Theorem pcdiv 17074

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.

Step	Hyp	Ref	Expression
1		simp1 1154	. . 3 ⊢ ((𝑃 ∈ ℙ ∧ (𝐴 ∈ ℤ ∧ 𝐴 ≠ 0) ∧ 𝐵 ∈ ℕ) → 𝑃 ∈ ℙ)
2		simp2l 1218	. . . 4 ⊢ ((𝑃 ∈ ℙ ∧ (𝐴 ∈ ℤ ∧ 𝐴 ≠ 0) ∧ 𝐵 ∈ ℕ) → 𝐴 ∈ ℤ)
3		simp3 1156	. . . 4 ⊢ ((𝑃 ∈ ℙ ∧ (𝐴 ∈ ℤ ∧ 𝐴 ≠ 0) ∧ 𝐵 ∈ ℕ) → 𝐵 ∈ ℕ)
4		znq 13119	. . . 4 ⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℕ) → (𝐴 / 𝐵) ∈ ℚ)
5	2, 3, 4	syl2anc 596	. . 3 ⊢ ((𝑃 ∈ ℙ ∧ (𝐴 ∈ ℤ ∧ 𝐴 ≠ 0) ∧ 𝐵 ∈ ℕ) → (𝐴 / 𝐵) ∈ ℚ)
6	2	zcnd 12844	. . . 4 ⊢ ((𝑃 ∈ ℙ ∧ (𝐴 ∈ ℤ ∧ 𝐴 ≠ 0) ∧ 𝐵 ∈ ℕ) → 𝐴 ∈ ℂ)
7	3	nncnd 12391	. . . 4 ⊢ ((𝑃 ∈ ℙ ∧ (𝐴 ∈ ℤ ∧ 𝐴 ≠ 0) ∧ 𝐵 ∈ ℕ) → 𝐵 ∈ ℂ)
8		simp2r 1219	. . . 4 ⊢ ((𝑃 ∈ ℙ ∧ (𝐴 ∈ ℤ ∧ 𝐴 ≠ 0) ∧ 𝐵 ∈ ℕ) → 𝐴 ≠ 0)
9	3	nnne0d 12428	. . . 4 ⊢ ((𝑃 ∈ ℙ ∧ (𝐴 ∈ ℤ ∧ 𝐴 ≠ 0) ∧ 𝐵 ∈ ℕ) → 𝐵 ≠ 0)
10	6, 7, 8, 9	divne0d 12149	. . 3 ⊢ ((𝑃 ∈ ℙ ∧ (𝐴 ∈ ℤ ∧ 𝐴 ≠ 0) ∧ 𝐵 ∈ ℕ) → (𝐴 / 𝐵) ≠ 0)
11		eqid 2763	. . . 4 ⊢ sup({𝑛 ∈ ℕ0 ∣ (𝑃↑𝑛) ∥ 𝑥}, ℝ, < ) = sup({𝑛 ∈ ℕ0 ∣ (𝑃↑𝑛) ∥ 𝑥}, ℝ, < )
12		eqid 2763	. . . 4 ⊢ sup({𝑛 ∈ ℕ0 ∣ (𝑃↑𝑛) ∥ 𝑦}, ℝ, < ) = sup({𝑛 ∈ ℕ0 ∣ (𝑃↑𝑛) ∥ 𝑦}, ℝ, < )
13	11, 12	pcval 17066	. . 3 ⊢ ((𝑃 ∈ ℙ ∧ ((𝐴 / 𝐵) ∈ ℚ ∧ (𝐴 / 𝐵) ≠ 0)) → (𝑃 pCnt (𝐴 / 𝐵)) = (℩𝑧∃𝑥 ∈ ℤ ∃𝑦 ∈ ℕ ((𝐴 / 𝐵) = (𝑥 / 𝑦) ∧ 𝑧 = (sup({𝑛 ∈ ℕ0 ∣ (𝑃↑𝑛) ∥ 𝑥}, ℝ, < ) − sup({𝑛 ∈ ℕ0 ∣ (𝑃↑𝑛) ∥ 𝑦}, ℝ, < )))))
14	1, 5, 10, 13	syl12anc 850	. 2 ⊢ ((𝑃 ∈ ℙ ∧ (𝐴 ∈ ℤ ∧ 𝐴 ≠ 0) ∧ 𝐵 ∈ ℕ) → (𝑃 pCnt (𝐴 / 𝐵)) = (℩𝑧∃𝑥 ∈ ℤ ∃𝑦 ∈ ℕ ((𝐴 / 𝐵) = (𝑥 / 𝑦) ∧ 𝑧 = (sup({𝑛 ∈ ℕ0 ∣ (𝑃↑𝑛) ∥ 𝑥}, ℝ, < ) − sup({𝑛 ∈ ℕ0 ∣ (𝑃↑𝑛) ∥ 𝑦}, ℝ, < )))))
