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Metamath Proof Explorer

Theorem dicssdvh

Description: The partial isomorphism C maps to a set of vectors in full vector space H. (Contributed by NM, 19-Jan-2014)

Ref Expression
Hypotheses dicssdvh.l ⊢ ≤ = ( le ‘ 𝐾 )
dicssdvh.a ⊢ 𝐴 = ( Atoms ‘ 𝐾 )
dicssdvh.h ⊢ 𝐻 = ( LHyp ‘ 𝐾 )
dicssdvh.i ⊢ 𝐼 = ( ( DIsoC ‘ 𝐾 ) ‘ 𝑊 )
dicssdvh.u ⊢ 𝑈 = ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 )
dicssdvh.v ⊢ 𝑉 = ( Base ‘ 𝑈 )
Assertion dicssdvh ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) → ( 𝐼 ‘ 𝑄 ) ⊆ 𝑉 )

Proof

Step Hyp Ref Expression
1 dicssdvh.l ⊢ ≤ = ( le ‘ 𝐾 )
2 dicssdvh.a ⊢ 𝐴 = ( Atoms ‘ 𝐾 )
3 dicssdvh.h ⊢ 𝐻 = ( LHyp ‘ 𝐾 )
4 dicssdvh.i ⊢ 𝐼 = ( ( DIsoC ‘ 𝐾 ) ‘ 𝑊 )
5 dicssdvh.u ⊢ 𝑈 = ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 )
6 dicssdvh.v ⊢ 𝑉 = ( Base ‘ 𝑈 )
7 simprl ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( 𝑓 = ( 𝑠 ‘ ( ℩ 𝑔 ∈ ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) ( 𝑔 ‘ ( ( oc ‘ 𝐾 ) ‘ 𝑊 ) ) = 𝑄 ) ) ∧ 𝑠 ∈ ( ( TEndo ‘ 𝐾 ) ‘ 𝑊 ) ) ) → 𝑓 = ( 𝑠 ‘ ( ℩ 𝑔 ∈ ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) ( 𝑔 ‘ ( ( oc ‘ 𝐾 ) ‘ 𝑊 ) ) = 𝑄 ) ) )
8 simpll ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( 𝑓 = ( 𝑠 ‘ ( ℩ 𝑔 ∈ ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) ( 𝑔 ‘ ( ( oc ‘ 𝐾 ) ‘ 𝑊 ) ) = 𝑄 ) ) ∧ 𝑠 ∈ ( ( TEndo ‘ 𝐾 ) ‘ 𝑊 ) ) ) → ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) )
9 simprr ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( 𝑓 = ( 𝑠 ‘ ( ℩ 𝑔 ∈ ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) ( 𝑔 ‘ ( ( oc ‘ 𝐾 ) ‘ 𝑊 ) ) = 𝑄 ) ) ∧ 𝑠 ∈ ( ( TEndo ‘ 𝐾 ) ‘ 𝑊 ) ) ) → 𝑠 ∈ ( ( TEndo ‘ 𝐾 ) ‘ 𝑊 ) )
10 eqid ⊢ ( oc ‘ 𝐾 ) = ( oc ‘ 𝐾 )
11 1 10 2 3 lhpocnel ⊢ ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) → ( ( ( oc ‘ 𝐾 ) ‘ 𝑊 ) ∈ 𝐴 ∧ ¬ ( ( oc ‘ 𝐾 ) ‘ 𝑊 ) ≤ 𝑊 ) )
12 11 ad2antrr ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( 𝑓 = ( 𝑠 ‘ ( ℩ 𝑔 ∈ ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) ( 𝑔 ‘ ( ( oc ‘ 𝐾 ) ‘ 𝑊 ) ) = 𝑄 ) ) ∧ 𝑠 ∈ ( ( TEndo ‘ 𝐾 ) ‘ 𝑊 ) ) ) → ( ( ( oc ‘ 𝐾 ) ‘ 𝑊 ) ∈ 𝐴 ∧ ¬ ( ( oc ‘ 𝐾 ) ‘ 𝑊 ) ≤ 𝑊 ) )
13 simplr ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( 𝑓 = ( 𝑠 ‘ ( ℩ 𝑔 ∈ ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) ( 𝑔 ‘ ( ( oc ‘ 𝐾 ) ‘ 𝑊 ) ) = 𝑄 ) ) ∧ 𝑠 ∈ ( ( TEndo ‘ 𝐾 ) ‘ 𝑊 ) ) ) → ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) )
