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Theorem pcdiv 16899
Description: Division property of the prime power function. (Contributed by Mario Carneiro, 1-Mar-2014.)
Assertion
Ref Expression
pcdiv ((𝑃 ∈ ℙ ∧ (𝐴 ∈ ℤ ∧ 𝐴 ≠ 0) ∧ 𝐵 ∈ ℕ) → (𝑃 pCnt (𝐴 / 𝐵)) = ((𝑃 pCnt 𝐴) − (𝑃 pCnt 𝐵)))

Proof of Theorem pcdiv
Dummy variables 𝑥 𝑛 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 simp1 1136 . . 3 ((𝑃 ∈ ℙ ∧ (𝐴 ∈ ℤ ∧ 𝐴 ≠ 0) ∧ 𝐵 ∈ ℕ) → 𝑃 ∈ ℙ)
2 simp2l 1199 . . . 4 ((𝑃 ∈ ℙ ∧ (𝐴 ∈ ℤ ∧ 𝐴 ≠ 0) ∧ 𝐵 ∈ ℕ) → 𝐴 ∈ ℤ)
3 simp3 1138 . . . 4 ((𝑃 ∈ ℙ ∧ (𝐴 ∈ ℤ ∧ 𝐴 ≠ 0) ∧ 𝐵 ∈ ℕ) → 𝐵 ∈ ℕ)
4 znq 13017 . . . 4 ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℕ) → (𝐴 / 𝐵) ∈ ℚ)
52, 3, 4syl2anc 583 . . 3 ((𝑃 ∈ ℙ ∧ (𝐴 ∈ ℤ ∧ 𝐴 ≠ 0) ∧ 𝐵 ∈ ℕ) → (𝐴 / 𝐵) ∈ ℚ)
62zcnd 12748 . . . 4 ((𝑃 ∈ ℙ ∧ (𝐴 ∈ ℤ ∧ 𝐴 ≠ 0) ∧ 𝐵 ∈ ℕ) → 𝐴 ∈ ℂ)
73nncnd 12309 . . . 4 ((𝑃 ∈ ℙ ∧ (𝐴 ∈ ℤ ∧ 𝐴 ≠ 0) ∧ 𝐵 ∈ ℕ) → 𝐵 ∈ ℂ)
8 simp2r 1200 . . . 4 ((𝑃 ∈ ℙ ∧ (𝐴 ∈ ℤ ∧ 𝐴 ≠ 0) ∧ 𝐵 ∈ ℕ) → 𝐴 ≠ 0)
93nnne0d 12343 . . . 4 ((𝑃 ∈ ℙ ∧ (𝐴 ∈ ℤ ∧ 𝐴 ≠ 0) ∧ 𝐵 ∈ ℕ) → 𝐵 ≠ 0)
106, 7, 8, 9divne0d 12086 . . 3 ((𝑃 ∈ ℙ ∧ (𝐴 ∈ ℤ ∧ 𝐴 ≠ 0) ∧ 𝐵 ∈ ℕ) → (𝐴 / 𝐵) ≠ 0)
11 eqid 2740 . . . 4 sup({𝑛 ∈ ℕ0 ∣ (𝑃𝑛) ∥ 𝑥}, ℝ, < ) = sup({𝑛 ∈ ℕ0 ∣ (𝑃𝑛) ∥ 𝑥}, ℝ, < )
12 eqid 2740 . . . 4 sup({𝑛 ∈ ℕ0 ∣ (𝑃𝑛) ∥ 𝑦}, ℝ, < ) = sup({𝑛 ∈ ℕ0 ∣ (𝑃𝑛) ∥ 𝑦}, ℝ, < )
1311, 12pcval 16891 . . 3 ((𝑃 ∈ ℙ ∧ ((𝐴 / 𝐵) ∈ ℚ ∧ (𝐴 / 𝐵) ≠ 0)) → (𝑃 pCnt (𝐴 / 𝐵)) = (℩𝑧𝑥 ∈ ℤ ∃𝑦 ∈ ℕ ((𝐴 / 𝐵) = (𝑥 / 𝑦) ∧ 𝑧 = (sup({𝑛 ∈ ℕ0 ∣ (𝑃𝑛) ∥ 𝑥}, ℝ, < ) − sup({𝑛 ∈ ℕ0 ∣ (𝑃𝑛) ∥ 𝑦}, ℝ, < )))))
