Metamath Proof Explorer


Theorem dibopelval3

Description: Member of the partial isomorphism B. (Contributed by NM, 3-Mar-2014)

Ref Expression
Hypotheses dibval3.b 𝐵 = ( Base ‘ 𝐾 )
dibval3.l = ( le ‘ 𝐾 )
dibval3.h 𝐻 = ( LHyp ‘ 𝐾 )
dibval3.t 𝑇 = ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 )
dibval3.r 𝑅 = ( ( trL ‘ 𝐾 ) ‘ 𝑊 )
dibval3.o 0 = ( 𝑔𝑇 ↦ ( I ↾ 𝐵 ) )
dibval3.i 𝐼 = ( ( DIsoB ‘ 𝐾 ) ‘ 𝑊 )
Assertion dibopelval3 ( ( ( 𝐾𝑉𝑊𝐻 ) ∧ ( 𝑋𝐵𝑋 𝑊 ) ) → ( ⟨ 𝐹 , 𝑆 ⟩ ∈ ( 𝐼𝑋 ) ↔ ( ( 𝐹𝑇 ∧ ( 𝑅𝐹 ) 𝑋 ) ∧ 𝑆 = 0 ) ) )

Proof

Step Hyp Ref Expression
1 dibval3.b 𝐵 = ( Base ‘ 𝐾 )
2 dibval3.l = ( le ‘ 𝐾 )
3 dibval3.h 𝐻 = ( LHyp ‘ 𝐾 )
4 dibval3.t 𝑇 = ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 )
5 dibval3.r 𝑅 = ( ( trL ‘ 𝐾 ) ‘ 𝑊 )
6 dibval3.o 0 = ( 𝑔𝑇 ↦ ( I ↾ 𝐵 ) )
7 dibval3.i 𝐼 = ( ( DIsoB ‘ 𝐾 ) ‘ 𝑊 )
8 eqid ( ( DIsoA ‘ 𝐾 ) ‘ 𝑊 ) = ( ( DIsoA ‘ 𝐾 ) ‘ 𝑊 )
9 1 2 3 4 6 8 7 dibopelval2 ( ( ( 𝐾𝑉𝑊𝐻 ) ∧ ( 𝑋𝐵𝑋 𝑊 ) ) → ( ⟨ 𝐹 , 𝑆 ⟩ ∈ ( 𝐼𝑋 ) ↔ ( 𝐹 ∈ ( ( ( DIsoA ‘ 𝐾 ) ‘ 𝑊 ) ‘ 𝑋 ) ∧ 𝑆 = 0 ) ) )
10 1 2 3 4 5 8 diaelval ( ( ( 𝐾𝑉𝑊𝐻 ) ∧ ( 𝑋𝐵𝑋 𝑊 ) ) → ( 𝐹 ∈ ( ( ( DIsoA ‘ 𝐾 ) ‘ 𝑊 ) ‘ 𝑋 ) ↔ ( 𝐹𝑇 ∧ ( 𝑅𝐹 ) 𝑋 ) ) )
11 10 anbi1d ( ( ( 𝐾𝑉𝑊𝐻 ) ∧ ( 𝑋𝐵𝑋 𝑊 ) ) → ( ( 𝐹 ∈ ( ( ( DIsoA ‘ 𝐾 ) ‘ 𝑊 ) ‘ 𝑋 ) ∧ 𝑆 = 0 ) ↔ ( ( 𝐹𝑇 ∧ ( 𝑅𝐹 ) 𝑋 ) ∧ 𝑆 = 0 ) ) )
12 9 11 bitrd ( ( ( 𝐾𝑉𝑊𝐻 ) ∧ ( 𝑋𝐵𝑋 𝑊 ) ) → ( ⟨ 𝐹 , 𝑆 ⟩ ∈ ( 𝐼𝑋 ) ↔ ( ( 𝐹𝑇 ∧ ( 𝑅𝐹 ) 𝑋 ) ∧ 𝑆 = 0 ) ) )