Step | Hyp | Ref
| Expression |
1 | | pcadd.2 |
. . 3
⊢ (𝜑 → 𝐴 ∈ ℚ) |
2 | | elq 12432 |
. . 3
⊢ (𝐴 ∈ ℚ ↔
∃𝑥 ∈ ℤ
∃𝑦 ∈ ℕ
𝐴 = (𝑥 / 𝑦)) |
3 | 1, 2 | sylib 221 |
. 2
⊢ (𝜑 → ∃𝑥 ∈ ℤ ∃𝑦 ∈ ℕ 𝐴 = (𝑥 / 𝑦)) |
4 | | pcadd.3 |
. . 3
⊢ (𝜑 → 𝐵 ∈ ℚ) |
5 | | elq 12432 |
. . 3
⊢ (𝐵 ∈ ℚ ↔
∃𝑧 ∈ ℤ
∃𝑤 ∈ ℕ
𝐵 = (𝑧 / 𝑤)) |
6 | 4, 5 | sylib 221 |
. 2
⊢ (𝜑 → ∃𝑧 ∈ ℤ ∃𝑤 ∈ ℕ 𝐵 = (𝑧 / 𝑤)) |
7 | | pcadd.1 |
. . . . . . . 8
⊢ (𝜑 → 𝑃 ∈ ℙ) |
8 | | pcxcl 16297 |
. . . . . . . 8
⊢ ((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℚ) → (𝑃 pCnt 𝐴) ∈
ℝ*) |
9 | 7, 1, 8 | syl2anc 587 |
. . . . . . 7
⊢ (𝜑 → (𝑃 pCnt 𝐴) ∈
ℝ*) |
10 | 9 | xrleidd 12628 |
. . . . . 6
⊢ (𝜑 → (𝑃 pCnt 𝐴) ≤ (𝑃 pCnt 𝐴)) |
11 | 10 | adantr 484 |
. . . . 5
⊢ ((𝜑 ∧ 𝐵 = 0) → (𝑃 pCnt 𝐴) ≤ (𝑃 pCnt 𝐴)) |
12 | | oveq2 7178 |
. . . . . . 7
⊢ (𝐵 = 0 → (𝐴 + 𝐵) = (𝐴 + 0)) |
13 | | qcn 12445 |
. . . . . . . . 9
⊢ (𝐴 ∈ ℚ → 𝐴 ∈
ℂ) |
14 | 1, 13 | syl 17 |
. . . . . . . 8
⊢ (𝜑 → 𝐴 ∈ ℂ) |
15 | 14 | addid1d 10918 |
. . . . . . 7
⊢ (𝜑 → (𝐴 + 0) = 𝐴) |
16 | 12, 15 | sylan9eqr 2795 |
. . . . . 6
⊢ ((𝜑 ∧ 𝐵 = 0) → (𝐴 + 𝐵) = 𝐴) |
17 | 16 | oveq2d 7186 |
. . . . 5
⊢ ((𝜑 ∧ 𝐵 = 0) → (𝑃 pCnt (𝐴 + 𝐵)) = (𝑃 pCnt 𝐴)) |
18 | 11, 17 | breqtrrd 5058 |
. . . 4
⊢ ((𝜑 ∧ 𝐵 = 0) → (𝑃 pCnt 𝐴) ≤ (𝑃 pCnt (𝐴 + 𝐵))) |
19 | 18 | a1d 25 |
. . 3
⊢ ((𝜑 ∧ 𝐵 = 0) → ((∃𝑥 ∈ ℤ ∃𝑦 ∈ ℕ 𝐴 = (𝑥 / 𝑦) ∧ ∃𝑧 ∈ ℤ ∃𝑤 ∈ ℕ 𝐵 = (𝑧 / 𝑤)) → (𝑃 pCnt 𝐴) ≤ (𝑃 pCnt (𝐴 + 𝐵)))) |
20 | | reeanv 3270 |
. . . 4
⊢
(∃𝑥 ∈
ℤ ∃𝑧 ∈
ℤ (∃𝑦 ∈
ℕ 𝐴 = (𝑥 / 𝑦) ∧ ∃𝑤 ∈ ℕ 𝐵 = (𝑧 / 𝑤)) ↔ (∃𝑥 ∈ ℤ ∃𝑦 ∈ ℕ 𝐴 = (𝑥 / 𝑦) ∧ ∃𝑧 ∈ ℤ ∃𝑤 ∈ ℕ 𝐵 = (𝑧 / 𝑤))) |
21 | | reeanv 3270 |
. . . . . 6
⊢
(∃𝑦 ∈
ℕ ∃𝑤 ∈
ℕ (𝐴 = (𝑥 / 𝑦) ∧ 𝐵 = (𝑧 / 𝑤)) ↔ (∃𝑦 ∈ ℕ 𝐴 = (𝑥 / 𝑦) ∧ ∃𝑤 ∈ ℕ 𝐵 = (𝑧 / 𝑤))) |
22 | 7 | ad3antrrr 730 |
. . . . . . . . 9
⊢ ((((𝜑 ∧ 𝐵 ≠ 0) ∧ (𝑥 ∈ ℤ ∧ 𝑧 ∈ ℤ)) ∧ ((𝑦 ∈ ℕ ∧ 𝑤 ∈ ℕ) ∧ (𝐴 = (𝑥 / 𝑦) ∧ 𝐵 = (𝑧 / 𝑤)))) → 𝑃 ∈ ℙ) |
23 | | prmnn 16115 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑃 ∈ ℙ → 𝑃 ∈
ℕ) |
24 | 22, 23 | syl 17 |
. . . . . . . . . . . . . . . 16
⊢ ((((𝜑 ∧ 𝐵 ≠ 0) ∧ (𝑥 ∈ ℤ ∧ 𝑧 ∈ ℤ)) ∧ ((𝑦 ∈ ℕ ∧ 𝑤 ∈ ℕ) ∧ (𝐴 = (𝑥 / 𝑦) ∧ 𝐵 = (𝑧 / 𝑤)))) → 𝑃 ∈ ℕ) |
25 | | simplrl 777 |
. . . . . . . . . . . . . . . . 17
⊢ ((((𝜑 ∧ 𝐵 ≠ 0) ∧ (𝑥 ∈ ℤ ∧ 𝑧 ∈ ℤ)) ∧ ((𝑦 ∈ ℕ ∧ 𝑤 ∈ ℕ) ∧ (𝐴 = (𝑥 / 𝑦) ∧ 𝐵 = (𝑧 / 𝑤)))) → 𝑥 ∈ ℤ) |
26 | | simprrl 781 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((((𝜑 ∧ 𝐵 ≠ 0) ∧ (𝑥 ∈ ℤ ∧ 𝑧 ∈ ℤ)) ∧ ((𝑦 ∈ ℕ ∧ 𝑤 ∈ ℕ) ∧ (𝐴 = (𝑥 / 𝑦) ∧ 𝐵 = (𝑧 / 𝑤)))) → 𝐴 = (𝑥 / 𝑦)) |
27 | | pc0 16291 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (𝑃 ∈ ℙ → (𝑃 pCnt 0) =
+∞) |
28 | 22, 27 | syl 17 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ((((𝜑 ∧ 𝐵 ≠ 0) ∧ (𝑥 ∈ ℤ ∧ 𝑧 ∈ ℤ)) ∧ ((𝑦 ∈ ℕ ∧ 𝑤 ∈ ℕ) ∧ (𝐴 = (𝑥 / 𝑦) ∧ 𝐵 = (𝑧 / 𝑤)))) → (𝑃 pCnt 0) = +∞) |
29 | 4 | ad3antrrr 730 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ ((((𝜑 ∧ 𝐵 ≠ 0) ∧ (𝑥 ∈ ℤ ∧ 𝑧 ∈ ℤ)) ∧ ((𝑦 ∈ ℕ ∧ 𝑤 ∈ ℕ) ∧ (𝐴 = (𝑥 / 𝑦) ∧ 𝐵 = (𝑧 / 𝑤)))) → 𝐵 ∈ ℚ) |
30 | | simpllr 776 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ ((((𝜑 ∧ 𝐵 ≠ 0) ∧ (𝑥 ∈ ℤ ∧ 𝑧 ∈ ℤ)) ∧ ((𝑦 ∈ ℕ ∧ 𝑤 ∈ ℕ) ∧ (𝐴 = (𝑥 / 𝑦) ∧ 𝐵 = (𝑧 / 𝑤)))) → 𝐵 ≠ 0) |
31 | | pcqcl 16293 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ ((𝑃 ∈ ℙ ∧ (𝐵 ∈ ℚ ∧ 𝐵 ≠ 0)) → (𝑃 pCnt 𝐵) ∈ ℤ) |
32 | 22, 29, 30, 31 | syl12anc 836 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ ((((𝜑 ∧ 𝐵 ≠ 0) ∧ (𝑥 ∈ ℤ ∧ 𝑧 ∈ ℤ)) ∧ ((𝑦 ∈ ℕ ∧ 𝑤 ∈ ℕ) ∧ (𝐴 = (𝑥 / 𝑦) ∧ 𝐵 = (𝑧 / 𝑤)))) → (𝑃 pCnt 𝐵) ∈ ℤ) |
33 | 32 | zred 12168 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ ((((𝜑 ∧ 𝐵 ≠ 0) ∧ (𝑥 ∈ ℤ ∧ 𝑧 ∈ ℤ)) ∧ ((𝑦 ∈ ℕ ∧ 𝑤 ∈ ℕ) ∧ (𝐴 = (𝑥 / 𝑦) ∧ 𝐵 = (𝑧 / 𝑤)))) → (𝑃 pCnt 𝐵) ∈ ℝ) |
34 | | ltpnf 12598 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ ((𝑃 pCnt 𝐵) ∈ ℝ → (𝑃 pCnt 𝐵) < +∞) |
35 | | rexr 10765 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ ((𝑃 pCnt 𝐵) ∈ ℝ → (𝑃 pCnt 𝐵) ∈
ℝ*) |
36 | | pnfxr 10773 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ +∞
∈ ℝ* |
37 | | xrltnle 10786 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ (((𝑃 pCnt 𝐵) ∈ ℝ* ∧ +∞
∈ ℝ*) → ((𝑃 pCnt 𝐵) < +∞ ↔ ¬ +∞ ≤
(𝑃 pCnt 𝐵))) |
38 | 35, 36, 37 | sylancl 589 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ ((𝑃 pCnt 𝐵) ∈ ℝ → ((𝑃 pCnt 𝐵) < +∞ ↔ ¬ +∞ ≤
(𝑃 pCnt 𝐵))) |
