| Step | Hyp | Ref | Expression | 
|---|
| 1 |  | zq 12996 | . . 3
⊢ (𝑁 ∈ ℤ → 𝑁 ∈
ℚ) | 
| 2 |  | eqid 2737 | . . . 4
⊢
sup({𝑛 ∈
ℕ0 ∣ (𝑃↑𝑛) ∥ 𝑥}, ℝ, < ) = sup({𝑛 ∈ ℕ0 ∣ (𝑃↑𝑛) ∥ 𝑥}, ℝ, < ) | 
| 3 |  | eqid 2737 | . . . 4
⊢
sup({𝑛 ∈
ℕ0 ∣ (𝑃↑𝑛) ∥ 𝑦}, ℝ, < ) = sup({𝑛 ∈ ℕ0 ∣ (𝑃↑𝑛) ∥ 𝑦}, ℝ, < ) | 
| 4 | 2, 3 | pcval 16882 | . . 3
⊢ ((𝑃 ∈ ℙ ∧ (𝑁 ∈ ℚ ∧ 𝑁 ≠ 0)) → (𝑃 pCnt 𝑁) = (℩𝑧∃𝑥 ∈ ℤ ∃𝑦 ∈ ℕ (𝑁 = (𝑥 / 𝑦) ∧ 𝑧 = (sup({𝑛 ∈ ℕ0 ∣ (𝑃↑𝑛) ∥ 𝑥}, ℝ, < ) − sup({𝑛 ∈ ℕ0
∣ (𝑃↑𝑛) ∥ 𝑦}, ℝ, < ))))) | 
| 5 | 1, 4 | sylanr1 682 | . 2
⊢ ((𝑃 ∈ ℙ ∧ (𝑁 ∈ ℤ ∧ 𝑁 ≠ 0)) → (𝑃 pCnt 𝑁) = (℩𝑧∃𝑥 ∈ ℤ ∃𝑦 ∈ ℕ (𝑁 = (𝑥 / 𝑦) ∧ 𝑧 = (sup({𝑛 ∈ ℕ0 ∣ (𝑃↑𝑛) ∥ 𝑥}, ℝ, < ) − sup({𝑛 ∈ ℕ0
∣ (𝑃↑𝑛) ∥ 𝑦}, ℝ, < ))))) | 
| 6 |  | simprl 771 | . . . 4
⊢ ((𝑃 ∈ ℙ ∧ (𝑁 ∈ ℤ ∧ 𝑁 ≠ 0)) → 𝑁 ∈
ℤ) | 
| 7 | 6 | zcnd 12723 | . . . . . 6
⊢ ((𝑃 ∈ ℙ ∧ (𝑁 ∈ ℤ ∧ 𝑁 ≠ 0)) → 𝑁 ∈
ℂ) | 
| 8 | 7 | div1d 12035 | . . . . 5
⊢ ((𝑃 ∈ ℙ ∧ (𝑁 ∈ ℤ ∧ 𝑁 ≠ 0)) → (𝑁 / 1) = 𝑁) | 
| 9 | 8 | eqcomd 2743 | . . . 4
⊢ ((𝑃 ∈ ℙ ∧ (𝑁 ∈ ℤ ∧ 𝑁 ≠ 0)) → 𝑁 = (𝑁 / 1)) | 
| 10 |  | prmuz2 16733 | . . . . . . . 8
⊢ (𝑃 ∈ ℙ → 𝑃 ∈
(ℤ≥‘2)) | 
| 11 |  | eqid 2737 | . . . . . . . 8
⊢ 1 =
1 | 
| 12 |  | eqid 2737 | . . . . . . . . 9
⊢ {𝑛 ∈ ℕ0
∣ (𝑃↑𝑛) ∥ 1} = {𝑛 ∈ ℕ0
∣ (𝑃↑𝑛) ∥ 1} | 
| 13 |  | eqid 2737 | . . . . . . . . 9
⊢
sup({𝑛 ∈
