Proof of Theorem nmparlem
| Step | Hyp | Ref
| Expression |
| 1 | | nmpar.h |
. . . . 5
⊢ , =
(·𝑖‘𝑊) |
| 2 | | nmpar.v |
. . . . 5
⊢ 𝑉 = (Base‘𝑊) |
| 3 | | nmpar.p |
. . . . 5
⊢ + =
(+g‘𝑊) |
| 4 | | nmpar.1 |
. . . . 5
⊢ (𝜑 → 𝑊 ∈ ℂPreHil) |
| 5 | | nmpar.2 |
. . . . 5
⊢ (𝜑 → 𝐴 ∈ 𝑉) |
| 6 | | nmpar.3 |
. . . . 5
⊢ (𝜑 → 𝐵 ∈ 𝑉) |
| 7 | 1, 2, 3, 4, 5, 6, 5, 6 | cph2di 25241 |
. . . 4
⊢ (𝜑 → ((𝐴 + 𝐵) , (𝐴 + 𝐵)) = (((𝐴 , 𝐴) + (𝐵 , 𝐵)) + ((𝐴 , 𝐵) + (𝐵 , 𝐴)))) |
| 8 | | nmpar.m |
. . . . 5
⊢ − =
(-g‘𝑊) |
| 9 | 1, 2, 8, 4, 5, 6, 5, 6 | cph2subdi 25244 |
. . . 4
⊢ (𝜑 → ((𝐴 − 𝐵) , (𝐴 − 𝐵)) = (((𝐴 , 𝐴) + (𝐵 , 𝐵)) − ((𝐴 , 𝐵) + (𝐵 , 𝐴)))) |
| 10 | 7, 9 | oveq12d 7449 |
. . 3
⊢ (𝜑 → (((𝐴 + 𝐵) , (𝐴 + 𝐵)) + ((𝐴 − 𝐵) , (𝐴 − 𝐵))) = ((((𝐴 , 𝐴) + (𝐵 , 𝐵)) + ((𝐴 , 𝐵) + (𝐵 , 𝐴))) + (((𝐴 , 𝐴) + (𝐵 , 𝐵)) − ((𝐴 , 𝐵) + (𝐵 , 𝐴))))) |
| 11 | | cphclm 25223 |
. . . . . . 7
⊢ (𝑊 ∈ ℂPreHil →
𝑊 ∈
ℂMod) |
| 12 | 4, 11 | syl 17 |
. . . . . 6
⊢ (𝜑 → 𝑊 ∈ ℂMod) |
| 13 | | nmpar.f |
. . . . . . 7
⊢ 𝐹 = (Scalar‘𝑊) |
| 14 | | nmpar.k |
. . . . . . 7
⊢ 𝐾 = (Base‘𝐹) |
| 15 | 13, 14 | clmsscn 25112 |
. . . . . 6
⊢ (𝑊 ∈ ℂMod → 𝐾 ⊆
ℂ) |
| 16 | 12, 15 | syl 17 |
. . . . 5
⊢ (𝜑 → 𝐾 ⊆ ℂ) |
| 17 | | cphphl 25205 |
. . . . . . . 8
⊢ (𝑊 ∈ ℂPreHil →
𝑊 ∈
PreHil) |
| 18 | 4, 17 | syl 17 |
. . . . . . 7
⊢ (𝜑 → 𝑊 ∈ PreHil) |
| 19 | 13, 1, 2, 14 | ipcl 21651 |
. . . . . . 7
⊢ ((𝑊 ∈ PreHil ∧ 𝐴 ∈ 𝑉 ∧ 𝐴 ∈ 𝑉) → (𝐴 , 𝐴) ∈ 𝐾) |
| 20 | 18, 5, 5, 19 | syl3anc 1373 |
. . . . . 6
⊢ (𝜑 → (𝐴 , 𝐴) ∈ 𝐾) |
| 21 | 13, 1, 2, 14 | ipcl 21651 |
. . . . . . 7
⊢ ((𝑊 ∈ PreHil ∧ 𝐵 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) → (𝐵 , 𝐵) ∈ 𝐾) |
| 22 | 18, 6, 6, 21 | syl3anc 1373 |
