Theorem cdlemksv

Description: Part of proof of Lemma K of Crawley p. 118. Value of the sigma(p) function. (Contributed by NM, 26-Jun-2013)

	Ref	Expression
Hypotheses	cdlemk.b	⊢ 𝐵 = ( Base ‘ 𝐾 )
	cdlemk.l	⊢ ≤ = ( le ‘ 𝐾 )
	cdlemk.j	⊢ ∨ = ( join ‘ 𝐾 )
	cdlemk.a	⊢ 𝐴 = ( Atoms ‘ 𝐾 )
	cdlemk.h	⊢ 𝐻 = ( LHyp ‘ 𝐾 )
	cdlemk.t	⊢ 𝑇 = ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 )
	cdlemk.r	⊢ 𝑅 = ( ( trL ‘ 𝐾 ) ‘ 𝑊 )
	cdlemk.m	⊢ ∧ = ( meet ‘ 𝐾 )
	cdlemk.s	⊢ 𝑆 = ( 𝑓 ∈ 𝑇 ↦ ( ℩ 𝑖 ∈ 𝑇 ( 𝑖 ‘ 𝑃 ) = ( ( 𝑃 ∨ ( 𝑅 ‘ 𝑓 ) ) ∧ ( ( 𝑁 ‘ 𝑃 ) ∨ ( 𝑅 ‘ ( 𝑓 ∘ ◡ 𝐹 ) ) ) ) ) )
Assertion	cdlemksv	⊢ ( 𝐺 ∈ 𝑇 → ( 𝑆 ‘ 𝐺 ) = ( ℩ 𝑖 ∈ 𝑇 ( 𝑖 ‘ 𝑃 ) = ( ( 𝑃 ∨ ( 𝑅 ‘ 𝐺 ) ) ∧ ( ( 𝑁 ‘ 𝑃 ) ∨ ( 𝑅 ‘ ( 𝐺 ∘ ◡ 𝐹 ) ) ) ) ) )

