Step |
Hyp |
Ref |
Expression |
1 |
|
2sq.1 |
⊢ 𝑆 = ran ( 𝑤 ∈ ℤ[i] ↦ ( ( abs ‘ 𝑤 ) ↑ 2 ) ) |
2 |
|
2sqlem6.1 |
⊢ ( 𝜑 → 𝐴 ∈ ℕ ) |
3 |
|
2sqlem6.2 |
⊢ ( 𝜑 → 𝐵 ∈ ℕ ) |
4 |
|
2sqlem6.3 |
⊢ ( 𝜑 → ∀ 𝑝 ∈ ℙ ( 𝑝 ∥ 𝐵 → 𝑝 ∈ 𝑆 ) ) |
5 |
|
2sqlem6.4 |
⊢ ( 𝜑 → ( 𝐴 · 𝐵 ) ∈ 𝑆 ) |
6 |
|
breq2 |
⊢ ( 𝑥 = 1 → ( 𝑝 ∥ 𝑥 ↔ 𝑝 ∥ 1 ) ) |
7 |
6
|
imbi1d |
⊢ ( 𝑥 = 1 → ( ( 𝑝 ∥ 𝑥 → 𝑝 ∈ 𝑆 ) ↔ ( 𝑝 ∥ 1 → 𝑝 ∈ 𝑆 ) ) ) |
8 |
7
|
ralbidv |
⊢ ( 𝑥 = 1 → ( ∀ 𝑝 ∈ ℙ ( 𝑝 ∥ 𝑥 → 𝑝 ∈ 𝑆 ) ↔ ∀ 𝑝 ∈ ℙ ( 𝑝 ∥ 1 → 𝑝 ∈ 𝑆 ) ) ) |
9 |
|
oveq2 |
⊢ ( 𝑥 = 1 → ( 𝑚 · 𝑥 ) = ( 𝑚 · 1 ) ) |
10 |
9
|
eleq1d |
⊢ ( 𝑥 = 1 → ( ( 𝑚 · 𝑥 ) ∈ 𝑆 ↔ ( 𝑚 · 1 ) ∈ 𝑆 ) ) |
11 |
10
|
imbi1d |
⊢ ( 𝑥 = 1 → ( ( ( 𝑚 · 𝑥 ) ∈ 𝑆 → 𝑚 ∈ 𝑆 ) ↔ ( ( 𝑚 · 1 ) ∈ 𝑆 → 𝑚 ∈ 𝑆 ) ) ) |
12 |
11
|
ralbidv |
⊢ ( 𝑥 = 1 → ( ∀ 𝑚 ∈ ℕ ( ( 𝑚 · 𝑥 ) ∈ 𝑆 → 𝑚 ∈ 𝑆 ) ↔ ∀ 𝑚 ∈ ℕ ( ( 𝑚 · 1 ) ∈ 𝑆 → 𝑚 ∈ 𝑆 ) ) ) |
13 |
8 12
|
imbi12d |
⊢ ( 𝑥 = 1 → ( ( ∀ 𝑝 ∈ ℙ ( 𝑝 ∥ 𝑥 → 𝑝 ∈ 𝑆 ) → ∀ 𝑚 ∈ ℕ ( ( 𝑚 · 𝑥 ) ∈ 𝑆 → 𝑚 ∈ 𝑆 ) ) ↔ ( ∀ 𝑝 ∈ ℙ ( 𝑝 ∥ 1 → 𝑝 ∈ 𝑆 ) → ∀ 𝑚 ∈ ℕ ( ( 𝑚 · 1 ) ∈ 𝑆 → 𝑚 ∈ 𝑆 ) ) ) ) |
14 |
|
breq2 |
⊢ ( 𝑥 = 𝑦 → ( 𝑝 ∥ 𝑥 ↔ 𝑝 ∥ 𝑦 ) ) |
15 |
14
|
imbi1d |
⊢ ( 𝑥 = 𝑦 → ( ( 𝑝 ∥ 𝑥 → 𝑝 ∈ 𝑆 ) ↔ ( 𝑝 ∥ 𝑦 → 𝑝 ∈ 𝑆 ) ) ) |
16 |
15
|
ralbidv |
⊢ ( 𝑥 = 𝑦 → ( ∀ 𝑝 ∈ ℙ ( 𝑝 ∥ 𝑥 → 𝑝 ∈ 𝑆 ) ↔ ∀ 𝑝 ∈ ℙ ( 𝑝 ∥ 𝑦 → 𝑝 ∈ 𝑆 ) ) ) |
17 |
|
oveq2 |
⊢ ( 𝑥 = 𝑦 → ( 𝑚 · 𝑥 ) = ( 𝑚 · 𝑦 ) ) |
18 |
17
|
eleq1d |
⊢ ( 𝑥 = 𝑦 → ( ( 𝑚 · 𝑥 ) ∈ 𝑆 ↔ ( 𝑚 · 𝑦 ) ∈ 𝑆 ) ) |
19 |
18
|
imbi1d |
⊢ ( 𝑥 = 𝑦 → ( ( ( 𝑚 · 𝑥 ) ∈ 𝑆 → 𝑚 ∈ 𝑆 ) ↔ ( ( 𝑚 · 𝑦 ) ∈ 𝑆 → 𝑚 ∈ 𝑆 ) ) ) |
20 |
19
|
ralbidv |
⊢ ( 𝑥 = 𝑦 → ( ∀ 𝑚 ∈ ℕ ( ( 𝑚 · 𝑥 ) ∈ 𝑆 → 𝑚 ∈ 𝑆 ) ↔ ∀ 𝑚 ∈ ℕ ( ( 𝑚 · 𝑦 ) ∈ 𝑆 → 𝑚 ∈ 𝑆 ) ) ) |
21 |
16 20
|
imbi12d |
⊢ ( 𝑥 = 𝑦 → ( ( ∀ 𝑝 ∈ ℙ ( 𝑝 ∥ 𝑥 → 𝑝 ∈ 𝑆 ) → ∀ 𝑚 ∈ ℕ ( ( 𝑚 · 𝑥 ) ∈ 𝑆 → 𝑚 ∈ 𝑆 ) ) ↔ ( ∀ 𝑝 ∈ ℙ ( 𝑝 ∥ 𝑦 → 𝑝 ∈ 𝑆 ) → ∀ 𝑚 ∈ ℕ ( ( 𝑚 · 𝑦 ) ∈ 𝑆 → 𝑚 ∈ 𝑆 ) ) ) ) |
22 |
|
breq2 |
⊢ ( 𝑥 = 𝑧 → ( 𝑝 ∥ 𝑥 ↔ 𝑝 ∥ 𝑧 ) ) |
23 |
