This is an inofficial mirror of http://metamath.tirix.org for personal testing of a visualizer extension only.

Metamath Proof Explorer

Theorem sspg

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

Ref Expression
Hypotheses sspg.y ⊢ 𝑌 = ( BaseSet ‘ 𝑊 )
sspg.g ⊢ 𝐺 = ( +𝑣 ‘ 𝑈 )
sspg.f ⊢ 𝐹 = ( +𝑣 ‘ 𝑊 )
sspg.h ⊢ 𝐻 = ( SubSp ‘ 𝑈 )
Assertion sspg ( ( 𝑈 ∈ NrmCVec ∧ 𝑊 ∈ 𝐻 ) → 𝐹 = ( 𝐺 ↾ ( 𝑌 × 𝑌 ) ) )

Proof

Step Hyp Ref Expression
1 sspg.y ⊢ 𝑌 = ( BaseSet ‘ 𝑊 )
2 sspg.g ⊢ 𝐺 = ( +𝑣 ‘ 𝑈 )
3 sspg.f ⊢ 𝐹 = ( +𝑣 ‘ 𝑊 )
4 sspg.h ⊢ 𝐻 = ( SubSp ‘ 𝑈 )
5 eqid ⊢ ( BaseSet ‘ 𝑈 ) = ( BaseSet ‘ 𝑈 )
6 5 2 nvgf ⊢ ( 𝑈 ∈ NrmCVec → 𝐺 : ( ( BaseSet ‘ 𝑈 ) × ( BaseSet ‘ 𝑈 ) ) ⟶ ( BaseSet ‘ 𝑈 ) )
7 6 ffund ⊢ ( 𝑈 ∈ NrmCVec → Fun 𝐺 )
8 7 funresd ⊢ ( 𝑈 ∈ NrmCVec → Fun ( 𝐺 ↾ ( 𝑌 × 𝑌 ) ) )
9 8 adantr ⊢ ( ( 𝑈 ∈ NrmCVec ∧ 𝑊 ∈ 𝐻 ) → Fun ( 𝐺 ↾ ( 𝑌 × 𝑌 ) ) )
10 4 sspnv ⊢ ( ( 𝑈 ∈ NrmCVec ∧ 𝑊 ∈ 𝐻 ) → 𝑊 ∈ NrmCVec )
11 1 3 nvgf ⊢ ( 𝑊 ∈ NrmCVec → 𝐹 : ( 𝑌 × 𝑌 ) ⟶ 𝑌 )
12 10 11 syl ⊢ ( ( 𝑈 ∈ NrmCVec ∧ 𝑊 ∈ 𝐻 ) → 𝐹 : ( 𝑌 × 𝑌 ) ⟶ 𝑌 )
13 12 ffnd ⊢ ( ( 𝑈 ∈ NrmCVec ∧ 𝑊 ∈ 𝐻 ) → 𝐹 Fn ( 𝑌 × 𝑌 ) )
14 fnresdm ⊢ ( 𝐹 Fn ( 𝑌 × 𝑌 ) → ( 𝐹 ↾ ( 𝑌 × 𝑌 ) ) = 𝐹 )
15 13 14 syl ⊢ ( ( 𝑈 ∈ NrmCVec ∧ 𝑊 ∈ 𝐻 ) → ( 𝐹 ↾ ( 𝑌 × 𝑌 ) ) = 𝐹 )
16 eqid ⊢ ( ·𝑠OLD ‘ 𝑈 ) = ( ·𝑠OLD ‘ 𝑈 )
17 eqid ⊢ ( ·𝑠OLD ‘ 𝑊 ) = ( ·𝑠OLD ‘ 𝑊 )
