Step |
Hyp |
Ref |
Expression |
1 |
|
2z |
⊢ 2 ∈ ℤ |
2 |
|
nnz |
⊢ ( 𝐴 ∈ ℕ → 𝐴 ∈ ℤ ) |
3 |
2
|
3ad2ant1 |
⊢ ( ( 𝐴 ∈ ℕ ∧ 𝐵 ∈ ( 0 ... 𝐴 ) ∧ 𝐶 ∈ ( 0 ... 𝐴 ) ) → 𝐴 ∈ ℤ ) |
4 |
|
zmulcl |
⊢ ( ( 2 ∈ ℤ ∧ 𝐴 ∈ ℤ ) → ( 2 · 𝐴 ) ∈ ℤ ) |
5 |
1 3 4
|
sylancr |
⊢ ( ( 𝐴 ∈ ℕ ∧ 𝐵 ∈ ( 0 ... 𝐴 ) ∧ 𝐶 ∈ ( 0 ... 𝐴 ) ) → ( 2 · 𝐴 ) ∈ ℤ ) |
6 |
|
elfzelz |
⊢ ( 𝐵 ∈ ( 0 ... 𝐴 ) → 𝐵 ∈ ℤ ) |
7 |
6
|
3ad2ant2 |
⊢ ( ( 𝐴 ∈ ℕ ∧ 𝐵 ∈ ( 0 ... 𝐴 ) ∧ 𝐶 ∈ ( 0 ... 𝐴 ) ) → 𝐵 ∈ ℤ ) |
8 |
|
congid |
⊢ ( ( ( 2 · 𝐴 ) ∈ ℤ ∧ 𝐵 ∈ ℤ ) → ( 2 · 𝐴 ) ∥ ( 𝐵 − 𝐵 ) ) |
9 |
5 7 8
|
syl2anc |
⊢ ( ( 𝐴 ∈ ℕ ∧ 𝐵 ∈ ( 0 ... 𝐴 ) ∧ 𝐶 ∈ ( 0 ... 𝐴 ) ) → ( 2 · 𝐴 ) ∥ ( 𝐵 − 𝐵 ) ) |
10 |
9
|
adantr |
⊢ ( ( ( 𝐴 ∈ ℕ ∧ 𝐵 ∈ ( 0 ... 𝐴 ) ∧ 𝐶 ∈ ( 0 ... 𝐴 ) ) ∧ 𝐵 = 𝐶 ) → ( 2 · 𝐴 ) ∥ ( 𝐵 − 𝐵 ) ) |
11 |
|
oveq2 |
⊢ ( 𝐵 = 𝐶 → ( 𝐵 − 𝐵 ) = ( 𝐵 − 𝐶 ) ) |
12 |
11
|
adantl |
⊢ ( ( ( 𝐴 ∈ ℕ ∧ 𝐵 ∈ ( 0 ... 𝐴 ) ∧ 𝐶 ∈ ( 0 ... 𝐴 ) ) ∧ 𝐵 = 𝐶 ) → ( 𝐵 − 𝐵 ) = ( 𝐵 − 𝐶 ) ) |
13 |
10 12
|
breqtrd |
⊢ ( ( ( 𝐴 ∈ ℕ ∧ 𝐵 ∈ ( 0 ... 𝐴 ) ∧ 𝐶 ∈ ( 0 ... 𝐴 ) ) ∧ 𝐵 = 𝐶 ) → ( 2 · 𝐴 ) ∥ ( 𝐵 − 𝐶 ) ) |
14 |
13
|
orcd |
⊢ ( ( ( 𝐴 ∈ ℕ ∧ 𝐵 ∈ ( 0 ... 𝐴 ) ∧ 𝐶 ∈ ( 0 ... 𝐴 ) ) ∧ 𝐵 = 𝐶 ) → ( ( 2 · 𝐴 ) ∥ ( 𝐵 − 𝐶 ) ∨ ( 2 · 𝐴 ) ∥ ( 𝐵 − - 𝐶 ) ) ) |
15 |
|
elfzelz |
⊢ ( 𝐶 ∈ ( 0 ... 𝐴 ) → 𝐶 ∈ ℤ ) |
16 |
15
|
3ad2ant3 |
⊢ ( ( 𝐴 ∈ ℕ ∧ 𝐵 ∈ ( 0 ... 𝐴 ) ∧ 𝐶 ∈ ( 0 ... 𝐴 ) ) → 𝐶 ∈ ℤ ) |
17 |
7 16
|
zsubcld |
⊢ ( ( 𝐴 ∈ ℕ ∧ 𝐵 ∈ ( 0 ... 𝐴 ) ∧ 𝐶 ∈ ( 0 ... 𝐴 ) ) → ( 𝐵 − 𝐶 ) ∈ ℤ ) |
18 |
17
|
zcnd |
⊢ ( ( 𝐴 ∈ ℕ ∧ 𝐵 ∈ ( 0 ... 𝐴 ) ∧ 𝐶 ∈ ( 0 ... 𝐴 ) ) → ( 𝐵 − 𝐶 ) ∈ ℂ ) |
19 |
18
|
abscld |
⊢ ( ( 𝐴 ∈ ℕ ∧ 𝐵 ∈ ( 0 ... 𝐴 ) ∧ 𝐶 ∈ ( 0 ... 𝐴 ) ) → ( abs ‘ ( 𝐵 − 𝐶 ) ) ∈ ℝ ) |
20 |
|
nnre |
⊢ ( 𝐴 ∈ ℕ → 𝐴 ∈ ℝ ) |
21 |
20
|
3ad2ant1 |
⊢ ( ( 𝐴 ∈ ℕ ∧ 𝐵 ∈ ( 0 ... 𝐴 ) ∧ 𝐶 ∈ ( 0 ... 𝐴 ) ) → 𝐴 ∈ ℝ ) |
22 |
|
0re |
⊢ 0 ∈ ℝ |
23 |
|
resubcl |
⊢ ( ( 𝐴 ∈ ℝ ∧ 0 ∈ ℝ ) → ( 𝐴 − 0 ) ∈ ℝ ) |
24 |
21 22 23
|
sylancl |
⊢ ( ( 𝐴 ∈ ℕ ∧ 𝐵 ∈ ( 0 ... 𝐴 ) ∧ 𝐶 ∈ ( 0 ... 𝐴 ) ) → ( 𝐴 − 0 ) ∈ ℝ ) |
25 |
|
2re |
⊢ 2 ∈ ℝ |
26 |
|
remulcl |
⊢ ( ( 2 ∈ ℝ ∧ 𝐴 ∈ ℝ ) → ( 2 · 𝐴 ) ∈ ℝ ) |
27 |
25 21 26
|
sylancr |
⊢ ( ( 𝐴 ∈ ℕ ∧ 𝐵 ∈ ( 0 ... 𝐴 ) ∧ 𝐶 ∈ ( 0 ... 𝐴 ) ) → ( 2 · 𝐴 ) ∈ ℝ ) |
28 |
|
simp2 |
⊢ ( ( 𝐴 ∈ ℕ ∧ 𝐵 ∈ ( 0 ... 𝐴 ) ∧ 𝐶 ∈ ( 0 ... 𝐴 ) ) → 𝐵 ∈ ( 0 ... 𝐴 ) ) |
29 |
|
simp3 |
⊢ ( ( 𝐴 ∈ ℕ ∧ 𝐵 ∈ ( 0 ... 𝐴 ) ∧ 𝐶 ∈ ( 0 ... 𝐴 ) ) → 𝐶 ∈ ( 0 ... 𝐴 ) ) |
30 |
24
|
leidd |
⊢ ( ( 𝐴 ∈ ℕ ∧ 𝐵 ∈ ( 0 ... 𝐴 ) ∧ 𝐶 ∈ ( 0 ... 𝐴 ) ) → ( 𝐴 − 0 ) ≤ ( 𝐴 − 0 ) ) |
31 |
|
fzmaxdif |
⊢ ( ( ( 𝐴 ∈ ℤ ∧ 𝐵 ∈ ( 0 ... 𝐴 ) ) ∧ ( 𝐴 ∈ ℤ ∧ 𝐶 ∈ ( 0 ... 𝐴 ) ) ∧ ( 𝐴 − 0 ) ≤ ( 𝐴 − 0 ) ) → ( abs ‘ ( 𝐵 − 𝐶 ) ) ≤ ( 𝐴 − 0 ) ) |
32 |
3 28 3 29 30 31
|
syl221anc |