22
|
imbi1d |
⊢ ( 𝑥 = 𝑧 → ( ( 𝑝 ∥ 𝑥 → 𝑝 ∈ 𝑆 ) ↔ ( 𝑝 ∥ 𝑧 → 𝑝 ∈ 𝑆 ) ) ) |
24 |
23
|
ralbidv |
⊢ ( 𝑥 = 𝑧 → ( ∀ 𝑝 ∈ ℙ ( 𝑝 ∥ 𝑥 → 𝑝 ∈ 𝑆 ) ↔ ∀ 𝑝 ∈ ℙ ( 𝑝 ∥ 𝑧 → 𝑝 ∈ 𝑆 ) ) ) |
25 |
|
oveq2 |
⊢ ( 𝑥 = 𝑧 → ( 𝑚 · 𝑥 ) = ( 𝑚 · 𝑧 ) ) |
26 |
25
|
eleq1d |
⊢ ( 𝑥 = 𝑧 → ( ( 𝑚 · 𝑥 ) ∈ 𝑆 ↔ ( 𝑚 · 𝑧 ) ∈ 𝑆 ) ) |
27 |
26
|
imbi1d |
⊢ ( 𝑥 = 𝑧 → ( ( ( 𝑚 · 𝑥 ) ∈ 𝑆 → 𝑚 ∈ 𝑆 ) ↔ ( ( 𝑚 · 𝑧 ) ∈ 𝑆 → 𝑚 ∈ 𝑆 ) ) ) |
28 |
27
|
ralbidv |
⊢ ( 𝑥 = 𝑧 → ( ∀ 𝑚 ∈ ℕ ( ( 𝑚 · 𝑥 ) ∈ 𝑆 → 𝑚 ∈ 𝑆 ) ↔ ∀ 𝑚 ∈ ℕ ( ( 𝑚 · 𝑧 ) ∈ 𝑆 → 𝑚 ∈ 𝑆 ) ) ) |
29 |
24 28
|
imbi12d |
⊢ ( 𝑥 = 𝑧 → ( ( ∀ 𝑝 ∈ ℙ ( 𝑝 ∥ 𝑥 → 𝑝 ∈ 𝑆 ) → ∀ 𝑚 ∈ ℕ ( ( 𝑚 · 𝑥 ) ∈ 𝑆 → 𝑚 ∈ 𝑆 ) ) ↔ ( ∀ 𝑝 ∈ ℙ ( 𝑝 ∥ 𝑧 → 𝑝 ∈ 𝑆 ) → ∀ 𝑚 ∈ ℕ ( ( 𝑚 · 𝑧 ) ∈ 𝑆 → 𝑚 ∈ 𝑆 ) ) ) ) |
30 |
|
breq2 |
⊢ ( 𝑥 = ( 𝑦 · 𝑧 ) → ( 𝑝 ∥ 𝑥 ↔ 𝑝 ∥ ( 𝑦 · 𝑧 ) ) ) |
31 |
30
|
imbi1d |
⊢ ( 𝑥 = ( 𝑦 · 𝑧 ) → ( ( 𝑝 ∥ 𝑥 → 𝑝 ∈ 𝑆 ) ↔ ( 𝑝 ∥ ( 𝑦 · 𝑧 ) → 𝑝 ∈ 𝑆 ) ) ) |
32 |
31
|
ralbidv |
⊢ ( 𝑥 = ( 𝑦 · 𝑧 ) → ( ∀ 𝑝 ∈ ℙ ( 𝑝 ∥ 𝑥 → 𝑝 ∈ 𝑆 ) ↔ ∀ 𝑝 ∈ ℙ ( 𝑝 ∥ ( 𝑦 · 𝑧 ) → 𝑝 ∈ 𝑆 ) ) ) |
33 |
|
oveq2 |
⊢ ( 𝑥 = ( 𝑦 · 𝑧 ) → ( 𝑚 · 𝑥 ) = ( 𝑚 · ( 𝑦 · 𝑧 ) ) ) |
34 |
33
|
eleq1d |
⊢ ( 𝑥 = ( 𝑦 · 𝑧 ) → ( ( 𝑚 · 𝑥 ) ∈ 𝑆 ↔ ( 𝑚 · ( 𝑦 · 𝑧 ) ) ∈ 𝑆 ) ) |
35 |
34
|
imbi1d |
⊢ ( 𝑥 = ( 𝑦 · 𝑧 ) → ( ( ( 𝑚 · 𝑥 ) ∈ 𝑆 → 𝑚 ∈ 𝑆 ) ↔ ( ( 𝑚 · ( 𝑦 · 𝑧 ) ) ∈ 𝑆 → 𝑚 ∈ 𝑆 ) ) ) |
36 |
35
|
ralbidv |
⊢ ( 𝑥 = ( 𝑦 · 𝑧 ) → ( ∀ 𝑚 ∈ ℕ ( ( 𝑚 · 𝑥 ) ∈ 𝑆 → 𝑚 ∈ 𝑆 ) ↔ ∀ 𝑚 ∈ ℕ ( ( 𝑚 · ( 𝑦 · 𝑧 ) ) ∈ 𝑆 → 𝑚 ∈ 𝑆 ) ) ) |
37 |
32 36
|
imbi12d |
⊢ ( 𝑥 = ( 𝑦 · 𝑧 ) → ( ( ∀ 𝑝 ∈ ℙ ( 𝑝 ∥ 𝑥 → 𝑝 ∈ 𝑆 ) → ∀ 𝑚 ∈ ℕ ( ( 𝑚 · 𝑥 ) ∈ 𝑆 → 𝑚 ∈ 𝑆 ) ) ↔ ( ∀ 𝑝 ∈ ℙ ( 𝑝 ∥ ( 𝑦 · 𝑧 ) → 𝑝 ∈ 𝑆 ) → ∀ 𝑚 ∈ ℕ ( ( 𝑚 · ( 𝑦 · 𝑧 ) ) ∈ 𝑆 → 𝑚 ∈ 𝑆 ) ) ) ) |
38 |
|
breq2 |
⊢ ( 𝑥 = 𝐵 → ( 𝑝 ∥ 𝑥 ↔ 𝑝 ∥ 𝐵 ) ) |
39 |
38
|
imbi1d |
⊢ ( 𝑥 = 𝐵 → ( ( 𝑝 ∥ 𝑥 → 𝑝 ∈ 𝑆 ) ↔ ( 𝑝 ∥ 𝐵 → 𝑝 ∈ 𝑆 ) ) ) |
40 |
39
|
ralbidv |
⊢ ( 𝑥 = 𝐵 → ( ∀ 𝑝 ∈ ℙ ( 𝑝 ∥ 𝑥 → 𝑝 ∈ 𝑆 ) ↔ ∀ 𝑝 ∈ ℙ ( 𝑝 ∥ 𝐵 → 𝑝 ∈ 𝑆 ) ) ) |
41 |
|
oveq2 |
