Step | Hyp | Ref
| Expression |
1 | | ecgrtg.a |
. . . 4
⊢ (𝜑 → 𝐴 ∈ 𝑃) |
2 | | ecgrtg.1 |
. . . . . 6
⊢ (𝜑 → 𝑁 ∈ ℕ) |
3 | | eengbas 27349 |
. . . . . 6
⊢ (𝑁 ∈ ℕ →
(𝔼‘𝑁) =
(Base‘(EEG‘𝑁))) |
4 | 2, 3 | syl 17 |
. . . . 5
⊢ (𝜑 → (𝔼‘𝑁) = (Base‘(EEG‘𝑁))) |
5 | | ecgrtg.2 |
. . . . 5
⊢ 𝑃 = (Base‘(EEG‘𝑁)) |
6 | 4, 5 | eqtr4di 2796 |
. . . 4
⊢ (𝜑 → (𝔼‘𝑁) = 𝑃) |
7 | 1, 6 | eleqtrrd 2842 |
. . 3
⊢ (𝜑 → 𝐴 ∈ (𝔼‘𝑁)) |
8 | | ecgrtg.b |
. . . 4
⊢ (𝜑 → 𝐵 ∈ 𝑃) |
9 | 8, 6 | eleqtrrd 2842 |
. . 3
⊢ (𝜑 → 𝐵 ∈ (𝔼‘𝑁)) |
10 | | ecgrtg.c |
. . . 4
⊢ (𝜑 → 𝐶 ∈ 𝑃) |
11 | 10, 6 | eleqtrrd 2842 |
. . 3
⊢ (𝜑 → 𝐶 ∈ (𝔼‘𝑁)) |
12 | | ecgrtg.d |
. . . 4
⊢ (𝜑 → 𝐷 ∈ 𝑃) |
13 | 12, 6 | eleqtrrd 2842 |
. . 3
⊢ (𝜑 → 𝐷 ∈ (𝔼‘𝑁)) |
14 | | brcgr 27268 |
. . 3
⊢ (((𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) ∧ (𝐶 ∈ (𝔼‘𝑁) ∧ 𝐷 ∈ (𝔼‘𝑁))) → (〈𝐴, 𝐵〉Cgr〈𝐶, 𝐷〉 ↔ Σ𝑖 ∈ (1...𝑁)(((𝐴‘𝑖) − (𝐵‘𝑖))↑2) = Σ𝑖 ∈ (1...𝑁)(((𝐶‘𝑖) − (𝐷‘𝑖))↑2))) |
15 | 7, 9, 11, 13, 14 | syl22anc 836 |
. 2
⊢ (𝜑 → (〈𝐴, 𝐵〉Cgr〈𝐶, 𝐷〉 ↔ Σ𝑖 ∈ (1...𝑁)(((𝐴‘𝑖) − (𝐵‘𝑖))↑2) = Σ𝑖 ∈ (1...𝑁)(((𝐶‘𝑖) − (𝐷‘𝑖))↑2))) |
16 | | ecgrtg.3 |
. . . . . 6
⊢ − =
(dist‘(EEG‘𝑁)) |
17 | | dsid 17096 |
. . . . . . 7
⊢ dist =
Slot (dist‘ndx) |
18 | | fvexd 6789 |
. . . . . . 7
⊢ (𝜑 → (EEG‘𝑁) ∈ V) |
19 | | eengstr 27348 |
. . . . . . . . . 10
⊢ (𝑁 ∈ ℕ →
(EEG‘𝑁) Struct
〈1, ;17〉) |
20 | 2, 19 | syl 17 |
. . . . . . . . 9
⊢ (𝜑 → (EEG‘𝑁) Struct 〈1, ;17〉) |