⊢ ( ( 𝐴 ∈ ℕ ∧ 𝐵 ∈ ( 0 ... 𝐴 ) ∧ 𝐶 ∈ ( 0 ... 𝐴 ) ) → ( abs ‘ ( 𝐵 − 𝐶 ) ) ≤ ( 𝐴 − 0 ) ) |
33 |
|
nnrp |
⊢ ( 𝐴 ∈ ℕ → 𝐴 ∈ ℝ+ ) |
34 |
33
|
3ad2ant1 |
⊢ ( ( 𝐴 ∈ ℕ ∧ 𝐵 ∈ ( 0 ... 𝐴 ) ∧ 𝐶 ∈ ( 0 ... 𝐴 ) ) → 𝐴 ∈ ℝ+ ) |
35 |
21 34
|
ltaddrpd |
⊢ ( ( 𝐴 ∈ ℕ ∧ 𝐵 ∈ ( 0 ... 𝐴 ) ∧ 𝐶 ∈ ( 0 ... 𝐴 ) ) → 𝐴 < ( 𝐴 + 𝐴 ) ) |
36 |
21
|
recnd |
⊢ ( ( 𝐴 ∈ ℕ ∧ 𝐵 ∈ ( 0 ... 𝐴 ) ∧ 𝐶 ∈ ( 0 ... 𝐴 ) ) → 𝐴 ∈ ℂ ) |
37 |
36
|
subid1d |
⊢ ( ( 𝐴 ∈ ℕ ∧ 𝐵 ∈ ( 0 ... 𝐴 ) ∧ 𝐶 ∈ ( 0 ... 𝐴 ) ) → ( 𝐴 − 0 ) = 𝐴 ) |
38 |
36
|
2timesd |
⊢ ( ( 𝐴 ∈ ℕ ∧ 𝐵 ∈ ( 0 ... 𝐴 ) ∧ 𝐶 ∈ ( 0 ... 𝐴 ) ) → ( 2 · 𝐴 ) = ( 𝐴 + 𝐴 ) ) |
39 |
35 37 38
|
3brtr4d |
⊢ ( ( 𝐴 ∈ ℕ ∧ 𝐵 ∈ ( 0 ... 𝐴 ) ∧ 𝐶 ∈ ( 0 ... 𝐴 ) ) → ( 𝐴 − 0 ) < ( 2 · 𝐴 ) ) |
40 |
19 24 27 32 39
|
lelttrd |
⊢ ( ( 𝐴 ∈ ℕ ∧ 𝐵 ∈ ( 0 ... 𝐴 ) ∧ 𝐶 ∈ ( 0 ... 𝐴 ) ) → ( abs ‘ ( 𝐵 − 𝐶 ) ) < ( 2 · 𝐴 ) ) |
41 |
40
|
adantr |
⊢ ( ( ( 𝐴 ∈ ℕ ∧ 𝐵 ∈ ( 0 ... 𝐴 ) ∧ 𝐶 ∈ ( 0 ... 𝐴 ) ) ∧ ( 2 · 𝐴 ) ∥ ( 𝐵 − 𝐶 ) ) → ( abs ‘ ( 𝐵 − 𝐶 ) ) < ( 2 · 𝐴 ) ) |
42 |
|
2nn |
⊢ 2 ∈ ℕ |
43 |
|
simpl1 |
⊢ ( ( ( 𝐴 ∈ ℕ ∧ 𝐵 ∈ ( 0 ... 𝐴 ) ∧ 𝐶 ∈ ( 0 ... 𝐴 ) ) ∧ ( 2 · 𝐴 ) ∥ ( 𝐵 − 𝐶 ) ) → 𝐴 ∈ ℕ ) |
44 |
|
nnmulcl |
⊢ ( ( 2 ∈ ℕ ∧ 𝐴 ∈ ℕ ) → ( 2 · 𝐴 ) ∈ ℕ ) |
45 |
42 43 44
|
sylancr |
⊢ ( ( ( 𝐴 ∈ ℕ ∧ 𝐵 ∈ ( 0 ... 𝐴 ) ∧ 𝐶 ∈ ( 0 ... 𝐴 ) ) ∧ ( 2 · 𝐴 ) ∥ ( 𝐵 − 𝐶 ) ) → ( 2 · 𝐴 ) ∈ ℕ ) |
46 |
|
simpl2 |
⊢ ( ( ( 𝐴 ∈ ℕ ∧ 𝐵 ∈ ( 0 ... 𝐴 ) ∧ 𝐶 ∈ ( 0 ... 𝐴 ) ) ∧ ( 2 · 𝐴 ) ∥ ( 𝐵 − 𝐶 ) ) → 𝐵 ∈ ( 0 ... 𝐴 ) ) |
47 |
46 6
|
syl |
⊢ ( ( ( 𝐴 ∈ ℕ ∧ 𝐵 ∈ ( 0 ... 𝐴 ) ∧ 𝐶 ∈ ( 0 ... 𝐴 ) ) ∧ ( 2 · 𝐴 ) ∥ ( 𝐵 − 𝐶 ) ) → 𝐵 ∈ ℤ ) |
48 |
|
simpl3 |
⊢ ( ( ( 𝐴 ∈ ℕ ∧ 𝐵 ∈ ( 0 ... 𝐴 ) ∧ 𝐶 ∈ ( 0 ... 𝐴 ) ) ∧ ( 2 · 𝐴 ) ∥ ( 𝐵 − 𝐶 ) ) → 𝐶 ∈ ( 0 ... 𝐴 ) ) |
49 |
48 15
|
syl |
⊢ ( ( ( 𝐴 ∈ ℕ ∧ 𝐵 ∈ ( 0 ... 𝐴 ) ∧ 𝐶 ∈ ( 0 ... 𝐴 ) ) ∧ ( 2 · 𝐴 ) ∥ ( 𝐵 − 𝐶 ) ) → 𝐶 ∈ ℤ ) |
50 |
|
simpr |
⊢ ( ( ( 𝐴 ∈ ℕ ∧ 𝐵 ∈ ( 0 ... 𝐴 ) ∧ 𝐶 ∈ ( 0 ... 𝐴 ) ) ∧ ( 2 · 𝐴 ) ∥ ( 𝐵 − 𝐶 ) ) → ( 2 · 𝐴 ) ∥ ( 𝐵 − 𝐶 ) ) |
51 |
|
congabseq |
⊢ ( ( ( ( 2 · 𝐴 ) ∈ ℕ ∧ 𝐵 ∈ ℤ ∧ 𝐶 ∈ ℤ ) ∧ ( 2 · 𝐴 ) ∥ ( 𝐵 − 𝐶 ) ) → ( ( abs ‘ ( 𝐵 − 𝐶 ) ) < ( 2 · 𝐴 ) ↔ 𝐵 = 𝐶 ) ) |
52 |
45 47 49 50 51
|
syl31anc |
⊢ ( ( ( 𝐴 ∈ ℕ ∧ 𝐵 ∈ ( 0 ... 𝐴 ) ∧ 𝐶 ∈ ( 0 ... 𝐴 ) ) ∧ ( 2 · 𝐴 ) ∥ ( 𝐵 − 𝐶 ) ) → ( ( abs ‘ ( 𝐵 − 𝐶 ) ) < ( 2 · 𝐴 ) ↔ 𝐵 = 𝐶 ) ) |
53 |
41 52
|
mpbid |
⊢ ( ( ( 𝐴 ∈ ℕ ∧ 𝐵 ∈ ( 0 ... 𝐴 ) ∧ 𝐶 ∈ ( 0 ... 𝐴 ) ) ∧ ( 2 · 𝐴 ) ∥ ( 𝐵 − 𝐶 ) ) → 𝐵 = 𝐶 ) |
54 |
|
simpll2 |
⊢ ( ( ( ( 𝐴 ∈ ℕ ∧ 𝐵 ∈ ( 0 ... 𝐴 ) ∧ 𝐶 ∈ ( 0 ... 𝐴 ) ) ∧ ( 2 · 𝐴 ) ∥ ( 𝐵 − - 𝐶 ) ) ∧ 𝐶 ∈ ( 0 ... ( 𝐴 − 1 ) ) ) → 𝐵 ∈ ( 0 ... 𝐴 ) ) |
55 |
|
elfzle1 |
⊢ ( 𝐵 ∈ ( 0 ... 𝐴 ) → 0 ≤ 𝐵 ) |
56 |
54 55
|
syl |
⊢ ( ( ( ( 𝐴 ∈ ℕ ∧ 𝐵 ∈ ( 0 ... 𝐴 ) ∧ 𝐶 ∈ ( 0 ... 𝐴 ) ) ∧ ( 2 · 𝐴 ) ∥ ( 𝐵 − - 𝐶 ) ) ∧ 𝐶 ∈ ( 0 ... ( 𝐴 − 1 ) ) ) → 0 ≤ 𝐵 ) |
