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Theorem pcqmul 16945

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 12999	. . 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 12999	. . 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 3234	. . 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 3234	. . . . 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 521	. . . . . . . 8 ⊢ ((𝑃 ∈ ℙ ∧ (𝐴 ∈ ℚ ∧ 𝐴 ≠ 0) ∧ (𝐵 ∈ ℚ ∧ 𝐵 ≠ 0)) → (𝐴 ≠ 0 ∧ 𝐵 ≠ 0))
12	11	ad2antrr 739	. . . . . . 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 783	. . . . . . . . . . . . . 14 ⊢ (((𝑃 ∈ ℙ ∧ (𝑥 ∈ ℤ ∧ 𝑧 ∈ ℤ)) ∧ (𝑦 ∈ ℕ ∧ 𝑤 ∈ ℕ)) → 𝑦 ∈ ℕ)
15	14	nncnd 12273	. . . . . . . . . . . . 13 ⊢ (((𝑃 ∈ ℙ ∧ (𝑥 ∈ ℤ ∧ 𝑧 ∈ ℤ)) ∧ (𝑦 ∈ ℕ ∧ 𝑤 ∈ ℕ)) → 𝑦 ∈ ℂ)
16	14	nnne0d 12310	. . . . . . . . . . . . 13 ⊢ (((𝑃 ∈ ℙ ∧ (𝑥 ∈ ℤ ∧ 𝑧 ∈ ℤ)) ∧ (𝑦 ∈ ℕ ∧ 𝑤 ∈ ℕ)) → 𝑦 ≠ 0)
17	15, 16	div0d 12014	. . . . . . . . . . . 12 ⊢ (((𝑃 ∈ ℙ ∧ (𝑥 ∈ ℤ ∧ 𝑧 ∈ ℤ)) ∧ (𝑦 ∈ ℕ ∧ 𝑤 ∈ ℕ)) → (0 / 𝑦) = 0)
18		oveq1 7420	. . . . . . . . . . . . 13 ⊢ (𝑥 = 0 → (𝑥 / 𝑦) = (0 / 𝑦))
19	18	eqeq1d 2762	. . . . . . . . . . . 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 2976	. . . . . . . . . 10 ⊢ (((𝑃 ∈ ℙ ∧ (𝑥 ∈ ℤ ∧ 𝑧 ∈ ℤ)) ∧ (𝑦 ∈ ℕ ∧ 𝑤 ∈ ℕ)) → ((𝑥 / 𝑦) ≠ 0 → 𝑥 ≠ 0))
22		simprr 785	. . . . . . . . . . . . . 14 ⊢ (((𝑃 ∈ ℙ ∧ (𝑥 ∈ ℤ ∧ 𝑧 ∈ ℤ)) ∧ (𝑦 ∈ ℕ ∧ 𝑤 ∈ ℕ)) → 𝑤 ∈ ℕ)
23	22	nncnd 12273	. . . . . . . . . . . . 13 ⊢ (((𝑃 ∈ ℙ ∧ (𝑥 ∈ ℤ ∧ 𝑧 ∈ ℤ)) ∧ (𝑦 ∈ ℕ ∧ 𝑤 ∈ ℕ)) → 𝑤 ∈ ℂ)
24	22	nnne0d 12310	. . . . . . . . . . . . 13 ⊢ (((𝑃 ∈ ℙ ∧ (𝑥 ∈ ℤ ∧ 𝑧 ∈ ℤ)) ∧ (𝑦 ∈ ℕ ∧ 𝑤 ∈ ℕ)) → 𝑤 ≠ 0)
25	23, 24	div0d 12014	. . . . . . . . . . . 12 ⊢ (((𝑃 ∈ ℙ ∧ (𝑥 ∈ ℤ ∧ 𝑧 ∈ ℤ)) ∧ (𝑦 ∈ ℕ ∧ 𝑤 ∈ ℕ)) → (0 / 𝑤) = 0)
26		oveq1 7420	. . . . . . . . . . . . 13 ⊢ (𝑧 = 0 → (𝑧 / 𝑤) = (0 / 𝑤))
27	26	eqeq1d 2762	. . . . . . . . . . . 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 2976	. . . . . . . . . 10 ⊢ (((𝑃 ∈ ℙ ∧ (𝑥 ∈ ℤ ∧ 𝑧 ∈ ℤ)) ∧ (𝑦 ∈ ℕ ∧ 𝑤 ∈ ℕ)) → ((𝑧 / 𝑤) ≠ 0 → 𝑧 ≠ 0))
30		simpll 779	. . . . . . . . . . . . . 14 ⊢ (((𝑃 ∈ ℙ ∧ (𝑥 ∈ ℤ ∧ 𝑧 ∈ ℤ)) ∧ ((𝑦 ∈ ℕ ∧ 𝑤 ∈ ℕ) ∧ (𝑥 ≠ 0 ∧ 𝑧 ≠ 0))) → 𝑃 ∈ ℙ)
