Theorem pcdiv 16890

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 1137	. . 3 ⊢ ((𝑃 ∈ ℙ ∧ (𝐴 ∈ ℤ ∧ 𝐴 ≠ 0) ∧ 𝐵 ∈ ℕ) → 𝑃 ∈ ℙ)
2		simp2l 1200	. . . 4 ⊢ ((𝑃 ∈ ℙ ∧ (𝐴 ∈ ℤ ∧ 𝐴 ≠ 0) ∧ 𝐵 ∈ ℕ) → 𝐴 ∈ ℤ)
3		simp3 1139	. . . 4 ⊢ ((𝑃 ∈ ℙ ∧ (𝐴 ∈ ℤ ∧ 𝐴 ≠ 0) ∧ 𝐵 ∈ ℕ) → 𝐵 ∈ ℕ)
4		znq 12994	. . . 4 ⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℕ) → (𝐴 / 𝐵) ∈ ℚ)
5	2, 3, 4	syl2anc 584	. . 3 ⊢ ((𝑃 ∈ ℙ ∧ (𝐴 ∈ ℤ ∧ 𝐴 ≠ 0) ∧ 𝐵 ∈ ℕ) → (𝐴 / 𝐵) ∈ ℚ)
6	2	zcnd 12723	. . . 4 ⊢ ((𝑃 ∈ ℙ ∧ (𝐴 ∈ ℤ ∧ 𝐴 ≠ 0) ∧ 𝐵 ∈ ℕ) → 𝐴 ∈ ℂ)
7	3	nncnd 12282	. . . 4 ⊢ ((𝑃 ∈ ℙ ∧ (𝐴 ∈ ℤ ∧ 𝐴 ≠ 0) ∧ 𝐵 ∈ ℕ) → 𝐵 ∈ ℂ)
8		simp2r 1201	. . . 4 ⊢ ((𝑃 ∈ ℙ ∧ (𝐴 ∈ ℤ ∧ 𝐴 ≠ 0) ∧ 𝐵 ∈ ℕ) → 𝐴 ≠ 0)
9	3	nnne0d 12316	. . . 4 ⊢ ((𝑃 ∈ ℙ ∧ (𝐴 ∈ ℤ ∧ 𝐴 ≠ 0) ∧ 𝐵 ∈ ℕ) → 𝐵 ≠ 0)
10	6, 7, 8, 9	divne0d 12059	. . 3 ⊢ ((𝑃 ∈ ℙ ∧ (𝐴 ∈ ℤ ∧ 𝐴 ≠ 0) ∧ 𝐵 ∈ ℕ) → (𝐴 / 𝐵) ≠ 0)
11		eqid 2737	. . . 4 ⊢ sup({𝑛 ∈ ℕ0 ∣ (𝑃↑𝑛) ∥ 𝑥}, ℝ, < ) = sup({𝑛 ∈ ℕ0 ∣ (𝑃↑𝑛) ∥ 𝑥}, ℝ, < )
12		eqid 2737	. . . 4 ⊢ sup({𝑛 ∈ ℕ0 ∣ (𝑃↑𝑛) ∥ 𝑦}, ℝ, < ) = sup({𝑛 ∈ ℕ0 ∣ (𝑃↑𝑛) ∥ 𝑦}, ℝ, < )
13	11, 12	pcval 16882	. . 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 837	. 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 2737	. . . . . . . 8 ⊢ sup({𝑛 ∈ ℕ0 ∣ (𝑃↑𝑛) ∥ 𝐴}, ℝ, < ) = sup({𝑛 ∈ ℕ0 ∣ (𝑃↑𝑛) ∥ 𝐴}, ℝ, < )
16	15	pczpre 16885	. . . . . . 7 ⊢ ((𝑃 ∈ ℙ ∧ (𝐴 ∈ ℤ ∧ 𝐴 ≠ 0)) → (𝑃 pCnt 𝐴) = sup({𝑛 ∈ ℕ0 ∣ (𝑃↑𝑛) ∥ 𝐴}, ℝ, < ))
17	16	3adant3 1133	. . . . . 6 ⊢ ((𝑃 ∈ ℙ ∧ (𝐴 ∈ ℤ ∧ 𝐴 ≠ 0) ∧ 𝐵 ∈ ℕ) → (𝑃 pCnt 𝐴) = sup({𝑛 ∈ ℕ0 ∣ (𝑃↑𝑛) ∥ 𝐴}, ℝ, < ))
18		nnz 12634	. . . . . . . . 9 ⊢ (𝐵 ∈ ℕ → 𝐵 ∈ ℤ)
19		nnne0 12300	. . . . . . . . 9 ⊢ (𝐵 ∈ ℕ → 𝐵 ≠ 0)
20	18, 19	jca 511	. . . . . . . 8 ⊢ (𝐵 ∈ ℕ → (𝐵 ∈ ℤ ∧ 𝐵 ≠ 0))
21		eqid 2737	. . . . . . . . 9 ⊢ sup({𝑛 ∈ ℕ0 ∣ (𝑃↑𝑛) ∥ 𝐵}, ℝ, < ) = sup({𝑛 ∈ ℕ0 ∣ (𝑃↑𝑛) ∥ 𝐵}, ℝ, < )
22	21	pczpre 16885	. . . . . . . 8 ⊢ ((𝑃 ∈ ℙ ∧ (𝐵 ∈ ℤ ∧ 𝐵 ≠ 0)) → (𝑃 pCnt 𝐵) = sup({𝑛 ∈ ℕ0 ∣ (𝑃↑𝑛) ∥ 𝐵}, ℝ, < ))
23	20, 22	sylan2 593	. . . . . . 7 ⊢ ((𝑃 ∈ ℙ ∧ 𝐵 ∈ ℕ) → (𝑃 pCnt 𝐵) = sup({𝑛 ∈ ℕ0 ∣ (𝑃↑𝑛) ∥ 𝐵}, ℝ, < ))
24	23	3adant2 1132	. . . . . 6 ⊢ ((𝑃 ∈ ℙ ∧ (𝐴 ∈ ℤ ∧ 𝐴 ≠ 0) ∧ 𝐵 ∈ ℕ) → (𝑃 pCnt 𝐵) = sup({𝑛 ∈ ℕ0 ∣ (𝑃↑𝑛) ∥ 𝐵}, ℝ, < ))