57 |
7
|
zred |
⊢ ( ( 𝐴 ∈ ℕ ∧ 𝐵 ∈ ( 0 ... 𝐴 ) ∧ 𝐶 ∈ ( 0 ... 𝐴 ) ) → 𝐵 ∈ ℝ ) |
58 |
16
|
zred |
⊢ ( ( 𝐴 ∈ ℕ ∧ 𝐵 ∈ ( 0 ... 𝐴 ) ∧ 𝐶 ∈ ( 0 ... 𝐴 ) ) → 𝐶 ∈ ℝ ) |
59 |
58
|
renegcld |
⊢ ( ( 𝐴 ∈ ℕ ∧ 𝐵 ∈ ( 0 ... 𝐴 ) ∧ 𝐶 ∈ ( 0 ... 𝐴 ) ) → - 𝐶 ∈ ℝ ) |
60 |
57 59
|
resubcld |
⊢ ( ( 𝐴 ∈ ℕ ∧ 𝐵 ∈ ( 0 ... 𝐴 ) ∧ 𝐶 ∈ ( 0 ... 𝐴 ) ) → ( 𝐵 − - 𝐶 ) ∈ ℝ ) |
61 |
60
|
recnd |
⊢ ( ( 𝐴 ∈ ℕ ∧ 𝐵 ∈ ( 0 ... 𝐴 ) ∧ 𝐶 ∈ ( 0 ... 𝐴 ) ) → ( 𝐵 − - 𝐶 ) ∈ ℂ ) |
62 |
61
|
abscld |
⊢ ( ( 𝐴 ∈ ℕ ∧ 𝐵 ∈ ( 0 ... 𝐴 ) ∧ 𝐶 ∈ ( 0 ... 𝐴 ) ) → ( abs ‘ ( 𝐵 − - 𝐶 ) ) ∈ ℝ ) |
63 |
62
|
ad2antrr |
⊢ ( ( ( ( 𝐴 ∈ ℕ ∧ 𝐵 ∈ ( 0 ... 𝐴 ) ∧ 𝐶 ∈ ( 0 ... 𝐴 ) ) ∧ ( 2 · 𝐴 ) ∥ ( 𝐵 − - 𝐶 ) ) ∧ 𝐶 ∈ ( 0 ... ( 𝐴 − 1 ) ) ) → ( abs ‘ ( 𝐵 − - 𝐶 ) ) ∈ ℝ ) |
64 |
|
1re |
⊢ 1 ∈ ℝ |
65 |
|
resubcl |
⊢ ( ( 𝐴 ∈ ℝ ∧ 1 ∈ ℝ ) → ( 𝐴 − 1 ) ∈ ℝ ) |
66 |
21 64 65
|
sylancl |
⊢ ( ( 𝐴 ∈ ℕ ∧ 𝐵 ∈ ( 0 ... 𝐴 ) ∧ 𝐶 ∈ ( 0 ... 𝐴 ) ) → ( 𝐴 − 1 ) ∈ ℝ ) |
67 |
66
|
renegcld |
⊢ ( ( 𝐴 ∈ ℕ ∧ 𝐵 ∈ ( 0 ... 𝐴 ) ∧ 𝐶 ∈ ( 0 ... 𝐴 ) ) → - ( 𝐴 − 1 ) ∈ ℝ ) |
68 |
21 67
|
resubcld |
⊢ ( ( 𝐴 ∈ ℕ ∧ 𝐵 ∈ ( 0 ... 𝐴 ) ∧ 𝐶 ∈ ( 0 ... 𝐴 ) ) → ( 𝐴 − - ( 𝐴 − 1 ) ) ∈ ℝ ) |
69 |
68
|
ad2antrr |
⊢ ( ( ( ( 𝐴 ∈ ℕ ∧ 𝐵 ∈ ( 0 ... 𝐴 ) ∧ 𝐶 ∈ ( 0 ... 𝐴 ) ) ∧ ( 2 · 𝐴 ) ∥ ( 𝐵 − - 𝐶 ) ) ∧ 𝐶 ∈ ( 0 ... ( 𝐴 − 1 ) ) ) → ( 𝐴 − - ( 𝐴 − 1 ) ) ∈ ℝ ) |
70 |
27
|
ad2antrr |
⊢ ( ( ( ( 𝐴 ∈ ℕ ∧ 𝐵 ∈ ( 0 ... 𝐴 ) ∧ 𝐶 ∈ ( 0 ... 𝐴 ) ) ∧ ( 2 · 𝐴 ) ∥ ( 𝐵 − - 𝐶 ) ) ∧ 𝐶 ∈ ( 0 ... ( 𝐴 − 1 ) ) ) → ( 2 · 𝐴 ) ∈ ℝ ) |
71 |
7
|
ad2antrr |
⊢ ( ( ( ( 𝐴 ∈ ℕ ∧ 𝐵 ∈ ( 0 ... 𝐴 ) ∧ 𝐶 ∈ ( 0 ... 𝐴 ) ) ∧ ( 2 · 𝐴 ) ∥ ( 𝐵 − - 𝐶 ) ) ∧ 𝐶 ∈ ( 0 ... ( 𝐴 − 1 ) ) ) → 𝐵 ∈ ℤ ) |
72 |
71
|
zcnd |
⊢ ( ( ( ( 𝐴 ∈ ℕ ∧ 𝐵 ∈ ( 0 ... 𝐴 ) ∧ 𝐶 ∈ ( 0 ... 𝐴 ) ) ∧ ( 2 · 𝐴 ) ∥ ( 𝐵 − - 𝐶 ) ) ∧ 𝐶 ∈ ( 0 ... ( 𝐴 − 1 ) ) ) → 𝐵 ∈ ℂ ) |
73 |
16
|
znegcld |
⊢ ( ( 𝐴 ∈ ℕ ∧ 𝐵 ∈ ( 0 ... 𝐴 ) ∧ 𝐶 ∈ ( 0 ... 𝐴 ) ) → - 𝐶 ∈ ℤ ) |
74 |
73
|
ad2antrr |
⊢ ( ( ( ( 𝐴 ∈ ℕ ∧ 𝐵 ∈ ( 0 ... 𝐴 ) ∧ 𝐶 ∈ ( 0 ... 𝐴 ) ) ∧ ( 2 · 𝐴 ) ∥ ( 𝐵 − - 𝐶 ) ) ∧ 𝐶 ∈ ( 0 ... ( 𝐴 − 1 ) ) ) → - 𝐶 ∈ ℤ ) |
75 |
74
|
zcnd |
⊢ ( ( ( ( 𝐴 ∈ ℕ ∧ 𝐵 ∈ ( 0 ... 𝐴 ) ∧ 𝐶 ∈ ( 0 ... 𝐴 ) ) ∧ ( 2 · 𝐴 ) ∥ ( 𝐵 − - 𝐶 ) ) ∧ 𝐶 ∈ ( 0 ... ( 𝐴 − 1 ) ) ) → - 𝐶 ∈ ℂ ) |
76 |
72 75
|
abssubd |
⊢ ( ( ( ( 𝐴 ∈ ℕ ∧ 𝐵 ∈ ( 0 ... 𝐴 ) ∧ 𝐶 ∈ ( 0 ... 𝐴 ) ) ∧ ( 2 · 𝐴 ) ∥ ( 𝐵 − - 𝐶 ) ) ∧ 𝐶 ∈ ( 0 ... ( 𝐴 − 1 ) ) ) → ( abs ‘ ( 𝐵 − - 𝐶 ) ) = ( abs ‘ ( - 𝐶 − 𝐵 ) ) ) |
77 |
|
0zd |
⊢ ( ( ( ( 𝐴 ∈ ℕ ∧ 𝐵 ∈ ( 0 ... 𝐴 ) ∧ 𝐶 ∈ ( 0 ... 𝐴 ) ) ∧ ( 2 · 𝐴 ) ∥ ( 𝐵 − - 𝐶 ) ) ∧ 𝐶 ∈ ( 0 ... ( 𝐴 − 1 ) ) ) → 0 ∈ ℤ ) |
78 |
|
simpr |
⊢ ( ( ( ( 𝐴 ∈ ℕ ∧ 𝐵 ∈ ( 0 ... 𝐴 ) ∧ 𝐶 ∈ ( 0 ... 𝐴 ) ) ∧ ( 2 · 𝐴 ) ∥ ( 𝐵 − - 𝐶 ) ) ∧ 𝐶 ∈ ( 0 ... ( 𝐴 − 1 ) ) ) → 𝐶 ∈ ( 0 ... ( 𝐴 − 1 ) ) ) |
79 |