21 | | structn0fun 16852 |
. . . . . . . . 9
⊢
((EEG‘𝑁)
Struct 〈1, ;17〉 →
Fun ((EEG‘𝑁) ∖
{∅})) |
22 | 20, 21 | syl 17 |
. . . . . . . 8
⊢ (𝜑 → Fun ((EEG‘𝑁) ∖
{∅})) |
23 | | structcnvcnv 16854 |
. . . . . . . . . 10
⊢
((EEG‘𝑁)
Struct 〈1, ;17〉 →
◡◡(EEG‘𝑁) = ((EEG‘𝑁) ∖ {∅})) |
24 | 20, 23 | syl 17 |
. . . . . . . . 9
⊢ (𝜑 → ◡◡(EEG‘𝑁) = ((EEG‘𝑁) ∖ {∅})) |
25 | 24 | funeqd 6456 |
. . . . . . . 8
⊢ (𝜑 → (Fun ◡◡(EEG‘𝑁) ↔ Fun ((EEG‘𝑁) ∖ {∅}))) |
26 | 22, 25 | mpbird 256 |
. . . . . . 7
⊢ (𝜑 → Fun ◡◡(EEG‘𝑁)) |
27 | | opex 5379 |
. . . . . . . . . 10
⊢
〈(dist‘ndx), (𝑥 ∈ (𝔼‘𝑁), 𝑦 ∈ (𝔼‘𝑁) ↦ Σ𝑖 ∈ (1...𝑁)(((𝑥‘𝑖) − (𝑦‘𝑖))↑2))〉 ∈ V |
28 | 27 | prid2 4699 |
. . . . . . . . 9
⊢
〈(dist‘ndx), (𝑥 ∈ (𝔼‘𝑁), 𝑦 ∈ (𝔼‘𝑁) ↦ Σ𝑖 ∈ (1...𝑁)(((𝑥‘𝑖) − (𝑦‘𝑖))↑2))〉 ∈
{〈(Base‘ndx), (𝔼‘𝑁)〉, 〈(dist‘ndx), (𝑥 ∈ (𝔼‘𝑁), 𝑦 ∈ (𝔼‘𝑁) ↦ Σ𝑖 ∈ (1...𝑁)(((𝑥‘𝑖) − (𝑦‘𝑖))↑2))〉} |
29 | | elun1 4110 |
. . . . . . . . 9
⊢
(〈(dist‘ndx), (𝑥 ∈ (𝔼‘𝑁), 𝑦 ∈ (𝔼‘𝑁) ↦ Σ𝑖 ∈ (1...𝑁)(((𝑥‘𝑖) − (𝑦‘𝑖))↑2))〉 ∈
{〈(Base‘ndx), (𝔼‘𝑁)〉, 〈(dist‘ndx), (𝑥 ∈ (𝔼‘𝑁), 𝑦 ∈ (𝔼‘𝑁) ↦ Σ𝑖 ∈ (1...𝑁)(((𝑥‘𝑖) − (𝑦‘𝑖))↑2))〉} →
〈(dist‘ndx), (𝑥
∈ (𝔼‘𝑁),
𝑦 ∈
(𝔼‘𝑁) ↦
Σ𝑖 ∈ (1...𝑁)(((𝑥‘𝑖) − (𝑦‘𝑖))↑2))〉 ∈
({〈(Base‘ndx), (𝔼‘𝑁)〉, 〈(dist‘ndx), (𝑥 ∈ (𝔼‘𝑁), 𝑦 ∈ (𝔼‘𝑁) ↦ Σ𝑖 ∈ (1...𝑁)(((𝑥‘𝑖) − (𝑦‘𝑖))↑2))〉} ∪
{〈(Itv‘ndx), (𝑥
∈ (𝔼‘𝑁),
𝑦 ∈
(𝔼‘𝑁) ↦
{𝑧 ∈
(𝔼‘𝑁) ∣
𝑧 Btwn 〈𝑥, 𝑦〉})〉, 〈(LineG‘ndx),
(𝑥 ∈
(𝔼‘𝑁), 𝑦 ∈ ((𝔼‘𝑁) ∖ {𝑥}) ↦ {𝑧 ∈ (𝔼‘𝑁) ∣ (𝑧 Btwn 〈𝑥, 𝑦〉 ∨ 𝑥 Btwn 〈𝑧, 𝑦〉 ∨ 𝑦 Btwn 〈𝑥, 𝑧〉)})〉})) |