⊢ ( 𝑥 = 𝐵 → ( 𝑚 · 𝑥 ) = ( 𝑚 · 𝐵 ) ) |
42 |
41
|
eleq1d |
⊢ ( 𝑥 = 𝐵 → ( ( 𝑚 · 𝑥 ) ∈ 𝑆 ↔ ( 𝑚 · 𝐵 ) ∈ 𝑆 ) ) |
43 |
42
|
imbi1d |
⊢ ( 𝑥 = 𝐵 → ( ( ( 𝑚 · 𝑥 ) ∈ 𝑆 → 𝑚 ∈ 𝑆 ) ↔ ( ( 𝑚 · 𝐵 ) ∈ 𝑆 → 𝑚 ∈ 𝑆 ) ) ) |
44 |
43
|
ralbidv |
⊢ ( 𝑥 = 𝐵 → ( ∀ 𝑚 ∈ ℕ ( ( 𝑚 · 𝑥 ) ∈ 𝑆 → 𝑚 ∈ 𝑆 ) ↔ ∀ 𝑚 ∈ ℕ ( ( 𝑚 · 𝐵 ) ∈ 𝑆 → 𝑚 ∈ 𝑆 ) ) ) |
45 |
40 44
|
imbi12d |
⊢ ( 𝑥 = 𝐵 → ( ( ∀ 𝑝 ∈ ℙ ( 𝑝 ∥ 𝑥 → 𝑝 ∈ 𝑆 ) → ∀ 𝑚 ∈ ℕ ( ( 𝑚 · 𝑥 ) ∈ 𝑆 → 𝑚 ∈ 𝑆 ) ) ↔ ( ∀ 𝑝 ∈ ℙ ( 𝑝 ∥ 𝐵 → 𝑝 ∈ 𝑆 ) → ∀ 𝑚 ∈ ℕ ( ( 𝑚 · 𝐵 ) ∈ 𝑆 → 𝑚 ∈ 𝑆 ) ) ) ) |
46 |
|
nncn |
⊢ ( 𝑚 ∈ ℕ → 𝑚 ∈ ℂ ) |
47 |
46
|
mulid1d |
⊢ ( 𝑚 ∈ ℕ → ( 𝑚 · 1 ) = 𝑚 ) |
48 |
47
|
eleq1d |
⊢ ( 𝑚 ∈ ℕ → ( ( 𝑚 · 1 ) ∈ 𝑆 ↔ 𝑚 ∈ 𝑆 ) ) |
49 |
48
|
biimpd |
⊢ ( 𝑚 ∈ ℕ → ( ( 𝑚 · 1 ) ∈ 𝑆 → 𝑚 ∈ 𝑆 ) ) |
50 |
49
|
rgen |
⊢ ∀ 𝑚 ∈ ℕ ( ( 𝑚 · 1 ) ∈ 𝑆 → 𝑚 ∈ 𝑆 ) |
51 |
50
|
a1i |
⊢ ( ∀ 𝑝 ∈ ℙ ( 𝑝 ∥ 1 → 𝑝 ∈ 𝑆 ) → ∀ 𝑚 ∈ ℕ ( ( 𝑚 · 1 ) ∈ 𝑆 → 𝑚 ∈ 𝑆 ) ) |
52 |
|
breq1 |
⊢ ( 𝑝 = 𝑥 → ( 𝑝 ∥ 𝑥 ↔ 𝑥 ∥ 𝑥 ) ) |
53 |
|
eleq1 |
⊢ ( 𝑝 = 𝑥 → ( 𝑝 ∈ 𝑆 ↔ 𝑥 ∈ 𝑆 ) ) |
54 |
52 53
|
imbi12d |
⊢ ( 𝑝 = 𝑥 → ( ( 𝑝 ∥ 𝑥 → 𝑝 ∈ 𝑆 ) ↔ ( 𝑥 ∥ 𝑥 → 𝑥 ∈ 𝑆 ) ) ) |
55 |
54
|
rspcv |
⊢ ( 𝑥 ∈ ℙ → ( ∀ 𝑝 ∈ ℙ ( 𝑝 ∥ 𝑥 → 𝑝 ∈ 𝑆 ) → ( 𝑥 ∥ 𝑥 → 𝑥 ∈ 𝑆 ) ) ) |
56 |
|
prmz |
⊢ ( 𝑥 ∈ ℙ → 𝑥 ∈ ℤ ) |
57 |
|
iddvds |
⊢ ( 𝑥 ∈ ℤ → 𝑥 ∥ 𝑥 ) |
58 |
56 57
|
syl |
⊢ ( 𝑥 ∈ ℙ → 𝑥 ∥ 𝑥 ) |
59 |
|
simprl |
⊢ ( ( ( 𝑥 ∈ ℙ ∧ 𝑥 ∈ 𝑆 ) ∧ ( 𝑚 ∈ ℕ ∧ ( 𝑚 · 𝑥 ) ∈ 𝑆 ) ) → 𝑚 ∈ ℕ ) |
60 |
|
simpll |
⊢ ( ( ( 𝑥 ∈ ℙ ∧ 𝑥 ∈ 𝑆 ) ∧ ( 𝑚 ∈ ℕ ∧ ( 𝑚 · 𝑥 ) ∈ 𝑆 ) ) → 𝑥 ∈ ℙ ) |
61 |
|
simprr |
⊢ ( ( ( 𝑥 ∈ ℙ ∧ 𝑥 ∈ 𝑆 ) ∧ ( 𝑚 ∈ ℕ ∧ ( 𝑚 · 𝑥 ) ∈ 𝑆 ) ) → ( 𝑚 · 𝑥 ) ∈ 𝑆 ) |
62 |
|
simplr |
⊢ ( ( ( 𝑥 ∈ ℙ ∧ 𝑥 ∈ 𝑆 ) ∧ ( 𝑚 ∈ ℕ ∧ ( 𝑚 · 𝑥 ) ∈ 𝑆 ) ) → 𝑥 ∈ 𝑆 ) |
63 |
1 59 60 61 62
|
2sqlem5 |
⊢ ( ( ( 𝑥 ∈ ℙ ∧ 𝑥 ∈ 𝑆 ) ∧ ( 𝑚 ∈ ℕ ∧ ( 𝑚 · 𝑥 ) ∈ 𝑆 ) ) → 𝑚 ∈ 𝑆 ) |
64 |
63
|
expr |
⊢ ( ( ( 𝑥 ∈ ℙ ∧ 𝑥 ∈ 𝑆 ) ∧ 𝑚 ∈ ℕ ) → ( ( 𝑚 · 𝑥 ) ∈ 𝑆 → 𝑚 ∈ 𝑆 ) ) |
65 |
64
|
ralrimiva |
