Step |
Hyp |
Ref |
Expression |
1 |
|
2sphere.i |
⊢ 𝐼 = { 1 , 2 } |
2 |
|
2sphere.e |
⊢ 𝐸 = ( ℝ^ ‘ 𝐼 ) |
3 |
|
2sphere.p |
⊢ 𝑃 = ( ℝ ↑m 𝐼 ) |
4 |
|
2sphere.s |
⊢ 𝑆 = ( Sphere ‘ 𝐸 ) |
5 |
|
2sphere.c |
⊢ 𝐶 = { 𝑝 ∈ 𝑃 ∣ ( ( ( ( 𝑝 ‘ 1 ) − ( 𝑀 ‘ 1 ) ) ↑ 2 ) + ( ( ( 𝑝 ‘ 2 ) − ( 𝑀 ‘ 2 ) ) ↑ 2 ) ) = ( 𝑅 ↑ 2 ) } |
6 |
|
prfi |
⊢ { 1 , 2 } ∈ Fin |
7 |
1 6
|
eqeltri |
⊢ 𝐼 ∈ Fin |
8 |
|
simpl |
⊢ ( ( 𝑀 ∈ 𝑃 ∧ 𝑅 ∈ ( 0 [,) +∞ ) ) → 𝑀 ∈ 𝑃 ) |
9 |
|
elrege0 |
⊢ ( 𝑅 ∈ ( 0 [,) +∞ ) ↔ ( 𝑅 ∈ ℝ ∧ 0 ≤ 𝑅 ) ) |
10 |
9
|
simplbi |
⊢ ( 𝑅 ∈ ( 0 [,) +∞ ) → 𝑅 ∈ ℝ ) |
11 |
10
|
adantl |
⊢ ( ( 𝑀 ∈ 𝑃 ∧ 𝑅 ∈ ( 0 [,) +∞ ) ) → 𝑅 ∈ ℝ ) |
12 |
|
eqid |
⊢ ( dist ‘ 𝐸 ) = ( dist ‘ 𝐸 ) |
13 |
2 3 12 4
|
rrxsphere |
⊢ ( ( 𝐼 ∈ Fin ∧ 𝑀 ∈ 𝑃 ∧ 𝑅 ∈ ℝ ) → ( 𝑀 𝑆 𝑅 ) = { 𝑝 ∈ 𝑃 ∣ ( 𝑝 ( dist ‘ 𝐸 ) 𝑀 ) = 𝑅 } ) |
14 |
7 8 11 13
|
mp3an2i |
⊢ ( ( 𝑀 ∈ 𝑃 ∧ 𝑅 ∈ ( 0 [,) +∞ ) ) → ( 𝑀 𝑆 𝑅 ) = { 𝑝 ∈ 𝑃 ∣ ( 𝑝 ( dist ‘ 𝐸 ) 𝑀 ) = 𝑅 } ) |
15 |
9
|
biimpi |
⊢ ( 𝑅 ∈ ( 0 [,) +∞ ) → ( 𝑅 ∈ ℝ ∧ 0 ≤ 𝑅 ) ) |
16 |
15
|
ad2antlr |
⊢ ( ( ( 𝑀 ∈ 𝑃 ∧ 𝑅 ∈ ( 0 [,) +∞ ) ) ∧ 𝑝 ∈ 𝑃 ) → ( 𝑅 ∈ ℝ ∧ 0 ≤ 𝑅 ) ) |
17 |
|
sqrtsq |
⊢ ( ( 𝑅 ∈ ℝ ∧ 0 ≤ 𝑅 ) → ( √ ‘ ( 𝑅 ↑ 2 ) ) = 𝑅 ) |
18 |
16 17
|
syl |
⊢ ( ( ( 𝑀 ∈ 𝑃 ∧ 𝑅 ∈ ( 0 [,) +∞ ) ) ∧ 𝑝 ∈ 𝑃 ) → ( √ ‘ ( 𝑅 ↑ 2 ) ) = 𝑅 ) |
19 |
18
|
eqeq2d |
