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Theorem pcdiv 16181
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 1130 . . 3 ((𝑃 ∈ ℙ ∧ (𝐴 ∈ ℤ ∧ 𝐴 ≠ 0) ∧ 𝐵 ∈ ℕ) → 𝑃 ∈ ℙ)
2 simp2l 1193 . . . 4 ((𝑃 ∈ ℙ ∧ (𝐴 ∈ ℤ ∧ 𝐴 ≠ 0) ∧ 𝐵 ∈ ℕ) → 𝐴 ∈ ℤ)
3 simp3 1132 . . . 4 ((𝑃 ∈ ℙ ∧ (𝐴 ∈ ℤ ∧ 𝐴 ≠ 0) ∧ 𝐵 ∈ ℕ) → 𝐵 ∈ ℕ)
4 znq 12344 . . . 4 ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℕ) → (𝐴 / 𝐵) ∈ ℚ)
52, 3, 4syl2anc 586 . . 3 ((𝑃 ∈ ℙ ∧ (𝐴 ∈ ℤ ∧ 𝐴 ≠ 0) ∧ 𝐵 ∈ ℕ) → (𝐴 / 𝐵) ∈ ℚ)
62zcnd 12080 . . . 4 ((𝑃 ∈ ℙ ∧ (𝐴 ∈ ℤ ∧ 𝐴 ≠ 0) ∧ 𝐵 ∈ ℕ) → 𝐴 ∈ ℂ)
73nncnd 11646 . . . 4 ((𝑃 ∈ ℙ ∧ (𝐴 ∈ ℤ ∧ 𝐴 ≠ 0) ∧ 𝐵 ∈ ℕ) → 𝐵 ∈ ℂ)
8 simp2r 1194 . . . 4 ((𝑃 ∈ ℙ ∧ (𝐴 ∈ ℤ ∧ 𝐴 ≠ 0) ∧ 𝐵 ∈ ℕ) → 𝐴 ≠ 0)
93nnne0d 11679 . . . 4 ((𝑃 ∈ ℙ ∧ (𝐴 ∈ ℤ ∧ 𝐴 ≠ 0) ∧ 𝐵 ∈ ℕ) → 𝐵 ≠ 0)
106, 7, 8, 9divne0d 11424 . . 3 ((𝑃 ∈ ℙ ∧ (𝐴 ∈ ℤ ∧ 𝐴 ≠ 0) ∧ 𝐵 ∈ ℕ) → (𝐴 / 𝐵) ≠ 0)
11 eqid 2819 . . . 4 sup({𝑛 ∈ ℕ0 ∣ (𝑃𝑛) ∥ 𝑥}, ℝ, < ) = sup({𝑛 ∈ ℕ0 ∣ (𝑃𝑛) ∥ 𝑥}, ℝ, < )
12 eqid 2819 . . . 4 sup({𝑛 ∈ ℕ0 ∣ (𝑃𝑛) ∥ 𝑦}, ℝ, < ) = sup({𝑛 ∈ ℕ0 ∣ (𝑃𝑛) ∥ 𝑦}, ℝ, < )
1311, 12pcval 16173 . . 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 834 . 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 2819 . . . . . . . 8 sup({𝑛 ∈ ℕ0 ∣ (𝑃𝑛) ∥ 𝐴}, ℝ, < ) = sup({𝑛 ∈ ℕ0 ∣ (𝑃𝑛) ∥ 𝐴}, ℝ, < )
1615pczpre 16176 . . . . . . 7 ((𝑃 ∈ ℙ ∧ (𝐴 ∈ ℤ ∧ 𝐴 ≠ 0)) → (𝑃 pCnt 𝐴) = sup({𝑛 ∈ ℕ0 ∣ (𝑃𝑛) ∥ 𝐴}, ℝ, < ))
17163adant3 1126 . . . . . 6 ((𝑃 ∈ ℙ ∧ (𝐴 ∈ ℤ ∧ 𝐴 ≠ 0) ∧ 𝐵 ∈ ℕ) → (𝑃 pCnt 𝐴) = sup({𝑛 ∈ ℕ0 ∣ (𝑃𝑛) ∥ 𝐴}, ℝ, < ))
18 nnz 11996 . . . . . . . . 9 (𝐵 ∈ ℕ → 𝐵 ∈ ℤ)
19 nnne0 11663 . . . . . . . . 9 (𝐵 ∈ ℕ → 𝐵 ≠ 0)
2018, 19jca 514 . . . . . . . 8 (𝐵 ∈ ℕ → (𝐵 ∈ ℤ ∧ 𝐵 ≠ 0))
21 eqid 2819 . . . . . . . . 9 sup({𝑛 ∈ ℕ0 ∣ (𝑃𝑛) ∥ 𝐵}, ℝ, < ) = sup({𝑛 ∈ ℕ0 ∣ (𝑃𝑛) ∥ 𝐵}, ℝ, < )
2221pczpre 16176 . . . . . . . 8 ((𝑃 ∈ ℙ ∧ (𝐵 ∈ ℤ ∧ 𝐵 ≠ 0)) → (𝑃 pCnt 𝐵) = sup({𝑛 ∈ ℕ0 ∣ (𝑃𝑛) ∥ 𝐵}, ℝ, < ))
2320, 22sylan2 594 . . . . . . 7 ((𝑃 ∈ ℙ ∧ 𝐵 ∈ ℕ) → (𝑃 pCnt 𝐵) = sup({𝑛 ∈ ℕ0 ∣ (𝑃𝑛) ∥ 𝐵}, ℝ, < ))
24233adant2 1125 . . . . . 6 ((𝑃 ∈ ℙ ∧ (𝐴 ∈ ℤ ∧ 𝐴 ≠ 0) ∧ 𝐵 ∈ ℕ) → (𝑃 pCnt 𝐵) = sup({𝑛 ∈ ℕ0 ∣ (𝑃𝑛) ∥ 𝐵}, ℝ, < ))
2517, 24oveq12d 7166 . . . . 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 2819 . . . . 5 (𝐴 / 𝐵) = (𝐴 / 𝐵)
2725, 26jctil 522 . . . 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 7155 . . . . . . 7 (𝑥 = 𝐴 → (𝑥 / 𝑦) = (𝐴 / 𝑦))