ℕ0 ∣ (𝑃↑𝑛) ∥ 1}, ℝ, < ) = sup({𝑛 ∈ ℕ0
∣ (𝑃↑𝑛) ∥ 1}, ℝ, <
) | 
| 14 | 12, 13 | pcpre1 16880 | . . . . . . . 8
⊢ ((𝑃 ∈
(ℤ≥‘2) ∧ 1 = 1) → sup({𝑛 ∈ ℕ0 ∣ (𝑃↑𝑛) ∥ 1}, ℝ, < ) =
0) | 
| 15 | 10, 11, 14 | sylancl 586 | . . . . . . 7
⊢ (𝑃 ∈ ℙ →
sup({𝑛 ∈
ℕ0 ∣ (𝑃↑𝑛) ∥ 1}, ℝ, < ) =
0) | 
| 16 | 15 | adantr 480 | . . . . . 6
⊢ ((𝑃 ∈ ℙ ∧ (𝑁 ∈ ℤ ∧ 𝑁 ≠ 0)) → sup({𝑛 ∈ ℕ0
∣ (𝑃↑𝑛) ∥ 1}, ℝ, < ) =
0) | 
| 17 | 16 | oveq2d 7447 | . . . . 5
⊢ ((𝑃 ∈ ℙ ∧ (𝑁 ∈ ℤ ∧ 𝑁 ≠ 0)) → (𝑆 − sup({𝑛 ∈ ℕ0 ∣ (𝑃↑𝑛) ∥ 1}, ℝ, < )) = (𝑆 − 0)) | 
| 18 |  | eqid 2737 | . . . . . . . . . 10
⊢ {𝑛 ∈ ℕ0
∣ (𝑃↑𝑛) ∥ 𝑁} = {𝑛 ∈ ℕ0 ∣ (𝑃↑𝑛) ∥ 𝑁} | 
| 19 |  | pczpre.1 | . . . . . . . . . 10
⊢ 𝑆 = sup({𝑛 ∈ ℕ0 ∣ (𝑃↑𝑛) ∥ 𝑁}, ℝ, < ) | 
| 20 | 18, 19 | pcprecl 16877 | . . . . . . . . 9
⊢ ((𝑃 ∈
(ℤ≥‘2) ∧ (𝑁 ∈ ℤ ∧ 𝑁 ≠ 0)) → (𝑆 ∈ ℕ0 ∧ (𝑃↑𝑆) ∥ 𝑁)) | 
| 21 | 10, 20 | sylan 580 | . . . . . . . 8
⊢ ((𝑃 ∈ ℙ ∧ (𝑁 ∈ ℤ ∧ 𝑁 ≠ 0)) → (𝑆 ∈ ℕ0
∧ (𝑃↑𝑆) ∥ 𝑁)) | 
| 22 | 21 | simpld 494 | . . . . . . 7
⊢ ((𝑃 ∈ ℙ ∧ (𝑁 ∈ ℤ ∧ 𝑁 ≠ 0)) → 𝑆 ∈
ℕ0) | 
| 23 | 22 | nn0cnd 12589 | . . . . . 6
⊢ ((𝑃 ∈ ℙ ∧ (𝑁 ∈ ℤ ∧ 𝑁 ≠ 0)) → 𝑆 ∈
ℂ) | 
| 24 | 23 | subid1d 11609 | . . . . 5
⊢ ((𝑃 ∈ ℙ ∧ (𝑁 ∈ ℤ ∧ 𝑁 ≠ 0)) → (𝑆 − 0) = 𝑆) | 
| 25 | 17, 24 | eqtr2d 2778 | . . . 4
⊢ ((𝑃 ∈ ℙ ∧ (𝑁 ∈ ℤ ∧ 𝑁 ≠ 0)) → 𝑆 = (𝑆 − sup({𝑛 ∈ ℕ0 ∣ (𝑃↑𝑛) ∥ 1}, ℝ, <