⊢ ( ( ( 𝑀 ∈ 𝑃 ∧ 𝑅 ∈ ( 0 [,) +∞ ) ) ∧ 𝑝 ∈ 𝑃 ) → ( ( √ ‘ ( ( ( ( 𝑝 ‘ 1 ) − ( 𝑀 ‘ 1 ) ) ↑ 2 ) + ( ( ( 𝑝 ‘ 2 ) − ( 𝑀 ‘ 2 ) ) ↑ 2 ) ) ) = ( √ ‘ ( 𝑅 ↑ 2 ) ) ↔ ( √ ‘ ( ( ( ( 𝑝 ‘ 1 ) − ( 𝑀 ‘ 1 ) ) ↑ 2 ) + ( ( ( 𝑝 ‘ 2 ) − ( 𝑀 ‘ 2 ) ) ↑ 2 ) ) ) = 𝑅 ) ) |
20 |
1 3
|
rrx2pxel |
⊢ ( 𝑝 ∈ 𝑃 → ( 𝑝 ‘ 1 ) ∈ ℝ ) |
21 |
20
|
adantl |
⊢ ( ( 𝑀 ∈ 𝑃 ∧ 𝑝 ∈ 𝑃 ) → ( 𝑝 ‘ 1 ) ∈ ℝ ) |
22 |
1 3
|
rrx2pxel |
⊢ ( 𝑀 ∈ 𝑃 → ( 𝑀 ‘ 1 ) ∈ ℝ ) |
23 |
22
|
adantr |
⊢ ( ( 𝑀 ∈ 𝑃 ∧ 𝑝 ∈ 𝑃 ) → ( 𝑀 ‘ 1 ) ∈ ℝ ) |
24 |
21 23
|
resubcld |
⊢ ( ( 𝑀 ∈ 𝑃 ∧ 𝑝 ∈ 𝑃 ) → ( ( 𝑝 ‘ 1 ) − ( 𝑀 ‘ 1 ) ) ∈ ℝ ) |
25 |
24
|
resqcld |
⊢ ( ( 𝑀 ∈ 𝑃 ∧ 𝑝 ∈ 𝑃 ) → ( ( ( 𝑝 ‘ 1 ) − ( 𝑀 ‘ 1 ) ) ↑ 2 ) ∈ ℝ ) |
26 |
1 3
|
rrx2pyel |
⊢ ( 𝑝 ∈ 𝑃 → ( 𝑝 ‘ 2 ) ∈ ℝ ) |
27 |
26
|
adantl |
⊢ ( ( 𝑀 ∈ 𝑃 ∧ 𝑝 ∈ 𝑃 ) → ( 𝑝 ‘ 2 ) ∈ ℝ ) |
28 |
1 3
|
rrx2pyel |
⊢ ( 𝑀 ∈ 𝑃 → ( 𝑀 ‘ 2 ) ∈ ℝ ) |
29 |
28
|
adantr |
⊢ ( ( 𝑀 ∈ 𝑃 ∧ 𝑝 ∈ 𝑃 ) → ( 𝑀 ‘ 2 ) ∈ ℝ ) |
30 |
27 29
|
resubcld |
⊢ ( ( 𝑀 ∈ 𝑃 ∧ 𝑝 ∈ 𝑃 ) → ( ( 𝑝 ‘ 2 ) − ( 𝑀 ‘ 2 ) ) ∈ ℝ ) |
31 |
30
|
resqcld |
⊢ ( ( 𝑀 ∈ 𝑃 ∧ 𝑝 ∈ 𝑃 ) → ( ( ( 𝑝 ‘ 2 ) − ( 𝑀 ‘ 2 ) ) ↑ 2 ) ∈ ℝ ) |
32 |
25 31
|
readdcld |
⊢ ( ( 𝑀 ∈ 𝑃 ∧ 𝑝 ∈ 𝑃 ) → ( ( ( ( 𝑝 ‘ 1 ) − ( 𝑀 ‘ 1 ) ) ↑ 2 ) + ( ( ( 𝑝 ‘ 2 ) − ( 𝑀 ‘ 2 ) ) ↑ 2 ) ) ∈ ℝ ) |
33 |
24
|
sqge0d |