2928eqeq2d 2830 . . . . . 6 (𝑥 = 𝐴 → ((𝐴 / 𝐵) = (𝑥 / 𝑦) ↔ (𝐴 / 𝐵) = (𝐴 / 𝑦)))
30 breq2 5061 . . . . . . . . . 10 (𝑥 = 𝐴 → ((𝑃𝑛) ∥ 𝑥 ↔ (𝑃𝑛) ∥ 𝐴))
3130rabbidv 3479 . . . . . . . . 9 (𝑥 = 𝐴 → {𝑛 ∈ ℕ0 ∣ (𝑃𝑛) ∥ 𝑥} = {𝑛 ∈ ℕ0 ∣ (𝑃𝑛) ∥ 𝐴})
3231supeq1d 8902 . . . . . . . 8 (𝑥 = 𝐴 → sup({𝑛 ∈ ℕ0 ∣ (𝑃𝑛) ∥ 𝑥}, ℝ, < ) = sup({𝑛 ∈ ℕ0 ∣ (𝑃𝑛) ∥ 𝐴}, ℝ, < ))
3332oveq1d 7163 . . . . . . 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 2830 . . . . . 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 632 . . . . 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 7156 . . . . . . 7 (𝑦 = 𝐵 → (𝐴 / 𝑦) = (𝐴 / 𝐵))
3736eqeq2d 2830 . . . . . 6 (𝑦 = 𝐵 → ((𝐴 / 𝐵) = (𝐴 / 𝑦) ↔ (𝐴 / 𝐵) = (𝐴 / 𝐵)))
38 breq2 5061 . . . . . . . . . 10 (𝑦 = 𝐵 → ((𝑃𝑛) ∥ 𝑦 ↔ (𝑃𝑛) ∥ 𝐵))
3938rabbidv 3479 . . . . . . . . 9 (𝑦 = 𝐵 → {𝑛 ∈ ℕ0 ∣ (𝑃𝑛) ∥ 𝑦} = {𝑛 ∈ ℕ0 ∣ (𝑃𝑛) ∥ 𝐵})
4039supeq1d 8902 . . . . . . . 8 (𝑦 = 𝐵 → sup({𝑛 ∈ ℕ0 ∣ (𝑃𝑛) ∥ 𝑦}, ℝ, < ) = sup({𝑛 ∈ ℕ0 ∣ (𝑃𝑛) ∥ 𝐵}, ℝ, < ))
4140oveq2d 7164 . . . . . . 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 2830 . . . . . 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 632 . . . . 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 3633 . . . 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 1365 . . 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 7181 . . . 4 ((𝑃 pCnt 𝐴) − (𝑃 pCnt 𝐵)) ∈ V
4711, 12pceu 16175 . . . . 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 834 . . . 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 2823 . . . . . . 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 630 . . . . . 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 3298 . . . . 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 6337 . . . 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 589 . . 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 234 . 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 2854 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 208  wa 398  w3a 1081   = wceq 1530  wcel 2107  ∃!weu 2647  wne 3014  wrex 3137  {crab 3140  Vcvv 3493   class class class wbr 5057  cio 6305  (class class class)co 7148  supcsup 8896  cr 10528  0cc0 10529   < clt 10667  cmin 10862   / cdiv 11289  cn 11630  0cn0 11889  cz 11973  cq 12340  cexp 13421  cdvds 15599  cprime 16007   pCnt cpc 16165
