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 |
|
funres |
⊢ ( Fun 𝑆 → Fun ( 𝑆 ↾ ( ℂ × 𝑌 ) ) ) |
9 |
7 8
|
syl |
⊢ ( 𝑈 ∈ NrmCVec → Fun ( 𝑆 ↾ ( ℂ × 𝑌 ) ) ) |
10 |
9
|
adantr |
⊢ ( ( 𝑈 ∈ NrmCVec ∧ 𝑊 ∈ 𝐻 ) → Fun ( 𝑆 ↾ ( ℂ × 𝑌 ) ) ) |
11 |
4
|
sspnv |
⊢ ( ( 𝑈 ∈ NrmCVec ∧ 𝑊 ∈ 𝐻 ) → 𝑊 ∈ NrmCVec ) |
12 |
1 3
|
nvsf |
⊢ ( 𝑊 ∈ NrmCVec → 𝑅 : ( ℂ × 𝑌 ) ⟶ 𝑌 ) |
13 |
11 12
|
syl |
⊢ ( ( 𝑈 ∈ NrmCVec ∧ 𝑊 ∈ 𝐻 ) → 𝑅 : ( ℂ × 𝑌 ) ⟶ 𝑌 ) |
14 |
13
|
ffnd |
⊢ ( ( 𝑈 ∈ NrmCVec ∧ 𝑊 ∈ 𝐻 ) → 𝑅 Fn ( ℂ × 𝑌 ) ) |
15 |
|
fnresdm |
⊢ ( 𝑅 Fn ( ℂ × 𝑌 ) → ( 𝑅 ↾ ( ℂ × 𝑌 ) ) = 𝑅 ) |
16 |
14 15
|
syl |
⊢ ( ( 𝑈 ∈ NrmCVec ∧ 𝑊 ∈ 𝐻 ) → ( 𝑅 ↾ ( ℂ × 𝑌 ) ) = 𝑅 ) |
17 |
|
eqid |
⊢ ( +𝑣 ‘ 𝑈 ) = ( +𝑣 ‘ 𝑈 ) |
18 |
|
eqid |
⊢ ( +𝑣 ‘ 𝑊 ) = ( +𝑣 ‘ 𝑊 ) |
19 |
|
eqid |
⊢ ( normCV ‘ 𝑈 ) = ( normCV ‘ 𝑈 ) |
20 |
|
eqid |
⊢ ( normCV ‘ 𝑊 ) = ( normCV ‘ 𝑊 ) |
21 |
17 18 2 3 19 20 4
|
isssp |
⊢ ( 𝑈 ∈ NrmCVec → ( 𝑊 ∈ 𝐻 ↔ ( 𝑊 ∈ NrmCVec ∧ ( ( +𝑣 ‘ 𝑊 ) ⊆ ( +𝑣 ‘ 𝑈 ) ∧ 𝑅 ⊆ 𝑆 ∧ ( normCV ‘ 𝑊 ) ⊆ ( normCV ‘ 𝑈 ) ) ) ) ) |
22 |
21
|
simplbda |
⊢ ( ( 𝑈 ∈ NrmCVec ∧ 𝑊 ∈ 𝐻 ) → ( ( +𝑣 ‘ 𝑊 ) ⊆ ( +𝑣 ‘ 𝑈 ) ∧ 𝑅 ⊆ 𝑆 ∧ ( normCV ‘ 𝑊 ) ⊆ ( normCV ‘ 𝑈 ) ) ) |
23 |
22
|
simp2d |
⊢ ( ( 𝑈 ∈ NrmCVec ∧ 𝑊 ∈ 𝐻 ) → 𝑅 ⊆ 𝑆 ) |
24 |
|
ssres |
⊢ ( 𝑅 ⊆ 𝑆 → ( 𝑅 ↾ ( ℂ × 𝑌 ) ) ⊆ ( 𝑆 ↾ ( ℂ × 𝑌 ) ) ) |
25 |
23 24
|
syl |
⊢ ( ( 𝑈 ∈ NrmCVec ∧ 𝑊 ∈ 𝐻 ) → ( 𝑅 ↾ ( ℂ × 𝑌 ) ) ⊆ ( 𝑆 ↾ ( ℂ × 𝑌 ) ) ) |
26 |
16 25
|
eqsstrrd |
⊢ ( ( 𝑈 ∈ NrmCVec ∧ 𝑊 ∈ 𝐻 ) → 𝑅 ⊆ ( 𝑆 ↾ ( ℂ × 𝑌 ) ) ) |
27 |
10 14 26
|
3jca |
⊢ ( ( 𝑈 ∈ NrmCVec ∧ 𝑊 ∈ 𝐻 ) → ( Fun ( 𝑆 ↾ ( ℂ × 𝑌 ) ) ∧ 𝑅 Fn ( ℂ × 𝑌 ) ∧ 𝑅 ⊆ ( 𝑆 ↾ ( ℂ × 𝑌 ) ) ) ) |
28 |
|
