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Metamath Proof Explorer

Theorem ssps

Description: Scalar multiplication on a subspace is a restriction of scalar multiplication on the parent space. (Contributed by NM, 28-Jan-2008) (New usage is discouraged.)

Ref Expression
Hypotheses ssps.y ⊢ 𝑌 = ( BaseSet ‘ 𝑊 )
ssps.s ⊢ 𝑆 = ( ·𝑠OLD ‘ 𝑈 )
ssps.r ⊢ 𝑅 = ( ·𝑠OLD ‘ 𝑊 )
ssps.h ⊢ 𝐻 = ( SubSp ‘ 𝑈 )
Assertion ssps ( ( 𝑈 ∈ NrmCVec ∧ 𝑊 ∈ 𝐻 ) → 𝑅 = ( 𝑆 ↾ ( ℂ × 𝑌 ) ) )

Proof

Step Hyp Ref Expression
1 ssps.y ⊢ 𝑌 = ( BaseSet ‘ 𝑊 )
2 ssps.s ⊢ 𝑆 = ( ·𝑠OLD ‘ 𝑈 )
3 ssps.r ⊢ 𝑅 = ( ·𝑠OLD ‘ 𝑊 )
4 ssps.h ⊢ 𝐻 = ( SubSp ‘ 𝑈 )
5 eqid ⊢ ( BaseSet ‘ 𝑈 ) = ( BaseSet ‘ 𝑈 )
6 5 2 nvsf ⊢ ( 𝑈 ∈ NrmCVec → 𝑆 : ( ℂ × ( BaseSet ‘ 𝑈 ) ) ⟶ ( BaseSet ‘ 𝑈 ) )
7 6 ffund ⊢ ( 𝑈 ∈ NrmCVec → Fun 𝑆 )
8 7 funresd ⊢ ( 𝑈 ∈ NrmCVec → Fun ( 𝑆 ↾ ( ℂ × 𝑌 ) ) )
9 8 adantr ⊢ ( ( 𝑈 ∈ NrmCVec ∧ 𝑊 ∈ 𝐻 ) → Fun ( 𝑆 ↾ ( ℂ × 𝑌 ) ) )
10 4 sspnv ⊢ ( ( 𝑈 ∈ NrmCVec ∧ 𝑊 ∈ 𝐻 ) → 𝑊 ∈ NrmCVec )
11 1 3 nvsf ⊢ ( 𝑊 ∈ NrmCVec → 𝑅 : ( ℂ × 𝑌 ) ⟶ 𝑌 )
12 10 11 syl ⊢ ( ( 𝑈 ∈ NrmCVec ∧ 𝑊 ∈ 𝐻 ) → 𝑅 : ( ℂ × 𝑌 ) ⟶ 𝑌 )
13 12 ffnd ⊢ ( ( 𝑈 ∈ NrmCVec ∧ 𝑊 ∈ 𝐻 ) → 𝑅 Fn ( ℂ × 𝑌 ) )
14 fnresdm ⊢ ( 𝑅 Fn ( ℂ × 𝑌 ) → ( 𝑅 ↾ ( ℂ × 𝑌 ) ) = 𝑅 )
15 13 14 syl ⊢ ( ( 𝑈 ∈ NrmCVec ∧ 𝑊 ∈ 𝐻 ) → ( 𝑅 ↾ ( ℂ × 𝑌 ) ) = 𝑅 )
