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Theorem pcqmul 16908

Description: Multiplication property of the prime power function. (Contributed by Mario Carneiro, 9-Sep-2014.)

**Assertion**
Ref	Expression
pcqmul	⊢ ((𝑃 ∈ ℙ ∧ (𝐴 ∈ ℚ ∧ 𝐴 ≠ 0) ∧ (𝐵 ∈ ℚ ∧ 𝐵 ≠ 0)) → (𝑃 pCnt (𝐴 · 𝐵)) = ((𝑃 pCnt 𝐴) + (𝑃 pCnt 𝐵)))

Proof of Theorem pcqmul Dummy variables 𝑥 𝑤 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		simp2l 1218	. . 3 ⊢ ((𝑃 ∈ ℙ ∧ (𝐴 ∈ ℚ ∧ 𝐴 ≠ 0) ∧ (𝐵 ∈ ℚ ∧ 𝐵 ≠ 0)) → 𝐴 ∈ ℚ)
2		elq 12969	. . 3 ⊢ (𝐴 ∈ ℚ ↔ ∃𝑥 ∈ ℤ ∃𝑦 ∈ ℕ 𝐴 = (𝑥 / 𝑦))
3	1, 2	sylib 221	. 2 ⊢ ((𝑃 ∈ ℙ ∧ (𝐴 ∈ ℚ ∧ 𝐴 ≠ 0) ∧ (𝐵 ∈ ℚ ∧ 𝐵 ≠ 0)) → ∃𝑥 ∈ ℤ ∃𝑦 ∈ ℕ 𝐴 = (𝑥 / 𝑦))
4		simp3l 1220	. . 3 ⊢ ((𝑃 ∈ ℙ ∧ (𝐴 ∈ ℚ ∧ 𝐴 ≠ 0) ∧ (𝐵 ∈ ℚ ∧ 𝐵 ≠ 0)) → 𝐵 ∈ ℚ)
5		elq 12969	. . 3 ⊢ (𝐵 ∈ ℚ ↔ ∃𝑧 ∈ ℤ ∃𝑤 ∈ ℕ 𝐵 = (𝑧 / 𝑤))
6	4, 5	sylib 221	. 2 ⊢ ((𝑃 ∈ ℙ ∧ (𝐴 ∈ ℚ ∧ 𝐴 ≠ 0) ∧ (𝐵 ∈ ℚ ∧ 𝐵 ≠ 0)) → ∃𝑧 ∈ ℤ ∃𝑤 ∈ ℕ 𝐵 = (𝑧 / 𝑤))
7		reeanv 3237	. . 3 ⊢ (∃𝑥 ∈ ℤ ∃𝑧 ∈ ℤ (∃𝑦 ∈ ℕ 𝐴 = (𝑥 / 𝑦) ∧ ∃𝑤 ∈ ℕ 𝐵 = (𝑧 / 𝑤)) ↔ (∃𝑥 ∈ ℤ ∃𝑦 ∈ ℕ 𝐴 = (𝑥 / 𝑦) ∧ ∃𝑧 ∈ ℤ ∃𝑤 ∈ ℕ 𝐵 = (𝑧 / 𝑤)))
8		reeanv 3237	. . . . 5 ⊢ (∃𝑦 ∈ ℕ ∃𝑤 ∈ ℕ (𝐴 = (𝑥 / 𝑦) ∧ 𝐵 = (𝑧 / 𝑤)) ↔ (∃𝑦 ∈ ℕ 𝐴 = (𝑥 / 𝑦) ∧ ∃𝑤 ∈ ℕ 𝐵 = (𝑧 / 𝑤)))
9		simp2r 1219	. . . . . . . . 9 ⊢ ((𝑃 ∈ ℙ ∧ (𝐴 ∈ ℚ ∧ 𝐴 ≠ 0) ∧ (𝐵 ∈ ℚ ∧ 𝐵 ≠ 0)) → 𝐴 ≠ 0)
10		simp3r 1221	. . . . . . . . 9 ⊢ ((𝑃 ∈ ℙ ∧ (𝐴 ∈ ℚ ∧ 𝐴 ≠ 0) ∧ (𝐵 ∈ ℚ ∧ 𝐵 ≠ 0)) → 𝐵 ≠ 0)
11	9, 10	jca 520	. . . . . . . 8 ⊢ ((𝑃 ∈ ℙ ∧ (𝐴 ∈ ℚ ∧ 𝐴 ≠ 0) ∧ (𝐵 ∈ ℚ ∧ 𝐵 ≠ 0)) → (𝐴 ≠ 0 ∧ 𝐵 ≠ 0))
12	11	ad2antrr 738	. . . . . . 7 ⊢ ((((𝑃 ∈ ℙ ∧ (𝐴 ∈ ℚ ∧ 𝐴 ≠ 0) ∧ (𝐵 ∈ ℚ ∧ 𝐵 ≠ 0)) ∧ (𝑥 ∈ ℤ ∧ 𝑧 ∈ ℤ)) ∧ (𝑦 ∈ ℕ ∧ 𝑤 ∈ ℕ)) → (𝐴 ≠ 0 ∧ 𝐵 ≠ 0))
13		simp1 1154	. . . . . . . 8 ⊢ ((𝑃 ∈ ℙ ∧ (𝐴 ∈ ℚ ∧ 𝐴 ≠ 0) ∧ (𝐵 ∈ ℚ ∧ 𝐵 ≠ 0)) → 𝑃 ∈ ℙ)
14		simprl 782	. . . . . . . . . . . . . 14 ⊢ (((𝑃 ∈ ℙ ∧ (𝑥 ∈ ℤ ∧ 𝑧 ∈ ℤ)) ∧ (𝑦 ∈ ℕ ∧ 𝑤 ∈ ℕ)) → 𝑦 ∈ ℕ)
15	14	nncnd 12244	. . . . . . . . . . . . 13 ⊢ (((𝑃 ∈ ℙ ∧ (𝑥 ∈ ℤ ∧ 𝑧 ∈ ℤ)) ∧ (𝑦 ∈ ℕ ∧ 𝑤 ∈ ℕ)) → 𝑦 ∈ ℂ)
16	14	nnne0d 12281	. . . . . . . . . . . . 13 ⊢ (((𝑃 ∈ ℙ ∧ (𝑥 ∈ ℤ ∧ 𝑧 ∈ ℤ)) ∧ (𝑦 ∈ ℕ ∧ 𝑤 ∈ ℕ)) → 𝑦 ≠ 0)