141, 5, 10, 13syl12anc 836 . 2 ((𝑃 ∈ ℙ ∧ (𝐴 ∈ ℤ ∧ 𝐴 ≠ 0) ∧ 𝐵 ∈ ℕ) → (𝑃 pCnt (𝐴 / 𝐵)) = (℩𝑧𝑥 ∈ ℤ ∃𝑦 ∈ ℕ ((𝐴 / 𝐵) = (𝑥 / 𝑦) ∧ 𝑧 = (sup({𝑛 ∈ ℕ0 ∣ (𝑃𝑛) ∥ 𝑥}, ℝ, < ) − sup({𝑛 ∈ ℕ0 ∣ (𝑃𝑛) ∥ 𝑦}, ℝ, < )))))
15 eqid 2740 . . . . . . . 8 sup({𝑛 ∈ ℕ0 ∣ (𝑃𝑛) ∥ 𝐴}, ℝ, < ) = sup({𝑛 ∈ ℕ0 ∣ (𝑃𝑛) ∥ 𝐴}, ℝ, < )
1615pczpre 16894 . . . . . . 7 ((𝑃 ∈ ℙ ∧ (𝐴 ∈ ℤ ∧ 𝐴 ≠ 0)) → (𝑃 pCnt 𝐴) = sup({𝑛 ∈ ℕ0 ∣ (𝑃𝑛) ∥ 𝐴}, ℝ, < ))
17163adant3 1132 . . . . . 6 ((𝑃 ∈ ℙ ∧ (𝐴 ∈ ℤ ∧ 𝐴 ≠ 0) ∧ 𝐵 ∈ ℕ) → (𝑃 pCnt 𝐴) = sup({𝑛 ∈ ℕ0 ∣ (𝑃𝑛) ∥ 𝐴}, ℝ, < ))
18 nnz 12660 . . . . . . . . 9 (𝐵 ∈ ℕ → 𝐵 ∈ ℤ)
19 nnne0 12327 . . . . . . . . 9 (𝐵 ∈ ℕ → 𝐵 ≠ 0)
2018, 19jca 511 . . . . . . . 8 (𝐵 ∈ ℕ → (𝐵 ∈ ℤ ∧ 𝐵 ≠ 0))
21 eqid 2740 . . . . . . . . 9 sup({𝑛 ∈ ℕ0 ∣ (𝑃𝑛) ∥ 𝐵}, ℝ, < ) = sup({𝑛 ∈ ℕ0 ∣ (𝑃𝑛) ∥ 𝐵}, ℝ, < )
2221pczpre 16894 . . . . . . . 8 ((𝑃 ∈ ℙ ∧ (𝐵 ∈ ℤ ∧ 𝐵 ≠ 0)) → (𝑃 pCnt 𝐵) = sup({𝑛 ∈ ℕ0 ∣ (𝑃𝑛) ∥ 𝐵}, ℝ, < ))
2320, 22sylan2 592 . . . . . . 7 ((𝑃 ∈ ℙ ∧ 𝐵 ∈ ℕ) → (𝑃 pCnt 𝐵) = sup({𝑛 ∈ ℕ0 ∣ (𝑃𝑛) ∥ 𝐵}, ℝ, < ))
24233adant2 1131 . . . . . 6 ((𝑃 ∈ ℙ ∧ (𝐴 ∈ ℤ ∧ 𝐴 ≠ 0) ∧ 𝐵 ∈ ℕ) → (𝑃 pCnt 𝐵) = sup({𝑛 ∈ ℕ0 ∣ (𝑃𝑛) ∥ 𝐵}, ℝ, < ))
2517, 24oveq12d 7466 . . . . 5 ((𝑃 ∈ ℙ ∧ (𝐴 ∈ ℤ ∧ 𝐴 ≠ 0) ∧ 𝐵 ∈ ℕ) → ((𝑃 pCnt 𝐴) − (𝑃 pCnt 𝐵)) = (sup({𝑛 ∈ ℕ0 ∣ (𝑃𝑛) ∥ 𝐴}, ℝ, < ) − sup({𝑛 ∈ ℕ0 ∣ (𝑃𝑛) ∥ 𝐵}, ℝ, < )))
26 eqid 2740 . . . . 5 (𝐴 / 𝐵) = (𝐴 / 𝐵)
2725, 26jctil 519 . . . 4 ((𝑃 ∈ ℙ ∧ (𝐴 ∈ ℤ ∧ 𝐴 ≠ 0) ∧ 𝐵 ∈ ℕ) → ((𝐴 / 𝐵) = (𝐴 / 𝐵) ∧ ((𝑃 pCnt 𝐴) − (𝑃 pCnt 𝐵)) = (sup({𝑛 ∈ ℕ0 ∣ (𝑃𝑛) ∥ 𝐴}, ℝ, < ) − sup({𝑛 ∈ ℕ0 ∣ (𝑃𝑛) ∥ 𝐵}, ℝ, < ))))
28 oveq1 7455 . . . . . . 7 (𝑥 = 𝐴 → (𝑥 / 𝑦) = (𝐴 / 𝑦))
