Step |
Hyp |
Ref |
Expression |
1 |
|
elznn0nn |
⊢ ( 𝑍 ∈ ℤ ↔ ( 𝑍 ∈ ℕ0 ∨ ( 𝑍 ∈ ℝ ∧ - 𝑍 ∈ ℕ ) ) ) |
2 |
|
elnn0 |
⊢ ( 𝑍 ∈ ℕ0 ↔ ( 𝑍 ∈ ℕ ∨ 𝑍 = 0 ) ) |
3 |
|
elnn1uz2 |
⊢ ( 𝑍 ∈ ℕ ↔ ( 𝑍 = 1 ∨ 𝑍 ∈ ( ℤ≥ ‘ 2 ) ) ) |
4 |
|
oveq1 |
⊢ ( 𝑍 = 1 → ( 𝑍 · 𝐴 ) = ( 1 · 𝐴 ) ) |
5 |
4
|
adantr |
⊢ ( ( 𝑍 = 1 ∧ ( 𝐴 ∈ ℙ ∧ 𝐵 ∈ ℙ ) ) → ( 𝑍 · 𝐴 ) = ( 1 · 𝐴 ) ) |
6 |
5
|
eqeq1d |
⊢ ( ( 𝑍 = 1 ∧ ( 𝐴 ∈ ℙ ∧ 𝐵 ∈ ℙ ) ) → ( ( 𝑍 · 𝐴 ) = 𝐵 ↔ ( 1 · 𝐴 ) = 𝐵 ) ) |
7 |
|
prmz |
⊢ ( 𝐴 ∈ ℙ → 𝐴 ∈ ℤ ) |
8 |
7
|
zcnd |
⊢ ( 𝐴 ∈ ℙ → 𝐴 ∈ ℂ ) |
9 |
8
|
mulid2d |
⊢ ( 𝐴 ∈ ℙ → ( 1 · 𝐴 ) = 𝐴 ) |
10 |
9
|
adantr |
⊢ ( ( 𝐴 ∈ ℙ ∧ 𝐵 ∈ ℙ ) → ( 1 · 𝐴 ) = 𝐴 ) |
11 |
10
|
eqeq1d |
⊢ ( ( 𝐴 ∈ ℙ ∧ 𝐵 ∈ ℙ ) → ( ( 1 · 𝐴 ) = 𝐵 ↔ 𝐴 = 𝐵 ) ) |
12 |
11
|
biimpd |
⊢ ( ( 𝐴 ∈ ℙ ∧ 𝐵 ∈ ℙ ) → ( ( 1 · 𝐴 ) = 𝐵 → 𝐴 = 𝐵 ) ) |
13 |
12
|
adantl |
⊢ ( ( 𝑍 = 1 ∧ ( 𝐴 ∈ ℙ ∧ 𝐵 ∈ ℙ ) ) → ( ( 1 · 𝐴 ) = 𝐵 → 𝐴 = 𝐵 ) ) |
14 |
6 13
|
sylbid |
⊢ ( ( 𝑍 = 1 ∧ ( 𝐴 ∈ ℙ ∧ 𝐵 ∈ ℙ ) ) → ( ( 𝑍 · 𝐴 ) = 𝐵 → 𝐴 = 𝐵 ) ) |
15 |
14
|
ex |
⊢ ( 𝑍 = 1 → ( ( 𝐴 ∈ ℙ ∧ 𝐵 ∈ ℙ ) → ( ( 𝑍 · 𝐴 ) = 𝐵 → 𝐴 = 𝐵 ) ) ) |
16 |
|
prmuz2 |
⊢ ( 𝐴 ∈ ℙ → 𝐴 ∈ ( ℤ≥ ‘ 2 ) ) |
17 |
16
|
adantr |
⊢ ( ( 𝐴 ∈ ℙ ∧ 𝐵 ∈ ℙ ) → 𝐴 ∈ ( ℤ≥ ‘ 2 ) ) |
18 |
|
nprm |
⊢ ( ( 𝑍 ∈ ( ℤ≥ ‘ 2 ) ∧ 𝐴 ∈ ( ℤ≥ ‘ 2 ) ) → ¬ ( 𝑍 · 𝐴 ) ∈ ℙ ) |
19 |
17 18
|
sylan2 |
⊢ ( ( 𝑍 ∈ ( ℤ≥ ‘ 2 ) ∧ ( 𝐴 ∈ ℙ ∧ 𝐵 ∈ ℙ ) ) → ¬ ( 𝑍 · 𝐴 ) ∈ ℙ ) |
20 |