))) | 
| 26 |  | 1nn 12277 | . . . . 5
⊢ 1 ∈
ℕ | 
| 27 |  | oveq1 7438 | . . . . . . . 8
⊢ (𝑥 = 𝑁 → (𝑥 / 𝑦) = (𝑁 / 𝑦)) | 
| 28 | 27 | eqeq2d 2748 | . . . . . . 7
⊢ (𝑥 = 𝑁 → (𝑁 = (𝑥 / 𝑦) ↔ 𝑁 = (𝑁 / 𝑦))) | 
| 29 |  | breq2 5147 | . . . . . . . . . . . 12
⊢ (𝑥 = 𝑁 → ((𝑃↑𝑛) ∥ 𝑥 ↔ (𝑃↑𝑛) ∥ 𝑁)) | 
| 30 | 29 | rabbidv 3444 | . . . . . . . . . . 11
⊢ (𝑥 = 𝑁 → {𝑛 ∈ ℕ0 ∣ (𝑃↑𝑛) ∥ 𝑥} = {𝑛 ∈ ℕ0 ∣ (𝑃↑𝑛) ∥ 𝑁}) | 
| 31 | 30 | supeq1d 9486 | . . . . . . . . . 10
⊢ (𝑥 = 𝑁 → sup({𝑛 ∈ ℕ0 ∣ (𝑃↑𝑛) ∥ 𝑥}, ℝ, < ) = sup({𝑛 ∈ ℕ0 ∣ (𝑃↑𝑛) ∥ 𝑁}, ℝ, < )) | 
| 32 | 31, 19 | eqtr4di 2795 | . . . . . . . . 9
⊢ (𝑥 = 𝑁 → sup({𝑛 ∈ ℕ0 ∣ (𝑃↑𝑛) ∥ 𝑥}, ℝ, < ) = 𝑆) | 
| 33 | 32 | oveq1d 7446 | . . . . . . . 8
⊢ (𝑥 = 𝑁 → (sup({𝑛 ∈ ℕ0 ∣ (𝑃↑𝑛) ∥ 𝑥}, ℝ, < ) − sup({𝑛 ∈ ℕ0
∣ (𝑃↑𝑛) ∥ 𝑦}, ℝ, < )) = (𝑆 − sup({𝑛 ∈ ℕ0 ∣ (𝑃↑𝑛) ∥ 𝑦}, ℝ, < ))) | 
| 34 | 33 | eqeq2d 2748 | . . . . . . 7
⊢ (𝑥 = 𝑁 → (𝑆 = (sup({𝑛 ∈ ℕ0 ∣ (𝑃↑𝑛) ∥ 𝑥}, ℝ, < ) − sup({𝑛 ∈ ℕ0
∣ (𝑃↑𝑛) ∥ 𝑦}, ℝ, < )) ↔ 𝑆 = (𝑆 − sup({𝑛 ∈ ℕ0 ∣ (𝑃↑𝑛) ∥ 𝑦}, ℝ, < )))) | 
| 35 | 28, 34 | anbi12d 632 | . . . . . 6
⊢ (𝑥 = 𝑁 → ((𝑁 = (𝑥 / 𝑦) ∧ 𝑆 = (sup({𝑛 ∈ ℕ0 ∣ (𝑃↑𝑛) ∥ 𝑥}, ℝ, < ) − sup({𝑛 ∈ ℕ0
∣ (𝑃↑𝑛) ∥ 𝑦}, ℝ, < ))) ↔ (𝑁 = (𝑁 / 𝑦) ∧ 𝑆 = (𝑆 − sup({𝑛 ∈ ℕ0 ∣ (𝑃↑𝑛) ∥ 𝑦}, ℝ, < ))))) | 
| 36 |  | oveq2 7439 | . . . . . . . 8