⊢ ( ( 𝑥 ∈ ℙ ∧ 𝑥 ∈ 𝑆 ) → ∀ 𝑚 ∈ ℕ ( ( 𝑚 · 𝑥 ) ∈ 𝑆 → 𝑚 ∈ 𝑆 ) ) |
66 |
65
|
ex |
⊢ ( 𝑥 ∈ ℙ → ( 𝑥 ∈ 𝑆 → ∀ 𝑚 ∈ ℕ ( ( 𝑚 · 𝑥 ) ∈ 𝑆 → 𝑚 ∈ 𝑆 ) ) ) |
67 |
58 66
|
embantd |
⊢ ( 𝑥 ∈ ℙ → ( ( 𝑥 ∥ 𝑥 → 𝑥 ∈ 𝑆 ) → ∀ 𝑚 ∈ ℕ ( ( 𝑚 · 𝑥 ) ∈ 𝑆 → 𝑚 ∈ 𝑆 ) ) ) |
68 |
55 67
|
syld |
⊢ ( 𝑥 ∈ ℙ → ( ∀ 𝑝 ∈ ℙ ( 𝑝 ∥ 𝑥 → 𝑝 ∈ 𝑆 ) → ∀ 𝑚 ∈ ℕ ( ( 𝑚 · 𝑥 ) ∈ 𝑆 → 𝑚 ∈ 𝑆 ) ) ) |
69 |
|
anim12 |
⊢ ( ( ( ∀ 𝑝 ∈ ℙ ( 𝑝 ∥ 𝑦 → 𝑝 ∈ 𝑆 ) → ∀ 𝑚 ∈ ℕ ( ( 𝑚 · 𝑦 ) ∈ 𝑆 → 𝑚 ∈ 𝑆 ) ) ∧ ( ∀ 𝑝 ∈ ℙ ( 𝑝 ∥ 𝑧 → 𝑝 ∈ 𝑆 ) → ∀ 𝑚 ∈ ℕ ( ( 𝑚 · 𝑧 ) ∈ 𝑆 → 𝑚 ∈ 𝑆 ) ) ) → ( ( ∀ 𝑝 ∈ ℙ ( 𝑝 ∥ 𝑦 → 𝑝 ∈ 𝑆 ) ∧ ∀ 𝑝 ∈ ℙ ( 𝑝 ∥ 𝑧 → 𝑝 ∈ 𝑆 ) ) → ( ∀ 𝑚 ∈ ℕ ( ( 𝑚 · 𝑦 ) ∈ 𝑆 → 𝑚 ∈ 𝑆 ) ∧ ∀ 𝑚 ∈ ℕ ( ( 𝑚 · 𝑧 ) ∈ 𝑆 → 𝑚 ∈ 𝑆 ) ) ) ) |
70 |
|
simpr |
⊢ ( ( ( 𝑦 ∈ ( ℤ≥ ‘ 2 ) ∧ 𝑧 ∈ ( ℤ≥ ‘ 2 ) ) ∧ 𝑝 ∈ ℙ ) → 𝑝 ∈ ℙ ) |
71 |
|
eluzelz |
⊢ ( 𝑦 ∈ ( ℤ≥ ‘ 2 ) → 𝑦 ∈ ℤ ) |
72 |
71
|
ad2antrr |
⊢ ( ( ( 𝑦 ∈ ( ℤ≥ ‘ 2 ) ∧ 𝑧 ∈ ( ℤ≥ ‘ 2 ) ) ∧ 𝑝 ∈ ℙ ) → 𝑦 ∈ ℤ ) |
73 |
|
eluzelz |
⊢ ( 𝑧 ∈ ( ℤ≥ ‘ 2 ) → 𝑧 ∈ ℤ ) |
74 |
73
|
ad2antlr |
⊢ ( ( ( 𝑦 ∈ ( ℤ≥ ‘ 2 ) ∧ 𝑧 ∈ ( ℤ≥ ‘ 2 ) ) ∧ 𝑝 ∈ ℙ ) → 𝑧 ∈ ℤ ) |
75 |
|
euclemma |
⊢ ( ( 𝑝 ∈ ℙ ∧ 𝑦 ∈ ℤ ∧ 𝑧 ∈ ℤ ) → ( 𝑝 ∥ ( 𝑦 · 𝑧 ) ↔ ( 𝑝 ∥ 𝑦 ∨ 𝑝 ∥ 𝑧 ) ) ) |
76 |
70 72 74 75
|
syl3anc |
⊢ ( ( ( 𝑦 ∈ ( ℤ≥ ‘ 2 ) ∧ 𝑧 ∈ ( ℤ≥ ‘ 2 ) ) ∧ 𝑝 ∈ ℙ ) → ( 𝑝 ∥ ( 𝑦 · 𝑧 ) ↔ ( 𝑝 ∥ 𝑦 ∨ 𝑝 ∥ 𝑧 ) ) ) |
77 |
76
|
imbi1d |
⊢ ( ( ( 𝑦 ∈ ( ℤ≥ ‘ 2 ) ∧ 𝑧 ∈ ( ℤ≥ ‘ 2 ) ) ∧ 𝑝 ∈ ℙ ) → ( ( 𝑝 ∥ ( 𝑦 · 𝑧 ) → 𝑝 ∈ 𝑆 ) ↔ ( ( 𝑝 ∥ 𝑦 ∨ 𝑝 ∥ 𝑧 ) → 𝑝 ∈ 𝑆 ) ) ) |
78 |
|
jaob |
⊢ ( ( ( 𝑝 ∥ 𝑦 ∨ 𝑝 ∥ 𝑧 ) → 𝑝 ∈ 𝑆 ) ↔ ( ( 𝑝 ∥ 𝑦 → 𝑝 ∈ 𝑆 ) ∧ ( 𝑝 ∥ 𝑧 → 𝑝 ∈ 𝑆 ) ) ) |
79 |
77 78
|
bitrdi |
⊢ ( ( ( 𝑦 ∈ ( ℤ≥ ‘ 2 ) ∧ 𝑧 ∈ ( ℤ≥ ‘ 2 ) ) ∧ 𝑝 ∈ ℙ ) → ( ( 𝑝 ∥ ( 𝑦 · 𝑧 ) → 𝑝 ∈ 𝑆 ) ↔ ( ( 𝑝 ∥ 𝑦 → 𝑝 ∈ 𝑆 ) ∧ ( 𝑝 ∥ 𝑧 → 𝑝 ∈ 𝑆 ) ) ) ) |
80 |
79
|
ralbidva |
⊢ ( ( 𝑦 ∈ ( ℤ≥ ‘ 2 ) ∧ 𝑧 ∈ ( ℤ≥ ‘ 2 ) ) → ( ∀ 𝑝 ∈ ℙ ( 𝑝 ∥ ( 𝑦 · 𝑧 ) → 𝑝 ∈ 𝑆 ) ↔ ∀ 𝑝 ∈ ℙ ( ( 𝑝 ∥ 𝑦 → 𝑝 ∈ 𝑆 ) ∧ ( 𝑝 ∥ 𝑧 → 𝑝 ∈ 𝑆 ) ) ) ) |
81 |
|
