Theorem cdlemk13

Description: Part of proof of Lemma K of Crawley p. 118. Line 13 on p. 119. O , D are k_1, f_1. (Contributed by NM, 1-Jul-2013)

	Ref	Expression
Hypotheses	cdlemk1.b	⊢ 𝐵 = ( Base ‘ 𝐾 )
	cdlemk1.l	⊢ ≤ = ( le ‘ 𝐾 )
	cdlemk1.j	⊢ ∨ = ( join ‘ 𝐾 )
	cdlemk1.m	⊢ ∧ = ( meet ‘ 𝐾 )
	cdlemk1.a	⊢ 𝐴 = ( Atoms ‘ 𝐾 )
	cdlemk1.h	⊢ 𝐻 = ( LHyp ‘ 𝐾 )
	cdlemk1.t	⊢ 𝑇 = ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 )
	cdlemk1.r	⊢ 𝑅 = ( ( trL ‘ 𝐾 ) ‘ 𝑊 )
	cdlemk1.s	⊢ 𝑆 = ( 𝑓 ∈ 𝑇 ↦ ( ℩ 𝑖 ∈ 𝑇 ( 𝑖 ‘ 𝑃 ) = ( ( 𝑃 ∨ ( 𝑅 ‘ 𝑓 ) ) ∧ ( ( 𝑁 ‘ 𝑃 ) ∨ ( 𝑅 ‘ ( 𝑓 ∘ ◡ 𝐹 ) ) ) ) ) )
	cdlemk1.o	⊢ 𝑂 = ( 𝑆 ‘ 𝐷 )
Assertion	cdlemk13	⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ∧ 𝐷 ∈ 𝑇 ) ∧ ( 𝑁 ∈ 𝑇 ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑅 ‘ 𝐹 ) = ( 𝑅 ‘ 𝑁 ) ) ∧ ( 𝐹 ≠ ( I ↾ 𝐵 ) ∧ 𝐷 ≠ ( I ↾ 𝐵 ) ∧ ( 𝑅 ‘ 𝐷 ) ≠ ( 𝑅 ‘ 𝐹 ) ) ) → ( 𝑂 ‘ 𝑃 ) = ( ( 𝑃 ∨ ( 𝑅 ‘ 𝐷 ) ) ∧ ( ( 𝑁 ‘ 𝑃 ) ∨ ( 𝑅 ‘ ( 𝐷 ∘ ◡ 𝐹 ) ) ) ) )

Proof

Step	Hyp	Ref	Expression
1		cdlemk1.b	⊢ 𝐵 = ( Base ‘ 𝐾 )
2		cdlemk1.l	⊢ ≤ = ( le ‘ 𝐾 )
3		cdlemk1.j	⊢ ∨ = ( join ‘ 𝐾 )
4		cdlemk1.m	⊢ ∧ = ( meet ‘ 𝐾 )
5		cdlemk1.a	⊢ 𝐴 = ( Atoms ‘ 𝐾 )
6		cdlemk1.h	⊢ 𝐻 = ( LHyp ‘ 𝐾 )
7		cdlemk1.t	⊢ 𝑇 = ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 )
8		cdlemk1.r	⊢ 𝑅 = ( ( trL ‘ 𝐾 ) ‘ 𝑊 )
9		cdlemk1.s	⊢ 𝑆 = ( 𝑓 ∈ 𝑇 ↦ ( ℩ 𝑖 ∈ 𝑇 ( 𝑖 ‘ 𝑃 ) = ( ( 𝑃 ∨ ( 𝑅 ‘ 𝑓 ) ) ∧ ( ( 𝑁 ‘ 𝑃 ) ∨ ( 𝑅 ‘ ( 𝑓 ∘ ◡ 𝐹 ) ) ) ) ) )
10		cdlemk1.o	⊢ 𝑂 = ( 𝑆 ‘ 𝐷 )
11	10	fveq1i	⊢ ( 𝑂 ‘ 𝑃 ) = ( ( 𝑆 ‘ 𝐷 ) ‘ 𝑃 )
12	1 2 3 5 6 7 8 4 9	cdlemksv2	⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ∧ 𝐷 ∈ 𝑇 ) ∧ ( 𝑁 ∈ 𝑇 ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑅 ‘ 𝐹 ) = ( 𝑅 ‘ 𝑁 ) ) ∧ ( 𝐹 ≠ ( I ↾ 𝐵 ) ∧ 𝐷 ≠ ( I ↾ 𝐵 ) ∧ ( 𝑅 ‘ 𝐷 ) ≠ ( 𝑅 ‘ 𝐹 ) ) ) → ( ( 𝑆 ‘ 𝐷 ) ‘ 𝑃 ) = ( ( 𝑃 ∨ ( 𝑅 ‘ 𝐷 ) ) ∧ ( ( 𝑁 ‘ 𝑃 ) ∨ ( 𝑅 ‘ ( 𝐷 ∘ ◡ 𝐹 ) ) ) ) )
13	11 12	syl5eq	⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ∧ 𝐷 ∈ 𝑇 ) ∧ ( 𝑁 ∈ 𝑇 ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑅 ‘ 𝐹 ) = ( 𝑅 ‘ 𝑁 ) ) ∧ ( 𝐹 ≠ ( I ↾ 𝐵 ) ∧ 𝐷 ≠ ( I ↾ 𝐵 ) ∧ ( 𝑅 ‘ 𝐷 ) ≠ ( 𝑅 ‘ 𝐹 ) ) ) → ( 𝑂 ‘ 𝑃 ) = ( ( 𝑃 ∨ ( 𝑅 ‘ 𝐷 ) ) ∧ ( ( 𝑁 ‘ 𝑃 ) ∨ ( 𝑅 ‘ ( 𝐷 ∘ ◡ 𝐹 ) ) ) ) )