⊢ (𝑦 = 1 → (𝑁 / 𝑦) = (𝑁 / 1)) | 
| 37 | 36 | eqeq2d 2748 | . . . . . . 7
⊢ (𝑦 = 1 → (𝑁 = (𝑁 / 𝑦) ↔ 𝑁 = (𝑁 / 1))) | 
| 38 |  | breq2 5147 | . . . . . . . . . . 11
⊢ (𝑦 = 1 → ((𝑃↑𝑛) ∥ 𝑦 ↔ (𝑃↑𝑛) ∥ 1)) | 
| 39 | 38 | rabbidv 3444 | . . . . . . . . . 10
⊢ (𝑦 = 1 → {𝑛 ∈ ℕ0 ∣ (𝑃↑𝑛) ∥ 𝑦} = {𝑛 ∈ ℕ0 ∣ (𝑃↑𝑛) ∥ 1}) | 
| 40 | 39 | supeq1d 9486 | . . . . . . . . 9
⊢ (𝑦 = 1 → sup({𝑛 ∈ ℕ0
∣ (𝑃↑𝑛) ∥ 𝑦}, ℝ, < ) = sup({𝑛 ∈ ℕ0 ∣ (𝑃↑𝑛) ∥ 1}, ℝ, <
)) | 
| 41 | 40 | oveq2d 7447 | . . . . . . . 8
⊢ (𝑦 = 1 → (𝑆 − sup({𝑛 ∈ ℕ0 ∣ (𝑃↑𝑛) ∥ 𝑦}, ℝ, < )) = (𝑆 − sup({𝑛 ∈ ℕ0 ∣ (𝑃↑𝑛) ∥ 1}, ℝ, <
))) | 
| 42 | 41 | eqeq2d 2748 | . . . . . . 7
⊢ (𝑦 = 1 → (𝑆 = (𝑆 − sup({𝑛 ∈ ℕ0 ∣ (𝑃↑𝑛) ∥ 𝑦}, ℝ, < )) ↔ 𝑆 = (𝑆 − sup({𝑛 ∈ ℕ0 ∣ (𝑃↑𝑛) ∥ 1}, ℝ, <
)))) | 
| 43 | 37, 42 | anbi12d 632 | . . . . . 6
⊢ (𝑦 = 1 → ((𝑁 = (𝑁 / 𝑦) ∧ 𝑆 = (𝑆 − sup({𝑛 ∈ ℕ0 ∣ (𝑃↑𝑛) ∥ 𝑦}, ℝ, < ))) ↔ (𝑁 = (𝑁 / 1) ∧ 𝑆 = (𝑆 − sup({𝑛 ∈ ℕ0 ∣ (𝑃↑𝑛) ∥ 1}, ℝ, <
))))) | 
| 44 | 35, 43 | rspc2ev 3635 | . . . . 5
⊢ ((𝑁 ∈ ℤ ∧ 1 ∈
ℕ ∧ (𝑁 = (𝑁 / 1) ∧ 𝑆 = (𝑆 − sup({𝑛 ∈ ℕ0 ∣ (𝑃↑𝑛) ∥ 1}, ℝ, < )))) →
∃𝑥 ∈ ℤ
∃𝑦 ∈ ℕ
(𝑁 = (𝑥 / 𝑦) ∧ 𝑆 = (sup({𝑛 ∈ ℕ0 ∣ (𝑃↑𝑛) ∥ 𝑥}, ℝ, < ) − sup({𝑛 ∈ ℕ0
∣ (𝑃↑𝑛) ∥ 𝑦}, ℝ, < )))) | 
| 45 | 26, 44 | mp3an2 1451 | . . . 4
⊢ ((𝑁 ∈ ℤ ∧ (𝑁 = (𝑁 / 1) ∧ 𝑆 = (𝑆 − sup({𝑛 ∈ ℕ0 ∣ (𝑃↑𝑛) ∥ 1}, ℝ, < )))) →
