Theorem pcdiv 16875

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 1133	. . 3 ⊢ ((𝑃 ∈ ℙ ∧ (𝐴 ∈ ℤ ∧ 𝐴 ≠ 0) ∧ 𝐵 ∈ ℕ) → 𝑃 ∈ ℙ)
2		simp2l 1196	. . . 4 ⊢ ((𝑃 ∈ ℙ ∧ (𝐴 ∈ ℤ ∧ 𝐴 ≠ 0) ∧ 𝐵 ∈ ℕ) → 𝐴 ∈ ℤ)
3		simp3 1135	. . . 4 ⊢ ((𝑃 ∈ ℙ ∧ (𝐴 ∈ ℤ ∧ 𝐴 ≠ 0) ∧ 𝐵 ∈ ℕ) → 𝐵 ∈ ℕ)
4		znq 12998	. . . 4 ⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℕ) → (𝐴 / 𝐵) ∈ ℚ)
5	2, 3, 4	syl2anc 582	. . 3 ⊢ ((𝑃 ∈ ℙ ∧ (𝐴 ∈ ℤ ∧ 𝐴 ≠ 0) ∧ 𝐵 ∈ ℕ) → (𝐴 / 𝐵) ∈ ℚ)
6	2	zcnd 12729	. . . 4 ⊢ ((𝑃 ∈ ℙ ∧ (𝐴 ∈ ℤ ∧ 𝐴 ≠ 0) ∧ 𝐵 ∈ ℕ) → 𝐴 ∈ ℂ)
7	3	nncnd 12290	. . . 4 ⊢ ((𝑃 ∈ ℙ ∧ (𝐴 ∈ ℤ ∧ 𝐴 ≠ 0) ∧ 𝐵 ∈ ℕ) → 𝐵 ∈ ℂ)
8		simp2r 1197	. . . 4 ⊢ ((𝑃 ∈ ℙ ∧ (𝐴 ∈ ℤ ∧ 𝐴 ≠ 0) ∧ 𝐵 ∈ ℕ) → 𝐴 ≠ 0)
9	3	nnne0d 12324	. . . 4 ⊢ ((𝑃 ∈ ℙ ∧ (𝐴 ∈ ℤ ∧ 𝐴 ≠ 0) ∧ 𝐵 ∈ ℕ) → 𝐵 ≠ 0)
10	6, 7, 8, 9	divne0d 12067	. . 3 ⊢ ((𝑃 ∈ ℙ ∧ (𝐴 ∈ ℤ ∧ 𝐴 ≠ 0) ∧ 𝐵 ∈ ℕ) → (𝐴 / 𝐵) ≠ 0)
11		eqid 2729	. . . 4 ⊢ sup({𝑛 ∈ ℕ0 ∣ (𝑃↑𝑛) ∥ 𝑥}, ℝ, < ) = sup({𝑛 ∈ ℕ0 ∣ (𝑃↑𝑛) ∥ 𝑥}, ℝ, < )
12		eqid 2729	. . . 4 ⊢ sup({𝑛 ∈ ℕ0 ∣ (𝑃↑𝑛) ∥ 𝑦}, ℝ, < ) = sup({𝑛 ∈ ℕ0 ∣ (𝑃↑𝑛) ∥ 𝑦}, ℝ, < )
13	11, 12	pcval 16867	. . 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 835	. 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 2729	. . . . . . . 8 ⊢ sup({𝑛 ∈ ℕ0 ∣ (𝑃↑𝑛) ∥ 𝐴}, ℝ, < ) = sup({𝑛 ∈ ℕ0 ∣ (𝑃↑𝑛) ∥ 𝐴}, ℝ, < )
16	15	pczpre 16870	. . . . . . 7 ⊢ ((𝑃 ∈ ℙ ∧ (𝐴 ∈ ℤ ∧ 𝐴 ≠ 0)) → (𝑃 pCnt 𝐴) = sup({𝑛 ∈ ℕ0 ∣ (𝑃↑𝑛) ∥ 𝐴}, ℝ, < ))
17	16	3adant3 1129	. . . . . 6 ⊢ ((𝑃 ∈ ℙ ∧ (𝐴 ∈ ℤ ∧ 𝐴 ≠ 0) ∧ 𝐵 ∈ ℕ) → (𝑃 pCnt 𝐴) = sup({𝑛 ∈ ℕ0 ∣ (𝑃↑𝑛) ∥ 𝐴}, ℝ, < ))
18		nnz 12641	. . . . . . . . 9 ⊢ (𝐵 ∈ ℕ → 𝐵 ∈ ℤ)
19		nnne0 12308	. . . . . . . . 9 ⊢ (𝐵 ∈ ℕ → 𝐵 ≠ 0)
20	18, 19	jca 510	. . . . . . . 8 ⊢ (𝐵 ∈ ℕ → (𝐵 ∈ ℤ ∧ 𝐵 ≠ 0))
21		eqid 2729	. . . . . . . . 9 ⊢ sup({𝑛 ∈ ℕ0 ∣ (𝑃↑𝑛) ∥ 𝐵}, ℝ, < ) = sup({𝑛 ∈ ℕ0 ∣ (𝑃↑𝑛) ∥ 𝐵}, ℝ, < )
22	21	pczpre 16870	. . . . . . . 8 ⊢ ((𝑃 ∈ ℙ ∧ (𝐵 ∈ ℤ ∧ 𝐵 ≠ 0)) → (𝑃 pCnt 𝐵) = sup({𝑛 ∈ ℕ0 ∣ (𝑃↑𝑛) ∥ 𝐵}, ℝ, < ))
23	20, 22	sylan2 591	. . . . . . 7 ⊢ ((𝑃 ∈ ℙ ∧ 𝐵 ∈ ℕ) → (𝑃 pCnt 𝐵) = sup({𝑛 ∈ ℕ0 ∣ (𝑃↑𝑛) ∥ 𝐵}, ℝ, < ))
24	23	3adant2 1128	. . . . . 6 ⊢ ((𝑃 ∈ ℙ ∧ (𝐴 ∈ ℤ ∧ 𝐴 ≠ 0) ∧ 𝐵 ∈ ℕ) → (𝑃 pCnt 𝐵) = sup({𝑛 ∈ ℕ0 ∣ (𝑃↑𝑛) ∥ 𝐵}, ℝ, < ))
25	17, 24	oveq12d 7448	. . . . 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 2729	. . . . 5 ⊢ (𝐴 / 𝐵) = (𝐴 / 𝐵)
27	25, 26	jctil 518	. . . 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 7437	. . . . . . 7 ⊢ (𝑥 = 𝐴 → (𝑥 / 𝑦) = (𝐴 / 𝑦))
