Metamath Proof Explorer

Theorem dvadiaN

Description: Any closed subspace is a member of the range of partial isomorphism A, showing the isomorphism maps onto the set of closed subspaces of partial vector space A. (Contributed by NM, 17-Jan-2014) (New usage is discouraged.)

Ref Expression
Hypotheses dvadia.h ⊢ 𝐻 = ( LHyp ‘ 𝐾 )
dvadia.u ⊢ 𝑈 = ( ( DVecA ‘ 𝐾 ) ‘ 𝑊 )
dvadia.i ⊢ 𝐼 = ( ( DIsoA ‘ 𝐾 ) ‘ 𝑊 )
dvadia.n ⊢ ⊥ = ( ( ocA ‘ 𝐾 ) ‘ 𝑊 )
dvadia.s ⊢ 𝑆 = ( LSubSp ‘ 𝑈 )
Assertion dvadiaN ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝑆 ∧ ( ⊥ ‘ ( ⊥ ‘ 𝑋 ) ) = 𝑋 ) ) → 𝑋 ∈ ran 𝐼 )

Proof

Step Hyp Ref Expression
1 dvadia.h ⊢ 𝐻 = ( LHyp ‘ 𝐾 )
2 dvadia.u ⊢ 𝑈 = ( ( DVecA ‘ 𝐾 ) ‘ 𝑊 )
3 dvadia.i ⊢ 𝐼 = ( ( DIsoA ‘ 𝐾 ) ‘ 𝑊 )
4 dvadia.n ⊢ ⊥ = ( ( ocA ‘ 𝐾 ) ‘ 𝑊 )
5 dvadia.s ⊢ 𝑆 = ( LSubSp ‘ 𝑈 )
6 simprr ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝑆 ∧ ( ⊥ ‘ ( ⊥ ‘ 𝑋 ) ) = 𝑋 ) ) → ( ⊥ ‘ ( ⊥ ‘ 𝑋 ) ) = 𝑋 )
7 eqid ⊢ ( Base ‘ 𝑈 ) = ( Base ‘ 𝑈 )
8 7 5 lssss ⊢ ( 𝑋 ∈ 𝑆 → 𝑋 ⊆ ( Base ‘ 𝑈 ) )
9 8 ad2antrl ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝑆 ∧ ( ⊥ ‘ ( ⊥ ‘ 𝑋 ) ) = 𝑋 ) ) → 𝑋 ⊆ ( Base ‘ 𝑈 ) )
10 eqid ⊢ ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) = ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 )
11 1 10 2 7 dvavbase ⊢ ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) → ( Base ‘ 𝑈 ) = ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) )
12 11 adantr ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝑆 ∧ ( ⊥ ‘ ( ⊥ ‘ 𝑋 ) ) = 𝑋 ) ) → ( Base ‘ 𝑈 ) = ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) )
13 9 12 sseqtrd ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝑆 ∧ ( ⊥ ‘ ( ⊥ ‘ 𝑋 ) ) = 𝑋 ) ) → 𝑋 ⊆ ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) )
14 1 10 3 4 docaclN ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑋 ⊆ ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) ) → ( ⊥ ‘ 𝑋 ) ∈ ran 𝐼 )
15 13 14 syldan ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝑆 ∧ ( ⊥ ‘ ( ⊥ ‘ 𝑋 ) ) = 𝑋 ) ) → ( ⊥ ‘ 𝑋 ) ∈ ran 𝐼 )
16 1 10 3 diaelrnN ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ⊥ ‘ 𝑋 ) ∈ ran 𝐼 ) → ( ⊥ ‘ 𝑋 ) ⊆ ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) )
17 15 16 syldan ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝑆 ∧ ( ⊥ ‘ ( ⊥ ‘ 𝑋 ) ) = 𝑋 ) ) → ( ⊥ ‘ 𝑋 ) ⊆ ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) )
18 1 10 3 4 docaclN ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ⊥ ‘ 𝑋 ) ⊆ ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) ) → ( ⊥ ‘ ( ⊥ ‘ 𝑋 ) ) ∈ ran 𝐼 )
19 17 18 syldan ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝑆 ∧ ( ⊥ ‘ ( ⊥ ‘ 𝑋 ) ) = 𝑋 ) ) → ( ⊥ ‘ ( ⊥ ‘ 𝑋 ) ) ∈ ran 𝐼 )
20 6 19 eqeltrrd ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝑆 ∧ ( ⊥ ‘ ( ⊥ ‘ 𝑋 ) ) = 𝑋 ) ) → 𝑋 ∈ ran 𝐼 )