Proof

Step	Hyp	Ref	Expression
1		cdlemk.b	⊢ 𝐵 = ( Base ‘ 𝐾 )
2		cdlemk.l	⊢ ≤ = ( le ‘ 𝐾 )
3		cdlemk.j	⊢ ∨ = ( join ‘ 𝐾 )
4		cdlemk.a	⊢ 𝐴 = ( Atoms ‘ 𝐾 )
5		cdlemk.h	⊢ 𝐻 = ( LHyp ‘ 𝐾 )
6		cdlemk.t	⊢ 𝑇 = ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 )
7		cdlemk.r	⊢ 𝑅 = ( ( trL ‘ 𝐾 ) ‘ 𝑊 )
8		cdlemk.m	⊢ ∧ = ( meet ‘ 𝐾 )
9		cdlemk.s	⊢ 𝑆 = ( 𝑓 ∈ 𝑇 ↦ ( ℩ 𝑖 ∈ 𝑇 ( 𝑖 ‘ 𝑃 ) = ( ( 𝑃 ∨ ( 𝑅 ‘ 𝑓 ) ) ∧ ( ( 𝑁 ‘ 𝑃 ) ∨ ( 𝑅 ‘ ( 𝑓 ∘ ◡ 𝐹 ) ) ) ) ) )
10		fveq2	⊢ ( 𝑓 = 𝐺 → ( 𝑅 ‘ 𝑓 ) = ( 𝑅 ‘ 𝐺 ) )
11	10	oveq2d	⊢ ( 𝑓 = 𝐺 → ( 𝑃 ∨ ( 𝑅 ‘ 𝑓 ) ) = ( 𝑃 ∨ ( 𝑅 ‘ 𝐺 ) ) )
12		coeq1	⊢ ( 𝑓 = 𝐺 → ( 𝑓 ∘ ◡ 𝐹 ) = ( 𝐺 ∘ ◡ 𝐹 ) )
13	12	fveq2d	⊢ ( 𝑓 = 𝐺 → ( 𝑅 ‘ ( 𝑓 ∘ ◡ 𝐹 ) ) = ( 𝑅 ‘ ( 𝐺 ∘ ◡ 𝐹 ) ) )
14	13	oveq2d	⊢ ( 𝑓 = 𝐺 → ( ( 𝑁 ‘ 𝑃 ) ∨ ( 𝑅 ‘ ( 𝑓 ∘ ◡ 𝐹 ) ) ) = ( ( 𝑁 ‘ 𝑃 ) ∨ ( 𝑅 ‘ ( 𝐺 ∘ ◡ 𝐹 ) ) ) )
15	11 14	oveq12d	⊢ ( 𝑓 = 𝐺 → ( ( 𝑃 ∨ ( 𝑅 ‘ 𝑓 ) ) ∧ ( ( 𝑁 ‘ 𝑃 ) ∨ ( 𝑅 ‘ ( 𝑓 ∘ ◡ 𝐹 ) ) ) ) = ( ( 𝑃 ∨ ( 𝑅 ‘ 𝐺 ) ) ∧ ( ( 𝑁 ‘ 𝑃 ) ∨ ( 𝑅 ‘ ( 𝐺 ∘ ◡ 𝐹 ) ) ) ) )
16	15	eqeq2d	⊢ ( 𝑓 = 𝐺 → ( ( 𝑖 ‘ 𝑃 ) = ( ( 𝑃 ∨ ( 𝑅 ‘ 𝑓 ) ) ∧ ( ( 𝑁 ‘ 𝑃 ) ∨ ( 𝑅 ‘ ( 𝑓 ∘ ◡ 𝐹 ) ) ) ) ↔ ( 𝑖 ‘ 𝑃 ) = ( ( 𝑃 ∨ ( 𝑅 ‘ 𝐺 ) ) ∧ ( ( 𝑁 ‘ 𝑃 ) ∨ ( 𝑅 ‘ ( 𝐺 ∘ ◡ 𝐹 ) ) ) ) ) )
17	16	riotabidv	⊢ ( 𝑓 = 𝐺 → ( ℩ 𝑖 ∈ 𝑇 ( 𝑖 ‘ 𝑃 ) = ( ( 𝑃 ∨ ( 𝑅 ‘ 𝑓 ) ) ∧ ( ( 𝑁 ‘ 𝑃 ) ∨ ( 𝑅 ‘ ( 𝑓 ∘ ◡ 𝐹 ) ) ) ) ) = ( ℩ 𝑖 ∈ 𝑇 ( 𝑖 ‘ 𝑃 ) = ( ( 𝑃 ∨ ( 𝑅 ‘ 𝐺 ) ) ∧ ( ( 𝑁 ‘ 𝑃 ) ∨ ( 𝑅 ‘ ( 𝐺 ∘ ◡ 𝐹 ) ) ) ) ) )
18		riotaex	⊢ ( ℩ 𝑖 ∈ 𝑇 ( 𝑖 ‘ 𝑃 ) = ( ( 𝑃 ∨ ( 𝑅 ‘ 𝐺 ) ) ∧ ( ( 𝑁 ‘ 𝑃 ) ∨ ( 𝑅 ‘ ( 𝐺 ∘ ◡ 𝐹 ) ) ) ) ) ∈ V
19	17 9 18	fvmpt	⊢ ( 𝐺 ∈ 𝑇 → ( 𝑆 ‘ 𝐺 ) = ( ℩ 𝑖 ∈ 𝑇 ( 𝑖 ‘ 𝑃 ) = ( ( 𝑃 ∨ ( 𝑅 ‘ 𝐺 ) ) ∧ ( ( 𝑁 ‘ 𝑃 ) ∨ ( 𝑅 ‘ ( 𝐺 ∘ ◡ 𝐹 ) ) ) ) ) )