18 eqid ⊢ ( normCV ‘ 𝑈 ) = ( normCV ‘ 𝑈 )
19 eqid ⊢ ( normCV ‘ 𝑊 ) = ( normCV ‘ 𝑊 )
20 2 3 16 17 18 19 4 isssp ⊢ ( 𝑈 ∈ NrmCVec → ( 𝑊 ∈ 𝐻 ↔ ( 𝑊 ∈ NrmCVec ∧ ( 𝐹 ⊆ 𝐺 ∧ ( ·𝑠OLD ‘ 𝑊 ) ⊆ ( ·𝑠OLD ‘ 𝑈 ) ∧ ( normCV ‘ 𝑊 ) ⊆ ( normCV ‘ 𝑈 ) ) ) ) )
21 20 simplbda ⊢ ( ( 𝑈 ∈ NrmCVec ∧ 𝑊 ∈ 𝐻 ) → ( 𝐹 ⊆ 𝐺 ∧ ( ·𝑠OLD ‘ 𝑊 ) ⊆ ( ·𝑠OLD ‘ 𝑈 ) ∧ ( normCV ‘ 𝑊 ) ⊆ ( normCV ‘ 𝑈 ) ) )
22 21 simp1d ⊢ ( ( 𝑈 ∈ NrmCVec ∧ 𝑊 ∈ 𝐻 ) → 𝐹 ⊆ 𝐺 )
23 ssres ⊢ ( 𝐹 ⊆ 𝐺 → ( 𝐹 ↾ ( 𝑌 × 𝑌 ) ) ⊆ ( 𝐺 ↾ ( 𝑌 × 𝑌 ) ) )
24 22 23 syl ⊢ ( ( 𝑈 ∈ NrmCVec ∧ 𝑊 ∈ 𝐻 ) → ( 𝐹 ↾ ( 𝑌 × 𝑌 ) ) ⊆ ( 𝐺 ↾ ( 𝑌 × 𝑌 ) ) )
25 15 24 eqsstrrd ⊢ ( ( 𝑈 ∈ NrmCVec ∧ 𝑊 ∈ 𝐻 ) → 𝐹 ⊆ ( 𝐺 ↾ ( 𝑌 × 𝑌 ) ) )
26 9 13 25 3jca ⊢ ( ( 𝑈 ∈ NrmCVec ∧ 𝑊 ∈ 𝐻 ) → ( Fun ( 𝐺 ↾ ( 𝑌 × 𝑌 ) ) ∧ 𝐹 Fn ( 𝑌 × 𝑌 ) ∧ 𝐹 ⊆ ( 𝐺 ↾ ( 𝑌 × 𝑌 ) ) ) )
27 oprssov ⊢ ( ( ( Fun ( 𝐺 ↾ ( 𝑌 × 𝑌 ) ) ∧ 𝐹 Fn ( 𝑌 × 𝑌 ) ∧ 𝐹 ⊆ ( 𝐺 ↾ ( 𝑌 × 𝑌 ) ) ) ∧ ( 𝑥 ∈ 𝑌 ∧ 𝑦 ∈ 𝑌 ) ) → ( 𝑥 ( 𝐺 ↾ ( 𝑌 × 𝑌 ) ) 𝑦 ) = ( 𝑥 𝐹 𝑦 ) )
28 26 27 sylan ⊢ ( ( ( 𝑈 ∈ NrmCVec ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑥 ∈ 𝑌 ∧ 𝑦 ∈ 𝑌 ) ) → ( 𝑥 ( 𝐺 ↾ ( 𝑌 × 𝑌 ) ) 𝑦 ) = ( 𝑥 𝐹 𝑦 ) )
29 28 eqcomd ⊢ ( ( ( 𝑈 ∈ NrmCVec ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑥 ∈ 𝑌 ∧ 𝑦 ∈ 𝑌 ) ) → ( 𝑥 𝐹 𝑦 ) = ( 𝑥 ( 𝐺 ↾ ( 𝑌 × 𝑌 ) ) 𝑦 ) )
30 29 ralrimivva ⊢ ( ( 𝑈 ∈ NrmCVec ∧ 𝑊 ∈ 𝐻 ) → ∀ 𝑥 ∈ 𝑌 ∀ 𝑦 ∈ 𝑌 ( 𝑥 𝐹 𝑦 ) = ( 𝑥 ( 𝐺 ↾ ( 𝑌 × 𝑌 ) ) 𝑦 ) )