|
0zd |
⊢ ( ( 𝐴 ∈ ℕ ∧ 𝐵 ∈ ( 0 ... 𝐴 ) ∧ 𝐶 ∈ ( 0 ... 𝐴 ) ) → 0 ∈ ℤ ) |
80 |
|
1z |
⊢ 1 ∈ ℤ |
81 |
|
zsubcl |
⊢ ( ( 𝐴 ∈ ℤ ∧ 1 ∈ ℤ ) → ( 𝐴 − 1 ) ∈ ℤ ) |
82 |
3 80 81
|
sylancl |
⊢ ( ( 𝐴 ∈ ℕ ∧ 𝐵 ∈ ( 0 ... 𝐴 ) ∧ 𝐶 ∈ ( 0 ... 𝐴 ) ) → ( 𝐴 − 1 ) ∈ ℤ ) |
83 |
|
fzneg |
⊢ ( ( 𝐶 ∈ ℤ ∧ 0 ∈ ℤ ∧ ( 𝐴 − 1 ) ∈ ℤ ) → ( 𝐶 ∈ ( 0 ... ( 𝐴 − 1 ) ) ↔ - 𝐶 ∈ ( - ( 𝐴 − 1 ) ... - 0 ) ) ) |
84 |
16 79 82 83
|
syl3anc |
⊢ ( ( 𝐴 ∈ ℕ ∧ 𝐵 ∈ ( 0 ... 𝐴 ) ∧ 𝐶 ∈ ( 0 ... 𝐴 ) ) → ( 𝐶 ∈ ( 0 ... ( 𝐴 − 1 ) ) ↔ - 𝐶 ∈ ( - ( 𝐴 − 1 ) ... - 0 ) ) ) |
85 |
84
|
ad2antrr |
⊢ ( ( ( ( 𝐴 ∈ ℕ ∧ 𝐵 ∈ ( 0 ... 𝐴 ) ∧ 𝐶 ∈ ( 0 ... 𝐴 ) ) ∧ ( 2 · 𝐴 ) ∥ ( 𝐵 − - 𝐶 ) ) ∧ 𝐶 ∈ ( 0 ... ( 𝐴 − 1 ) ) ) → ( 𝐶 ∈ ( 0 ... ( 𝐴 − 1 ) ) ↔ - 𝐶 ∈ ( - ( 𝐴 − 1 ) ... - 0 ) ) ) |
86 |
78 85
|
mpbid |
⊢ ( ( ( ( 𝐴 ∈ ℕ ∧ 𝐵 ∈ ( 0 ... 𝐴 ) ∧ 𝐶 ∈ ( 0 ... 𝐴 ) ) ∧ ( 2 · 𝐴 ) ∥ ( 𝐵 − - 𝐶 ) ) ∧ 𝐶 ∈ ( 0 ... ( 𝐴 − 1 ) ) ) → - 𝐶 ∈ ( - ( 𝐴 − 1 ) ... - 0 ) ) |
87 |
|
neg0 |
⊢ - 0 = 0 |
88 |
87
|
a1i |
⊢ ( ( ( ( 𝐴 ∈ ℕ ∧ 𝐵 ∈ ( 0 ... 𝐴 ) ∧ 𝐶 ∈ ( 0 ... 𝐴 ) ) ∧ ( 2 · 𝐴 ) ∥ ( 𝐵 − - 𝐶 ) ) ∧ 𝐶 ∈ ( 0 ... ( 𝐴 − 1 ) ) ) → - 0 = 0 ) |
89 |
88
|
oveq2d |
⊢ ( ( ( ( 𝐴 ∈ ℕ ∧ 𝐵 ∈ ( 0 ... 𝐴 ) ∧ 𝐶 ∈ ( 0 ... 𝐴 ) ) ∧ ( 2 · 𝐴 ) ∥ ( 𝐵 − - 𝐶 ) ) ∧ 𝐶 ∈ ( 0 ... ( 𝐴 − 1 ) ) ) → ( - ( 𝐴 − 1 ) ... - 0 ) = ( - ( 𝐴 − 1 ) ... 0 ) ) |
90 |
86 89
|
eleqtrd |
⊢ ( ( ( ( 𝐴 ∈ ℕ ∧ 𝐵 ∈ ( 0 ... 𝐴 ) ∧ 𝐶 ∈ ( 0 ... 𝐴 ) ) ∧ ( 2 · 𝐴 ) ∥ ( 𝐵 − - 𝐶 ) ) ∧ 𝐶 ∈ ( 0 ... ( 𝐴 − 1 ) ) ) → - 𝐶 ∈ ( - ( 𝐴 − 1 ) ... 0 ) ) |
91 |
3
|
ad2antrr |
⊢ ( ( ( ( 𝐴 ∈ ℕ ∧ 𝐵 ∈ ( 0 ... 𝐴 ) ∧ 𝐶 ∈ ( 0 ... 𝐴 ) ) ∧ ( 2 · 𝐴 ) ∥ ( 𝐵 − - 𝐶 ) ) ∧ 𝐶 ∈ ( 0 ... ( 𝐴 − 1 ) ) ) → 𝐴 ∈ ℤ ) |
92 |
|
simp1 |
⊢ ( ( 𝐴 ∈ ℕ ∧ 𝐵 ∈ ( 0 ... 𝐴 ) ∧ 𝐶 ∈ ( 0 ... 𝐴 ) ) → 𝐴 ∈ ℕ ) |
93 |
42 92 44
|
sylancr |
⊢ ( ( 𝐴 ∈ ℕ ∧ 𝐵 ∈ ( 0 ... 𝐴 ) ∧ 𝐶 ∈ ( 0 ... 𝐴 ) ) → ( 2 · 𝐴 ) ∈ ℕ ) |
94 |
|
nnm1nn0 |
⊢ ( ( 2 · 𝐴 ) ∈ ℕ → ( ( 2 · 𝐴 ) − 1 ) ∈ ℕ0 ) |
95 |
93 94
|
syl |
⊢ ( ( 𝐴 ∈ ℕ ∧ 𝐵 ∈ ( 0 ... 𝐴 ) ∧ 𝐶 ∈ ( 0 ... 𝐴 ) ) → ( ( 2 · 𝐴 ) − 1 ) ∈ ℕ0 ) |
96 |
95
|
nn0ge0d |
⊢ ( ( 𝐴 ∈ ℕ ∧ 𝐵 ∈ ( 0 ... 𝐴 ) ∧ 𝐶 ∈ ( 0 ... 𝐴 ) ) → 0 ≤ ( ( 2 · 𝐴 ) − 1 ) ) |
97 |
|
0m0e0 |
⊢ ( 0 − 0 ) = 0 |
98 |
97
|
a1i |
⊢ ( ( 𝐴 ∈ ℕ ∧ 𝐵 ∈ ( 0 ... 𝐴 ) ∧ 𝐶 ∈ ( 0 ... 𝐴 ) ) → ( 0 − 0 ) = 0 ) |
99 |
|
1cnd |
⊢ ( ( 𝐴 ∈ ℕ ∧ 𝐵 ∈ ( 0 ... 𝐴 ) ∧ 𝐶 ∈ ( 0 ... 𝐴 ) ) → 1 ∈ ℂ ) |
100 |
36 36 99
|
addsubassd |
⊢ ( ( 𝐴 ∈ ℕ ∧ 𝐵 ∈ ( 0 ... 𝐴 ) ∧ 𝐶 ∈ ( 0 ... 𝐴 ) ) → ( ( 𝐴 + 𝐴 ) − 1 ) = ( 𝐴 + ( 𝐴 − 1 ) ) ) |
101 |
38
|
oveq1d |
⊢ ( ( 𝐴 ∈ ℕ ∧ 𝐵 ∈ ( 0 ... 𝐴 ) ∧ 𝐶 ∈ ( 0 ... 𝐴 ) ) → ( ( 2 · 𝐴 ) − 1 ) = ( ( 𝐴 + 𝐴 ) − 1 ) ) |
102 |
|
ax-1cn |
⊢ 1 ∈ ℂ |
103 |
|
subcl |
⊢ ( ( 𝐴 ∈ ℂ ∧ 1 ∈ ℂ ) → ( 𝐴 − 1 ) ∈ ℂ ) |
104 |
36 102 103
|
sylancl |
