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Metamath Proof Explorer

Theorem dibelval1st

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

Ref Expression
Hypotheses dibelval1.b ⊢ 𝐵 = ( Base ‘ 𝐾 )
dibelval1.l ⊢ ≤ = ( le ‘ 𝐾 )
dibelval1.h ⊢ 𝐻 = ( LHyp ‘ 𝐾 )
dibelval1.j ⊢ 𝐽 = ( ( DIsoA ‘ 𝐾 ) ‘ 𝑊 )
dibelval1.i ⊢ 𝐼 = ( ( DIsoB ‘ 𝐾 ) ‘ 𝑊 )
Assertion dibelval1st ( ( ( 𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊 ) ∧ 𝑌 ∈ ( 𝐼 ‘ 𝑋 ) ) → ( 1st ‘ 𝑌 ) ∈ ( 𝐽 ‘ 𝑋 ) )

Proof

Step Hyp Ref Expression
1 dibelval1.b ⊢ 𝐵 = ( Base ‘ 𝐾 )
2 dibelval1.l ⊢ ≤ = ( le ‘ 𝐾 )
3 dibelval1.h ⊢ 𝐻 = ( LHyp ‘ 𝐾 )
4 dibelval1.j ⊢ 𝐽 = ( ( DIsoA ‘ 𝐾 ) ‘ 𝑊 )
5 dibelval1.i ⊢ 𝐼 = ( ( DIsoB ‘ 𝐾 ) ‘ 𝑊 )
6 eqid ⊢ ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) = ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 )
7 eqid ⊢ ( 𝑓 ∈ ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) ↦ ( I ↾ 𝐵 ) ) = ( 𝑓 ∈ ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) ↦ ( I ↾ 𝐵 ) )
8 1 2 3 6 7 4 5 dibval2 ⊢ ( ( ( 𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊 ) ) → ( 𝐼 ‘ 𝑋 ) = ( ( 𝐽 ‘ 𝑋 ) × { ( 𝑓 ∈ ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) ↦ ( I ↾ 𝐵 ) ) } ) )
9 8 eleq2d ⊢ ( ( ( 𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊 ) ) → ( 𝑌 ∈ ( 𝐼 ‘ 𝑋 ) ↔ 𝑌 ∈ ( ( 𝐽 ‘ 𝑋 ) × { ( 𝑓 ∈ ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) ↦ ( I ↾ 𝐵 ) ) } ) ) )
10 9 biimp3a ⊢ ( ( ( 𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊 ) ∧ 𝑌 ∈ ( 𝐼 ‘ 𝑋 ) ) → 𝑌 ∈ ( ( 𝐽 ‘ 𝑋 ) × { ( 𝑓 ∈ ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) ↦ ( I ↾ 𝐵 ) ) } ) )
11 xp1st ⊢ ( 𝑌 ∈ ( ( 𝐽 ‘ 𝑋 ) × { ( 𝑓 ∈ ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) ↦ ( I ↾ 𝐵 ) ) } ) → ( 1st ‘ 𝑌 ) ∈ ( 𝐽 ‘ 𝑋 ) )
12 10 11 syl ⊢ ( ( ( 𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊 ) ∧ 𝑌 ∈ ( 𝐼 ‘ 𝑋 ) ) → ( 1st ‘ 𝑌 ) ∈ ( 𝐽 ‘ 𝑋 ) )