Theorem dicelval1stN

Description: Membership in value of the partial isomorphism C for a lattice K . (Contributed by NM, 16-Feb-2014) (New usage is discouraged.)

	Ref	Expression
Hypotheses	dicelval1st.l	⊢ ≤ = ( le ‘ 𝐾 )
	dicelval1st.a	⊢ 𝐴 = ( Atoms ‘ 𝐾 )
	dicelval1st.h	⊢ 𝐻 = ( LHyp ‘ 𝐾 )
	dicelval1st.t	⊢ 𝑇 = ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 )
	dicelval1st.i	⊢ 𝐼 = ( ( DIsoC ‘ 𝐾 ) ‘ 𝑊 )
Assertion	dicelval1stN	⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ 𝑌 ∈ ( 𝐼 ‘ 𝑄 ) ) → ( 1st ‘ 𝑌 ) ∈ 𝑇 )

Proof

Step	Hyp	Ref	Expression
1		dicelval1st.l	⊢ ≤ = ( le ‘ 𝐾 )
2		dicelval1st.a	⊢ 𝐴 = ( Atoms ‘ 𝐾 )
3		dicelval1st.h	⊢ 𝐻 = ( LHyp ‘ 𝐾 )
4		dicelval1st.t	⊢ 𝑇 = ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 )
5		dicelval1st.i	⊢ 𝐼 = ( ( DIsoC ‘ 𝐾 ) ‘ 𝑊 )
6		eqid	⊢ ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 ) = ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 )
7		eqid	⊢ ( Base ‘ ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 ) ) = ( Base ‘ ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 ) )
8	1 2 3 5 6 7	dicssdvh	⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) → ( 𝐼 ‘ 𝑄 ) ⊆ ( Base ‘ ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 ) ) )
9		eqid	⊢ ( ( TEndo ‘ 𝐾 ) ‘ 𝑊 ) = ( ( TEndo ‘ 𝐾 ) ‘ 𝑊 )
10	3 4 9 6 7	dvhvbase	⊢ ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) → ( Base ‘ ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 ) ) = ( 𝑇 × ( ( TEndo ‘ 𝐾 ) ‘ 𝑊 ) ) )
11	10	adantr	⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) → ( Base ‘ ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 ) ) = ( 𝑇 × ( ( TEndo ‘ 𝐾 ) ‘ 𝑊 ) ) )
12	8 11	sseqtrd	⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) → ( 𝐼 ‘ 𝑄 ) ⊆ ( 𝑇 × ( ( TEndo ‘ 𝐾 ) ‘ 𝑊 ) ) )
13	12	sseld	⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) → ( 𝑌 ∈ ( 𝐼 ‘ 𝑄 ) → 𝑌 ∈ ( 𝑇 × ( ( TEndo ‘ 𝐾 ) ‘ 𝑊 ) ) ) )
14	13	3impia	⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ 𝑌 ∈ ( 𝐼 ‘ 𝑄 ) ) → 𝑌 ∈ ( 𝑇 × ( ( TEndo ‘ 𝐾 ) ‘ 𝑊 ) ) )
15		xp1st	⊢ ( 𝑌 ∈ ( 𝑇 × ( ( TEndo ‘ 𝐾 ) ‘ 𝑊 ) ) → ( 1st ‘ 𝑌 ) ∈ 𝑇 )
16	14 15	syl	⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ 𝑌 ∈ ( 𝐼 ‘ 𝑄 ) ) → ( 1st ‘ 𝑌 ) ∈ 𝑇 )