17	15, 16	div0d 11985	. . . . . . . . . . . 12 ⊢ (((𝑃 ∈ ℙ ∧ (𝑥 ∈ ℤ ∧ 𝑧 ∈ ℤ)) ∧ (𝑦 ∈ ℕ ∧ 𝑤 ∈ ℕ)) → (0 / 𝑦) = 0)
18		oveq1 7417	. . . . . . . . . . . . 13 ⊢ (𝑥 = 0 → (𝑥 / 𝑦) = (0 / 𝑦))
19	18	eqeq1d 2765	. . . . . . . . . . . 12 ⊢ (𝑥 = 0 → ((𝑥 / 𝑦) = 0 ↔ (0 / 𝑦) = 0))
20	17, 19	syl5ibrcom 250	. . . . . . . . . . 11 ⊢ (((𝑃 ∈ ℙ ∧ (𝑥 ∈ ℤ ∧ 𝑧 ∈ ℤ)) ∧ (𝑦 ∈ ℕ ∧ 𝑤 ∈ ℕ)) → (𝑥 = 0 → (𝑥 / 𝑦) = 0))
21	20	necon3d 2979	. . . . . . . . . 10 ⊢ (((𝑃 ∈ ℙ ∧ (𝑥 ∈ ℤ ∧ 𝑧 ∈ ℤ)) ∧ (𝑦 ∈ ℕ ∧ 𝑤 ∈ ℕ)) → ((𝑥 / 𝑦) ≠ 0 → 𝑥 ≠ 0))
22		simprr 784	. . . . . . . . . . . . . 14 ⊢ (((𝑃 ∈ ℙ ∧ (𝑥 ∈ ℤ ∧ 𝑧 ∈ ℤ)) ∧ (𝑦 ∈ ℕ ∧ 𝑤 ∈ ℕ)) → 𝑤 ∈ ℕ)
23	22	nncnd 12244	. . . . . . . . . . . . 13 ⊢ (((𝑃 ∈ ℙ ∧ (𝑥 ∈ ℤ ∧ 𝑧 ∈ ℤ)) ∧ (𝑦 ∈ ℕ ∧ 𝑤 ∈ ℕ)) → 𝑤 ∈ ℂ)
24	22	nnne0d 12281	. . . . . . . . . . . . 13 ⊢ (((𝑃 ∈ ℙ ∧ (𝑥 ∈ ℤ ∧ 𝑧 ∈ ℤ)) ∧ (𝑦 ∈ ℕ ∧ 𝑤 ∈ ℕ)) → 𝑤 ≠ 0)
25	23, 24	div0d 11985	. . . . . . . . . . . 12 ⊢ (((𝑃 ∈ ℙ ∧ (𝑥 ∈ ℤ ∧ 𝑧 ∈ ℤ)) ∧ (𝑦 ∈ ℕ ∧ 𝑤 ∈ ℕ)) → (0 / 𝑤) = 0)
26		oveq1 7417	. . . . . . . . . . . . 13 ⊢ (𝑧 = 0 → (𝑧 / 𝑤) = (0 / 𝑤))
27	26	eqeq1d 2765	. . . . . . . . . . . 12 ⊢ (𝑧 = 0 → ((𝑧 / 𝑤) = 0 ↔ (0 / 𝑤) = 0))
28	25, 27	syl5ibrcom 250	. . . . . . . . . . 11 ⊢ (((𝑃 ∈ ℙ ∧ (𝑥 ∈ ℤ ∧ 𝑧 ∈ ℤ)) ∧ (𝑦 ∈ ℕ ∧ 𝑤 ∈ ℕ)) → (𝑧 = 0 → (𝑧 / 𝑤) = 0))
29	28	necon3d 2979	. . . . . . . . . 10 ⊢ (((𝑃 ∈ ℙ ∧ (𝑥 ∈ ℤ ∧ 𝑧 ∈ ℤ)) ∧ (𝑦 ∈ ℕ ∧ 𝑤 ∈ ℕ)) → ((𝑧 / 𝑤) ≠ 0 → 𝑧 ≠ 0))
30		simpll 778	. . . . . . . . . . . . . 14 ⊢ (((𝑃 ∈ ℙ ∧ (𝑥 ∈ ℤ ∧ 𝑧 ∈ ℤ)) ∧ ((𝑦 ∈ ℕ ∧ 𝑤 ∈ ℕ) ∧ (𝑥 ≠ 0 ∧ 𝑧 ≠ 0))) → 𝑃 ∈ ℙ)
31		simplrl 788	. . . . . . . . . . . . . . 15 ⊢ (((𝑃 ∈ ℙ ∧ (𝑥 ∈ ℤ ∧ 𝑧 ∈ ℤ)) ∧ ((𝑦 ∈ ℕ ∧ 𝑤 ∈ ℕ) ∧ (𝑥 ≠ 0 ∧ 𝑧 ≠ 0))) → 𝑥 ∈ ℤ)
32		simplrr 789	. . . . . . . . . . . . . . 15 ⊢ (((𝑃 ∈ ℙ ∧ (𝑥 ∈ ℤ ∧ 𝑧 ∈ ℤ)) ∧ ((𝑦 ∈ ℕ ∧ 𝑤 ∈ ℕ) ∧ (𝑥 ≠ 0 ∧ 𝑧 ≠ 0))) → 𝑧 ∈ ℤ)
33	31, 32	zmulcld 12701	. . . . . . . . . . . . . 14 ⊢ (((𝑃 ∈ ℙ ∧ (𝑥 ∈ ℤ ∧ 𝑧 ∈ ℤ)) ∧ ((𝑦 ∈ ℕ ∧ 𝑤 ∈ ℕ) ∧ (𝑥 ≠ 0 ∧ 𝑧 ≠ 0))) → (𝑥 · 𝑧) ∈ ℤ)
34	31	zcnd 12696	. . . . . . . . . . . . . . 15 ⊢ (((𝑃 ∈ ℙ ∧ (𝑥 ∈ ℤ ∧ 𝑧 ∈ ℤ)) ∧ ((𝑦 ∈ ℕ ∧ 𝑤 ∈ ℕ) ∧ (𝑥 ≠ 0 ∧ 𝑧 ≠ 0))) → 𝑥 ∈ ℂ)