30 | 28, 29 | ax-mp 5 |
. . . . . . . 8
⊢
〈(dist‘ndx), (𝑥 ∈ (𝔼‘𝑁), 𝑦 ∈ (𝔼‘𝑁) ↦ Σ𝑖 ∈ (1...𝑁)(((𝑥‘𝑖) − (𝑦‘𝑖))↑2))〉 ∈
({〈(Base‘ndx), (𝔼‘𝑁)〉, 〈(dist‘ndx), (𝑥 ∈ (𝔼‘𝑁), 𝑦 ∈ (𝔼‘𝑁) ↦ Σ𝑖 ∈ (1...𝑁)(((𝑥‘𝑖) − (𝑦‘𝑖))↑2))〉} ∪
{〈(Itv‘ndx), (𝑥
∈ (𝔼‘𝑁),
𝑦 ∈
(𝔼‘𝑁) ↦
{𝑧 ∈
(𝔼‘𝑁) ∣
𝑧 Btwn 〈𝑥, 𝑦〉})〉, 〈(LineG‘ndx),
(𝑥 ∈
(𝔼‘𝑁), 𝑦 ∈ ((𝔼‘𝑁) ∖ {𝑥}) ↦ {𝑧 ∈ (𝔼‘𝑁) ∣ (𝑧 Btwn 〈𝑥, 𝑦〉 ∨ 𝑥 Btwn 〈𝑧, 𝑦〉 ∨ 𝑦 Btwn 〈𝑥, 𝑧〉)})〉}) |
31 | | eengv 27347 |
. . . . . . . . 9
⊢ (𝑁 ∈ ℕ →
(EEG‘𝑁) =
({〈(Base‘ndx), (𝔼‘𝑁)〉, 〈(dist‘ndx), (𝑥 ∈ (𝔼‘𝑁), 𝑦 ∈ (𝔼‘𝑁) ↦ Σ𝑖 ∈ (1...𝑁)(((𝑥‘𝑖) − (𝑦‘𝑖))↑2))〉} ∪
{〈(Itv‘ndx), (𝑥
∈ (𝔼‘𝑁),
𝑦 ∈
(𝔼‘𝑁) ↦
{𝑧 ∈
(𝔼‘𝑁) ∣
𝑧 Btwn 〈𝑥, 𝑦〉})〉, 〈(LineG‘ndx),
(𝑥 ∈
(𝔼‘𝑁), 𝑦 ∈ ((𝔼‘𝑁) ∖ {𝑥}) ↦ {𝑧 ∈ (𝔼‘𝑁) ∣ (𝑧 Btwn 〈𝑥, 𝑦〉 ∨ 𝑥 Btwn 〈𝑧, 𝑦〉 ∨ 𝑦 Btwn 〈𝑥, 𝑧〉)})〉})) |
32 | 2, 31 | syl 17 |
. . . . . . . 8
⊢ (𝜑 → (EEG‘𝑁) = ({〈(Base‘ndx),
(𝔼‘𝑁)〉,
〈(dist‘ndx), (𝑥
∈ (𝔼‘𝑁),
𝑦 ∈
(𝔼‘𝑁) ↦
Σ𝑖 ∈ (1...𝑁)(((𝑥‘𝑖) − (𝑦‘𝑖))↑2))〉} ∪
{〈(Itv‘ndx), (𝑥
∈ (𝔼‘𝑁),
𝑦 ∈
(𝔼‘𝑁) ↦
{𝑧 ∈
(𝔼‘𝑁) ∣
𝑧 Btwn 〈𝑥, 𝑦〉})〉, 〈(LineG‘ndx),
(𝑥 ∈
(𝔼‘𝑁), 𝑦 ∈ ((𝔼‘𝑁) ∖ {𝑥}) ↦ {𝑧 ∈ (𝔼‘𝑁) ∣ (𝑧 Btwn 〈𝑥, 𝑦〉 ∨ 𝑥 Btwn 〈𝑧, 𝑦〉 ∨ 𝑦 Btwn 〈𝑥, 𝑧〉)})〉})) |
33 | 30, 32 | eleqtrrid 2846 |
. . . . . . 7
⊢ (𝜑 → 〈(dist‘ndx),
(𝑥 ∈
(𝔼‘𝑁), 𝑦 ∈ (𝔼‘𝑁) ↦ Σ𝑖 ∈ (1...𝑁)(((𝑥‘𝑖) − (𝑦‘𝑖))↑2))〉 ∈ (EEG‘𝑁)) |
34 | | fvex 6787 |
. . . . . . . . 9
⊢
(𝔼‘𝑁)
∈ V |
35 | 34, 34 | mpoex 7920 |
. . . . . . . 8
⊢ (𝑥 ∈ (𝔼‘𝑁), 𝑦 ∈ (𝔼‘𝑁) ↦ Σ𝑖 ∈ (1...𝑁)(((𝑥‘𝑖) − (𝑦‘𝑖))↑2)) ∈ V |