∃𝑥 ∈ ℤ
∃𝑦 ∈ ℕ
(𝑁 = (𝑥 / 𝑦) ∧ 𝑆 = (sup({𝑛 ∈ ℕ0 ∣ (𝑃↑𝑛) ∥ 𝑥}, ℝ, < ) − sup({𝑛 ∈ ℕ0
∣ (𝑃↑𝑛) ∥ 𝑦}, ℝ, < )))) | 
| 46 | 6, 9, 25, 45 | syl12anc 837 | . . 3
⊢ ((𝑃 ∈ ℙ ∧ (𝑁 ∈ ℤ ∧ 𝑁 ≠ 0)) → ∃𝑥 ∈ ℤ ∃𝑦 ∈ ℕ (𝑁 = (𝑥 / 𝑦) ∧ 𝑆 = (sup({𝑛 ∈ ℕ0 ∣ (𝑃↑𝑛) ∥ 𝑥}, ℝ, < ) − sup({𝑛 ∈ ℕ0
∣ (𝑃↑𝑛) ∥ 𝑦}, ℝ, < )))) | 
| 47 |  | ltso 11341 | . . . . . 6
⊢  < Or
ℝ | 
| 48 | 47 | supex 9503 | . . . . 5
⊢
sup({𝑛 ∈
ℕ0 ∣ (𝑃↑𝑛) ∥ 𝑁}, ℝ, < ) ∈ V | 
| 49 | 19, 48 | eqeltri 2837 | . . . 4
⊢ 𝑆 ∈ V | 
| 50 | 2, 3 | pceu 16884 | . . . . 5
⊢ ((𝑃 ∈ ℙ ∧ (𝑁 ∈ ℚ ∧ 𝑁 ≠ 0)) → ∃!𝑧∃𝑥 ∈ ℤ ∃𝑦 ∈ ℕ (𝑁 = (𝑥 / 𝑦) ∧ 𝑧 = (sup({𝑛 ∈ ℕ0 ∣ (𝑃↑𝑛) ∥ 𝑥}, ℝ, < ) − sup({𝑛 ∈ ℕ0
∣ (𝑃↑𝑛) ∥ 𝑦}, ℝ, < )))) | 
| 51 | 1, 50 | sylanr1 682 | . . . 4
⊢ ((𝑃 ∈ ℙ ∧ (𝑁 ∈ ℤ ∧ 𝑁 ≠ 0)) → ∃!𝑧∃𝑥 ∈ ℤ ∃𝑦 ∈ ℕ (𝑁 = (𝑥 / 𝑦) ∧ 𝑧 = (sup({𝑛 ∈ ℕ0 ∣ (𝑃↑𝑛) ∥ 𝑥}, ℝ, < ) − sup({𝑛 ∈ ℕ0
∣ (𝑃↑𝑛) ∥ 𝑦}, ℝ, < )))) | 
| 52 |  | eqeq1 2741 | . . . . . . 7
⊢ (𝑧 = 𝑆 → (𝑧 = (sup({𝑛 ∈ ℕ0 ∣ (𝑃↑𝑛) ∥ 𝑥}, ℝ, < ) − sup({𝑛 ∈ ℕ0
∣ (𝑃↑𝑛) ∥ 𝑦}, ℝ, < )) ↔ 𝑆 = (sup({𝑛 ∈ ℕ0 ∣ (𝑃↑𝑛) ∥ 𝑥}, ℝ, < ) − sup({𝑛 ∈ ℕ0
∣ (𝑃↑𝑛) ∥ 𝑦}, ℝ, < )))) | 
| 53 | 52 | anbi2d 630 | . . . . . 6
⊢ (𝑧 = 𝑆 → ((𝑁 = (𝑥 / 𝑦) ∧ 𝑧 = (sup({𝑛 ∈ ℕ0 ∣ (𝑃↑𝑛) ∥ 𝑥}, ℝ, < ) − sup({𝑛 ∈ ℕ0
∣ (𝑃↑𝑛) ∥ 𝑦}, ℝ, < ))) ↔ (𝑁 = (𝑥 / 𝑦) ∧ 𝑆 = (sup({𝑛 ∈ ℕ0 ∣ (𝑃↑𝑛) ∥ 𝑥}, ℝ, < ) − sup({𝑛 ∈ ℕ0
