Metamath Proof Explorer


Theorem cdleme50laut

Description: Part of proof of Lemma D in Crawley p. 113. F is a lattice automorphism. 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 ) )
cdleme50laut.i 𝐼 = ( LAut ‘ 𝐾 )
Assertion cdleme50laut ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ) → 𝐹𝐼 )

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 cdleme50laut.i 𝐼 = ( LAut ‘ 𝐾 )
12 1 2 3 4 5 6 7 8 9 10 cdleme50f1o ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ) → 𝐹 : 𝐵1-1-onto𝐵 )
13 1 2 3 4 5 6 7 8 9 10 cdleme50lebi ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ) ∧ ( 𝑑𝐵𝑒𝐵 ) ) → ( 𝑑 𝑒 ↔ ( 𝐹𝑑 ) ( 𝐹𝑒 ) ) )
14 13 ralrimivva ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ) → ∀ 𝑑𝐵𝑒𝐵 ( 𝑑 𝑒 ↔ ( 𝐹𝑑 ) ( 𝐹𝑒 ) ) )
15 simp1l ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ) → 𝐾 ∈ HL )
16 1 2 11 islaut ( 𝐾 ∈ HL → ( 𝐹𝐼 ↔ ( 𝐹 : 𝐵1-1-onto𝐵 ∧ ∀ 𝑑𝐵𝑒𝐵 ( 𝑑 𝑒 ↔ ( 𝐹𝑑 ) ( 𝐹𝑒 ) ) ) ) )
17 15 16 syl ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ) → ( 𝐹𝐼 ↔ ( 𝐹 : 𝐵1-1-onto𝐵 ∧ ∀ 𝑑𝐵𝑒𝐵 ( 𝑑 𝑒 ↔ ( 𝐹𝑑 ) ( 𝐹𝑒 ) ) ) ) )
18 12 14 17 mpbir2and ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ) → 𝐹𝐼 )