14 eqid ⊢ ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) = ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 )
15 eqid ⊢ ( ℩ 𝑔 ∈ ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) ( 𝑔 ‘ ( ( oc ‘ 𝐾 ) ‘ 𝑊 ) ) = 𝑄 ) = ( ℩ 𝑔 ∈ ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) ( 𝑔 ‘ ( ( oc ‘ 𝐾 ) ‘ 𝑊 ) ) = 𝑄 )
16 1 2 3 14 15 ltrniotacl ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( ( oc ‘ 𝐾 ) ‘ 𝑊 ) ∈ 𝐴 ∧ ¬ ( ( oc ‘ 𝐾 ) ‘ 𝑊 ) ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) → ( ℩ 𝑔 ∈ ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) ( 𝑔 ‘ ( ( oc ‘ 𝐾 ) ‘ 𝑊 ) ) = 𝑄 ) ∈ ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) )
17 8 12 13 16 syl3anc ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( 𝑓 = ( 𝑠 ‘ ( ℩ 𝑔 ∈ ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) ( 𝑔 ‘ ( ( oc ‘ 𝐾 ) ‘ 𝑊 ) ) = 𝑄 ) ) ∧ 𝑠 ∈ ( ( TEndo ‘ 𝐾 ) ‘ 𝑊 ) ) ) → ( ℩ 𝑔 ∈ ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) ( 𝑔 ‘ ( ( oc ‘ 𝐾 ) ‘ 𝑊 ) ) = 𝑄 ) ∈ ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) )
18 eqid ⊢ ( ( TEndo ‘ 𝐾 ) ‘ 𝑊 ) = ( ( TEndo ‘ 𝐾 ) ‘ 𝑊 )
19 3 14 18 tendocl ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑠 ∈ ( ( TEndo ‘ 𝐾 ) ‘ 𝑊 ) ∧ ( ℩ 𝑔 ∈ ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) ( 𝑔 ‘ ( ( oc ‘ 𝐾 ) ‘ 𝑊 ) ) = 𝑄 ) ∈ ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) ) → ( 𝑠 ‘ ( ℩ 𝑔 ∈ ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) ( 𝑔 ‘ ( ( oc ‘ 𝐾 ) ‘ 𝑊 ) ) = 𝑄 ) ) ∈ ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) )
20 8 9 17 19 syl3anc ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( 𝑓 = ( 𝑠 ‘ ( ℩ 𝑔 ∈ ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) ( 𝑔 ‘ ( ( oc ‘ 𝐾 ) ‘ 𝑊 ) ) = 𝑄 ) ) ∧ 𝑠 ∈ ( ( TEndo ‘ 𝐾 ) ‘ 𝑊 ) ) ) → ( 𝑠 ‘ ( ℩ 𝑔 ∈ ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) ( 𝑔 ‘ ( ( oc ‘ 𝐾 ) ‘ 𝑊 ) ) = 𝑄 ) ) ∈ ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) )
21 7 20 eqeltrd ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( 𝑓 = ( 𝑠 ‘ ( ℩ 𝑔 ∈ ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) ( 𝑔 ‘ ( ( oc ‘ 𝐾 ) ‘ 𝑊 ) ) = 𝑄 ) ) ∧ 𝑠 ∈ ( ( TEndo ‘ 𝐾 ) ‘ 𝑊 ) ) ) → 𝑓 ∈ ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) )