2928eqeq2d 2751 . . . . . 6 (𝑥 = 𝐴 → ((𝐴 / 𝐵) = (𝑥 / 𝑦) ↔ (𝐴 / 𝐵) = (𝐴 / 𝑦)))
30 breq2 5170 . . . . . . . . . 10 (𝑥 = 𝐴 → ((𝑃𝑛) ∥ 𝑥 ↔ (𝑃𝑛) ∥ 𝐴))
3130rabbidv 3451 . . . . . . . . 9 (𝑥 = 𝐴 → {𝑛 ∈ ℕ0 ∣ (𝑃𝑛) ∥ 𝑥} = {𝑛 ∈ ℕ0 ∣ (𝑃𝑛) ∥ 𝐴})
3231supeq1d 9515 . . . . . . . 8 (𝑥 = 𝐴 → sup({𝑛 ∈ ℕ0 ∣ (𝑃𝑛) ∥ 𝑥}, ℝ, < ) = sup({𝑛 ∈ ℕ0 ∣ (𝑃𝑛) ∥ 𝐴}, ℝ, < ))
3332oveq1d 7463 . . . . . . 7 (𝑥 = 𝐴 → (sup({𝑛 ∈ ℕ0 ∣ (𝑃𝑛) ∥ 𝑥}, ℝ, < ) − sup({𝑛 ∈ ℕ0 ∣ (𝑃𝑛) ∥ 𝑦}, ℝ, < )) = (sup({𝑛 ∈ ℕ0 ∣ (𝑃𝑛) ∥ 𝐴}, ℝ, < ) − sup({𝑛 ∈ ℕ0 ∣ (𝑃𝑛) ∥ 𝑦}, ℝ, < )))
3433eqeq2d 2751 . . . . . 6 (𝑥 = 𝐴 → (((𝑃 pCnt 𝐴) − (𝑃 pCnt 𝐵)) = (sup({𝑛 ∈ ℕ0 ∣ (𝑃𝑛) ∥ 𝑥}, ℝ, < ) − sup({𝑛 ∈ ℕ0 ∣ (𝑃𝑛) ∥ 𝑦}, ℝ, < )) ↔ ((𝑃 pCnt 𝐴) − (𝑃 pCnt 𝐵)) = (sup({𝑛 ∈ ℕ0 ∣ (𝑃𝑛) ∥ 𝐴}, ℝ, < ) − sup({𝑛 ∈ ℕ0 ∣ (𝑃𝑛) ∥ 𝑦}, ℝ, < ))))
3529, 34anbi12d 631 . . . . 5 (𝑥 = 𝐴 → (((𝐴 / 𝐵) = (𝑥 / 𝑦) ∧ ((𝑃 pCnt 𝐴) − (𝑃 pCnt 𝐵)) = (sup({𝑛 ∈ ℕ0 ∣ (𝑃𝑛) ∥ 𝑥}, ℝ, < ) − sup({𝑛 ∈ ℕ0 ∣ (𝑃𝑛) ∥ 𝑦}, ℝ, < ))) ↔ ((𝐴 / 𝐵) = (𝐴 / 𝑦) ∧ ((𝑃 pCnt 𝐴) − (𝑃 pCnt 𝐵)) = (sup({𝑛 ∈ ℕ0 ∣ (𝑃𝑛) ∥ 𝐴}, ℝ, < ) − sup({𝑛 ∈ ℕ0 ∣ (𝑃𝑛) ∥ 𝑦}, ℝ, < )))))
36 oveq2 7456 . . . . . . 7 (𝑦 = 𝐵 → (𝐴 / 𝑦) = (𝐴 / 𝐵))
3736eqeq2d 2751 . . . . . 6 (𝑦 = 𝐵 → ((𝐴 / 𝐵) = (𝐴 / 𝑦) ↔ (𝐴 / 𝐵) = (𝐴 / 𝐵)))
38 breq2 5170 . . . . . . . . . 10 (𝑦 = 𝐵 → ((𝑃𝑛) ∥ 𝑦 ↔ (𝑃𝑛) ∥ 𝐵))
3938rabbidv 3451 . . . . . . . . 9 (𝑦 = 𝐵 → {𝑛 ∈ ℕ0 ∣ (𝑃𝑛) ∥ 𝑦} = {𝑛 ∈ ℕ0 ∣ (𝑃𝑛) ∥ 𝐵})
4039supeq1d 9515 . . . . . . . 8 (𝑦 = 𝐵 → sup({𝑛 ∈ ℕ0 ∣ (𝑃𝑛) ∥ 𝑦}, ℝ, < ) = sup({𝑛 ∈ ℕ0 ∣ (𝑃𝑛) ∥ 𝐵}, ℝ, < ))
4140oveq2d 7464 . . . . . . 7 (𝑦 = 𝐵 → (sup({𝑛 ∈ ℕ0 ∣ (𝑃𝑛) ∥ 𝐴}, ℝ, < ) − sup({𝑛 ∈ ℕ0 ∣ (𝑃𝑛) ∥ 𝑦}, ℝ, < )) = (sup({𝑛 ∈ ℕ0 ∣ (𝑃𝑛) ∥ 𝐴}, ℝ, < ) − sup({𝑛 ∈ ℕ0 ∣ (𝑃𝑛) ∥ 𝐵}, ℝ, < )))