⊢ ( ( 𝐴 ∈ ℕ ∧ 𝐵 ∈ ( 0 ... 𝐴 ) ∧ 𝐶 ∈ ( 0 ... 𝐴 ) ) → ( 𝐴 − 1 ) ∈ ℂ ) |
105 |
36 104
|
subnegd |
⊢ ( ( 𝐴 ∈ ℕ ∧ 𝐵 ∈ ( 0 ... 𝐴 ) ∧ 𝐶 ∈ ( 0 ... 𝐴 ) ) → ( 𝐴 − - ( 𝐴 − 1 ) ) = ( 𝐴 + ( 𝐴 − 1 ) ) ) |
106 |
100 101 105
|
3eqtr4rd |
⊢ ( ( 𝐴 ∈ ℕ ∧ 𝐵 ∈ ( 0 ... 𝐴 ) ∧ 𝐶 ∈ ( 0 ... 𝐴 ) ) → ( 𝐴 − - ( 𝐴 − 1 ) ) = ( ( 2 · 𝐴 ) − 1 ) ) |
107 |
96 98 106
|
3brtr4d |
⊢ ( ( 𝐴 ∈ ℕ ∧ 𝐵 ∈ ( 0 ... 𝐴 ) ∧ 𝐶 ∈ ( 0 ... 𝐴 ) ) → ( 0 − 0 ) ≤ ( 𝐴 − - ( 𝐴 − 1 ) ) ) |
108 |
107
|
ad2antrr |
⊢ ( ( ( ( 𝐴 ∈ ℕ ∧ 𝐵 ∈ ( 0 ... 𝐴 ) ∧ 𝐶 ∈ ( 0 ... 𝐴 ) ) ∧ ( 2 · 𝐴 ) ∥ ( 𝐵 − - 𝐶 ) ) ∧ 𝐶 ∈ ( 0 ... ( 𝐴 − 1 ) ) ) → ( 0 − 0 ) ≤ ( 𝐴 − - ( 𝐴 − 1 ) ) ) |
109 |
|
fzmaxdif |
⊢ ( ( ( 0 ∈ ℤ ∧ - 𝐶 ∈ ( - ( 𝐴 − 1 ) ... 0 ) ) ∧ ( 𝐴 ∈ ℤ ∧ 𝐵 ∈ ( 0 ... 𝐴 ) ) ∧ ( 0 − 0 ) ≤ ( 𝐴 − - ( 𝐴 − 1 ) ) ) → ( abs ‘ ( - 𝐶 − 𝐵 ) ) ≤ ( 𝐴 − - ( 𝐴 − 1 ) ) ) |
110 |
77 90 91 54 108 109
|
syl221anc |
⊢ ( ( ( ( 𝐴 ∈ ℕ ∧ 𝐵 ∈ ( 0 ... 𝐴 ) ∧ 𝐶 ∈ ( 0 ... 𝐴 ) ) ∧ ( 2 · 𝐴 ) ∥ ( 𝐵 − - 𝐶 ) ) ∧ 𝐶 ∈ ( 0 ... ( 𝐴 − 1 ) ) ) → ( abs ‘ ( - 𝐶 − 𝐵 ) ) ≤ ( 𝐴 − - ( 𝐴 − 1 ) ) ) |
111 |
76 110
|
eqbrtrd |
⊢ ( ( ( ( 𝐴 ∈ ℕ ∧ 𝐵 ∈ ( 0 ... 𝐴 ) ∧ 𝐶 ∈ ( 0 ... 𝐴 ) ) ∧ ( 2 · 𝐴 ) ∥ ( 𝐵 − - 𝐶 ) ) ∧ 𝐶 ∈ ( 0 ... ( 𝐴 − 1 ) ) ) → ( abs ‘ ( 𝐵 − - 𝐶 ) ) ≤ ( 𝐴 − - ( 𝐴 − 1 ) ) ) |
112 |
27
|
ltm1d |
⊢ ( ( 𝐴 ∈ ℕ ∧ 𝐵 ∈ ( 0 ... 𝐴 ) ∧ 𝐶 ∈ ( 0 ... 𝐴 ) ) → ( ( 2 · 𝐴 ) − 1 ) < ( 2 · 𝐴 ) ) |
113 |
106 112
|
eqbrtrd |
⊢ ( ( 𝐴 ∈ ℕ ∧ 𝐵 ∈ ( 0 ... 𝐴 ) ∧ 𝐶 ∈ ( 0 ... 𝐴 ) ) → ( 𝐴 − - ( 𝐴 − 1 ) ) < ( 2 · 𝐴 ) ) |
114 |
113
|
ad2antrr |
⊢ ( ( ( ( 𝐴 ∈ ℕ ∧ 𝐵 ∈ ( 0 ... 𝐴 ) ∧ 𝐶 ∈ ( 0 ... 𝐴 ) ) ∧ ( 2 · 𝐴 ) ∥ ( 𝐵 − - 𝐶 ) ) ∧ 𝐶 ∈ ( 0 ... ( 𝐴 − 1 ) ) ) → ( 𝐴 − - ( 𝐴 − 1 ) ) < ( 2 · 𝐴 ) ) |
115 |
63 69 70 111 114
|
lelttrd |
⊢ ( ( ( ( 𝐴 ∈ ℕ ∧ 𝐵 ∈ ( 0 ... 𝐴 ) ∧ 𝐶 ∈ ( 0 ... 𝐴 ) ) ∧ ( 2 · 𝐴 ) ∥ ( 𝐵 − - 𝐶 ) ) ∧ 𝐶 ∈ ( 0 ... ( 𝐴 − 1 ) ) ) → ( abs ‘ ( 𝐵 − - 𝐶 ) ) < ( 2 · 𝐴 ) ) |
116 |
93
|
ad2antrr |
⊢ ( ( ( ( 𝐴 ∈ ℕ ∧ 𝐵 ∈ ( 0 ... 𝐴 ) ∧ 𝐶 ∈ ( 0 ... 𝐴 ) ) ∧ ( 2 · 𝐴 ) ∥ ( 𝐵 − - 𝐶 ) ) ∧ 𝐶 ∈ ( 0 ... ( 𝐴 − 1 ) ) ) → ( 2 · 𝐴 ) ∈ ℕ ) |
117 |
|
simplr |
⊢ ( ( ( ( 𝐴 ∈ ℕ ∧ 𝐵 ∈ ( 0 ... 𝐴 ) ∧ 𝐶 ∈ ( 0 ... 𝐴 ) ) ∧ ( 2 · 𝐴 ) ∥ ( 𝐵 − - 𝐶 ) ) ∧ 𝐶 ∈ ( 0 ... ( 𝐴 − 1 ) ) ) → ( 2 · 𝐴 ) ∥ ( 𝐵 − - 𝐶 ) ) |
118 |
|
congabseq |
⊢ ( ( ( ( 2 · 𝐴 ) ∈ ℕ ∧ 𝐵 ∈ ℤ ∧ - 𝐶 ∈ ℤ ) ∧ ( 2 · 𝐴 ) ∥ ( 𝐵 − - 𝐶 ) ) → ( ( abs ‘ ( 𝐵 − - 𝐶 ) ) < ( 2 · 𝐴 ) ↔ 𝐵 = - 𝐶 ) ) |
119 |
116 71 74 117 118
|
syl31anc |
⊢ ( ( ( ( 𝐴 ∈ ℕ ∧ 𝐵 ∈ ( 0 ... 𝐴 ) ∧ 𝐶 ∈ ( 0 ... 𝐴 ) ) ∧ ( 2 · 𝐴 ) ∥ ( 𝐵 − - 𝐶 ) ) ∧ 𝐶 ∈ ( 0 ... ( 𝐴 − 1 ) ) ) → ( ( abs ‘ ( 𝐵 − - 𝐶 ) ) < ( 2 · 𝐴 ) ↔ 𝐵 = - 𝐶 ) ) |
120 |
115 119
|
mpbid |
⊢ ( ( ( ( 𝐴 ∈ ℕ ∧ 𝐵 ∈ ( 0 ... 𝐴 ) ∧ 𝐶 ∈ ( 0 ... 𝐴 ) ) ∧ ( 2 · 𝐴 ) ∥ ( 𝐵 − - 𝐶 ) ) ∧ 𝐶 ∈ ( 0 ... ( 𝐴 − 1 ) ) ) → 𝐵 = - 𝐶 ) |
