Step |
Hyp |
Ref |
Expression |
1 |
|
ecgrtg.1 |
⊢ ( 𝜑 → 𝑁 ∈ ℕ ) |
2 |
|
ecgrtg.2 |
⊢ 𝑃 = ( Base ‘ ( EEG ‘ 𝑁 ) ) |
3 |
|
ecgrtg.3 |
⊢ − = ( dist ‘ ( EEG ‘ 𝑁 ) ) |
4 |
|
ecgrtg.a |
⊢ ( 𝜑 → 𝐴 ∈ 𝑃 ) |
5 |
|
ecgrtg.b |
⊢ ( 𝜑 → 𝐵 ∈ 𝑃 ) |
6 |
|
ecgrtg.c |
⊢ ( 𝜑 → 𝐶 ∈ 𝑃 ) |
7 |
|
ecgrtg.d |
⊢ ( 𝜑 → 𝐷 ∈ 𝑃 ) |
8 |
|
eengbas |
⊢ ( 𝑁 ∈ ℕ → ( 𝔼 ‘ 𝑁 ) = ( Base ‘ ( EEG ‘ 𝑁 ) ) ) |
9 |
1 8
|
syl |
⊢ ( 𝜑 → ( 𝔼 ‘ 𝑁 ) = ( Base ‘ ( EEG ‘ 𝑁 ) ) ) |
10 |
9 2
|
eqtr4di |
⊢ ( 𝜑 → ( 𝔼 ‘ 𝑁 ) = 𝑃 ) |
11 |
4 10
|
eleqtrrd |
⊢ ( 𝜑 → 𝐴 ∈ ( 𝔼 ‘ 𝑁 ) ) |
12 |
5 10
|
eleqtrrd |
⊢ ( 𝜑 → 𝐵 ∈ ( 𝔼 ‘ 𝑁 ) ) |
13 |
6 10
|
eleqtrrd |
⊢ ( 𝜑 → 𝐶 ∈ ( 𝔼 ‘ 𝑁 ) ) |
14 |
7 10
|
eleqtrrd |
⊢ ( 𝜑 → 𝐷 ∈ ( 𝔼 ‘ 𝑁 ) ) |
15 |
|
brcgr |
⊢ ( ( ( 𝐴 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐵 ∈ ( 𝔼 ‘ 𝑁 ) ) ∧ ( 𝐶 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐷 ∈ ( 𝔼 ‘ 𝑁 ) ) ) → ( 〈 𝐴 , 𝐵 〉 Cgr 〈 𝐶 , 𝐷 〉 ↔ Σ 𝑖 ∈ ( 1 ... 𝑁 ) ( ( ( 𝐴 ‘ 𝑖 ) − ( 𝐵 ‘ 𝑖 ) ) ↑ 2 ) = Σ 𝑖 ∈ ( 1 ... 𝑁 ) ( ( ( 𝐶 ‘ 𝑖 ) − ( 𝐷 ‘ 𝑖 ) ) ↑ 2 ) ) ) |
16 |
11 12 13 14 15
|
syl22anc |
⊢ ( 𝜑 → ( 〈 𝐴 , 𝐵 〉 Cgr 〈 𝐶 , 𝐷 〉 ↔ Σ 𝑖 ∈ ( 1 ... 𝑁 ) ( ( ( 𝐴 ‘ 𝑖 ) − ( 𝐵 ‘ 𝑖 ) ) ↑ 2 ) = Σ 𝑖 ∈ ( 1 ... 𝑁 ) ( ( ( 𝐶 ‘ 𝑖 ) − ( 𝐷 ‘ 𝑖 ) ) ↑ 2 ) ) ) |
17 |
|
dsid |
⊢ dist = Slot ( dist ‘ ndx ) |
18 |
|
fvexd |
⊢ ( 𝜑 → ( EEG ‘ 𝑁 ) ∈ V ) |
19 |
|
eengstr |
⊢ ( 𝑁 ∈ ℕ → ( EEG ‘ 𝑁 ) Struct 〈 1 , ; 1 7 〉 ) |
20 |
1 19
|
syl |
