| Step | Hyp | Ref | Expression | 
|---|
| 1 |  | phpar.2 | . . . . . 6
⊢ 𝐺 = ( +𝑣
‘𝑈) | 
| 2 | 1 | fvexi 6919 | . . . . 5
⊢ 𝐺 ∈ V | 
| 3 |  | phpar.4 | . . . . . 6
⊢ 𝑆 = (
·𝑠OLD ‘𝑈) | 
| 4 | 3 | fvexi 6919 | . . . . 5
⊢ 𝑆 ∈ V | 
| 5 |  | phpar.6 | . . . . . 6
⊢ 𝑁 =
(normCV‘𝑈) | 
| 6 | 5 | fvexi 6919 | . . . . 5
⊢ 𝑁 ∈ V | 
| 7 | 2, 4, 6 | 3pm3.2i 1339 | . . . 4
⊢ (𝐺 ∈ V ∧ 𝑆 ∈ V ∧ 𝑁 ∈ V) | 
| 8 | 1, 3, 5 | phop 30838 | . . . . . 6
⊢ (𝑈 ∈ CPreHilOLD
→ 𝑈 =
〈〈𝐺, 𝑆〉, 𝑁〉) | 
| 9 | 8 | eleq1d 2825 | . . . . 5
⊢ (𝑈 ∈ CPreHilOLD
→ (𝑈 ∈
CPreHilOLD ↔ 〈〈𝐺, 𝑆〉, 𝑁〉 ∈
CPreHilOLD)) | 
| 10 | 9 | ibi 267 | . . . 4
⊢ (𝑈 ∈ CPreHilOLD
→ 〈〈𝐺, 𝑆〉, 𝑁〉 ∈
CPreHilOLD) | 
| 11 |  | phpar.1 | . . . . . . 7
⊢ 𝑋 = (BaseSet‘𝑈) | 
| 12 | 11, 1 | bafval 30624 | . . . . . 6
⊢ 𝑋 = ran 𝐺 | 
| 13 | 12 | isphg 30837 | . . . . 5
⊢ ((𝐺 ∈ V ∧ 𝑆 ∈ V ∧ 𝑁 ∈ V) →
(〈〈𝐺, 𝑆〉, 𝑁〉 ∈ CPreHilOLD ↔
(〈〈𝐺, 𝑆〉, 𝑁〉 ∈ NrmCVec ∧ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (((𝑁‘(𝑥𝐺𝑦))↑2) + ((𝑁‘(𝑥𝐺(-1𝑆𝑦)))↑2)) = (2 · (((𝑁‘𝑥)↑2) + ((𝑁‘𝑦)↑2)))))) | 
| 14 | 13 | simplbda 499 | . . . 4
⊢ (((𝐺 ∈ V ∧ 𝑆 ∈ V ∧ 𝑁 ∈ V) ∧
〈〈𝐺, 𝑆〉, 𝑁〉 ∈ CPreHilOLD) →
∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (((𝑁‘(𝑥𝐺𝑦))↑2) + ((𝑁‘(𝑥𝐺(-1𝑆𝑦)))↑2)) = (2 · (((𝑁‘𝑥)↑2) + ((𝑁‘𝑦)↑2)))) | 
| 15 | 7, 10, 14 | sylancr 587 | . . 3
⊢ (𝑈 ∈ CPreHilOLD
→ ∀𝑥 ∈
𝑋 ∀𝑦 ∈ 𝑋 (((𝑁‘(𝑥𝐺𝑦))↑2) + ((𝑁‘(𝑥𝐺(-1𝑆𝑦)))↑2)) = (2 · (((𝑁‘𝑥)↑2) + ((𝑁‘𝑦)↑2)))) | 
| 16 | 15 | 3ad2ant1 1133 | . 2
⊢ ((𝑈 ∈ CPreHilOLD
∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (((𝑁‘(𝑥𝐺𝑦))↑2) + ((𝑁‘(𝑥𝐺(-1𝑆𝑦)))↑2)) = (2 · (((𝑁‘𝑥)↑2) + ((𝑁‘𝑦)↑2)))) | 
