Metamath Proof Explorer

Theorem diclss

Description: The value of partial isomorphism C is a subspace of partial vector space H. (Contributed by NM, 16-Feb-2014)

Ref Expression
Hypotheses diclss.l ⊢ ≤ = ( le ‘ 𝐾 )
diclss.a ⊢ 𝐴 = ( Atoms ‘ 𝐾 )
diclss.h ⊢ 𝐻 = ( LHyp ‘ 𝐾 )
diclss.u ⊢ 𝑈 = ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 )
diclss.i ⊢ 𝐼 = ( ( DIsoC ‘ 𝐾 ) ‘ 𝑊 )
diclss.s ⊢ 𝑆 = ( LSubSp ‘ 𝑈 )
Assertion diclss ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) → ( 𝐼 ‘ 𝑄 ) ∈ 𝑆 )

Proof

Step Hyp Ref Expression
1 diclss.l ⊢ ≤ = ( le ‘ 𝐾 )
2 diclss.a ⊢ 𝐴 = ( Atoms ‘ 𝐾 )
3 diclss.h ⊢ 𝐻 = ( LHyp ‘ 𝐾 )
4 diclss.u ⊢ 𝑈 = ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 )
5 diclss.i ⊢ 𝐼 = ( ( DIsoC ‘ 𝐾 ) ‘ 𝑊 )
6 diclss.s ⊢ 𝑆 = ( LSubSp ‘ 𝑈 )
7 eqidd ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) → ( Scalar ‘ 𝑈 ) = ( Scalar ‘ 𝑈 ) )
8 eqid ⊢ ( ( TEndo ‘ 𝐾 ) ‘ 𝑊 ) = ( ( TEndo ‘ 𝐾 ) ‘ 𝑊 )
9 eqid ⊢ ( Scalar ‘ 𝑈 ) = ( Scalar ‘ 𝑈 )
10 eqid ⊢ ( Base ‘ ( Scalar ‘ 𝑈 ) ) = ( Base ‘ ( Scalar ‘ 𝑈 ) )
11 3 8 4 9 10 dvhbase ⊢ ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) → ( Base ‘ ( Scalar ‘ 𝑈 ) ) = ( ( TEndo ‘ 𝐾 ) ‘ 𝑊 ) )
12 11 eqcomd ⊢ ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) → ( ( TEndo ‘ 𝐾 ) ‘ 𝑊 ) = ( Base ‘ ( Scalar ‘ 𝑈 ) ) )
13 12 adantr ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) → ( ( TEndo ‘ 𝐾 ) ‘ 𝑊 ) = ( Base ‘ ( Scalar ‘ 𝑈 ) ) )
14 eqid ⊢ ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) = ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 )
15 eqid ⊢ ( Base ‘ 𝑈 ) = ( Base ‘ 𝑈 )
16 3 14 8 4 15 dvhvbase ⊢ ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) → ( Base ‘ 𝑈 ) = ( ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) × ( ( TEndo ‘ 𝐾 ) ‘ 𝑊 ) ) )
17 16 eqcomd ⊢ ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) → ( ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) × ( ( TEndo ‘ 𝐾 ) ‘ 𝑊 ) ) = ( Base ‘ 𝑈 ) )
18 17 adantr ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) → ( ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) × ( ( TEndo ‘ 𝐾 ) ‘ 𝑊 ) ) = ( Base ‘ 𝑈 ) )
19 eqidd ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) → ( +g ‘ 𝑈 ) = ( +g ‘ 𝑈 ) )