39 | 34, 38 | mpbid 235 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ ((𝑃 pCnt 𝐵) ∈ ℝ → ¬ +∞ ≤
(𝑃 pCnt 𝐵)) |
40 | 33, 39 | syl 17 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ((((𝜑 ∧ 𝐵 ≠ 0) ∧ (𝑥 ∈ ℤ ∧ 𝑧 ∈ ℤ)) ∧ ((𝑦 ∈ ℕ ∧ 𝑤 ∈ ℕ) ∧ (𝐴 = (𝑥 / 𝑦) ∧ 𝐵 = (𝑧 / 𝑤)))) → ¬ +∞ ≤ (𝑃 pCnt 𝐵)) |
41 | 28, 40 | eqnbrtrd 5048 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((((𝜑 ∧ 𝐵 ≠ 0) ∧ (𝑥 ∈ ℤ ∧ 𝑧 ∈ ℤ)) ∧ ((𝑦 ∈ ℕ ∧ 𝑤 ∈ ℕ) ∧ (𝐴 = (𝑥 / 𝑦) ∧ 𝐵 = (𝑧 / 𝑤)))) → ¬ (𝑃 pCnt 0) ≤ (𝑃 pCnt 𝐵)) |
42 | | pcadd.4 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (𝜑 → (𝑃 pCnt 𝐴) ≤ (𝑃 pCnt 𝐵)) |
43 | 42 | ad3antrrr 730 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ ((((𝜑 ∧ 𝐵 ≠ 0) ∧ (𝑥 ∈ ℤ ∧ 𝑧 ∈ ℤ)) ∧ ((𝑦 ∈ ℕ ∧ 𝑤 ∈ ℕ) ∧ (𝐴 = (𝑥 / 𝑦) ∧ 𝐵 = (𝑧 / 𝑤)))) → (𝑃 pCnt 𝐴) ≤ (𝑃 pCnt 𝐵)) |
44 | | oveq2 7178 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (𝐴 = 0 → (𝑃 pCnt 𝐴) = (𝑃 pCnt 0)) |
45 | 44 | breq1d 5040 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (𝐴 = 0 → ((𝑃 pCnt 𝐴) ≤ (𝑃 pCnt 𝐵) ↔ (𝑃 pCnt 0) ≤ (𝑃 pCnt 𝐵))) |
46 | 43, 45 | syl5ibcom 248 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ((((𝜑 ∧ 𝐵 ≠ 0) ∧ (𝑥 ∈ ℤ ∧ 𝑧 ∈ ℤ)) ∧ ((𝑦 ∈ ℕ ∧ 𝑤 ∈ ℕ) ∧ (𝐴 = (𝑥 / 𝑦) ∧ 𝐵 = (𝑧 / 𝑤)))) → (𝐴 = 0 → (𝑃 pCnt 0) ≤ (𝑃 pCnt 𝐵))) |
47 | 46 | necon3bd 2948 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((((𝜑 ∧ 𝐵 ≠ 0) ∧ (𝑥 ∈ ℤ ∧ 𝑧 ∈ ℤ)) ∧ ((𝑦 ∈ ℕ ∧ 𝑤 ∈ ℕ) ∧ (𝐴 = (𝑥 / 𝑦) ∧ 𝐵 = (𝑧 / 𝑤)))) → (¬ (𝑃 pCnt 0) ≤ (𝑃 pCnt 𝐵) → 𝐴 ≠ 0)) |
48 | 41, 47 | mpd 15 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((((𝜑 ∧ 𝐵 ≠ 0) ∧ (𝑥 ∈ ℤ ∧ 𝑧 ∈ ℤ)) ∧ ((𝑦 ∈ ℕ ∧ 𝑤 ∈ ℕ) ∧ (𝐴 = (𝑥 / 𝑦) ∧ 𝐵 = (𝑧 / 𝑤)))) → 𝐴 ≠ 0) |
49 | 26, 48 | eqnetrrd 3002 |
. . . . . . . . . . . . . . . . . 18
⊢ ((((𝜑 ∧ 𝐵 ≠ 0) ∧ (𝑥 ∈ ℤ ∧ 𝑧 ∈ ℤ)) ∧ ((𝑦 ∈ ℕ ∧ 𝑤 ∈ ℕ) ∧ (𝐴 = (𝑥 / 𝑦) ∧ 𝐵 = (𝑧 / 𝑤)))) → (𝑥 / 𝑦) ≠ 0) |
50 | | simprll 779 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ ((((𝜑 ∧ 𝐵 ≠ 0) ∧ (𝑥 ∈ ℤ ∧ 𝑧 ∈ ℤ)) ∧ ((𝑦 ∈ ℕ ∧ 𝑤 ∈ ℕ) ∧ (𝐴 = (𝑥 / 𝑦) ∧ 𝐵 = (𝑧 / 𝑤)))) → 𝑦 ∈ ℕ) |
51 | 50 | nncnd 11732 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ((((𝜑 ∧ 𝐵 ≠ 0) ∧ (𝑥 ∈ ℤ ∧ 𝑧 ∈ ℤ)) ∧ ((𝑦 ∈ ℕ ∧ 𝑤 ∈ ℕ) ∧ (𝐴 = (𝑥 / 𝑦) ∧ 𝐵 = (𝑧 / 𝑤)))) → 𝑦 ∈ ℂ) |
52 | 50 | nnne0d 11766 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ((((𝜑 ∧ 𝐵 ≠ 0) ∧ (𝑥 ∈ ℤ ∧ 𝑧 ∈ ℤ)) ∧ ((𝑦 ∈ ℕ ∧ 𝑤 ∈ ℕ) ∧ (𝐴 = (𝑥 / 𝑦) ∧ 𝐵 = (𝑧 / 𝑤)))) → 𝑦 ≠ 0) |
53 | 51, 52 | div0d 11493 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((((𝜑 ∧ 𝐵 ≠ 0) ∧ (𝑥 ∈ ℤ ∧ 𝑧 ∈ ℤ)) ∧ ((𝑦 ∈ ℕ ∧ 𝑤 ∈ ℕ) ∧ (𝐴 = (𝑥 / 𝑦) ∧ 𝐵 = (𝑧 / 𝑤)))) → (0 / 𝑦) = 0) |
54 | | oveq1 7177 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝑥 = 0 → (𝑥 / 𝑦) = (0 / 𝑦)) |
55 | 54 | eqeq1d 2740 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑥 = 0 → ((𝑥 / 𝑦) = 0 ↔ (0 / 𝑦) = 0)) |
56 | 53, 55 | syl5ibrcom 250 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((((𝜑 ∧ 𝐵 ≠ 0) ∧ (𝑥 ∈ ℤ ∧ 𝑧 ∈ ℤ)) ∧ ((𝑦 ∈ ℕ ∧ 𝑤 ∈ ℕ) ∧ (𝐴 = (𝑥 / 𝑦) ∧ 𝐵 = (𝑧 / 𝑤)))) → (𝑥 = 0 → (𝑥 / 𝑦) = 0)) |
57 | 56 | necon3d 2955 |
. . . . . . . . . . . . . . . . . 18
⊢ ((((𝜑 ∧ 𝐵 ≠ 0) ∧ (𝑥 ∈ ℤ ∧ 𝑧 ∈ ℤ)) ∧ ((𝑦 ∈ ℕ ∧ 𝑤 ∈ ℕ) ∧ (𝐴 = (𝑥 / 𝑦) ∧ 𝐵 = (𝑧 / 𝑤)))) → ((𝑥 / 𝑦) ≠ 0 → 𝑥 ≠ 0)) |
58 | 49, 57 | mpd 15 |
. . . . . . . . . . . . . . . . 17
⊢ ((((𝜑 ∧ 𝐵 ≠ 0) ∧ (𝑥 ∈ ℤ ∧ 𝑧 ∈ ℤ)) ∧ ((𝑦 ∈ ℕ ∧ 𝑤 ∈ ℕ) ∧ (𝐴 = (𝑥 / 𝑦) ∧ 𝐵 = (𝑧 / 𝑤)))) → 𝑥 ≠ 0) |
59 | | pczcl 16285 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝑃 ∈ ℙ ∧ (𝑥 ∈ ℤ ∧ 𝑥 ≠ 0)) → (𝑃 pCnt 𝑥) ∈
ℕ0) |
60 | 22, 25, 58, 59 | syl12anc 836 |
. . . . . . . . . . . . . . . 16
⊢ ((((𝜑 ∧ 𝐵 ≠ 0) ∧ (𝑥 ∈ ℤ ∧ 𝑧 ∈ ℤ)) ∧ ((𝑦 ∈ ℕ ∧ 𝑤 ∈ ℕ) ∧ (𝐴 = (𝑥 / 𝑦) ∧ 𝐵 = (𝑧 / 𝑤)))) → (𝑃 pCnt 𝑥) ∈
ℕ0) |
61 | 24, 60 | nnexpcld 13698 |
. . . . . . . . . . . . . . 15
⊢ ((((𝜑 ∧ 𝐵 ≠ 0) ∧ (𝑥 ∈ ℤ ∧ 𝑧 ∈ ℤ)) ∧ ((𝑦 ∈ ℕ ∧ 𝑤 ∈ ℕ) ∧ (𝐴 = (𝑥 / 𝑦) ∧ 𝐵 = (𝑧 / 𝑤)))) → (𝑃↑(𝑃 pCnt 𝑥)) ∈ ℕ) |
62 | 61 | nncnd 11732 |
. . . . . . . . . . . . . 14
⊢ ((((𝜑 ∧ 𝐵 ≠ 0) ∧ (𝑥 ∈ ℤ ∧ 𝑧 ∈ ℤ)) ∧ ((𝑦 ∈ ℕ ∧ 𝑤 ∈ ℕ) ∧ (𝐴 = (𝑥 / 𝑦) ∧ 𝐵 = (𝑧 / 𝑤)))) → (𝑃↑(𝑃 pCnt 𝑥)) ∈ ℂ) |
63 | 62, 51 | mulcomd 10740 |
. . . . . . . . . . . . 13
⊢ ((((𝜑 ∧ 𝐵 ≠ 0) ∧ (𝑥 ∈ ℤ ∧ 𝑧 ∈ ℤ)) ∧ ((𝑦 ∈ ℕ ∧ 𝑤 ∈ ℕ) ∧ (𝐴 = (𝑥 / 𝑦) ∧ 𝐵 = (𝑧 / 𝑤)))) → ((𝑃↑(𝑃 pCnt 𝑥)) · 𝑦) = (𝑦 · (𝑃↑(𝑃 pCnt 𝑥)))) |
64 | 63 | oveq2d 7186 |
. . . . . . . . . . . 12