29	28	eqeq2d 2740	. . . . . 6 ⊢ (𝑥 = 𝐴 → ((𝐴 / 𝐵) = (𝑥 / 𝑦) ↔ (𝐴 / 𝐵) = (𝐴 / 𝑦)))
30		breq2 5160	. . . . . . . . . 10 ⊢ (𝑥 = 𝐴 → ((𝑃↑𝑛) ∥ 𝑥 ↔ (𝑃↑𝑛) ∥ 𝐴))
31	30	rabbidv 3436	. . . . . . . . 9 ⊢ (𝑥 = 𝐴 → {𝑛 ∈ ℕ0 ∣ (𝑃↑𝑛) ∥ 𝑥} = {𝑛 ∈ ℕ0 ∣ (𝑃↑𝑛) ∥ 𝐴})
32	31	supeq1d 9496	. . . . . . . 8 ⊢ (𝑥 = 𝐴 → sup({𝑛 ∈ ℕ0 ∣ (𝑃↑𝑛) ∥ 𝑥}, ℝ, < ) = sup({𝑛 ∈ ℕ0 ∣ (𝑃↑𝑛) ∥ 𝐴}, ℝ, < ))
33	32	oveq1d 7445	. . . . . . 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 2740	. . . . . 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 630	. . . . 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 7438	. . . . . . 7 ⊢ (𝑦 = 𝐵 → (𝐴 / 𝑦) = (𝐴 / 𝐵))
37	36	eqeq2d 2740	. . . . . 6 ⊢ (𝑦 = 𝐵 → ((𝐴 / 𝐵) = (𝐴 / 𝑦) ↔ (𝐴 / 𝐵) = (𝐴 / 𝐵)))
38		breq2 5160	. . . . . . . . . 10 ⊢ (𝑦 = 𝐵 → ((𝑃↑𝑛) ∥ 𝑦 ↔ (𝑃↑𝑛) ∥ 𝐵))
39	38	rabbidv 3436	. . . . . . . . 9 ⊢ (𝑦 = 𝐵 → {𝑛 ∈ ℕ0 ∣ (𝑃↑𝑛) ∥ 𝑦} = {𝑛 ∈ ℕ0 ∣ (𝑃↑𝑛) ∥ 𝐵})
40	39	supeq1d 9496	. . . . . . . 8 ⊢ (𝑦 = 𝐵 → sup({𝑛 ∈ ℕ0 ∣ (𝑃↑𝑛) ∥ 𝑦}, ℝ, < ) = sup({𝑛 ∈ ℕ0 ∣ (𝑃↑𝑛) ∥ 𝐵}, ℝ, < ))
41	40	oveq2d 7446	. . . . . . 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 2740	. . . . . 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 630	. . . . 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 3633	. . . 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 1368	. . 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 7463	. . . 4 ⊢ ((𝑃 pCnt 𝐴) − (𝑃 pCnt 𝐵)) ∈ V
47	11, 12	pceu 16869	. . . . 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 835	. . . 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 2733	. . . . . . 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 628	. . . . . 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 3214	. . . . 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 6547	. . . 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 585	. . 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 231	. 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 2769	1 ⊢ ((𝑃 ∈ ℙ ∧ (𝐴 ∈ ℤ ∧ 𝐴 ≠ 0) ∧ 𝐵 ∈ ℕ) → (𝑃 pCnt (𝐴 / 𝐵)) = ((𝑃 pCnt 𝐴) − (𝑃 pCnt 𝐵)))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 394   ∧ w3a 1084   = wceq 1534   ∈ wcel 2100  ∃!weu 2560   ≠ wne 2933  ∃wrex 3063  {crab 3428  Vcvv 3472   class class class wbr 5156  ℩cio 6508  (class class class)co 7430  supcsup 9490  ℝcr 11164  0cc0 11165   < clt 11305   − cmin 11501   / cdiv 11928  ℕcn 12274  ℕ₀cn0 12534  ℤcz 12620  ℚcq 12994  ↑cexp 14092   ∥ cdvds 16277  ℙcprime 16693   pCnt cpc 16859