31		simplrl 789	. . . . . . . . . . . . . . 15 ⊢ (((𝑃 ∈ ℙ ∧ (𝑥 ∈ ℤ ∧ 𝑧 ∈ ℤ)) ∧ ((𝑦 ∈ ℕ ∧ 𝑤 ∈ ℕ) ∧ (𝑥 ≠ 0 ∧ 𝑧 ≠ 0))) → 𝑥 ∈ ℤ)
32		simplrr 790	. . . . . . . . . . . . . . 15 ⊢ (((𝑃 ∈ ℙ ∧ (𝑥 ∈ ℤ ∧ 𝑧 ∈ ℤ)) ∧ ((𝑦 ∈ ℕ ∧ 𝑤 ∈ ℕ) ∧ (𝑥 ≠ 0 ∧ 𝑧 ≠ 0))) → 𝑧 ∈ ℤ)
33	31, 32	zmulcld 12731	. . . . . . . . . . . . . 14 ⊢ (((𝑃 ∈ ℙ ∧ (𝑥 ∈ ℤ ∧ 𝑧 ∈ ℤ)) ∧ ((𝑦 ∈ ℕ ∧ 𝑤 ∈ ℕ) ∧ (𝑥 ≠ 0 ∧ 𝑧 ≠ 0))) → (𝑥 · 𝑧) ∈ ℤ)
34	31	zcnd 12726	. . . . . . . . . . . . . . 15 ⊢ (((𝑃 ∈ ℙ ∧ (𝑥 ∈ ℤ ∧ 𝑧 ∈ ℤ)) ∧ ((𝑦 ∈ ℕ ∧ 𝑤 ∈ ℕ) ∧ (𝑥 ≠ 0 ∧ 𝑧 ≠ 0))) → 𝑥 ∈ ℂ)
35	32	zcnd 12726	. . . . . . . . . . . . . . 15 ⊢ (((𝑃 ∈ ℙ ∧ (𝑥 ∈ ℤ ∧ 𝑧 ∈ ℤ)) ∧ ((𝑦 ∈ ℕ ∧ 𝑤 ∈ ℕ) ∧ (𝑥 ≠ 0 ∧ 𝑧 ≠ 0))) → 𝑧 ∈ ℂ)
36		simprrl 793	. . . . . . . . . . . . . . 15 ⊢ (((𝑃 ∈ ℙ ∧ (𝑥 ∈ ℤ ∧ 𝑧 ∈ ℤ)) ∧ ((𝑦 ∈ ℕ ∧ 𝑤 ∈ ℕ) ∧ (𝑥 ≠ 0 ∧ 𝑧 ≠ 0))) → 𝑥 ≠ 0)
37		simprrr 794	. . . . . . . . . . . . . . 15 ⊢ (((𝑃 ∈ ℙ ∧ (𝑥 ∈ ℤ ∧ 𝑧 ∈ ℤ)) ∧ ((𝑦 ∈ ℕ ∧ 𝑤 ∈ ℕ) ∧ (𝑥 ≠ 0 ∧ 𝑧 ≠ 0))) → 𝑧 ≠ 0)
38	34, 35, 36, 37	mulne0d 11890	. . . . . . . . . . . . . 14 ⊢ (((𝑃 ∈ ℙ ∧ (𝑥 ∈ ℤ ∧ 𝑧 ∈ ℤ)) ∧ ((𝑦 ∈ ℕ ∧ 𝑤 ∈ ℕ) ∧ (𝑥 ≠ 0 ∧ 𝑧 ≠ 0))) → (𝑥 · 𝑧) ≠ 0)
39	14	adantrr 730	. . . . . . . . . . . . . . 15 ⊢ (((𝑃 ∈ ℙ ∧ (𝑥 ∈ ℤ ∧ 𝑧 ∈ ℤ)) ∧ ((𝑦 ∈ ℕ ∧ 𝑤 ∈ ℕ) ∧ (𝑥 ≠ 0 ∧ 𝑧 ≠ 0))) → 𝑦 ∈ ℕ)
40	22	adantrr 730	. . . . . . . . . . . . . . 15 ⊢ (((𝑃 ∈ ℙ ∧ (𝑥 ∈ ℤ ∧ 𝑧 ∈ ℤ)) ∧ ((𝑦 ∈ ℕ ∧ 𝑤 ∈ ℕ) ∧ (𝑥 ≠ 0 ∧ 𝑧 ≠ 0))) → 𝑤 ∈ ℕ)
41	39, 40	nnmulcld 12313	. . . . . . . . . . . . . 14 ⊢ (((𝑃 ∈ ℙ ∧ (𝑥 ∈ ℤ ∧ 𝑧 ∈ ℤ)) ∧ ((𝑦 ∈ ℕ ∧ 𝑤 ∈ ℕ) ∧ (𝑥 ≠ 0 ∧ 𝑧 ≠ 0))) → (𝑦 · 𝑤) ∈ ℕ)
42		pcdiv 16944	. . . . . . . . . . . . . 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 16943	. . . . . . . . . . . . . . 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 12641	. . . . . . . . . . . . . . 15 ⊢ (((𝑃 ∈ ℙ ∧ (𝑥 ∈ ℤ ∧ 𝑧 ∈ ℤ)) ∧ ((𝑦 ∈ ℕ ∧ 𝑤 ∈ ℕ) ∧ (𝑥 ≠ 0 ∧ 𝑧 ≠ 0))) → 𝑦 ∈ ℤ)
47	16	adantrr 730	. . . . . . . . . . . . . . 15 ⊢ (((𝑃 ∈ ℙ ∧ (𝑥 ∈ ℤ ∧ 𝑧 ∈ ℤ)) ∧ ((𝑦 ∈ ℕ ∧ 𝑤 ∈ ℕ) ∧ (𝑥 ≠ 0 ∧ 𝑧 ≠ 0))) → 𝑦 ≠ 0)
48	40	nnzd 12641	. . . . . . . . . . . . . . 15 ⊢ (((𝑃 ∈ ℙ ∧ (𝑥 ∈ ℤ ∧ 𝑧 ∈ ℤ)) ∧ ((𝑦 ∈ ℕ ∧ 𝑤 ∈ ℕ) ∧ (𝑥 ≠ 0 ∧ 𝑧 ≠ 0))) → 𝑤 ∈ ℤ)