31 eqid ⊢ ( 𝑌 × 𝑌 ) = ( 𝑌 × 𝑌 )
32 30 31 jctil ⊢ ( ( 𝑈 ∈ NrmCVec ∧ 𝑊 ∈ 𝐻 ) → ( ( 𝑌 × 𝑌 ) = ( 𝑌 × 𝑌 ) ∧ ∀ 𝑥 ∈ 𝑌 ∀ 𝑦 ∈ 𝑌 ( 𝑥 𝐹 𝑦 ) = ( 𝑥 ( 𝐺 ↾ ( 𝑌 × 𝑌 ) ) 𝑦 ) ) )
33 6 ffnd ⊢ ( 𝑈 ∈ NrmCVec → 𝐺 Fn ( ( BaseSet ‘ 𝑈 ) × ( BaseSet ‘ 𝑈 ) ) )
34 33 adantr ⊢ ( ( 𝑈 ∈ NrmCVec ∧ 𝑊 ∈ 𝐻 ) → 𝐺 Fn ( ( BaseSet ‘ 𝑈 ) × ( BaseSet ‘ 𝑈 ) ) )
35 5 1 4 sspba ⊢ ( ( 𝑈 ∈ NrmCVec ∧ 𝑊 ∈ 𝐻 ) → 𝑌 ⊆ ( BaseSet ‘ 𝑈 ) )
36 xpss12 ⊢ ( ( 𝑌 ⊆ ( BaseSet ‘ 𝑈 ) ∧ 𝑌 ⊆ ( BaseSet ‘ 𝑈 ) ) → ( 𝑌 × 𝑌 ) ⊆ ( ( BaseSet ‘ 𝑈 ) × ( BaseSet ‘ 𝑈 ) ) )
37 35 35 36 syl2anc ⊢ ( ( 𝑈 ∈ NrmCVec ∧ 𝑊 ∈ 𝐻 ) → ( 𝑌 × 𝑌 ) ⊆ ( ( BaseSet ‘ 𝑈 ) × ( BaseSet ‘ 𝑈 ) ) )
38 fnssres ⊢ ( ( 𝐺 Fn ( ( BaseSet ‘ 𝑈 ) × ( BaseSet ‘ 𝑈 ) ) ∧ ( 𝑌 × 𝑌 ) ⊆ ( ( BaseSet ‘ 𝑈 ) × ( BaseSet ‘ 𝑈 ) ) ) → ( 𝐺 ↾ ( 𝑌 × 𝑌 ) ) Fn ( 𝑌 × 𝑌 ) )
39 34 37 38 syl2anc ⊢ ( ( 𝑈 ∈ NrmCVec ∧ 𝑊 ∈ 𝐻 ) → ( 𝐺 ↾ ( 𝑌 × 𝑌 ) ) Fn ( 𝑌 × 𝑌 ) )
40 eqfnov ⊢ ( ( 𝐹 Fn ( 𝑌 × 𝑌 ) ∧ ( 𝐺 ↾ ( 𝑌 × 𝑌 ) ) Fn ( 𝑌 × 𝑌 ) ) → ( 𝐹 = ( 𝐺 ↾ ( 𝑌 × 𝑌 ) ) ↔ ( ( 𝑌 × 𝑌 ) = ( 𝑌 × 𝑌 ) ∧ ∀ 𝑥 ∈ 𝑌 ∀ 𝑦 ∈ 𝑌 ( 𝑥 𝐹 𝑦 ) = ( 𝑥 ( 𝐺 ↾ ( 𝑌 × 𝑌 ) ) 𝑦 ) ) ) )
41 13 39 40 syl2anc ⊢ ( ( 𝑈 ∈ NrmCVec ∧ 𝑊 ∈ 𝐻 ) → ( 𝐹 = ( 𝐺 ↾ ( 𝑌 × 𝑌 ) ) ↔ ( ( 𝑌 × 𝑌 ) = ( 𝑌 × 𝑌 ) ∧ ∀ 𝑥 ∈ 𝑌 ∀ 𝑦 ∈ 𝑌 ( 𝑥 𝐹 𝑦 ) = ( 𝑥 ( 𝐺 ↾ ( 𝑌 × 𝑌 ) ) 𝑦 ) ) ) )
42 32 41 mpbird ⊢ ( ( 𝑈 ∈ NrmCVec ∧ 𝑊 ∈ 𝐻 ) → 𝐹 = ( 𝐺 ↾ ( 𝑌 × 𝑌 ) ) )