121 |
56 120
|
breqtrd |
⊢ ( ( ( ( 𝐴 ∈ ℕ ∧ 𝐵 ∈ ( 0 ... 𝐴 ) ∧ 𝐶 ∈ ( 0 ... 𝐴 ) ) ∧ ( 2 · 𝐴 ) ∥ ( 𝐵 − - 𝐶 ) ) ∧ 𝐶 ∈ ( 0 ... ( 𝐴 − 1 ) ) ) → 0 ≤ - 𝐶 ) |
122 |
|
elfzelz |
⊢ ( 𝐶 ∈ ( 0 ... ( 𝐴 − 1 ) ) → 𝐶 ∈ ℤ ) |
123 |
122
|
zred |
⊢ ( 𝐶 ∈ ( 0 ... ( 𝐴 − 1 ) ) → 𝐶 ∈ ℝ ) |
124 |
123
|
adantl |
⊢ ( ( ( ( 𝐴 ∈ ℕ ∧ 𝐵 ∈ ( 0 ... 𝐴 ) ∧ 𝐶 ∈ ( 0 ... 𝐴 ) ) ∧ ( 2 · 𝐴 ) ∥ ( 𝐵 − - 𝐶 ) ) ∧ 𝐶 ∈ ( 0 ... ( 𝐴 − 1 ) ) ) → 𝐶 ∈ ℝ ) |
125 |
124
|
le0neg1d |
⊢ ( ( ( ( 𝐴 ∈ ℕ ∧ 𝐵 ∈ ( 0 ... 𝐴 ) ∧ 𝐶 ∈ ( 0 ... 𝐴 ) ) ∧ ( 2 · 𝐴 ) ∥ ( 𝐵 − - 𝐶 ) ) ∧ 𝐶 ∈ ( 0 ... ( 𝐴 − 1 ) ) ) → ( 𝐶 ≤ 0 ↔ 0 ≤ - 𝐶 ) ) |
126 |
121 125
|
mpbird |
⊢ ( ( ( ( 𝐴 ∈ ℕ ∧ 𝐵 ∈ ( 0 ... 𝐴 ) ∧ 𝐶 ∈ ( 0 ... 𝐴 ) ) ∧ ( 2 · 𝐴 ) ∥ ( 𝐵 − - 𝐶 ) ) ∧ 𝐶 ∈ ( 0 ... ( 𝐴 − 1 ) ) ) → 𝐶 ≤ 0 ) |
127 |
|
elfzle1 |
⊢ ( 𝐶 ∈ ( 0 ... ( 𝐴 − 1 ) ) → 0 ≤ 𝐶 ) |
128 |
127
|
adantl |
⊢ ( ( ( ( 𝐴 ∈ ℕ ∧ 𝐵 ∈ ( 0 ... 𝐴 ) ∧ 𝐶 ∈ ( 0 ... 𝐴 ) ) ∧ ( 2 · 𝐴 ) ∥ ( 𝐵 − - 𝐶 ) ) ∧ 𝐶 ∈ ( 0 ... ( 𝐴 − 1 ) ) ) → 0 ≤ 𝐶 ) |
129 |
|
letri3 |
⊢ ( ( 𝐶 ∈ ℝ ∧ 0 ∈ ℝ ) → ( 𝐶 = 0 ↔ ( 𝐶 ≤ 0 ∧ 0 ≤ 𝐶 ) ) ) |
130 |
124 22 129
|
sylancl |
⊢ ( ( ( ( 𝐴 ∈ ℕ ∧ 𝐵 ∈ ( 0 ... 𝐴 ) ∧ 𝐶 ∈ ( 0 ... 𝐴 ) ) ∧ ( 2 · 𝐴 ) ∥ ( 𝐵 − - 𝐶 ) ) ∧ 𝐶 ∈ ( 0 ... ( 𝐴 − 1 ) ) ) → ( 𝐶 = 0 ↔ ( 𝐶 ≤ 0 ∧ 0 ≤ 𝐶 ) ) ) |
131 |
126 128 130
|
mpbir2and |
⊢ ( ( ( ( 𝐴 ∈ ℕ ∧ 𝐵 ∈ ( 0 ... 𝐴 ) ∧ 𝐶 ∈ ( 0 ... 𝐴 ) ) ∧ ( 2 · 𝐴 ) ∥ ( 𝐵 − - 𝐶 ) ) ∧ 𝐶 ∈ ( 0 ... ( 𝐴 − 1 ) ) ) → 𝐶 = 0 ) |
132 |
131
|
negeqd |
⊢ ( ( ( ( 𝐴 ∈ ℕ ∧ 𝐵 ∈ ( 0 ... 𝐴 ) ∧ 𝐶 ∈ ( 0 ... 𝐴 ) ) ∧ ( 2 · 𝐴 ) ∥ ( 𝐵 − - 𝐶 ) ) ∧ 𝐶 ∈ ( 0 ... ( 𝐴 − 1 ) ) ) → - 𝐶 = - 0 ) |
133 |
132 88
|
eqtrd |
⊢ ( ( ( ( 𝐴 ∈ ℕ ∧ 𝐵 ∈ ( 0 ... 𝐴 ) ∧ 𝐶 ∈ ( 0 ... 𝐴 ) ) ∧ ( 2 · 𝐴 ) ∥ ( 𝐵 − - 𝐶 ) ) ∧ 𝐶 ∈ ( 0 ... ( 𝐴 − 1 ) ) ) → - 𝐶 = 0 ) |
134 |
133 120 131
|
3eqtr4d |
⊢ ( ( ( ( 𝐴 ∈ ℕ ∧ 𝐵 ∈ ( 0 ... 𝐴 ) ∧ 𝐶 ∈ ( 0 ... 𝐴 ) ) ∧ ( 2 · 𝐴 ) ∥ ( 𝐵 − - 𝐶 ) ) ∧ 𝐶 ∈ ( 0 ... ( 𝐴 − 1 ) ) ) → 𝐵 = 𝐶 ) |
135 |
|
oveq2 |
⊢ ( 𝐶 = 𝐴 → ( 𝐵 − 𝐶 ) = ( 𝐵 − 𝐴 ) ) |
136 |
135
|
adantl |
⊢ ( ( ( ( 𝐴 ∈ ℕ ∧ 𝐵 ∈ ( 0 ... 𝐴 ) ∧ 𝐶 ∈ ( 0 ... 𝐴 ) ) ∧ ( 2 · 𝐴 ) ∥ ( 𝐵 − - 𝐶 ) ) ∧ 𝐶 = 𝐴 ) → ( 𝐵 − 𝐶 ) = ( 𝐵 − 𝐴 ) ) |
137 |
136
|
fveq2d |
⊢ ( ( ( ( 𝐴 ∈ ℕ ∧ 𝐵 ∈ ( 0 ... 𝐴 ) ∧ 𝐶 ∈ ( 0 ... 𝐴 ) ) ∧ ( 2 · 𝐴 ) ∥ ( 𝐵 − - 𝐶 ) ) ∧ 𝐶 = 𝐴 ) → ( abs ‘ ( 𝐵 − 𝐶 ) ) = ( abs ‘ ( 𝐵 − 𝐴 ) ) ) |
138 |
40
|
ad2antrr |
⊢ ( ( ( ( 𝐴 ∈ ℕ ∧ 𝐵 ∈ ( 0 ... 𝐴 ) ∧ 𝐶 ∈ ( 0 ... 𝐴 ) ) ∧ ( 2 · 𝐴 ) ∥ ( 𝐵 − - 𝐶 ) ) ∧ 𝐶 = 𝐴 ) → ( abs ‘ ( 𝐵 − 𝐶 ) ) < ( 2 · 𝐴 ) ) |
139 |
137 138
|
eqbrtrrd |
⊢ ( ( ( ( 𝐴 ∈ ℕ ∧ 𝐵 ∈ ( 0 ... 𝐴 ) ∧ 𝐶 ∈ ( 0 ... 𝐴 ) ) ∧ ( 2 · 𝐴 ) ∥ ( 𝐵 − - 𝐶 ) ) ∧ 𝐶 = 𝐴 ) → ( abs ‘ ( 𝐵 − 𝐴 ) ) < ( 2 · 𝐴 ) ) |
140 |
93
|
ad2antrr |