49	24	adantrr 730	. . . . . . . . . . . . . . 15 ⊢ (((𝑃 ∈ ℙ ∧ (𝑥 ∈ ℤ ∧ 𝑧 ∈ ℤ)) ∧ ((𝑦 ∈ ℕ ∧ 𝑤 ∈ ℕ) ∧ (𝑥 ≠ 0 ∧ 𝑧 ≠ 0))) → 𝑤 ≠ 0)
50		pcmul 16943	. . . . . . . . . . . . . . 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 7431	. . . . . . . . . . . . 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 16940	. . . . . . . . . . . . . . . 16 ⊢ ((𝑃 ∈ ℙ ∧ (𝑥 ∈ ℤ ∧ 𝑥 ≠ 0)) → (𝑃 pCnt 𝑥) ∈ ℕ₀)
54	30, 31, 36, 53	syl12anc 850	. . . . . . . . . . . . . . 15 ⊢ (((𝑃 ∈ ℙ ∧ (𝑥 ∈ ℤ ∧ 𝑧 ∈ ℤ)) ∧ ((𝑦 ∈ ℕ ∧ 𝑤 ∈ ℕ) ∧ (𝑥 ≠ 0 ∧ 𝑧 ≠ 0))) → (𝑃 pCnt 𝑥) ∈ ℕ₀)
55	54	nn0cnd 12591	. . . . . . . . . . . . . 14 ⊢ (((𝑃 ∈ ℙ ∧ (𝑥 ∈ ℤ ∧ 𝑧 ∈ ℤ)) ∧ ((𝑦 ∈ ℕ ∧ 𝑤 ∈ ℕ) ∧ (𝑥 ≠ 0 ∧ 𝑧 ≠ 0))) → (𝑃 pCnt 𝑥) ∈ ℂ)
56		pczcl 16940	. . . . . . . . . . . . . . . 16 ⊢ ((𝑃 ∈ ℙ ∧ (𝑧 ∈ ℤ ∧ 𝑧 ≠ 0)) → (𝑃 pCnt 𝑧) ∈ ℕ₀)
57	30, 32, 37, 56	syl12anc 850	. . . . . . . . . . . . . . 15 ⊢ (((𝑃 ∈ ℙ ∧ (𝑥 ∈ ℤ ∧ 𝑧 ∈ ℤ)) ∧ ((𝑦 ∈ ℕ ∧ 𝑤 ∈ ℕ) ∧ (𝑥 ≠ 0 ∧ 𝑧 ≠ 0))) → (𝑃 pCnt 𝑧) ∈ ℕ₀)
58	57	nn0cnd 12591	. . . . . . . . . . . . . 14 ⊢ (((𝑃 ∈ ℙ ∧ (𝑥 ∈ ℤ ∧ 𝑧 ∈ ℤ)) ∧ ((𝑦 ∈ ℕ ∧ 𝑤 ∈ ℕ) ∧ (𝑥 ≠ 0 ∧ 𝑧 ≠ 0))) → (𝑃 pCnt 𝑧) ∈ ℂ)
59	30, 39	pccld 16942	. . . . . . . . . . . . . . 15 ⊢ (((𝑃 ∈ ℙ ∧ (𝑥 ∈ ℤ ∧ 𝑧 ∈ ℤ)) ∧ ((𝑦 ∈ ℕ ∧ 𝑤 ∈ ℕ) ∧ (𝑥 ≠ 0 ∧ 𝑧 ≠ 0))) → (𝑃 pCnt 𝑦) ∈ ℕ₀)
60	59	nn0cnd 12591	. . . . . . . . . . . . . 14 ⊢ (((𝑃 ∈ ℙ ∧ (𝑥 ∈ ℤ ∧ 𝑧 ∈ ℤ)) ∧ ((𝑦 ∈ ℕ ∧ 𝑤 ∈ ℕ) ∧ (𝑥 ≠ 0 ∧ 𝑧 ≠ 0))) → (𝑃 pCnt 𝑦) ∈ ℂ)
61	30, 40	pccld 16942	. . . . . . . . . . . . . . 15 ⊢ (((𝑃 ∈ ℙ ∧ (𝑥 ∈ ℤ ∧ 𝑧 ∈ ℤ)) ∧ ((𝑦 ∈ ℕ ∧ 𝑤 ∈ ℕ) ∧ (𝑥 ≠ 0 ∧ 𝑧 ≠ 0))) → (𝑃 pCnt 𝑤) ∈ ℕ₀)
62	61	nn0cnd 12591	. . . . . . . . . . . . . 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 11640	. . . . . . . . . . . . 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 2799	. . . . . . . . . . . 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 730	. . . . . . . . . . . . . 14 ⊢ (((𝑃 ∈ ℙ ∧ (𝑥 ∈ ℤ ∧ 𝑧 ∈ ℤ)) ∧ ((𝑦 ∈ ℕ ∧ 𝑤 ∈ ℕ) ∧ (𝑥 ≠ 0 ∧ 𝑧 ≠ 0))) → 𝑦 ∈ ℂ)
