| Step | Hyp | Ref
| Expression |
| 1 | | eqid 2737 |
. . . . . . . . . . 11
⊢
(BaseSet‘𝑈) =
(BaseSet‘𝑈) |
| 2 | | ssps.s |
. . . . . . . . . . 11
⊢ 𝑆 = (
·𝑠OLD ‘𝑈) |
| 3 | 1, 2 | nvsf 30638 |
. . . . . . . . . 10
⊢ (𝑈 ∈ NrmCVec → 𝑆:(ℂ ×
(BaseSet‘𝑈))⟶(BaseSet‘𝑈)) |
| 4 | 3 | ffund 6740 |
. . . . . . . . 9
⊢ (𝑈 ∈ NrmCVec → Fun 𝑆) |
| 5 | 4 | funresd 6609 |
. . . . . . . 8
⊢ (𝑈 ∈ NrmCVec → Fun
(𝑆 ↾ (ℂ ×
𝑌))) |
| 6 | 5 | adantr 480 |
. . . . . . 7
⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ 𝐻) → Fun (𝑆 ↾ (ℂ × 𝑌))) |
| 7 | | ssps.h |
. . . . . . . . . 10
⊢ 𝐻 = (SubSp‘𝑈) |
| 8 | 7 | sspnv 30745 |
. . . . . . . . 9
⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ 𝐻) → 𝑊 ∈ NrmCVec) |
| 9 | | ssps.y |
. . . . . . . . . 10
⊢ 𝑌 = (BaseSet‘𝑊) |
| 10 | | ssps.r |
. . . . . . . . . 10
⊢ 𝑅 = (
·𝑠OLD ‘𝑊) |
| 11 | 9, 10 | nvsf 30638 |
. . . . . . . . 9
⊢ (𝑊 ∈ NrmCVec → 𝑅:(ℂ × 𝑌)⟶𝑌) |
| 12 | 8, 11 | syl 17 |
. . . . . . . 8
⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ 𝐻) → 𝑅:(ℂ × 𝑌)⟶𝑌) |
| 13 | 12 | ffnd 6737 |
. . . . . . 7
⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ 𝐻) → 𝑅 Fn (ℂ × 𝑌)) |
| 14 | | fnresdm 6687 |
. . . . . . . . 9
⊢ (𝑅 Fn (ℂ × 𝑌) → (𝑅 ↾ (ℂ × 𝑌)) = 𝑅) |
| 15 | 13, 14 | syl 17 |
. . . . . . . 8
⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ 𝐻) → (𝑅 ↾ (ℂ × 𝑌)) = 𝑅) |
| 16 | | eqid 2737 |
. . . . . . . . . . . 12
⊢ (
+𝑣 ‘𝑈) = ( +𝑣 ‘𝑈) |
| 17 | | eqid 2737 |
. . . . . . . . . . . 12
⊢ (
+𝑣 ‘𝑊) = ( +𝑣 ‘𝑊) |
| 18 | | eqid 2737 |
. . . . . . . . . . . 12
⊢
(normCV‘𝑈) = (normCV‘𝑈) |
| 19 | | eqid 2737 |
. . . . . . . . . . . 12
⊢
(normCV‘𝑊) = (normCV‘𝑊) |
| 20 | 16, 17, 2, 10, 18, 19, 7 | isssp 30743 |
. . . . . . . . . . 11
⊢ (𝑈 ∈ NrmCVec → (𝑊 ∈ 𝐻 ↔ (𝑊 ∈ NrmCVec ∧ ((
+𝑣 ‘𝑊) ⊆ ( +𝑣
‘𝑈) ∧ 𝑅 ⊆ 𝑆 ∧ (normCV‘𝑊) ⊆
(normCV‘𝑈))))) |
| 21 | 20 | simplbda 499 |
. . . . . . . . . 10
⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ 𝐻) → (( +𝑣
‘𝑊) ⊆ (
+𝑣 ‘𝑈) ∧ 𝑅 ⊆ 𝑆 ∧ (normCV‘𝑊) ⊆
(normCV‘𝑈))) |
| 22 | 21 | simp2d 1144 |
. . . . . . . . 9
⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ 𝐻) → 𝑅 ⊆ 𝑆) |
| 23 | | ssres 6021 |
. . . . . . . . 9
⊢ (𝑅 ⊆ 𝑆 → (𝑅 ↾ (ℂ × 𝑌)) ⊆ (𝑆 ↾ (ℂ × 𝑌))) |
| 24 | 22, 23 | syl 17 |
. . . . . . . 8
⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ 𝐻) → (𝑅 ↾ (ℂ × 𝑌)) ⊆ (𝑆 ↾ (ℂ × 𝑌))) |
| 25 | 15, 24 | eqsstrrd 4019 |