15		eqid 2763	. . . . . . . 8 ⊢ sup({𝑛 ∈ ℕ0 ∣ (𝑃↑𝑛) ∥ 𝐴}, ℝ, < ) = sup({𝑛 ∈ ℕ0 ∣ (𝑃↑𝑛) ∥ 𝐴}, ℝ, < )
16	15	pczpre 17069	. . . . . . 7 ⊢ ((𝑃 ∈ ℙ ∧ (𝐴 ∈ ℤ ∧ 𝐴 ≠ 0)) → (𝑃 pCnt 𝐴) = sup({𝑛 ∈ ℕ0 ∣ (𝑃↑𝑛) ∥ 𝐴}, ℝ, < ))
17	16	3adant3 1150	. . . . . 6 ⊢ ((𝑃 ∈ ℙ ∧ (𝐴 ∈ ℤ ∧ 𝐴 ≠ 0) ∧ 𝐵 ∈ ℕ) → (𝑃 pCnt 𝐴) = sup({𝑛 ∈ ℕ0 ∣ (𝑃↑𝑛) ∥ 𝐴}, ℝ, < ))
18		nnz 12754	. . . . . . . . 9 ⊢ (𝐵 ∈ ℕ → 𝐵 ∈ ℤ)
19		nnne0 12412	. . . . . . . . 9 ⊢ (𝐵 ∈ ℕ → 𝐵 ≠ 0)
20	18, 19	jca 521	. . . . . . . 8 ⊢ (𝐵 ∈ ℕ → (𝐵 ∈ ℤ ∧ 𝐵 ≠ 0))
21		eqid 2763	. . . . . . . . 9 ⊢ sup({𝑛 ∈ ℕ0 ∣ (𝑃↑𝑛) ∥ 𝐵}, ℝ, < ) = sup({𝑛 ∈ ℕ0 ∣ (𝑃↑𝑛) ∥ 𝐵}, ℝ, < )
22	21	pczpre 17069	. . . . . . . 8 ⊢ ((𝑃 ∈ ℙ ∧ (𝐵 ∈ ℤ ∧ 𝐵 ≠ 0)) → (𝑃 pCnt 𝐵) = sup({𝑛 ∈ ℕ0 ∣ (𝑃↑𝑛) ∥ 𝐵}, ℝ, < ))
23	20, 22	sylan2 605	. . . . . . 7 ⊢ ((𝑃 ∈ ℙ ∧ 𝐵 ∈ ℕ) → (𝑃 pCnt 𝐵) = sup({𝑛 ∈ ℕ0 ∣ (𝑃↑𝑛) ∥ 𝐵}, ℝ, < ))
24	23	3adant2 1149	. . . . . 6 ⊢ ((𝑃 ∈ ℙ ∧ (𝐴 ∈ ℤ ∧ 𝐴 ≠ 0) ∧ 𝐵 ∈ ℕ) → (𝑃 pCnt 𝐵) = sup({𝑛 ∈ ℕ0 ∣ (𝑃↑𝑛) ∥ 𝐵}, ℝ, < ))
25	17, 24	oveq12d 7469	. . . . 5 ⊢ ((𝑃 ∈ ℙ ∧ (𝐴 ∈ ℤ ∧ 𝐴 ≠ 0) ∧ 𝐵 ∈ ℕ) → ((𝑃 pCnt 𝐴) − (𝑃 pCnt 𝐵)) = (sup({𝑛 ∈ ℕ0 ∣ (𝑃↑𝑛) ∥ 𝐴}, ℝ, < ) − sup({𝑛 ∈ ℕ0 ∣ (𝑃↑𝑛) ∥ 𝐵}, ℝ, < )))
26		eqid 2763	. . . . 5 ⊢ (𝐴 / 𝐵) = (𝐴 / 𝐵)
27	25, 26	jctil 529	. . . 4 ⊢ ((𝑃 ∈ ℙ ∧ (𝐴 ∈ ℤ ∧ 𝐴 ≠ 0) ∧ 𝐵 ∈ ℕ) → ((𝐴 / 𝐵) = (𝐴 / 𝐵) ∧ ((𝑃 pCnt 𝐴) − (𝑃 pCnt 𝐵)) = (sup({𝑛 ∈ ℕ0 ∣ (𝑃↑𝑛) ∥ 𝐴}, ℝ, < ) − sup({𝑛 ∈ ℕ0 ∣ (𝑃↑𝑛) ∥ 𝐵}, ℝ, < ))))
28		oveq1 7458	. . . . . . 7 ⊢ (𝑥 = 𝐴 → (𝑥 / 𝑦) = (𝐴 / 𝑦))
29	28	eqeq2d 2774	. . . . . 6 ⊢ (𝑥 = 𝐴 → ((𝐴 / 𝐵) = (𝑥 / 𝑦) ↔ (𝐴 / 𝐵) = (𝐴 / 𝑦)))