⊢ ((((𝜑 ∧ 𝐵 ≠ 0) ∧ (𝑥 ∈ ℤ ∧ 𝑧 ∈ ℤ)) ∧ ((𝑦 ∈ ℕ ∧ 𝑤 ∈ ℕ) ∧ (𝐴 = (𝑥 / 𝑦) ∧ 𝐵 = (𝑧 / 𝑤)))) → ((𝑥 · (𝑃↑(𝑃 pCnt 𝑦))) / ((𝑃↑(𝑃 pCnt 𝑥)) · 𝑦)) = ((𝑥 · (𝑃↑(𝑃 pCnt 𝑦))) / (𝑦 · (𝑃↑(𝑃 pCnt 𝑥))))) |
65 | 25 | zcnd 12169 |
. . . . . . . . . . . . 13
⊢ ((((𝜑 ∧ 𝐵 ≠ 0) ∧ (𝑥 ∈ ℤ ∧ 𝑧 ∈ ℤ)) ∧ ((𝑦 ∈ ℕ ∧ 𝑤 ∈ ℕ) ∧ (𝐴 = (𝑥 / 𝑦) ∧ 𝐵 = (𝑧 / 𝑤)))) → 𝑥 ∈ ℂ) |
66 | 22, 50 | pccld 16287 |
. . . . . . . . . . . . . . 15
⊢ ((((𝜑 ∧ 𝐵 ≠ 0) ∧ (𝑥 ∈ ℤ ∧ 𝑧 ∈ ℤ)) ∧ ((𝑦 ∈ ℕ ∧ 𝑤 ∈ ℕ) ∧ (𝐴 = (𝑥 / 𝑦) ∧ 𝐵 = (𝑧 / 𝑤)))) → (𝑃 pCnt 𝑦) ∈
ℕ0) |
67 | 24, 66 | nnexpcld 13698 |
. . . . . . . . . . . . . 14
⊢ ((((𝜑 ∧ 𝐵 ≠ 0) ∧ (𝑥 ∈ ℤ ∧ 𝑧 ∈ ℤ)) ∧ ((𝑦 ∈ ℕ ∧ 𝑤 ∈ ℕ) ∧ (𝐴 = (𝑥 / 𝑦) ∧ 𝐵 = (𝑧 / 𝑤)))) → (𝑃↑(𝑃 pCnt 𝑦)) ∈ ℕ) |
68 | 67 | nncnd 11732 |
. . . . . . . . . . . . 13
⊢ ((((𝜑 ∧ 𝐵 ≠ 0) ∧ (𝑥 ∈ ℤ ∧ 𝑧 ∈ ℤ)) ∧ ((𝑦 ∈ ℕ ∧ 𝑤 ∈ ℕ) ∧ (𝐴 = (𝑥 / 𝑦) ∧ 𝐵 = (𝑧 / 𝑤)))) → (𝑃↑(𝑃 pCnt 𝑦)) ∈ ℂ) |
69 | 61 | nnne0d 11766 |
. . . . . . . . . . . . 13
⊢ ((((𝜑 ∧ 𝐵 ≠ 0) ∧ (𝑥 ∈ ℤ ∧ 𝑧 ∈ ℤ)) ∧ ((𝑦 ∈ ℕ ∧ 𝑤 ∈ ℕ) ∧ (𝐴 = (𝑥 / 𝑦) ∧ 𝐵 = (𝑧 / 𝑤)))) → (𝑃↑(𝑃 pCnt 𝑥)) ≠ 0) |
70 | 67 | nnne0d 11766 |
. . . . . . . . . . . . 13
⊢ ((((𝜑 ∧ 𝐵 ≠ 0) ∧ (𝑥 ∈ ℤ ∧ 𝑧 ∈ ℤ)) ∧ ((𝑦 ∈ ℕ ∧ 𝑤 ∈ ℕ) ∧ (𝐴 = (𝑥 / 𝑦) ∧ 𝐵 = (𝑧 / 𝑤)))) → (𝑃↑(𝑃 pCnt 𝑦)) ≠ 0) |
71 | 65, 62, 51, 68, 69, 70, 52 | divdivdivd 11541 |
. . . . . . . . . . . 12
⊢ ((((𝜑 ∧ 𝐵 ≠ 0) ∧ (𝑥 ∈ ℤ ∧ 𝑧 ∈ ℤ)) ∧ ((𝑦 ∈ ℕ ∧ 𝑤 ∈ ℕ) ∧ (𝐴 = (𝑥 / 𝑦) ∧ 𝐵 = (𝑧 / 𝑤)))) → ((𝑥 / (𝑃↑(𝑃 pCnt 𝑥))) / (𝑦 / (𝑃↑(𝑃 pCnt 𝑦)))) = ((𝑥 · (𝑃↑(𝑃 pCnt 𝑦))) / ((𝑃↑(𝑃 pCnt 𝑥)) · 𝑦))) |
72 | 26 | oveq2d 7186 |
. . . . . . . . . . . . . . . . 17
⊢ ((((𝜑 ∧ 𝐵 ≠ 0) ∧ (𝑥 ∈ ℤ ∧ 𝑧 ∈ ℤ)) ∧ ((𝑦 ∈ ℕ ∧ 𝑤 ∈ ℕ) ∧ (𝐴 = (𝑥 / 𝑦) ∧ 𝐵 = (𝑧 / 𝑤)))) → (𝑃 pCnt 𝐴) = (𝑃 pCnt (𝑥 / 𝑦))) |
73 | | pcdiv 16289 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝑃 ∈ ℙ ∧ (𝑥 ∈ ℤ ∧ 𝑥 ≠ 0) ∧ 𝑦 ∈ ℕ) → (𝑃 pCnt (𝑥 / 𝑦)) = ((𝑃 pCnt 𝑥) − (𝑃 pCnt 𝑦))) |
74 | 22, 25, 58, 50, 73 | syl121anc 1376 |
. . . . . . . . . . . . . . . . 17
⊢ ((((𝜑 ∧ 𝐵 ≠ 0) ∧ (𝑥 ∈ ℤ ∧ 𝑧 ∈ ℤ)) ∧ ((𝑦 ∈ ℕ ∧ 𝑤 ∈ ℕ) ∧ (𝐴 = (𝑥 / 𝑦) ∧ 𝐵 = (𝑧 / 𝑤)))) → (𝑃 pCnt (𝑥 / 𝑦)) = ((𝑃 pCnt 𝑥) − (𝑃 pCnt 𝑦))) |
75 | 72, 74 | eqtrd 2773 |
. . . . . . . . . . . . . . . 16
⊢ ((((𝜑 ∧ 𝐵 ≠ 0) ∧ (𝑥 ∈ ℤ ∧ 𝑧 ∈ ℤ)) ∧ ((𝑦 ∈ ℕ ∧ 𝑤 ∈ ℕ) ∧ (𝐴 = (𝑥 / 𝑦) ∧ 𝐵 = (𝑧 / 𝑤)))) → (𝑃 pCnt 𝐴) = ((𝑃 pCnt 𝑥) − (𝑃 pCnt 𝑦))) |
76 | 75 | oveq2d 7186 |
. . . . . . . . . . . . . . 15
⊢ ((((𝜑 ∧ 𝐵 ≠ 0) ∧ (𝑥 ∈ ℤ ∧ 𝑧 ∈ ℤ)) ∧ ((𝑦 ∈ ℕ ∧ 𝑤 ∈ ℕ) ∧ (𝐴 = (𝑥 / 𝑦) ∧ 𝐵 = (𝑧 / 𝑤)))) → (𝑃↑(𝑃 pCnt 𝐴)) = (𝑃↑((𝑃 pCnt 𝑥) − (𝑃 pCnt 𝑦)))) |
77 | 24 | nncnd 11732 |
. . . . . . . . . . . . . . . 16
⊢ ((((𝜑 ∧ 𝐵 ≠ 0) ∧ (𝑥 ∈ ℤ ∧ 𝑧 ∈ ℤ)) ∧ ((𝑦 ∈ ℕ ∧ 𝑤 ∈ ℕ) ∧ (𝐴 = (𝑥 / 𝑦) ∧ 𝐵 = (𝑧 / 𝑤)))) → 𝑃 ∈ ℂ) |
78 | 24 | nnne0d 11766 |
. . . . . . . . . . . . . . . 16
⊢ ((((𝜑 ∧ 𝐵 ≠ 0) ∧ (𝑥 ∈ ℤ ∧ 𝑧 ∈ ℤ)) ∧ ((𝑦 ∈ ℕ ∧ 𝑤 ∈ ℕ) ∧ (𝐴 = (𝑥 / 𝑦) ∧ 𝐵 = (𝑧 / 𝑤)))) → 𝑃 ≠ 0) |
79 | 66 | nn0zd 12166 |
. . . . . . . . . . . . . . . 16
⊢ ((((𝜑 ∧ 𝐵 ≠ 0) ∧ (𝑥 ∈ ℤ ∧ 𝑧 ∈ ℤ)) ∧ ((𝑦 ∈ ℕ ∧ 𝑤 ∈ ℕ) ∧ (𝐴 = (𝑥 / 𝑦) ∧ 𝐵 = (𝑧 / 𝑤)))) → (𝑃 pCnt 𝑦) ∈ ℤ) |
80 | 60 | nn0zd 12166 |
. . . . . . . . . . . . . . . 16
⊢ ((((𝜑 ∧ 𝐵 ≠ 0) ∧ (𝑥 ∈ ℤ ∧ 𝑧 ∈ ℤ)) ∧ ((𝑦 ∈ ℕ ∧ 𝑤 ∈ ℕ) ∧ (𝐴 = (𝑥 / 𝑦) ∧ 𝐵 = (𝑧 / 𝑤)))) → (𝑃 pCnt 𝑥) ∈ ℤ) |
81 | 77, 78, 79, 80 | expsubd 13613 |
. . . . . . . . . . . . . . 15
⊢ ((((𝜑 ∧ 𝐵 ≠ 0) ∧ (𝑥 ∈ ℤ ∧ 𝑧 ∈ ℤ)) ∧ ((𝑦 ∈ ℕ ∧ 𝑤 ∈ ℕ) ∧ (𝐴 = (𝑥 / 𝑦) ∧ 𝐵 = (𝑧 / 𝑤)))) → (𝑃↑((𝑃 pCnt 𝑥) − (𝑃 pCnt 𝑦))) = ((𝑃↑(𝑃 pCnt 𝑥)) / (𝑃↑(𝑃 pCnt 𝑦)))) |
82 | 76, 81 | eqtrd 2773 |
. . . . . . . . . . . . . 14
⊢ ((((𝜑 ∧ 𝐵 ≠ 0) ∧ (𝑥 ∈ ℤ ∧ 𝑧 ∈ ℤ)) ∧ ((𝑦 ∈ ℕ ∧ 𝑤 ∈ ℕ) ∧ (𝐴 = (𝑥 / 𝑦) ∧ 𝐵 = (𝑧 / 𝑤)))) → (𝑃↑(𝑃 pCnt 𝐴)) = ((𝑃↑(𝑃 pCnt 𝑥)) / (𝑃↑(𝑃 pCnt 𝑦)))) |
83 | 82 | oveq2d 7186 |
. . . . . . . . . . . . 13
⊢ ((((𝜑 ∧ 𝐵 ≠ 0) ∧ (𝑥 ∈ ℤ ∧ 𝑧 ∈ ℤ)) ∧ ((𝑦 ∈ ℕ ∧ 𝑤 ∈ ℕ) ∧ (𝐴 = (𝑥 / 𝑦) ∧ 𝐵 = (𝑧 / 𝑤)))) → (𝐴 / (𝑃↑(𝑃 pCnt 𝐴))) = (𝐴 / ((𝑃↑(𝑃 pCnt 𝑥)) / (𝑃↑(𝑃 pCnt 𝑦))))) |
84 | 26 | oveq1d 7185 |
. . . . . . . . . . . . 13
⊢ ((((𝜑 ∧ 𝐵 ≠ 0) ∧ (𝑥 ∈ ℤ ∧ 𝑧 ∈ ℤ)) ∧ ((𝑦 ∈ ℕ ∧ 𝑤 ∈ ℕ) ∧ (𝐴 = (𝑥 / 𝑦) ∧ 𝐵 = (𝑧 / 𝑤)))) → (𝐴 / ((𝑃↑(𝑃 pCnt 𝑥)) / (𝑃↑(𝑃 pCnt 𝑦)))) = ((𝑥 / 𝑦) / ((𝑃↑(𝑃 pCnt 𝑥)) / (𝑃↑(𝑃 pCnt 𝑦))))) |
85 | 65, 51, 62, 68, 52, 70, 69 | divdivdivd 11541 |
. . . . . . . . . . . . 13
⊢ ((((𝜑 ∧ 𝐵 ≠ 0) ∧ (𝑥 ∈ ℤ ∧ 𝑧 ∈ ℤ)) ∧ ((𝑦 ∈ ℕ ∧ 𝑤 ∈ ℕ) ∧ (𝐴 = (𝑥 / 𝑦) ∧ 𝐵 = (𝑧 / 𝑤)))) → ((𝑥 / 𝑦) / ((𝑃↑(𝑃 pCnt 𝑥)) / (𝑃↑(𝑃 pCnt 𝑦)))) = ((𝑥 · (𝑃↑(𝑃 pCnt 𝑦))) / (𝑦 · (𝑃↑(𝑃 pCnt 𝑥))))) |