4241eqeq2d 2751 . . . . . 6 (𝑦 = 𝐵 → (((𝑃 pCnt 𝐴) − (𝑃 pCnt 𝐵)) = (sup({𝑛 ∈ ℕ0 ∣ (𝑃𝑛) ∥ 𝐴}, ℝ, < ) − sup({𝑛 ∈ ℕ0 ∣ (𝑃𝑛) ∥ 𝑦}, ℝ, < )) ↔ ((𝑃 pCnt 𝐴) − (𝑃 pCnt 𝐵)) = (sup({𝑛 ∈ ℕ0 ∣ (𝑃𝑛) ∥ 𝐴}, ℝ, < ) − sup({𝑛 ∈ ℕ0 ∣ (𝑃𝑛) ∥ 𝐵}, ℝ, < ))))
4337, 42anbi12d 631 . . . . 5 (𝑦 = 𝐵 → (((𝐴 / 𝐵) = (𝐴 / 𝑦) ∧ ((𝑃 pCnt 𝐴) − (𝑃 pCnt 𝐵)) = (sup({𝑛 ∈ ℕ0 ∣ (𝑃𝑛) ∥ 𝐴}, ℝ, < ) − sup({𝑛 ∈ ℕ0 ∣ (𝑃𝑛) ∥ 𝑦}, ℝ, < ))) ↔ ((𝐴 / 𝐵) = (𝐴 / 𝐵) ∧ ((𝑃 pCnt 𝐴) − (𝑃 pCnt 𝐵)) = (sup({𝑛 ∈ ℕ0 ∣ (𝑃𝑛) ∥ 𝐴}, ℝ, < ) − sup({𝑛 ∈ ℕ0 ∣ (𝑃𝑛) ∥ 𝐵}, ℝ, < )))))
4435, 43rspc2ev 3648 . . . 4 ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℕ ∧ ((𝐴 / 𝐵) = (𝐴 / 𝐵) ∧ ((𝑃 pCnt 𝐴) − (𝑃 pCnt 𝐵)) = (sup({𝑛 ∈ ℕ0 ∣ (𝑃𝑛) ∥ 𝐴}, ℝ, < ) − sup({𝑛 ∈ ℕ0 ∣ (𝑃𝑛) ∥ 𝐵}, ℝ, < )))) → ∃𝑥 ∈ ℤ ∃𝑦 ∈ ℕ ((𝐴 / 𝐵) = (𝑥 / 𝑦) ∧ ((𝑃 pCnt 𝐴) − (𝑃 pCnt 𝐵)) = (sup({𝑛 ∈ ℕ0 ∣ (𝑃𝑛) ∥ 𝑥}, ℝ, < ) − sup({𝑛 ∈ ℕ0 ∣ (𝑃𝑛) ∥ 𝑦}, ℝ, < ))))
452, 3, 27, 44syl3anc 1371 . . 3 ((𝑃 ∈ ℙ ∧ (𝐴 ∈ ℤ ∧ 𝐴 ≠ 0) ∧ 𝐵 ∈ ℕ) → ∃𝑥 ∈ ℤ ∃𝑦 ∈ ℕ ((𝐴 / 𝐵) = (𝑥 / 𝑦) ∧ ((𝑃 pCnt 𝐴) − (𝑃 pCnt 𝐵)) = (sup({𝑛 ∈ ℕ0 ∣ (𝑃𝑛) ∥ 𝑥}, ℝ, < ) − sup({𝑛 ∈ ℕ0 ∣ (𝑃𝑛) ∥ 𝑦}, ℝ, < ))))
46 ovex 7481 . . . 4 ((𝑃 pCnt 𝐴) − (𝑃 pCnt 𝐵)) ∈ V
4711, 12pceu 16893 . . . . 5 ((𝑃 ∈ ℙ ∧ ((𝐴 / 𝐵) ∈ ℚ ∧ (𝐴 / 𝐵) ≠ 0)) → ∃!𝑧𝑥 ∈ ℤ ∃𝑦 ∈ ℕ ((𝐴 / 𝐵) = (𝑥 / 𝑦) ∧ 𝑧 = (sup({𝑛 ∈ ℕ0 ∣ (𝑃𝑛) ∥ 𝑥}, ℝ, < ) − sup({𝑛 ∈ ℕ0 ∣ (𝑃𝑛) ∥ 𝑦}, ℝ, < ))))
481, 5, 10, 47syl12anc 836 . . . 4 ((𝑃 ∈ ℙ ∧ (𝐴 ∈ ℤ ∧ 𝐴 ≠ 0) ∧ 𝐵 ∈ ℕ) → ∃!𝑧𝑥 ∈ ℤ ∃𝑦 ∈ ℕ ((𝐴 / 𝐵) = (𝑥 / 𝑦) ∧ 𝑧 = (sup({𝑛 ∈ ℕ0 ∣ (𝑃𝑛) ∥ 𝑥}, ℝ, < ) − sup({𝑛 ∈ ℕ0 ∣ (𝑃𝑛) ∥ 𝑦}, ℝ, < ))))
