Metamath Proof Explorer

Theorem cdleme50ldil

Description: Part of proof of Lemma D in Crawley p. 113. F is a lattice dilation. TODO: fix comment. (Contributed by NM, 9-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 ) )
cdleme50ldil.i ⊢ 𝐶 = ( ( LDil ‘ 𝐾 ) ‘ 𝑊 )
Assertion cdleme50ldil ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) → 𝐹 ∈ 𝐶 )

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 cdleme50ldil.i ⊢ 𝐶 = ( ( LDil ‘ 𝐾 ) ‘ 𝑊 )
12 eqid ⊢ ( LAut ‘ 𝐾 ) = ( LAut ‘ 𝐾 )
13 1 2 3 4 5 6 7 8 9 10 12 cdleme50laut ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) → 𝐹 ∈ ( LAut ‘ 𝐾 ) )
14 simpr ⊢ ( ( 𝑃 ≠ 𝑄 ∧ ¬ 𝑒 ≤ 𝑊 ) → ¬ 𝑒 ≤ 𝑊 )
15 14 con2i ⊢ ( 𝑒 ≤ 𝑊 → ¬ ( 𝑃 ≠ 𝑄 ∧ ¬ 𝑒 ≤ 𝑊 ) )
16 10 cdleme31fv2 ⊢ ( ( 𝑒 ∈ 𝐵 ∧ ¬ ( 𝑃 ≠ 𝑄 ∧ ¬ 𝑒 ≤ 𝑊 ) ) → ( 𝐹 ‘ 𝑒 ) = 𝑒 )
17 15 16 sylan2 ⊢ ( ( 𝑒 ∈ 𝐵 ∧ 𝑒 ≤ 𝑊 ) → ( 𝐹 ‘ 𝑒 ) = 𝑒 )
18 17 ex ⊢ ( 𝑒 ∈ 𝐵 → ( 𝑒 ≤ 𝑊 → ( 𝐹 ‘ 𝑒 ) = 𝑒 ) )
19 18 rgen ⊢ ∀ 𝑒 ∈ 𝐵 ( 𝑒 ≤ 𝑊 → ( 𝐹 ‘ 𝑒 ) = 𝑒 )
20 19 a1i ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) → ∀ 𝑒 ∈ 𝐵 ( 𝑒 ≤ 𝑊 → ( 𝐹 ‘ 𝑒 ) = 𝑒 ) )
21 1 2 6 12 11 isldil ⊢ ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) → ( 𝐹 ∈ 𝐶 ↔ ( 𝐹 ∈ ( LAut ‘ 𝐾 ) ∧ ∀ 𝑒 ∈ 𝐵 ( 𝑒 ≤ 𝑊 → ( 𝐹 ‘ 𝑒 ) = 𝑒 ) ) ) )
22 21 3ad2ant1 ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) → ( 𝐹 ∈ 𝐶 ↔ ( 𝐹 ∈ ( LAut ‘ 𝐾 ) ∧ ∀ 𝑒 ∈ 𝐵 ( 𝑒 ≤ 𝑊 → ( 𝐹 ‘ 𝑒 ) = 𝑒 ) ) ) )
23 13 20 22 mpbir2and ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) → 𝐹 ∈ 𝐶 )