⊢ ( 𝜑 → ( EEG ‘ 𝑁 ) Struct 〈 1 , ; 1 7 〉 ) |
21 |
|
structn0fun |
⊢ ( ( EEG ‘ 𝑁 ) Struct 〈 1 , ; 1 7 〉 → Fun ( ( EEG ‘ 𝑁 ) ∖ { ∅ } ) ) |
22 |
20 21
|
syl |
⊢ ( 𝜑 → Fun ( ( EEG ‘ 𝑁 ) ∖ { ∅ } ) ) |
23 |
|
structcnvcnv |
⊢ ( ( EEG ‘ 𝑁 ) Struct 〈 1 , ; 1 7 〉 → ◡ ◡ ( EEG ‘ 𝑁 ) = ( ( EEG ‘ 𝑁 ) ∖ { ∅ } ) ) |
24 |
20 23
|
syl |
⊢ ( 𝜑 → ◡ ◡ ( EEG ‘ 𝑁 ) = ( ( EEG ‘ 𝑁 ) ∖ { ∅ } ) ) |
25 |
24
|
funeqd |
⊢ ( 𝜑 → ( Fun ◡ ◡ ( EEG ‘ 𝑁 ) ↔ Fun ( ( EEG ‘ 𝑁 ) ∖ { ∅ } ) ) ) |
26 |
22 25
|
mpbird |
⊢ ( 𝜑 → Fun ◡ ◡ ( EEG ‘ 𝑁 ) ) |
27 |
|
opex |
⊢ 〈 ( dist ‘ ndx ) , ( 𝑥 ∈ ( 𝔼 ‘ 𝑁 ) , 𝑦 ∈ ( 𝔼 ‘ 𝑁 ) ↦ Σ 𝑖 ∈ ( 1 ... 𝑁 ) ( ( ( 𝑥 ‘ 𝑖 ) − ( 𝑦 ‘ 𝑖 ) ) ↑ 2 ) ) 〉 ∈ V |
28 |
27
|
prid2 |
⊢ 〈 ( dist ‘ ndx ) , ( 𝑥 ∈ ( 𝔼 ‘ 𝑁 ) , 𝑦 ∈ ( 𝔼 ‘ 𝑁 ) ↦ Σ 𝑖 ∈ ( 1 ... 𝑁 ) ( ( ( 𝑥 ‘ 𝑖 ) − ( 𝑦 ‘ 𝑖 ) ) ↑ 2 ) ) 〉 ∈ { 〈 ( Base ‘ ndx ) , ( 𝔼 ‘ 𝑁 ) 〉 , 〈 ( dist ‘ ndx ) , ( 𝑥 ∈ ( 𝔼 ‘ 𝑁 ) , 𝑦 ∈ ( 𝔼 ‘ 𝑁 ) ↦ Σ 𝑖 ∈ ( 1 ... 𝑁 ) ( ( ( 𝑥 ‘ 𝑖 ) − ( 𝑦 ‘ 𝑖 ) ) ↑ 2 ) ) 〉 } |
29 |
|
elun1 |
⊢ ( 〈 ( dist ‘ ndx ) , ( 𝑥 ∈ ( 𝔼 ‘ 𝑁 ) , 𝑦 ∈ ( 𝔼 ‘ 𝑁 ) ↦ Σ 𝑖 ∈ ( 1 ... 𝑁 ) ( ( ( 𝑥 ‘ 𝑖 ) − ( 𝑦 ‘ 𝑖 ) ) ↑ 2 ) ) 〉 ∈ { 〈 ( Base ‘ ndx ) , ( 𝔼 ‘ 𝑁 ) 〉 , 〈 ( dist ‘ ndx ) , ( 𝑥 ∈ ( 𝔼 ‘ 𝑁 ) , 𝑦 ∈ ( 𝔼 ‘ 𝑁 ) ↦ Σ 𝑖 ∈ ( 1 ... 𝑁 ) ( ( ( 𝑥 ‘ 𝑖 ) − ( 𝑦 ‘ 𝑖 ) ) ↑ 2 ) ) 〉 } → 〈 ( dist ‘ ndx ) , ( 𝑥 ∈ ( 𝔼 ‘ 𝑁 ) , 𝑦 ∈ ( 𝔼 ‘ 𝑁 ) ↦ Σ 𝑖 ∈ ( 1 ... 𝑁 ) ( ( ( 𝑥 ‘ 𝑖 ) − ( 𝑦 ‘ 𝑖 ) ) ↑ 2 ) ) 〉 ∈ ( { 〈 ( Base ‘ ndx ) , ( 𝔼 ‘ 𝑁 ) 〉 , 〈 ( dist ‘ ndx ) , ( 𝑥 ∈ ( 𝔼 ‘ 𝑁 ) , 𝑦 ∈ ( 𝔼 ‘ 𝑁 ) ↦ Σ 𝑖 ∈ ( 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 |
