Theorem dihvalc

Description: Value of isomorphism H for a lattice K when -. X .<_ W . (Contributed by NM, 4-Mar-2014)

	Ref	Expression
Hypotheses	dihval.b	⊢ 𝐵 = ( Base ‘ 𝐾 )
	dihval.l	⊢ ≤ = ( le ‘ 𝐾 )
	dihval.j	⊢ ∨ = ( join ‘ 𝐾 )
	dihval.m	⊢ ∧ = ( meet ‘ 𝐾 )
	dihval.a	⊢ 𝐴 = ( Atoms ‘ 𝐾 )
	dihval.h	⊢ 𝐻 = ( LHyp ‘ 𝐾 )
	dihval.i	⊢ 𝐼 = ( ( DIsoH ‘ 𝐾 ) ‘ 𝑊 )
	dihval.d	⊢ 𝐷 = ( ( DIsoB ‘ 𝐾 ) ‘ 𝑊 )
	dihval.c	⊢ 𝐶 = ( ( DIsoC ‘ 𝐾 ) ‘ 𝑊 )
	dihval.u	⊢ 𝑈 = ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 )
	dihval.s	⊢ 𝑆 = ( LSubSp ‘ 𝑈 )
	dihval.p	⊢ ⊕ = ( LSSum ‘ 𝑈 )
Assertion	dihvalc	⊢ ( ( ( 𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊 ) ) → ( 𝐼 ‘ 𝑋 ) = ( ℩ 𝑢 ∈ 𝑆 ∀ 𝑞 ∈ 𝐴 ( ( ¬ 𝑞 ≤ 𝑊 ∧ ( 𝑞 ∨ ( 𝑋 ∧ 𝑊 ) ) = 𝑋 ) → 𝑢 = ( ( 𝐶 ‘ 𝑞 ) ⊕ ( 𝐷 ‘ ( 𝑋 ∧ 𝑊 ) ) ) ) ) )

Proof

Step	Hyp	Ref	Expression
1		dihval.b	⊢ 𝐵 = ( Base ‘ 𝐾 )
2		dihval.l	⊢ ≤ = ( le ‘ 𝐾 )
3		dihval.j	⊢ ∨ = ( join ‘ 𝐾 )
4		dihval.m	⊢ ∧ = ( meet ‘ 𝐾 )
5		dihval.a	⊢ 𝐴 = ( Atoms ‘ 𝐾 )
6		dihval.h	⊢ 𝐻 = ( LHyp ‘ 𝐾 )
7		dihval.i	⊢ 𝐼 = ( ( DIsoH ‘ 𝐾 ) ‘ 𝑊 )
8		dihval.d	⊢ 𝐷 = ( ( DIsoB ‘ 𝐾 ) ‘ 𝑊 )
9		dihval.c	⊢ 𝐶 = ( ( DIsoC ‘ 𝐾 ) ‘ 𝑊 )
10		dihval.u	⊢ 𝑈 = ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 )
11		dihval.s	⊢ 𝑆 = ( LSubSp ‘ 𝑈 )
12		dihval.p	⊢ ⊕ = ( LSSum ‘ 𝑈 )
13	1 2 3 4 5 6 7 8 9 10 11 12	dihval	⊢ ( ( ( 𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑋 ∈ 𝐵 ) → ( 𝐼 ‘ 𝑋 ) = if ( 𝑋 ≤ 𝑊 , ( 𝐷 ‘ 𝑋 ) , ( ℩ 𝑢 ∈ 𝑆 ∀ 𝑞 ∈ 𝐴 ( ( ¬ 𝑞 ≤ 𝑊 ∧ ( 𝑞 ∨ ( 𝑋 ∧ 𝑊 ) ) = 𝑋 ) → 𝑢 = ( ( 𝐶 ‘ 𝑞 ) ⊕ ( 𝐷 ‘ ( 𝑋 ∧ 𝑊 ) ) ) ) ) ) )
14		iffalse	⊢ ( ¬ 𝑋 ≤ 𝑊 → if ( 𝑋 ≤ 𝑊 , ( 𝐷 ‘ 𝑋 ) , ( ℩ 𝑢 ∈ 𝑆 ∀ 𝑞 ∈ 𝐴 ( ( ¬ 𝑞 ≤ 𝑊 ∧ ( 𝑞 ∨ ( 𝑋 ∧ 𝑊 ) ) = 𝑋 ) → 𝑢 = ( ( 𝐶 ‘ 𝑞 ) ⊕ ( 𝐷 ‘ ( 𝑋 ∧ 𝑊 ) ) ) ) ) ) = ( ℩ 𝑢 ∈ 𝑆 ∀ 𝑞 ∈ 𝐴 ( ( ¬ 𝑞 ≤ 𝑊 ∧ ( 𝑞 ∨ ( 𝑋 ∧ 𝑊 ) ) = 𝑋 ) → 𝑢 = ( ( 𝐶 ‘ 𝑞 ) ⊕ ( 𝐷 ‘ ( 𝑋 ∧ 𝑊 ) ) ) ) ) )
15	13 14	sylan9eq	⊢ ( ( ( ( 𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑋 ∈ 𝐵 ) ∧ ¬ 𝑋 ≤ 𝑊 ) → ( 𝐼 ‘ 𝑋 ) = ( ℩ 𝑢 ∈ 𝑆 ∀ 𝑞 ∈ 𝐴 ( ( ¬ 𝑞 ≤ 𝑊 ∧ ( 𝑞 ∨ ( 𝑋 ∧ 𝑊 ) ) = 𝑋 ) → 𝑢 = ( ( 𝐶 ‘ 𝑞 ) ⊕ ( 𝐷 ‘ ( 𝑋 ∧ 𝑊 ) ) ) ) ) )
16	15	anasss	⊢ ( ( ( 𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊 ) ) → ( 𝐼 ‘ 𝑋 ) = ( ℩ 𝑢 ∈ 𝑆 ∀ 𝑞 ∈ 𝐴 ( ( ¬ 𝑞 ≤ 𝑊 ∧ ( 𝑞 ∨ ( 𝑋 ∧ 𝑊 ) ) = 𝑋 ) → 𝑢 = ( ( 𝐶 ‘ 𝑞 ) ⊕ ( 𝐷 ‘ ( 𝑋 ∧ 𝑊 ) ) ) ) ) )