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Metamath Proof Explorer

Theorem dihopelvalcqat

Description: Ordered pair member of the partial isomorphism H for atom argument not under W . TODO: remove .t hypothesis. (Contributed by NM, 30-Mar-2014)

Ref Expression
Hypotheses dihelval2.l ⊢ ≤ = ( le ‘ 𝐾 )
dihelval2.a ⊢ 𝐴 = ( Atoms ‘ 𝐾 )
dihelval2.h ⊢ 𝐻 = ( LHyp ‘ 𝐾 )
dihelval2.p ⊢ 𝑃 = ( ( oc ‘ 𝐾 ) ‘ 𝑊 )
dihelval2.t ⊢ 𝑇 = ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 )
dihelval2.e ⊢ 𝐸 = ( ( TEndo ‘ 𝐾 ) ‘ 𝑊 )
dihelval2.i ⊢ 𝐼 = ( ( DIsoH ‘ 𝐾 ) ‘ 𝑊 )
dihelval2.g ⊢ 𝐺 = ( ℩ 𝑔 ∈ 𝑇 ( 𝑔 ‘ 𝑃 ) = 𝑄 )
dihelval2.f ⊢ 𝐹 ∈ V
dihelval2.s ⊢ 𝑆 ∈ V
Assertion dihopelvalcqat ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) → ( ⟨ 𝐹 , 𝑆 ⟩ ∈ ( 𝐼 ‘ 𝑄 ) ↔ ( 𝐹 = ( 𝑆 ‘ 𝐺 ) ∧ 𝑆 ∈ 𝐸 ) ) )

Proof

Step Hyp Ref Expression
1 dihelval2.l ⊢ ≤ = ( le ‘ 𝐾 )
2 dihelval2.a ⊢ 𝐴 = ( Atoms ‘ 𝐾 )
3 dihelval2.h ⊢ 𝐻 = ( LHyp ‘ 𝐾 )
4 dihelval2.p ⊢ 𝑃 = ( ( oc ‘ 𝐾 ) ‘ 𝑊 )
5 dihelval2.t ⊢ 𝑇 = ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 )
6 dihelval2.e ⊢ 𝐸 = ( ( TEndo ‘ 𝐾 ) ‘ 𝑊 )
7 dihelval2.i ⊢ 𝐼 = ( ( DIsoH ‘ 𝐾 ) ‘ 𝑊 )
8 dihelval2.g ⊢ 𝐺 = ( ℩ 𝑔 ∈ 𝑇 ( 𝑔 ‘ 𝑃 ) = 𝑄 )
9 dihelval2.f ⊢ 𝐹 ∈ V
10 dihelval2.s ⊢ 𝑆 ∈ V
11 eqid ⊢ ( ( DIsoC ‘ 𝐾 ) ‘ 𝑊 ) = ( ( DIsoC ‘ 𝐾 ) ‘ 𝑊 )
12 1 2 3 11 7 dihvalcqat ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) → ( 𝐼 ‘ 𝑄 ) = ( ( ( DIsoC ‘ 𝐾 ) ‘ 𝑊 ) ‘ 𝑄 ) )
13 12 eleq2d ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) → ( ⟨ 𝐹 , 𝑆 ⟩ ∈ ( 𝐼 ‘ 𝑄 ) ↔ ⟨ 𝐹 , 𝑆 ⟩ ∈ ( ( ( DIsoC ‘ 𝐾 ) ‘ 𝑊 ) ‘ 𝑄 ) ) )
14 1 2 3 4 5 6 11 8 9 10 dicopelval2 ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) → ( ⟨ 𝐹 , 𝑆 ⟩ ∈ ( ( ( DIsoC ‘ 𝐾 ) ‘ 𝑊 ) ‘ 𝑄 ) ↔ ( 𝐹 = ( 𝑆 ‘ 𝐺 ) ∧ 𝑆 ∈ 𝐸 ) ) )
15 13 14 bitrd ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) → ( ⟨ 𝐹 , 𝑆 ⟩ ∈ ( 𝐼 ‘ 𝑄 ) ↔ ( 𝐹 = ( 𝑆 ‘ 𝐺 ) ∧ 𝑆 ∈ 𝐸 ) ) )