⊢ 〈 ( dist ‘ ndx ) , ( 𝑥 ∈ ( 𝔼 ‘ 𝑁 ) , 𝑦 ∈ ( 𝔼 ‘ 𝑁 ) ↦ Σ 𝑖 ∈ ( 1 ... 𝑁 ) ( ( ( 𝑥 ‘ 𝑖 ) − ( 𝑦 ‘ 𝑖 ) ) ↑ 2 ) ) 〉 ∈ ( { 〈 ( Base ‘ ndx ) , ( 𝔼 ‘ 𝑁 ) 〉 , 〈 ( dist ‘ ndx ) , ( 𝑥 ∈ ( 𝔼 ‘ 𝑁 ) , 𝑦 ∈ ( 𝔼 ‘ 𝑁 ) ↦ Σ 𝑖 ∈ ( 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 |
⊢ ( 𝑁 ∈ ℕ → ( EEG ‘ 𝑁 ) = ( { 〈 ( Base ‘ ndx ) , ( 𝔼 ‘ 𝑁 ) 〉 , 〈 ( dist ‘ ndx ) , ( 𝑥 ∈ ( 𝔼 ‘ 𝑁 ) , 𝑦 ∈ ( 𝔼 ‘ 𝑁 ) ↦ Σ 𝑖 ∈ ( 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 |
1 31
|
syl |
⊢ ( 𝜑 → ( EEG ‘ 𝑁 ) = ( { 〈 ( Base ‘ ndx ) , ( 𝔼 ‘ 𝑁 ) 〉 , 〈 ( dist ‘ ndx ) , ( 𝑥 ∈ ( 𝔼 ‘ 𝑁 ) , 𝑦 ∈ ( 𝔼 ‘ 𝑁 ) ↦ Σ 𝑖 ∈ ( 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 |
⊢ ( 𝜑 → 〈 ( dist ‘ ndx ) , ( 𝑥 ∈ ( 𝔼 ‘ 𝑁 ) , 𝑦 ∈ ( 𝔼 ‘ 𝑁 ) ↦ Σ 𝑖 ∈ ( 1 ... 𝑁 ) ( ( ( 𝑥 ‘ 𝑖 ) − ( 𝑦 ‘ 𝑖 ) ) ↑ 2 ) ) 〉 ∈ ( EEG ‘ 𝑁 ) ) |
34 |
|
fvex |
⊢ ( 𝔼 ‘ 𝑁 ) ∈ V |
35 |
34 34
|
mpoex |
⊢ ( 𝑥 ∈ ( 𝔼 ‘ 𝑁 ) , 𝑦 ∈ ( 𝔼 ‘ 𝑁 ) ↦ Σ 𝑖 ∈ ( 1 ... 𝑁 ) ( ( ( 𝑥 ‘ 𝑖 ) − ( 𝑦 ‘ 𝑖 ) ) ↑ 2 ) ) ∈ V |
36 |
35
|
a1i |
⊢ ( 𝜑 → ( 𝑥 ∈ ( 𝔼 ‘ 𝑁 ) , 𝑦 ∈ ( 𝔼 ‘ 𝑁 ) ↦ Σ 𝑖 ∈ ( 1 ... 𝑁 ) ( ( ( 𝑥 ‘ 𝑖 ) − ( 𝑦 ‘ 𝑖 ) ) ↑ 2 ) ) ∈ V ) |
37 |
17 18 26 33 36
|
strfv2d |
⊢ ( 𝜑 → ( 𝑥 ∈ ( 𝔼 ‘ 𝑁 ) , 𝑦 ∈ ( 𝔼 ‘ 𝑁 ) ↦ Σ 𝑖 ∈ ( 1 ... 𝑁 ) ( ( ( 𝑥 ‘ 𝑖 ) − ( 𝑦 ‘ 𝑖 ) ) ↑ 2 ) ) = ( dist ‘ ( EEG ‘ 𝑁 ) ) ) |
38 |
3 37
|
eqtr4id |