r19.26 |
⊢ ( ∀ 𝑝 ∈ ℙ ( ( 𝑝 ∥ 𝑦 → 𝑝 ∈ 𝑆 ) ∧ ( 𝑝 ∥ 𝑧 → 𝑝 ∈ 𝑆 ) ) ↔ ( ∀ 𝑝 ∈ ℙ ( 𝑝 ∥ 𝑦 → 𝑝 ∈ 𝑆 ) ∧ ∀ 𝑝 ∈ ℙ ( 𝑝 ∥ 𝑧 → 𝑝 ∈ 𝑆 ) ) ) |
82 |
80 81
|
bitrdi |
⊢ ( ( 𝑦 ∈ ( ℤ≥ ‘ 2 ) ∧ 𝑧 ∈ ( ℤ≥ ‘ 2 ) ) → ( ∀ 𝑝 ∈ ℙ ( 𝑝 ∥ ( 𝑦 · 𝑧 ) → 𝑝 ∈ 𝑆 ) ↔ ( ∀ 𝑝 ∈ ℙ ( 𝑝 ∥ 𝑦 → 𝑝 ∈ 𝑆 ) ∧ ∀ 𝑝 ∈ ℙ ( 𝑝 ∥ 𝑧 → 𝑝 ∈ 𝑆 ) ) ) ) |
83 |
82
|
biimpa |
⊢ ( ( ( 𝑦 ∈ ( ℤ≥ ‘ 2 ) ∧ 𝑧 ∈ ( ℤ≥ ‘ 2 ) ) ∧ ∀ 𝑝 ∈ ℙ ( 𝑝 ∥ ( 𝑦 · 𝑧 ) → 𝑝 ∈ 𝑆 ) ) → ( ∀ 𝑝 ∈ ℙ ( 𝑝 ∥ 𝑦 → 𝑝 ∈ 𝑆 ) ∧ ∀ 𝑝 ∈ ℙ ( 𝑝 ∥ 𝑧 → 𝑝 ∈ 𝑆 ) ) ) |
84 |
|
oveq1 |
⊢ ( 𝑚 = 𝑛 → ( 𝑚 · 𝑦 ) = ( 𝑛 · 𝑦 ) ) |
85 |
84
|
eleq1d |
⊢ ( 𝑚 = 𝑛 → ( ( 𝑚 · 𝑦 ) ∈ 𝑆 ↔ ( 𝑛 · 𝑦 ) ∈ 𝑆 ) ) |
86 |
|
eleq1 |
⊢ ( 𝑚 = 𝑛 → ( 𝑚 ∈ 𝑆 ↔ 𝑛 ∈ 𝑆 ) ) |
87 |
85 86
|
imbi12d |
⊢ ( 𝑚 = 𝑛 → ( ( ( 𝑚 · 𝑦 ) ∈ 𝑆 → 𝑚 ∈ 𝑆 ) ↔ ( ( 𝑛 · 𝑦 ) ∈ 𝑆 → 𝑛 ∈ 𝑆 ) ) ) |
88 |
87
|
cbvralvw |
⊢ ( ∀ 𝑚 ∈ ℕ ( ( 𝑚 · 𝑦 ) ∈ 𝑆 → 𝑚 ∈ 𝑆 ) ↔ ∀ 𝑛 ∈ ℕ ( ( 𝑛 · 𝑦 ) ∈ 𝑆 → 𝑛 ∈ 𝑆 ) ) |
89 |
46
|
adantl |
⊢ ( ( ( ( ( 𝑦 ∈ ( ℤ≥ ‘ 2 ) ∧ 𝑧 ∈ ( ℤ≥ ‘ 2 ) ) ∧ ∀ 𝑝 ∈ ℙ ( 𝑝 ∥ ( 𝑦 · 𝑧 ) → 𝑝 ∈ 𝑆 ) ) ∧ ∀ 𝑛 ∈ ℕ ( ( 𝑛 · 𝑦 ) ∈ 𝑆 → 𝑛 ∈ 𝑆 ) ) ∧ 𝑚 ∈ ℕ ) → 𝑚 ∈ ℂ ) |
90 |
|
uzssz |
⊢ ( ℤ≥ ‘ 2 ) ⊆ ℤ |
91 |
|
zsscn |
⊢ ℤ ⊆ ℂ |
92 |
90 91
|
sstri |
⊢ ( ℤ≥ ‘ 2 ) ⊆ ℂ |
93 |
|
simpll |
⊢ ( ( ( 𝑦 ∈ ( ℤ≥ ‘ 2 ) ∧ 𝑧 ∈ ( ℤ≥ ‘ 2 ) ) ∧ ∀ 𝑝 ∈ ℙ ( 𝑝 ∥ ( 𝑦 · 𝑧 ) → 𝑝 ∈ 𝑆 ) ) → 𝑦 ∈ ( ℤ≥ ‘ 2 ) ) |
94 |
93
|
ad2antrr |
⊢ ( ( ( ( ( 𝑦 ∈ ( ℤ≥ ‘ 2 ) ∧ 𝑧 ∈ ( ℤ≥ ‘ 2 ) ) ∧ ∀ 𝑝 ∈ ℙ ( 𝑝 ∥ ( 𝑦 · 𝑧 ) → 𝑝 ∈ 𝑆 ) ) ∧ ∀ 𝑛 ∈ ℕ ( ( 𝑛 · 𝑦 ) ∈ 𝑆 → 𝑛 ∈ 𝑆 ) ) ∧ 𝑚 ∈ ℕ ) → 𝑦 ∈ ( ℤ≥ ‘ 2 ) ) |
95 |
92 94
|
sselid |
⊢ ( ( ( ( ( 𝑦 ∈ ( ℤ≥ ‘ 2 ) ∧ 𝑧 ∈ ( ℤ≥ ‘ 2 ) ) ∧ ∀ 𝑝 ∈ ℙ ( 𝑝 ∥ ( 𝑦 · 𝑧 ) → 𝑝 ∈ 𝑆 ) ) ∧ ∀ 𝑛 ∈ ℕ ( ( 𝑛 · 𝑦 ) ∈ 𝑆 → 𝑛 ∈ 𝑆 ) ) ∧ 𝑚 ∈ ℕ ) → 𝑦 ∈ ℂ ) |
96 |
|
simplr |
⊢ ( ( ( 𝑦 ∈ ( ℤ≥ ‘ 2 ) ∧ 𝑧 ∈ ( ℤ≥ ‘ 2 ) ) ∧ ∀ 𝑝 ∈ ℙ ( 𝑝 ∥ ( 𝑦 · 𝑧 ) → 𝑝 ∈ 𝑆 ) ) → 𝑧 ∈ ( ℤ≥ ‘ 2 ) ) |
97 |
96
|
ad2antrr |
⊢ ( ( ( ( ( 𝑦 ∈ ( ℤ≥ ‘ 2 ) ∧ 𝑧 ∈ ( ℤ≥ ‘ 2 ) ) ∧ ∀ 𝑝 ∈ ℙ ( 𝑝 ∥ ( 𝑦 · 𝑧 ) → 𝑝 ∈ 𝑆 ) ) ∧ ∀ 𝑛 ∈ ℕ ( ( 𝑛 · 𝑦 ) ∈ 𝑆 → 𝑛 ∈ 𝑆 ) ) ∧ 𝑚 ∈ ℕ ) → 𝑧 ∈ ( ℤ≥ ‘ 2 ) ) |
