| Step | Hyp | Ref | Expression | 
|---|
| 1 |  | sspn.h | . . . . 5
⊢ 𝐻 = (SubSp‘𝑈) | 
| 2 | 1 | sspnv 30745 | . . . 4
⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ 𝐻) → 𝑊 ∈ NrmCVec) | 
| 3 |  | sspn.y | . . . . 5
⊢ 𝑌 = (BaseSet‘𝑊) | 
| 4 |  | sspn.m | . . . . 5
⊢ 𝑀 =
(normCV‘𝑊) | 
| 5 | 3, 4 | nvf 30679 | . . . 4
⊢ (𝑊 ∈ NrmCVec → 𝑀:𝑌⟶ℝ) | 
| 6 | 2, 5 | syl 17 | . . 3
⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ 𝐻) → 𝑀:𝑌⟶ℝ) | 
| 7 | 6 | ffnd 6737 | . 2
⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ 𝐻) → 𝑀 Fn 𝑌) | 
| 8 |  | eqid 2737 | . . . . . 6
⊢
(BaseSet‘𝑈) =
(BaseSet‘𝑈) | 
| 9 |  | sspn.n | . . . . . 6
⊢ 𝑁 =
(normCV‘𝑈) | 
| 10 | 8, 9 | nvf 30679 | . . . . 5
⊢ (𝑈 ∈ NrmCVec → 𝑁:(BaseSet‘𝑈)⟶ℝ) | 
| 11 | 10 | ffnd 6737 | . . . 4
⊢ (𝑈 ∈ NrmCVec → 𝑁 Fn (BaseSet‘𝑈)) | 
| 12 | 11 | adantr 480 | . . 3
⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ 𝐻) → 𝑁 Fn (BaseSet‘𝑈)) | 
| 13 | 8, 3, 1 | sspba 30746 | . . 3
⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ 𝐻) → 𝑌 ⊆ (BaseSet‘𝑈)) | 
| 14 |  | fnssres 6691 | . . 3
⊢ ((𝑁 Fn (BaseSet‘𝑈) ∧ 𝑌 ⊆ (BaseSet‘𝑈)) → (𝑁 ↾ 𝑌) Fn 𝑌) | 
| 15 | 12, 13, 14 | syl2anc 584 | . 2
⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ 𝐻) → (𝑁 ↾ 𝑌) Fn 𝑌) | 
| 16 | 10 | ffund 6740 | . . . . . 6
⊢ (𝑈 ∈ NrmCVec → Fun 𝑁) | 
| 17 | 16 | funresd 6609 | . . . . 5
⊢ (𝑈 ∈ NrmCVec → Fun
(𝑁 ↾ 𝑌)) | 
| 18 | 17 | ad2antrr 726 | . . . 4
⊢ (((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ 𝐻) ∧ 𝑥 ∈ 𝑌) → Fun (𝑁 ↾ 𝑌)) | 
| 19 |  | fnresdm 6687 | . . . . . . 7
⊢ (𝑀 Fn 𝑌 → (𝑀 ↾ 𝑌) = 𝑀) | 
| 20 | 7, 19 | syl 17 | . . . . . 6
⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ 𝐻) → (𝑀 ↾ 𝑌) = 𝑀) | 
| 21 |  | eqid 2737 | . . . . . . . . . 10
⊢ (
+𝑣 ‘𝑈) = ( +𝑣 ‘𝑈) | 
| 22 |  | eqid 2737 | . . . . . . . . . 10
⊢ (
+𝑣 ‘𝑊) = ( +𝑣 ‘𝑊) | 
| 23 |  | eqid 2737 | . . . . . . . . . 10
⊢ (
·𝑠OLD ‘𝑈) = ( ·𝑠OLD
‘𝑈) | 
| 24 |  | eqid 2737 | . . . . . . . . . 10
⊢ (
·𝑠OLD ‘𝑊) = ( ·𝑠OLD
‘𝑊) | 
| 25 | 21, 22, 23, 24, 9, 4, 1 | isssp 30743 | . . . . . . . . 9
⊢ (𝑈 ∈ NrmCVec → (𝑊 ∈ 𝐻 ↔ (𝑊 ∈ NrmCVec ∧ ((
+𝑣 ‘𝑊) ⊆ ( +𝑣
‘𝑈) ∧ (
·𝑠OLD ‘𝑊) ⊆ (
·𝑠OLD ‘𝑈) ∧ 𝑀 ⊆ 𝑁)))) | 
| 26 | 25 | simplbda 499 | . . . . . . . 8
⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ 𝐻) → (( +𝑣
‘𝑊) ⊆ (
+𝑣 ‘𝑈) ∧ (
·𝑠OLD ‘𝑊) ⊆ (
·𝑠OLD ‘𝑈) ∧ 𝑀 ⊆ 𝑁)) | 
| 27 | 26 | simp3d 1145 | . . . . . . 7
⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ 𝐻) → 𝑀 ⊆ 𝑁) | 
| 28 |  | ssres 6021 | . . . . . . 7
⊢ (𝑀 ⊆ 𝑁 → (𝑀 ↾ 𝑌) ⊆ (𝑁 ↾ 𝑌)) | 
| 29 | 27, 28 | syl 17 | . . . . . 6
⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ 𝐻) → (𝑀 ↾ 𝑌) ⊆ (𝑁 ↾ 𝑌)) | 
| 30 | 20, 29 | eqsstrrd 4019 | . . . . 5
⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ 𝐻) → 𝑀 ⊆ (𝑁 ↾ 𝑌)) | 
| 31 | 30 | adantr 480 | . . . 4
⊢ (((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ 𝐻) ∧ 𝑥 ∈ 𝑌) → 𝑀 ⊆ (𝑁 ↾ 𝑌)) | 
| 32 | 5 | fdmd 6746 | . . . . . . 7
⊢ (𝑊 ∈ NrmCVec → dom 𝑀 = 𝑌) | 
| 33 | 32 | eleq2d 2827 | . . . . . 6
⊢ (𝑊 ∈ NrmCVec → (𝑥 ∈ dom 𝑀 ↔ 𝑥 ∈ 𝑌)) | 
| 34 | 33 | biimpar 477 | . . . . 5
⊢ ((𝑊 ∈ NrmCVec ∧ 𝑥 ∈ 𝑌) → 𝑥 ∈ dom 𝑀) | 
| 35 | 2, 34 | sylan 580 | . . . 4
⊢ (((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ 𝐻) ∧ 𝑥 ∈ 𝑌) → 𝑥 ∈ dom 𝑀) | 
| 36 |  | funssfv 6927 | . . . 4
⊢ ((Fun
(𝑁 ↾ 𝑌) ∧ 𝑀 ⊆ (𝑁 ↾ 𝑌) ∧ 𝑥 ∈ dom 𝑀) → ((𝑁 ↾ 𝑌)‘𝑥) = (𝑀‘𝑥)) | 
| 37 | 18, 31, 35, 36 | syl3anc 1373 | . . 3
⊢ (((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ 𝐻) ∧ 𝑥 ∈ 𝑌) → ((𝑁 ↾ 𝑌)‘𝑥) = (𝑀‘𝑥)) | 
| 38 | 37 | eqcomd 2743 | . 2
⊢ (((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ 𝐻) ∧ 𝑥 ∈ 𝑌) → (𝑀‘𝑥) = ((𝑁 ↾ 𝑌)‘𝑥)) | 
| 39 | 7, 15, 38 | eqfnfvd 7054 | 1
⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ 𝐻) → 𝑀 = (𝑁 ↾ 𝑌)) |