36 | 35 | a1i 11 |
. . . . . . 7
⊢ (𝜑 → (𝑥 ∈ (𝔼‘𝑁), 𝑦 ∈ (𝔼‘𝑁) ↦ Σ𝑖 ∈ (1...𝑁)(((𝑥‘𝑖) − (𝑦‘𝑖))↑2)) ∈ V) |
37 | 17, 18, 26, 33, 36 | strfv2d 16903 |
. . . . . 6
⊢ (𝜑 → (𝑥 ∈ (𝔼‘𝑁), 𝑦 ∈ (𝔼‘𝑁) ↦ Σ𝑖 ∈ (1...𝑁)(((𝑥‘𝑖) − (𝑦‘𝑖))↑2)) = (dist‘(EEG‘𝑁))) |
38 | 16, 37 | eqtr4id 2797 |
. . . . 5
⊢ (𝜑 → − = (𝑥 ∈ (𝔼‘𝑁), 𝑦 ∈ (𝔼‘𝑁) ↦ Σ𝑖 ∈ (1...𝑁)(((𝑥‘𝑖) − (𝑦‘𝑖))↑2))) |
39 | | simplrl 774 |
. . . . . . . . 9
⊢ (((𝜑 ∧ (𝑥 = 𝐴 ∧ 𝑦 = 𝐵)) ∧ 𝑖 ∈ (1...𝑁)) → 𝑥 = 𝐴) |
40 | 39 | fveq1d 6776 |
. . . . . . . 8
⊢ (((𝜑 ∧ (𝑥 = 𝐴 ∧ 𝑦 = 𝐵)) ∧ 𝑖 ∈ (1...𝑁)) → (𝑥‘𝑖) = (𝐴‘𝑖)) |
41 | | simplrr 775 |
. . . . . . . . 9
⊢ (((𝜑 ∧ (𝑥 = 𝐴 ∧ 𝑦 = 𝐵)) ∧ 𝑖 ∈ (1...𝑁)) → 𝑦 = 𝐵) |
42 | 41 | fveq1d 6776 |
. . . . . . . 8
⊢ (((𝜑 ∧ (𝑥 = 𝐴 ∧ 𝑦 = 𝐵)) ∧ 𝑖 ∈ (1...𝑁)) → (𝑦‘𝑖) = (𝐵‘𝑖)) |
43 | 40, 42 | oveq12d 7293 |
. . . . . . 7
⊢ (((𝜑 ∧ (𝑥 = 𝐴 ∧ 𝑦 = 𝐵)) ∧ 𝑖 ∈ (1...𝑁)) → ((𝑥‘𝑖) − (𝑦‘𝑖)) = ((𝐴‘𝑖) − (𝐵‘𝑖))) |
44 | 43 | oveq1d 7290 |
. . . . . 6
⊢ (((𝜑 ∧ (𝑥 = 𝐴 ∧ 𝑦 = 𝐵)) ∧ 𝑖 ∈ (1...𝑁)) → (((𝑥‘𝑖) − (𝑦‘𝑖))↑2) = (((𝐴‘𝑖) − (𝐵‘𝑖))↑2)) |
45 | 44 | sumeq2dv 15415 |
. . . . 5
⊢ ((𝜑 ∧ (𝑥 = 𝐴 ∧ 𝑦 = 𝐵)) → Σ𝑖 ∈ (1...𝑁)(((𝑥‘𝑖) − (𝑦‘𝑖))↑2) = Σ𝑖 ∈ (1...𝑁)(((𝐴‘𝑖) − (𝐵‘𝑖))↑2)) |
46 | | sumex 15399 |
. . . . . 6
⊢
Σ𝑖 ∈
(1...𝑁)(((𝐴‘𝑖) − (𝐵‘𝑖))↑2) ∈ V |
47 | 46 | a1i 11 |
. . . . 5
⊢ (𝜑 → Σ𝑖 ∈ (1...𝑁)(((𝐴‘𝑖) − (𝐵‘𝑖))↑2) ∈ V) |
48 | 38, 45, 7, 9, 47 | ovmpod 7425 |
. . . 4
⊢ (𝜑 → (𝐴 − 𝐵) = Σ𝑖 ∈ (1...𝑁)(((𝐴‘𝑖) − (𝐵‘𝑖))↑2)) |
