Proof of Theorem pcdvdstr
Step | Hyp | Ref
| Expression |
1 | | 0z 11265 |
. . . . . . 7
⊢ 0 ∈
ℤ |
2 | | zq 11670 |
. . . . . . 7
⊢ (0 ∈
ℤ → 0 ∈ ℚ) |
3 | 1, 2 | ax-mp 5 |
. . . . . 6
⊢ 0 ∈
ℚ |
4 | | pcxcl 15403 |
. . . . . 6
⊢ ((𝑃 ∈ ℙ ∧ 0 ∈
ℚ) → (𝑃 pCnt 0)
∈ ℝ*) |
5 | 3, 4 | mpan2 703 |
. . . . 5
⊢ (𝑃 ∈ ℙ → (𝑃 pCnt 0) ∈
ℝ*) |
6 | | xrleid 11859 |
. . . . 5
⊢ ((𝑃 pCnt 0) ∈
ℝ* → (𝑃 pCnt 0) ≤ (𝑃 pCnt 0)) |
7 | 5, 6 | syl 17 |
. . . 4
⊢ (𝑃 ∈ ℙ → (𝑃 pCnt 0) ≤ (𝑃 pCnt 0)) |
8 | 7 | ad2antrr 758 |
. . 3
⊢ (((𝑃 ∈ ℙ ∧ (𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝐴 ∥ 𝐵)) ∧ 𝐴 = 0) → (𝑃 pCnt 0) ≤ (𝑃 pCnt 0)) |
9 | | simpr 476 |
. . . 4
⊢ (((𝑃 ∈ ℙ ∧ (𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝐴 ∥ 𝐵)) ∧ 𝐴 = 0) → 𝐴 = 0) |
10 | 9 | oveq2d 6565 |
. . 3
⊢ (((𝑃 ∈ ℙ ∧ (𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝐴 ∥ 𝐵)) ∧ 𝐴 = 0) → (𝑃 pCnt 𝐴) = (𝑃 pCnt 0)) |
11 | | simplr3 1098 |
. . . . . 6
⊢ (((𝑃 ∈ ℙ ∧ (𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝐴 ∥ 𝐵)) ∧ 𝐴 = 0) → 𝐴 ∥ 𝐵) |
12 | 9, 11 | eqbrtrrd 4607 |
. . . . 5
⊢ (((𝑃 ∈ ℙ ∧ (𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝐴 ∥ 𝐵)) ∧ 𝐴 = 0) → 0 ∥ 𝐵) |
13 | | simplr2 1097 |
. . . . . 6
⊢ (((𝑃 ∈ ℙ ∧ (𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝐴 ∥ 𝐵)) ∧ 𝐴 = 0) → 𝐵 ∈ ℤ) |
14 | | 0dvds 14840 |
. . . . . 6
⊢ (𝐵 ∈ ℤ → (0
∥ 𝐵 ↔ 𝐵 = 0)) |
15 | 13, 14 | syl 17 |
. . . . 5
⊢ (((𝑃 ∈ ℙ ∧ (𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝐴 ∥ 𝐵)) ∧ 𝐴 = 0) → (0 ∥ 𝐵 ↔ 𝐵 = 0)) |
16 | 12, 15 | mpbid 221 |
. . . 4
⊢ (((𝑃 ∈ ℙ ∧ (𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝐴 ∥ 𝐵)) ∧ 𝐴 = 0) → 𝐵 = 0) |
17 | 16 | oveq2d 6565 |
. . 3
⊢ (((𝑃 ∈ ℙ ∧ (𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝐴 ∥ 𝐵)) ∧ 𝐴 = 0) → (𝑃 pCnt 𝐵) = (𝑃 pCnt 0)) |
18 | 8, 10, 17 | 3brtr4d 4615 |
. 2
⊢ (((𝑃 ∈ ℙ ∧ (𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝐴 ∥ 𝐵)) ∧ 𝐴 = 0) → (𝑃 pCnt 𝐴) ≤ (𝑃 pCnt 𝐵)) |
19 | | simpll 786 |
. . . . 5
⊢ (((𝑃 ∈ ℙ ∧ (𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝐴 ∥ 𝐵)) ∧ 𝐴 ≠ 0) → 𝑃 ∈ ℙ) |
20 | | simplr1 1096 |
. . . . 5
⊢ (((𝑃 ∈ ℙ ∧ (𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝐴 ∥ 𝐵)) ∧ 𝐴 ≠ 0) → 𝐴 ∈ ℤ) |
21 | | simpr 476 |
. . . . 5
⊢ (((𝑃 ∈ ℙ ∧ (𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝐴 ∥ 𝐵)) ∧ 𝐴 ≠ 0) → 𝐴 ≠ 0) |
22 | | pczdvds 15405 |
