Theorem dihvalb

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

	Ref	Expression
Hypotheses	dihvalb.b	⊢ 𝐵 = ( Base ‘ 𝐾 )
	dihvalb.l	⊢ ≤ = ( le ‘ 𝐾 )
	dihvalb.h	⊢ 𝐻 = ( LHyp ‘ 𝐾 )
	dihvalb.i	⊢ 𝐼 = ( ( DIsoH ‘ 𝐾 ) ‘ 𝑊 )
	dihvalb.d	⊢ 𝐷 = ( ( DIsoB ‘ 𝐾 ) ‘ 𝑊 )
Assertion	dihvalb	⊢ ( ( ( 𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊 ) ) → ( 𝐼 ‘ 𝑋 ) = ( 𝐷 ‘ 𝑋 ) )

Proof

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