# Metamath Proof Explorer

## Theorem diasslssN

Description: The partial isomorphism A maps to subspaces of partial vector space A. (Contributed by NM, 17-Jan-2014) (New usage is discouraged.)

Ref Expression
Hypotheses diasslss.h 𝐻 = ( LHyp ‘ 𝐾 )
diasslss.u 𝑈 = ( ( DVecA ‘ 𝐾 ) ‘ 𝑊 )
diasslss.i 𝐼 = ( ( DIsoA ‘ 𝐾 ) ‘ 𝑊 )
diasslss.s 𝑆 = ( LSubSp ‘ 𝑈 )
Assertion diasslssN ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) → ran 𝐼𝑆 )

### Proof

Step Hyp Ref Expression
1 diasslss.h 𝐻 = ( LHyp ‘ 𝐾 )
2 diasslss.u 𝑈 = ( ( DVecA ‘ 𝐾 ) ‘ 𝑊 )
3 diasslss.i 𝐼 = ( ( DIsoA ‘ 𝐾 ) ‘ 𝑊 )
4 diasslss.s 𝑆 = ( LSubSp ‘ 𝑈 )
5 1 3 diaf11N ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) → 𝐼 : dom 𝐼1-1-onto→ ran 𝐼 )
6 f1ocnvfv2 ( ( 𝐼 : dom 𝐼1-1-onto→ ran 𝐼𝑥 ∈ ran 𝐼 ) → ( 𝐼 ‘ ( 𝐼𝑥 ) ) = 𝑥 )
7 5 6 sylan ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ 𝑥 ∈ ran 𝐼 ) → ( 𝐼 ‘ ( 𝐼𝑥 ) ) = 𝑥 )
8 1 3 diacnvclN ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ 𝑥 ∈ ran 𝐼 ) → ( 𝐼𝑥 ) ∈ dom 𝐼 )
9 eqid ( Base ‘ 𝐾 ) = ( Base ‘ 𝐾 )
10 eqid ( le ‘ 𝐾 ) = ( le ‘ 𝐾 )
11 9 10 1 3 diaeldm ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) → ( ( 𝐼𝑥 ) ∈ dom 𝐼 ↔ ( ( 𝐼𝑥 ) ∈ ( Base ‘ 𝐾 ) ∧ ( 𝐼𝑥 ) ( le ‘ 𝐾 ) 𝑊 ) ) )
12 11 adantr ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ 𝑥 ∈ ran 𝐼 ) → ( ( 𝐼𝑥 ) ∈ dom 𝐼 ↔ ( ( 𝐼𝑥 ) ∈ ( Base ‘ 𝐾 ) ∧ ( 𝐼𝑥 ) ( le ‘ 𝐾 ) 𝑊 ) ) )
13 8 12 mpbid ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ 𝑥 ∈ ran 𝐼 ) → ( ( 𝐼𝑥 ) ∈ ( Base ‘ 𝐾 ) ∧ ( 𝐼𝑥 ) ( le ‘ 𝐾 ) 𝑊 ) )
14 9 10 1 2 3 4 dialss ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( ( 𝐼𝑥 ) ∈ ( Base ‘ 𝐾 ) ∧ ( 𝐼𝑥 ) ( le ‘ 𝐾 ) 𝑊 ) ) → ( 𝐼 ‘ ( 𝐼𝑥 ) ) ∈ 𝑆 )
15 13 14 syldan ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ 𝑥 ∈ ran 𝐼 ) → ( 𝐼 ‘ ( 𝐼𝑥 ) ) ∈ 𝑆 )
16 7 15 eqeltrrd ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ 𝑥 ∈ ran 𝐼 ) → 𝑥𝑆 )
17 16 ex ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) → ( 𝑥 ∈ ran 𝐼𝑥𝑆 ) )
18 17 ssrdv ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) → ran 𝐼𝑆 )