|
eleq1 |
⊢ ( ( 𝑍 · 𝐴 ) = 𝐵 → ( ( 𝑍 · 𝐴 ) ∈ ℙ ↔ 𝐵 ∈ ℙ ) ) |
21 |
20
|
notbid |
⊢ ( ( 𝑍 · 𝐴 ) = 𝐵 → ( ¬ ( 𝑍 · 𝐴 ) ∈ ℙ ↔ ¬ 𝐵 ∈ ℙ ) ) |
22 |
|
pm2.24 |
⊢ ( 𝐵 ∈ ℙ → ( ¬ 𝐵 ∈ ℙ → 𝐴 = 𝐵 ) ) |
23 |
22
|
adantl |
⊢ ( ( 𝐴 ∈ ℙ ∧ 𝐵 ∈ ℙ ) → ( ¬ 𝐵 ∈ ℙ → 𝐴 = 𝐵 ) ) |
24 |
23
|
adantl |
⊢ ( ( 𝑍 ∈ ( ℤ≥ ‘ 2 ) ∧ ( 𝐴 ∈ ℙ ∧ 𝐵 ∈ ℙ ) ) → ( ¬ 𝐵 ∈ ℙ → 𝐴 = 𝐵 ) ) |
25 |
24
|
com12 |
⊢ ( ¬ 𝐵 ∈ ℙ → ( ( 𝑍 ∈ ( ℤ≥ ‘ 2 ) ∧ ( 𝐴 ∈ ℙ ∧ 𝐵 ∈ ℙ ) ) → 𝐴 = 𝐵 ) ) |
26 |
21 25
|
syl6bi |
⊢ ( ( 𝑍 · 𝐴 ) = 𝐵 → ( ¬ ( 𝑍 · 𝐴 ) ∈ ℙ → ( ( 𝑍 ∈ ( ℤ≥ ‘ 2 ) ∧ ( 𝐴 ∈ ℙ ∧ 𝐵 ∈ ℙ ) ) → 𝐴 = 𝐵 ) ) ) |
27 |
26
|
com3l |
⊢ ( ¬ ( 𝑍 · 𝐴 ) ∈ ℙ → ( ( 𝑍 ∈ ( ℤ≥ ‘ 2 ) ∧ ( 𝐴 ∈ ℙ ∧ 𝐵 ∈ ℙ ) ) → ( ( 𝑍 · 𝐴 ) = 𝐵 → 𝐴 = 𝐵 ) ) ) |
28 |
19 27
|
mpcom |
⊢ ( ( 𝑍 ∈ ( ℤ≥ ‘ 2 ) ∧ ( 𝐴 ∈ ℙ ∧ 𝐵 ∈ ℙ ) ) → ( ( 𝑍 · 𝐴 ) = 𝐵 → 𝐴 = 𝐵 ) ) |
29 |
28
|
ex |
⊢ ( 𝑍 ∈ ( ℤ≥ ‘ 2 ) → ( ( 𝐴 ∈ ℙ ∧ 𝐵 ∈ ℙ ) → ( ( 𝑍 · 𝐴 ) = 𝐵 → 𝐴 = 𝐵 ) ) ) |
30 |
15 29
|
jaoi |
⊢ ( ( 𝑍 = 1 ∨ 𝑍 ∈ ( ℤ≥ ‘ 2 ) ) → ( ( 𝐴 ∈ ℙ ∧ 𝐵 ∈ ℙ ) → ( ( 𝑍 · 𝐴 ) = 𝐵 → 𝐴 = 𝐵 ) ) ) |
31 |
3 30
|
sylbi |
⊢ ( 𝑍 ∈ ℕ → ( ( 𝐴 ∈ ℙ ∧ 𝐵 ∈ ℙ ) → ( ( 𝑍 · 𝐴 ) = 𝐵 → 𝐴 = 𝐵 ) ) ) |
32 |
|
oveq1 |
⊢ ( 𝑍 = 0 → ( 𝑍 · 𝐴 ) = ( 0 · 𝐴 ) ) |
33 |
32
|
eqeq1d |
⊢ ( 𝑍 = 0 → ( ( 𝑍 · 𝐴 ) = 𝐵 ↔ ( 0 · 𝐴 ) = 𝐵 ) ) |
34 |
|
prmnn |
⊢ ( 𝐴 ∈ ℙ → 𝐴 ∈ ℕ ) |
35 |
34
|
nnred |
⊢ ( 𝐴 ∈ ℙ → 𝐴 ∈ ℝ ) |
36 |
|
mul02lem2 |
⊢ ( 𝐴 ∈ ℝ → ( 0 · 𝐴 ) = 0 ) |