. . . . . 6
⊢ (𝜑 → (𝐵 , 𝐵) ∈ 𝐾) |
| 23 | 13, 14 | clmacl 25117 |
. . . . . 6
⊢ ((𝑊 ∈ ℂMod ∧ (𝐴 , 𝐴) ∈ 𝐾 ∧ (𝐵 , 𝐵) ∈ 𝐾) → ((𝐴 , 𝐴) + (𝐵 , 𝐵)) ∈ 𝐾) |
| 24 | 12, 20, 22, 23 | syl3anc 1373 |
. . . . 5
⊢ (𝜑 → ((𝐴 , 𝐴) + (𝐵 , 𝐵)) ∈ 𝐾) |
| 25 | 16, 24 | sseldd 3984 |
. . . 4
⊢ (𝜑 → ((𝐴 , 𝐴) + (𝐵 , 𝐵)) ∈ ℂ) |
| 26 | 13, 1, 2, 14 | ipcl 21651 |
. . . . . . 7
⊢ ((𝑊 ∈ PreHil ∧ 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) → (𝐴 , 𝐵) ∈ 𝐾) |
| 27 | 18, 5, 6, 26 | syl3anc 1373 |
. . . . . 6
⊢ (𝜑 → (𝐴 , 𝐵) ∈ 𝐾) |
| 28 | 13, 1, 2, 14 | ipcl 21651 |
. . . . . . 7
⊢ ((𝑊 ∈ PreHil ∧ 𝐵 ∈ 𝑉 ∧ 𝐴 ∈ 𝑉) → (𝐵 , 𝐴) ∈ 𝐾) |
| 29 | 18, 6, 5, 28 | syl3anc 1373 |
. . . . . 6
⊢ (𝜑 → (𝐵 , 𝐴) ∈ 𝐾) |
| 30 | 13, 14 | clmacl 25117 |
. . . . . 6
⊢ ((𝑊 ∈ ℂMod ∧ (𝐴 , 𝐵) ∈ 𝐾 ∧ (𝐵 , 𝐴) ∈ 𝐾) → ((𝐴 , 𝐵) + (𝐵 , 𝐴)) ∈ 𝐾) |
| 31 | 12, 27, 29, 30 | syl3anc 1373 |
. . . . 5
⊢ (𝜑 → ((𝐴 , 𝐵) + (𝐵 , 𝐴)) ∈ 𝐾) |
| 32 | 16, 31 | sseldd 3984 |
. . . 4
⊢ (𝜑 → ((𝐴 , 𝐵) + (𝐵 , 𝐴)) ∈ ℂ) |
| 33 | 25, 32, 25 | ppncand 11660 |
. . 3
⊢ (𝜑 → ((((𝐴 , 𝐴) + (𝐵 , 𝐵)) + ((𝐴 , 𝐵) + (𝐵 , 𝐴))) + (((𝐴 , 𝐴) + (𝐵 , 𝐵)) − ((𝐴 , 𝐵) + (𝐵 , 𝐴)))) = (((𝐴 , 𝐴) + (𝐵 , 𝐵)) + ((𝐴 , 𝐴) + (𝐵 , 𝐵)))) |
| 34 | 10, 33 | eqtrd 2777 |
. 2
⊢ (𝜑 → (((𝐴 + 𝐵) , (𝐴 + 𝐵)) + ((𝐴 − 𝐵) , (𝐴 − 𝐵))) = (((𝐴 , 𝐴) + (𝐵 , 𝐵)) + ((𝐴 , 𝐴) + (𝐵 , 𝐵)))) |
| 35 | | cphlmod 25208 |
. . . . . 6
⊢ (𝑊 ∈ ℂPreHil →
𝑊 ∈
LMod) |
| 36 | 4, 35 | syl 17 |
. . . . 5
⊢ (𝜑 → 𝑊 ∈ LMod) |
| 37 | 2, 3 | lmodvacl 20873 |
. . . . 5
⊢ ((𝑊 ∈ LMod ∧ 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) → (𝐴 + 𝐵) ∈ 𝑉) |
| 38 | 36, 5, 6, 37 | syl3anc 1373 |
. . . 4
⊢ (𝜑 → (𝐴 + 𝐵) ∈ 𝑉) |
| 39 | | nmpar.n |