86 | 83, 84, 85 | 3eqtrd 2777 |
. . . . . . . . . . . 12
⊢ ((((𝜑 ∧ 𝐵 ≠ 0) ∧ (𝑥 ∈ ℤ ∧ 𝑧 ∈ ℤ)) ∧ ((𝑦 ∈ ℕ ∧ 𝑤 ∈ ℕ) ∧ (𝐴 = (𝑥 / 𝑦) ∧ 𝐵 = (𝑧 / 𝑤)))) → (𝐴 / (𝑃↑(𝑃 pCnt 𝐴))) = ((𝑥 · (𝑃↑(𝑃 pCnt 𝑦))) / (𝑦 · (𝑃↑(𝑃 pCnt 𝑥))))) |
87 | 64, 71, 86 | 3eqtr4d 2783 |
. . . . . . . . . . 11
⊢ ((((𝜑 ∧ 𝐵 ≠ 0) ∧ (𝑥 ∈ ℤ ∧ 𝑧 ∈ ℤ)) ∧ ((𝑦 ∈ ℕ ∧ 𝑤 ∈ ℕ) ∧ (𝐴 = (𝑥 / 𝑦) ∧ 𝐵 = (𝑧 / 𝑤)))) → ((𝑥 / (𝑃↑(𝑃 pCnt 𝑥))) / (𝑦 / (𝑃↑(𝑃 pCnt 𝑦)))) = (𝐴 / (𝑃↑(𝑃 pCnt 𝐴)))) |
88 | 87 | oveq2d 7186 |
. . . . . . . . . 10
⊢ ((((𝜑 ∧ 𝐵 ≠ 0) ∧ (𝑥 ∈ ℤ ∧ 𝑧 ∈ ℤ)) ∧ ((𝑦 ∈ ℕ ∧ 𝑤 ∈ ℕ) ∧ (𝐴 = (𝑥 / 𝑦) ∧ 𝐵 = (𝑧 / 𝑤)))) → ((𝑃↑(𝑃 pCnt 𝐴)) · ((𝑥 / (𝑃↑(𝑃 pCnt 𝑥))) / (𝑦 / (𝑃↑(𝑃 pCnt 𝑦))))) = ((𝑃↑(𝑃 pCnt 𝐴)) · (𝐴 / (𝑃↑(𝑃 pCnt 𝐴))))) |
89 | 1 | ad3antrrr 730 |
. . . . . . . . . . . 12
⊢ ((((𝜑 ∧ 𝐵 ≠ 0) ∧ (𝑥 ∈ ℤ ∧ 𝑧 ∈ ℤ)) ∧ ((𝑦 ∈ ℕ ∧ 𝑤 ∈ ℕ) ∧ (𝐴 = (𝑥 / 𝑦) ∧ 𝐵 = (𝑧 / 𝑤)))) → 𝐴 ∈ ℚ) |
90 | 89, 13 | syl 17 |
. . . . . . . . . . 11
⊢ ((((𝜑 ∧ 𝐵 ≠ 0) ∧ (𝑥 ∈ ℤ ∧ 𝑧 ∈ ℤ)) ∧ ((𝑦 ∈ ℕ ∧ 𝑤 ∈ ℕ) ∧ (𝐴 = (𝑥 / 𝑦) ∧ 𝐵 = (𝑧 / 𝑤)))) → 𝐴 ∈ ℂ) |
91 | | pcqcl 16293 |
. . . . . . . . . . . . 13
⊢ ((𝑃 ∈ ℙ ∧ (𝐴 ∈ ℚ ∧ 𝐴 ≠ 0)) → (𝑃 pCnt 𝐴) ∈ ℤ) |
92 | 22, 89, 48, 91 | syl12anc 836 |
. . . . . . . . . . . 12
⊢ ((((𝜑 ∧ 𝐵 ≠ 0) ∧ (𝑥 ∈ ℤ ∧ 𝑧 ∈ ℤ)) ∧ ((𝑦 ∈ ℕ ∧ 𝑤 ∈ ℕ) ∧ (𝐴 = (𝑥 / 𝑦) ∧ 𝐵 = (𝑧 / 𝑤)))) → (𝑃 pCnt 𝐴) ∈ ℤ) |
93 | 77, 78, 92 | expclzd 13607 |
. . . . . . . . . . 11
⊢ ((((𝜑 ∧ 𝐵 ≠ 0) ∧ (𝑥 ∈ ℤ ∧ 𝑧 ∈ ℤ)) ∧ ((𝑦 ∈ ℕ ∧ 𝑤 ∈ ℕ) ∧ (𝐴 = (𝑥 / 𝑦) ∧ 𝐵 = (𝑧 / 𝑤)))) → (𝑃↑(𝑃 pCnt 𝐴)) ∈ ℂ) |
94 | 77, 78, 92 | expne0d 13608 |
. . . . . . . . . . 11
⊢ ((((𝜑 ∧ 𝐵 ≠ 0) ∧ (𝑥 ∈ ℤ ∧ 𝑧 ∈ ℤ)) ∧ ((𝑦 ∈ ℕ ∧ 𝑤 ∈ ℕ) ∧ (𝐴 = (𝑥 / 𝑦) ∧ 𝐵 = (𝑧 / 𝑤)))) → (𝑃↑(𝑃 pCnt 𝐴)) ≠ 0) |
95 | 90, 93, 94 | divcan2d 11496 |
. . . . . . . . . 10
⊢ ((((𝜑 ∧ 𝐵 ≠ 0) ∧ (𝑥 ∈ ℤ ∧ 𝑧 ∈ ℤ)) ∧ ((𝑦 ∈ ℕ ∧ 𝑤 ∈ ℕ) ∧ (𝐴 = (𝑥 / 𝑦) ∧ 𝐵 = (𝑧 / 𝑤)))) → ((𝑃↑(𝑃 pCnt 𝐴)) · (𝐴 / (𝑃↑(𝑃 pCnt 𝐴)))) = 𝐴) |
96 | 88, 95 | eqtr2d 2774 |
. . . . . . . . 9
⊢ ((((𝜑 ∧ 𝐵 ≠ 0) ∧ (𝑥 ∈ ℤ ∧ 𝑧 ∈ ℤ)) ∧ ((𝑦 ∈ ℕ ∧ 𝑤 ∈ ℕ) ∧ (𝐴 = (𝑥 / 𝑦) ∧ 𝐵 = (𝑧 / 𝑤)))) → 𝐴 = ((𝑃↑(𝑃 pCnt 𝐴)) · ((𝑥 / (𝑃↑(𝑃 pCnt 𝑥))) / (𝑦 / (𝑃↑(𝑃 pCnt 𝑦)))))) |
97 | | simplrr 778 |
. . . . . . . . . . . . . . . . 17
⊢ ((((𝜑 ∧ 𝐵 ≠ 0) ∧ (𝑥 ∈ ℤ ∧ 𝑧 ∈ ℤ)) ∧ ((𝑦 ∈ ℕ ∧ 𝑤 ∈ ℕ) ∧ (𝐴 = (𝑥 / 𝑦) ∧ 𝐵 = (𝑧 / 𝑤)))) → 𝑧 ∈ ℤ) |
98 | | simprrr 782 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((((𝜑 ∧ 𝐵 ≠ 0) ∧ (𝑥 ∈ ℤ ∧ 𝑧 ∈ ℤ)) ∧ ((𝑦 ∈ ℕ ∧ 𝑤 ∈ ℕ) ∧ (𝐴 = (𝑥 / 𝑦) ∧ 𝐵 = (𝑧 / 𝑤)))) → 𝐵 = (𝑧 / 𝑤)) |
99 | 98, 30 | eqnetrrd 3002 |
. . . . . . . . . . . . . . . . . 18
⊢ ((((𝜑 ∧ 𝐵 ≠ 0) ∧ (𝑥 ∈ ℤ ∧ 𝑧 ∈ ℤ)) ∧ ((𝑦 ∈ ℕ ∧ 𝑤 ∈ ℕ) ∧ (𝐴 = (𝑥 / 𝑦) ∧ 𝐵 = (𝑧 / 𝑤)))) → (𝑧 / 𝑤) ≠ 0) |
100 | | simprlr 780 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ ((((𝜑 ∧ 𝐵 ≠ 0) ∧ (𝑥 ∈ ℤ ∧ 𝑧 ∈ ℤ)) ∧ ((𝑦 ∈ ℕ ∧ 𝑤 ∈ ℕ) ∧ (𝐴 = (𝑥 / 𝑦) ∧ 𝐵 = (𝑧 / 𝑤)))) → 𝑤 ∈ ℕ) |
101 | 100 | nncnd 11732 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ((((𝜑 ∧ 𝐵 ≠ 0) ∧ (𝑥 ∈ ℤ ∧ 𝑧 ∈ ℤ)) ∧ ((𝑦 ∈ ℕ ∧ 𝑤 ∈ ℕ) ∧ (𝐴 = (𝑥 / 𝑦) ∧ 𝐵 = (𝑧 / 𝑤)))) → 𝑤 ∈ ℂ) |
102 | 100 | nnne0d 11766 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ((((𝜑 ∧ 𝐵 ≠ 0) ∧ (𝑥 ∈ ℤ ∧ 𝑧 ∈ ℤ)) ∧ ((𝑦 ∈ ℕ ∧ 𝑤 ∈ ℕ) ∧ (𝐴 = (𝑥 / 𝑦) ∧ 𝐵 = (𝑧 / 𝑤)))) → 𝑤 ≠ 0) |
103 | 101, 102 | div0d 11493 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((((𝜑 ∧ 𝐵 ≠ 0) ∧ (𝑥 ∈ ℤ ∧ 𝑧 ∈ ℤ)) ∧ ((𝑦 ∈ ℕ ∧ 𝑤 ∈ ℕ) ∧ (𝐴 = (𝑥 / 𝑦) ∧ 𝐵 = (𝑧 / 𝑤)))) → (0 / 𝑤) = 0) |
104 | | oveq1 7177 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝑧 = 0 → (𝑧 / 𝑤) = (0 / 𝑤)) |
105 | 104 | eqeq1d 2740 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑧 = 0 → ((𝑧 / 𝑤) = 0 ↔ (0 / 𝑤) = 0)) |
106 | 103, 105 | syl5ibrcom 250 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((((𝜑 ∧ 𝐵 ≠ 0) ∧ (𝑥 ∈ ℤ ∧ 𝑧 ∈ ℤ)) ∧ ((𝑦 ∈ ℕ ∧ 𝑤 ∈ ℕ) ∧ (𝐴 = (𝑥 / 𝑦) ∧ 𝐵 = (𝑧 / 𝑤)))) → (𝑧 = 0 → (𝑧 / 𝑤) = 0)) |
107 | 106 | necon3d 2955 |
. . . . . . . . . . . . . . . . . 18
⊢ ((((𝜑 ∧ 𝐵 ≠ 0) ∧ (𝑥 ∈ ℤ ∧ 𝑧 ∈ ℤ)) ∧ ((𝑦 ∈ ℕ ∧ 𝑤 ∈ ℕ) ∧ (𝐴 = (𝑥 / 𝑦) ∧ 𝐵 = (𝑧 / 𝑤)))) → ((𝑧 / 𝑤) ≠ 0 → 𝑧 ≠ 0)) |
108 | 99, 107 | mpd 15 |
. . . . . . . . . . . . . . . . 17
⊢ ((((𝜑 ∧ 𝐵 ≠ 0) ∧ (𝑥 ∈ ℤ ∧ 𝑧 ∈ ℤ)) ∧ ((𝑦 ∈ ℕ ∧ 𝑤 ∈ ℕ) ∧ (𝐴 = (𝑥 / 𝑦) ∧ 𝐵 = (𝑧 / 𝑤)))) → 𝑧 ≠ 0) |
109 | | pczcl 16285 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝑃 ∈ ℙ ∧ (𝑧 ∈ ℤ ∧ 𝑧 ≠ 0)) → (𝑃 pCnt 𝑧) ∈
