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 = ( 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 ) )
	cdleme50laut.i	⊢ 𝐼 = ( LAut ‘ 𝐾 )
Assertion	cdleme50laut	⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) → 𝐹 ∈ 𝐼 )

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		cdleme50laut.i	⊢ 𝐼 = ( LAut ‘ 𝐾 )
12	1 2 3 4 5 6 7 8 9 10	cdleme50f1o	⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) → 𝐹 : 𝐵 –1-1-onto→ 𝐵 )
13	1 2 3 4 5 6 7 8 9 10	cdleme50lebi	⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( 𝑑 ∈ 𝐵 ∧ 𝑒 ∈ 𝐵 ) ) → ( 𝑑 ≤ 𝑒 ↔ ( 𝐹 ‘ 𝑑 ) ≤ ( 𝐹 ‘ 𝑒 ) ) )
14	13	ralrimivva	⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) → ∀ 𝑑 ∈ 𝐵 ∀ 𝑒 ∈ 𝐵 ( 𝑑 ≤ 𝑒 ↔ ( 𝐹 ‘ 𝑑 ) ≤ ( 𝐹 ‘ 𝑒 ) ) )
15		simp1l	⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) → 𝐾 ∈ HL )
16	1 2 11	islaut	⊢ ( 𝐾 ∈ HL → ( 𝐹 ∈ 𝐼 ↔ ( 𝐹 : 𝐵 –1-1-onto→ 𝐵 ∧ ∀ 𝑑 ∈ 𝐵 ∀ 𝑒 ∈ 𝐵 ( 𝑑 ≤ 𝑒 ↔ ( 𝐹 ‘ 𝑑 ) ≤ ( 𝐹 ‘ 𝑒 ) ) ) ) )
17	15 16	syl	⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) → ( 𝐹 ∈ 𝐼 ↔ ( 𝐹 : 𝐵 –1-1-onto→ 𝐵 ∧ ∀ 𝑑 ∈ 𝐵 ∀ 𝑒 ∈ 𝐵 ( 𝑑 ≤ 𝑒 ↔ ( 𝐹 ‘ 𝑑 ) ≤ ( 𝐹 ‘ 𝑒 ) ) ) ) )
18	12 14 17	mpbir2and	⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) → 𝐹 ∈ 𝐼 )