98 |
92 97
|
sselid |
⊢ ( ( ( ( ( 𝑦 ∈ ( ℤ≥ ‘ 2 ) ∧ 𝑧 ∈ ( ℤ≥ ‘ 2 ) ) ∧ ∀ 𝑝 ∈ ℙ ( 𝑝 ∥ ( 𝑦 · 𝑧 ) → 𝑝 ∈ 𝑆 ) ) ∧ ∀ 𝑛 ∈ ℕ ( ( 𝑛 · 𝑦 ) ∈ 𝑆 → 𝑛 ∈ 𝑆 ) ) ∧ 𝑚 ∈ ℕ ) → 𝑧 ∈ ℂ ) |
99 |
|
mul32 |
⊢ ( ( 𝑚 ∈ ℂ ∧ 𝑦 ∈ ℂ ∧ 𝑧 ∈ ℂ ) → ( ( 𝑚 · 𝑦 ) · 𝑧 ) = ( ( 𝑚 · 𝑧 ) · 𝑦 ) ) |
100 |
|
mulass |
⊢ ( ( 𝑚 ∈ ℂ ∧ 𝑦 ∈ ℂ ∧ 𝑧 ∈ ℂ ) → ( ( 𝑚 · 𝑦 ) · 𝑧 ) = ( 𝑚 · ( 𝑦 · 𝑧 ) ) ) |
101 |
99 100
|
eqtr3d |
⊢ ( ( 𝑚 ∈ ℂ ∧ 𝑦 ∈ ℂ ∧ 𝑧 ∈ ℂ ) → ( ( 𝑚 · 𝑧 ) · 𝑦 ) = ( 𝑚 · ( 𝑦 · 𝑧 ) ) ) |
102 |
89 95 98 101
|
syl3anc |
⊢ ( ( ( ( ( 𝑦 ∈ ( ℤ≥ ‘ 2 ) ∧ 𝑧 ∈ ( ℤ≥ ‘ 2 ) ) ∧ ∀ 𝑝 ∈ ℙ ( 𝑝 ∥ ( 𝑦 · 𝑧 ) → 𝑝 ∈ 𝑆 ) ) ∧ ∀ 𝑛 ∈ ℕ ( ( 𝑛 · 𝑦 ) ∈ 𝑆 → 𝑛 ∈ 𝑆 ) ) ∧ 𝑚 ∈ ℕ ) → ( ( 𝑚 · 𝑧 ) · 𝑦 ) = ( 𝑚 · ( 𝑦 · 𝑧 ) ) ) |
103 |
102
|
eleq1d |
⊢ ( ( ( ( ( 𝑦 ∈ ( ℤ≥ ‘ 2 ) ∧ 𝑧 ∈ ( ℤ≥ ‘ 2 ) ) ∧ ∀ 𝑝 ∈ ℙ ( 𝑝 ∥ ( 𝑦 · 𝑧 ) → 𝑝 ∈ 𝑆 ) ) ∧ ∀ 𝑛 ∈ ℕ ( ( 𝑛 · 𝑦 ) ∈ 𝑆 → 𝑛 ∈ 𝑆 ) ) ∧ 𝑚 ∈ ℕ ) → ( ( ( 𝑚 · 𝑧 ) · 𝑦 ) ∈ 𝑆 ↔ ( 𝑚 · ( 𝑦 · 𝑧 ) ) ∈ 𝑆 ) ) |
104 |
|
simpr |
⊢ ( ( ( ( ( 𝑦 ∈ ( ℤ≥ ‘ 2 ) ∧ 𝑧 ∈ ( ℤ≥ ‘ 2 ) ) ∧ ∀ 𝑝 ∈ ℙ ( 𝑝 ∥ ( 𝑦 · 𝑧 ) → 𝑝 ∈ 𝑆 ) ) ∧ ∀ 𝑛 ∈ ℕ ( ( 𝑛 · 𝑦 ) ∈ 𝑆 → 𝑛 ∈ 𝑆 ) ) ∧ 𝑚 ∈ ℕ ) → 𝑚 ∈ ℕ ) |
105 |
|
eluz2nn |
⊢ ( 𝑧 ∈ ( ℤ≥ ‘ 2 ) → 𝑧 ∈ ℕ ) |
106 |
97 105
|
syl |
⊢ ( ( ( ( ( 𝑦 ∈ ( ℤ≥ ‘ 2 ) ∧ 𝑧 ∈ ( ℤ≥ ‘ 2 ) ) ∧ ∀ 𝑝 ∈ ℙ ( 𝑝 ∥ ( 𝑦 · 𝑧 ) → 𝑝 ∈ 𝑆 ) ) ∧ ∀ 𝑛 ∈ ℕ ( ( 𝑛 · 𝑦 ) ∈ 𝑆 → 𝑛 ∈ 𝑆 ) ) ∧ 𝑚 ∈ ℕ ) → 𝑧 ∈ ℕ ) |
107 |
104 106
|
nnmulcld |
⊢ ( ( ( ( ( 𝑦 ∈ ( ℤ≥ ‘ 2 ) ∧ 𝑧 ∈ ( ℤ≥ ‘ 2 ) ) ∧ ∀ 𝑝 ∈ ℙ ( 𝑝 ∥ ( 𝑦 · 𝑧 ) → 𝑝 ∈ 𝑆 ) ) ∧ ∀ 𝑛 ∈ ℕ ( ( 𝑛 · 𝑦 ) ∈ 𝑆 → 𝑛 ∈ 𝑆 ) ) ∧ 𝑚 ∈ ℕ ) → ( 𝑚 · 𝑧 ) ∈ ℕ ) |
108 |
|
simplr |
⊢ ( ( ( ( ( 𝑦 ∈ ( ℤ≥ ‘ 2 ) ∧ 𝑧 ∈ ( ℤ≥ ‘ 2 ) ) ∧ ∀ 𝑝 ∈ ℙ ( 𝑝 ∥ ( 𝑦 · 𝑧 ) → 𝑝 ∈ 𝑆 ) ) ∧ ∀ 𝑛 ∈ ℕ ( ( 𝑛 · 𝑦 ) ∈ 𝑆 → 𝑛 ∈ 𝑆 ) ) ∧ 𝑚 ∈ ℕ ) → ∀ 𝑛 ∈ ℕ ( ( 𝑛 · 𝑦 ) ∈ 𝑆 → 𝑛 ∈ 𝑆 ) ) |
109 |
|
oveq1 |
⊢ ( 𝑛 = ( 𝑚 · 𝑧 ) → ( 𝑛 · 𝑦 ) = ( ( 𝑚 · 𝑧 ) · 𝑦 ) ) |
110 |
109