⊢ ( ( 𝑀 ∈ 𝑃 ∧ 𝑝 ∈ 𝑃 ) → 0 ≤ ( ( ( 𝑝 ‘ 1 ) − ( 𝑀 ‘ 1 ) ) ↑ 2 ) ) |
34 |
30
|
sqge0d |
⊢ ( ( 𝑀 ∈ 𝑃 ∧ 𝑝 ∈ 𝑃 ) → 0 ≤ ( ( ( 𝑝 ‘ 2 ) − ( 𝑀 ‘ 2 ) ) ↑ 2 ) ) |
35 |
25 31 33 34
|
addge0d |
⊢ ( ( 𝑀 ∈ 𝑃 ∧ 𝑝 ∈ 𝑃 ) → 0 ≤ ( ( ( ( 𝑝 ‘ 1 ) − ( 𝑀 ‘ 1 ) ) ↑ 2 ) + ( ( ( 𝑝 ‘ 2 ) − ( 𝑀 ‘ 2 ) ) ↑ 2 ) ) ) |
36 |
32 35
|
jca |
⊢ ( ( 𝑀 ∈ 𝑃 ∧ 𝑝 ∈ 𝑃 ) → ( ( ( ( ( 𝑝 ‘ 1 ) − ( 𝑀 ‘ 1 ) ) ↑ 2 ) + ( ( ( 𝑝 ‘ 2 ) − ( 𝑀 ‘ 2 ) ) ↑ 2 ) ) ∈ ℝ ∧ 0 ≤ ( ( ( ( 𝑝 ‘ 1 ) − ( 𝑀 ‘ 1 ) ) ↑ 2 ) + ( ( ( 𝑝 ‘ 2 ) − ( 𝑀 ‘ 2 ) ) ↑ 2 ) ) ) ) |
37 |
36
|
adantlr |
⊢ ( ( ( 𝑀 ∈ 𝑃 ∧ 𝑅 ∈ ( 0 [,) +∞ ) ) ∧ 𝑝 ∈ 𝑃 ) → ( ( ( ( ( 𝑝 ‘ 1 ) − ( 𝑀 ‘ 1 ) ) ↑ 2 ) + ( ( ( 𝑝 ‘ 2 ) − ( 𝑀 ‘ 2 ) ) ↑ 2 ) ) ∈ ℝ ∧ 0 ≤ ( ( ( ( 𝑝 ‘ 1 ) − ( 𝑀 ‘ 1 ) ) ↑ 2 ) + ( ( ( 𝑝 ‘ 2 ) − ( 𝑀 ‘ 2 ) ) ↑ 2 ) ) ) ) |
38 |
|
resqcl |
⊢ ( 𝑅 ∈ ℝ → ( 𝑅 ↑ 2 ) ∈ ℝ ) |
39 |
|
sqge0 |
⊢ ( 𝑅 ∈ ℝ → 0 ≤ ( 𝑅 ↑ 2 ) ) |
40 |
38 39
|
jca |
⊢ ( 𝑅 ∈ ℝ → ( ( 𝑅 ↑ 2 ) ∈ ℝ ∧ 0 ≤ ( 𝑅 ↑ 2 ) ) ) |
41 |
10 40
|
syl |
⊢ ( 𝑅 ∈ ( 0 [,) +∞ ) → ( ( 𝑅 ↑ 2 ) ∈ ℝ ∧ 0 ≤ ( 𝑅 ↑ 2 ) ) ) |
42 |
41
|
ad2antlr |
⊢ ( ( ( 𝑀 ∈ 𝑃 ∧ 𝑅 ∈ ( 0 [,) +∞ ) ) ∧ 𝑝 ∈ 𝑃 ) → ( ( 𝑅 ↑ 2 ) ∈ ℝ ∧ 0 ≤ ( 𝑅 ↑ 2 ) ) ) |
43 |
|
sqrt11 |
