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Metamath Proof Explorer

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 ) )