⊢ ( ( ( ( 𝐴 ∈ ℕ ∧ 𝐵 ∈ ( 0 ... 𝐴 ) ∧ 𝐶 ∈ ( 0 ... 𝐴 ) ) ∧ ( 2 · 𝐴 ) ∥ ( 𝐵 − - 𝐶 ) ) ∧ 𝐶 = 𝐴 ) → ( 2 · 𝐴 ) ∈ ℕ ) |
141 |
7
|
ad2antrr |
⊢ ( ( ( ( 𝐴 ∈ ℕ ∧ 𝐵 ∈ ( 0 ... 𝐴 ) ∧ 𝐶 ∈ ( 0 ... 𝐴 ) ) ∧ ( 2 · 𝐴 ) ∥ ( 𝐵 − - 𝐶 ) ) ∧ 𝐶 = 𝐴 ) → 𝐵 ∈ ℤ ) |
142 |
3
|
ad2antrr |
⊢ ( ( ( ( 𝐴 ∈ ℕ ∧ 𝐵 ∈ ( 0 ... 𝐴 ) ∧ 𝐶 ∈ ( 0 ... 𝐴 ) ) ∧ ( 2 · 𝐴 ) ∥ ( 𝐵 − - 𝐶 ) ) ∧ 𝐶 = 𝐴 ) → 𝐴 ∈ ℤ ) |
143 |
|
simplr |
⊢ ( ( ( ( 𝐴 ∈ ℕ ∧ 𝐵 ∈ ( 0 ... 𝐴 ) ∧ 𝐶 ∈ ( 0 ... 𝐴 ) ) ∧ ( 2 · 𝐴 ) ∥ ( 𝐵 − - 𝐶 ) ) ∧ 𝐶 = 𝐴 ) → ( 2 · 𝐴 ) ∥ ( 𝐵 − - 𝐶 ) ) |
144 |
7
|
zcnd |
⊢ ( ( 𝐴 ∈ ℕ ∧ 𝐵 ∈ ( 0 ... 𝐴 ) ∧ 𝐶 ∈ ( 0 ... 𝐴 ) ) → 𝐵 ∈ ℂ ) |
145 |
36 36 144
|
ppncand |
⊢ ( ( 𝐴 ∈ ℕ ∧ 𝐵 ∈ ( 0 ... 𝐴 ) ∧ 𝐶 ∈ ( 0 ... 𝐴 ) ) → ( ( 𝐴 + 𝐴 ) + ( 𝐵 − 𝐴 ) ) = ( 𝐴 + 𝐵 ) ) |
146 |
36 144
|
addcomd |
⊢ ( ( 𝐴 ∈ ℕ ∧ 𝐵 ∈ ( 0 ... 𝐴 ) ∧ 𝐶 ∈ ( 0 ... 𝐴 ) ) → ( 𝐴 + 𝐵 ) = ( 𝐵 + 𝐴 ) ) |
147 |
145 146
|
eqtrd |
⊢ ( ( 𝐴 ∈ ℕ ∧ 𝐵 ∈ ( 0 ... 𝐴 ) ∧ 𝐶 ∈ ( 0 ... 𝐴 ) ) → ( ( 𝐴 + 𝐴 ) + ( 𝐵 − 𝐴 ) ) = ( 𝐵 + 𝐴 ) ) |
148 |
147
|
ad2antrr |
⊢ ( ( ( ( 𝐴 ∈ ℕ ∧ 𝐵 ∈ ( 0 ... 𝐴 ) ∧ 𝐶 ∈ ( 0 ... 𝐴 ) ) ∧ ( 2 · 𝐴 ) ∥ ( 𝐵 − - 𝐶 ) ) ∧ 𝐶 = 𝐴 ) → ( ( 𝐴 + 𝐴 ) + ( 𝐵 − 𝐴 ) ) = ( 𝐵 + 𝐴 ) ) |
149 |
|
oveq2 |
⊢ ( 𝐶 = 𝐴 → ( 𝐵 + 𝐶 ) = ( 𝐵 + 𝐴 ) ) |
150 |
149
|
adantl |
⊢ ( ( ( ( 𝐴 ∈ ℕ ∧ 𝐵 ∈ ( 0 ... 𝐴 ) ∧ 𝐶 ∈ ( 0 ... 𝐴 ) ) ∧ ( 2 · 𝐴 ) ∥ ( 𝐵 − - 𝐶 ) ) ∧ 𝐶 = 𝐴 ) → ( 𝐵 + 𝐶 ) = ( 𝐵 + 𝐴 ) ) |
151 |
148 150
|
eqtr4d |
⊢ ( ( ( ( 𝐴 ∈ ℕ ∧ 𝐵 ∈ ( 0 ... 𝐴 ) ∧ 𝐶 ∈ ( 0 ... 𝐴 ) ) ∧ ( 2 · 𝐴 ) ∥ ( 𝐵 − - 𝐶 ) ) ∧ 𝐶 = 𝐴 ) → ( ( 𝐴 + 𝐴 ) + ( 𝐵 − 𝐴 ) ) = ( 𝐵 + 𝐶 ) ) |
152 |
38
|
oveq1d |
⊢ ( ( 𝐴 ∈ ℕ ∧ 𝐵 ∈ ( 0 ... 𝐴 ) ∧ 𝐶 ∈ ( 0 ... 𝐴 ) ) → ( ( 2 · 𝐴 ) + ( 𝐵 − 𝐴 ) ) = ( ( 𝐴 + 𝐴 ) + ( 𝐵 − 𝐴 ) ) ) |
153 |
152
|
ad2antrr |
⊢ ( ( ( ( 𝐴 ∈ ℕ ∧ 𝐵 ∈ ( 0 ... 𝐴 ) ∧ 𝐶 ∈ ( 0 ... 𝐴 ) ) ∧ ( 2 · 𝐴 ) ∥ ( 𝐵 − - 𝐶 ) ) ∧ 𝐶 = 𝐴 ) → ( ( 2 · 𝐴 ) + ( 𝐵 − 𝐴 ) ) = ( ( 𝐴 + 𝐴 ) + ( 𝐵 − 𝐴 ) ) ) |
154 |
16
|
zcnd |
⊢ ( ( 𝐴 ∈ ℕ ∧ 𝐵 ∈ ( 0 ... 𝐴 ) ∧ 𝐶 ∈ ( 0 ... 𝐴 ) ) → 𝐶 ∈ ℂ ) |
155 |
144 154
|
subnegd |
⊢ ( ( 𝐴 ∈ ℕ ∧ 𝐵 ∈ ( 0 ... 𝐴 ) ∧ 𝐶 ∈ ( 0 ... 𝐴 ) ) → ( 𝐵 − - 𝐶 ) = ( 𝐵 + 𝐶 ) ) |
156 |
155
|
ad2antrr |
⊢ ( ( ( ( 𝐴 ∈ ℕ ∧ 𝐵 ∈ ( 0 ... 𝐴 ) ∧ 𝐶 ∈ ( 0 ... 𝐴 ) ) ∧ ( 2 · 𝐴 ) ∥ ( 𝐵 − - 𝐶 ) ) ∧ 𝐶 = 𝐴 ) → ( 𝐵 − - 𝐶 ) = ( 𝐵 + 𝐶 ) ) |
157 |
151 153 156
|
3eqtr4d |
⊢ ( ( ( ( 𝐴 ∈ ℕ ∧ 𝐵 ∈ ( 0 ... 𝐴 ) ∧ 𝐶 ∈ ( 0 ... 𝐴 ) ) ∧ ( 2 · 𝐴 ) ∥ ( 𝐵 − - 𝐶 ) ) ∧ 𝐶 = 𝐴 ) → ( ( 2 · 𝐴 ) + ( 𝐵 − 𝐴 ) ) = ( 𝐵 − - 𝐶 ) ) |
158 |
143 157
|
breqtrrd |
⊢ ( ( ( ( 𝐴 ∈ ℕ ∧ 𝐵 ∈ ( 0 ... 𝐴 ) ∧ 𝐶 ∈ ( 0 ... 𝐴 ) ) ∧ ( 2 · 𝐴 ) ∥ ( 𝐵 − - 𝐶 ) ) ∧ 𝐶 = 𝐴 ) → ( 2 · 𝐴 ) ∥ ( ( 2 · 𝐴 ) + ( 𝐵 − 𝐴 ) ) ) |
159 |
5
|
ad2antrr |
⊢ ( ( ( ( 𝐴 ∈ ℕ ∧ 𝐵 ∈ ( 0 ... 𝐴 ) ∧ 𝐶 ∈ ( 0 ... 𝐴 ) ) ∧ ( 2 · 𝐴 ) ∥ ( 𝐵 − - 𝐶 ) ) ∧ 𝐶 = 𝐴 ) → ( 2 · 𝐴 ) ∈ ℤ ) |
160 |
7 3