ℕ0) |
110 | 22, 97, 108, 109 | syl12anc 836 |
. . . . . . . . . . . . . . . 16
⊢ ((((𝜑 ∧ 𝐵 ≠ 0) ∧ (𝑥 ∈ ℤ ∧ 𝑧 ∈ ℤ)) ∧ ((𝑦 ∈ ℕ ∧ 𝑤 ∈ ℕ) ∧ (𝐴 = (𝑥 / 𝑦) ∧ 𝐵 = (𝑧 / 𝑤)))) → (𝑃 pCnt 𝑧) ∈
ℕ0) |
111 | 24, 110 | nnexpcld 13698 |
. . . . . . . . . . . . . . 15
⊢ ((((𝜑 ∧ 𝐵 ≠ 0) ∧ (𝑥 ∈ ℤ ∧ 𝑧 ∈ ℤ)) ∧ ((𝑦 ∈ ℕ ∧ 𝑤 ∈ ℕ) ∧ (𝐴 = (𝑥 / 𝑦) ∧ 𝐵 = (𝑧 / 𝑤)))) → (𝑃↑(𝑃 pCnt 𝑧)) ∈ ℕ) |
112 | 111 | nncnd 11732 |
. . . . . . . . . . . . . 14
⊢ ((((𝜑 ∧ 𝐵 ≠ 0) ∧ (𝑥 ∈ ℤ ∧ 𝑧 ∈ ℤ)) ∧ ((𝑦 ∈ ℕ ∧ 𝑤 ∈ ℕ) ∧ (𝐴 = (𝑥 / 𝑦) ∧ 𝐵 = (𝑧 / 𝑤)))) → (𝑃↑(𝑃 pCnt 𝑧)) ∈ ℂ) |
113 | 112, 101 | mulcomd 10740 |
. . . . . . . . . . . . 13
⊢ ((((𝜑 ∧ 𝐵 ≠ 0) ∧ (𝑥 ∈ ℤ ∧ 𝑧 ∈ ℤ)) ∧ ((𝑦 ∈ ℕ ∧ 𝑤 ∈ ℕ) ∧ (𝐴 = (𝑥 / 𝑦) ∧ 𝐵 = (𝑧 / 𝑤)))) → ((𝑃↑(𝑃 pCnt 𝑧)) · 𝑤) = (𝑤 · (𝑃↑(𝑃 pCnt 𝑧)))) |
114 | 113 | oveq2d 7186 |
. . . . . . . . . . . 12
⊢ ((((𝜑 ∧ 𝐵 ≠ 0) ∧ (𝑥 ∈ ℤ ∧ 𝑧 ∈ ℤ)) ∧ ((𝑦 ∈ ℕ ∧ 𝑤 ∈ ℕ) ∧ (𝐴 = (𝑥 / 𝑦) ∧ 𝐵 = (𝑧 / 𝑤)))) → ((𝑧 · (𝑃↑(𝑃 pCnt 𝑤))) / ((𝑃↑(𝑃 pCnt 𝑧)) · 𝑤)) = ((𝑧 · (𝑃↑(𝑃 pCnt 𝑤))) / (𝑤 · (𝑃↑(𝑃 pCnt 𝑧))))) |
115 | 97 | zcnd 12169 |
. . . . . . . . . . . . 13
⊢ ((((𝜑 ∧ 𝐵 ≠ 0) ∧ (𝑥 ∈ ℤ ∧ 𝑧 ∈ ℤ)) ∧ ((𝑦 ∈ ℕ ∧ 𝑤 ∈ ℕ) ∧ (𝐴 = (𝑥 / 𝑦) ∧ 𝐵 = (𝑧 / 𝑤)))) → 𝑧 ∈ ℂ) |
116 | 22, 100 | pccld 16287 |
. . . . . . . . . . . . . . 15
⊢ ((((𝜑 ∧ 𝐵 ≠ 0) ∧ (𝑥 ∈ ℤ ∧ 𝑧 ∈ ℤ)) ∧ ((𝑦 ∈ ℕ ∧ 𝑤 ∈ ℕ) ∧ (𝐴 = (𝑥 / 𝑦) ∧ 𝐵 = (𝑧 / 𝑤)))) → (𝑃 pCnt 𝑤) ∈
ℕ0) |
117 | 24, 116 | nnexpcld 13698 |
. . . . . . . . . . . . . 14
⊢ ((((𝜑 ∧ 𝐵 ≠ 0) ∧ (𝑥 ∈ ℤ ∧ 𝑧 ∈ ℤ)) ∧ ((𝑦 ∈ ℕ ∧ 𝑤 ∈ ℕ) ∧ (𝐴 = (𝑥 / 𝑦) ∧ 𝐵 = (𝑧 / 𝑤)))) → (𝑃↑(𝑃 pCnt 𝑤)) ∈ ℕ) |
118 | 117 | nncnd 11732 |
. . . . . . . . . . . . 13
⊢ ((((𝜑 ∧ 𝐵 ≠ 0) ∧ (𝑥 ∈ ℤ ∧ 𝑧 ∈ ℤ)) ∧ ((𝑦 ∈ ℕ ∧ 𝑤 ∈ ℕ) ∧ (𝐴 = (𝑥 / 𝑦) ∧ 𝐵 = (𝑧 / 𝑤)))) → (𝑃↑(𝑃 pCnt 𝑤)) ∈ ℂ) |
119 | 111 | nnne0d 11766 |
. . . . . . . . . . . . 13
⊢ ((((𝜑 ∧ 𝐵 ≠ 0) ∧ (𝑥 ∈ ℤ ∧ 𝑧 ∈ ℤ)) ∧ ((𝑦 ∈ ℕ ∧ 𝑤 ∈ ℕ) ∧ (𝐴 = (𝑥 / 𝑦) ∧ 𝐵 = (𝑧 / 𝑤)))) → (𝑃↑(𝑃 pCnt 𝑧)) ≠ 0) |
120 | 117 | nnne0d 11766 |
. . . . . . . . . . . . 13
⊢ ((((𝜑 ∧ 𝐵 ≠ 0) ∧ (𝑥 ∈ ℤ ∧ 𝑧 ∈ ℤ)) ∧ ((𝑦 ∈ ℕ ∧ 𝑤 ∈ ℕ) ∧ (𝐴 = (𝑥 / 𝑦) ∧ 𝐵 = (𝑧 / 𝑤)))) → (𝑃↑(𝑃 pCnt 𝑤)) ≠ 0) |
121 | 115, 112,
101, 118, 119, 120, 102 | divdivdivd 11541 |
. . . . . . . . . . . 12
⊢ ((((𝜑 ∧ 𝐵 ≠ 0) ∧ (𝑥 ∈ ℤ ∧ 𝑧 ∈ ℤ)) ∧ ((𝑦 ∈ ℕ ∧ 𝑤 ∈ ℕ) ∧ (𝐴 = (𝑥 / 𝑦) ∧ 𝐵 = (𝑧 / 𝑤)))) → ((𝑧 / (𝑃↑(𝑃 pCnt 𝑧))) / (𝑤 / (𝑃↑(𝑃 pCnt 𝑤)))) = ((𝑧 · (𝑃↑(𝑃 pCnt 𝑤))) / ((𝑃↑(𝑃 pCnt 𝑧)) · 𝑤))) |
122 | 98 | oveq2d 7186 |
. . . . . . . . . . . . . . . . 17
⊢ ((((𝜑 ∧ 𝐵 ≠ 0) ∧ (𝑥 ∈ ℤ ∧ 𝑧 ∈ ℤ)) ∧ ((𝑦 ∈ ℕ ∧ 𝑤 ∈ ℕ) ∧ (𝐴 = (𝑥 / 𝑦) ∧ 𝐵 = (𝑧 / 𝑤)))) → (𝑃 pCnt 𝐵) = (𝑃 pCnt (𝑧 / 𝑤))) |
123 | | pcdiv 16289 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝑃 ∈ ℙ ∧ (𝑧 ∈ ℤ ∧ 𝑧 ≠ 0) ∧ 𝑤 ∈ ℕ) → (𝑃 pCnt (𝑧 / 𝑤)) = ((𝑃 pCnt 𝑧) − (𝑃 pCnt 𝑤))) |
124 | 22, 97, 108, 100, 123 | syl121anc 1376 |
. . . . . . . . . . . . . . . . 17
⊢ ((((𝜑 ∧ 𝐵 ≠ 0) ∧ (𝑥 ∈ ℤ ∧ 𝑧 ∈ ℤ)) ∧ ((𝑦 ∈ ℕ ∧ 𝑤 ∈ ℕ) ∧ (𝐴 = (𝑥 / 𝑦) ∧ 𝐵 = (𝑧 / 𝑤)))) → (𝑃 pCnt (𝑧 / 𝑤)) = ((𝑃 pCnt 𝑧) − (𝑃 pCnt 𝑤))) |
125 | 122, 124 | eqtrd 2773 |
. . . . . . . . . . . . . . . 16
⊢ ((((𝜑 ∧ 𝐵 ≠ 0) ∧ (𝑥 ∈ ℤ ∧ 𝑧 ∈ ℤ)) ∧ ((𝑦 ∈ ℕ ∧ 𝑤 ∈ ℕ) ∧ (𝐴 = (𝑥 / 𝑦) ∧ 𝐵 = (𝑧 / 𝑤)))) → (𝑃 pCnt 𝐵) = ((𝑃 pCnt 𝑧) − (𝑃 pCnt 𝑤))) |
126 | 125 | oveq2d 7186 |
. . . . . . . . . . . . . . 15
⊢ ((((𝜑 ∧ 𝐵 ≠ 0) ∧ (𝑥 ∈ ℤ ∧ 𝑧 ∈ ℤ)) ∧ ((𝑦 ∈ ℕ ∧ 𝑤 ∈ ℕ) ∧ (𝐴 = (𝑥 / 𝑦) ∧ 𝐵 = (𝑧 / 𝑤)))) → (𝑃↑(𝑃 pCnt 𝐵)) = (𝑃↑((𝑃 pCnt 𝑧) − (𝑃 pCnt 𝑤)))) |
127 | 116 | nn0zd 12166 |
. . . . . . . . . . . . . . . 16
⊢ ((((𝜑 ∧ 𝐵 ≠ 0) ∧ (𝑥 ∈ ℤ ∧ 𝑧 ∈ ℤ)) ∧ ((𝑦 ∈ ℕ ∧ 𝑤 ∈ ℕ) ∧ (𝐴 = (𝑥 / 𝑦) ∧ 𝐵 = (𝑧 / 𝑤)))) → (𝑃 pCnt 𝑤) ∈ ℤ) |
128 | 110 | nn0zd 12166 |
. . . . . . . . . . . . . . . 16
⊢ ((((𝜑 ∧ 𝐵 ≠ 0) ∧ (𝑥 ∈ ℤ ∧ 𝑧 ∈ ℤ)) ∧ ((𝑦 ∈ ℕ ∧ 𝑤 ∈ ℕ) ∧ (𝐴 = (𝑥 / 𝑦) ∧ 𝐵 = (𝑧 / 𝑤)))) → (𝑃 pCnt 𝑧) ∈ ℤ) |
129 | 77, 78, 127, 128 | expsubd 13613 |
. . . . . . . . . . . . . . 15
⊢ ((((𝜑 ∧ 𝐵 ≠ 0) ∧ (𝑥 ∈ ℤ ∧ 𝑧 ∈ ℤ)) ∧ ((𝑦 ∈ ℕ ∧ 𝑤 ∈ ℕ) ∧ (𝐴 = (𝑥 / 𝑦) ∧ 𝐵 = (𝑧 / 𝑤)))) → (𝑃↑((𝑃 pCnt 𝑧) − (𝑃 pCnt 𝑤))) = ((𝑃↑(𝑃 pCnt 𝑧)) / (𝑃↑(𝑃 pCnt 𝑤)))) |
130 | 126, 129 | eqtrd 2773 |
. . . . . . . . . . . . . 14
⊢ ((((𝜑 ∧ 𝐵 ≠ 0) ∧ (𝑥 ∈ ℤ ∧ 𝑧 ∈ ℤ)) ∧ ((𝑦 ∈ ℕ ∧ 𝑤 ∈ ℕ) ∧ (𝐴 = (𝑥 / 𝑦) ∧ 𝐵 = (𝑧 / 𝑤)))) → (𝑃↑(𝑃 pCnt 𝐵)) = ((𝑃↑(𝑃 pCnt 𝑧)) / (𝑃↑(𝑃 pCnt 𝑤)))) |
131 | 130 | oveq2d 7186 |
. . . . . . . . . . . . 13