30		breq2 5112	. . . . . . . . . 10 ⊢ (𝑥 = 𝐴 → ((𝑃↑𝑛) ∥ 𝑥 ↔ (𝑃↑𝑛) ∥ 𝐴))
31	30	rabbidv 3423	. . . . . . . . 9 ⊢ (𝑥 = 𝐴 → {𝑛 ∈ ℕ0 ∣ (𝑃↑𝑛) ∥ 𝑥} = {𝑛 ∈ ℕ0 ∣ (𝑃↑𝑛) ∥ 𝐴})
32	31	supeq1d 9475	. . . . . . . 8 ⊢ (𝑥 = 𝐴 → sup({𝑛 ∈ ℕ0 ∣ (𝑃↑𝑛) ∥ 𝑥}, ℝ, < ) = sup({𝑛 ∈ ℕ0 ∣ (𝑃↑𝑛) ∥ 𝐴}, ℝ, < ))
33	32	oveq1d 7466	. . . . . . 7 ⊢ (𝑥 = 𝐴 → (sup({𝑛 ∈ ℕ0 ∣ (𝑃↑𝑛) ∥ 𝑥}, ℝ, < ) − sup({𝑛 ∈ ℕ0 ∣ (𝑃↑𝑛) ∥ 𝑦}, ℝ, < )) = (sup({𝑛 ∈ ℕ0 ∣ (𝑃↑𝑛) ∥ 𝐴}, ℝ, < ) − sup({𝑛 ∈ ℕ0 ∣ (𝑃↑𝑛) ∥ 𝑦}, ℝ, < )))
34	33	eqeq2d 2774	. . . . . 6 ⊢ (𝑥 = 𝐴 → (((𝑃 pCnt 𝐴) − (𝑃 pCnt 𝐵)) = (sup({𝑛 ∈ ℕ0 ∣ (𝑃↑𝑛) ∥ 𝑥}, ℝ, < ) − sup({𝑛 ∈ ℕ0 ∣ (𝑃↑𝑛) ∥ 𝑦}, ℝ, < )) ↔ ((𝑃 pCnt 𝐴) − (𝑃 pCnt 𝐵)) = (sup({𝑛 ∈ ℕ0 ∣ (𝑃↑𝑛) ∥ 𝐴}, ℝ, < ) − sup({𝑛 ∈ ℕ0 ∣ (𝑃↑𝑛) ∥ 𝑦}, ℝ, < ))))
35	29, 34	anbi12d 644	. . . . 5 ⊢ (𝑥 = 𝐴 → (((𝐴 / 𝐵) = (𝑥 / 𝑦) ∧ ((𝑃 pCnt 𝐴) − (𝑃 pCnt 𝐵)) = (sup({𝑛 ∈ ℕ0 ∣ (𝑃↑𝑛) ∥ 𝑥}, ℝ, < ) − sup({𝑛 ∈ ℕ0 ∣ (𝑃↑𝑛) ∥ 𝑦}, ℝ, < ))) ↔ ((𝐴 / 𝐵) = (𝐴 / 𝑦) ∧ ((𝑃 pCnt 𝐴) − (𝑃 pCnt 𝐵)) = (sup({𝑛 ∈ ℕ0 ∣ (𝑃↑𝑛) ∥ 𝐴}, ℝ, < ) − sup({𝑛 ∈ ℕ0 ∣ (𝑃↑𝑛) ∥ 𝑦}, ℝ, < )))))
36		oveq2 7459	. . . . . . 7 ⊢ (𝑦 = 𝐵 → (𝐴 / 𝑦) = (𝐴 / 𝐵))
37	36	eqeq2d 2774	. . . . . 6 ⊢ (𝑦 = 𝐵 → ((𝐴 / 𝐵) = (𝐴 / 𝑦) ↔ (𝐴 / 𝐵) = (𝐴 / 𝐵)))
38		breq2 5112	. . . . . . . . . 10 ⊢ (𝑦 = 𝐵 → ((𝑃↑𝑛) ∥ 𝑦 ↔ (𝑃↑𝑛) ∥ 𝐵))
39	38	rabbidv 3423	. . . . . . . . 9 ⊢ (𝑦 = 𝐵 → {𝑛 ∈ ℕ0 ∣ (𝑃↑𝑛) ∥ 𝑦} = {𝑛 ∈ ℕ0 ∣ (𝑃↑𝑛) ∥ 𝐵})
40	39	supeq1d 9475	. . . . . . . 8 ⊢ (𝑦 = 𝐵 → sup({𝑛 ∈ ℕ0 ∣ (𝑃↑𝑛) ∥ 𝑦}, ℝ, < ) = sup({𝑛 ∈ ℕ0 ∣ (𝑃↑𝑛) ∥ 𝐵}, ℝ, < ))
