Metamath Proof Explorer


Theorem dibopelval2

Description: Member of the partial isomorphism B. (Contributed by NM, 3-Mar-2014) (Revised by Mario Carneiro, 6-May-2015)

Ref Expression
Hypotheses dibval2.b 𝐵 = ( Base ‘ 𝐾 )
dibval2.l = ( le ‘ 𝐾 )
dibval2.h 𝐻 = ( LHyp ‘ 𝐾 )
dibval2.t 𝑇 = ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 )
dibval2.o 0 = ( 𝑓𝑇 ↦ ( I ↾ 𝐵 ) )
dibval2.j 𝐽 = ( ( DIsoA ‘ 𝐾 ) ‘ 𝑊 )
dibval2.i 𝐼 = ( ( DIsoB ‘ 𝐾 ) ‘ 𝑊 )
Assertion dibopelval2 ( ( ( 𝐾𝑉𝑊𝐻 ) ∧ ( 𝑋𝐵𝑋 𝑊 ) ) → ( ⟨ 𝐹 , 𝑆 ⟩ ∈ ( 𝐼𝑋 ) ↔ ( 𝐹 ∈ ( 𝐽𝑋 ) ∧ 𝑆 = 0 ) ) )

Proof

Step Hyp Ref Expression
1 dibval2.b 𝐵 = ( Base ‘ 𝐾 )
2 dibval2.l = ( le ‘ 𝐾 )
3 dibval2.h 𝐻 = ( LHyp ‘ 𝐾 )
4 dibval2.t 𝑇 = ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 )
5 dibval2.o 0 = ( 𝑓𝑇 ↦ ( I ↾ 𝐵 ) )
6 dibval2.j 𝐽 = ( ( DIsoA ‘ 𝐾 ) ‘ 𝑊 )
7 dibval2.i 𝐼 = ( ( DIsoB ‘ 𝐾 ) ‘ 𝑊 )
8 1 2 3 4 5 6 7 dibval2 ( ( ( 𝐾𝑉𝑊𝐻 ) ∧ ( 𝑋𝐵𝑋 𝑊 ) ) → ( 𝐼𝑋 ) = ( ( 𝐽𝑋 ) × { 0 } ) )
9 8 eleq2d ( ( ( 𝐾𝑉𝑊𝐻 ) ∧ ( 𝑋𝐵𝑋 𝑊 ) ) → ( ⟨ 𝐹 , 𝑆 ⟩ ∈ ( 𝐼𝑋 ) ↔ ⟨ 𝐹 , 𝑆 ⟩ ∈ ( ( 𝐽𝑋 ) × { 0 } ) ) )
10 opelxp ( ⟨ 𝐹 , 𝑆 ⟩ ∈ ( ( 𝐽𝑋 ) × { 0 } ) ↔ ( 𝐹 ∈ ( 𝐽𝑋 ) ∧ 𝑆 ∈ { 0 } ) )
11 4 fvexi 𝑇 ∈ V
12 11 mptex ( 𝑓𝑇 ↦ ( I ↾ 𝐵 ) ) ∈ V
13 5 12 eqeltri 0 ∈ V
14 13 elsn2 ( 𝑆 ∈ { 0 } ↔ 𝑆 = 0 )
15 14 anbi2i ( ( 𝐹 ∈ ( 𝐽𝑋 ) ∧ 𝑆 ∈ { 0 } ) ↔ ( 𝐹 ∈ ( 𝐽𝑋 ) ∧ 𝑆 = 0 ) )
16 10 15 bitri ( ⟨ 𝐹 , 𝑆 ⟩ ∈ ( ( 𝐽𝑋 ) × { 0 } ) ↔ ( 𝐹 ∈ ( 𝐽𝑋 ) ∧ 𝑆 = 0 ) )
17 9 16 bitrdi ( ( ( 𝐾𝑉𝑊𝐻 ) ∧ ( 𝑋𝐵𝑋 𝑊 ) ) → ( ⟨ 𝐹 , 𝑆 ⟩ ∈ ( 𝐼𝑋 ) ↔ ( 𝐹 ∈ ( 𝐽𝑋 ) ∧ 𝑆 = 0 ) ) )