∣ (𝑃↑𝑛) ∥ 𝑦}, ℝ, < ))))) | 
| 54 | 53 | 2rexbidv 3222 | . . . . 5
⊢ (𝑧 = 𝑆 → (∃𝑥 ∈ ℤ ∃𝑦 ∈ ℕ (𝑁 = (𝑥 / 𝑦) ∧ 𝑧 = (sup({𝑛 ∈ ℕ0 ∣ (𝑃↑𝑛) ∥ 𝑥}, ℝ, < ) − sup({𝑛 ∈ ℕ0
∣ (𝑃↑𝑛) ∥ 𝑦}, ℝ, < ))) ↔ ∃𝑥 ∈ ℤ ∃𝑦 ∈ ℕ (𝑁 = (𝑥 / 𝑦) ∧ 𝑆 = (sup({𝑛 ∈ ℕ0 ∣ (𝑃↑𝑛) ∥ 𝑥}, ℝ, < ) − sup({𝑛 ∈ ℕ0
∣ (𝑃↑𝑛) ∥ 𝑦}, ℝ, < ))))) | 
| 55 | 54 | iota2 6550 | . . . 4
⊢ ((𝑆 ∈ V ∧ ∃!𝑧∃𝑥 ∈ ℤ ∃𝑦 ∈ ℕ (𝑁 = (𝑥 / 𝑦) ∧ 𝑧 = (sup({𝑛 ∈ ℕ0 ∣ (𝑃↑𝑛) ∥ 𝑥}, ℝ, < ) − sup({𝑛 ∈ ℕ0
∣ (𝑃↑𝑛) ∥ 𝑦}, ℝ, < )))) → (∃𝑥 ∈ ℤ ∃𝑦 ∈ ℕ (𝑁 = (𝑥 / 𝑦) ∧ 𝑆 = (sup({𝑛 ∈ ℕ0 ∣ (𝑃↑𝑛) ∥ 𝑥}, ℝ, < ) − sup({𝑛 ∈ ℕ0
∣ (𝑃↑𝑛) ∥ 𝑦}, ℝ, < ))) ↔ (℩𝑧∃𝑥 ∈ ℤ ∃𝑦 ∈ ℕ (𝑁 = (𝑥 / 𝑦) ∧ 𝑧 = (sup({𝑛 ∈ ℕ0 ∣ (𝑃↑𝑛) ∥ 𝑥}, ℝ, < ) − sup({𝑛 ∈ ℕ0
∣ (𝑃↑𝑛) ∥ 𝑦}, ℝ, < )))) = 𝑆)) | 
| 56 | 49, 51, 55 | sylancr 587 | . . 3
⊢ ((𝑃 ∈ ℙ ∧ (𝑁 ∈ ℤ ∧ 𝑁 ≠ 0)) → (∃𝑥 ∈ ℤ ∃𝑦 ∈ ℕ (𝑁 = (𝑥 / 𝑦) ∧ 𝑆 = (sup({𝑛 ∈ ℕ0 ∣ (𝑃↑𝑛) ∥ 𝑥}, ℝ, < ) − sup({𝑛 ∈ ℕ0
∣ (𝑃↑𝑛) ∥ 𝑦}, ℝ, < ))) ↔ (℩𝑧∃𝑥 ∈ ℤ ∃𝑦 ∈ ℕ (𝑁 = (𝑥 / 𝑦) ∧ 𝑧 = (sup({𝑛 ∈ ℕ0 ∣ (𝑃↑𝑛) ∥ 𝑥}, ℝ, < ) − sup({𝑛 ∈ ℕ0
∣ (𝑃↑𝑛) ∥ 𝑦}, ℝ, < )))) = 𝑆)) | 
| 57 | 46, 56 | mpbid 232 | . 2
⊢ ((𝑃 ∈ ℙ ∧ (𝑁 ∈ ℤ ∧ 𝑁 ≠ 0)) → (℩𝑧∃𝑥 ∈ ℤ ∃𝑦 ∈ ℕ (𝑁 = (𝑥 / 𝑦) ∧ 𝑧 = (sup({𝑛 ∈ ℕ0 ∣ (𝑃↑𝑛) ∥ 𝑥}, ℝ, < ) − sup({𝑛 ∈ ℕ0
∣ (𝑃↑𝑛) ∥ 𝑦}, ℝ, < )))) = 𝑆) | 
| 58 | 5, 57 | eqtrd 2777 | 1
⊢ ((𝑃 ∈ ℙ ∧ (𝑁 ∈ ℤ ∧ 𝑁 ≠ 0)) → (𝑃 pCnt 𝑁) = 𝑆) |