37 |
35 36
|
syl |
⊢ ( 𝐴 ∈ ℙ → ( 0 · 𝐴 ) = 0 ) |
38 |
37
|
adantr |
⊢ ( ( 𝐴 ∈ ℙ ∧ 𝐵 ∈ ℙ ) → ( 0 · 𝐴 ) = 0 ) |
39 |
38
|
eqeq1d |
⊢ ( ( 𝐴 ∈ ℙ ∧ 𝐵 ∈ ℙ ) → ( ( 0 · 𝐴 ) = 𝐵 ↔ 0 = 𝐵 ) ) |
40 |
|
prmnn |
⊢ ( 𝐵 ∈ ℙ → 𝐵 ∈ ℕ ) |
41 |
|
elnnne0 |
⊢ ( 𝐵 ∈ ℕ ↔ ( 𝐵 ∈ ℕ0 ∧ 𝐵 ≠ 0 ) ) |
42 |
|
eqneqall |
⊢ ( 𝐵 = 0 → ( 𝐵 ≠ 0 → 𝐴 = 𝐵 ) ) |
43 |
42
|
eqcoms |
⊢ ( 0 = 𝐵 → ( 𝐵 ≠ 0 → 𝐴 = 𝐵 ) ) |
44 |
43
|
com12 |
⊢ ( 𝐵 ≠ 0 → ( 0 = 𝐵 → 𝐴 = 𝐵 ) ) |
45 |
44
|
adantl |
⊢ ( ( 𝐵 ∈ ℕ0 ∧ 𝐵 ≠ 0 ) → ( 0 = 𝐵 → 𝐴 = 𝐵 ) ) |
46 |
41 45
|
sylbi |
⊢ ( 𝐵 ∈ ℕ → ( 0 = 𝐵 → 𝐴 = 𝐵 ) ) |
47 |
40 46
|
syl |
⊢ ( 𝐵 ∈ ℙ → ( 0 = 𝐵 → 𝐴 = 𝐵 ) ) |
48 |
47
|
adantl |
⊢ ( ( 𝐴 ∈ ℙ ∧ 𝐵 ∈ ℙ ) → ( 0 = 𝐵 → 𝐴 = 𝐵 ) ) |
49 |
39 48
|
sylbid |
⊢ ( ( 𝐴 ∈ ℙ ∧ 𝐵 ∈ ℙ ) → ( ( 0 · 𝐴 ) = 𝐵 → 𝐴 = 𝐵 ) ) |
50 |
49
|
com12 |
⊢ ( ( 0 · 𝐴 ) = 𝐵 → ( ( 𝐴 ∈ ℙ ∧ 𝐵 ∈ ℙ ) → 𝐴 = 𝐵 ) ) |
51 |
33 50
|
syl6bi |
⊢ ( 𝑍 = 0 → ( ( 𝑍 · 𝐴 ) = 𝐵 → ( ( 𝐴 ∈ ℙ ∧ 𝐵 ∈ ℙ ) → 𝐴 = 𝐵 ) ) ) |
52 |
51
|
com23 |
⊢ ( 𝑍 = 0 → ( ( 𝐴 ∈ ℙ ∧ 𝐵 ∈ ℙ ) → ( ( 𝑍 · 𝐴 ) = 𝐵 → 𝐴 = 𝐵 ) ) ) |
53 |
31 52
|
jaoi |
⊢ ( ( 𝑍 ∈ ℕ ∨ 𝑍 = 0 ) → ( ( 𝐴 ∈ ℙ ∧ 𝐵 ∈ ℙ ) → ( ( 𝑍 · 𝐴 ) = 𝐵 → 𝐴 = 𝐵 ) ) ) |
54 |
2 53
|
sylbi |
⊢ ( 𝑍 ∈ ℕ0 → ( ( 𝐴 ∈ ℙ ∧ 𝐵 ∈ ℙ ) → ( ( 𝑍 · 𝐴 ) = 𝐵 → 𝐴 = 𝐵 ) ) ) |
55 |
|
elnnz |
⊢ ( - 𝑍 ∈ ℕ ↔ ( - 𝑍 ∈ ℤ ∧ 0 < - 𝑍 ) ) |
56 |
|
lt0neg1 |
⊢ ( 𝑍 ∈ ℝ → ( 𝑍 < 0 ↔ 0 < - 𝑍 ) ) |