25	17, 24	oveq12d 7449	. . . . 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 2737	. . . . 5 ⊢ (𝐴 / 𝐵) = (𝐴 / 𝐵)
27	25, 26	jctil 519	. . . 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 7438	. . . . . . 7 ⊢ (𝑥 = 𝐴 → (𝑥 / 𝑦) = (𝐴 / 𝑦))
29	28	eqeq2d 2748	. . . . . 6 ⊢ (𝑥 = 𝐴 → ((𝐴 / 𝐵) = (𝑥 / 𝑦) ↔ (𝐴 / 𝐵) = (𝐴 / 𝑦)))
30		breq2 5147	. . . . . . . . . 10 ⊢ (𝑥 = 𝐴 → ((𝑃↑𝑛) ∥ 𝑥 ↔ (𝑃↑𝑛) ∥ 𝐴))
31	30	rabbidv 3444	. . . . . . . . 9 ⊢ (𝑥 = 𝐴 → {𝑛 ∈ ℕ0 ∣ (𝑃↑𝑛) ∥ 𝑥} = {𝑛 ∈ ℕ0 ∣ (𝑃↑𝑛) ∥ 𝐴})
32	31	supeq1d 9486	. . . . . . . 8 ⊢ (𝑥 = 𝐴 → sup({𝑛 ∈ ℕ0 ∣ (𝑃↑𝑛) ∥ 𝑥}, ℝ, < ) = sup({𝑛 ∈ ℕ0 ∣ (𝑃↑𝑛) ∥ 𝐴}, ℝ, < ))
33	32	oveq1d 7446	. . . . . . 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 2748	. . . . . 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 632	. . . . 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 7439	. . . . . . 7 ⊢ (𝑦 = 𝐵 → (𝐴 / 𝑦) = (𝐴 / 𝐵))
37	36	eqeq2d 2748	. . . . . 6 ⊢ (𝑦 = 𝐵 → ((𝐴 / 𝐵) = (𝐴 / 𝑦) ↔ (𝐴 / 𝐵) = (𝐴 / 𝐵)))
38		breq2 5147	. . . . . . . . . 10 ⊢ (𝑦 = 𝐵 → ((𝑃↑𝑛) ∥ 𝑦 ↔ (𝑃↑𝑛) ∥ 𝐵))
39	38	rabbidv 3444	. . . . . . . . 9 ⊢ (𝑦 = 𝐵 → {𝑛 ∈ ℕ0 ∣ (𝑃↑𝑛) ∥ 𝑦} = {𝑛 ∈ ℕ0 ∣ (𝑃↑𝑛) ∥ 𝐵})
40	39	supeq1d 9486	. . . . . . . 8 ⊢ (𝑦 = 𝐵 → sup({𝑛 ∈ ℕ0 ∣ (𝑃↑𝑛) ∥ 𝑦}, ℝ, < ) = sup({𝑛 ∈ ℕ0 ∣ (𝑃↑𝑛) ∥ 𝐵}, ℝ, < ))
41	40	oveq2d 7447	. . . . . . 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 2748	. . . . . 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 632	. . . . 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 3635	. . . 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 1373	. . 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 7464	. . . 4 ⊢ ((𝑃 pCnt 𝐴) − (𝑃 pCnt 𝐵)) ∈ V
47	11, 12	pceu 16884	. . . . 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 837	. . . 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 2741	. . . . . . 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 630	. . . . . 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 3222	. . . . 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 6550	. . . 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 587	. . 