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2102  ax-9 2110  ax-10 2133  ax-11 2150  ax-12 2170  ax-ext 2700  ax-sep 5307  ax-nul 5314  ax-pow 5373  ax-pr 5437  ax-un 7751  ax-cnex 11221  ax-resscn 11222  ax-1cn 11223  ax-icn 11224  ax-addcl 11225  ax-addrcl 11226  ax-mulcl 11227  ax-mulrcl 11228  ax-mulcom 11229  ax-addass 11230  ax-mulass 11231  ax-distr 11232  ax-i2m1 11233  ax-1ne0 11234  ax-1rid 11235  ax-rnegex 11236  ax-rrecex 11237  ax-cnre 11238  ax-pre-lttri 11239  ax-pre-lttrn 11240  ax-pre-ltadd 11241  ax-pre-mulgt0 11242  ax-pre-sup 11243

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2062  df-mo 2532  df-eu 2561  df-clab 2707  df-cleq 2721  df-clel 2806  df-nfc 2881  df-ne 2934  df-nel 3040  df-ral 3055  df-rex 3064  df-rmo 3373  df-reu 3374  df-rab 3429  df-v 3474  df-sbc 3789  df-csb 3905  df-dif 3962  df-un 3964  df-in 3966  df-ss 3976  df-pss 3979  df-nul 4336  df-if 4537  df-pw 4612  df-sn 4637  df-pr 4639  df-op 4643  df-uni 4919  df-iun 5008  df-br 5157  df-opab 5219  df-mpt 5240  df-tr 5274  df-id 5584  df-eprel 5590  df-po 5598  df-so 5599  df-fr 5641  df-we 5643  df-xp 5692  df-rel 5693  df-cnv 5694  df-co 5695  df-dm 5696  df-rn 5697  df-res 5698  df-ima 5699  df-pred 6317  df-ord 6383  df-on 6384  df-lim 6385  df-suc 6386  df-iota 6510  df-fun 6560  df-fn 6561  df-f 6562  df-f1 6563  df-fo 6564  df-f1o 6565  df-fv 6566  df-riota 7386  df-ov 7433  df-oprab 7434  df-mpo 7435  df-om 7885  df-1st 8011  df-2nd 8012  df-frecs 8304  df-wrecs 8335  df-recs 8409  df-rdg 8448  df-1o 8504  df-2o 8505  df-er 8742  df-en 8983  df-dom 8984  df-sdom 8985  df-fin 8986  df-sup 9492  df-inf 9493  df-pnf 11307  df-mnf 11308  df-xr 11309  df-ltxr 11310  df-le 11311  df-sub 11503  df-neg 11504  df-div 11929  df-nn 12275  df-2 12337  df-3 12338  df-n0 12535  df-z 12621  df-uz 12885  df-q 12995  df-rp 13039  df-fl 13823  df-mod 13901  df-seq 14033  df-exp 14093  df-cj 15125  df-re 15126  df-im 15127  df-sqrt 15261  df-abs 15262  df-dvds 16278  df-gcd 16516  df-prm 16694  df-pc 16860

This theorem is referenced by:  pcqmul  16876  pcqcl  16879  pcid  16896  pcneg  16897  pc2dvds  16902  pcz  16904  pcaddlem  16911  pcadd  16912  pcmpt2  16916  pcbc  16923  sylow1lem1  19618  chtublem  27263