49 | 48 | eqcomd 2744 |
. . 3
⊢ (𝜑 → Σ𝑖 ∈ (1...𝑁)(((𝐴‘𝑖) − (𝐵‘𝑖))↑2) = (𝐴 − 𝐵)) |
50 | | simplrl 774 |
. . . . . . . . 9
⊢ (((𝜑 ∧ (𝑥 = 𝐶 ∧ 𝑦 = 𝐷)) ∧ 𝑖 ∈ (1...𝑁)) → 𝑥 = 𝐶) |
51 | 50 | fveq1d 6776 |
. . . . . . . 8
⊢ (((𝜑 ∧ (𝑥 = 𝐶 ∧ 𝑦 = 𝐷)) ∧ 𝑖 ∈ (1...𝑁)) → (𝑥‘𝑖) = (𝐶‘𝑖)) |
52 | | simplrr 775 |
. . . . . . . . 9
⊢ (((𝜑 ∧ (𝑥 = 𝐶 ∧ 𝑦 = 𝐷)) ∧ 𝑖 ∈ (1...𝑁)) → 𝑦 = 𝐷) |
53 | 52 | fveq1d 6776 |
. . . . . . . 8
⊢ (((𝜑 ∧ (𝑥 = 𝐶 ∧ 𝑦 = 𝐷)) ∧ 𝑖 ∈ (1...𝑁)) → (𝑦‘𝑖) = (𝐷‘𝑖)) |
54 | 51, 53 | oveq12d 7293 |
. . . . . . 7
⊢ (((𝜑 ∧ (𝑥 = 𝐶 ∧ 𝑦 = 𝐷)) ∧ 𝑖 ∈ (1...𝑁)) → ((𝑥‘𝑖) − (𝑦‘𝑖)) = ((𝐶‘𝑖) − (𝐷‘𝑖))) |
55 | 54 | oveq1d 7290 |
. . . . . 6
⊢ (((𝜑 ∧ (𝑥 = 𝐶 ∧ 𝑦 = 𝐷)) ∧ 𝑖 ∈ (1...𝑁)) → (((𝑥‘𝑖) − (𝑦‘𝑖))↑2) = (((𝐶‘𝑖) − (𝐷‘𝑖))↑2)) |
56 | 55 | sumeq2dv 15415 |
. . . . 5
⊢ ((𝜑 ∧ (𝑥 = 𝐶 ∧ 𝑦 = 𝐷)) → Σ𝑖 ∈ (1...𝑁)(((𝑥‘𝑖) − (𝑦‘𝑖))↑2) = Σ𝑖 ∈ (1...𝑁)(((𝐶‘𝑖) − (𝐷‘𝑖))↑2)) |
57 | | sumex 15399 |
. . . . . 6
⊢
Σ𝑖 ∈
(1...𝑁)(((𝐶‘𝑖) − (𝐷‘𝑖))↑2) ∈ V |
58 | 57 | a1i 11 |
. . . . 5
⊢ (𝜑 → Σ𝑖 ∈ (1...𝑁)(((𝐶‘𝑖) − (𝐷‘𝑖))↑2) ∈ V) |
59 | 38, 56, 11, 13, 58 | ovmpod 7425 |
. . . 4
⊢ (𝜑 → (𝐶 − 𝐷) = Σ𝑖 ∈ (1...𝑁)(((𝐶‘𝑖) − (𝐷‘𝑖))↑2)) |
60 | 59 | eqcomd 2744 |
. . 3
⊢ (𝜑 → Σ𝑖 ∈ (1...𝑁)(((𝐶‘𝑖) − (𝐷‘𝑖))↑2) = (𝐶 − 𝐷)) |
61 | 49, 60 | eqeq12d 2754 |
. 2
⊢ (𝜑 → (Σ𝑖 ∈ (1...𝑁)(((𝐴‘𝑖) − (𝐵‘𝑖))↑2) = Σ𝑖 ∈ (1...𝑁)(((𝐶‘𝑖) − (𝐷‘𝑖))↑2) ↔ (𝐴 − 𝐵) = (𝐶 − 𝐷))) |
62 | 15, 61 | bitrd 278 |
1
⊢ (𝜑 → (〈𝐴, 𝐵〉Cgr〈𝐶, 𝐷〉 ↔ (𝐴 − 𝐵) = (𝐶 − 𝐷))) |