. . . . 5
⊢ ((𝑃 ∈ ℙ ∧ (𝐴 ∈ ℤ ∧ 𝐴 ≠ 0)) → (𝑃↑(𝑃 pCnt 𝐴)) ∥ 𝐴) |
23 | 19, 20, 21, 22 | syl12anc 1316 |
. . . 4
⊢ (((𝑃 ∈ ℙ ∧ (𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝐴 ∥ 𝐵)) ∧ 𝐴 ≠ 0) → (𝑃↑(𝑃 pCnt 𝐴)) ∥ 𝐴) |
24 | | simplr3 1098 |
. . . 4
⊢ (((𝑃 ∈ ℙ ∧ (𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝐴 ∥ 𝐵)) ∧ 𝐴 ≠ 0) → 𝐴 ∥ 𝐵) |
25 | | prmnn 15226 |
. . . . . . . 8
⊢ (𝑃 ∈ ℙ → 𝑃 ∈
ℕ) |
26 | 19, 25 | syl 17 |
. . . . . . 7
⊢ (((𝑃 ∈ ℙ ∧ (𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝐴 ∥ 𝐵)) ∧ 𝐴 ≠ 0) → 𝑃 ∈ ℕ) |
27 | | pczcl 15391 |
. . . . . . . 8
⊢ ((𝑃 ∈ ℙ ∧ (𝐴 ∈ ℤ ∧ 𝐴 ≠ 0)) → (𝑃 pCnt 𝐴) ∈
ℕ0) |
28 | 19, 20, 21, 27 | syl12anc 1316 |
. . . . . . 7
⊢ (((𝑃 ∈ ℙ ∧ (𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝐴 ∥ 𝐵)) ∧ 𝐴 ≠ 0) → (𝑃 pCnt 𝐴) ∈
ℕ0) |
29 | 26, 28 | nnexpcld 12892 |
. . . . . 6
⊢ (((𝑃 ∈ ℙ ∧ (𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝐴 ∥ 𝐵)) ∧ 𝐴 ≠ 0) → (𝑃↑(𝑃 pCnt 𝐴)) ∈ ℕ) |
30 | 29 | nnzd 11357 |
. . . . 5
⊢ (((𝑃 ∈ ℙ ∧ (𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝐴 ∥ 𝐵)) ∧ 𝐴 ≠ 0) → (𝑃↑(𝑃 pCnt 𝐴)) ∈ ℤ) |
31 | | simplr2 1097 |
. . . . 5
⊢ (((𝑃 ∈ ℙ ∧ (𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝐴 ∥ 𝐵)) ∧ 𝐴 ≠ 0) → 𝐵 ∈ ℤ) |
32 | | dvdstr 14856 |
. . . . 5
⊢ (((𝑃↑(𝑃 pCnt 𝐴)) ∈ ℤ ∧ 𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → (((𝑃↑(𝑃 pCnt 𝐴)) ∥ 𝐴 ∧ 𝐴 ∥ 𝐵) → (𝑃↑(𝑃 pCnt 𝐴)) ∥ 𝐵)) |
33 | 30, 20, 31, 32 | syl3anc 1318 |
. . . 4
⊢ (((𝑃 ∈ ℙ ∧ (𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝐴 ∥ 𝐵)) ∧ 𝐴 ≠ 0) → (((𝑃↑(𝑃 pCnt 𝐴)) ∥ 𝐴 ∧ 𝐴 ∥ 𝐵) → (𝑃↑(𝑃 pCnt 𝐴)) ∥ 𝐵)) |
34 | 23, 24, 33 | mp2and 711 |
. . 3
⊢ (((𝑃 ∈ ℙ ∧ (𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝐴 ∥ 𝐵)) ∧ 𝐴 ≠ 0) → (𝑃↑(𝑃 pCnt 𝐴)) ∥ 𝐵) |
35 | | pcdvdsb 15411 |
. . . 4
⊢ ((𝑃 ∈ ℙ ∧ 𝐵 ∈ ℤ ∧ (𝑃 pCnt 𝐴) ∈ ℕ0) → ((𝑃 pCnt 𝐴) ≤ (𝑃 pCnt 𝐵) ↔ (𝑃↑(𝑃 pCnt 𝐴)) ∥ 𝐵)) |
36 | 19, 31, 28, 35 | syl3anc 1318 |
. . 3
⊢ (((𝑃 ∈ ℙ ∧ (𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝐴 ∥ 𝐵)) ∧ 𝐴 ≠ 0) → ((𝑃 pCnt 𝐴) ≤ (𝑃 pCnt 𝐵) ↔ (𝑃↑(𝑃 pCnt 𝐴)) ∥ 𝐵)) |
37 | 34, 36 | mpbird 246 |
. 2
⊢ (((𝑃 ∈ ℙ ∧ (𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝐴 ∥ 𝐵)) ∧ 𝐴 ≠ 0) → (𝑃 pCnt 𝐴) ≤ (𝑃 pCnt 𝐵)) |
38 | 18, 37 | pm2.61dane 2869 |
1
⊢ ((𝑃 ∈ ℙ ∧ (𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝐴 ∥ 𝐵)) → (𝑃 pCnt 𝐴) ≤ (𝑃 pCnt 𝐵)) |