| 17 |  | fvoveq1 7455 | . . . . . . 7
⊢ (𝑥 = 𝐴 → (𝑁‘(𝑥𝐺𝑦)) = (𝑁‘(𝐴𝐺𝑦))) | 
| 18 | 17 | oveq1d 7447 | . . . . . 6
⊢ (𝑥 = 𝐴 → ((𝑁‘(𝑥𝐺𝑦))↑2) = ((𝑁‘(𝐴𝐺𝑦))↑2)) | 
| 19 |  | fvoveq1 7455 | . . . . . . 7
⊢ (𝑥 = 𝐴 → (𝑁‘(𝑥𝐺(-1𝑆𝑦))) = (𝑁‘(𝐴𝐺(-1𝑆𝑦)))) | 
| 20 | 19 | oveq1d 7447 | . . . . . 6
⊢ (𝑥 = 𝐴 → ((𝑁‘(𝑥𝐺(-1𝑆𝑦)))↑2) = ((𝑁‘(𝐴𝐺(-1𝑆𝑦)))↑2)) | 
| 21 | 18, 20 | oveq12d 7450 | . . . . 5
⊢ (𝑥 = 𝐴 → (((𝑁‘(𝑥𝐺𝑦))↑2) + ((𝑁‘(𝑥𝐺(-1𝑆𝑦)))↑2)) = (((𝑁‘(𝐴𝐺𝑦))↑2) + ((𝑁‘(𝐴𝐺(-1𝑆𝑦)))↑2))) | 
| 22 |  | fveq2 6905 | . . . . . . . 8
⊢ (𝑥 = 𝐴 → (𝑁‘𝑥) = (𝑁‘𝐴)) | 
| 23 | 22 | oveq1d 7447 | . . . . . . 7
⊢ (𝑥 = 𝐴 → ((𝑁‘𝑥)↑2) = ((𝑁‘𝐴)↑2)) | 
| 24 | 23 | oveq1d 7447 | . . . . . 6
⊢ (𝑥 = 𝐴 → (((𝑁‘𝑥)↑2) + ((𝑁‘𝑦)↑2)) = (((𝑁‘𝐴)↑2) + ((𝑁‘𝑦)↑2))) | 
| 25 | 24 | oveq2d 7448 | . . . . 5
⊢ (𝑥 = 𝐴 → (2 · (((𝑁‘𝑥)↑2) + ((𝑁‘𝑦)↑2))) = (2 · (((𝑁‘𝐴)↑2) + ((𝑁‘𝑦)↑2)))) | 
| 26 | 21, 25 | eqeq12d 2752 | . . . 4
⊢ (𝑥 = 𝐴 → ((((𝑁‘(𝑥𝐺𝑦))↑2) + ((𝑁‘(𝑥𝐺(-1𝑆𝑦)))↑2)) = (2 · (((𝑁‘𝑥)↑2) + ((𝑁‘𝑦)↑2))) ↔ (((𝑁‘(𝐴𝐺𝑦))↑2) + ((𝑁‘(𝐴𝐺(-1𝑆𝑦)))↑2)) = (2 · (((𝑁‘𝐴)↑2) + ((𝑁‘𝑦)↑2))))) | 
| 27 |  | oveq2 7440 | . . . . . . . 8
⊢ (𝑦 = 𝐵 → (𝐴𝐺𝑦) = (𝐴𝐺𝐵)) | 
| 28 | 27 | fveq2d 6909 | . . . . . . 7
⊢ (𝑦 = 𝐵 → (𝑁‘(𝐴𝐺𝑦)) = (𝑁‘(𝐴𝐺𝐵))) | 
| 29 | 28 | oveq1d 7447 | . . . . . 6
⊢ (𝑦 = 𝐵 → ((𝑁‘(𝐴𝐺𝑦))↑2) = ((𝑁‘(𝐴𝐺𝐵))↑2)) | 
| 30 |  | oveq2 7440 | . . . . . . . . 9
⊢ (𝑦 = 𝐵 → (-1𝑆𝑦) = (-1𝑆𝐵)) | 
| 31 | 30 | oveq2d 7448 | . . . . . . . 8
⊢ (𝑦 = 𝐵 → (𝐴𝐺(-1𝑆𝑦)) = (𝐴𝐺(-1𝑆𝐵))) | 
| 32 | 31 | fveq2d 6909 | . . . . . . 7