49 eqeq1 2744 . . . . . . 7 (𝑧 = ((𝑃 pCnt 𝐴) − (𝑃 pCnt 𝐵)) → (𝑧 = (sup({𝑛 ∈ ℕ0 ∣ (𝑃𝑛) ∥ 𝑥}, ℝ, < ) − sup({𝑛 ∈ ℕ0 ∣ (𝑃𝑛) ∥ 𝑦}, ℝ, < )) ↔ ((𝑃 pCnt 𝐴) − (𝑃 pCnt 𝐵)) = (sup({𝑛 ∈ ℕ0 ∣ (𝑃𝑛) ∥ 𝑥}, ℝ, < ) − sup({𝑛 ∈ ℕ0 ∣ (𝑃𝑛) ∥ 𝑦}, ℝ, < ))))
5049anbi2d 629 . . . . . 6 (𝑧 = ((𝑃 pCnt 𝐴) − (𝑃 pCnt 𝐵)) → (((𝐴 / 𝐵) = (𝑥 / 𝑦) ∧ 𝑧 = (sup({𝑛 ∈ ℕ0 ∣ (𝑃𝑛) ∥ 𝑥}, ℝ, < ) − sup({𝑛 ∈ ℕ0 ∣ (𝑃𝑛) ∥ 𝑦}, ℝ, < ))) ↔ ((𝐴 / 𝐵) = (𝑥 / 𝑦) ∧ ((𝑃 pCnt 𝐴) − (𝑃 pCnt 𝐵)) = (sup({𝑛 ∈ ℕ0 ∣ (𝑃𝑛) ∥ 𝑥}, ℝ, < ) − sup({𝑛 ∈ ℕ0 ∣ (𝑃𝑛) ∥ 𝑦}, ℝ, < )))))
51502rexbidv 3228 . . . . 5 (𝑧 = ((𝑃 pCnt 𝐴) − (𝑃 pCnt 𝐵)) → (∃𝑥 ∈ ℤ ∃𝑦 ∈ ℕ ((𝐴 / 𝐵) = (𝑥 / 𝑦) ∧ 𝑧 = (sup({𝑛 ∈ ℕ0 ∣ (𝑃𝑛) ∥ 𝑥}, ℝ, < ) − sup({𝑛 ∈ ℕ0 ∣ (𝑃𝑛) ∥ 𝑦}, ℝ, < ))) ↔ ∃𝑥 ∈ ℤ ∃𝑦 ∈ ℕ ((𝐴 / 𝐵) = (𝑥 / 𝑦) ∧ ((𝑃 pCnt 𝐴) − (𝑃 pCnt 𝐵)) = (sup({𝑛 ∈ ℕ0 ∣ (𝑃𝑛) ∥ 𝑥}, ℝ, < ) − sup({𝑛 ∈ ℕ0 ∣ (𝑃𝑛) ∥ 𝑦}, ℝ, < )))))
5251iota2 6562 . . . 4 ((((𝑃 pCnt 𝐴) − (𝑃 pCnt 𝐵)) ∈ V ∧ ∃!𝑧𝑥 ∈ ℤ ∃𝑦 ∈ ℕ ((𝐴 / 𝐵) = (𝑥 / 𝑦) ∧ 𝑧 = (sup({𝑛 ∈ ℕ0 ∣ (𝑃𝑛) ∥ 𝑥}, ℝ, < ) − sup({𝑛 ∈ ℕ0 ∣ (𝑃𝑛) ∥ 𝑦}, ℝ, < )))) → (∃𝑥 ∈ ℤ ∃𝑦 ∈ ℕ ((𝐴 / 𝐵) = (𝑥 / 𝑦) ∧ ((𝑃 pCnt 𝐴) − (𝑃 pCnt 𝐵)) = (sup({𝑛 ∈ ℕ0 ∣ (𝑃𝑛) ∥ 𝑥}, ℝ, < ) − sup({𝑛 ∈ ℕ0 ∣ (𝑃𝑛) ∥ 𝑦}, ℝ, < ))) ↔ (℩𝑧𝑥 ∈ ℤ ∃𝑦 ∈ ℕ ((𝐴 / 𝐵) = (𝑥 / 𝑦) ∧ 𝑧 = (sup({𝑛 ∈ ℕ0 ∣ (𝑃𝑛) ∥ 𝑥}, ℝ, < ) − sup({𝑛 ∈ ℕ0 ∣ (𝑃𝑛) ∥ 𝑦}, ℝ, < )))) = ((𝑃 pCnt 𝐴) − (𝑃 pCnt 𝐵))))
5346, 48, 52sylancr 586 . . 3 ((𝑃 ∈ ℙ ∧ (𝐴 ∈ ℤ ∧ 𝐴 ≠ 0) ∧ 𝐵 ∈ ℕ) → (∃𝑥 ∈ ℤ ∃𝑦 ∈ ℕ ((𝐴 / 𝐵) = (𝑥 / 𝑦) ∧ ((𝑃 pCnt 𝐴) − (𝑃 pCnt 𝐵)) = (sup({𝑛 ∈ ℕ0 ∣ (𝑃𝑛) ∥ 𝑥}, ℝ, < ) − sup({𝑛 ∈ ℕ0 ∣ (𝑃𝑛) ∥ 𝑦}, ℝ, < ))) ↔ (℩𝑧𝑥 ∈ ℤ ∃𝑦 ∈ ℕ ((𝐴 / 𝐵) = (𝑥 / 𝑦) ∧ 𝑧 = (sup({𝑛 ∈ ℕ0 ∣ (𝑃𝑛) ∥ 𝑥}, ℝ, < ) − sup({𝑛 ∈ ℕ0 ∣ (𝑃𝑛) ∥ 𝑦}, ℝ, < )))) = ((𝑃 pCnt 𝐴) − (𝑃 pCnt 𝐵))))