35	32	zcnd 12696	. . . . . . . . . . . . . . 15 ⊢ (((𝑃 ∈ ℙ ∧ (𝑥 ∈ ℤ ∧ 𝑧 ∈ ℤ)) ∧ ((𝑦 ∈ ℕ ∧ 𝑤 ∈ ℕ) ∧ (𝑥 ≠ 0 ∧ 𝑧 ≠ 0))) → 𝑧 ∈ ℂ)
36		simprrl 792	. . . . . . . . . . . . . . 15 ⊢ (((𝑃 ∈ ℙ ∧ (𝑥 ∈ ℤ ∧ 𝑧 ∈ ℤ)) ∧ ((𝑦 ∈ ℕ ∧ 𝑤 ∈ ℕ) ∧ (𝑥 ≠ 0 ∧ 𝑧 ≠ 0))) → 𝑥 ≠ 0)
37		simprrr 793	. . . . . . . . . . . . . . 15 ⊢ (((𝑃 ∈ ℙ ∧ (𝑥 ∈ ℤ ∧ 𝑧 ∈ ℤ)) ∧ ((𝑦 ∈ ℕ ∧ 𝑤 ∈ ℕ) ∧ (𝑥 ≠ 0 ∧ 𝑧 ≠ 0))) → 𝑧 ≠ 0)
38	34, 35, 36, 37	mulne0d 11861	. . . . . . . . . . . . . 14 ⊢ (((𝑃 ∈ ℙ ∧ (𝑥 ∈ ℤ ∧ 𝑧 ∈ ℤ)) ∧ ((𝑦 ∈ ℕ ∧ 𝑤 ∈ ℕ) ∧ (𝑥 ≠ 0 ∧ 𝑧 ≠ 0))) → (𝑥 · 𝑧) ≠ 0)
39	14	adantrr 729	. . . . . . . . . . . . . . 15 ⊢ (((𝑃 ∈ ℙ ∧ (𝑥 ∈ ℤ ∧ 𝑧 ∈ ℤ)) ∧ ((𝑦 ∈ ℕ ∧ 𝑤 ∈ ℕ) ∧ (𝑥 ≠ 0 ∧ 𝑧 ≠ 0))) → 𝑦 ∈ ℕ)
40	22	adantrr 729	. . . . . . . . . . . . . . 15 ⊢ (((𝑃 ∈ ℙ ∧ (𝑥 ∈ ℤ ∧ 𝑧 ∈ ℤ)) ∧ ((𝑦 ∈ ℕ ∧ 𝑤 ∈ ℕ) ∧ (𝑥 ≠ 0 ∧ 𝑧 ≠ 0))) → 𝑤 ∈ ℕ)
41	39, 40	nnmulcld 12284	. . . . . . . . . . . . . 14 ⊢ (((𝑃 ∈ ℙ ∧ (𝑥 ∈ ℤ ∧ 𝑧 ∈ ℤ)) ∧ ((𝑦 ∈ ℕ ∧ 𝑤 ∈ ℕ) ∧ (𝑥 ≠ 0 ∧ 𝑧 ≠ 0))) → (𝑦 · 𝑤) ∈ ℕ)
42		pcdiv 16907	. . . . . . . . . . . . . 14 ⊢ ((𝑃 ∈ ℙ ∧ ((𝑥 · 𝑧) ∈ ℤ ∧ (𝑥 · 𝑧) ≠ 0) ∧ (𝑦 · 𝑤) ∈ ℕ) → (𝑃 pCnt ((𝑥 · 𝑧) / (𝑦 · 𝑤))) = ((𝑃 pCnt (𝑥 · 𝑧)) − (𝑃 pCnt (𝑦 · 𝑤))))
43	30, 33, 38, 41, 42	syl121anc 1402	. . . . . . . . . . . . 13 ⊢ (((𝑃 ∈ ℙ ∧ (𝑥 ∈ ℤ ∧ 𝑧 ∈ ℤ)) ∧ ((𝑦 ∈ ℕ ∧ 𝑤 ∈ ℕ) ∧ (𝑥 ≠ 0 ∧ 𝑧 ≠ 0))) → (𝑃 pCnt ((𝑥 · 𝑧) / (𝑦 · 𝑤))) = ((𝑃 pCnt (𝑥 · 𝑧)) − (𝑃 pCnt (𝑦 · 𝑤))))
44		pcmul 16906	. . . . . . . . . . . . . . 15 ⊢ ((𝑃 ∈ ℙ ∧ (𝑥 ∈ ℤ ∧ 𝑥 ≠ 0) ∧ (𝑧 ∈ ℤ ∧ 𝑧 ≠ 0)) → (𝑃 pCnt (𝑥 · 𝑧)) = ((𝑃 pCnt 𝑥) + (𝑃 pCnt 𝑧)))
45	30, 31, 36, 32, 37, 44	syl122anc 1406	. . . . . . . . . . . . . 14 ⊢ (((𝑃 ∈ ℙ ∧ (𝑥 ∈ ℤ ∧ 𝑧 ∈ ℤ)) ∧ ((𝑦 ∈ ℕ ∧ 𝑤 ∈ ℕ) ∧ (𝑥 ≠ 0 ∧ 𝑧 ≠ 0))) → (𝑃 pCnt (𝑥 · 𝑧)) = ((𝑃 pCnt 𝑥) + (𝑃 pCnt 𝑧)))
46	39	nnzd 12612	. . . . . . . . . . . . . . 15 ⊢ (((𝑃 ∈ ℙ ∧ (𝑥 ∈ ℤ ∧ 𝑧 ∈ ℤ)) ∧ ((𝑦 ∈ ℕ ∧ 𝑤 ∈ ℕ) ∧ (𝑥 ≠ 0 ∧ 𝑧 ≠ 0))) → 𝑦 ∈ ℤ)
47	16	adantrr 729	. . . . . . . . . . . . . . 15 ⊢ (((𝑃 ∈ ℙ ∧ (𝑥 ∈ ℤ ∧ 𝑧 ∈ ℤ)) ∧ ((𝑦 ∈ ℕ ∧ 𝑤 ∈ ℕ) ∧ (𝑥 ≠ 0 ∧ 𝑧 ≠ 0))) → 𝑦 ≠ 0)
48	40	nnzd 12612	. . . . . . . . . . . . . . 15 ⊢ (((𝑃 ∈ ℙ ∧ (𝑥 ∈ ℤ ∧ 𝑧 ∈ ℤ)) ∧ ((𝑦 ∈ ℕ ∧ 𝑤 ∈ ℕ) ∧ (𝑥 ≠ 0 ∧ 𝑧 ≠ 0))) → 𝑤 ∈ ℤ)