⊢ ( 𝜑 → − = ( 𝑥 ∈ ( 𝔼 ‘ 𝑁 ) , 𝑦 ∈ ( 𝔼 ‘ 𝑁 ) ↦ Σ 𝑖 ∈ ( 1 ... 𝑁 ) ( ( ( 𝑥 ‘ 𝑖 ) − ( 𝑦 ‘ 𝑖 ) ) ↑ 2 ) ) ) |
39 |
|
simplrl |
⊢ ( ( ( 𝜑 ∧ ( 𝑥 = 𝐴 ∧ 𝑦 = 𝐵 ) ) ∧ 𝑖 ∈ ( 1 ... 𝑁 ) ) → 𝑥 = 𝐴 ) |
40 |
39
|
fveq1d |
⊢ ( ( ( 𝜑 ∧ ( 𝑥 = 𝐴 ∧ 𝑦 = 𝐵 ) ) ∧ 𝑖 ∈ ( 1 ... 𝑁 ) ) → ( 𝑥 ‘ 𝑖 ) = ( 𝐴 ‘ 𝑖 ) ) |
41 |
|
simplrr |
⊢ ( ( ( 𝜑 ∧ ( 𝑥 = 𝐴 ∧ 𝑦 = 𝐵 ) ) ∧ 𝑖 ∈ ( 1 ... 𝑁 ) ) → 𝑦 = 𝐵 ) |
42 |
41
|
fveq1d |
⊢ ( ( ( 𝜑 ∧ ( 𝑥 = 𝐴 ∧ 𝑦 = 𝐵 ) ) ∧ 𝑖 ∈ ( 1 ... 𝑁 ) ) → ( 𝑦 ‘ 𝑖 ) = ( 𝐵 ‘ 𝑖 ) ) |
43 |
40 42
|
oveq12d |
⊢ ( ( ( 𝜑 ∧ ( 𝑥 = 𝐴 ∧ 𝑦 = 𝐵 ) ) ∧ 𝑖 ∈ ( 1 ... 𝑁 ) ) → ( ( 𝑥 ‘ 𝑖 ) − ( 𝑦 ‘ 𝑖 ) ) = ( ( 𝐴 ‘ 𝑖 ) − ( 𝐵 ‘ 𝑖 ) ) ) |
44 |
43
|
oveq1d |
⊢ ( ( ( 𝜑 ∧ ( 𝑥 = 𝐴 ∧ 𝑦 = 𝐵 ) ) ∧ 𝑖 ∈ ( 1 ... 𝑁 ) ) → ( ( ( 𝑥 ‘ 𝑖 ) − ( 𝑦 ‘ 𝑖 ) ) ↑ 2 ) = ( ( ( 𝐴 ‘ 𝑖 ) − ( 𝐵 ‘ 𝑖 ) ) ↑ 2 ) ) |
45 |
44
|
sumeq2dv |
⊢ ( ( 𝜑 ∧ ( 𝑥 = 𝐴 ∧ 𝑦 = 𝐵 ) ) → Σ 𝑖 ∈ ( 1 ... 𝑁 ) ( ( ( 𝑥 ‘ 𝑖 ) − ( 𝑦 ‘ 𝑖 ) ) ↑ 2 ) = Σ 𝑖 ∈ ( 1 ... 𝑁 ) ( ( ( 𝐴 ‘ 𝑖 ) − ( 𝐵 ‘ 𝑖 ) ) ↑ 2 ) ) |
46 |
|
sumex |
⊢ Σ 𝑖 ∈ ( 1 ... 𝑁 ) ( ( ( 𝐴 ‘ 𝑖 ) − ( 𝐵 ‘ 𝑖 ) ) ↑ 2 ) ∈ V |
47 |
46
|
a1i |
⊢ ( 𝜑 → Σ 𝑖 ∈ ( 1 ... 𝑁 ) ( ( ( 𝐴 ‘ 𝑖 ) − ( 𝐵 ‘ 𝑖 ) ) ↑ 2 ) ∈ V ) |
48 |
38 45 11 12 47
|
ovmpod |
⊢ ( 𝜑 → ( 𝐴 − 𝐵 ) = Σ 𝑖 ∈ ( 1 ... 𝑁 ) ( ( ( 𝐴 ‘ 𝑖 ) − ( 𝐵 ‘ 𝑖 ) ) ↑ 2 ) ) |
49 |
48
|
eqcomd |
⊢ ( 𝜑 → Σ 𝑖 ∈ ( 1 ... 𝑁 ) ( ( ( 𝐴 ‘ 𝑖 ) − ( 𝐵 ‘ 𝑖 ) ) ↑ 2 ) = ( 𝐴 − 𝐵 ) ) |
50 |
|
simplrl |
⊢ ( ( ( 𝜑 ∧ ( 𝑥 = 𝐶 ∧ 𝑦 = 𝐷 ) ) ∧ 𝑖 ∈ ( 1 ... 𝑁 ) ) → 𝑥 = 𝐶 ) |
