Proof of Theorem ip2eq
Step | Hyp | Ref
| Expression |
1 | | oveq2 7279 |
. . 3
⊢ (𝐴 = 𝐵 → (𝑥 , 𝐴) = (𝑥 , 𝐵)) |
2 | 1 | ralrimivw 3111 |
. 2
⊢ (𝐴 = 𝐵 → ∀𝑥 ∈ 𝑉 (𝑥 , 𝐴) = (𝑥 , 𝐵)) |
3 | | phllmod 20833 |
. . . . 5
⊢ (𝑊 ∈ PreHil → 𝑊 ∈ LMod) |
4 | | ip2eq.v |
. . . . . 6
⊢ 𝑉 = (Base‘𝑊) |
5 | | eqid 2740 |
. . . . . 6
⊢
(-g‘𝑊) = (-g‘𝑊) |
6 | 4, 5 | lmodvsubcl 20166 |
. . . . 5
⊢ ((𝑊 ∈ LMod ∧ 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) → (𝐴(-g‘𝑊)𝐵) ∈ 𝑉) |
7 | 3, 6 | syl3an1 1162 |
. . . 4
⊢ ((𝑊 ∈ PreHil ∧ 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) → (𝐴(-g‘𝑊)𝐵) ∈ 𝑉) |
8 | | oveq1 7278 |
. . . . . 6
⊢ (𝑥 = (𝐴(-g‘𝑊)𝐵) → (𝑥 , 𝐴) = ((𝐴(-g‘𝑊)𝐵) , 𝐴)) |
9 | | oveq1 7278 |
. . . . . 6
⊢ (𝑥 = (𝐴(-g‘𝑊)𝐵) → (𝑥 , 𝐵) = ((𝐴(-g‘𝑊)𝐵) , 𝐵)) |
10 | 8, 9 | eqeq12d 2756 |
. . . . 5
⊢ (𝑥 = (𝐴(-g‘𝑊)𝐵) → ((𝑥 , 𝐴) = (𝑥 , 𝐵) ↔ ((𝐴(-g‘𝑊)𝐵) , 𝐴) = ((𝐴(-g‘𝑊)𝐵) , 𝐵))) |
11 | 10 | rspcv 3556 |
. . . 4
⊢ ((𝐴(-g‘𝑊)𝐵) ∈ 𝑉 → (∀𝑥 ∈ 𝑉 (𝑥 , 𝐴) = (𝑥 , 𝐵) → ((𝐴(-g‘𝑊)𝐵) , 𝐴) = ((𝐴(-g‘𝑊)𝐵) , 𝐵))) |
12 | 7, 11 | syl 17 |
. . 3
⊢ ((𝑊 ∈ PreHil ∧ 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) → (∀𝑥 ∈ 𝑉 (𝑥 , 𝐴) = (𝑥 , 𝐵) → ((𝐴(-g‘𝑊)𝐵) , 𝐴) = ((𝐴(-g‘𝑊)𝐵) , 𝐵))) |
13 | | simp1 1135 |
. . . . . . 7
⊢ ((𝑊 ∈ PreHil ∧ 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) → 𝑊 ∈ PreHil) |
14 | | simp2 1136 |
. . . . . . 7
⊢ ((𝑊 ∈ PreHil ∧ 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) → 𝐴 ∈ 𝑉) |
15 | | simp3 1137 |
. . . . . . 7
⊢ ((𝑊 ∈ PreHil ∧ 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) → 𝐵 ∈ 𝑉) |
16 | | eqid 2740 |
. . . . . . . 8
⊢
(Scalar‘𝑊) =
(Scalar‘𝑊) |
17 | | ip2eq.h |
. . . . . . . 8
⊢ , =
(·𝑖‘𝑊) |
18 | | eqid 2740 |
. . . . . . . 8
⊢
(-g‘(Scalar‘𝑊)) =
(-g‘(Scalar‘𝑊)) |
19 | 16, 17, 4, 5, 18 | ipsubdi 20846 |
. . . . . . 7