41	40	oveq2d 7467	. . . . . . 7 ⊢ (𝑦 = 𝐵 → (sup({𝑛 ∈ ℕ0 ∣ (𝑃↑𝑛) ∥ 𝐴}, ℝ, < ) − sup({𝑛 ∈ ℕ0 ∣ (𝑃↑𝑛) ∥ 𝑦}, ℝ, < )) = (sup({𝑛 ∈ ℕ0 ∣ (𝑃↑𝑛) ∥ 𝐴}, ℝ, < ) − sup({𝑛 ∈ ℕ0 ∣ (𝑃↑𝑛) ∥ 𝐵}, ℝ, < )))
42	41	eqeq2d 2774	. . . . . 6 ⊢ (𝑦 = 𝐵 → (((𝑃 pCnt 𝐴) − (𝑃 pCnt 𝐵)) = (sup({𝑛 ∈ ℕ0 ∣ (𝑃↑𝑛) ∥ 𝐴}, ℝ, < ) − sup({𝑛 ∈ ℕ0 ∣ (𝑃↑𝑛) ∥ 𝑦}, ℝ, < )) ↔ ((𝑃 pCnt 𝐴) − (𝑃 pCnt 𝐵)) = (sup({𝑛 ∈ ℕ0 ∣ (𝑃↑𝑛) ∥ 𝐴}, ℝ, < ) − sup({𝑛 ∈ ℕ0 ∣ (𝑃↑𝑛) ∥ 𝐵}, ℝ, < ))))
43	37, 42	anbi12d 644	. . . . 5 ⊢ (𝑦 = 𝐵 → (((𝐴 / 𝐵) = (𝐴 / 𝑦) ∧ ((𝑃 pCnt 𝐴) − (𝑃 pCnt 𝐵)) = (sup({𝑛 ∈ ℕ0 ∣ (𝑃↑𝑛) ∥ 𝐴}, ℝ, < ) − sup({𝑛 ∈ ℕ0 ∣ (𝑃↑𝑛) ∥ 𝑦}, ℝ, < ))) ↔ ((𝐴 / 𝐵) = (𝐴 / 𝐵) ∧ ((𝑃 pCnt 𝐴) − (𝑃 pCnt 𝐵)) = (sup({𝑛 ∈ ℕ0 ∣ (𝑃↑𝑛) ∥ 𝐴}, ℝ, < ) − sup({𝑛 ∈ ℕ0 ∣ (𝑃↑𝑛) ∥ 𝐵}, ℝ, < )))))
44	35, 43	rspc2ev 3592	. . . 4 ⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℕ ∧ ((𝐴 / 𝐵) = (𝐴 / 𝐵) ∧ ((𝑃 pCnt 𝐴) − (𝑃 pCnt 𝐵)) = (sup({𝑛 ∈ ℕ0 ∣ (𝑃↑𝑛) ∥ 𝐴}, ℝ, < ) − sup({𝑛 ∈ ℕ0 ∣ (𝑃↑𝑛) ∥ 𝐵}, ℝ, < )))) → ∃𝑥 ∈ ℤ ∃𝑦 ∈ ℕ ((𝐴 / 𝐵) = (𝑥 / 𝑦) ∧ ((𝑃 pCnt 𝐴) − (𝑃 pCnt 𝐵)) = (sup({𝑛 ∈ ℕ0 ∣ (𝑃↑𝑛) ∥ 𝑥}, ℝ, < ) − sup({𝑛 ∈ ℕ0 ∣ (𝑃↑𝑛) ∥ 𝑦}, ℝ, < ))))
45	2, 3, 27, 44	syl3anc 1399	. . 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 ∣ (𝑃↑𝑛) ∥ 𝑥}, ℝ, < ) − sup({𝑛 ∈ ℕ0 ∣ (𝑃↑𝑛) ∥ 𝑦}, ℝ, < ))))
46		ovex 7484	. . . 4 ⊢ ((𝑃 pCnt 𝐴) − (𝑃 pCnt 𝐵)) ∈ V
47	11, 12	pceu 17068	. . . . 5 ⊢ ((𝑃 ∈ ℙ ∧ ((𝐴 / 𝐵) ∈ ℚ ∧ (𝐴 / 𝐵) ≠ 0)) → ∃!𝑧∃𝑥 ∈ ℤ ∃𝑦 ∈ ℕ ((𝐴 / 𝐵) = (𝑥 / 𝑦) ∧ 𝑧 = (sup({𝑛 ∈ ℕ0 ∣ (𝑃↑𝑛) ∥ 𝑥}, ℝ, < ) − sup({𝑛 ∈ ℕ0 ∣ (𝑃↑𝑛) ∥ 𝑦}, ℝ, < ))))
48	1, 5, 10, 47	syl12anc 850	. . . 4 ⊢ ((𝑃 ∈ ℙ ∧ (𝐴 ∈ ℤ ∧ 𝐴 ≠ 0) ∧ 𝐵 ∈ ℕ) → ∃!𝑧∃𝑥 ∈ ℤ ∃𝑦 ∈ ℕ ((𝐴 / 𝐵) = (𝑥 / 𝑦) ∧ 𝑧 = (sup({𝑛 ∈ ℕ0 ∣ (𝑃↑𝑛) ∥ 𝑥}, ℝ, < ) − sup({𝑛 ∈ ℕ0 ∣ (𝑃↑𝑛) ∥ 𝑦}, ℝ, < ))))