⊢ (𝑦 = 𝐵 → (𝑁‘(𝐴𝐺(-1𝑆𝑦))) = (𝑁‘(𝐴𝐺(-1𝑆𝐵)))) | 
| 33 | 32 | oveq1d 7447 | . . . . . 6
⊢ (𝑦 = 𝐵 → ((𝑁‘(𝐴𝐺(-1𝑆𝑦)))↑2) = ((𝑁‘(𝐴𝐺(-1𝑆𝐵)))↑2)) | 
| 34 | 29, 33 | oveq12d 7450 | . . . . 5
⊢ (𝑦 = 𝐵 → (((𝑁‘(𝐴𝐺𝑦))↑2) + ((𝑁‘(𝐴𝐺(-1𝑆𝑦)))↑2)) = (((𝑁‘(𝐴𝐺𝐵))↑2) + ((𝑁‘(𝐴𝐺(-1𝑆𝐵)))↑2))) | 
| 35 |  | fveq2 6905 | . . . . . . . 8
⊢ (𝑦 = 𝐵 → (𝑁‘𝑦) = (𝑁‘𝐵)) | 
| 36 | 35 | oveq1d 7447 | . . . . . . 7
⊢ (𝑦 = 𝐵 → ((𝑁‘𝑦)↑2) = ((𝑁‘𝐵)↑2)) | 
| 37 | 36 | oveq2d 7448 | . . . . . 6
⊢ (𝑦 = 𝐵 → (((𝑁‘𝐴)↑2) + ((𝑁‘𝑦)↑2)) = (((𝑁‘𝐴)↑2) + ((𝑁‘𝐵)↑2))) | 
| 38 | 37 | oveq2d 7448 | . . . . 5
⊢ (𝑦 = 𝐵 → (2 · (((𝑁‘𝐴)↑2) + ((𝑁‘𝑦)↑2))) = (2 · (((𝑁‘𝐴)↑2) + ((𝑁‘𝐵)↑2)))) | 
| 39 | 34, 38 | eqeq12d 2752 | . . . 4
⊢ (𝑦 = 𝐵 → ((((𝑁‘(𝐴𝐺𝑦))↑2) + ((𝑁‘(𝐴𝐺(-1𝑆𝑦)))↑2)) = (2 · (((𝑁‘𝐴)↑2) + ((𝑁‘𝑦)↑2))) ↔ (((𝑁‘(𝐴𝐺𝐵))↑2) + ((𝑁‘(𝐴𝐺(-1𝑆𝐵)))↑2)) = (2 · (((𝑁‘𝐴)↑2) + ((𝑁‘𝐵)↑2))))) | 
| 40 | 26, 39 | rspc2v 3632 | . . 3
⊢ ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (((𝑁‘(𝑥𝐺𝑦))↑2) + ((𝑁‘(𝑥𝐺(-1𝑆𝑦)))↑2)) = (2 · (((𝑁‘𝑥)↑2) + ((𝑁‘𝑦)↑2))) → (((𝑁‘(𝐴𝐺𝐵))↑2) + ((𝑁‘(𝐴𝐺(-1𝑆𝐵)))↑2)) = (2 · (((𝑁‘𝐴)↑2) + ((𝑁‘𝐵)↑2))))) | 
| 41 | 40 | 3adant1 1130 | . 2
⊢ ((𝑈 ∈ CPreHilOLD
∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (((𝑁‘(𝑥𝐺𝑦))↑2) + ((𝑁‘(𝑥𝐺(-1𝑆𝑦)))↑2)) = (2 · (((𝑁‘𝑥)↑2) + ((𝑁‘𝑦)↑2))) → (((𝑁‘(𝐴𝐺𝐵))↑2) + ((𝑁‘(𝐴𝐺(-1𝑆𝐵)))↑2)) = (2 · (((𝑁‘𝐴)↑2) + ((𝑁‘𝐵)↑2))))) | 
| 42 | 16, 41 | mpd 15 | 1
⊢ ((𝑈 ∈ CPreHilOLD
∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (((𝑁‘(𝐴𝐺𝐵))↑2) + ((𝑁‘(𝐴𝐺(-1𝑆𝐵)))↑2)) = (2 · (((𝑁‘𝐴)↑2) + ((𝑁‘𝐵)↑2)))) |