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Metamath Proof Explorer

Theorem dibelval1st2N

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

Ref Expression
Hypotheses dibelval1st2.b ⊢ 𝐵 = ( Base ‘ 𝐾 )
dibelval1st2.l ⊢ ≤ = ( le ‘ 𝐾 )
dibelval1st2.h ⊢ 𝐻 = ( LHyp ‘ 𝐾 )
dibelval1st2.t ⊢ 𝑇 = ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 )
dibelval1st2.r ⊢ 𝑅 = ( ( trL ‘ 𝐾 ) ‘ 𝑊 )
dibelval1st2.i ⊢ 𝐼 = ( ( DIsoB ‘ 𝐾 ) ‘ 𝑊 )
Assertion dibelval1st2N ( ( ( 𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊 ) ∧ 𝑌 ∈ ( 𝐼 ‘ 𝑋 ) ) → ( 𝑅 ‘ ( 1st ‘ 𝑌 ) ) ≤ 𝑋 )

Proof

Step Hyp Ref Expression
1 dibelval1st2.b ⊢ 𝐵 = ( Base ‘ 𝐾 )
2 dibelval1st2.l ⊢ ≤ = ( le ‘ 𝐾 )
3 dibelval1st2.h ⊢ 𝐻 = ( LHyp ‘ 𝐾 )
4 dibelval1st2.t ⊢ 𝑇 = ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 )
5 dibelval1st2.r ⊢ 𝑅 = ( ( trL ‘ 𝐾 ) ‘ 𝑊 )
6 dibelval1st2.i ⊢ 𝐼 = ( ( DIsoB ‘ 𝐾 ) ‘ 𝑊 )
7 eqid ⊢ ( ( DIsoA ‘ 𝐾 ) ‘ 𝑊 ) = ( ( DIsoA ‘ 𝐾 ) ‘ 𝑊 )
8 1 2 3 7 6 dibelval1st ⊢ ( ( ( 𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊 ) ∧ 𝑌 ∈ ( 𝐼 ‘ 𝑋 ) ) → ( 1st ‘ 𝑌 ) ∈ ( ( ( DIsoA ‘ 𝐾 ) ‘ 𝑊 ) ‘ 𝑋 ) )
9 1 2 3 4 5 7 diatrl ⊢ ( ( ( 𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊 ) ∧ ( 1st ‘ 𝑌 ) ∈ ( ( ( DIsoA ‘ 𝐾 ) ‘ 𝑊 ) ‘ 𝑋 ) ) → ( 𝑅 ‘ ( 1st ‘ 𝑌 ) ) ≤ 𝑋 )
10 8 9 syld3an3 ⊢ ( ( ( 𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊 ) ∧ 𝑌 ∈ ( 𝐼 ‘ 𝑋 ) ) → ( 𝑅 ‘ ( 1st ‘ 𝑌 ) ) ≤ 𝑋 )