57 |
34
|
nngt0d |
⊢ ( 𝐴 ∈ ℙ → 0 < 𝐴 ) |
58 |
57
|
adantr |
⊢ ( ( 𝐴 ∈ ℙ ∧ 𝐵 ∈ ℙ ) → 0 < 𝐴 ) |
59 |
|
simpr |
⊢ ( ( 𝑍 ∈ ℝ ∧ 𝑍 < 0 ) → 𝑍 < 0 ) |
60 |
58 59
|
anim12ci |
⊢ ( ( ( 𝐴 ∈ ℙ ∧ 𝐵 ∈ ℙ ) ∧ ( 𝑍 ∈ ℝ ∧ 𝑍 < 0 ) ) → ( 𝑍 < 0 ∧ 0 < 𝐴 ) ) |
61 |
60
|
orcd |
⊢ ( ( ( 𝐴 ∈ ℙ ∧ 𝐵 ∈ ℙ ) ∧ ( 𝑍 ∈ ℝ ∧ 𝑍 < 0 ) ) → ( ( 𝑍 < 0 ∧ 0 < 𝐴 ) ∨ ( 0 < 𝑍 ∧ 𝐴 < 0 ) ) ) |
62 |
|
simprl |
⊢ ( ( ( 𝐴 ∈ ℙ ∧ 𝐵 ∈ ℙ ) ∧ ( 𝑍 ∈ ℝ ∧ 𝑍 < 0 ) ) → 𝑍 ∈ ℝ ) |
63 |
35
|
adantr |
⊢ ( ( 𝐴 ∈ ℙ ∧ 𝐵 ∈ ℙ ) → 𝐴 ∈ ℝ ) |
64 |
63
|
adantr |
⊢ ( ( ( 𝐴 ∈ ℙ ∧ 𝐵 ∈ ℙ ) ∧ ( 𝑍 ∈ ℝ ∧ 𝑍 < 0 ) ) → 𝐴 ∈ ℝ ) |
65 |
62 64
|
mul2lt0bi |
⊢ ( ( ( 𝐴 ∈ ℙ ∧ 𝐵 ∈ ℙ ) ∧ ( 𝑍 ∈ ℝ ∧ 𝑍 < 0 ) ) → ( ( 𝑍 · 𝐴 ) < 0 ↔ ( ( 𝑍 < 0 ∧ 0 < 𝐴 ) ∨ ( 0 < 𝑍 ∧ 𝐴 < 0 ) ) ) ) |
66 |
61 65
|
mpbird |
⊢ ( ( ( 𝐴 ∈ ℙ ∧ 𝐵 ∈ ℙ ) ∧ ( 𝑍 ∈ ℝ ∧ 𝑍 < 0 ) ) → ( 𝑍 · 𝐴 ) < 0 ) |
67 |
66
|
ex |
⊢ ( ( 𝐴 ∈ ℙ ∧ 𝐵 ∈ ℙ ) → ( ( 𝑍 ∈ ℝ ∧ 𝑍 < 0 ) → ( 𝑍 · 𝐴 ) < 0 ) ) |
68 |
|
breq1 |
⊢ ( ( 𝑍 · 𝐴 ) = 𝐵 → ( ( 𝑍 · 𝐴 ) < 0 ↔ 𝐵 < 0 ) ) |
69 |
68
|
adantl |
⊢ ( ( ( 𝐴 ∈ ℙ ∧ 𝐵 ∈ ℙ ) ∧ ( 𝑍 · 𝐴 ) = 𝐵 ) → ( ( 𝑍 · 𝐴 ) < 0 ↔ 𝐵 < 0 ) ) |
70 |
|
nnnn0 |
⊢ ( 𝐵 ∈ ℕ → 𝐵 ∈ ℕ0 ) |
71 |
|
nn0nlt0 |
⊢ ( 𝐵 ∈ ℕ0 → ¬ 𝐵 < 0 ) |
72 |
71
|
pm2.21d |
⊢ ( 𝐵 ∈ ℕ0 → ( 𝐵 < 0 → 𝐴 = 𝐵 ) ) |
73 |
70 72
|
syl |
⊢ ( 𝐵 ∈ ℕ → ( 𝐵 < 0 → 𝐴 = 𝐵 ) ) |
74 |
40 73
|
syl |
⊢ ( 𝐵 ∈ ℙ → ( 𝐵 < 0 → 𝐴 = 𝐵 ) ) |
75 |
74
|
adantl |
