊢ 𝐸 = ( ( 𝑃 ∨ 𝑄 ) ∧ ( 𝐷 ∨ ( ( 𝑠 ∨ 𝑡 ) ∧ 𝑊 ) ) )
10 cdlemef46g.f ⊢ 𝐹 = ( 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 ) )
11 cdlemef46.v ⊢ 𝑉 = ( ( 𝑄 ∨ 𝑃 ) ∧ 𝑊 )
12 cdlemef46.n ⊢ 𝑁 = ( ( 𝑣 ∨ 𝑉 ) ∧ ( 𝑃 ∨ ( ( 𝑄 ∨ 𝑣 ) ∧ 𝑊 ) ) )
13 cdlemefs46.o ⊢ 𝑂 = ( ( 𝑄 ∨ 𝑃 ) ∧ ( 𝑁 ∨ ( ( 𝑢 ∨ 𝑣 ) ∧ 𝑊 ) ) )
14 cdlemef46.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 ) )
15 simpll ⊢ ( ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ 𝑋 ∈ 𝐵 ) ∧ ( 𝑃 ≠ 𝑄 ∧ ¬ 𝑋 ≤ 𝑊 ) ) → ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) )
16 simprl ⊢ ( ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ 𝑋 ∈ 𝐵 ) ∧ ( 𝑃 ≠ 𝑄 ∧ ¬ 𝑋 ≤ 𝑊 ) ) → 𝑃 ≠ 𝑄 )
17 simplr ⊢ ( ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ 𝑋 ∈ 𝐵 ) ∧ ( 𝑃 ≠ 𝑄 ∧ ¬ 𝑋 ≤ 𝑊 ) ) → 𝑋 ∈ 𝐵 )
18 simprr ⊢ ( ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ 𝑋 ∈ 𝐵 ) ∧ ( 𝑃 ≠ 𝑄 ∧ ¬ 𝑋 ≤ 𝑊 ) ) → ¬ 𝑋 ≤ 𝑊 )
19 17 18 jca ⊢ ( ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ 𝑋 ∈ 𝐵 ) ∧ ( 𝑃 ≠ 𝑄 ∧ ¬ 𝑋 ≤ 𝑊 ) ) → ( 𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊 ) )
20 1 2 3 4 5 6 7 8 9 10 11 12 13 14 cdleme48gfv1 ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( 𝑃 ≠ 𝑄 ∧ ( 𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊 ) ) ) → ( 𝐺 ‘ ( 𝐹 ‘ 𝑋 ) ) = 𝑋 )
21 15 16 19 20 syl12anc ⊢ ( ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ 𝑋 ∈ 𝐵 ) ∧ ( 𝑃 ≠ 𝑄 ∧ ¬ 𝑋 ≤ 𝑊 ) ) → ( 𝐺 ‘ ( 𝐹 ‘ 𝑋 ) ) = 𝑋 )
22 10 cdleme31fv2 ⊢ ( ( 𝑋 ∈ 𝐵 ∧ ¬ ( 𝑃 ≠ 𝑄 ∧ ¬ 𝑋 ≤ 𝑊 ) ) → ( 𝐹 ‘ 𝑋 ) = 𝑋 )
23 22 adantll ⊢ ( ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ 𝑋 ∈ 𝐵 ) ∧ ¬ ( 𝑃 ≠ 𝑄 ∧ ¬ 𝑋 ≤ 𝑊 ) ) → ( 𝐹 ‘ 𝑋 ) = 𝑋 )