16 eqid ⊢ ( +𝑣 ‘ 𝑈 ) = ( +𝑣 ‘ 𝑈 )
17 eqid ⊢ ( +𝑣 ‘ 𝑊 ) = ( +𝑣 ‘ 𝑊 )
18 eqid ⊢ ( normCV ‘ 𝑈 ) = ( normCV ‘ 𝑈 )
19 eqid ⊢ ( normCV ‘ 𝑊 ) = ( normCV ‘ 𝑊 )
20 16 17 2 3 18 19 4 isssp ⊢ ( 𝑈 ∈ NrmCVec → ( 𝑊 ∈ 𝐻 ↔ ( 𝑊 ∈ NrmCVec ∧ ( ( +𝑣 ‘ 𝑊 ) ⊆ ( +𝑣 ‘ 𝑈 ) ∧ 𝑅 ⊆ 𝑆 ∧ ( normCV ‘ 𝑊 ) ⊆ ( normCV ‘ 𝑈 ) ) ) ) )
21 20 simplbda ⊢ ( ( 𝑈 ∈ NrmCVec ∧ 𝑊 ∈ 𝐻 ) → ( ( +𝑣 ‘ 𝑊 ) ⊆ ( +𝑣 ‘ 𝑈 ) ∧ 𝑅 ⊆ 𝑆 ∧ ( normCV ‘ 𝑊 ) ⊆ ( normCV ‘ 𝑈 ) ) )
22 21 simp2d ⊢ ( ( 𝑈 ∈ NrmCVec ∧ 𝑊 ∈ 𝐻 ) → 𝑅 ⊆ 𝑆 )
23 ssres ⊢ ( 𝑅 ⊆ 𝑆 → ( 𝑅 ↾ ( ℂ × 𝑌 ) ) ⊆ ( 𝑆 ↾ ( ℂ × 𝑌 ) ) )
24 22 23 syl ⊢ ( ( 𝑈 ∈ NrmCVec ∧ 𝑊 ∈ 𝐻 ) → ( 𝑅 ↾ ( ℂ × 𝑌 ) ) ⊆ ( 𝑆 ↾ ( ℂ × 𝑌 ) ) )
25 15 24 eqsstrrd ⊢ ( ( 𝑈 ∈ NrmCVec ∧ 𝑊 ∈ 𝐻 ) → 𝑅 ⊆ ( 𝑆 ↾ ( ℂ × 𝑌 ) ) )
26 9 13 25 3jca ⊢ ( ( 𝑈 ∈ NrmCVec ∧ 𝑊 ∈ 𝐻 ) → ( Fun ( 𝑆 ↾ ( ℂ × 𝑌 ) ) ∧ 𝑅 Fn ( ℂ × 𝑌 ) ∧ 𝑅 ⊆ ( 𝑆 ↾ ( ℂ × 𝑌 ) ) ) )
27 oprssov ⊢ ( ( ( Fun ( 𝑆 ↾ ( ℂ × 𝑌 ) ) ∧ 𝑅 Fn ( ℂ × 𝑌 ) ∧ 𝑅 ⊆ ( 𝑆 ↾ ( ℂ × 𝑌 ) ) ) ∧ ( 𝑥 ∈ ℂ ∧ 𝑦 ∈ 𝑌 ) ) → ( 𝑥 ( 𝑆 ↾ ( ℂ × 𝑌 ) ) 𝑦 ) = ( 𝑥 𝑅 𝑦 ) )
28 26 27 sylan ⊢ ( ( ( 𝑈 ∈ NrmCVec ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑥 ∈ ℂ ∧ 𝑦 ∈ 𝑌 ) ) → ( 𝑥 ( 𝑆 ↾ ( ℂ × 𝑌 ) ) 𝑦 ) = ( 𝑥 𝑅 𝑦 ) )
29 28 eqcomd ⊢ ( ( ( 𝑈 ∈ NrmCVec ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑥 ∈ ℂ ∧ 𝑦 ∈ 𝑌 ) ) → ( 𝑥 𝑅 𝑦 ) = ( 𝑥 ( 𝑆 ↾ ( ℂ × 𝑌 ) ) 𝑦 ) )
