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 = ( 𝑥𝐵 ↦ if ( ( 𝑃𝑄 ∧ ¬ 𝑥 𝑊 ) , ( 𝑧𝐵𝑠𝐴 ( ( ¬ 𝑠 𝑊 ∧ ( 𝑠 ( 𝑥 𝑊 ) ) = 𝑥 ) → 𝑧 = ( if ( 𝑠 ( 𝑃 𝑄 ) , ( 𝑦𝐵𝑡𝐴 ( ( ¬ 𝑡 𝑊 ∧ ¬ 𝑡 ( 𝑃 𝑄 ) ) → 𝑦 = 𝐸 ) ) , 𝑠 / 𝑡 𝐷 ) ( 𝑥 𝑊 ) ) ) ) , 𝑥 ) )
cdleme50ldil.i 𝐶 = ( ( LDil ‘ 𝐾 ) ‘ 𝑊 )
Assertion cdleme50ldil ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ) → 𝐹𝐶 )

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 = ( 𝑥𝐵 ↦ if ( ( 𝑃𝑄 ∧ ¬ 𝑥 𝑊 ) , ( 𝑧𝐵𝑠𝐴 ( ( ¬ 𝑠 𝑊 ∧ ( 𝑠 ( 𝑥 𝑊 ) ) = 𝑥 ) → 𝑧 = ( if ( 𝑠 ( 𝑃 𝑄 ) , ( 𝑦𝐵𝑡𝐴 ( ( ¬ 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 ∧ 𝑊𝐻 ) ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ) → 𝐹 ∈ ( LAut ‘ 𝐾 ) )
14 simpr ( ( 𝑃𝑄 ∧ ¬ 𝑒 𝑊 ) → ¬ 𝑒 𝑊 )
15 14 con2i ( 𝑒 𝑊 → ¬ ( 𝑃𝑄 ∧ ¬ 𝑒 𝑊 ) )
16 10 cdleme31fv2 ( ( 𝑒𝐵 ∧ ¬ ( 𝑃𝑄 ∧ ¬ 𝑒 𝑊 ) ) → ( 𝐹𝑒 ) = 𝑒 )
17 15 16 sylan2 ( ( 𝑒𝐵𝑒 𝑊 ) → ( 𝐹𝑒 ) = 𝑒 )
18 17 ex ( 𝑒𝐵 → ( 𝑒 𝑊 → ( 𝐹𝑒 ) = 𝑒 ) )
19 18 rgen 𝑒𝐵 ( 𝑒 𝑊 → ( 𝐹𝑒 ) = 𝑒 )
20 19 a1i ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ) → ∀ 𝑒𝐵 ( 𝑒 𝑊 → ( 𝐹𝑒 ) = 𝑒 ) )
21 1 2 6 12 11 isldil ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) → ( 𝐹𝐶 ↔ ( 𝐹 ∈ ( LAut ‘ 𝐾 ) ∧ ∀ 𝑒𝐵 ( 𝑒 𝑊 → ( 𝐹𝑒 ) = 𝑒 ) ) ) )
22 21 3ad2ant1 ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ) → ( 𝐹𝐶 ↔ ( 𝐹 ∈ ( LAut ‘ 𝐾 ) ∧ ∀ 𝑒𝐵 ( 𝑒 𝑊 → ( 𝐹𝑒 ) = 𝑒 ) ) ) )
23 13 20 22 mpbir2and ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ) → 𝐹𝐶 )