|
zsubcld |
⊢ ( ( 𝐴 ∈ ℕ ∧ 𝐵 ∈ ( 0 ... 𝐴 ) ∧ 𝐶 ∈ ( 0 ... 𝐴 ) ) → ( 𝐵 − 𝐴 ) ∈ ℤ ) |
161 |
160
|
ad2antrr |
⊢ ( ( ( ( 𝐴 ∈ ℕ ∧ 𝐵 ∈ ( 0 ... 𝐴 ) ∧ 𝐶 ∈ ( 0 ... 𝐴 ) ) ∧ ( 2 · 𝐴 ) ∥ ( 𝐵 − - 𝐶 ) ) ∧ 𝐶 = 𝐴 ) → ( 𝐵 − 𝐴 ) ∈ ℤ ) |
162 |
|
dvdsadd |
⊢ ( ( ( 2 · 𝐴 ) ∈ ℤ ∧ ( 𝐵 − 𝐴 ) ∈ ℤ ) → ( ( 2 · 𝐴 ) ∥ ( 𝐵 − 𝐴 ) ↔ ( 2 · 𝐴 ) ∥ ( ( 2 · 𝐴 ) + ( 𝐵 − 𝐴 ) ) ) ) |
163 |
159 161 162
|
syl2anc |
⊢ ( ( ( ( 𝐴 ∈ ℕ ∧ 𝐵 ∈ ( 0 ... 𝐴 ) ∧ 𝐶 ∈ ( 0 ... 𝐴 ) ) ∧ ( 2 · 𝐴 ) ∥ ( 𝐵 − - 𝐶 ) ) ∧ 𝐶 = 𝐴 ) → ( ( 2 · 𝐴 ) ∥ ( 𝐵 − 𝐴 ) ↔ ( 2 · 𝐴 ) ∥ ( ( 2 · 𝐴 ) + ( 𝐵 − 𝐴 ) ) ) ) |
164 |
158 163
|
mpbird |
⊢ ( ( ( ( 𝐴 ∈ ℕ ∧ 𝐵 ∈ ( 0 ... 𝐴 ) ∧ 𝐶 ∈ ( 0 ... 𝐴 ) ) ∧ ( 2 · 𝐴 ) ∥ ( 𝐵 − - 𝐶 ) ) ∧ 𝐶 = 𝐴 ) → ( 2 · 𝐴 ) ∥ ( 𝐵 − 𝐴 ) ) |
165 |
|
congabseq |
⊢ ( ( ( ( 2 · 𝐴 ) ∈ ℕ ∧ 𝐵 ∈ ℤ ∧ 𝐴 ∈ ℤ ) ∧ ( 2 · 𝐴 ) ∥ ( 𝐵 − 𝐴 ) ) → ( ( abs ‘ ( 𝐵 − 𝐴 ) ) < ( 2 · 𝐴 ) ↔ 𝐵 = 𝐴 ) ) |
166 |
140 141 142 164 165
|
syl31anc |
⊢ ( ( ( ( 𝐴 ∈ ℕ ∧ 𝐵 ∈ ( 0 ... 𝐴 ) ∧ 𝐶 ∈ ( 0 ... 𝐴 ) ) ∧ ( 2 · 𝐴 ) ∥ ( 𝐵 − - 𝐶 ) ) ∧ 𝐶 = 𝐴 ) → ( ( abs ‘ ( 𝐵 − 𝐴 ) ) < ( 2 · 𝐴 ) ↔ 𝐵 = 𝐴 ) ) |
167 |
139 166
|
mpbid |
⊢ ( ( ( ( 𝐴 ∈ ℕ ∧ 𝐵 ∈ ( 0 ... 𝐴 ) ∧ 𝐶 ∈ ( 0 ... 𝐴 ) ) ∧ ( 2 · 𝐴 ) ∥ ( 𝐵 − - 𝐶 ) ) ∧ 𝐶 = 𝐴 ) → 𝐵 = 𝐴 ) |
168 |
|
simpr |
⊢ ( ( ( ( 𝐴 ∈ ℕ ∧ 𝐵 ∈ ( 0 ... 𝐴 ) ∧ 𝐶 ∈ ( 0 ... 𝐴 ) ) ∧ ( 2 · 𝐴 ) ∥ ( 𝐵 − - 𝐶 ) ) ∧ 𝐶 = 𝐴 ) → 𝐶 = 𝐴 ) |
169 |
167 168
|
eqtr4d |
⊢ ( ( ( ( 𝐴 ∈ ℕ ∧ 𝐵 ∈ ( 0 ... 𝐴 ) ∧ 𝐶 ∈ ( 0 ... 𝐴 ) ) ∧ ( 2 · 𝐴 ) ∥ ( 𝐵 − - 𝐶 ) ) ∧ 𝐶 = 𝐴 ) → 𝐵 = 𝐶 ) |
170 |
|
nnnn0 |
⊢ ( 𝐴 ∈ ℕ → 𝐴 ∈ ℕ0 ) |
171 |
170
|
3ad2ant1 |
⊢ ( ( 𝐴 ∈ ℕ ∧ 𝐵 ∈ ( 0 ... 𝐴 ) ∧ 𝐶 ∈ ( 0 ... 𝐴 ) ) → 𝐴 ∈ ℕ0 ) |
172 |
|
nn0uz |
⊢ ℕ0 = ( ℤ≥ ‘ 0 ) |
173 |
171 172
|
eleqtrdi |
⊢ ( ( 𝐴 ∈ ℕ ∧ 𝐵 ∈ ( 0 ... 𝐴 ) ∧ 𝐶 ∈ ( 0 ... 𝐴 ) ) → 𝐴 ∈ ( ℤ≥ ‘ 0 ) ) |
174 |
|
fzm1 |
⊢ ( 𝐴 ∈ ( ℤ≥ ‘ 0 ) → ( 𝐶 ∈ ( 0 ... 𝐴 ) ↔ ( 𝐶 ∈ ( 0 ... ( 𝐴 − 1 ) ) ∨ 𝐶 = 𝐴 ) ) ) |
175 |
174
|
biimpa |
⊢ ( ( 𝐴 ∈ ( ℤ≥ ‘ 0 ) ∧ 𝐶 ∈ ( 0 ... 𝐴 ) ) → ( 𝐶 ∈ ( 0 ... ( 𝐴 − 1 ) ) ∨ 𝐶 = 𝐴 ) ) |
176 |
173 29 175
|
syl2anc |
⊢ ( ( 𝐴 ∈ ℕ ∧ 𝐵 ∈ ( 0 ... 𝐴 ) ∧ 𝐶 ∈ ( 0 ... 𝐴 ) ) → ( 𝐶 ∈ ( 0 ... ( 𝐴 − 1 ) ) ∨ 𝐶 = 𝐴 ) ) |
177 |
176
|
adantr |
⊢ ( ( ( 𝐴 ∈ ℕ ∧ 𝐵 ∈ ( 0 ... 𝐴 ) ∧ 𝐶 ∈ ( 0 ... 𝐴 ) ) ∧ ( 2 · 𝐴 ) ∥ ( 𝐵 − - 𝐶 ) ) → ( 𝐶 ∈ ( 0 ... ( 𝐴 − 1 ) ) ∨ 𝐶 = 𝐴 ) ) |
178 |
134 169 177
|
mpjaodan |
⊢ ( ( ( 𝐴 ∈ ℕ ∧ 𝐵 ∈ ( 0 ... 𝐴 ) ∧ 𝐶 ∈ ( 0 ... 𝐴 ) ) ∧ ( 2 · 𝐴 ) ∥ ( 𝐵 − - 𝐶 ) ) → 𝐵 = 𝐶 ) |
179 |
53 178
|
jaodan |
⊢ ( ( ( 𝐴 ∈ ℕ ∧ 𝐵 ∈ ( 0 ... 𝐴 ) ∧ 𝐶 ∈ ( 0 ... 𝐴 ) ) ∧ ( ( 2 · 𝐴 ) ∥ ( 𝐵 − 𝐶 ) ∨ ( 2 · 𝐴 ) ∥ ( 𝐵 − - 𝐶 ) ) ) → 𝐵 = 𝐶 ) |
180 |
14 179
|
impbida |
⊢ ( ( 𝐴 ∈ ℕ ∧ 𝐵 ∈ ( 0 ... 𝐴 ) ∧ 𝐶 ∈ ( 0 ... 𝐴 ) ) → ( 𝐵 = 𝐶 ↔ ( ( 2 · 𝐴 ) ∥ ( 𝐵 − 𝐶 ) ∨ ( 2 · 𝐴 ) ∥ ( 𝐵 − - 𝐶 ) ) ) ) |