Theorem dicelvalN

Description: Membership in value of the partial isomorphism C for a lattice K . (Contributed by NM, 25-Feb-2014) (New usage is discouraged.)

	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 )
Assertion	dicelvalN	⊢ ( ( ( 𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) → ( 𝑌 ∈ ( 𝐼 ‘ 𝑄 ) ↔ ( 𝑌 ∈ ( V × V ) ∧ ( ( 1st ‘ 𝑌 ) = ( ( 2nd ‘ 𝑌 ) ‘ ( ℩ 𝑔 ∈ 𝑇 ( 𝑔 ‘ 𝑃 ) = 𝑄 ) ) ∧ ( 2nd ‘ 𝑌 ) ∈ 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	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 ) } )
9	8	eleq2d	⊢ ( ( ( 𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) → ( 𝑌 ∈ ( 𝐼 ‘ 𝑄 ) ↔ 𝑌 ∈ { ⟨ 𝑓 , 𝑠 ⟩ ∣ ( 𝑓 = ( 𝑠 ‘ ( ℩ 𝑔 ∈ 𝑇 ( 𝑔 ‘ 𝑃 ) = 𝑄 ) ) ∧ 𝑠 ∈ 𝐸 ) } ) )
10		vex	⊢ 𝑓 ∈ V
11		vex	⊢ 𝑠 ∈ V
12	10 11	op1std	⊢ ( 𝑌 = ⟨ 𝑓 , 𝑠 ⟩ → ( 1st ‘ 𝑌 ) = 𝑓 )
13	10 11	op2ndd	⊢ ( 𝑌 = ⟨ 𝑓 , 𝑠 ⟩ → ( 2nd ‘ 𝑌 ) = 𝑠 )
14	13	fveq1d	⊢ ( 𝑌 = ⟨ 𝑓 , 𝑠 ⟩ → ( ( 2nd ‘ 𝑌 ) ‘ ( ℩ 𝑔 ∈ 𝑇 ( 𝑔 ‘ 𝑃 ) = 𝑄 ) ) = ( 𝑠 ‘ ( ℩ 𝑔 ∈ 𝑇 ( 𝑔 ‘ 𝑃 ) = 𝑄 ) ) )
15	12 14	eqeq12d	⊢ ( 𝑌 = ⟨ 𝑓 , 𝑠 ⟩ → ( ( 1st ‘ 𝑌 ) = ( ( 2nd ‘ 𝑌 ) ‘ ( ℩ 𝑔 ∈ 𝑇 ( 𝑔 ‘ 𝑃 ) = 𝑄 ) ) ↔ 𝑓 = ( 𝑠 ‘ ( ℩ 𝑔 ∈ 𝑇 ( 𝑔 ‘ 𝑃 ) = 𝑄 ) ) ) )
16	13	eleq1d	⊢ ( 𝑌 = ⟨ 𝑓 , 𝑠 ⟩ → ( ( 2nd ‘ 𝑌 ) ∈ 𝐸 ↔ 𝑠 ∈ 𝐸 ) )
17	15 16	anbi12d	⊢ ( 𝑌 = ⟨ 𝑓 , 𝑠 ⟩ → ( ( ( 1st ‘ 𝑌 ) = ( ( 2nd ‘ 𝑌 ) ‘ ( ℩ 𝑔 ∈ 𝑇 ( 𝑔 ‘ 𝑃 ) = 𝑄 ) ) ∧ ( 2nd ‘ 𝑌 ) ∈ 𝐸 ) ↔ ( 𝑓 = ( 𝑠 ‘ ( ℩ 𝑔 ∈ 𝑇 ( 𝑔 ‘ 𝑃 ) = 𝑄 ) ) ∧ 𝑠 ∈ 𝐸 ) ) )
18	17	elopaba	⊢ ( 𝑌 ∈ { ⟨ 𝑓 , 𝑠 ⟩ ∣ ( 𝑓 = ( 𝑠 ‘ ( ℩ 𝑔 ∈ 𝑇 ( 𝑔 ‘ 𝑃 ) = 𝑄 ) ) ∧ 𝑠 ∈ 𝐸 ) } ↔ ( 𝑌 ∈ ( V × V ) ∧ ( ( 1st ‘ 𝑌 ) = ( ( 2nd ‘ 𝑌 ) ‘ ( ℩ 𝑔 ∈ 𝑇 ( 𝑔 ‘ 𝑃 ) = 𝑄 ) ) ∧ ( 2nd ‘ 𝑌 ) ∈ 𝐸 ) ) )
19	9 18	bitrdi	⊢ ( ( ( 𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) → ( 𝑌 ∈ ( 𝐼 ‘ 𝑄 ) ↔ ( 𝑌 ∈ ( V × V ) ∧ ( ( 1st ‘ 𝑌 ) = ( ( 2nd ‘ 𝑌 ) ‘ ( ℩ 𝑔 ∈ 𝑇 ( 𝑔 ‘ 𝑃 ) = 𝑄 ) ) ∧ ( 2nd ‘ 𝑌 ) ∈ 𝐸 ) ) ) )