⊢ ((𝑊 ∈ PreHil ∧ ((𝐴(-g‘𝑊)𝐵) ∈ 𝑉 ∧ 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉)) → ((𝐴(-g‘𝑊)𝐵) , (𝐴(-g‘𝑊)𝐵)) = (((𝐴(-g‘𝑊)𝐵) , 𝐴)(-g‘(Scalar‘𝑊))((𝐴(-g‘𝑊)𝐵) , 𝐵))) |
20 | 13, 7, 14, 15, 19 | syl13anc 1371 |
. . . . . 6
⊢ ((𝑊 ∈ PreHil ∧ 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) → ((𝐴(-g‘𝑊)𝐵) , (𝐴(-g‘𝑊)𝐵)) = (((𝐴(-g‘𝑊)𝐵) , 𝐴)(-g‘(Scalar‘𝑊))((𝐴(-g‘𝑊)𝐵) , 𝐵))) |
21 | 20 | eqeq1d 2742 |
. . . . 5
⊢ ((𝑊 ∈ PreHil ∧ 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) → (((𝐴(-g‘𝑊)𝐵) , (𝐴(-g‘𝑊)𝐵)) =
(0g‘(Scalar‘𝑊)) ↔ (((𝐴(-g‘𝑊)𝐵) , 𝐴)(-g‘(Scalar‘𝑊))((𝐴(-g‘𝑊)𝐵) , 𝐵)) =
(0g‘(Scalar‘𝑊)))) |
22 | | eqid 2740 |
. . . . . . 7
⊢
(0g‘(Scalar‘𝑊)) =
(0g‘(Scalar‘𝑊)) |
23 | | eqid 2740 |
. . . . . . 7
⊢
(0g‘𝑊) = (0g‘𝑊) |
24 | 16, 17, 4, 22, 23 | ipeq0 20841 |
. . . . . 6
⊢ ((𝑊 ∈ PreHil ∧ (𝐴(-g‘𝑊)𝐵) ∈ 𝑉) → (((𝐴(-g‘𝑊)𝐵) , (𝐴(-g‘𝑊)𝐵)) =
(0g‘(Scalar‘𝑊)) ↔ (𝐴(-g‘𝑊)𝐵) = (0g‘𝑊))) |
25 | 13, 7, 24 | syl2anc 584 |
. . . . 5
⊢ ((𝑊 ∈ PreHil ∧ 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) → (((𝐴(-g‘𝑊)𝐵) , (𝐴(-g‘𝑊)𝐵)) =
(0g‘(Scalar‘𝑊)) ↔ (𝐴(-g‘𝑊)𝐵) = (0g‘𝑊))) |
26 | 21, 25 | bitr3d 280 |
. . . 4
⊢ ((𝑊 ∈ PreHil ∧ 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) → ((((𝐴(-g‘𝑊)𝐵) , 𝐴)(-g‘(Scalar‘𝑊))((𝐴(-g‘𝑊)𝐵) , 𝐵)) =
(0g‘(Scalar‘𝑊)) ↔ (𝐴(-g‘𝑊)𝐵) = (0g‘𝑊))) |
27 | 3 | 3ad2ant1 1132 |
. . . . . 6
⊢ ((𝑊 ∈ PreHil ∧ 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) → 𝑊 ∈ LMod) |
28 | 16 | lmodfgrp 20130 |
. . . . . 6
⊢ (𝑊 ∈ LMod →
(Scalar‘𝑊) ∈
Grp) |
29 | 27, 28 | syl 17 |
. . . . 5
⊢ ((𝑊 ∈ PreHil ∧ 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) → (Scalar‘𝑊) ∈ Grp) |
30 | | eqid 2740 |
. . . . . . 7
⊢
(Base‘(Scalar‘𝑊)) = (Base‘(Scalar‘𝑊)) |
31 | 16, 17, 4, 30 | ipcl 20836 |
. . . . . 6
⊢ ((𝑊 ∈ PreHil ∧ (𝐴(-g‘𝑊)𝐵) ∈ 𝑉 ∧ 𝐴 ∈ 𝑉) → ((𝐴(-g‘𝑊)𝐵) , 𝐴) ∈ (Base‘(Scalar‘𝑊))) |
32 | 13, 7, 14, 31 | syl3anc 1370 |