66	23	adantrr 730	. . . . . . . . . . . . . 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 12056	. . . . . . . . . . . . 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 7429	. . . . . . . . . . . 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 16944	. . . . . . . . . . . . . 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 16944	. . . . . . . . . . . . . 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 7431	. . . . . . . . . . . 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 2805	. . . . . . . . . . 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 462	. . . . . . . . . 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 620	. . . . . . . . 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 3017	. . . . . . . . . . 11 ⊢ (𝐴 = (𝑥 / 𝑦) → (𝐴 ≠ 0 ↔ (𝑥 / 𝑦) ≠ 0))
78		neeq1 3017	. . . . . . . . . . 11 ⊢ (𝐵 = (𝑧 / 𝑤) → (𝐵 ≠ 0 ↔ (𝑧 / 𝑤) ≠ 0))
79	77, 78	bi2anan9 650	. . . . . . . . . 10 ⊢ ((𝐴 = (𝑥 / 𝑦) ∧ 𝐵 = (𝑧 / 𝑤)) → ((𝐴 ≠ 0 ∧ 𝐵 ≠ 0) ↔ ((𝑥 / 𝑦) ≠ 0 ∧ (𝑧 / 𝑤) ≠ 0)))
80		oveq12 7422	. . . . . . . . . . . 12 ⊢ ((𝐴 = (𝑥 / 𝑦) ∧ 𝐵 = (𝑧 / 𝑤)) → (𝐴 · 𝐵) = ((𝑥 / 𝑦) · (𝑧 / 𝑤)))
81	80	oveq2d 7429	. . . . . . . . . . 11 ⊢ ((𝐴 = (𝑥 / 𝑦) ∧ 𝐵 = (𝑧 / 𝑤)) → (𝑃 pCnt (𝐴 · 𝐵)) = (𝑃 pCnt ((𝑥 / 𝑦) · (𝑧 / 𝑤))))
82		oveq2 7421	. . . . . . . . . . . 12 ⊢ (𝐴 = (𝑥 / 𝑦) → (𝑃 pCnt 𝐴) = (𝑃 pCnt (𝑥 / 𝑦)))
83		oveq2 7421	. . . . . . . . . . . 12 ⊢ (𝐵 = (𝑧 / 𝑤) → (𝑃 pCnt 𝐵) = (𝑃 pCnt (𝑧 / 𝑤)))
84	82, 83	oveqan12d 7432	. . . . . . . . . . 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 2776	. . . . . . . . . 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 693	. . . . . . 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 3219	. . . . 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 3219	. . 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 712	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

This proof depends on syntax axioms: → wi 4 ∧ wa 401 ∧ w3a 1103 = wceq 1570 ∈ wcel 2145 ≠ wne 2955 ∃wrex 3086 (class class class)co 7413 ℂcc 11122 0cc0 11124 + caddc 11127 · cmul 11129 − cmin 11465 / cdiv 11895 ℕcn 12257 ℕ₀cn0 12528 ℤcz 12615 ℚcq 12997 ℙcprime 16761 pCnt cpc 16928