49		eqeq1 2767	. . . . . . 7 ⊢ (𝑧 = ((𝑃 pCnt 𝐴) − (𝑃 pCnt 𝐵)) → (𝑧 = (sup({𝑛 ∈ ℕ0 ∣ (𝑃↑𝑛) ∥ 𝑥}, ℝ, < ) − sup({𝑛 ∈ ℕ0 ∣ (𝑃↑𝑛) ∥ 𝑦}, ℝ, < )) ↔ ((𝑃 pCnt 𝐴) − (𝑃 pCnt 𝐵)) = (sup({𝑛 ∈ ℕ0 ∣ (𝑃↑𝑛) ∥ 𝑥}, ℝ, < ) − sup({𝑛 ∈ ℕ0 ∣ (𝑃↑𝑛) ∥ 𝑦}, ℝ, < ))))
50	49	anbi2d 642	. . . . . 6 ⊢ (𝑧 = ((𝑃 pCnt 𝐴) − (𝑃 pCnt 𝐵)) → (((𝐴 / 𝐵) = (𝑥 / 𝑦) ∧ 𝑧 = (sup({𝑛 ∈ ℕ0 ∣ (𝑃↑𝑛) ∥ 𝑥}, ℝ, < ) − sup({𝑛 ∈ ℕ0 ∣ (𝑃↑𝑛) ∥ 𝑦}, ℝ, < ))) ↔ ((𝐴 / 𝐵) = (𝑥 / 𝑦) ∧ ((𝑃 pCnt 𝐴) − (𝑃 pCnt 𝐵)) = (sup({𝑛 ∈ ℕ0 ∣ (𝑃↑𝑛) ∥ 𝑥}, ℝ, < ) − sup({𝑛 ∈ ℕ0 ∣ (𝑃↑𝑛) ∥ 𝑦}, ℝ, < )))))
51	50	2rexbidv 3231	. . . . 5 ⊢ (𝑧 = ((𝑃 pCnt 𝐴) − (𝑃 pCnt 𝐵)) → (∃𝑥 ∈ ℤ ∃𝑦 ∈ ℕ ((𝐴 / 𝐵) = (𝑥 / 𝑦) ∧ 𝑧 = (sup({𝑛 ∈ ℕ0 ∣ (𝑃↑𝑛) ∥ 𝑥}, ℝ, < ) − sup({𝑛 ∈ ℕ0 ∣ (𝑃↑𝑛) ∥ 𝑦}, ℝ, < ))) ↔ ∃𝑥 ∈ ℤ ∃𝑦 ∈ ℕ ((𝐴 / 𝐵) = (𝑥 / 𝑦) ∧ ((𝑃 pCnt 𝐴) − (𝑃 pCnt 𝐵)) = (sup({𝑛 ∈ ℕ0 ∣ (𝑃↑𝑛) ∥ 𝑥}, ℝ, < ) − sup({𝑛 ∈ ℕ0 ∣ (𝑃↑𝑛) ∥ 𝑦}, ℝ, < )))))
52	51	iota2 6554	. . . 4 ⊢ ((((𝑃 pCnt 𝐴) − (𝑃 pCnt 𝐵)) ∈ V ∧ ∃!𝑧∃𝑥 ∈ ℤ ∃𝑦 ∈ ℕ ((𝐴 / 𝐵) = (𝑥 / 𝑦) ∧ 𝑧 = (sup({𝑛 ∈ ℕ0 ∣ (𝑃↑𝑛) ∥ 𝑥}, ℝ, < ) − sup({𝑛 ∈ ℕ0 ∣ (𝑃↑𝑛) ∥ 𝑦}, ℝ, < )))) → (∃𝑥 ∈ ℤ ∃𝑦 ∈ ℕ ((𝐴 / 𝐵) = (𝑥 / 𝑦) ∧ ((𝑃 pCnt 𝐴) − (𝑃 pCnt 𝐵)) = (sup({𝑛 ∈ ℕ0 ∣ (𝑃↑𝑛) ∥ 𝑥}, ℝ, < ) − sup({𝑛 ∈ ℕ0 ∣ (𝑃↑𝑛) ∥ 𝑦}, ℝ, < ))) ↔ (℩𝑧∃𝑥 ∈ ℤ ∃𝑦 ∈ ℕ ((𝐴 / 𝐵) = (𝑥 / 𝑦) ∧ 𝑧 = (sup({𝑛 ∈ ℕ0 ∣ (𝑃↑𝑛) ∥ 𝑥}, ℝ, < ) − sup({𝑛 ∈ ℕ0 ∣ (𝑃↑𝑛) ∥ 𝑦}, ℝ, < )))) = ((𝑃 pCnt 𝐴) − (𝑃 pCnt 𝐵))))
