Metamath Proof Explorer

Theorem cdleme50ltrn

Description: Part of proof of Lemma E in Crawley p. 113. F is a lattice translation. TODO: fix comment. (Contributed by NM, 10-Apr-2013)

Ref Expression
Hypotheses cdlemef50.b ⊢ 𝐵 = ( Base ‘ 𝐾 )
cdlemef50.l ⊢ ≤ = ( le ‘ 𝐾 )
cdlemef50.j ⊢ ∨ = ( join ‘ 𝐾 )
cdlemef50.m ⊢ ∧ = ( meet ‘ 𝐾 )
cdlemef50.a ⊢ 𝐴 = ( Atoms ‘ 𝐾 )
cdlemef50.h ⊢ 𝐻 = ( LHyp ‘ 𝐾 )
cdlemef50.u ⊢ 𝑈 = ( ( 𝑃 ∨ 𝑄 ) ∧ 𝑊 )
cdlemef50.d ⊢ 𝐷 = ( ( 𝑡 ∨ 𝑈 ) ∧ ( 𝑄 ∨ ( ( 𝑃 ∨ 𝑡 ) ∧ 𝑊 ) ) )
cdlemefs50.e ⊢ 𝐸 = ( ( 𝑃 ∨ 𝑄 ) ∧ ( 𝐷 ∨ ( ( 𝑠 ∨ 𝑡 ) ∧ 𝑊 ) ) )
cdlemef50.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 ) )
cdleme50ltrn.t ⊢ 𝑇 = ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 )
Assertion cdleme50ltrn ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) → 𝐹 ∈ 𝑇 )

