Theorem cdlemkuel-2N

Description: Part of proof of Lemma K of Crawley p. 118. Conditions for the sigma_2 (p) function to be a translation. TODO: combine cdlemkj ? (Contributed by NM, 2-Jul-2013) (New usage is discouraged.)

	Ref	Expression
Hypotheses	cdlemk2.b	⊢ 𝐵 = ( Base ‘ 𝐾 )
	cdlemk2.l	⊢ ≤ = ( le ‘ 𝐾 )
	cdlemk2.j	⊢ ∨ = ( join ‘ 𝐾 )
	cdlemk2.m	⊢ ∧ = ( meet ‘ 𝐾 )
	cdlemk2.a	⊢ 𝐴 = ( Atoms ‘ 𝐾 )
	cdlemk2.h	⊢ 𝐻 = ( LHyp ‘ 𝐾 )
	cdlemk2.t	⊢ 𝑇 = ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 )
	cdlemk2.r	⊢ 𝑅 = ( ( trL ‘ 𝐾 ) ‘ 𝑊 )
	cdlemk2.s	⊢ 𝑆 = ( 𝑓 ∈ 𝑇 ↦ ( ℩ 𝑖 ∈ 𝑇 ( 𝑖 ‘ 𝑃 ) = ( ( 𝑃 ∨ ( 𝑅 ‘ 𝑓 ) ) ∧ ( ( 𝑁 ‘ 𝑃 ) ∨ ( 𝑅 ‘ ( 𝑓 ∘ ◡ 𝐹 ) ) ) ) ) )
	cdlemk2.q	⊢ 𝑄 = ( 𝑆 ‘ 𝐶 )
	cdlemk2.v	⊢ 𝑉 = ( 𝑑 ∈ 𝑇 ↦ ( ℩ 𝑘 ∈ 𝑇 ( 𝑘 ‘ 𝑃 ) = ( ( 𝑃 ∨ ( 𝑅 ‘ 𝑑 ) ) ∧ ( ( 𝑄 ‘ 𝑃 ) ∨ ( 𝑅 ‘ ( 𝑑 ∘ ◡ 𝐶 ) ) ) ) ) )
Assertion	cdlemkuel-2N	⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑅 ‘ 𝐹 ) = ( 𝑅 ‘ 𝑁 ) ∧ 𝐺 ∈ 𝑇 ) ∧ ( 𝐹 ∈ 𝑇 ∧ 𝐶 ∈ 𝑇 ∧ 𝑁 ∈ 𝑇 ) ∧ ( ( ( 𝑅 ‘ 𝐶 ) ≠ ( 𝑅 ‘ 𝐹 ) ∧ ( 𝑅 ‘ 𝐶 ) ≠ ( 𝑅 ‘ 𝐺 ) ) ∧ ( 𝐹 ≠ ( I ↾ 𝐵 ) ∧ 𝐺 ≠ ( I ↾ 𝐵 ) ∧ 𝐶 ≠ ( I ↾ 𝐵 ) ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ) ) → ( 𝑉 ‘ 𝐺 ) ∈ 𝑇 )

Proof

Step	Hyp	Ref	Expression
1		cdlemk2.b	⊢ 𝐵 = ( Base ‘ 𝐾 )
2		cdlemk2.l	⊢ ≤ = ( le ‘ 𝐾 )
3		cdlemk2.j	⊢ ∨ = ( join ‘ 𝐾 )
4		cdlemk2.m	⊢ ∧ = ( meet ‘ 𝐾 )
5		cdlemk2.a	⊢ 𝐴 = ( Atoms ‘ 𝐾 )
6		cdlemk2.h	⊢ 𝐻 = ( LHyp ‘ 𝐾 )
7		cdlemk2.t	⊢ 𝑇 = ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 )
8		cdlemk2.r	⊢ 𝑅 = ( ( trL ‘ 𝐾 ) ‘ 𝑊 )
9		cdlemk2.s	⊢ 𝑆 = ( 𝑓 ∈ 𝑇 ↦ ( ℩ 𝑖 ∈ 𝑇 ( 𝑖 ‘ 𝑃 ) = ( ( 𝑃 ∨ ( 𝑅 ‘ 𝑓 ) ) ∧ ( ( 𝑁 ‘ 𝑃 ) ∨ ( 𝑅 ‘ ( 𝑓 ∘ ◡ 𝐹 ) ) ) ) ) )
10		cdlemk2.q	⊢ 𝑄 = ( 𝑆 ‘ 𝐶 )
11		cdlemk2.v	⊢ 𝑉 = ( 𝑑 ∈ 𝑇 ↦ ( ℩ 𝑘 ∈ 𝑇 ( 𝑘 ‘ 𝑃 ) = ( ( 𝑃 ∨ ( 𝑅 ‘ 𝑑 ) ) ∧ ( ( 𝑄 ‘ 𝑃 ) ∨ ( 𝑅 ‘ ( 𝑑 ∘ ◡ 𝐶 ) ) ) ) ) )
12	1 2 3 4 5 6 7 8 9 10 11	cdlemkuel	⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑅 ‘ 𝐹 ) = ( 𝑅 ‘ 𝑁 ) ∧ 𝐺 ∈ 𝑇 ) ∧ ( 𝐹 ∈ 𝑇 ∧ 𝐶 ∈ 𝑇 ∧ 𝑁 ∈ 𝑇 ) ∧ ( ( ( 𝑅 ‘ 𝐶 ) ≠ ( 𝑅 ‘ 𝐹 ) ∧ ( 𝑅 ‘ 𝐶 ) ≠ ( 𝑅 ‘ 𝐺 ) ) ∧ ( 𝐹 ≠ ( I ↾ 𝐵 ) ∧ 𝐺 ≠ ( I ↾ 𝐵 ) ∧ 𝐶 ≠ ( I ↾ 𝐵 ) ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ) ) → ( 𝑉 ‘ 𝐺 ) ∈ 𝑇 )