. . . . . . 7
⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ 𝐻) → 𝑅 ⊆ (𝑆 ↾ (ℂ × 𝑌))) |
| 26 | 6, 13, 25 | 3jca 1129 |
. . . . . 6
⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ 𝐻) → (Fun (𝑆 ↾ (ℂ × 𝑌)) ∧ 𝑅 Fn (ℂ × 𝑌) ∧ 𝑅 ⊆ (𝑆 ↾ (ℂ × 𝑌)))) |
| 27 | | oprssov 7602 |
. . . . . 6
⊢ (((Fun
(𝑆 ↾ (ℂ ×
𝑌)) ∧ 𝑅 Fn (ℂ × 𝑌) ∧ 𝑅 ⊆ (𝑆 ↾ (ℂ × 𝑌))) ∧ (𝑥 ∈ ℂ ∧ 𝑦 ∈ 𝑌)) → (𝑥(𝑆 ↾ (ℂ × 𝑌))𝑦) = (𝑥𝑅𝑦)) |
| 28 | 26, 27 | sylan 580 |
. . . . 5
⊢ (((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ 𝐻) ∧ (𝑥 ∈ ℂ ∧ 𝑦 ∈ 𝑌)) → (𝑥(𝑆 ↾ (ℂ × 𝑌))𝑦) = (𝑥𝑅𝑦)) |
| 29 | 28 | eqcomd 2743 |
. . . 4
⊢ (((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ 𝐻) ∧ (𝑥 ∈ ℂ ∧ 𝑦 ∈ 𝑌)) → (𝑥𝑅𝑦) = (𝑥(𝑆 ↾ (ℂ × 𝑌))𝑦)) |
| 30 | 29 | ralrimivva 3202 |
. . 3
⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ 𝐻) → ∀𝑥 ∈ ℂ ∀𝑦 ∈ 𝑌 (𝑥𝑅𝑦) = (𝑥(𝑆 ↾ (ℂ × 𝑌))𝑦)) |
| 31 | | eqid 2737 |
. . 3
⊢ (ℂ
× 𝑌) = (ℂ
× 𝑌) |
| 32 | 30, 31 | jctil 519 |
. 2
⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ 𝐻) → ((ℂ × 𝑌) = (ℂ × 𝑌) ∧ ∀𝑥 ∈ ℂ ∀𝑦 ∈ 𝑌 (𝑥𝑅𝑦) = (𝑥(𝑆 ↾ (ℂ × 𝑌))𝑦))) |
| 33 | 3 | ffnd 6737 |
. . . . 5
⊢ (𝑈 ∈ NrmCVec → 𝑆 Fn (ℂ ×
(BaseSet‘𝑈))) |
| 34 | 33 | adantr 480 |
. . . 4
⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ 𝐻) → 𝑆 Fn (ℂ × (BaseSet‘𝑈))) |
| 35 | | ssid 4006 |
. . . . 5
⊢ ℂ
⊆ ℂ |
| 36 | 1, 9, 7 | sspba 30746 |
. . . . 5
⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ 𝐻) → 𝑌 ⊆ (BaseSet‘𝑈)) |
| 37 | | xpss12 5700 |
. . . . 5
⊢ ((ℂ
⊆ ℂ ∧ 𝑌
⊆ (BaseSet‘𝑈))
→ (ℂ × 𝑌)
⊆ (ℂ × (BaseSet‘𝑈))) |
| 38 | 35, 36, 37 | sylancr 587 |
. . . 4
⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ 𝐻) → (ℂ × 𝑌) ⊆ (ℂ ×
(BaseSet‘𝑈))) |
| 39 | | fnssres 6691 |
. . . 4
⊢ ((𝑆 Fn (ℂ ×
(BaseSet‘𝑈)) ∧
(ℂ × 𝑌) ⊆
(ℂ × (BaseSet‘𝑈))) → (𝑆 ↾ (ℂ × 𝑌)) Fn (ℂ × 𝑌)) |
| 40 | 34, 38, 39 | syl2anc 584 |
. . 3
⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ 𝐻) → (𝑆 ↾ (ℂ × 𝑌)) Fn (ℂ × 𝑌)) |
| 41 | | eqfnov 7562 |
. . 3
⊢ ((𝑅 Fn (ℂ × 𝑌) ∧ (𝑆 ↾ (ℂ × 𝑌)) Fn (ℂ × 𝑌)) → (𝑅 = (𝑆 ↾ (ℂ × 𝑌)) ↔ ((ℂ × 𝑌) = (ℂ × 𝑌) ∧ ∀𝑥 ∈ ℂ ∀𝑦 ∈ 𝑌 (𝑥𝑅𝑦) = (𝑥(𝑆 ↾ (ℂ × 𝑌))𝑦)))) |
| 42 | 13, 40, 41 | syl2anc 584 |
. 2
⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ 𝐻) → (𝑅 = (𝑆 ↾ (ℂ × 𝑌)) ↔ ((ℂ × 𝑌) = (ℂ × 𝑌) ∧ ∀𝑥 ∈ ℂ ∀𝑦 ∈ 𝑌 (𝑥𝑅𝑦) = (𝑥(𝑆 ↾ (ℂ × 𝑌))𝑦)))) |
| 43 | 32, 42 | mpbird 257 |
1
⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ 𝐻) → 𝑅 = (𝑆 ↾ (ℂ × 𝑌))) |