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 232	. 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 2777	1 ⊢ ((𝑃 ∈ ℙ ∧ (𝐴 ∈ ℤ ∧ 𝐴 ≠ 0) ∧ 𝐵 ∈ ℕ) → (𝑃 pCnt (𝐴 / 𝐵)) = ((𝑃 pCnt 𝐴) − (𝑃 pCnt 𝐵)))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1087   = wceq 1540   ∈ wcel 2108  ∃!weu 2568   ≠ wne 2940  ∃wrex 3070  {crab 3436  Vcvv 3480   class class class wbr 5143  ℩cio 6512  (class class class)co 7431  supcsup 9480  ℝcr 11154  0cc0 11155   < clt 11295   − cmin 11492   / cdiv 11920  ℕcn 12266  ℕ₀cn0 12526  ℤcz 12613  ℚcq 12990  ↑cexp 14102   ∥ cdvds 16290  ℙcprime 16708   pCnt cpc 16874

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755  ax-cnex 11211  ax-resscn 11212  ax-1cn 11213  ax-icn 11214  ax-addcl 11215  ax-addrcl 11216  ax-mulcl 11217  ax-mulrcl 11218  ax-mulcom 11219  ax-addass 11220  ax-mulass 11221  ax-distr 11222  ax-i2m1 11223  ax-1ne0 11224  ax-1rid 11225  ax-rnegex 11226  ax-rrecex 11227  ax-cnre 11228  ax-pre-lttri 11229  ax-pre-lttrn 11230  ax-pre-ltadd 11231  ax-pre-mulgt0 11232  ax-pre-sup 11233

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-rmo 3380  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-pred 6321  df-ord 6387  df-on 6388  df-lim 6389  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-1st 8014  df-2nd 8015  df-frecs 8306  df-wrecs 8337  df-recs 8411  df-rdg 8450  df-1o 8506  df-2o 8507  df-er 8745  df-en 8986  df-dom 8987  df-sdom 8988  df-fin 8989  df-sup 9482  df-inf 9483  df-pnf 11297  df-mnf 11298  df-xr 11299  df-ltxr 11300  df-le 11301  df-sub 11494  df-neg 11495  df-div 11921  df-nn 12267  df-2 12329  df-3 12330  df-n0 12527  df-z 12614  df-uz 12879  df-q 12991  df-rp 13035  df-fl 13832  df-mod 13910  df-seq 14043  df-exp 14103  df-cj 15138  df-re 15139  df-im 15140  df-sqrt 15274  df-abs 15275  df-dvds 16291  df-gcd 16532  df-prm 16709  df-pc 16875

This theorem is referenced by:  pcqmul  16891  pcqcl  16894  pcid  16911  pcneg  16912  pc2dvds  16917  pcz  16919  pcaddlem  16926  pcadd  16927  pcmpt2  16931  pcbc  16938  sylow1lem1  19616  chtublem  27255