⊢ ((((𝜑 ∧ 𝐵 ≠ 0) ∧ (𝑥 ∈ ℤ ∧ 𝑧 ∈ ℤ)) ∧ ((𝑦 ∈ ℕ ∧ 𝑤 ∈ ℕ) ∧ (𝐴 = (𝑥 / 𝑦) ∧ 𝐵 = (𝑧 / 𝑤)))) → (𝐵 / (𝑃↑(𝑃 pCnt 𝐵))) = (𝐵 / ((𝑃↑(𝑃 pCnt 𝑧)) / (𝑃↑(𝑃 pCnt 𝑤))))) |
132 | 98 | oveq1d 7185 |
. . . . . . . . . . . . 13
⊢ ((((𝜑 ∧ 𝐵 ≠ 0) ∧ (𝑥 ∈ ℤ ∧ 𝑧 ∈ ℤ)) ∧ ((𝑦 ∈ ℕ ∧ 𝑤 ∈ ℕ) ∧ (𝐴 = (𝑥 / 𝑦) ∧ 𝐵 = (𝑧 / 𝑤)))) → (𝐵 / ((𝑃↑(𝑃 pCnt 𝑧)) / (𝑃↑(𝑃 pCnt 𝑤)))) = ((𝑧 / 𝑤) / ((𝑃↑(𝑃 pCnt 𝑧)) / (𝑃↑(𝑃 pCnt 𝑤))))) |
133 | 115, 101,
112, 118, 102, 120, 119 | divdivdivd 11541 |
. . . . . . . . . . . . 13
⊢ ((((𝜑 ∧ 𝐵 ≠ 0) ∧ (𝑥 ∈ ℤ ∧ 𝑧 ∈ ℤ)) ∧ ((𝑦 ∈ ℕ ∧ 𝑤 ∈ ℕ) ∧ (𝐴 = (𝑥 / 𝑦) ∧ 𝐵 = (𝑧 / 𝑤)))) → ((𝑧 / 𝑤) / ((𝑃↑(𝑃 pCnt 𝑧)) / (𝑃↑(𝑃 pCnt 𝑤)))) = ((𝑧 · (𝑃↑(𝑃 pCnt 𝑤))) / (𝑤 · (𝑃↑(𝑃 pCnt 𝑧))))) |
134 | 131, 132,
133 | 3eqtrd 2777 |
. . . . . . . . . . . 12
⊢ ((((𝜑 ∧ 𝐵 ≠ 0) ∧ (𝑥 ∈ ℤ ∧ 𝑧 ∈ ℤ)) ∧ ((𝑦 ∈ ℕ ∧ 𝑤 ∈ ℕ) ∧ (𝐴 = (𝑥 / 𝑦) ∧ 𝐵 = (𝑧 / 𝑤)))) → (𝐵 / (𝑃↑(𝑃 pCnt 𝐵))) = ((𝑧 · (𝑃↑(𝑃 pCnt 𝑤))) / (𝑤 · (𝑃↑(𝑃 pCnt 𝑧))))) |
135 | 114, 121,
134 | 3eqtr4d 2783 |
. . . . . . . . . . 11
⊢ ((((𝜑 ∧ 𝐵 ≠ 0) ∧ (𝑥 ∈ ℤ ∧ 𝑧 ∈ ℤ)) ∧ ((𝑦 ∈ ℕ ∧ 𝑤 ∈ ℕ) ∧ (𝐴 = (𝑥 / 𝑦) ∧ 𝐵 = (𝑧 / 𝑤)))) → ((𝑧 / (𝑃↑(𝑃 pCnt 𝑧))) / (𝑤 / (𝑃↑(𝑃 pCnt 𝑤)))) = (𝐵 / (𝑃↑(𝑃 pCnt 𝐵)))) |
136 | 135 | oveq2d 7186 |
. . . . . . . . . 10
⊢ ((((𝜑 ∧ 𝐵 ≠ 0) ∧ (𝑥 ∈ ℤ ∧ 𝑧 ∈ ℤ)) ∧ ((𝑦 ∈ ℕ ∧ 𝑤 ∈ ℕ) ∧ (𝐴 = (𝑥 / 𝑦) ∧ 𝐵 = (𝑧 / 𝑤)))) → ((𝑃↑(𝑃 pCnt 𝐵)) · ((𝑧 / (𝑃↑(𝑃 pCnt 𝑧))) / (𝑤 / (𝑃↑(𝑃 pCnt 𝑤))))) = ((𝑃↑(𝑃 pCnt 𝐵)) · (𝐵 / (𝑃↑(𝑃 pCnt 𝐵))))) |
137 | | qcn 12445 |
. . . . . . . . . . . 12
⊢ (𝐵 ∈ ℚ → 𝐵 ∈
ℂ) |
138 | 29, 137 | syl 17 |
. . . . . . . . . . 11
⊢ ((((𝜑 ∧ 𝐵 ≠ 0) ∧ (𝑥 ∈ ℤ ∧ 𝑧 ∈ ℤ)) ∧ ((𝑦 ∈ ℕ ∧ 𝑤 ∈ ℕ) ∧ (𝐴 = (𝑥 / 𝑦) ∧ 𝐵 = (𝑧 / 𝑤)))) → 𝐵 ∈ ℂ) |
139 | 77, 78, 32 | expclzd 13607 |
. . . . . . . . . . 11
⊢ ((((𝜑 ∧ 𝐵 ≠ 0) ∧ (𝑥 ∈ ℤ ∧ 𝑧 ∈ ℤ)) ∧ ((𝑦 ∈ ℕ ∧ 𝑤 ∈ ℕ) ∧ (𝐴 = (𝑥 / 𝑦) ∧ 𝐵 = (𝑧 / 𝑤)))) → (𝑃↑(𝑃 pCnt 𝐵)) ∈ ℂ) |
140 | 77, 78, 32 | expne0d 13608 |
. . . . . . . . . . 11
⊢ ((((𝜑 ∧ 𝐵 ≠ 0) ∧ (𝑥 ∈ ℤ ∧ 𝑧 ∈ ℤ)) ∧ ((𝑦 ∈ ℕ ∧ 𝑤 ∈ ℕ) ∧ (𝐴 = (𝑥 / 𝑦) ∧ 𝐵 = (𝑧 / 𝑤)))) → (𝑃↑(𝑃 pCnt 𝐵)) ≠ 0) |
141 | 138, 139,
140 | divcan2d 11496 |
. . . . . . . . . 10
⊢ ((((𝜑 ∧ 𝐵 ≠ 0) ∧ (𝑥 ∈ ℤ ∧ 𝑧 ∈ ℤ)) ∧ ((𝑦 ∈ ℕ ∧ 𝑤 ∈ ℕ) ∧ (𝐴 = (𝑥 / 𝑦) ∧ 𝐵 = (𝑧 / 𝑤)))) → ((𝑃↑(𝑃 pCnt 𝐵)) · (𝐵 / (𝑃↑(𝑃 pCnt 𝐵)))) = 𝐵) |
142 | 136, 141 | eqtr2d 2774 |
. . . . . . . . 9
⊢ ((((𝜑 ∧ 𝐵 ≠ 0) ∧ (𝑥 ∈ ℤ ∧ 𝑧 ∈ ℤ)) ∧ ((𝑦 ∈ ℕ ∧ 𝑤 ∈ ℕ) ∧ (𝐴 = (𝑥 / 𝑦) ∧ 𝐵 = (𝑧 / 𝑤)))) → 𝐵 = ((𝑃↑(𝑃 pCnt 𝐵)) · ((𝑧 / (𝑃↑(𝑃 pCnt 𝑧))) / (𝑤 / (𝑃↑(𝑃 pCnt 𝑤)))))) |
143 | | eluz 12338 |
. . . . . . . . . . 11
⊢ (((𝑃 pCnt 𝐴) ∈ ℤ ∧ (𝑃 pCnt 𝐵) ∈ ℤ) → ((𝑃 pCnt 𝐵) ∈
(ℤ≥‘(𝑃 pCnt 𝐴)) ↔ (𝑃 pCnt 𝐴) ≤ (𝑃 pCnt 𝐵))) |
144 | 92, 32, 143 | syl2anc 587 |
. . . . . . . . . 10
⊢ ((((𝜑 ∧ 𝐵 ≠ 0) ∧ (𝑥 ∈ ℤ ∧ 𝑧 ∈ ℤ)) ∧ ((𝑦 ∈ ℕ ∧ 𝑤 ∈ ℕ) ∧ (𝐴 = (𝑥 / 𝑦) ∧ 𝐵 = (𝑧 / 𝑤)))) → ((𝑃 pCnt 𝐵) ∈
(ℤ≥‘(𝑃 pCnt 𝐴)) ↔ (𝑃 pCnt 𝐴) ≤ (𝑃 pCnt 𝐵))) |
145 | 43, 144 | mpbird 260 |
. . . . . . . . 9
⊢ ((((𝜑 ∧ 𝐵 ≠ 0) ∧ (𝑥 ∈ ℤ ∧ 𝑧 ∈ ℤ)) ∧ ((𝑦 ∈ ℕ ∧ 𝑤 ∈ ℕ) ∧ (𝐴 = (𝑥 / 𝑦) ∧ 𝐵 = (𝑧 / 𝑤)))) → (𝑃 pCnt 𝐵) ∈
(ℤ≥‘(𝑃 pCnt 𝐴))) |
146 | | pczdvds 16299 |
. . . . . . . . . . . 12
⊢ ((𝑃 ∈ ℙ ∧ (𝑥 ∈ ℤ ∧ 𝑥 ≠ 0)) → (𝑃↑(𝑃 pCnt 𝑥)) ∥ 𝑥) |
147 | 22, 25, 58, 146 | syl12anc 836 |
. . . . . . . . . . 11
⊢ ((((𝜑 ∧ 𝐵 ≠ 0) ∧ (𝑥 ∈ ℤ ∧ 𝑧 ∈ ℤ)) ∧ ((𝑦 ∈ ℕ ∧ 𝑤 ∈ ℕ) ∧ (𝐴 = (𝑥 / 𝑦) ∧ 𝐵 = (𝑧 / 𝑤)))) → (𝑃↑(𝑃 pCnt 𝑥)) ∥ 𝑥) |
148 | 61 | nnzd 12167 |
. . . . . . . . . . . 12
⊢ ((((𝜑 ∧ 𝐵 ≠ 0) ∧ (𝑥 ∈ ℤ ∧ 𝑧 ∈ ℤ)) ∧ ((𝑦 ∈ ℕ ∧ 𝑤 ∈ ℕ) ∧ (𝐴 = (𝑥 / 𝑦) ∧ 𝐵 = (𝑧 / 𝑤)))) → (𝑃↑(𝑃 pCnt 𝑥)) ∈ ℤ) |
149 | | dvdsval2 15702 |
. . . . . . . . . . . 12
⊢ (((𝑃↑(𝑃 pCnt 𝑥)) ∈ ℤ ∧ (𝑃↑(𝑃 pCnt 𝑥)) ≠ 0 ∧ 𝑥 ∈ ℤ) → ((𝑃↑(𝑃 pCnt 𝑥)) ∥ 𝑥 ↔ (𝑥 / (𝑃↑(𝑃 pCnt 𝑥))) ∈ ℤ)) |
150 | 148, 69, 25, 149 | syl3anc 1372 |
. . . . . . . . . . 11
⊢ ((((𝜑 ∧ 𝐵 ≠ 0) ∧ (𝑥 ∈ ℤ ∧ 𝑧 ∈ ℤ)) ∧ ((𝑦 ∈ ℕ ∧ 𝑤 ∈ ℕ) ∧ (𝐴 = (𝑥 / 𝑦) ∧ 𝐵 = (𝑧 / 𝑤)))) → ((𝑃↑(𝑃 pCnt 𝑥)) ∥ 𝑥 ↔ (𝑥 / (𝑃↑(𝑃 pCnt 𝑥))) ∈ ℤ)) |
151 | 147, 150 | mpbid 235 |
. . . . . . . . . 10
⊢ ((((𝜑 ∧ 𝐵 ≠ 0) ∧ (𝑥 ∈ ℤ ∧ 𝑧 ∈ ℤ)) ∧ ((𝑦 ∈ ℕ ∧ 𝑤 ∈ ℕ) ∧ (𝐴 = (𝑥 / 𝑦) ∧ 𝐵 = (𝑧 / 𝑤)))) → (𝑥 / (𝑃↑(𝑃 pCnt 𝑥))) ∈ ℤ) |
152 | | pczndvds2 16303 |
. . . . . . . . . . 11
⊢ ((𝑃 ∈ ℙ ∧ (𝑥 ∈ ℤ ∧ 𝑥 ≠ 0)) → ¬ 𝑃 ∥ (𝑥 / (𝑃↑(𝑃 pCnt 𝑥)))) |
153 | 22, 25, 58, 152 | syl12anc 836 |
. . . . . . . . . 10
⊢ ((((𝜑 ∧ 𝐵 ≠ 0) ∧ (𝑥 ∈ ℤ ∧ 𝑧 ∈ ℤ)) ∧ ((𝑦 ∈ ℕ ∧ 𝑤 ∈ ℕ) ∧ (𝐴 = (𝑥 / 𝑦) ∧ 𝐵 = (𝑧 / 𝑤)))) → ¬ 𝑃 ∥ (𝑥 / (𝑃↑(𝑃 pCnt 𝑥)))) |
154 | 151, 153 | jca 515 |
. . . . . . . . 9
