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 ∧ 𝑊 ∈ 𝐻 ) → ran 𝐼 ⊆ 𝑆 )

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 ∧ 𝑊 ∈ 𝐻 ) → 𝐼 : dom 𝐼 –1-1-onto→ ran 𝐼 )
6		f1ocnvfv2	⊢ ( ( 𝐼 : dom 𝐼 –1-1-onto→ ran 𝐼 ∧ 𝑥 ∈ ran 𝐼 ) → ( 𝐼 ‘ ( ◡ 𝐼 ‘ 𝑥 ) ) = 𝑥 )
7	5 6	sylan	⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑥 ∈ ran 𝐼 ) → ( 𝐼 ‘ ( ◡ 𝐼 ‘ 𝑥 ) ) = 𝑥 )
8	1 3	diacnvclN	⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑥 ∈ ran 𝐼 ) → ( ◡ 𝐼 ‘ 𝑥 ) ∈ dom 𝐼 )
9		eqid	⊢ ( Base ‘ 𝐾 ) = ( Base ‘ 𝐾 )
10		eqid	⊢ ( le ‘ 𝐾 ) = ( le ‘ 𝐾 )
11	9 10 1 3	diaeldm	⊢ ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) → ( ( ◡ 𝐼 ‘ 𝑥 ) ∈ dom 𝐼 ↔ ( ( ◡ 𝐼 ‘ 𝑥 ) ∈ ( Base ‘ 𝐾 ) ∧ ( ◡ 𝐼 ‘ 𝑥 ) ( le ‘ 𝐾 ) 𝑊 ) ) )
12	11	adantr	⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑥 ∈ ran 𝐼 ) → ( ( ◡ 𝐼 ‘ 𝑥 ) ∈ dom 𝐼 ↔ ( ( ◡ 𝐼 ‘ 𝑥 ) ∈ ( Base ‘ 𝐾 ) ∧ ( ◡ 𝐼 ‘ 𝑥 ) ( le ‘ 𝐾 ) 𝑊 ) ) )
13	8 12	mpbid	⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑥 ∈ ran 𝐼 ) → ( ( ◡ 𝐼 ‘ 𝑥 ) ∈ ( Base ‘ 𝐾 ) ∧ ( ◡ 𝐼 ‘ 𝑥 ) ( le ‘ 𝐾 ) 𝑊 ) )
14	9 10 1 2 3 4	dialss	⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( ◡ 𝐼 ‘ 𝑥 ) ∈ ( Base ‘ 𝐾 ) ∧ ( ◡ 𝐼 ‘ 𝑥 ) ( le ‘ 𝐾 ) 𝑊 ) ) → ( 𝐼 ‘ ( ◡ 𝐼 ‘ 𝑥 ) ) ∈ 𝑆 )
15	13 14	syldan	⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑥 ∈ ran 𝐼 ) → ( 𝐼 ‘ ( ◡ 𝐼 ‘ 𝑥 ) ) ∈ 𝑆 )
16	7 15	eqeltrrd	⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑥 ∈ ran 𝐼 ) → 𝑥 ∈ 𝑆 )
17	16	ex	⊢ ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) → ( 𝑥 ∈ ran 𝐼 → 𝑥 ∈ 𝑆 ) )
18	17	ssrdv	⊢ ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) → ran 𝐼 ⊆ 𝑆 )