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Metamath Proof Explorer

Theorem dih1dimat

Description: Any 1-dimensional subspace is a value of isomorphism H. (Contributed by NM, 11-Apr-2014)

Ref Expression
Hypotheses dih1dimat.h ⊢ 𝐻 = ( LHyp ‘ 𝐾 )
dih1dimat.u ⊢ 𝑈 = ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 )
dih1dimat.i ⊢ 𝐼 = ( ( DIsoH ‘ 𝐾 ) ‘ 𝑊 )
dih1dimat.a ⊢ 𝐴 = ( LSAtoms ‘ 𝑈 )
Assertion dih1dimat ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑃 ∈ 𝐴 ) → 𝑃 ∈ ran 𝐼 )

Proof

Step Hyp Ref Expression
1 dih1dimat.h ⊢ 𝐻 = ( LHyp ‘ 𝐾 )
2 dih1dimat.u ⊢ 𝑈 = ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 )
3 dih1dimat.i ⊢ 𝐼 = ( ( DIsoH ‘ 𝐾 ) ‘ 𝑊 )
4 dih1dimat.a ⊢ 𝐴 = ( LSAtoms ‘ 𝑈 )
5 eqid ⊢ ( Base ‘ 𝐾 ) = ( Base ‘ 𝐾 )
6 eqid ⊢ ( le ‘ 𝐾 ) = ( le ‘ 𝐾 )
7 eqid ⊢ ( Atoms ‘ 𝐾 ) = ( Atoms ‘ 𝐾 )
8 eqid ⊢ ( ( oc ‘ 𝐾 ) ‘ 𝑊 ) = ( ( oc ‘ 𝐾 ) ‘ 𝑊 )
9 eqid ⊢ ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) = ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 )
10 eqid ⊢ ( ( trL ‘ 𝐾 ) ‘ 𝑊 ) = ( ( trL ‘ 𝐾 ) ‘ 𝑊 )
11 eqid ⊢ ( ( TEndo ‘ 𝐾 ) ‘ 𝑊 ) = ( ( TEndo ‘ 𝐾 ) ‘ 𝑊 )
12 eqid ⊢ ( ℎ ∈ ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) ↦ ( I ↾ ( Base ‘ 𝐾 ) ) ) = ( ℎ ∈ ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) ↦ ( I ↾ ( Base ‘ 𝐾 ) ) )
13 eqid ⊢ ( Scalar ‘ 𝑈 ) = ( Scalar ‘ 𝑈 )
14 eqid ⊢ ( invr ‘ ( Scalar ‘ 𝑈 ) ) = ( invr ‘ ( Scalar ‘ 𝑈 ) )
15 eqid ⊢ ( Base ‘ 𝑈 ) = ( Base ‘ 𝑈 )
16 eqid ⊢ ( ·𝑠 ‘ 𝑈 ) = ( ·𝑠 ‘ 𝑈 )
17 eqid ⊢ ( LSubSp ‘ 𝑈 ) = ( LSubSp ‘ 𝑈 )
18 eqid ⊢ ( LSpan ‘ 𝑈 ) = ( LSpan ‘ 𝑈 )
19 eqid ⊢ ( 0g ‘ 𝑈 ) = ( 0g ‘ 𝑈 )
20 eqid ⊢ ( ℩ ℎ ∈ ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) ( ℎ ‘ ( ( oc ‘ 𝐾 ) ‘ 𝑊 ) ) = ( ( ( ( invr ‘ ( Scalar ‘ 𝑈 ) ) ‘ 𝑠 ) ‘ 𝑓 ) ‘ ( ( oc ‘ 𝐾 ) ‘ 𝑊 ) ) ) = ( ℩ ℎ ∈ ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) ( ℎ ‘ ( ( oc ‘ 𝐾 ) ‘ 𝑊 ) ) = ( ( ( ( invr ‘ ( Scalar ‘ 𝑈 ) ) ‘ 𝑠 ) ‘ 𝑓 ) ‘ ( ( oc ‘ 𝐾 ) ‘ 𝑊 ) ) )
21 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 dih1dimatlem ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑃 ∈ 𝐴 ) → 𝑃 ∈ ran 𝐼 )