. . . . 5
⊢ 𝑁 = (norm‘𝑊) |
| 40 | 2, 1, 39 | nmsq 25228 |
. . . 4
⊢ ((𝑊 ∈ ℂPreHil ∧
(𝐴 + 𝐵) ∈ 𝑉) → ((𝑁‘(𝐴 + 𝐵))↑2) = ((𝐴 + 𝐵) , (𝐴 + 𝐵))) |
| 41 | 4, 38, 40 | syl2anc 584 |
. . 3
⊢ (𝜑 → ((𝑁‘(𝐴 + 𝐵))↑2) = ((𝐴 + 𝐵) , (𝐴 + 𝐵))) |
| 42 | 2, 8 | lmodvsubcl 20905 |
. . . . 5
⊢ ((𝑊 ∈ LMod ∧ 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) → (𝐴 − 𝐵) ∈ 𝑉) |
| 43 | 36, 5, 6, 42 | syl3anc 1373 |
. . . 4
⊢ (𝜑 → (𝐴 − 𝐵) ∈ 𝑉) |
| 44 | 2, 1, 39 | nmsq 25228 |
. . . 4
⊢ ((𝑊 ∈ ℂPreHil ∧
(𝐴 − 𝐵) ∈ 𝑉) → ((𝑁‘(𝐴 − 𝐵))↑2) = ((𝐴 − 𝐵) , (𝐴 − 𝐵))) |
| 45 | 4, 43, 44 | syl2anc 584 |
. . 3
⊢ (𝜑 → ((𝑁‘(𝐴 − 𝐵))↑2) = ((𝐴 − 𝐵) , (𝐴 − 𝐵))) |
| 46 | 41, 45 | oveq12d 7449 |
. 2
⊢ (𝜑 → (((𝑁‘(𝐴 + 𝐵))↑2) + ((𝑁‘(𝐴 − 𝐵))↑2)) = (((𝐴 + 𝐵) , (𝐴 + 𝐵)) + ((𝐴 − 𝐵) , (𝐴 − 𝐵)))) |
| 47 | 2, 1, 39 | nmsq 25228 |
. . . . . 6
⊢ ((𝑊 ∈ ℂPreHil ∧
𝐴 ∈ 𝑉) → ((𝑁‘𝐴)↑2) = (𝐴 , 𝐴)) |
| 48 | 4, 5, 47 | syl2anc 584 |
. . . . 5
⊢ (𝜑 → ((𝑁‘𝐴)↑2) = (𝐴 , 𝐴)) |
| 49 | 2, 1, 39 | nmsq 25228 |
. . . . . 6
⊢ ((𝑊 ∈ ℂPreHil ∧
𝐵 ∈ 𝑉) → ((𝑁‘𝐵)↑2) = (𝐵 , 𝐵)) |
| 50 | 4, 6, 49 | syl2anc 584 |
. . . . 5
⊢ (𝜑 → ((𝑁‘𝐵)↑2) = (𝐵 , 𝐵)) |
| 51 | 48, 50 | oveq12d 7449 |
. . . 4
⊢ (𝜑 → (((𝑁‘𝐴)↑2) + ((𝑁‘𝐵)↑2)) = ((𝐴 , 𝐴) + (𝐵 , 𝐵))) |
| 52 | 51 | oveq2d 7447 |
. . 3
⊢ (𝜑 → (2 · (((𝑁‘𝐴)↑2) + ((𝑁‘𝐵)↑2))) = (2 · ((𝐴 , 𝐴) + (𝐵 , 𝐵)))) |
| 53 | 25 | 2timesd 12509 |
. . 3
⊢ (𝜑 → (2 · ((𝐴 , 𝐴) + (𝐵 , 𝐵))) = (((𝐴 , 𝐴) + (𝐵 , 𝐵)) + ((𝐴 , 𝐴) + (𝐵 , 𝐵)))) |
| 54 | 52, 53 | eqtrd 2777 |
. 2
⊢ (𝜑 → (2 · (((𝑁‘𝐴)↑2) + ((𝑁‘𝐵)↑2))) = (((𝐴 , 𝐴) + (𝐵 , 𝐵)) + ((𝐴 , 𝐴) + (𝐵 , 𝐵)))) |
| 55 | 34, 46, 54 | 3eqtr4d 2787 |
1
⊢ (𝜑 → (((𝑁‘(𝐴 + 𝐵))↑2) + ((𝑁‘(𝐴 − 𝐵))↑2)) = (2 · (((𝑁‘𝐴)↑2) + ((𝑁‘𝐵)↑2)))) |