53	46, 48, 52	sylancr 599	. . 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 ∣ (𝑃↑𝑛) ∥ 𝑥}, ℝ, < ) − sup({𝑛 ∈ ℕ0 ∣ (𝑃↑𝑛) ∥ 𝑦}, ℝ, < ))) ↔ (℩𝑧∃𝑥 ∈ ℤ ∃𝑦 ∈ ℕ ((𝐴 / 𝐵) = (𝑥 / 𝑦) ∧ 𝑧 = (sup({𝑛 ∈ ℕ0 ∣ (𝑃↑𝑛) ∥ 𝑥}, ℝ, < ) − sup({𝑛 ∈ ℕ0 ∣ (𝑃↑𝑛) ∥ 𝑦}, ℝ, < )))) = ((𝑃 pCnt 𝐴) − (𝑃 pCnt 𝐵))))
54	45, 53	mpbid 235	. 2 ⊢ ((𝑃 ∈ ℙ ∧ (𝐴 ∈ ℤ ∧ 𝐴 ≠ 0) ∧ 𝐵 ∈ ℕ) → (℩𝑧∃𝑥 ∈ ℤ ∃𝑦 ∈ ℕ ((𝐴 / 𝐵) = (𝑥 / 𝑦) ∧ 𝑧 = (sup({𝑛 ∈ ℕ0 ∣ (𝑃↑𝑛) ∥ 𝑥}, ℝ, < ) − sup({𝑛 ∈ ℕ0 ∣ (𝑃↑𝑛) ∥ 𝑦}, ℝ, < )))) = ((𝑃 pCnt 𝐴) − (𝑃 pCnt 𝐵)))
55	14, 54	eqtrd 2798	1 ⊢ ((𝑃 ∈ ℙ ∧ (𝐴 ∈ ℤ ∧ 𝐴 ≠ 0) ∧ 𝐵 ∈ ℕ) → (𝑃 pCnt (𝐴 / 𝐵)) = ((𝑃 pCnt 𝐴) − (𝑃 pCnt 𝐵)))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   ∧ w3a 1103   = wceq 1570   ∈ wcel 2147  ∃!weu 2596   ≠ wne 2959  ∃wrex 3090  {crab 3416  Vcvv 3454   class class class wbr 5108  ℩cio 6518  (class class class)co 7451  supcsup 9469  ℝcr 11238  0cc0 11239   < clt 11382   − cmin 11580   / cdiv 12013  ℕcn 12375  ℕ₀cn0 12646  ℤcz 12733  ℚcq 13115  ↑cexp 14244   ∥ cdvds 16466  ℙcprime 16890   pCnt cpc 17058

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-sep 5255  ax-nul 5266  ax-pow 5334  ax-pr 5398  ax-un 7783  ax-cnex 11295  ax-resscn 11296  ax-1cn 11297  ax-icn 11298  ax-addcl 11299  ax-addrcl 11300  ax-mulcl 11301  ax-mulrcl 11302  ax-mulcom 11303  ax-addass 11304  ax-mulass 11305  ax-distr 11306  ax-i2m1 11307  ax-1ne0 11308  ax-1rid 11309  ax-rnegex 11310  ax-rrecex 11311  ax-cnre 11312  ax-pre-lttri 11313  ax-pre-lttrn 11314  ax-pre-ltadd 11315  ax-pre-mulgt0 11316  ax-pre-sup 11317

