| Step | Hyp | Ref | Expression | 
|---|
| 1 |  | isph.1 | . . . . 5
⊢ 𝑋 = (BaseSet‘𝑈) | 
| 2 |  | isph.2 | . . . . 5
⊢ 𝐺 = ( +𝑣
‘𝑈) | 
| 3 |  | isph.3 | . . . . 5
⊢ 𝑀 = ( −𝑣
‘𝑈) | 
| 4 |  | isph.6 | . . . . 5
⊢ 𝑁 =
(normCV‘𝑈) | 
| 5 | 1, 2, 3, 4 | isph 30841 | . . . 4
⊢ (𝑈 ∈ CPreHilOLD
↔ (𝑈 ∈ NrmCVec
∧ ∀𝑥 ∈
𝑋 ∀𝑦 ∈ 𝑋 (((𝑁‘(𝑥𝐺𝑦))↑2) + ((𝑁‘(𝑥𝑀𝑦))↑2)) = (2 · (((𝑁‘𝑥)↑2) + ((𝑁‘𝑦)↑2))))) | 
| 6 | 5 | simprbi 496 | . . 3
⊢ (𝑈 ∈ CPreHilOLD
→ ∀𝑥 ∈
𝑋 ∀𝑦 ∈ 𝑋 (((𝑁‘(𝑥𝐺𝑦))↑2) + ((𝑁‘(𝑥𝑀𝑦))↑2)) = (2 · (((𝑁‘𝑥)↑2) + ((𝑁‘𝑦)↑2)))) | 
| 7 | 6 | 3ad2ant1 1134 | . 2
⊢ ((𝑈 ∈ CPreHilOLD
∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (((𝑁‘(𝑥𝐺𝑦))↑2) + ((𝑁‘(𝑥𝑀𝑦))↑2)) = (2 · (((𝑁‘𝑥)↑2) + ((𝑁‘𝑦)↑2)))) | 
| 8 |  | fvoveq1 7454 | . . . . . . 7
⊢ (𝑥 = 𝐴 → (𝑁‘(𝑥𝐺𝑦)) = (𝑁‘(𝐴𝐺𝑦))) | 
| 9 | 8 | oveq1d 7446 | . . . . . 6
⊢ (𝑥 = 𝐴 → ((𝑁‘(𝑥𝐺𝑦))↑2) = ((𝑁‘(𝐴𝐺𝑦))↑2)) | 
| 10 |  | fvoveq1 7454 | . . . . . . 7
⊢ (𝑥 = 𝐴 → (𝑁‘(𝑥𝑀𝑦)) = (𝑁‘(𝐴𝑀𝑦))) | 
| 11 | 10 | oveq1d 7446 | . . . . . 6
⊢ (𝑥 = 𝐴 → ((𝑁‘(𝑥𝑀𝑦))↑2) = ((𝑁‘(𝐴𝑀𝑦))↑2)) | 
| 12 | 9, 11 | oveq12d 7449 | . . . . 5
⊢ (𝑥 = 𝐴 → (((𝑁‘(𝑥𝐺𝑦))↑2) + ((𝑁‘(𝑥𝑀𝑦))↑2)) = (((𝑁‘(𝐴𝐺𝑦))↑2) + ((𝑁‘(𝐴𝑀𝑦))↑2))) | 
| 13 |  | fveq2 6906 | . . . . . . . 8
⊢ (𝑥 = 𝐴 → (𝑁‘𝑥) = (𝑁‘𝐴)) | 
| 14 | 13 | oveq1d 7446 | . . . . . . 7
⊢ (𝑥 = 𝐴 → ((𝑁‘𝑥)↑2) = ((𝑁‘𝐴)↑2)) | 
| 15 | 14 | oveq1d 7446 | . . . . . 6
⊢ (𝑥 = 𝐴 → (((𝑁‘𝑥)↑2) + ((𝑁‘𝑦)↑2)) = (((𝑁‘𝐴)↑2) + ((𝑁‘𝑦)↑2))) | 
| 16 | 15 | oveq2d 7447 | . . . . 5
⊢ (𝑥 = 𝐴 → (2 · (((𝑁‘𝑥)↑2) + ((𝑁‘𝑦)↑2))) = (2 · (((𝑁‘𝐴)↑2) + ((𝑁‘𝑦)↑2)))) | 
| 17 | 12, 16 | eqeq12d 2753 | . . . 4
⊢ (𝑥 = 𝐴 → ((((𝑁‘(𝑥𝐺𝑦))↑2) + ((𝑁‘(𝑥𝑀𝑦))↑2)) = (2 · (((𝑁‘𝑥)↑2) + ((𝑁‘𝑦)↑2))) ↔ (((𝑁‘(𝐴𝐺𝑦))↑2) + ((𝑁‘(𝐴𝑀𝑦))↑2)) = (2 · (((𝑁‘𝐴)↑2) + ((𝑁‘𝑦)↑2))))) | 
| 18 |  | oveq2 7439 | . . . . . . . 8