20 eqidd ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) → ( ·𝑠 ‘ 𝑈 ) = ( ·𝑠 ‘ 𝑈 ) )
21 6 a1i ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) → 𝑆 = ( LSubSp ‘ 𝑈 ) )
22 1 2 3 5 4 15 dicssdvh ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) → ( 𝐼 ‘ 𝑄 ) ⊆ ( Base ‘ 𝑈 ) )
23 22 18 sseqtrrd ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) → ( 𝐼 ‘ 𝑄 ) ⊆ ( ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) × ( ( TEndo ‘ 𝐾 ) ‘ 𝑊 ) ) )
24 1 2 3 5 dicn0 ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) → ( 𝐼 ‘ 𝑄 ) ≠ ∅ )
25 simpll ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( 𝑥 ∈ ( ( TEndo ‘ 𝐾 ) ‘ 𝑊 ) ∧ 𝑎 ∈ ( 𝐼 ‘ 𝑄 ) ∧ 𝑏 ∈ ( 𝐼 ‘ 𝑄 ) ) ) → ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) )
26 simplr ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( 𝑥 ∈ ( ( TEndo ‘ 𝐾 ) ‘ 𝑊 ) ∧ 𝑎 ∈ ( 𝐼 ‘ 𝑄 ) ∧ 𝑏 ∈ ( 𝐼 ‘ 𝑄 ) ) ) → ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) )
27 simpr1 ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( 𝑥 ∈ ( ( TEndo ‘ 𝐾 ) ‘ 𝑊 ) ∧ 𝑎 ∈ ( 𝐼 ‘ 𝑄 ) ∧ 𝑏 ∈ ( 𝐼 ‘ 𝑄 ) ) ) → 𝑥 ∈ ( ( TEndo ‘ 𝐾 ) ‘ 𝑊 ) )
28 simpr2 ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( 𝑥 ∈ ( ( TEndo ‘ 𝐾 ) ‘ 𝑊 ) ∧ 𝑎 ∈ ( 𝐼 ‘ 𝑄 ) ∧ 𝑏 ∈ ( 𝐼 ‘ 𝑄 ) ) ) → 𝑎 ∈ ( 𝐼 ‘ 𝑄 ) )
29 eqid ⊢ ( ·𝑠 ‘ 𝑈 ) = ( ·𝑠 ‘ 𝑈 )
30 1 2 3 8 4 5 29 dicvscacl ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑥 ∈ ( ( TEndo ‘ 𝐾 ) ‘ 𝑊 ) ∧ 𝑎 ∈ ( 𝐼 ‘ 𝑄 ) ) ) → ( 𝑥 ( ·𝑠 ‘ 𝑈 ) 𝑎 ) ∈ ( 𝐼 ‘ 𝑄 ) )
31 25 26 27 28 30 syl112anc ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( 𝑥 ∈ ( ( TEndo ‘ 𝐾 ) ‘ 𝑊 ) ∧ 𝑎 ∈ ( 𝐼 ‘ 𝑄 ) ∧ 𝑏 ∈ ( 𝐼 ‘ 𝑄 ) ) ) → ( 𝑥 ( ·𝑠 ‘ 𝑈 ) 𝑎 ) ∈ ( 𝐼 ‘ 𝑄 ) )
32 simpr3 ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( 𝑥 ∈ ( ( TEndo ‘ 𝐾 ) ‘ 𝑊 ) ∧ 𝑎 ∈ ( 𝐼 ‘ 𝑄 ) ∧ 𝑏 ∈ ( 𝐼 ‘ 𝑄 ) ) ) → 𝑏 ∈ ( 𝐼 ‘ 𝑄 ) )
33 eqid ⊢ ( +g ‘ 𝑈 ) = ( +g ‘ 𝑈 )
34 1 2 3 4 5 33 dicvaddcl ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( ( 𝑥 ( ·𝑠 ‘ 𝑈 ) 𝑎 ) ∈ ( 𝐼 ‘ 𝑄 ) ∧ 𝑏 ∈ ( 𝐼 ‘ 𝑄 ) ) ) → ( ( 𝑥 ( ·𝑠 ‘ 𝑈 ) 𝑎 ) ( +g ‘ 𝑈 ) 𝑏 ) ∈ ( 𝐼 ‘ 𝑄 ) )
35 25 26 31 32 34 syl112anc ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( 𝑥 ∈ ( ( TEndo ‘ 𝐾 ) ‘ 𝑊 ) ∧ 𝑎 ∈ ( 𝐼 ‘ 𝑄 ) ∧ 𝑏 ∈ ( 𝐼 ‘ 𝑄 ) ) ) → ( ( 𝑥 ( ·𝑠 ‘ 𝑈 ) 𝑎 ) ( +g ‘ 𝑈 ) 𝑏 ) ∈ ( 𝐼 ‘ 𝑄 ) )
36 7 13 18 19 20 21 23 24 35 islssd ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) → ( 𝐼 ‘ 𝑄 ) ∈ 𝑆 )