5445, 53mpbid 232 . 2 ((𝑃 ∈ ℙ ∧ (𝐴 ∈ ℤ ∧ 𝐴 ≠ 0) ∧ 𝐵 ∈ ℕ) → (℩𝑧𝑥 ∈ ℤ ∃𝑦 ∈ ℕ ((𝐴 / 𝐵) = (𝑥 / 𝑦) ∧ 𝑧 = (sup({𝑛 ∈ ℕ0 ∣ (𝑃𝑛) ∥ 𝑥}, ℝ, < ) − sup({𝑛 ∈ ℕ0 ∣ (𝑃𝑛) ∥ 𝑦}, ℝ, < )))) = ((𝑃 pCnt 𝐴) − (𝑃 pCnt 𝐵)))
5514, 54eqtrd 2780 1 ((𝑃 ∈ ℙ ∧ (𝐴 ∈ ℤ ∧ 𝐴 ≠ 0) ∧ 𝐵 ∈ ℕ) → (𝑃 pCnt (𝐴 / 𝐵)) = ((𝑃 pCnt 𝐴) − (𝑃 pCnt 𝐵)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395  w3a 1087   = wceq 1537  wcel 2108  ∃!weu 2571  wne 2946  wrex 3076  {crab 3443  Vcvv 3488   class class class wbr 5166  cio 6523  (class class class)co 7448  supcsup 9509  cr 11183  0cc0 11184   < clt 11324  cmin 11520   / cdiv 11947  cn 12293  0cn0 12553  cz 12639  cq 13013  cexp 14112  cdvds 16302  cprime 16718   pCnt cpc 16883
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-sep 5317  ax-nul 5324  ax-pow 5383  ax-pr 5447  ax-un 7770  ax-cnex 11240  ax-resscn 11241  ax-1cn 11242  ax-icn 11243  ax-addcl 11244  ax-addrcl 11245  ax-mulcl 11246  ax-mulrcl 11247  ax-mulcom 11248  ax-addass 11249  ax-mulass 11250  ax-distr 11251  ax-i2m1 11252  ax-1ne0 11253  ax-1rid 11254  ax-rnegex 11255  ax-rrecex 11256  ax-cnre 11257  ax-pre-lttri 11258  ax-pre-lttrn 11259  ax-pre-ltadd 11260  ax-pre-mulgt0 11261  ax-pre-sup 11262