49	24	adantrr 729	. . . . . . . . . . . . . . 15 ⊢ (((𝑃 ∈ ℙ ∧ (𝑥 ∈ ℤ ∧ 𝑧 ∈ ℤ)) ∧ ((𝑦 ∈ ℕ ∧ 𝑤 ∈ ℕ) ∧ (𝑥 ≠ 0 ∧ 𝑧 ≠ 0))) → 𝑤 ≠ 0)
50		pcmul 16906	. . . . . . . . . . . . . . 15 ⊢ ((𝑃 ∈ ℙ ∧ (𝑦 ∈ ℤ ∧ 𝑦 ≠ 0) ∧ (𝑤 ∈ ℤ ∧ 𝑤 ≠ 0)) → (𝑃 pCnt (𝑦 · 𝑤)) = ((𝑃 pCnt 𝑦) + (𝑃 pCnt 𝑤)))
51	30, 46, 47, 48, 49, 50	syl122anc 1406	. . . . . . . . . . . . . 14 ⊢ (((𝑃 ∈ ℙ ∧ (𝑥 ∈ ℤ ∧ 𝑧 ∈ ℤ)) ∧ ((𝑦 ∈ ℕ ∧ 𝑤 ∈ ℕ) ∧ (𝑥 ≠ 0 ∧ 𝑧 ≠ 0))) → (𝑃 pCnt (𝑦 · 𝑤)) = ((𝑃 pCnt 𝑦) + (𝑃 pCnt 𝑤)))
52	45, 51	oveq12d 7428	. . . . . . . . . . . . 13 ⊢ (((𝑃 ∈ ℙ ∧ (𝑥 ∈ ℤ ∧ 𝑧 ∈ ℤ)) ∧ ((𝑦 ∈ ℕ ∧ 𝑤 ∈ ℕ) ∧ (𝑥 ≠ 0 ∧ 𝑧 ≠ 0))) → ((𝑃 pCnt (𝑥 · 𝑧)) − (𝑃 pCnt (𝑦 · 𝑤))) = (((𝑃 pCnt 𝑥) + (𝑃 pCnt 𝑧)) − ((𝑃 pCnt 𝑦) + (𝑃 pCnt 𝑤))))
53		pczcl 16903	. . . . . . . . . . . . . . . 16 ⊢ ((𝑃 ∈ ℙ ∧ (𝑥 ∈ ℤ ∧ 𝑥 ≠ 0)) → (𝑃 pCnt 𝑥) ∈ ℕ₀)
54	30, 31, 36, 53	syl12anc 849	. . . . . . . . . . . . . . 15 ⊢ (((𝑃 ∈ ℙ ∧ (𝑥 ∈ ℤ ∧ 𝑧 ∈ ℤ)) ∧ ((𝑦 ∈ ℕ ∧ 𝑤 ∈ ℕ) ∧ (𝑥 ≠ 0 ∧ 𝑧 ≠ 0))) → (𝑃 pCnt 𝑥) ∈ ℕ₀)
55	54	nn0cnd 12562	. . . . . . . . . . . . . 14 ⊢ (((𝑃 ∈ ℙ ∧ (𝑥 ∈ ℤ ∧ 𝑧 ∈ ℤ)) ∧ ((𝑦 ∈ ℕ ∧ 𝑤 ∈ ℕ) ∧ (𝑥 ≠ 0 ∧ 𝑧 ≠ 0))) → (𝑃 pCnt 𝑥) ∈ ℂ)
56		pczcl 16903	. . . . . . . . . . . . . . . 16 ⊢ ((𝑃 ∈ ℙ ∧ (𝑧 ∈ ℤ ∧ 𝑧 ≠ 0)) → (𝑃 pCnt 𝑧) ∈ ℕ₀)
57	30, 32, 37, 56	syl12anc 849	. . . . . . . . . . . . . . 15 ⊢ (((𝑃 ∈ ℙ ∧ (𝑥 ∈ ℤ ∧ 𝑧 ∈ ℤ)) ∧ ((𝑦 ∈ ℕ ∧ 𝑤 ∈ ℕ) ∧ (𝑥 ≠ 0 ∧ 𝑧 ≠ 0))) → (𝑃 pCnt 𝑧) ∈ ℕ₀)
58	57	nn0cnd 12562	. . . . . . . . . . . . . 14 ⊢ (((𝑃 ∈ ℙ ∧ (𝑥 ∈ ℤ ∧ 𝑧 ∈ ℤ)) ∧ ((𝑦 ∈ ℕ ∧ 𝑤 ∈ ℕ) ∧ (𝑥 ≠ 0 ∧ 𝑧 ≠ 0))) → (𝑃 pCnt 𝑧) ∈ ℂ)
59	30, 39	pccld 16905	. . . . . . . . . . . . . . 15 ⊢ (((𝑃 ∈ ℙ ∧ (𝑥 ∈ ℤ ∧ 𝑧 ∈ ℤ)) ∧ ((𝑦 ∈ ℕ ∧ 𝑤 ∈ ℕ) ∧ (𝑥 ≠ 0 ∧ 𝑧 ≠ 0))) → (𝑃 pCnt 𝑦) ∈ ℕ₀)
60	59	nn0cnd 12562	. . . . . . . . . . . . . 14 ⊢ (((𝑃 ∈ ℙ ∧ (𝑥 ∈ ℤ ∧ 𝑧 ∈ ℤ)) ∧ ((𝑦 ∈ ℕ ∧ 𝑤 ∈ ℕ) ∧ (𝑥 ≠ 0 ∧ 𝑧 ≠ 0))) → (𝑃 pCnt 𝑦) ∈ ℂ)
61	30, 40	pccld 16905	. . . . . . . . . . . . . . 15 ⊢ (((𝑃 ∈ ℙ ∧ (𝑥 ∈ ℤ ∧ 𝑧 ∈ ℤ)) ∧ ((𝑦 ∈ ℕ ∧ 𝑤 ∈ ℕ) ∧ (𝑥 ≠ 0 ∧ 𝑧 ≠ 0))) → (𝑃 pCnt 𝑤) ∈ ℕ₀)