51 |
50
|
fveq1d |
⊢ ( ( ( 𝜑 ∧ ( 𝑥 = 𝐶 ∧ 𝑦 = 𝐷 ) ) ∧ 𝑖 ∈ ( 1 ... 𝑁 ) ) → ( 𝑥 ‘ 𝑖 ) = ( 𝐶 ‘ 𝑖 ) ) |
52 |
|
simplrr |
⊢ ( ( ( 𝜑 ∧ ( 𝑥 = 𝐶 ∧ 𝑦 = 𝐷 ) ) ∧ 𝑖 ∈ ( 1 ... 𝑁 ) ) → 𝑦 = 𝐷 ) |
53 |
52
|
fveq1d |
⊢ ( ( ( 𝜑 ∧ ( 𝑥 = 𝐶 ∧ 𝑦 = 𝐷 ) ) ∧ 𝑖 ∈ ( 1 ... 𝑁 ) ) → ( 𝑦 ‘ 𝑖 ) = ( 𝐷 ‘ 𝑖 ) ) |
54 |
51 53
|
oveq12d |
⊢ ( ( ( 𝜑 ∧ ( 𝑥 = 𝐶 ∧ 𝑦 = 𝐷 ) ) ∧ 𝑖 ∈ ( 1 ... 𝑁 ) ) → ( ( 𝑥 ‘ 𝑖 ) − ( 𝑦 ‘ 𝑖 ) ) = ( ( 𝐶 ‘ 𝑖 ) − ( 𝐷 ‘ 𝑖 ) ) ) |
55 |
54
|
oveq1d |
⊢ ( ( ( 𝜑 ∧ ( 𝑥 = 𝐶 ∧ 𝑦 = 𝐷 ) ) ∧ 𝑖 ∈ ( 1 ... 𝑁 ) ) → ( ( ( 𝑥 ‘ 𝑖 ) − ( 𝑦 ‘ 𝑖 ) ) ↑ 2 ) = ( ( ( 𝐶 ‘ 𝑖 ) − ( 𝐷 ‘ 𝑖 ) ) ↑ 2 ) ) |
56 |
55
|
sumeq2dv |
⊢ ( ( 𝜑 ∧ ( 𝑥 = 𝐶 ∧ 𝑦 = 𝐷 ) ) → Σ 𝑖 ∈ ( 1 ... 𝑁 ) ( ( ( 𝑥 ‘ 𝑖 ) − ( 𝑦 ‘ 𝑖 ) ) ↑ 2 ) = Σ 𝑖 ∈ ( 1 ... 𝑁 ) ( ( ( 𝐶 ‘ 𝑖 ) − ( 𝐷 ‘ 𝑖 ) ) ↑ 2 ) ) |
57 |
|
sumex |
⊢ Σ 𝑖 ∈ ( 1 ... 𝑁 ) ( ( ( 𝐶 ‘ 𝑖 ) − ( 𝐷 ‘ 𝑖 ) ) ↑ 2 ) ∈ V |
58 |
57
|
a1i |
⊢ ( 𝜑 → Σ 𝑖 ∈ ( 1 ... 𝑁 ) ( ( ( 𝐶 ‘ 𝑖 ) − ( 𝐷 ‘ 𝑖 ) ) ↑ 2 ) ∈ V ) |
59 |
38 56 13 14 58
|
ovmpod |
⊢ ( 𝜑 → ( 𝐶 − 𝐷 ) = Σ 𝑖 ∈ ( 1 ... 𝑁 ) ( ( ( 𝐶 ‘ 𝑖 ) − ( 𝐷 ‘ 𝑖 ) ) ↑ 2 ) ) |
60 |
59
|
eqcomd |
⊢ ( 𝜑 → Σ 𝑖 ∈ ( 1 ... 𝑁 ) ( ( ( 𝐶 ‘ 𝑖 ) − ( 𝐷 ‘ 𝑖 ) ) ↑ 2 ) = ( 𝐶 − 𝐷 ) ) |
61 |
49 60
|
eqeq12d |
⊢ ( 𝜑 → ( Σ 𝑖 ∈ ( 1 ... 𝑁 ) ( ( ( 𝐴 ‘ 𝑖 ) − ( 𝐵 ‘ 𝑖 ) ) ↑ 2 ) = Σ 𝑖 ∈ ( 1 ... 𝑁 ) ( ( ( 𝐶 ‘ 𝑖 ) − ( 𝐷 ‘ 𝑖 ) ) ↑ 2 ) ↔ ( 𝐴 − 𝐵 ) = ( 𝐶 − 𝐷 ) ) ) |
62 |
16 61
|
bitrd |
⊢ ( 𝜑 → ( 〈 𝐴 , 𝐵 〉 Cgr 〈 𝐶 , 𝐷 〉 ↔ ( 𝐴 − 𝐵 ) = ( 𝐶 − 𝐷 ) ) ) |