Metamath Proof Explorer


Theorem dicelval2nd

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

Ref Expression
Hypotheses dicelval2nd.l = ( le ‘ 𝐾 )
dicelval2nd.a 𝐴 = ( Atoms ‘ 𝐾 )
dicelval2nd.h 𝐻 = ( LHyp ‘ 𝐾 )
dicelval2nd.e 𝐸 = ( ( TEndo ‘ 𝐾 ) ‘ 𝑊 )
dicelval2nd.i 𝐼 = ( ( DIsoC ‘ 𝐾 ) ‘ 𝑊 )
Assertion dicelval2nd ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ∧ 𝑌 ∈ ( 𝐼𝑄 ) ) → ( 2nd𝑌 ) ∈ 𝐸 )

Proof

Step Hyp Ref Expression
1 dicelval2nd.l = ( le ‘ 𝐾 )
2 dicelval2nd.a 𝐴 = ( Atoms ‘ 𝐾 )
3 dicelval2nd.h 𝐻 = ( LHyp ‘ 𝐾 )
4 dicelval2nd.e 𝐸 = ( ( TEndo ‘ 𝐾 ) ‘ 𝑊 )
5 dicelval2nd.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 ∧ 𝑊𝐻 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ) → ( 𝐼𝑄 ) ⊆ ( Base ‘ ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 ) ) )
9 eqid ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) = ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 )
10 3 9 4 6 7 dvhvbase ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) → ( Base ‘ ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 ) ) = ( ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) × 𝐸 ) )
11 10 adantr ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ) → ( Base ‘ ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 ) ) = ( ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) × 𝐸 ) )
12 8 11 sseqtrd ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ) → ( 𝐼𝑄 ) ⊆ ( ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) × 𝐸 ) )
13 12 sseld ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ) → ( 𝑌 ∈ ( 𝐼𝑄 ) → 𝑌 ∈ ( ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) × 𝐸 ) ) )
14 13 3impia ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ∧ 𝑌 ∈ ( 𝐼𝑄 ) ) → 𝑌 ∈ ( ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) × 𝐸 ) )
15 xp2nd ( 𝑌 ∈ ( ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) × 𝐸 ) → ( 2nd𝑌 ) ∈ 𝐸 )
16 14 15 syl ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ∧ 𝑌 ∈ ( 𝐼𝑄 ) ) → ( 2nd𝑌 ) ∈ 𝐸 )