62	61	nn0cnd 12562	. . . . . . . . . . . . . 14 ⊢ (((𝑃 ∈ ℙ ∧ (𝑥 ∈ ℤ ∧ 𝑧 ∈ ℤ)) ∧ ((𝑦 ∈ ℕ ∧ 𝑤 ∈ ℕ) ∧ (𝑥 ≠ 0 ∧ 𝑧 ≠ 0))) → (𝑃 pCnt 𝑤) ∈ ℂ)
63	55, 58, 60, 62	addsub4d 11611	. . . . . . . . . . . . 13 ⊢ (((𝑃 ∈ ℙ ∧ (𝑥 ∈ ℤ ∧ 𝑧 ∈ ℤ)) ∧ ((𝑦 ∈ ℕ ∧ 𝑤 ∈ ℕ) ∧ (𝑥 ≠ 0 ∧ 𝑧 ≠ 0))) → (((𝑃 pCnt 𝑥) + (𝑃 pCnt 𝑧)) − ((𝑃 pCnt 𝑦) + (𝑃 pCnt 𝑤))) = (((𝑃 pCnt 𝑥) − (𝑃 pCnt 𝑦)) + ((𝑃 pCnt 𝑧) − (𝑃 pCnt 𝑤))))
64	43, 52, 63	3eqtrd 2802	. . . . . . . . . . . 12 ⊢ (((𝑃 ∈ ℙ ∧ (𝑥 ∈ ℤ ∧ 𝑧 ∈ ℤ)) ∧ ((𝑦 ∈ ℕ ∧ 𝑤 ∈ ℕ) ∧ (𝑥 ≠ 0 ∧ 𝑧 ≠ 0))) → (𝑃 pCnt ((𝑥 · 𝑧) / (𝑦 · 𝑤))) = (((𝑃 pCnt 𝑥) − (𝑃 pCnt 𝑦)) + ((𝑃 pCnt 𝑧) − (𝑃 pCnt 𝑤))))
65	15	adantrr 729	. . . . . . . . . . . . . 14 ⊢ (((𝑃 ∈ ℙ ∧ (𝑥 ∈ ℤ ∧ 𝑧 ∈ ℤ)) ∧ ((𝑦 ∈ ℕ ∧ 𝑤 ∈ ℕ) ∧ (𝑥 ≠ 0 ∧ 𝑧 ≠ 0))) → 𝑦 ∈ ℂ)
66	23	adantrr 729	. . . . . . . . . . . . . 14 ⊢ (((𝑃 ∈ ℙ ∧ (𝑥 ∈ ℤ ∧ 𝑧 ∈ ℤ)) ∧ ((𝑦 ∈ ℕ ∧ 𝑤 ∈ ℕ) ∧ (𝑥 ≠ 0 ∧ 𝑧 ≠ 0))) → 𝑤 ∈ ℂ)
67	34, 65, 35, 66, 47, 49	divmuldivd 12027	. . . . . . . . . . . . 13 ⊢ (((𝑃 ∈ ℙ ∧ (𝑥 ∈ ℤ ∧ 𝑧 ∈ ℤ)) ∧ ((𝑦 ∈ ℕ ∧ 𝑤 ∈ ℕ) ∧ (𝑥 ≠ 0 ∧ 𝑧 ≠ 0))) → ((𝑥 / 𝑦) · (𝑧 / 𝑤)) = ((𝑥 · 𝑧) / (𝑦 · 𝑤)))
68	67	oveq2d 7426	. . . . . . . . . . . 12 ⊢ (((𝑃 ∈ ℙ ∧ (𝑥 ∈ ℤ ∧ 𝑧 ∈ ℤ)) ∧ ((𝑦 ∈ ℕ ∧ 𝑤 ∈ ℕ) ∧ (𝑥 ≠ 0 ∧ 𝑧 ≠ 0))) → (𝑃 pCnt ((𝑥 / 𝑦) · (𝑧 / 𝑤))) = (𝑃 pCnt ((𝑥 · 𝑧) / (𝑦 · 𝑤))))
69		pcdiv 16907	. . . . . . . . . . . . . 14 ⊢ ((𝑃 ∈ ℙ ∧ (𝑥 ∈ ℤ ∧ 𝑥 ≠ 0) ∧ 𝑦 ∈ ℕ) → (𝑃 pCnt (𝑥 / 𝑦)) = ((𝑃 pCnt 𝑥) − (𝑃 pCnt 𝑦)))
70	30, 31, 36, 39, 69	syl121anc 1402	. . . . . . . . . . . . 13 ⊢ (((𝑃 ∈ ℙ ∧ (𝑥 ∈ ℤ ∧ 𝑧 ∈ ℤ)) ∧ ((𝑦 ∈ ℕ ∧ 𝑤 ∈ ℕ) ∧ (𝑥 ≠ 0 ∧ 𝑧 ≠ 0))) → (𝑃 pCnt (𝑥 / 𝑦)) = ((𝑃 pCnt 𝑥) − (𝑃 pCnt 𝑦)))
71		pcdiv 16907	. . . . . . . . . . . . . 14 ⊢ ((𝑃 ∈ ℙ ∧ (𝑧 ∈ ℤ ∧ 𝑧 ≠ 0) ∧ 𝑤 ∈ ℕ) → (𝑃 pCnt (𝑧 / 𝑤)) = ((𝑃 pCnt 𝑧) − (𝑃 pCnt 𝑤)))
72	30, 32, 37, 40, 71	syl121anc 1402	. . . . . . . . . . . . 13 ⊢ (((𝑃 ∈ ℙ ∧ (𝑥 ∈ ℤ ∧ 𝑧 ∈ ℤ)) ∧ ((𝑦 ∈ ℕ ∧ 𝑤 ∈ ℕ) ∧ (𝑥 ≠ 0 ∧ 𝑧 ≠ 0))) → (𝑃 pCnt (𝑧 / 𝑤)) = ((𝑃 pCnt 𝑧) − (𝑃 pCnt 𝑤)))
73	70, 72	oveq12d 7428	. . . . . . . . . . . 12 ⊢ (((𝑃 ∈ ℙ ∧ (𝑥 ∈ ℤ ∧ 𝑧 ∈ ℤ)) ∧ ((𝑦 ∈ ℕ ∧ 𝑤 ∈ ℕ) ∧ (𝑥 ≠ 0 ∧ 𝑧 ≠ 0))) → ((𝑃 pCnt (𝑥 / 𝑦)) + (𝑃 pCnt (𝑧 / 𝑤))) = (((𝑃 pCnt 𝑥) − (𝑃 pCnt 𝑦)) + ((𝑃 pCnt 𝑧) − (𝑃 pCnt 𝑤))))