⊢ ( ( ( ( ( ( ( 𝑝 ‘ 1 ) − ( 𝑀 ‘ 1 ) ) ↑ 2 ) + ( ( ( 𝑝 ‘ 2 ) − ( 𝑀 ‘ 2 ) ) ↑ 2 ) ) ∈ ℝ ∧ 0 ≤ ( ( ( ( 𝑝 ‘ 1 ) − ( 𝑀 ‘ 1 ) ) ↑ 2 ) + ( ( ( 𝑝 ‘ 2 ) − ( 𝑀 ‘ 2 ) ) ↑ 2 ) ) ) ∧ ( ( 𝑅 ↑ 2 ) ∈ ℝ ∧ 0 ≤ ( 𝑅 ↑ 2 ) ) ) → ( ( √ ‘ ( ( ( ( 𝑝 ‘ 1 ) − ( 𝑀 ‘ 1 ) ) ↑ 2 ) + ( ( ( 𝑝 ‘ 2 ) − ( 𝑀 ‘ 2 ) ) ↑ 2 ) ) ) = ( √ ‘ ( 𝑅 ↑ 2 ) ) ↔ ( ( ( ( 𝑝 ‘ 1 ) − ( 𝑀 ‘ 1 ) ) ↑ 2 ) + ( ( ( 𝑝 ‘ 2 ) − ( 𝑀 ‘ 2 ) ) ↑ 2 ) ) = ( 𝑅 ↑ 2 ) ) ) |
44 |
37 42 43
|
syl2anc |
⊢ ( ( ( 𝑀 ∈ 𝑃 ∧ 𝑅 ∈ ( 0 [,) +∞ ) ) ∧ 𝑝 ∈ 𝑃 ) → ( ( √ ‘ ( ( ( ( 𝑝 ‘ 1 ) − ( 𝑀 ‘ 1 ) ) ↑ 2 ) + ( ( ( 𝑝 ‘ 2 ) − ( 𝑀 ‘ 2 ) ) ↑ 2 ) ) ) = ( √ ‘ ( 𝑅 ↑ 2 ) ) ↔ ( ( ( ( 𝑝 ‘ 1 ) − ( 𝑀 ‘ 1 ) ) ↑ 2 ) + ( ( ( 𝑝 ‘ 2 ) − ( 𝑀 ‘ 2 ) ) ↑ 2 ) ) = ( 𝑅 ↑ 2 ) ) ) |
45 |
8
|
anim1ci |
⊢ ( ( ( 𝑀 ∈ 𝑃 ∧ 𝑅 ∈ ( 0 [,) +∞ ) ) ∧ 𝑝 ∈ 𝑃 ) → ( 𝑝 ∈ 𝑃 ∧ 𝑀 ∈ 𝑃 ) ) |
46 |
|
2nn0 |
⊢ 2 ∈ ℕ0 |
47 |
|
eqid |
⊢ ( 𝔼hil ‘ 2 ) = ( 𝔼hil ‘ 2 ) |
48 |
47
|
ehlval |
⊢ ( 2 ∈ ℕ0 → ( 𝔼hil ‘ 2 ) = ( ℝ^ ‘ ( 1 ... 2 ) ) ) |
49 |
46 48
|
ax-mp |
⊢ ( 𝔼hil ‘ 2 ) = ( ℝ^ ‘ ( 1 ... 2 ) ) |
50 |
|
fz12pr |
⊢ ( 1 ... 2 ) = { 1 , 2 } |
51 |
50 1
|
eqtr4i |
⊢ ( 1 ... 2 ) = 𝐼 |
52 |
51
|
fveq2i |
⊢ ( ℝ^ ‘ ( 1 ... 2 ) ) = ( ℝ^ ‘ 𝐼 ) |
53 |
49 52
|
eqtri |
⊢ ( 𝔼hil ‘ 2 ) = ( ℝ^ ‘ 𝐼 ) |
54 |
2 53
|
eqtr4i |
⊢ 𝐸 = ( 𝔼hil ‘ 2 ) |
55 |
1
|
oveq2i |
⊢ ( ℝ ↑m 𝐼 ) = ( ℝ ↑m { 1 , 2 } ) |
56 |
3 55
|
eqtri |
⊢ 𝑃 = ( ℝ ↑m { 1 , 2 } ) |
57 |
54 56 12
|
ehl2eudisval |