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 ) ) |
29 |
27 28
|
sylan |
⊢ ( ( ( 𝑈 ∈ NrmCVec ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑥 ∈ ℂ ∧ 𝑦 ∈ 𝑌 ) ) → ( 𝑥 ( 𝑆 ↾ ( ℂ × 𝑌 ) ) 𝑦 ) = ( 𝑥 𝑅 𝑦 ) ) |
30 |
29
|
eqcomd |
⊢ ( ( ( 𝑈 ∈ NrmCVec ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑥 ∈ ℂ ∧ 𝑦 ∈ 𝑌 ) ) → ( 𝑥 𝑅 𝑦 ) = ( 𝑥 ( 𝑆 ↾ ( ℂ × 𝑌 ) ) 𝑦 ) ) |
31 |
30
|
ralrimivva |
⊢ ( ( 𝑈 ∈ NrmCVec ∧ 𝑊 ∈ 𝐻 ) → ∀ 𝑥 ∈ ℂ ∀ 𝑦 ∈ 𝑌 ( 𝑥 𝑅 𝑦 ) = ( 𝑥 ( 𝑆 ↾ ( ℂ × 𝑌 ) ) 𝑦 ) ) |
32 |
|
eqid |
⊢ ( ℂ × 𝑌 ) = ( ℂ × 𝑌 ) |
33 |
31 32
|
jctil |
⊢ ( ( 𝑈 ∈ NrmCVec ∧ 𝑊 ∈ 𝐻 ) → ( ( ℂ × 𝑌 ) = ( ℂ × 𝑌 ) ∧ ∀ 𝑥 ∈ ℂ ∀ 𝑦 ∈ 𝑌 ( 𝑥 𝑅 𝑦 ) = ( 𝑥 ( 𝑆 ↾ ( ℂ × 𝑌 ) ) 𝑦 ) ) ) |
34 |
6
|
ffnd |
⊢ ( 𝑈 ∈ NrmCVec → 𝑆 Fn ( ℂ × ( BaseSet ‘ 𝑈 ) ) ) |
35 |
34
|
adantr |
⊢ ( ( 𝑈 ∈ NrmCVec ∧ 𝑊 ∈ 𝐻 ) → 𝑆 Fn ( ℂ × ( BaseSet ‘ 𝑈 ) ) ) |
36 |
|
ssid |
⊢ ℂ ⊆ ℂ |
37 |
5 1 4
|
sspba |
⊢ ( ( 𝑈 ∈ NrmCVec ∧ 𝑊 ∈ 𝐻 ) → 𝑌 ⊆ ( BaseSet ‘ 𝑈 ) ) |
38 |
|
xpss12 |
⊢ ( ( ℂ ⊆ ℂ ∧ 𝑌 ⊆ ( BaseSet ‘ 𝑈 ) ) → ( ℂ × 𝑌 ) ⊆ ( ℂ × ( BaseSet ‘ 𝑈 ) ) ) |
39 |
36 37 38
|
sylancr |
⊢ ( ( 𝑈 ∈ NrmCVec ∧ 𝑊 ∈ 𝐻 ) → ( ℂ × 𝑌 ) ⊆ ( ℂ × ( BaseSet ‘ 𝑈 ) ) ) |
40 |
|
fnssres |
⊢ ( ( 𝑆 Fn ( ℂ × ( BaseSet ‘ 𝑈 ) ) ∧ ( ℂ × 𝑌 ) ⊆ ( ℂ × ( BaseSet ‘ 𝑈 ) ) ) → ( 𝑆 ↾ ( ℂ × 𝑌 ) ) Fn ( ℂ × 𝑌 ) ) |
41 |
35 39 40
|
syl2anc |
⊢ ( ( 𝑈 ∈ NrmCVec ∧ 𝑊 ∈ 𝐻 ) → ( 𝑆 ↾ ( ℂ × 𝑌 ) ) Fn ( ℂ × 𝑌 ) ) |
42 |
|
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 ) ) ) ) |
43 |
14 41 42
|
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 ) ) ) ) |
44 |
33 43
|
mpbird |
⊢ ( ( 𝑈 ∈ NrmCVec ∧ 𝑊 ∈ 𝐻 ) → 𝑅 = ( 𝑆 ↾ ( ℂ × 𝑌 ) ) ) |