24 simplr ⊢ ( ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ 𝑋 ∈ 𝐵 ) ∧ ¬ ( 𝑃 ≠ 𝑄 ∧ ¬ 𝑋 ≤ 𝑊 ) ) → 𝑋 ∈ 𝐵 )
25 23 24 eqeltrd ⊢ ( ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ 𝑋 ∈ 𝐵 ) ∧ ¬ ( 𝑃 ≠ 𝑄 ∧ ¬ 𝑋 ≤ 𝑊 ) ) → ( 𝐹 ‘ 𝑋 ) ∈ 𝐵 )
26 simpr ⊢ ( ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ 𝑋 ∈ 𝐵 ) ∧ ¬ ( 𝑃 ≠ 𝑄 ∧ ¬ 𝑋 ≤ 𝑊 ) ) → ¬ ( 𝑃 ≠ 𝑄 ∧ ¬ 𝑋 ≤ 𝑊 ) )
27 necom ⊢ ( 𝑄 ≠ 𝑃 ↔ 𝑃 ≠ 𝑄 )
28 27 a1i ⊢ ( ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ 𝑋 ∈ 𝐵 ) ∧ ¬ ( 𝑃 ≠ 𝑄 ∧ ¬ 𝑋 ≤ 𝑊 ) ) → ( 𝑄 ≠ 𝑃 ↔ 𝑃 ≠ 𝑄 ) )
29 23 breq1d ⊢ ( ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ 𝑋 ∈ 𝐵 ) ∧ ¬ ( 𝑃 ≠ 𝑄 ∧ ¬ 𝑋 ≤ 𝑊 ) ) → ( ( 𝐹 ‘ 𝑋 ) ≤ 𝑊 ↔ 𝑋 ≤ 𝑊 ) )
30 29 notbid ⊢ ( ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ 𝑋 ∈ 𝐵 ) ∧ ¬ ( 𝑃 ≠ 𝑄 ∧ ¬ 𝑋 ≤ 𝑊 ) ) → ( ¬ ( 𝐹 ‘ 𝑋 ) ≤ 𝑊 ↔ ¬ 𝑋 ≤ 𝑊 ) )
31 28 30 anbi12d ⊢ ( ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ 𝑋 ∈ 𝐵 ) ∧ ¬ ( 𝑃 ≠ 𝑄 ∧ ¬ 𝑋 ≤ 𝑊 ) ) → ( ( 𝑄 ≠ 𝑃 ∧ ¬ ( 𝐹 ‘ 𝑋 ) ≤ 𝑊 ) ↔ ( 𝑃 ≠ 𝑄 ∧ ¬ 𝑋 ≤ 𝑊 ) ) )
32 26 31 mtbird ⊢ ( ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ 𝑋 ∈ 𝐵 ) ∧ ¬ ( 𝑃 ≠ 𝑄 ∧ ¬ 𝑋 ≤ 𝑊 ) ) → ¬ ( 𝑄 ≠ 𝑃 ∧ ¬ ( 𝐹 ‘ 𝑋 ) ≤ 𝑊 ) )
33 14 cdleme31fv2 ⊢ ( ( ( 𝐹 ‘ 𝑋 ) ∈ 𝐵 ∧ ¬ ( 𝑄 ≠ 𝑃 ∧ ¬ ( 𝐹 ‘ 𝑋 ) ≤ 𝑊 ) ) → ( 𝐺 ‘ ( 𝐹 ‘ 𝑋 ) ) = ( 𝐹 ‘ 𝑋 ) )
34 25 32 33 syl2anc ⊢ ( ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ 𝑋 ∈ 𝐵 ) ∧ ¬ ( 𝑃 ≠ 𝑄 ∧ ¬ 𝑋 ≤ 𝑊 ) ) → ( 𝐺 ‘ ( 𝐹 ‘ 𝑋 ) ) = ( 𝐹 ‘ 𝑋 ) )
35 34 23 eqtrd ⊢ ( ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ 𝑋 ∈ 𝐵 ) ∧ ¬ ( 𝑃 ≠ 𝑄 ∧ ¬ 𝑋 ≤ 𝑊 ) ) → ( 𝐺 ‘ ( 𝐹 ‘ 𝑋 ) ) = 𝑋 )
36 21 35 pm2.61dan ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ 𝑋 ∈ 𝐵 ) → ( 𝐺 ‘ ( 𝐹 ‘ 𝑋 ) ) = 𝑋 )