. . . . 5
⊢ ((𝑊 ∈ PreHil ∧ 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) → ((𝐴(-g‘𝑊)𝐵) , 𝐴) ∈ (Base‘(Scalar‘𝑊))) |
33 | 16, 17, 4, 30 | ipcl 20836 |
. . . . . 6
⊢ ((𝑊 ∈ PreHil ∧ (𝐴(-g‘𝑊)𝐵) ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) → ((𝐴(-g‘𝑊)𝐵) , 𝐵) ∈ (Base‘(Scalar‘𝑊))) |
34 | 13, 7, 15, 33 | syl3anc 1370 |
. . . . 5
⊢ ((𝑊 ∈ PreHil ∧ 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) → ((𝐴(-g‘𝑊)𝐵) , 𝐵) ∈ (Base‘(Scalar‘𝑊))) |
35 | 30, 22, 18 | grpsubeq0 18659 |
. . . . 5
⊢
(((Scalar‘𝑊)
∈ Grp ∧ ((𝐴(-g‘𝑊)𝐵) , 𝐴) ∈ (Base‘(Scalar‘𝑊)) ∧ ((𝐴(-g‘𝑊)𝐵) , 𝐵) ∈ (Base‘(Scalar‘𝑊))) → ((((𝐴(-g‘𝑊)𝐵) , 𝐴)(-g‘(Scalar‘𝑊))((𝐴(-g‘𝑊)𝐵) , 𝐵)) =
(0g‘(Scalar‘𝑊)) ↔ ((𝐴(-g‘𝑊)𝐵) , 𝐴) = ((𝐴(-g‘𝑊)𝐵) , 𝐵))) |
36 | 29, 32, 34, 35 | syl3anc 1370 |
. . . 4
⊢ ((𝑊 ∈ PreHil ∧ 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) → ((((𝐴(-g‘𝑊)𝐵) , 𝐴)(-g‘(Scalar‘𝑊))((𝐴(-g‘𝑊)𝐵) , 𝐵)) =
(0g‘(Scalar‘𝑊)) ↔ ((𝐴(-g‘𝑊)𝐵) , 𝐴) = ((𝐴(-g‘𝑊)𝐵) , 𝐵))) |
37 | | lmodgrp 20128 |
. . . . . 6
⊢ (𝑊 ∈ LMod → 𝑊 ∈ Grp) |
38 | 3, 37 | syl 17 |
. . . . 5
⊢ (𝑊 ∈ PreHil → 𝑊 ∈ Grp) |
39 | 4, 23, 5 | grpsubeq0 18659 |
. . . . 5
⊢ ((𝑊 ∈ Grp ∧ 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) → ((𝐴(-g‘𝑊)𝐵) = (0g‘𝑊) ↔ 𝐴 = 𝐵)) |
40 | 38, 39 | syl3an1 1162 |
. . . 4
⊢ ((𝑊 ∈ PreHil ∧ 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) → ((𝐴(-g‘𝑊)𝐵) = (0g‘𝑊) ↔ 𝐴 = 𝐵)) |
41 | 26, 36, 40 | 3bitr3d 309 |
. . 3
⊢ ((𝑊 ∈ PreHil ∧ 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) → (((𝐴(-g‘𝑊)𝐵) , 𝐴) = ((𝐴(-g‘𝑊)𝐵) , 𝐵) ↔ 𝐴 = 𝐵)) |
42 | 12, 41 | sylibd 238 |
. 2
⊢ ((𝑊 ∈ PreHil ∧ 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) → (∀𝑥 ∈ 𝑉 (𝑥 , 𝐴) = (𝑥 , 𝐵) → 𝐴 = 𝐵)) |
43 | 2, 42 | impbid2 225 |
1
⊢ ((𝑊 ∈ PreHil ∧ 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) → (𝐴 = 𝐵 ↔ ∀𝑥 ∈ 𝑉 (𝑥 , 𝐴) = (𝑥 , 𝐵))) |