74	64, 68, 73	3eqtr4d 2808	. . . . . . . . . . 11 ⊢ (((𝑃 ∈ ℙ ∧ (𝑥 ∈ ℤ ∧ 𝑧 ∈ ℤ)) ∧ ((𝑦 ∈ ℕ ∧ 𝑤 ∈ ℕ) ∧ (𝑥 ≠ 0 ∧ 𝑧 ≠ 0))) → (𝑃 pCnt ((𝑥 / 𝑦) · (𝑧 / 𝑤))) = ((𝑃 pCnt (𝑥 / 𝑦)) + (𝑃 pCnt (𝑧 / 𝑤))))
75	74	expr 461	. . . . . . . . . 10 ⊢ (((𝑃 ∈ ℙ ∧ (𝑥 ∈ ℤ ∧ 𝑧 ∈ ℤ)) ∧ (𝑦 ∈ ℕ ∧ 𝑤 ∈ ℕ)) → ((𝑥 ≠ 0 ∧ 𝑧 ≠ 0) → (𝑃 pCnt ((𝑥 / 𝑦) · (𝑧 / 𝑤))) = ((𝑃 pCnt (𝑥 / 𝑦)) + (𝑃 pCnt (𝑧 / 𝑤)))))
76	21, 29, 75	syl2and 619	. . . . . . . . 9 ⊢ (((𝑃 ∈ ℙ ∧ (𝑥 ∈ ℤ ∧ 𝑧 ∈ ℤ)) ∧ (𝑦 ∈ ℕ ∧ 𝑤 ∈ ℕ)) → (((𝑥 / 𝑦) ≠ 0 ∧ (𝑧 / 𝑤) ≠ 0) → (𝑃 pCnt ((𝑥 / 𝑦) · (𝑧 / 𝑤))) = ((𝑃 pCnt (𝑥 / 𝑦)) + (𝑃 pCnt (𝑧 / 𝑤)))))
77		neeq1 3020	. . . . . . . . . . 11 ⊢ (𝐴 = (𝑥 / 𝑦) → (𝐴 ≠ 0 ↔ (𝑥 / 𝑦) ≠ 0))
78		neeq1 3020	. . . . . . . . . . 11 ⊢ (𝐵 = (𝑧 / 𝑤) → (𝐵 ≠ 0 ↔ (𝑧 / 𝑤) ≠ 0))
79	77, 78	bi2anan9 649	. . . . . . . . . 10 ⊢ ((𝐴 = (𝑥 / 𝑦) ∧ 𝐵 = (𝑧 / 𝑤)) → ((𝐴 ≠ 0 ∧ 𝐵 ≠ 0) ↔ ((𝑥 / 𝑦) ≠ 0 ∧ (𝑧 / 𝑤) ≠ 0)))
80		oveq12 7419	. . . . . . . . . . . 12 ⊢ ((𝐴 = (𝑥 / 𝑦) ∧ 𝐵 = (𝑧 / 𝑤)) → (𝐴 · 𝐵) = ((𝑥 / 𝑦) · (𝑧 / 𝑤)))
81	80	oveq2d 7426	. . . . . . . . . . 11 ⊢ ((𝐴 = (𝑥 / 𝑦) ∧ 𝐵 = (𝑧 / 𝑤)) → (𝑃 pCnt (𝐴 · 𝐵)) = (𝑃 pCnt ((𝑥 / 𝑦) · (𝑧 / 𝑤))))
82		oveq2 7418	. . . . . . . . . . . 12 ⊢ (𝐴 = (𝑥 / 𝑦) → (𝑃 pCnt 𝐴) = (𝑃 pCnt (𝑥 / 𝑦)))
83		oveq2 7418	. . . . . . . . . . . 12 ⊢ (𝐵 = (𝑧 / 𝑤) → (𝑃 pCnt 𝐵) = (𝑃 pCnt (𝑧 / 𝑤)))
84	82, 83	oveqan12d 7429	. . . . . . . . . . 11 ⊢ ((𝐴 = (𝑥 / 𝑦) ∧ 𝐵 = (𝑧 / 𝑤)) → ((𝑃 pCnt 𝐴) + (𝑃 pCnt 𝐵)) = ((𝑃 pCnt (𝑥 / 𝑦)) + (𝑃 pCnt (𝑧 / 𝑤))))
85	81, 84	eqeq12d 2779	. . . . . . . . . 10 ⊢ ((𝐴 = (𝑥 / 𝑦) ∧ 𝐵 = (𝑧 / 𝑤)) → ((𝑃 pCnt (𝐴 · 𝐵)) = ((𝑃 pCnt 𝐴) + (𝑃 pCnt 𝐵)) ↔ (𝑃 pCnt ((𝑥 / 𝑦) · (𝑧 / 𝑤))) = ((𝑃 pCnt (𝑥 / 𝑦)) + (𝑃 pCnt (𝑧 / 𝑤)))))
86	79, 85	imbi12d 347	. . . . . . . . 9 ⊢ ((𝐴 = (𝑥 / 𝑦) ∧ 𝐵 = (𝑧 / 𝑤)) → (((𝐴 ≠ 0 ∧ 𝐵 ≠ 0) → (𝑃 pCnt (𝐴 · 𝐵)) = ((𝑃 pCnt 𝐴) + (𝑃 pCnt 𝐵))) ↔ (((𝑥 / 𝑦) ≠ 0 ∧ (𝑧 / 𝑤) ≠ 0) → (𝑃 pCnt ((𝑥 / 𝑦) · (𝑧 / 𝑤))) = ((𝑃 pCnt (𝑥 / 𝑦)) + (𝑃 pCnt (𝑧 / 𝑤))))))
