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Metamath Proof Explorer

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 )