30 29 ralrimivva ⊢ ( ( 𝑈 ∈ NrmCVec ∧ 𝑊 ∈ 𝐻 ) → ∀ 𝑥 ∈ ℂ ∀ 𝑦 ∈ 𝑌 ( 𝑥 𝑅 𝑦 ) = ( 𝑥 ( 𝑆 ↾ ( ℂ × 𝑌 ) ) 𝑦 ) )
31 eqid ⊢ ( ℂ × 𝑌 ) = ( ℂ × 𝑌 )
32 30 31 jctil ⊢ ( ( 𝑈 ∈ NrmCVec ∧ 𝑊 ∈ 𝐻 ) → ( ( ℂ × 𝑌 ) = ( ℂ × 𝑌 ) ∧ ∀ 𝑥 ∈ ℂ ∀ 𝑦 ∈ 𝑌 ( 𝑥 𝑅 𝑦 ) = ( 𝑥 ( 𝑆 ↾ ( ℂ × 𝑌 ) ) 𝑦 ) ) )
33 6 ffnd ⊢ ( 𝑈 ∈ NrmCVec → 𝑆 Fn ( ℂ × ( BaseSet ‘ 𝑈 ) ) )
34 33 adantr ⊢ ( ( 𝑈 ∈ NrmCVec ∧ 𝑊 ∈ 𝐻 ) → 𝑆 Fn ( ℂ × ( BaseSet ‘ 𝑈 ) ) )
35 ssid ⊢ ℂ ⊆ ℂ
36 5 1 4 sspba ⊢ ( ( 𝑈 ∈ NrmCVec ∧ 𝑊 ∈ 𝐻 ) → 𝑌 ⊆ ( BaseSet ‘ 𝑈 ) )
37 xpss12 ⊢ ( ( ℂ ⊆ ℂ ∧ 𝑌 ⊆ ( BaseSet ‘ 𝑈 ) ) → ( ℂ × 𝑌 ) ⊆ ( ℂ × ( BaseSet ‘ 𝑈 ) ) )
38 35 36 37 sylancr ⊢ ( ( 𝑈 ∈ NrmCVec ∧ 𝑊 ∈ 𝐻 ) → ( ℂ × 𝑌 ) ⊆ ( ℂ × ( BaseSet ‘ 𝑈 ) ) )
39 fnssres ⊢ ( ( 𝑆 Fn ( ℂ × ( BaseSet ‘ 𝑈 ) ) ∧ ( ℂ × 𝑌 ) ⊆ ( ℂ × ( BaseSet ‘ 𝑈 ) ) ) → ( 𝑆 ↾ ( ℂ × 𝑌 ) ) Fn ( ℂ × 𝑌 ) )
40 34 38 39 syl2anc ⊢ ( ( 𝑈 ∈ NrmCVec ∧ 𝑊 ∈ 𝐻 ) → ( 𝑆 ↾ ( ℂ × 𝑌 ) ) Fn ( ℂ × 𝑌 ) )
41 eqfnov ⊢ ( ( 𝑅 Fn ( ℂ × 𝑌 ) ∧ ( 𝑆 ↾ ( ℂ × 𝑌 ) ) Fn ( ℂ × 𝑌 ) ) → ( 𝑅 = ( 𝑆 ↾ ( ℂ × 𝑌 ) ) ↔ ( ( ℂ × 𝑌 ) = ( ℂ × 𝑌 ) ∧ ∀ 𝑥 ∈ ℂ ∀ 𝑦 ∈ 𝑌 ( 𝑥 𝑅 𝑦 ) = ( 𝑥 ( 𝑆 ↾ ( ℂ × 𝑌 ) ) 𝑦 ) ) ) )
42 13 40 41 syl2anc ⊢ ( ( 𝑈 ∈ NrmCVec ∧ 𝑊 ∈ 𝐻 ) → ( 𝑅 = ( 𝑆 ↾ ( ℂ × 𝑌 ) ) ↔ ( ( ℂ × 𝑌 ) = ( ℂ × 𝑌 ) ∧ ∀ 𝑥 ∈ ℂ ∀ 𝑦 ∈ 𝑌 ( 𝑥 𝑅 𝑦 ) = ( 𝑥 ( 𝑆 ↾ ( ℂ × 𝑌 ) ) 𝑦 ) ) ) )
43 32 42 mpbird ⊢ ( ( 𝑈 ∈ NrmCVec ∧ 𝑊 ∈ 𝐻 ) → 𝑅 = ( 𝑆 ↾ ( ℂ × 𝑌 ) ) )