Metamath Proof Explorer


Theorem diaelval

Description: Member of the partial isomorphism A for a lattice K . (Contributed by NM, 3-Dec-2013)

Ref Expression
Hypotheses diaval.b 𝐵 = ( Base ‘ 𝐾 )
diaval.l = ( le ‘ 𝐾 )
diaval.h 𝐻 = ( LHyp ‘ 𝐾 )
diaval.t 𝑇 = ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 )
diaval.r 𝑅 = ( ( trL ‘ 𝐾 ) ‘ 𝑊 )
diaval.i 𝐼 = ( ( DIsoA ‘ 𝐾 ) ‘ 𝑊 )
Assertion diaelval ( ( ( 𝐾𝑉𝑊𝐻 ) ∧ ( 𝑋𝐵𝑋 𝑊 ) ) → ( 𝐹 ∈ ( 𝐼𝑋 ) ↔ ( 𝐹𝑇 ∧ ( 𝑅𝐹 ) 𝑋 ) ) )

Proof

Step Hyp Ref Expression
1 diaval.b 𝐵 = ( Base ‘ 𝐾 )
2 diaval.l = ( le ‘ 𝐾 )
3 diaval.h 𝐻 = ( LHyp ‘ 𝐾 )
4 diaval.t 𝑇 = ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 )
5 diaval.r 𝑅 = ( ( trL ‘ 𝐾 ) ‘ 𝑊 )
6 diaval.i 𝐼 = ( ( DIsoA ‘ 𝐾 ) ‘ 𝑊 )
7 1 2 3 4 5 6 diaval ( ( ( 𝐾𝑉𝑊𝐻 ) ∧ ( 𝑋𝐵𝑋 𝑊 ) ) → ( 𝐼𝑋 ) = { 𝑓𝑇 ∣ ( 𝑅𝑓 ) 𝑋 } )
8 7 eleq2d ( ( ( 𝐾𝑉𝑊𝐻 ) ∧ ( 𝑋𝐵𝑋 𝑊 ) ) → ( 𝐹 ∈ ( 𝐼𝑋 ) ↔ 𝐹 ∈ { 𝑓𝑇 ∣ ( 𝑅𝑓 ) 𝑋 } ) )
9 fveq2 ( 𝑓 = 𝐹 → ( 𝑅𝑓 ) = ( 𝑅𝐹 ) )
10 9 breq1d ( 𝑓 = 𝐹 → ( ( 𝑅𝑓 ) 𝑋 ↔ ( 𝑅𝐹 ) 𝑋 ) )
11 10 elrab ( 𝐹 ∈ { 𝑓𝑇 ∣ ( 𝑅𝑓 ) 𝑋 } ↔ ( 𝐹𝑇 ∧ ( 𝑅𝐹 ) 𝑋 ) )
12 8 11 bitrdi ( ( ( 𝐾𝑉𝑊𝐻 ) ∧ ( 𝑋𝐵𝑋 𝑊 ) ) → ( 𝐹 ∈ ( 𝐼𝑋 ) ↔ ( 𝐹𝑇 ∧ ( 𝑅𝐹 ) 𝑋 ) ) )