Theorem dicval2

Description: The partial isomorphism C for a lattice K . (Contributed by NM, 20-Feb-2014)

	Ref	Expression
Hypotheses	dicval.l	⊢ ≤ = ( le ‘ 𝐾 )
	dicval.a	⊢ 𝐴 = ( Atoms ‘ 𝐾 )
	dicval.h	⊢ 𝐻 = ( LHyp ‘ 𝐾 )
	dicval.p	⊢ 𝑃 = ( ( oc ‘ 𝐾 ) ‘ 𝑊 )
	dicval.t	⊢ 𝑇 = ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 )
	dicval.e	⊢ 𝐸 = ( ( TEndo ‘ 𝐾 ) ‘ 𝑊 )
	dicval.i	⊢ 𝐼 = ( ( DIsoC ‘ 𝐾 ) ‘ 𝑊 )
	dicval2.g	⊢ 𝐺 = ( ℩ 𝑔 ∈ 𝑇 ( 𝑔 ‘ 𝑃 ) = 𝑄 )
Assertion	dicval2	⊢ ( ( ( 𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) → ( 𝐼 ‘ 𝑄 ) = { ⟨ 𝑓 , 𝑠 ⟩ ∣ ( 𝑓 = ( 𝑠 ‘ 𝐺 ) ∧ 𝑠 ∈ 𝐸 ) } )

Proof

Step	Hyp	Ref	Expression
1		dicval.l	⊢ ≤ = ( le ‘ 𝐾 )
2		dicval.a	⊢ 𝐴 = ( Atoms ‘ 𝐾 )
3		dicval.h	⊢ 𝐻 = ( LHyp ‘ 𝐾 )
4		dicval.p	⊢ 𝑃 = ( ( oc ‘ 𝐾 ) ‘ 𝑊 )
5		dicval.t	⊢ 𝑇 = ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 )
6		dicval.e	⊢ 𝐸 = ( ( TEndo ‘ 𝐾 ) ‘ 𝑊 )
7		dicval.i	⊢ 𝐼 = ( ( DIsoC ‘ 𝐾 ) ‘ 𝑊 )
8		dicval2.g	⊢ 𝐺 = ( ℩ 𝑔 ∈ 𝑇 ( 𝑔 ‘ 𝑃 ) = 𝑄 )
9	1 2 3 4 5 6 7	dicval	⊢ ( ( ( 𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) → ( 𝐼 ‘ 𝑄 ) = { ⟨ 𝑓 , 𝑠 ⟩ ∣ ( 𝑓 = ( 𝑠 ‘ ( ℩ 𝑔 ∈ 𝑇 ( 𝑔 ‘ 𝑃 ) = 𝑄 ) ) ∧ 𝑠 ∈ 𝐸 ) } )
10	8	fveq2i	⊢ ( 𝑠 ‘ 𝐺 ) = ( 𝑠 ‘ ( ℩ 𝑔 ∈ 𝑇 ( 𝑔 ‘ 𝑃 ) = 𝑄 ) )
11	10	eqeq2i	⊢ ( 𝑓 = ( 𝑠 ‘ 𝐺 ) ↔ 𝑓 = ( 𝑠 ‘ ( ℩ 𝑔 ∈ 𝑇 ( 𝑔 ‘ 𝑃 ) = 𝑄 ) ) )
12	11	anbi1i	⊢ ( ( 𝑓 = ( 𝑠 ‘ 𝐺 ) ∧ 𝑠 ∈ 𝐸 ) ↔ ( 𝑓 = ( 𝑠 ‘ ( ℩ 𝑔 ∈ 𝑇 ( 𝑔 ‘ 𝑃 ) = 𝑄 ) ) ∧ 𝑠 ∈ 𝐸 ) )
13	12	opabbii	⊢ { ⟨ 𝑓 , 𝑠 ⟩ ∣ ( 𝑓 = ( 𝑠 ‘ 𝐺 ) ∧ 𝑠 ∈ 𝐸 ) } = { ⟨ 𝑓 , 𝑠 ⟩ ∣ ( 𝑓 = ( 𝑠 ‘ ( ℩ 𝑔 ∈ 𝑇 ( 𝑔 ‘ 𝑃 ) = 𝑄 ) ) ∧ 𝑠 ∈ 𝐸 ) }
14	9 13	eqtr4di	⊢ ( ( ( 𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) → ( 𝐼 ‘ 𝑄 ) = { ⟨ 𝑓 , 𝑠 ⟩ ∣ ( 𝑓 = ( 𝑠 ‘ 𝐺 ) ∧ 𝑠 ∈ 𝐸 ) } )