|
eleq1d |
⊢ ( 𝑛 = ( 𝑚 · 𝑧 ) → ( ( 𝑛 · 𝑦 ) ∈ 𝑆 ↔ ( ( 𝑚 · 𝑧 ) · 𝑦 ) ∈ 𝑆 ) ) |
111 |
|
eleq1 |
⊢ ( 𝑛 = ( 𝑚 · 𝑧 ) → ( 𝑛 ∈ 𝑆 ↔ ( 𝑚 · 𝑧 ) ∈ 𝑆 ) ) |
112 |
110 111
|
imbi12d |
⊢ ( 𝑛 = ( 𝑚 · 𝑧 ) → ( ( ( 𝑛 · 𝑦 ) ∈ 𝑆 → 𝑛 ∈ 𝑆 ) ↔ ( ( ( 𝑚 · 𝑧 ) · 𝑦 ) ∈ 𝑆 → ( 𝑚 · 𝑧 ) ∈ 𝑆 ) ) ) |
113 |
112
|
rspcv |
⊢ ( ( 𝑚 · 𝑧 ) ∈ ℕ → ( ∀ 𝑛 ∈ ℕ ( ( 𝑛 · 𝑦 ) ∈ 𝑆 → 𝑛 ∈ 𝑆 ) → ( ( ( 𝑚 · 𝑧 ) · 𝑦 ) ∈ 𝑆 → ( 𝑚 · 𝑧 ) ∈ 𝑆 ) ) ) |
114 |
107 108 113
|
sylc |
⊢ ( ( ( ( ( 𝑦 ∈ ( ℤ≥ ‘ 2 ) ∧ 𝑧 ∈ ( ℤ≥ ‘ 2 ) ) ∧ ∀ 𝑝 ∈ ℙ ( 𝑝 ∥ ( 𝑦 · 𝑧 ) → 𝑝 ∈ 𝑆 ) ) ∧ ∀ 𝑛 ∈ ℕ ( ( 𝑛 · 𝑦 ) ∈ 𝑆 → 𝑛 ∈ 𝑆 ) ) ∧ 𝑚 ∈ ℕ ) → ( ( ( 𝑚 · 𝑧 ) · 𝑦 ) ∈ 𝑆 → ( 𝑚 · 𝑧 ) ∈ 𝑆 ) ) |
115 |
103 114
|
sylbird |
⊢ ( ( ( ( ( 𝑦 ∈ ( ℤ≥ ‘ 2 ) ∧ 𝑧 ∈ ( ℤ≥ ‘ 2 ) ) ∧ ∀ 𝑝 ∈ ℙ ( 𝑝 ∥ ( 𝑦 · 𝑧 ) → 𝑝 ∈ 𝑆 ) ) ∧ ∀ 𝑛 ∈ ℕ ( ( 𝑛 · 𝑦 ) ∈ 𝑆 → 𝑛 ∈ 𝑆 ) ) ∧ 𝑚 ∈ ℕ ) → ( ( 𝑚 · ( 𝑦 · 𝑧 ) ) ∈ 𝑆 → ( 𝑚 · 𝑧 ) ∈ 𝑆 ) ) |
116 |
115
|
imim1d |
⊢ ( ( ( ( ( 𝑦 ∈ ( ℤ≥ ‘ 2 ) ∧ 𝑧 ∈ ( ℤ≥ ‘ 2 ) ) ∧ ∀ 𝑝 ∈ ℙ ( 𝑝 ∥ ( 𝑦 · 𝑧 ) → 𝑝 ∈ 𝑆 ) ) ∧ ∀ 𝑛 ∈ ℕ ( ( 𝑛 · 𝑦 ) ∈ 𝑆 → 𝑛 ∈ 𝑆 ) ) ∧ 𝑚 ∈ ℕ ) → ( ( ( 𝑚 · 𝑧 ) ∈ 𝑆 → 𝑚 ∈ 𝑆 ) → ( ( 𝑚 · ( 𝑦 · 𝑧 ) ) ∈ 𝑆 → 𝑚 ∈ 𝑆 ) ) ) |
117 |
116
|
ralimdva |
⊢ ( ( ( ( 𝑦 ∈ ( ℤ≥ ‘ 2 ) ∧ 𝑧 ∈ ( ℤ≥ ‘ 2 ) ) ∧ ∀ 𝑝 ∈ ℙ ( 𝑝 ∥ ( 𝑦 · 𝑧 ) → 𝑝 ∈ 𝑆 ) ) ∧ ∀ 𝑛 ∈ ℕ ( ( 𝑛 · 𝑦 ) ∈ 𝑆 → 𝑛 ∈ 𝑆 ) ) → ( ∀ 𝑚 ∈ ℕ ( ( 𝑚 · 𝑧 ) ∈ 𝑆 → 𝑚 ∈ 𝑆 ) → ∀ 𝑚 ∈ ℕ ( ( 𝑚 · ( 𝑦 · 𝑧 ) ) ∈ 𝑆 → 𝑚 ∈ 𝑆 ) ) ) |
118 |
88 117
|
sylan2b |
⊢ ( ( ( ( 𝑦 ∈ ( ℤ≥ ‘ 2 ) ∧ 𝑧 ∈ ( ℤ≥ ‘ 2 ) ) ∧ ∀ 𝑝 ∈ ℙ ( 𝑝 ∥ ( 𝑦 · 𝑧 ) → 𝑝 ∈ 𝑆 ) ) ∧ ∀ 𝑚 ∈ ℕ ( ( 𝑚 · 𝑦 ) ∈ 𝑆 → 𝑚 ∈ 𝑆 ) ) → ( ∀ 𝑚 ∈ ℕ ( ( 𝑚 · 𝑧 ) ∈ 𝑆 → 𝑚 ∈ 𝑆 ) → ∀ 𝑚 ∈ ℕ ( ( 𝑚 · ( 𝑦 · 𝑧 ) ) ∈ 𝑆 → 𝑚 ∈ 𝑆 ) ) ) |
119 |
118
|
expimpd |
⊢ ( ( ( 𝑦 ∈ ( ℤ≥ ‘ 2 ) ∧ 𝑧 ∈ ( ℤ≥ ‘ 2 ) ) ∧ ∀ 𝑝 ∈ ℙ ( 𝑝 ∥ ( 𝑦 · 𝑧 ) → 𝑝 ∈ 𝑆 ) ) → ( ( ∀ 𝑚 ∈ ℕ ( ( 𝑚 · 𝑦 ) ∈ 𝑆 → 𝑚 ∈ 𝑆 ) ∧ ∀ 𝑚 ∈ ℕ ( ( 𝑚 · 𝑧 ) ∈ 𝑆 → 𝑚 ∈ 𝑆 ) ) → ∀ 𝑚 ∈ ℕ ( ( 𝑚 · ( 𝑦 · 𝑧 ) ) ∈ 𝑆 → 𝑚 ∈ 𝑆 ) ) ) |
120 |
83 119
|
embantd |