⊢ (𝑦 = 𝐵 → (𝐴𝐺𝑦) = (𝐴𝐺𝐵)) | 
| 19 | 18 | fveq2d 6910 | . . . . . . 7
⊢ (𝑦 = 𝐵 → (𝑁‘(𝐴𝐺𝑦)) = (𝑁‘(𝐴𝐺𝐵))) | 
| 20 | 19 | oveq1d 7446 | . . . . . 6
⊢ (𝑦 = 𝐵 → ((𝑁‘(𝐴𝐺𝑦))↑2) = ((𝑁‘(𝐴𝐺𝐵))↑2)) | 
| 21 |  | oveq2 7439 | . . . . . . . 8
⊢ (𝑦 = 𝐵 → (𝐴𝑀𝑦) = (𝐴𝑀𝐵)) | 
| 22 | 21 | fveq2d 6910 | . . . . . . 7
⊢ (𝑦 = 𝐵 → (𝑁‘(𝐴𝑀𝑦)) = (𝑁‘(𝐴𝑀𝐵))) | 
| 23 | 22 | oveq1d 7446 | . . . . . 6
⊢ (𝑦 = 𝐵 → ((𝑁‘(𝐴𝑀𝑦))↑2) = ((𝑁‘(𝐴𝑀𝐵))↑2)) | 
| 24 | 20, 23 | oveq12d 7449 | . . . . 5
⊢ (𝑦 = 𝐵 → (((𝑁‘(𝐴𝐺𝑦))↑2) + ((𝑁‘(𝐴𝑀𝑦))↑2)) = (((𝑁‘(𝐴𝐺𝐵))↑2) + ((𝑁‘(𝐴𝑀𝐵))↑2))) | 
| 25 |  | fveq2 6906 | . . . . . . . 8
⊢ (𝑦 = 𝐵 → (𝑁‘𝑦) = (𝑁‘𝐵)) | 
| 26 | 25 | oveq1d 7446 | . . . . . . 7
⊢ (𝑦 = 𝐵 → ((𝑁‘𝑦)↑2) = ((𝑁‘𝐵)↑2)) | 
| 27 | 26 | oveq2d 7447 | . . . . . 6
⊢ (𝑦 = 𝐵 → (((𝑁‘𝐴)↑2) + ((𝑁‘𝑦)↑2)) = (((𝑁‘𝐴)↑2) + ((𝑁‘𝐵)↑2))) | 
| 28 | 27 | oveq2d 7447 | . . . . 5
⊢ (𝑦 = 𝐵 → (2 · (((𝑁‘𝐴)↑2) + ((𝑁‘𝑦)↑2))) = (2 · (((𝑁‘𝐴)↑2) + ((𝑁‘𝐵)↑2)))) | 
| 29 | 24, 28 | eqeq12d 2753 | . . . 4
⊢ (𝑦 = 𝐵 → ((((𝑁‘(𝐴𝐺𝑦))↑2) + ((𝑁‘(𝐴𝑀𝑦))↑2)) = (2 · (((𝑁‘𝐴)↑2) + ((𝑁‘𝑦)↑2))) ↔ (((𝑁‘(𝐴𝐺𝐵))↑2) + ((𝑁‘(𝐴𝑀𝐵))↑2)) = (2 · (((𝑁‘𝐴)↑2) + ((𝑁‘𝐵)↑2))))) | 
| 30 | 17, 29 | rspc2v 3633 | . . 3
⊢ ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (((𝑁‘(𝑥𝐺𝑦))↑2) + ((𝑁‘(𝑥𝑀𝑦))↑2)) = (2 · (((𝑁‘𝑥)↑2) + ((𝑁‘𝑦)↑2))) → (((𝑁‘(𝐴𝐺𝐵))↑2) + ((𝑁‘(𝐴𝑀𝐵))↑2)) = (2 · (((𝑁‘𝐴)↑2) + ((𝑁‘𝐵)↑2))))) | 
| 31 | 30 | 3adant1 1131 | . 2
⊢ ((𝑈 ∈ CPreHilOLD
∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (((𝑁‘(𝑥𝐺𝑦))↑2) + ((𝑁‘(𝑥𝑀𝑦))↑2)) = (2 · (((𝑁‘𝑥)↑2) + ((𝑁‘𝑦)↑2))) → (((𝑁‘(𝐴𝐺𝐵))↑2) + ((𝑁‘(𝐴𝑀𝐵))↑2)) = (2 · (((𝑁‘𝐴)↑2) + ((𝑁‘𝐵)↑2))))) | 
| 32 | 7, 31 | mpd 15 | 1
⊢ ((𝑈 ∈ CPreHilOLD
∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (((𝑁‘(𝐴𝐺𝐵))↑2) + ((𝑁‘(𝐴𝑀𝐵))↑2)) = (2 · (((𝑁‘𝐴)↑2) + ((𝑁‘𝐵)↑2)))) |