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2791  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7453  ax-cnex 10585  ax-resscn 10586  ax-1cn 10587  ax-icn 10588  ax-addcl 10589  ax-addrcl 10590  ax-mulcl 10591  ax-mulrcl 10592  ax-mulcom 10593  ax-addass 10594  ax-mulass 10595  ax-distr 10596  ax-i2m1 10597  ax-1ne0 10598  ax-1rid 10599  ax-rnegex 10600  ax-rrecex 10601  ax-cnre 10602  ax-pre-lttri 10603  ax-pre-lttrn 10604  ax-pre-ltadd 10605  ax-pre-mulgt0 10606  ax-pre-sup 10607
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1082  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2616  df-eu 2648  df-clab 2798  df-cleq 2812  df-clel 2891  df-nfc 2961  df-ne 3015  df-nel 3122  df-ral 3141  df-rex 3142  df-reu 3143  df-rmo 3144  df-rab 3145  df-v 3495  df-sbc 3771  df-csb 3882  df-dif 3937  df-un 3939  df-in 3941  df-ss 3950  df-pss 3952  df-nul 4290  df-if 4466  df-pw 4539  df-sn 4560  df-pr 4562  df-tp 4564  df-op 4566  df-uni 4831  df-iun 4912  df-br 5058  df-opab 5120  df-mpt 5138  df-tr 5164  df-id 5453  df-eprel 5458  df-po 5467  df-so 5468  df-fr 5507  df-we 5509  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-pred 6141  df-ord 6187  df-on 6188  df-lim 6189  df-suc 6190  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-f1 6353  df-fo 6354  df-f1o 6355  df-fv 6356  df-riota 7106  df-ov 7151  df-oprab 7152  df-mpo 7153  df-om 7573  df-1st 7681  df-2nd 7682  df-wrecs 7939  df-recs 8000  df-rdg 8038  df-1o 8094  df-2o 8095  df-er 8281  df-en 8502  df-dom 8503  df-sdom 8504  df-fin 8505  df-sup 8898  df-inf 8899  df-pnf 10669  df-mnf 10670  df-xr 10671  df-ltxr 10672  df-le 10673  df-sub 10864  df-neg 10865  df-div 11290  df-nn 11631  df-2 11692  df-3 11693  df-n0 11890  df-z 11974  df-uz 12236  df-q 12341  df-rp 12382  df-fl 13154  df-mod 13230  df-seq 13362  df-exp 13422  df-cj 14450  df-re 14451  df-im 14452  df-sqrt 14586  df-abs 14587  df-dvds 15600  df-gcd 15836  df-prm 16008  df-pc 16166
This theorem is referenced by:  pcqmul  16182  pcqcl  16185  pcid  16201  pcneg  16202  pc2dvds  16207  pcz  16209  pcaddlem  16216  pcadd  16217  pcmpt2  16221  pcbc  16228  sylow1lem1  18715  chtublem  25779
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