This is an inofficial mirror of http://metamath.tirix.org for personal testing of a visualizer extension only.

Metamath Proof Explorer

Theorem dibelval2nd

Description: Membership in value of the partial isomorphism B for a lattice K . (Contributed by NM, 13-Feb-2014)

Ref Expression
Hypotheses dibelval2nd.b ⊢ 𝐵 = ( Base ‘ 𝐾 )
dibelval2nd.l ⊢ ≤ = ( le ‘ 𝐾 )
dibelval2nd.h ⊢ 𝐻 = ( LHyp ‘ 𝐾 )
dibelval2nd.t ⊢ 𝑇 = ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 )
dibelval2nd.o ⊢ 0 = ( 𝑓 ∈ 𝑇 ↦ ( I ↾ 𝐵 ) )
dibelval2nd.i ⊢ 𝐼 = ( ( DIsoB ‘ 𝐾 ) ‘ 𝑊 )
Assertion dibelval2nd ( ( ( 𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊 ) ∧ 𝑌 ∈ ( 𝐼 ‘ 𝑋 ) ) → ( 2nd ‘ 𝑌 ) = 0 )

Proof

Step Hyp Ref Expression
1 dibelval2nd.b ⊢ 𝐵 = ( Base ‘ 𝐾 )
2 dibelval2nd.l ⊢ ≤ = ( le ‘ 𝐾 )
3 dibelval2nd.h ⊢ 𝐻 = ( LHyp ‘ 𝐾 )
4 dibelval2nd.t ⊢ 𝑇 = ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 )
5 dibelval2nd.o ⊢ 0 = ( 𝑓 ∈ 𝑇 ↦ ( I ↾ 𝐵 ) )
6 dibelval2nd.i ⊢ 𝐼 = ( ( DIsoB ‘ 𝐾 ) ‘ 𝑊 )
7 eqid ⊢ ( ( DIsoA ‘ 𝐾 ) ‘ 𝑊 ) = ( ( DIsoA ‘ 𝐾 ) ‘ 𝑊 )
8 1 2 3 4 5 7 6 dibval2 ⊢ ( ( ( 𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊 ) ) → ( 𝐼 ‘ 𝑋 ) = ( ( ( ( DIsoA ‘ 𝐾 ) ‘ 𝑊 ) ‘ 𝑋 ) × { 0 } ) )
9 8 eleq2d ⊢ ( ( ( 𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊 ) ) → ( 𝑌 ∈ ( 𝐼 ‘ 𝑋 ) ↔ 𝑌 ∈ ( ( ( ( DIsoA ‘ 𝐾 ) ‘ 𝑊 ) ‘ 𝑋 ) × { 0 } ) ) )
10 9 biimp3a ⊢ ( ( ( 𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊 ) ∧ 𝑌 ∈ ( 𝐼 ‘ 𝑋 ) ) → 𝑌 ∈ ( ( ( ( DIsoA ‘ 𝐾 ) ‘ 𝑊 ) ‘ 𝑋 ) × { 0 } ) )
11 xp2nd ⊢ ( 𝑌 ∈ ( ( ( ( DIsoA ‘ 𝐾 ) ‘ 𝑊 ) ‘ 𝑋 ) × { 0 } ) → ( 2nd ‘ 𝑌 ) ∈ { 0 } )
12 elsni ⊢ ( ( 2nd ‘ 𝑌 ) ∈ { 0 } → ( 2nd ‘ 𝑌 ) = 0 )
13 10 11 12 3syl ⊢ ( ( ( 𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊 ) ∧ 𝑌 ∈ ( 𝐼 ‘ 𝑋 ) ) → ( 2nd ‘ 𝑌 ) = 0 )