⊢ ( ( 𝐴 ∈ ℙ ∧ 𝐵 ∈ ℙ ) → ( 𝐵 < 0 → 𝐴 = 𝐵 ) ) |
76 |
75
|
adantr |
⊢ ( ( ( 𝐴 ∈ ℙ ∧ 𝐵 ∈ ℙ ) ∧ ( 𝑍 · 𝐴 ) = 𝐵 ) → ( 𝐵 < 0 → 𝐴 = 𝐵 ) ) |
77 |
69 76
|
sylbid |
⊢ ( ( ( 𝐴 ∈ ℙ ∧ 𝐵 ∈ ℙ ) ∧ ( 𝑍 · 𝐴 ) = 𝐵 ) → ( ( 𝑍 · 𝐴 ) < 0 → 𝐴 = 𝐵 ) ) |
78 |
77
|
ex |
⊢ ( ( 𝐴 ∈ ℙ ∧ 𝐵 ∈ ℙ ) → ( ( 𝑍 · 𝐴 ) = 𝐵 → ( ( 𝑍 · 𝐴 ) < 0 → 𝐴 = 𝐵 ) ) ) |
79 |
78
|
com23 |
⊢ ( ( 𝐴 ∈ ℙ ∧ 𝐵 ∈ ℙ ) → ( ( 𝑍 · 𝐴 ) < 0 → ( ( 𝑍 · 𝐴 ) = 𝐵 → 𝐴 = 𝐵 ) ) ) |
80 |
67 79
|
syldc |
⊢ ( ( 𝑍 ∈ ℝ ∧ 𝑍 < 0 ) → ( ( 𝐴 ∈ ℙ ∧ 𝐵 ∈ ℙ ) → ( ( 𝑍 · 𝐴 ) = 𝐵 → 𝐴 = 𝐵 ) ) ) |
81 |
80
|
ex |
⊢ ( 𝑍 ∈ ℝ → ( 𝑍 < 0 → ( ( 𝐴 ∈ ℙ ∧ 𝐵 ∈ ℙ ) → ( ( 𝑍 · 𝐴 ) = 𝐵 → 𝐴 = 𝐵 ) ) ) ) |
82 |
56 81
|
sylbird |
⊢ ( 𝑍 ∈ ℝ → ( 0 < - 𝑍 → ( ( 𝐴 ∈ ℙ ∧ 𝐵 ∈ ℙ ) → ( ( 𝑍 · 𝐴 ) = 𝐵 → 𝐴 = 𝐵 ) ) ) ) |
83 |
82
|
adantld |
⊢ ( 𝑍 ∈ ℝ → ( ( - 𝑍 ∈ ℤ ∧ 0 < - 𝑍 ) → ( ( 𝐴 ∈ ℙ ∧ 𝐵 ∈ ℙ ) → ( ( 𝑍 · 𝐴 ) = 𝐵 → 𝐴 = 𝐵 ) ) ) ) |
84 |
55 83
|
syl5bi |
⊢ ( 𝑍 ∈ ℝ → ( - 𝑍 ∈ ℕ → ( ( 𝐴 ∈ ℙ ∧ 𝐵 ∈ ℙ ) → ( ( 𝑍 · 𝐴 ) = 𝐵 → 𝐴 = 𝐵 ) ) ) ) |
85 |
84
|
imp |
⊢ ( ( 𝑍 ∈ ℝ ∧ - 𝑍 ∈ ℕ ) → ( ( 𝐴 ∈ ℙ ∧ 𝐵 ∈ ℙ ) → ( ( 𝑍 · 𝐴 ) = 𝐵 → 𝐴 = 𝐵 ) ) ) |
86 |
54 85
|
jaoi |
⊢ ( ( 𝑍 ∈ ℕ0 ∨ ( 𝑍 ∈ ℝ ∧ - 𝑍 ∈ ℕ ) ) → ( ( 𝐴 ∈ ℙ ∧ 𝐵 ∈ ℙ ) → ( ( 𝑍 · 𝐴 ) = 𝐵 → 𝐴 = 𝐵 ) ) ) |
87 |
1 86
|
sylbi |
⊢ ( 𝑍 ∈ ℤ → ( ( 𝐴 ∈ ℙ ∧ 𝐵 ∈ ℙ ) → ( ( 𝑍 · 𝐴 ) = 𝐵 → 𝐴 = 𝐵 ) ) ) |
88 |
87
|
3impib |
⊢ ( ( 𝑍 ∈ ℤ ∧ 𝐴 ∈ ℙ ∧ 𝐵 ∈ ℙ ) → ( ( 𝑍 · 𝐴 ) = 𝐵 → 𝐴 = 𝐵 ) ) |