87	76, 86	syl5ibrcom 250	. . . . . . . 8 ⊢ (((𝑃 ∈ ℙ ∧ (𝑥 ∈ ℤ ∧ 𝑧 ∈ ℤ)) ∧ (𝑦 ∈ ℕ ∧ 𝑤 ∈ ℕ)) → ((𝐴 = (𝑥 / 𝑦) ∧ 𝐵 = (𝑧 / 𝑤)) → ((𝐴 ≠ 0 ∧ 𝐵 ≠ 0) → (𝑃 pCnt (𝐴 · 𝐵)) = ((𝑃 pCnt 𝐴) + (𝑃 pCnt 𝐵)))))
88	13, 87	sylanl1 692	. . . . . . 7 ⊢ ((((𝑃 ∈ ℙ ∧ (𝐴 ∈ ℚ ∧ 𝐴 ≠ 0) ∧ (𝐵 ∈ ℚ ∧ 𝐵 ≠ 0)) ∧ (𝑥 ∈ ℤ ∧ 𝑧 ∈ ℤ)) ∧ (𝑦 ∈ ℕ ∧ 𝑤 ∈ ℕ)) → ((𝐴 = (𝑥 / 𝑦) ∧ 𝐵 = (𝑧 / 𝑤)) → ((𝐴 ≠ 0 ∧ 𝐵 ≠ 0) → (𝑃 pCnt (𝐴 · 𝐵)) = ((𝑃 pCnt 𝐴) + (𝑃 pCnt 𝐵)))))
89	12, 88	mpid 45	. . . . . 6 ⊢ ((((𝑃 ∈ ℙ ∧ (𝐴 ∈ ℚ ∧ 𝐴 ≠ 0) ∧ (𝐵 ∈ ℚ ∧ 𝐵 ≠ 0)) ∧ (𝑥 ∈ ℤ ∧ 𝑧 ∈ ℤ)) ∧ (𝑦 ∈ ℕ ∧ 𝑤 ∈ ℕ)) → ((𝐴 = (𝑥 / 𝑦) ∧ 𝐵 = (𝑧 / 𝑤)) → (𝑃 pCnt (𝐴 · 𝐵)) = ((𝑃 pCnt 𝐴) + (𝑃 pCnt 𝐵))))
90	89	rexlimdvva 3222	. . . . 5 ⊢ (((𝑃 ∈ ℙ ∧ (𝐴 ∈ ℚ ∧ 𝐴 ≠ 0) ∧ (𝐵 ∈ ℚ ∧ 𝐵 ≠ 0)) ∧ (𝑥 ∈ ℤ ∧ 𝑧 ∈ ℤ)) → (∃𝑦 ∈ ℕ ∃𝑤 ∈ ℕ (𝐴 = (𝑥 / 𝑦) ∧ 𝐵 = (𝑧 / 𝑤)) → (𝑃 pCnt (𝐴 · 𝐵)) = ((𝑃 pCnt 𝐴) + (𝑃 pCnt 𝐵))))
91	8, 90	biimtrrid 246	. . . 4 ⊢ (((𝑃 ∈ ℙ ∧ (𝐴 ∈ ℚ ∧ 𝐴 ≠ 0) ∧ (𝐵 ∈ ℚ ∧ 𝐵 ≠ 0)) ∧ (𝑥 ∈ ℤ ∧ 𝑧 ∈ ℤ)) → ((∃𝑦 ∈ ℕ 𝐴 = (𝑥 / 𝑦) ∧ ∃𝑤 ∈ ℕ 𝐵 = (𝑧 / 𝑤)) → (𝑃 pCnt (𝐴 · 𝐵)) = ((𝑃 pCnt 𝐴) + (𝑃 pCnt 𝐵))))
92	91	rexlimdvva 3222	. . 3 ⊢ ((𝑃 ∈ ℙ ∧ (𝐴 ∈ ℚ ∧ 𝐴 ≠ 0) ∧ (𝐵 ∈ ℚ ∧ 𝐵 ≠ 0)) → (∃𝑥 ∈ ℤ ∃𝑧 ∈ ℤ (∃𝑦 ∈ ℕ 𝐴 = (𝑥 / 𝑦) ∧ ∃𝑤 ∈ ℕ 𝐵 = (𝑧 / 𝑤)) → (𝑃 pCnt (𝐴 · 𝐵)) = ((𝑃 pCnt 𝐴) + (𝑃 pCnt 𝐵))))
93	7, 92	biimtrrid 246	. 2 ⊢ ((𝑃 ∈ ℙ ∧ (𝐴 ∈ ℚ ∧ 𝐴 ≠ 0) ∧ (𝐵 ∈ ℚ ∧ 𝐵 ≠ 0)) → ((∃𝑥 ∈ ℤ ∃𝑦 ∈ ℕ 𝐴 = (𝑥 / 𝑦) ∧ ∃𝑧 ∈ ℤ ∃𝑤 ∈ ℕ 𝐵 = (𝑧 / 𝑤)) → (𝑃 pCnt (𝐴 · 𝐵)) = ((𝑃 pCnt 𝐴) + (𝑃 pCnt 𝐵))))
94	3, 6, 93	mp2and 711	1 ⊢ ((𝑃 ∈ ℙ ∧ (𝐴 ∈ ℚ ∧ 𝐴 ≠ 0) ∧ (𝐵 ∈ ℚ ∧ 𝐵 ≠ 0)) → (𝑃 pCnt (𝐴 · 𝐵)) = ((𝑃 pCnt 𝐴) + (𝑃 pCnt 𝐵)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 400 ∧ w3a 1103 = wceq 1570 ∈ wcel 2143 ≠ wne 2958 ∃wrex 3089 (class class class)co 7410 ℂcc 11093 0cc0 11095 + caddc 11098 · cmul 11100 − cmin 11436 / cdiv 11866 ℕcn 12228 ℕ₀cn0 12499 ℤcz 12586 ℚcq 12967 ℙcprime 16724 pCnt cpc 16891