22 21 9 9 jca31 ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( 𝑓 = ( 𝑠 ‘ ( ℩ 𝑔 ∈ ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) ( 𝑔 ‘ ( ( oc ‘ 𝐾 ) ‘ 𝑊 ) ) = 𝑄 ) ) ∧ 𝑠 ∈ ( ( TEndo ‘ 𝐾 ) ‘ 𝑊 ) ) ) → ( ( 𝑓 ∈ ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) ∧ 𝑠 ∈ ( ( TEndo ‘ 𝐾 ) ‘ 𝑊 ) ) ∧ 𝑠 ∈ ( ( TEndo ‘ 𝐾 ) ‘ 𝑊 ) ) )
23 22 ex ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) → ( ( 𝑓 = ( 𝑠 ‘ ( ℩ 𝑔 ∈ ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) ( 𝑔 ‘ ( ( oc ‘ 𝐾 ) ‘ 𝑊 ) ) = 𝑄 ) ) ∧ 𝑠 ∈ ( ( TEndo ‘ 𝐾 ) ‘ 𝑊 ) ) → ( ( 𝑓 ∈ ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) ∧ 𝑠 ∈ ( ( TEndo ‘ 𝐾 ) ‘ 𝑊 ) ) ∧ 𝑠 ∈ ( ( TEndo ‘ 𝐾 ) ‘ 𝑊 ) ) ) )
24 23 ssopab2dv ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) → { ⟨ 𝑓 , 𝑠 ⟩ ∣ ( 𝑓 = ( 𝑠 ‘ ( ℩ 𝑔 ∈ ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) ( 𝑔 ‘ ( ( oc ‘ 𝐾 ) ‘ 𝑊 ) ) = 𝑄 ) ) ∧ 𝑠 ∈ ( ( TEndo ‘ 𝐾 ) ‘ 𝑊 ) ) } ⊆ { ⟨ 𝑓 , 𝑠 ⟩ ∣ ( ( 𝑓 ∈ ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) ∧ 𝑠 ∈ ( ( TEndo ‘ 𝐾 ) ‘ 𝑊 ) ) ∧ 𝑠 ∈ ( ( TEndo ‘ 𝐾 ) ‘ 𝑊 ) ) } )
25 opabssxp ⊢ { ⟨ 𝑓 , 𝑠 ⟩ ∣ ( ( 𝑓 ∈ ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) ∧ 𝑠 ∈ ( ( TEndo ‘ 𝐾 ) ‘ 𝑊 ) ) ∧ 𝑠 ∈ ( ( TEndo ‘ 𝐾 ) ‘ 𝑊 ) ) } ⊆ ( ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) × ( ( TEndo ‘ 𝐾 ) ‘ 𝑊 ) )
26 24 25 sstrdi ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) → { ⟨ 𝑓 , 𝑠 ⟩ ∣ ( 𝑓 = ( 𝑠 ‘ ( ℩ 𝑔 ∈ ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) ( 𝑔 ‘ ( ( oc ‘ 𝐾 ) ‘ 𝑊 ) ) = 𝑄 ) ) ∧ 𝑠 ∈ ( ( TEndo ‘ 𝐾 ) ‘ 𝑊 ) ) } ⊆ ( ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) × ( ( TEndo ‘ 𝐾 ) ‘ 𝑊 ) ) )
27 eqid ⊢ ( ( oc ‘ 𝐾 ) ‘ 𝑊 ) = ( ( oc ‘ 𝐾 ) ‘ 𝑊 )
28 1 2 3 27 14 18 4 dicval ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) → ( 𝐼 ‘ 𝑄 ) = { ⟨ 𝑓 , 𝑠 ⟩ ∣ ( 𝑓 = ( 𝑠 ‘ ( ℩ 𝑔 ∈ ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) ( 𝑔 ‘ ( ( oc ‘ 𝐾 ) ‘ 𝑊 ) ) = 𝑄 ) ) ∧ 𝑠 ∈ ( ( TEndo ‘ 𝐾 ) ‘ 𝑊 ) ) } )
29 3 14 18 5 6 dvhvbase ⊢ ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) → 𝑉 = ( ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) × ( ( TEndo ‘ 𝐾 ) ‘ 𝑊 ) ) )
30 29 adantr ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) → 𝑉 = ( ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) × ( ( TEndo ‘ 𝐾 ) ‘ 𝑊 ) ) )
31 26 28 30 3sstr4d ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) → ( 𝐼 ‘ 𝑄 ) ⊆ 𝑉 )