⊢ ( ( 𝑝 ∈ 𝑃 ∧ 𝑀 ∈ 𝑃 ) → ( 𝑝 ( dist ‘ 𝐸 ) 𝑀 ) = ( √ ‘ ( ( ( ( 𝑝 ‘ 1 ) − ( 𝑀 ‘ 1 ) ) ↑ 2 ) + ( ( ( 𝑝 ‘ 2 ) − ( 𝑀 ‘ 2 ) ) ↑ 2 ) ) ) ) |
58 |
45 57
|
syl |
⊢ ( ( ( 𝑀 ∈ 𝑃 ∧ 𝑅 ∈ ( 0 [,) +∞ ) ) ∧ 𝑝 ∈ 𝑃 ) → ( 𝑝 ( dist ‘ 𝐸 ) 𝑀 ) = ( √ ‘ ( ( ( ( 𝑝 ‘ 1 ) − ( 𝑀 ‘ 1 ) ) ↑ 2 ) + ( ( ( 𝑝 ‘ 2 ) − ( 𝑀 ‘ 2 ) ) ↑ 2 ) ) ) ) |
59 |
58
|
eqcomd |
⊢ ( ( ( 𝑀 ∈ 𝑃 ∧ 𝑅 ∈ ( 0 [,) +∞ ) ) ∧ 𝑝 ∈ 𝑃 ) → ( √ ‘ ( ( ( ( 𝑝 ‘ 1 ) − ( 𝑀 ‘ 1 ) ) ↑ 2 ) + ( ( ( 𝑝 ‘ 2 ) − ( 𝑀 ‘ 2 ) ) ↑ 2 ) ) ) = ( 𝑝 ( dist ‘ 𝐸 ) 𝑀 ) ) |
60 |
59
|
eqeq1d |
⊢ ( ( ( 𝑀 ∈ 𝑃 ∧ 𝑅 ∈ ( 0 [,) +∞ ) ) ∧ 𝑝 ∈ 𝑃 ) → ( ( √ ‘ ( ( ( ( 𝑝 ‘ 1 ) − ( 𝑀 ‘ 1 ) ) ↑ 2 ) + ( ( ( 𝑝 ‘ 2 ) − ( 𝑀 ‘ 2 ) ) ↑ 2 ) ) ) = 𝑅 ↔ ( 𝑝 ( dist ‘ 𝐸 ) 𝑀 ) = 𝑅 ) ) |
61 |
19 44 60
|
3bitr3d |
⊢ ( ( ( 𝑀 ∈ 𝑃 ∧ 𝑅 ∈ ( 0 [,) +∞ ) ) ∧ 𝑝 ∈ 𝑃 ) → ( ( ( ( ( 𝑝 ‘ 1 ) − ( 𝑀 ‘ 1 ) ) ↑ 2 ) + ( ( ( 𝑝 ‘ 2 ) − ( 𝑀 ‘ 2 ) ) ↑ 2 ) ) = ( 𝑅 ↑ 2 ) ↔ ( 𝑝 ( dist ‘ 𝐸 ) 𝑀 ) = 𝑅 ) ) |
62 |
61
|
rabbidva |
⊢ ( ( 𝑀 ∈ 𝑃 ∧ 𝑅 ∈ ( 0 [,) +∞ ) ) → { 𝑝 ∈ 𝑃 ∣ ( ( ( ( 𝑝 ‘ 1 ) − ( 𝑀 ‘ 1 ) ) ↑ 2 ) + ( ( ( 𝑝 ‘ 2 ) − ( 𝑀 ‘ 2 ) ) ↑ 2 ) ) = ( 𝑅 ↑ 2 ) } = { 𝑝 ∈ 𝑃 ∣ ( 𝑝 ( dist ‘ 𝐸 ) 𝑀 ) = 𝑅 } ) |
63 |
5 62
|
eqtr2id |
⊢ ( ( 𝑀 ∈ 𝑃 ∧ 𝑅 ∈ ( 0 [,) +∞ ) ) → { 𝑝 ∈ 𝑃 ∣ ( 𝑝 ( dist ‘ 𝐸 ) 𝑀 ) = 𝑅 } = 𝐶 ) |
64 |
14 63
|
eqtrd |
⊢ ( ( 𝑀 ∈ 𝑃 ∧ 𝑅 ∈ ( 0 [,) +∞ ) ) → ( 𝑀 𝑆 𝑅 ) = 𝐶 ) |