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1825 ax-4 1839 ax-5 1940 ax-6 1997 ax-7 2038 ax-8 2145 ax-9 2153 ax-10 2176 ax-11 2192 ax-12 2213 ax-ext 2735 ax-sep 5257 ax-nul 5269 ax-pow 5336 ax-pr 5404 ax-un 7732 ax-cnex 11151 ax-resscn 11152 ax-1cn 11153 ax-icn 11154 ax-addcl 11155 ax-addrcl 11156 ax-mulcl 11157 ax-mulrcl 11158 ax-mulcom 11159 ax-addass 11160 ax-mulass 11161 ax-distr 11162 ax-i2m1 11163 ax-1ne0 11164 ax-1rid 11165 ax-rnegex 11166 ax-rrecex 11167 ax-cnre 11168 ax-pre-lttri 11169 ax-pre-lttrn 11170 ax-pre-ltadd 11171 ax-pre-mulgt0 11172 ax-pre-sup 11173

This theorem depends on definitions: df-bi 210 df-an 401 df-or 861 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1810 df-nf 1814 df-sb 2097 df-mo 2567 df-eu 2597 df-clab 2742 df-cleq 2755 df-clel 2838 df-nfc 2912 df-ne 2959 df-nel 3065 df-ral 3080 df-rex 3090 df-rmo 3369 df-reu 3370 df-rab 3417 df-v 3457 df-sbc 3745 df-csb 3854 df-dif 3908 df-un 3910 df-in 3912 df-ss 3922 df-pss 3925 df-nul 4287 df-if 4488 df-pw 4564 df-sn 4590 df-pr 4592 df-op 4596 df-uni 4873 df-iun 4958 df-br 5110 df-opab 5174 df-mpt 5193 df-tr 5219 df-id 5556 df-eprel 5561 df-po 5569 df-so 5570 df-fr 5614 df-we 5616 df-xp 5667 df-rel 5668 df-cnv 5669 df-co 5670 df-dm 5671 df-rn 5672 df-res 5673 df-ima 5674 df-pred 6302 df-ord 6363 df-on 6364 df-lim 6365 df-suc 6366 df-iota 6492 df-fun 6538 df-fn 6539 df-f 6540 df-f1 6541 df-fo 6542 df-f1o 6543 df-fv 6544 df-riota 7367 df-ov 7413 df-oprab 7414 df-mpo 7415 df-om 7859 df-1st 7982 df-2nd 7983 df-frecs 8274 df-wrecs 8305 df-recs 8354 df-rdg 8393 df-1o 8449 df-2o 8450 df-er 8690 df-en 8940 df-dom 8941 df-sdom 8942 df-fin 8943 df-sup 9398 df-inf 9399 df-pnf 11240 df-mnf 11241 df-xr 11242 df-ltxr 11243 df-le 11244 df-sub 11438 df-neg 11439 df-div 11867 df-nn 12229 df-2 12298 df-3 12299 df-n0 12500 df-z 12587 df-uz 12858 df-q 12968 df-rp 13012 df-fl 13821 df-mod 13899 df-seq 14034 df-exp 14094 df-cj 15146 df-re 15147 df-im 15148 df-sqrt 15282 df-abs 15283 df-dvds 16306 df-gcd 16548 df-prm 16725 df-pc 16892

This theorem is referenced by: pcqdiv 16912 pcexp 16914 pcaddlem 16943 sylow1lem1 19663 padicabv 27794

W3C validator