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-nel 3066  df-ral 3081  df-rex 3091  df-rmo 3369  df-reu 3370  df-rab 3417  df-v 3456  df-sbc 3743  df-csb 3852  df-ss 3901  df-pss 4017  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-op 4841  df-uni 4873  df-iun 4958  df-br 5109  df-opab 5174  df-mpt 5193  df-tr 5219  df-id 5553  df-eprel 5558  df-po 5566  df-so 5567  df-fr 5611  df-we 5613  df-xp 5664  df-rel 5665  df-cnv 5666  df-co 5667  df-dm 5668  df-rn 5669  df-res 5670  df-ima 5671  df-pred 6330  df-ord 6391  df-on 6392  df-lim 6393  df-suc 6394  df-iota 6520  df-fun 6568  df-fn 6569  df-f 6570  df-f1 6571  df-fo 6572  df-f1o 6573  df-fv 6574  df-riota 7408  df-ov 7454  df-oprab 7455  df-mpo 7456  df-om 7912  df-1st 8035  df-2nd 8036  df-frecs 8337  df-wrecs 8368  df-recs 8417  df-rdg 8456  df-1o 8514  df-2o 8515  df-er 8755  df-en 9012  df-dom 9013  df-sdom 9014  df-fin 9015  df-sup 9471  df-inf 9472  df-pnf 11384  df-mnf 11385  df-xr 11386  df-ltxr 11387  df-le 11388  df-sub 11582  df-neg 11583  df-div 12014  df-nn 12376  df-2 12445  df-3 12446  df-n0 12647  df-z 12734  df-uz 13006  df-q 13116  df-rp 13161  df-fl 13972  df-mod 14050  df-seq 14185  df-exp 14245  df-cj 15309  df-re 15310  df-im 15311  df-sqrt 15445  df-abs 15446  df-dvds 16467  df-gcd 16709  df-prm 16891  df-pc 17059

This theorem is used by:  pcqmul  17075  pcqcl  17078  pcid  17095  pcneg  17096  pc2dvds  17101  pcz  17103  pcaddlem  17110  pcadd  17111  pcmpt2  17115  pcbc  17122  sylow1lem1  19863  chtublem  27595