⊢ ((((𝜑 ∧ 𝐵 ≠ 0) ∧ (𝑥 ∈ ℤ ∧ 𝑧 ∈ ℤ)) ∧ ((𝑦 ∈ ℕ ∧ 𝑤 ∈ ℕ) ∧ (𝐴 = (𝑥 / 𝑦) ∧ 𝐵 = (𝑧 / 𝑤)))) → ((𝑥 / (𝑃↑(𝑃 pCnt 𝑥))) ∈ ℤ ∧ ¬ 𝑃 ∥ (𝑥 / (𝑃↑(𝑃 pCnt 𝑥))))) |
155 | | pcdvds 16300 |
. . . . . . . . . . . . 13
⊢ ((𝑃 ∈ ℙ ∧ 𝑦 ∈ ℕ) → (𝑃↑(𝑃 pCnt 𝑦)) ∥ 𝑦) |
156 | 22, 50, 155 | syl2anc 587 |
. . . . . . . . . . . 12
⊢ ((((𝜑 ∧ 𝐵 ≠ 0) ∧ (𝑥 ∈ ℤ ∧ 𝑧 ∈ ℤ)) ∧ ((𝑦 ∈ ℕ ∧ 𝑤 ∈ ℕ) ∧ (𝐴 = (𝑥 / 𝑦) ∧ 𝐵 = (𝑧 / 𝑤)))) → (𝑃↑(𝑃 pCnt 𝑦)) ∥ 𝑦) |
157 | 67 | nnzd 12167 |
. . . . . . . . . . . . 13
⊢ ((((𝜑 ∧ 𝐵 ≠ 0) ∧ (𝑥 ∈ ℤ ∧ 𝑧 ∈ ℤ)) ∧ ((𝑦 ∈ ℕ ∧ 𝑤 ∈ ℕ) ∧ (𝐴 = (𝑥 / 𝑦) ∧ 𝐵 = (𝑧 / 𝑤)))) → (𝑃↑(𝑃 pCnt 𝑦)) ∈ ℤ) |
158 | 50 | nnzd 12167 |
. . . . . . . . . . . . 13
⊢ ((((𝜑 ∧ 𝐵 ≠ 0) ∧ (𝑥 ∈ ℤ ∧ 𝑧 ∈ ℤ)) ∧ ((𝑦 ∈ ℕ ∧ 𝑤 ∈ ℕ) ∧ (𝐴 = (𝑥 / 𝑦) ∧ 𝐵 = (𝑧 / 𝑤)))) → 𝑦 ∈ ℤ) |
159 | | dvdsval2 15702 |
. . . . . . . . . . . . 13
⊢ (((𝑃↑(𝑃 pCnt 𝑦)) ∈ ℤ ∧ (𝑃↑(𝑃 pCnt 𝑦)) ≠ 0 ∧ 𝑦 ∈ ℤ) → ((𝑃↑(𝑃 pCnt 𝑦)) ∥ 𝑦 ↔ (𝑦 / (𝑃↑(𝑃 pCnt 𝑦))) ∈ ℤ)) |
160 | 157, 70, 158, 159 | syl3anc 1372 |
. . . . . . . . . . . 12
⊢ ((((𝜑 ∧ 𝐵 ≠ 0) ∧ (𝑥 ∈ ℤ ∧ 𝑧 ∈ ℤ)) ∧ ((𝑦 ∈ ℕ ∧ 𝑤 ∈ ℕ) ∧ (𝐴 = (𝑥 / 𝑦) ∧ 𝐵 = (𝑧 / 𝑤)))) → ((𝑃↑(𝑃 pCnt 𝑦)) ∥ 𝑦 ↔ (𝑦 / (𝑃↑(𝑃 pCnt 𝑦))) ∈ ℤ)) |
161 | 156, 160 | mpbid 235 |
. . . . . . . . . . 11
⊢ ((((𝜑 ∧ 𝐵 ≠ 0) ∧ (𝑥 ∈ ℤ ∧ 𝑧 ∈ ℤ)) ∧ ((𝑦 ∈ ℕ ∧ 𝑤 ∈ ℕ) ∧ (𝐴 = (𝑥 / 𝑦) ∧ 𝐵 = (𝑧 / 𝑤)))) → (𝑦 / (𝑃↑(𝑃 pCnt 𝑦))) ∈ ℤ) |
162 | 50 | nnred 11731 |
. . . . . . . . . . . 12
⊢ ((((𝜑 ∧ 𝐵 ≠ 0) ∧ (𝑥 ∈ ℤ ∧ 𝑧 ∈ ℤ)) ∧ ((𝑦 ∈ ℕ ∧ 𝑤 ∈ ℕ) ∧ (𝐴 = (𝑥 / 𝑦) ∧ 𝐵 = (𝑧 / 𝑤)))) → 𝑦 ∈ ℝ) |
163 | 67 | nnred 11731 |
. . . . . . . . . . . 12
⊢ ((((𝜑 ∧ 𝐵 ≠ 0) ∧ (𝑥 ∈ ℤ ∧ 𝑧 ∈ ℤ)) ∧ ((𝑦 ∈ ℕ ∧ 𝑤 ∈ ℕ) ∧ (𝐴 = (𝑥 / 𝑦) ∧ 𝐵 = (𝑧 / 𝑤)))) → (𝑃↑(𝑃 pCnt 𝑦)) ∈ ℝ) |
164 | 50 | nngt0d 11765 |
. . . . . . . . . . . 12
⊢ ((((𝜑 ∧ 𝐵 ≠ 0) ∧ (𝑥 ∈ ℤ ∧ 𝑧 ∈ ℤ)) ∧ ((𝑦 ∈ ℕ ∧ 𝑤 ∈ ℕ) ∧ (𝐴 = (𝑥 / 𝑦) ∧ 𝐵 = (𝑧 / 𝑤)))) → 0 < 𝑦) |
165 | 67 | nngt0d 11765 |
. . . . . . . . . . . 12
⊢ ((((𝜑 ∧ 𝐵 ≠ 0) ∧ (𝑥 ∈ ℤ ∧ 𝑧 ∈ ℤ)) ∧ ((𝑦 ∈ ℕ ∧ 𝑤 ∈ ℕ) ∧ (𝐴 = (𝑥 / 𝑦) ∧ 𝐵 = (𝑧 / 𝑤)))) → 0 < (𝑃↑(𝑃 pCnt 𝑦))) |
166 | 162, 163,
164, 165 | divgt0d 11653 |
. . . . . . . . . . 11
⊢ ((((𝜑 ∧ 𝐵 ≠ 0) ∧ (𝑥 ∈ ℤ ∧ 𝑧 ∈ ℤ)) ∧ ((𝑦 ∈ ℕ ∧ 𝑤 ∈ ℕ) ∧ (𝐴 = (𝑥 / 𝑦) ∧ 𝐵 = (𝑧 / 𝑤)))) → 0 < (𝑦 / (𝑃↑(𝑃 pCnt 𝑦)))) |
167 | | elnnz 12072 |
. . . . . . . . . . 11
⊢ ((𝑦 / (𝑃↑(𝑃 pCnt 𝑦))) ∈ ℕ ↔ ((𝑦 / (𝑃↑(𝑃 pCnt 𝑦))) ∈ ℤ ∧ 0 < (𝑦 / (𝑃↑(𝑃 pCnt 𝑦))))) |
168 | 161, 166,
167 | sylanbrc 586 |
. . . . . . . . . 10
⊢ ((((𝜑 ∧ 𝐵 ≠ 0) ∧ (𝑥 ∈ ℤ ∧ 𝑧 ∈ ℤ)) ∧ ((𝑦 ∈ ℕ ∧ 𝑤 ∈ ℕ) ∧ (𝐴 = (𝑥 / 𝑦) ∧ 𝐵 = (𝑧 / 𝑤)))) → (𝑦 / (𝑃↑(𝑃 pCnt 𝑦))) ∈ ℕ) |
169 | | pcndvds2 16304 |
. . . . . . . . . . 11
⊢ ((𝑃 ∈ ℙ ∧ 𝑦 ∈ ℕ) → ¬
𝑃 ∥ (𝑦 / (𝑃↑(𝑃 pCnt 𝑦)))) |
170 | 22, 50, 169 | syl2anc 587 |
. . . . . . . . . 10
⊢ ((((𝜑 ∧ 𝐵 ≠ 0) ∧ (𝑥 ∈ ℤ ∧ 𝑧 ∈ ℤ)) ∧ ((𝑦 ∈ ℕ ∧ 𝑤 ∈ ℕ) ∧ (𝐴 = (𝑥 / 𝑦) ∧ 𝐵 = (𝑧 / 𝑤)))) → ¬ 𝑃 ∥ (𝑦 / (𝑃↑(𝑃 pCnt 𝑦)))) |
171 | 168, 170 | jca 515 |
. . . . . . . . 9
⊢ ((((𝜑 ∧ 𝐵 ≠ 0) ∧ (𝑥 ∈ ℤ ∧ 𝑧 ∈ ℤ)) ∧ ((𝑦 ∈ ℕ ∧ 𝑤 ∈ ℕ) ∧ (𝐴 = (𝑥 / 𝑦) ∧ 𝐵 = (𝑧 / 𝑤)))) → ((𝑦 / (𝑃↑(𝑃 pCnt 𝑦))) ∈ ℕ ∧ ¬ 𝑃 ∥ (𝑦 / (𝑃↑(𝑃 pCnt 𝑦))))) |
172 | | pczdvds 16299 |
. . . . . . . . . . . 12
⊢ ((𝑃 ∈ ℙ ∧ (𝑧 ∈ ℤ ∧ 𝑧 ≠ 0)) → (𝑃↑(𝑃 pCnt 𝑧)) ∥ 𝑧) |
173 | 22, 97, 108, 172 | syl12anc 836 |
. . . . . . . . . . 11
⊢ ((((𝜑 ∧ 𝐵 ≠ 0) ∧ (𝑥 ∈ ℤ ∧ 𝑧 ∈ ℤ)) ∧ ((𝑦 ∈ ℕ ∧ 𝑤 ∈ ℕ) ∧ (𝐴 = (𝑥 / 𝑦) ∧ 𝐵 = (𝑧 / 𝑤)))) → (𝑃↑(𝑃 pCnt 𝑧)) ∥ 𝑧) |
174 | 111 | nnzd 12167 |
. . . . . . . . . . . 12
⊢ ((((𝜑 ∧ 𝐵 ≠ 0) ∧ (𝑥 ∈ ℤ ∧ 𝑧 ∈ ℤ)) ∧ ((𝑦 ∈ ℕ ∧ 𝑤 ∈ ℕ) ∧ (𝐴 = (𝑥 / 𝑦) ∧ 𝐵 = (𝑧 / 𝑤)))) → (𝑃↑(𝑃 pCnt 𝑧)) ∈ ℤ) |
175 | | dvdsval2 15702 |
. . . . . . . . . . . 12
⊢ (((𝑃↑(𝑃 pCnt 𝑧)) ∈ ℤ ∧ (𝑃↑(𝑃 pCnt 𝑧)) ≠ 0 ∧ 𝑧 ∈ ℤ) → ((𝑃↑(𝑃 pCnt 𝑧)) ∥ 𝑧 ↔ (𝑧 / (𝑃↑(𝑃 pCnt 𝑧))) ∈ ℤ)) |
176 | 174, 119,
97, 175 | syl3anc 1372 |
. . . . . . . . . . 11
⊢ ((((𝜑 ∧ 𝐵 ≠ 0) ∧ (𝑥 ∈ ℤ ∧ 𝑧 ∈ ℤ)) ∧ ((𝑦 ∈ ℕ ∧ 𝑤 ∈ ℕ) ∧ (𝐴 = (𝑥 / 𝑦) ∧ 𝐵 = (𝑧 / 𝑤)))) → ((𝑃↑(𝑃 pCnt 𝑧)) ∥ 𝑧 ↔ (𝑧 / (𝑃↑(𝑃 pCnt 𝑧))) ∈ ℤ)) |
177 | 173, 176 | mpbid 235 |
. . . . . . . . . 10
⊢ ((((𝜑 ∧ 𝐵 ≠ 0) ∧ (𝑥 ∈ ℤ ∧ 𝑧 ∈ ℤ)) ∧ ((𝑦 ∈ ℕ ∧ 𝑤 ∈ ℕ) ∧ (𝐴 = (𝑥 / 𝑦) ∧ 𝐵 = (𝑧 / 𝑤)))) → (𝑧 / (𝑃↑(𝑃 pCnt 𝑧))) ∈ ℤ) |
178 | | pczndvds2 16303 |
. . . . . . . . . . 11
⊢ ((𝑃 ∈ ℙ ∧ (𝑧 ∈ ℤ ∧ 𝑧 ≠ 0)) → ¬ 𝑃 ∥ (𝑧 / (𝑃↑(𝑃 pCnt 𝑧)))) |
179 | 22, 97, 108, 178 | syl12anc 836 |
. . . . . . . . . 10
⊢ ((((𝜑 ∧ 𝐵 ≠ 0) ∧ (𝑥 ∈ ℤ ∧ 𝑧 ∈ ℤ)) ∧ ((𝑦 ∈ ℕ ∧ 𝑤 ∈ ℕ) ∧ (𝐴 = (𝑥 / 𝑦) ∧ 𝐵 = (𝑧 / 𝑤)))) → ¬ 𝑃 ∥ (𝑧 / (𝑃↑(𝑃 pCnt 𝑧)))) |