⊢ ( ( ( 𝑦 ∈ ( ℤ≥ ‘ 2 ) ∧ 𝑧 ∈ ( ℤ≥ ‘ 2 ) ) ∧ ∀ 𝑝 ∈ ℙ ( 𝑝 ∥ ( 𝑦 · 𝑧 ) → 𝑝 ∈ 𝑆 ) ) → ( ( ( ∀ 𝑝 ∈ ℙ ( 𝑝 ∥ 𝑦 → 𝑝 ∈ 𝑆 ) ∧ ∀ 𝑝 ∈ ℙ ( 𝑝 ∥ 𝑧 → 𝑝 ∈ 𝑆 ) ) → ( ∀ 𝑚 ∈ ℕ ( ( 𝑚 · 𝑦 ) ∈ 𝑆 → 𝑚 ∈ 𝑆 ) ∧ ∀ 𝑚 ∈ ℕ ( ( 𝑚 · 𝑧 ) ∈ 𝑆 → 𝑚 ∈ 𝑆 ) ) ) → ∀ 𝑚 ∈ ℕ ( ( 𝑚 · ( 𝑦 · 𝑧 ) ) ∈ 𝑆 → 𝑚 ∈ 𝑆 ) ) ) |
121 |
120
|
ex |
⊢ ( ( 𝑦 ∈ ( ℤ≥ ‘ 2 ) ∧ 𝑧 ∈ ( ℤ≥ ‘ 2 ) ) → ( ∀ 𝑝 ∈ ℙ ( 𝑝 ∥ ( 𝑦 · 𝑧 ) → 𝑝 ∈ 𝑆 ) → ( ( ( ∀ 𝑝 ∈ ℙ ( 𝑝 ∥ 𝑦 → 𝑝 ∈ 𝑆 ) ∧ ∀ 𝑝 ∈ ℙ ( 𝑝 ∥ 𝑧 → 𝑝 ∈ 𝑆 ) ) → ( ∀ 𝑚 ∈ ℕ ( ( 𝑚 · 𝑦 ) ∈ 𝑆 → 𝑚 ∈ 𝑆 ) ∧ ∀ 𝑚 ∈ ℕ ( ( 𝑚 · 𝑧 ) ∈ 𝑆 → 𝑚 ∈ 𝑆 ) ) ) → ∀ 𝑚 ∈ ℕ ( ( 𝑚 · ( 𝑦 · 𝑧 ) ) ∈ 𝑆 → 𝑚 ∈ 𝑆 ) ) ) ) |
122 |
121
|
com23 |
⊢ ( ( 𝑦 ∈ ( ℤ≥ ‘ 2 ) ∧ 𝑧 ∈ ( ℤ≥ ‘ 2 ) ) → ( ( ( ∀ 𝑝 ∈ ℙ ( 𝑝 ∥ 𝑦 → 𝑝 ∈ 𝑆 ) ∧ ∀ 𝑝 ∈ ℙ ( 𝑝 ∥ 𝑧 → 𝑝 ∈ 𝑆 ) ) → ( ∀ 𝑚 ∈ ℕ ( ( 𝑚 · 𝑦 ) ∈ 𝑆 → 𝑚 ∈ 𝑆 ) ∧ ∀ 𝑚 ∈ ℕ ( ( 𝑚 · 𝑧 ) ∈ 𝑆 → 𝑚 ∈ 𝑆 ) ) ) → ( ∀ 𝑝 ∈ ℙ ( 𝑝 ∥ ( 𝑦 · 𝑧 ) → 𝑝 ∈ 𝑆 ) → ∀ 𝑚 ∈ ℕ ( ( 𝑚 · ( 𝑦 · 𝑧 ) ) ∈ 𝑆 → 𝑚 ∈ 𝑆 ) ) ) ) |
123 |
69 122
|
syl5 |
⊢ ( ( 𝑦 ∈ ( ℤ≥ ‘ 2 ) ∧ 𝑧 ∈ ( ℤ≥ ‘ 2 ) ) → ( ( ( ∀ 𝑝 ∈ ℙ ( 𝑝 ∥ 𝑦 → 𝑝 ∈ 𝑆 ) → ∀ 𝑚 ∈ ℕ ( ( 𝑚 · 𝑦 ) ∈ 𝑆 → 𝑚 ∈ 𝑆 ) ) ∧ ( ∀ 𝑝 ∈ ℙ ( 𝑝 ∥ 𝑧 → 𝑝 ∈ 𝑆 ) → ∀ 𝑚 ∈ ℕ ( ( 𝑚 · 𝑧 ) ∈ 𝑆 → 𝑚 ∈ 𝑆 ) ) ) → ( ∀ 𝑝 ∈ ℙ ( 𝑝 ∥ ( 𝑦 · 𝑧 ) → 𝑝 ∈ 𝑆 ) → ∀ 𝑚 ∈ ℕ ( ( 𝑚 · ( 𝑦 · 𝑧 ) ) ∈ 𝑆 → 𝑚 ∈ 𝑆 ) ) ) ) |
124 |
13 21 29 37 45 51 68 123
|
prmind |
⊢ ( 𝐵 ∈ ℕ → ( ∀ 𝑝 ∈ ℙ ( 𝑝 ∥ 𝐵 → 𝑝 ∈ 𝑆 ) → ∀ 𝑚 ∈ ℕ ( ( 𝑚 · 𝐵 ) ∈ 𝑆 → 𝑚 ∈ 𝑆 ) ) ) |
125 |
3 4 124
|
sylc |
⊢ ( 𝜑 → ∀ 𝑚 ∈ ℕ ( ( 𝑚 · 𝐵 ) ∈ 𝑆 → 𝑚 ∈ 𝑆 ) ) |
126 |
|
oveq1 |
⊢ ( 𝑚 = 𝐴 → ( 𝑚 · 𝐵 ) = ( 𝐴 · 𝐵 ) ) |
127 |
126
|
eleq1d |
⊢ ( 𝑚 = 𝐴 → ( ( 𝑚 · 𝐵 ) ∈ 𝑆 ↔ ( 𝐴 · 𝐵 ) ∈ 𝑆 ) ) |
128 |
|
eleq1 |
⊢ ( 𝑚 = 𝐴 → ( 𝑚 ∈ 𝑆 ↔ 𝐴 ∈ 𝑆 ) ) |
129 |
127 128
|
imbi12d |
⊢ ( 𝑚 = 𝐴 → ( ( ( 𝑚 · 𝐵 ) ∈ 𝑆 → 𝑚 ∈ 𝑆 ) ↔ ( ( 𝐴 · 𝐵 ) ∈ 𝑆 → 𝐴 ∈ 𝑆 ) ) ) |
130 |
129
|
rspcv |
⊢ ( 𝐴 ∈ ℕ → ( ∀ 𝑚 ∈ ℕ ( ( 𝑚 · 𝐵 ) ∈ 𝑆 → 𝑚 ∈ 𝑆 ) → ( ( 𝐴 · 𝐵 ) ∈ 𝑆 → 𝐴 ∈ 𝑆 ) ) ) |
131 |
2 125 5 130
|
syl3c |
⊢ ( 𝜑 → 𝐴 ∈ 𝑆 ) |