Proof

Step Hyp Ref Expression
1 cdlemef50.b ⊢ 𝐵 = ( Base ‘ 𝐾 )
2 cdlemef50.l ⊢ ≤ = ( le ‘ 𝐾 )
3 cdlemef50.j ⊢ ∨ = ( join ‘ 𝐾 )
4 cdlemef50.m ⊢ ∧ = ( meet ‘ 𝐾 )
5 cdlemef50.a ⊢ 𝐴 = ( Atoms ‘ 𝐾 )
6 cdlemef50.h ⊢ 𝐻 = ( LHyp ‘ 𝐾 )
7 cdlemef50.u ⊢ 𝑈 = ( ( 𝑃 ∨ 𝑄 ) ∧ 𝑊 )
8 cdlemef50.d ⊢ 𝐷 = ( ( 𝑡 ∨ 𝑈 ) ∧ ( 𝑄 ∨ ( ( 𝑃 ∨ 𝑡 ) ∧ 𝑊 ) ) )
9 cdlemefs50.e ⊢ 𝐸 = ( ( 𝑃 ∨ 𝑄 ) ∧ ( 𝐷 ∨ ( ( 𝑠 ∨ 𝑡 ) ∧ 𝑊 ) ) )
10 cdlemef50.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 cdleme50ltrn.t ⊢ 𝑇 = ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 )
12 eqid ⊢ ( ( LDil ‘ 𝐾 ) ‘ 𝑊 ) = ( ( LDil ‘ 𝐾 ) ‘ 𝑊 )
13 1 2 3 4 5 6 7 8 9 10 12 cdleme50ldil ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) → 𝐹 ∈ ( ( LDil ‘ 𝐾 ) ‘ 𝑊 ) )
14 simp1 ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( 𝑑 ∈ 𝐴 ∧ 𝑒 ∈ 𝐴 ) ∧ ( ¬ 𝑑 ≤ 𝑊 ∧ ¬ 𝑒 ≤ 𝑊 ) ) → ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) )
15 simp2l ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( 𝑑 ∈ 𝐴 ∧ 𝑒 ∈ 𝐴 ) ∧ ( ¬ 𝑑 ≤ 𝑊 ∧ ¬ 𝑒 ≤ 𝑊 ) ) → 𝑑 ∈ 𝐴 )
16 simp3l ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( 𝑑 ∈ 𝐴 ∧ 𝑒 ∈ 𝐴 ) ∧ ( ¬ 𝑑 ≤ 𝑊 ∧ ¬ 𝑒 ≤ 𝑊 ) ) → ¬ 𝑑 ≤ 𝑊 )
17 1 2 3 4 5 6 7 8 9 10 cdleme50trn123 ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( 𝑑 ∈ 𝐴 ∧ ¬ 𝑑 ≤ 𝑊 ) ) → ( ( 𝑑 ∨ ( 𝐹 ‘ 𝑑 ) ) ∧ 𝑊 ) = 𝑈 )
18 14 15 16 17 syl12anc ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( 𝑑 ∈ 𝐴 ∧ 𝑒 ∈ 𝐴 ) ∧ ( ¬ 𝑑 ≤ 𝑊 ∧ ¬ 𝑒 ≤ 𝑊 ) ) → ( ( 𝑑 ∨ ( 𝐹 ‘ 𝑑 ) ) ∧ 𝑊 ) = 𝑈 )
19 simp2r ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( 𝑑 ∈ 𝐴 ∧ 𝑒 ∈ 𝐴 ) ∧ ( ¬ 𝑑 ≤ 𝑊 ∧ ¬ 𝑒 ≤ 𝑊 ) ) → 𝑒 ∈ 𝐴 )
20 simp3r ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( 𝑑 ∈ 𝐴 ∧ 𝑒 ∈ 𝐴 ) ∧ ( ¬ 𝑑 ≤ 𝑊 ∧ ¬ 𝑒 ≤ 𝑊 ) ) → ¬ 𝑒 ≤ 𝑊 )
21 1 2 3 4 5 6 7 8 9 10 cdleme50trn123 ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( 𝑒 ∈ 𝐴 ∧ ¬ 𝑒 ≤ 𝑊 ) ) → ( ( 𝑒 ∨ ( 𝐹 ‘ 𝑒 ) ) ∧ 𝑊 ) = 𝑈 )
22 14 19 20 21 syl12anc ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( 𝑑 ∈ 𝐴 ∧ 𝑒 ∈ 𝐴 ) ∧ ( ¬ 𝑑 ≤ 𝑊 ∧ ¬ 𝑒 ≤ 𝑊 ) ) → ( ( 𝑒 ∨ ( 𝐹 ‘ 𝑒 ) ) ∧ 𝑊 ) = 𝑈 )
23 18 22 eqtr4d ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( 𝑑 ∈ 𝐴 ∧ 𝑒 ∈ 𝐴 ) ∧ ( ¬ 𝑑 ≤ 𝑊 ∧ ¬ 𝑒 ≤ 𝑊 ) ) → ( ( 𝑑 ∨ ( 𝐹 ‘ 𝑑 ) ) ∧ 𝑊 ) = ( ( 𝑒 ∨ ( 𝐹 ‘ 𝑒 ) ) ∧ 𝑊 ) )
24 23 3exp ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) → ( ( 𝑑 ∈ 𝐴 ∧ 𝑒 ∈ 𝐴 ) → ( ( ¬ 𝑑 ≤ 𝑊 ∧ ¬ 𝑒 ≤ 𝑊 ) → ( ( 𝑑 ∨ ( 𝐹 ‘ 𝑑 ) ) ∧ 𝑊 ) = ( ( 𝑒 ∨ ( 𝐹 ‘ 𝑒 ) ) ∧ 𝑊 ) ) ) )
25 24 ralrimivv ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) → ∀ 𝑑 ∈ 𝐴 ∀ 𝑒 ∈ 𝐴 ( ( ¬ 𝑑 ≤ 𝑊 ∧ ¬ 𝑒 ≤ 𝑊 ) → ( ( 𝑑 ∨ ( 𝐹 ‘ 𝑑 ) ) ∧ 𝑊 ) = ( ( 𝑒 ∨ ( 𝐹 ‘ 𝑒 ) ) ∧ 𝑊 ) ) )
26 2 3 4 5 6 12 11 isltrn ⊢ ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) → ( 𝐹 ∈ 𝑇 ↔ ( 𝐹 ∈ ( ( LDil ‘ 𝐾 ) ‘ 𝑊 ) ∧ ∀ 𝑑 ∈ 𝐴 ∀ 𝑒 ∈ 𝐴 ( ( ¬ 𝑑 ≤ 𝑊 ∧ ¬ 𝑒 ≤ 𝑊 ) → ( ( 𝑑 ∨ ( 𝐹 ‘ 𝑑 ) ) ∧ 𝑊 ) = ( ( 𝑒 ∨ ( 𝐹 ‘ 𝑒 ) ) ∧ 𝑊 ) ) ) ) )
27 26 3ad2ant1 ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) → ( 𝐹 ∈ 𝑇 ↔ ( 𝐹 ∈ ( ( LDil ‘ 𝐾 ) ‘ 𝑊 ) ∧ ∀ 𝑑 ∈ 𝐴 ∀ 𝑒 ∈ 𝐴 ( ( ¬ 𝑑 ≤ 𝑊 ∧ ¬ 𝑒 ≤ 𝑊 ) → ( ( 𝑑 ∨ ( 𝐹 ‘ 𝑑 ) ) ∧ 𝑊 ) = ( ( 𝑒 ∨ ( 𝐹 ‘ 𝑒 ) ) ∧ 𝑊 ) ) ) ) )
28 13 25 27 mpbir2and ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) → 𝐹 ∈ 𝑇 )