180 | 177, 179 | jca 515 |
. . . . . . . . 9
⊢ ((((𝜑 ∧ 𝐵 ≠ 0) ∧ (𝑥 ∈ ℤ ∧ 𝑧 ∈ ℤ)) ∧ ((𝑦 ∈ ℕ ∧ 𝑤 ∈ ℕ) ∧ (𝐴 = (𝑥 / 𝑦) ∧ 𝐵 = (𝑧 / 𝑤)))) → ((𝑧 / (𝑃↑(𝑃 pCnt 𝑧))) ∈ ℤ ∧ ¬ 𝑃 ∥ (𝑧 / (𝑃↑(𝑃 pCnt 𝑧))))) |
181 | | pcdvds 16300 |
. . . . . . . . . . . . 13
⊢ ((𝑃 ∈ ℙ ∧ 𝑤 ∈ ℕ) → (𝑃↑(𝑃 pCnt 𝑤)) ∥ 𝑤) |
182 | 22, 100, 181 | syl2anc 587 |
. . . . . . . . . . . 12
⊢ ((((𝜑 ∧ 𝐵 ≠ 0) ∧ (𝑥 ∈ ℤ ∧ 𝑧 ∈ ℤ)) ∧ ((𝑦 ∈ ℕ ∧ 𝑤 ∈ ℕ) ∧ (𝐴 = (𝑥 / 𝑦) ∧ 𝐵 = (𝑧 / 𝑤)))) → (𝑃↑(𝑃 pCnt 𝑤)) ∥ 𝑤) |
183 | 117 | nnzd 12167 |
. . . . . . . . . . . . 13
⊢ ((((𝜑 ∧ 𝐵 ≠ 0) ∧ (𝑥 ∈ ℤ ∧ 𝑧 ∈ ℤ)) ∧ ((𝑦 ∈ ℕ ∧ 𝑤 ∈ ℕ) ∧ (𝐴 = (𝑥 / 𝑦) ∧ 𝐵 = (𝑧 / 𝑤)))) → (𝑃↑(𝑃 pCnt 𝑤)) ∈ ℤ) |
184 | 100 | nnzd 12167 |
. . . . . . . . . . . . 13
⊢ ((((𝜑 ∧ 𝐵 ≠ 0) ∧ (𝑥 ∈ ℤ ∧ 𝑧 ∈ ℤ)) ∧ ((𝑦 ∈ ℕ ∧ 𝑤 ∈ ℕ) ∧ (𝐴 = (𝑥 / 𝑦) ∧ 𝐵 = (𝑧 / 𝑤)))) → 𝑤 ∈ ℤ) |
185 | | dvdsval2 15702 |
. . . . . . . . . . . . 13
⊢ (((𝑃↑(𝑃 pCnt 𝑤)) ∈ ℤ ∧ (𝑃↑(𝑃 pCnt 𝑤)) ≠ 0 ∧ 𝑤 ∈ ℤ) → ((𝑃↑(𝑃 pCnt 𝑤)) ∥ 𝑤 ↔ (𝑤 / (𝑃↑(𝑃 pCnt 𝑤))) ∈ ℤ)) |
186 | 183, 120,
184, 185 | syl3anc 1372 |
. . . . . . . . . . . 12
⊢ ((((𝜑 ∧ 𝐵 ≠ 0) ∧ (𝑥 ∈ ℤ ∧ 𝑧 ∈ ℤ)) ∧ ((𝑦 ∈ ℕ ∧ 𝑤 ∈ ℕ) ∧ (𝐴 = (𝑥 / 𝑦) ∧ 𝐵 = (𝑧 / 𝑤)))) → ((𝑃↑(𝑃 pCnt 𝑤)) ∥ 𝑤 ↔ (𝑤 / (𝑃↑(𝑃 pCnt 𝑤))) ∈ ℤ)) |
187 | 182, 186 | mpbid 235 |
. . . . . . . . . . 11
⊢ ((((𝜑 ∧ 𝐵 ≠ 0) ∧ (𝑥 ∈ ℤ ∧ 𝑧 ∈ ℤ)) ∧ ((𝑦 ∈ ℕ ∧ 𝑤 ∈ ℕ) ∧ (𝐴 = (𝑥 / 𝑦) ∧ 𝐵 = (𝑧 / 𝑤)))) → (𝑤 / (𝑃↑(𝑃 pCnt 𝑤))) ∈ ℤ) |
188 | 100 | nnred 11731 |
. . . . . . . . . . . 12
⊢ ((((𝜑 ∧ 𝐵 ≠ 0) ∧ (𝑥 ∈ ℤ ∧ 𝑧 ∈ ℤ)) ∧ ((𝑦 ∈ ℕ ∧ 𝑤 ∈ ℕ) ∧ (𝐴 = (𝑥 / 𝑦) ∧ 𝐵 = (𝑧 / 𝑤)))) → 𝑤 ∈ ℝ) |
189 | 117 | nnred 11731 |
. . . . . . . . . . . 12
⊢ ((((𝜑 ∧ 𝐵 ≠ 0) ∧ (𝑥 ∈ ℤ ∧ 𝑧 ∈ ℤ)) ∧ ((𝑦 ∈ ℕ ∧ 𝑤 ∈ ℕ) ∧ (𝐴 = (𝑥 / 𝑦) ∧ 𝐵 = (𝑧 / 𝑤)))) → (𝑃↑(𝑃 pCnt 𝑤)) ∈ ℝ) |
190 | 100 | nngt0d 11765 |
. . . . . . . . . . . 12
⊢ ((((𝜑 ∧ 𝐵 ≠ 0) ∧ (𝑥 ∈ ℤ ∧ 𝑧 ∈ ℤ)) ∧ ((𝑦 ∈ ℕ ∧ 𝑤 ∈ ℕ) ∧ (𝐴 = (𝑥 / 𝑦) ∧ 𝐵 = (𝑧 / 𝑤)))) → 0 < 𝑤) |
191 | 117 | nngt0d 11765 |
. . . . . . . . . . . 12
⊢ ((((𝜑 ∧ 𝐵 ≠ 0) ∧ (𝑥 ∈ ℤ ∧ 𝑧 ∈ ℤ)) ∧ ((𝑦 ∈ ℕ ∧ 𝑤 ∈ ℕ) ∧ (𝐴 = (𝑥 / 𝑦) ∧ 𝐵 = (𝑧 / 𝑤)))) → 0 < (𝑃↑(𝑃 pCnt 𝑤))) |
192 | 188, 189,
190, 191 | divgt0d 11653 |
. . . . . . . . . . 11
⊢ ((((𝜑 ∧ 𝐵 ≠ 0) ∧ (𝑥 ∈ ℤ ∧ 𝑧 ∈ ℤ)) ∧ ((𝑦 ∈ ℕ ∧ 𝑤 ∈ ℕ) ∧ (𝐴 = (𝑥 / 𝑦) ∧ 𝐵 = (𝑧 / 𝑤)))) → 0 < (𝑤 / (𝑃↑(𝑃 pCnt 𝑤)))) |
193 | | elnnz 12072 |
. . . . . . . . . . 11
⊢ ((𝑤 / (𝑃↑(𝑃 pCnt 𝑤))) ∈ ℕ ↔ ((𝑤 / (𝑃↑(𝑃 pCnt 𝑤))) ∈ ℤ ∧ 0 < (𝑤 / (𝑃↑(𝑃 pCnt 𝑤))))) |
194 | 187, 192,
193 | sylanbrc 586 |
. . . . . . . . . 10
⊢ ((((𝜑 ∧ 𝐵 ≠ 0) ∧ (𝑥 ∈ ℤ ∧ 𝑧 ∈ ℤ)) ∧ ((𝑦 ∈ ℕ ∧ 𝑤 ∈ ℕ) ∧ (𝐴 = (𝑥 / 𝑦) ∧ 𝐵 = (𝑧 / 𝑤)))) → (𝑤 / (𝑃↑(𝑃 pCnt 𝑤))) ∈ ℕ) |
195 | | pcndvds2 16304 |
. . . . . . . . . . 11
⊢ ((𝑃 ∈ ℙ ∧ 𝑤 ∈ ℕ) → ¬
𝑃 ∥ (𝑤 / (𝑃↑(𝑃 pCnt 𝑤)))) |
196 | 22, 100, 195 | syl2anc 587 |
. . . . . . . . . 10
⊢ ((((𝜑 ∧ 𝐵 ≠ 0) ∧ (𝑥 ∈ ℤ ∧ 𝑧 ∈ ℤ)) ∧ ((𝑦 ∈ ℕ ∧ 𝑤 ∈ ℕ) ∧ (𝐴 = (𝑥 / 𝑦) ∧ 𝐵 = (𝑧 / 𝑤)))) → ¬ 𝑃 ∥ (𝑤 / (𝑃↑(𝑃 pCnt 𝑤)))) |
197 | 194, 196 | jca 515 |
. . . . . . . . 9
⊢ ((((𝜑 ∧ 𝐵 ≠ 0) ∧ (𝑥 ∈ ℤ ∧ 𝑧 ∈ ℤ)) ∧ ((𝑦 ∈ ℕ ∧ 𝑤 ∈ ℕ) ∧ (𝐴 = (𝑥 / 𝑦) ∧ 𝐵 = (𝑧 / 𝑤)))) → ((𝑤 / (𝑃↑(𝑃 pCnt 𝑤))) ∈ ℕ ∧ ¬ 𝑃 ∥ (𝑤 / (𝑃↑(𝑃 pCnt 𝑤))))) |
198 | 22, 96, 142, 145, 154, 171, 180, 197 | pcaddlem 16324 |
. . . . . . . 8
⊢ ((((𝜑 ∧ 𝐵 ≠ 0) ∧ (𝑥 ∈ ℤ ∧ 𝑧 ∈ ℤ)) ∧ ((𝑦 ∈ ℕ ∧ 𝑤 ∈ ℕ) ∧ (𝐴 = (𝑥 / 𝑦) ∧ 𝐵 = (𝑧 / 𝑤)))) → (𝑃 pCnt 𝐴) ≤ (𝑃 pCnt (𝐴 + 𝐵))) |
199 | 198 | expr 460 |
. . . . . . 7
⊢ ((((𝜑 ∧ 𝐵 ≠ 0) ∧ (𝑥 ∈ ℤ ∧ 𝑧 ∈ ℤ)) ∧ (𝑦 ∈ ℕ ∧ 𝑤 ∈ ℕ)) → ((𝐴 = (𝑥 / 𝑦) ∧ 𝐵 = (𝑧 / 𝑤)) → (𝑃 pCnt 𝐴) ≤ (𝑃 pCnt (𝐴 + 𝐵)))) |
200 | 199 | rexlimdvva 3204 |
. . . . . 6
⊢ (((𝜑 ∧ 𝐵 ≠ 0) ∧ (𝑥 ∈ ℤ ∧ 𝑧 ∈ ℤ)) → (∃𝑦 ∈ ℕ ∃𝑤 ∈ ℕ (𝐴 = (𝑥 / 𝑦) ∧ 𝐵 = (𝑧 / 𝑤)) → (𝑃 pCnt 𝐴) ≤ (𝑃 pCnt (𝐴 + 𝐵)))) |
201 | 21, 200 | syl5bir 246 |
. . . . 5
⊢ (((𝜑 ∧ 𝐵 ≠ 0) ∧ (𝑥 ∈ ℤ ∧ 𝑧 ∈ ℤ)) → ((∃𝑦 ∈ ℕ 𝐴 = (𝑥 / 𝑦) ∧ ∃𝑤 ∈ ℕ 𝐵 = (𝑧 / 𝑤)) → (𝑃 pCnt 𝐴) ≤ (𝑃 pCnt (𝐴 + 𝐵)))) |
202 | 201 | rexlimdvva 3204 |
. . . 4
⊢ ((𝜑 ∧ 𝐵 ≠ 0) → (∃𝑥 ∈ ℤ ∃𝑧 ∈ ℤ (∃𝑦 ∈ ℕ 𝐴 = (𝑥 / 𝑦) ∧ ∃𝑤 ∈ ℕ 𝐵 = (𝑧 / 𝑤)) → (𝑃 pCnt 𝐴) ≤ (𝑃 pCnt (𝐴 + 𝐵)))) |
203 | 20, 202 | syl5bir 246 |
. . 3
⊢ ((𝜑 ∧ 𝐵 ≠ 0) → ((∃𝑥 ∈ ℤ ∃𝑦 ∈ ℕ 𝐴 = (𝑥 / 𝑦) ∧ ∃𝑧 ∈ ℤ ∃𝑤 ∈ ℕ 𝐵 = (𝑧 / 𝑤)) → (𝑃 pCnt 𝐴) ≤ (𝑃 pCnt (𝐴 + 𝐵)))) |
204 | 19, 203 | pm2.61dane 3021 |
. 2
⊢ (𝜑 → ((∃𝑥 ∈ ℤ ∃𝑦 ∈ ℕ 𝐴 = (𝑥 / 𝑦) ∧ ∃𝑧 ∈ ℤ ∃𝑤 ∈ ℕ 𝐵 = (𝑧 / 𝑤)) → (𝑃 pCnt 𝐴) ≤ (𝑃 pCnt (𝐴 + 𝐵)))) |
205 | 3, 6, 204 | mp2and 699 |
1
⊢ (𝜑 → (𝑃 pCnt 𝐴) ≤ (𝑃 pCnt (𝐴 + 𝐵))) |