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 27677 |
. . . 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 480 |
. . 3
⊢ (𝑈 ∈ CPreHilOLD
→ ∀𝑥 ∈
𝑋 ∀𝑦 ∈ 𝑋 (((𝑁‘(𝑥𝐺𝑦))↑2) + ((𝑁‘(𝑥𝑀𝑦))↑2)) = (2 · (((𝑁‘𝑥)↑2) + ((𝑁‘𝑦)↑2)))) |
7 | 6 | 3ad2ant1 1082 |
. 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 | | oveq1 6657 |
. . . . . . . 8
⊢ (𝑥 = 𝐴 → (𝑥𝐺𝑦) = (𝐴𝐺𝑦)) |
9 | 8 | fveq2d 6195 |
. . . . . . 7
⊢ (𝑥 = 𝐴 → (𝑁‘(𝑥𝐺𝑦)) = (𝑁‘(𝐴𝐺𝑦))) |
10 | 9 | oveq1d 6665 |
. . . . . 6
⊢ (𝑥 = 𝐴 → ((𝑁‘(𝑥𝐺𝑦))↑2) = ((𝑁‘(𝐴𝐺𝑦))↑2)) |
11 | | oveq1 6657 |
. . . . . . . 8
⊢ (𝑥 = 𝐴 → (𝑥𝑀𝑦) = (𝐴𝑀𝑦)) |
12 | 11 | fveq2d 6195 |
. . . . . . 7
⊢ (𝑥 = 𝐴 → (𝑁‘(𝑥𝑀𝑦)) = (𝑁‘(𝐴𝑀𝑦))) |
13 | 12 | oveq1d 6665 |
. . . . . 6
⊢ (𝑥 = 𝐴 → ((𝑁‘(𝑥𝑀𝑦))↑2) = ((𝑁‘(𝐴𝑀𝑦))↑2)) |
14 | 10, 13 | oveq12d 6668 |
. . . . 5
⊢ (𝑥 = 𝐴 → (((𝑁‘(𝑥𝐺𝑦))↑2) + ((𝑁‘(𝑥𝑀𝑦))↑2)) = (((𝑁‘(𝐴𝐺𝑦))↑2) + ((𝑁‘(𝐴𝑀𝑦))↑2))) |
15 | | fveq2 6191 |
. . . . . . . 8
⊢ (𝑥 = 𝐴 → (𝑁‘𝑥) = (𝑁‘𝐴)) |
16 | 15 | oveq1d 6665 |
. . . . . . 7
⊢ (𝑥 = 𝐴 → ((𝑁‘𝑥)↑2) = ((𝑁‘𝐴)↑2)) |
17 | 16 | oveq1d 6665 |
. . . . . 6
⊢ (𝑥 = 𝐴 → (((𝑁‘𝑥)↑2) + ((𝑁‘𝑦)↑2)) = (((𝑁‘𝐴)↑2) + ((𝑁‘𝑦)↑2))) |
18 | 17 | oveq2d 6666 |
. . . . 5
⊢ (𝑥 = 𝐴 → (2 · (((𝑁‘𝑥)↑2) + ((𝑁‘𝑦)↑2))) = (2 · (((𝑁‘𝐴)↑2) + ((𝑁‘𝑦)↑2)))) |
19 | 14, 18 | eqeq12d 2637 |
. . . 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))))) |
20 | | oveq2 6658 |
. . . . . . . 8
⊢ (𝑦 = 𝐵 → (𝐴𝐺𝑦) = (𝐴𝐺𝐵)) |
21 | 20 | fveq2d 6195 |
. . . . . . 7
⊢ (𝑦 = 𝐵 → (𝑁‘(𝐴𝐺𝑦)) = (𝑁‘(𝐴𝐺𝐵))) |
22 | 21 | oveq1d 6665 |
. . . . . 6
⊢ (𝑦 = 𝐵 → ((𝑁‘(𝐴𝐺𝑦))↑2) = ((𝑁‘(𝐴𝐺𝐵))↑2)) |
23 | | oveq2 6658 |
. . . . . . . 8
⊢ (𝑦 = 𝐵 → (𝐴𝑀𝑦) = (𝐴𝑀𝐵)) |
24 | 23 | fveq2d 6195 |
. . . . . . 7
⊢ (𝑦 = 𝐵 → (𝑁‘(𝐴𝑀𝑦)) = (𝑁‘(𝐴𝑀𝐵))) |
25 | 24 | oveq1d 6665 |
. . . . . 6
⊢ (𝑦 = 𝐵 → ((𝑁‘(𝐴𝑀𝑦))↑2) = ((𝑁‘(𝐴𝑀𝐵))↑2)) |
26 | 22, 25 | oveq12d 6668 |
. . . . 5
⊢ (𝑦 = 𝐵 → (((𝑁‘(𝐴𝐺𝑦))↑2) + ((𝑁‘(𝐴𝑀𝑦))↑2)) = (((𝑁‘(𝐴𝐺𝐵))↑2) + ((𝑁‘(𝐴𝑀𝐵))↑2))) |
27 | | fveq2 6191 |
. . . . . . . 8
⊢ (𝑦 = 𝐵 → (𝑁‘𝑦) = (𝑁‘𝐵)) |
28 | 27 | oveq1d 6665 |
. . . . . . 7
⊢ (𝑦 = 𝐵 → ((𝑁‘𝑦)↑2) = ((𝑁‘𝐵)↑2)) |
29 | 28 | oveq2d 6666 |
. . . . . 6
⊢ (𝑦 = 𝐵 → (((𝑁‘𝐴)↑2) + ((𝑁‘𝑦)↑2)) = (((𝑁‘𝐴)↑2) + ((𝑁‘𝐵)↑2))) |
30 | 29 | oveq2d 6666 |
. . . . 5
⊢ (𝑦 = 𝐵 → (2 · (((𝑁‘𝐴)↑2) + ((𝑁‘𝑦)↑2))) = (2 · (((𝑁‘𝐴)↑2) + ((𝑁‘𝐵)↑2)))) |
31 | 26, 30 | eqeq12d 2637 |
. . . 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))))) |
32 | 19, 31 | rspc2v 3322 |
. . 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))))) |
33 | 32 | 3adant1 1079 |
. 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))))) |
34 | 7, 33 | mpd 15 |
1
⊢ ((𝑈 ∈ CPreHilOLD
∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (((𝑁‘(𝐴𝐺𝐵))↑2) + ((𝑁‘(𝐴𝑀𝐵))↑2)) = (2 · (((𝑁‘𝐴)↑2) + ((𝑁‘𝐵)↑2)))) |