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3or 1088  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-nel 3053  df-ral 3068  df-rex 3077  df-rmo 3388  df-reu 3389  df-rab 3444  df-v 3490  df-sbc 3805  df-csb 3922  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-pss 3996  df-nul 4353  df-if 4549  df-pw 4624  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-iun 5017  df-br 5167  df-opab 5229  df-mpt 5250  df-tr 5284  df-id 5593  df-eprel 5599  df-po 5607  df-so 5608  df-fr 5652  df-we 5654  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-rn 5711  df-res 5712  df-ima 5713  df-pred 6332  df-ord 6398  df-on 6399  df-lim 6400  df-suc 6401  df-iota 6525  df-fun 6575  df-fn 6576  df-f 6577  df-f1 6578  df-fo 6579  df-f1o 6580  df-fv 6581  df-riota 7404  df-ov 7451  df-oprab 7452  df-mpo 7453  df-om 7904  df-1st 8030  df-2nd 8031  df-frecs 8322  df-wrecs 8353  df-recs 8427  df-rdg 8466  df-1o 8522  df-2o 8523  df-er 8763  df-en 9004  df-dom 9005  df-sdom 9006  df-fin 9007  df-sup 9511  df-inf 9512  df-pnf 11326  df-mnf 11327  df-xr 11328  df-ltxr 11329  df-le 11330  df-sub 11522  df-neg 11523  df-div 11948  df-nn 12294  df-2 12356  df-3 12357  df-n0 12554  df-z 12640  df-uz 12904  df-q 13014  df-rp 13058  df-fl 13843  df-mod 13921  df-seq 14053  df-exp 14113  df-cj 15148  df-re 15149  df-im 15150  df-sqrt 15284  df-abs 15285  df-dvds 16303  df-gcd 16541  df-prm 16719  df-pc 16884
This theorem is referenced by:  pcqmul  16900  pcqcl  16903  pcid  16920  pcneg  16921  pc2dvds  16926  pcz  16928  pcaddlem  16935  pcadd  16936  pcmpt2  16940  pcbc  16947  sylow1lem1  19640  chtublem  27273
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