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 1943 ax-6 2000 ax-7 2041 ax-8 2147 ax-9 2155 ax-10 2178 ax-11 2194 ax-12 2213 ax-ext 2732 ax-sep 5251 ax-nul 5263 ax-pow 5330 ax-pr 5398 ax-un 7736 ax-cnex 11180 ax-resscn 11181 ax-1cn 11182 ax-icn 11183 ax-addcl 11184 ax-addrcl 11185 ax-mulcl 11186 ax-mulrcl 11187 ax-mulcom 11188 ax-addass 11189 ax-mulass 11190 ax-distr 11191 ax-i2m1 11192 ax-1ne0 11193 ax-1rid 11194 ax-rnegex 11195 ax-rrecex 11196 ax-cnre 11197 ax-pre-lttri 11198 ax-pre-lttrn 11199 ax-pre-ltadd 11200 ax-pre-mulgt0 11201 ax-pre-sup 11202

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 2100 df-mo 2564 df-eu 2594 df-clab 2739 df-cleq 2752 df-clel 2835 df-nfc 2909 df-ne 2956 df-nel 3062 df-ral 3077 df-rex 3087 df-rmo 3365 df-reu 3366 df-rab 3413 df-v 3452 df-sbc 3740 df-csb 3848 df-dif 3902 df-un 3904 df-in 3906 df-ss 3916 df-pss 3919 df-nul 4280 df-if 4483 df-pw 4559 df-sn 4585 df-pr 4587 df-op 4591 df-uni 4868 df-iun 4953 df-br 5104 df-opab 5168 df-mpt 5187 df-tr 5213 df-id 5550 df-eprel 5555 df-po 5563 df-so 5564 df-fr 5608 df-we 5610 df-xp 5661 df-rel 5662 df-cnv 5663 df-co 5664 df-dm 5665 df-rn 5666 df-res 5667 df-ima 5668 df-pred 6299 df-ord 6360 df-on 6361 df-lim 6362 df-suc 6363 df-iota 6489 df-fun 6535 df-fn 6536 df-f 6537 df-f1 6538 df-fo 6539 df-f1o 6540 df-fv 6541 df-riota 7370 df-ov 7416 df-oprab 7417 df-mpo 7418 df-om 7863 df-1st 7986 df-2nd 7987 df-frecs 8280 df-wrecs 8311 df-recs 8360 df-rdg 8399 df-1o 8455 df-2o 8456 df-er 8696 df-en 8953 df-dom 8954 df-sdom 8955 df-fin 8956 df-sup 9412 df-inf 9413 df-pnf 11269 df-mnf 11270 df-xr 11271 df-ltxr 11272 df-le 11273 df-sub 11467 df-neg 11468 df-div 11896 df-nn 12258 df-2 12327 df-3 12328 df-n0 12529 df-z 12616 df-uz 12888 df-q 12998 df-rp 13043 df-fl 13853 df-mod 13931 df-seq 14066 df-exp 14126 df-cj 15186 df-re 15187 df-im 15188 df-sqrt 15322 df-abs 15323 df-dvds 16343 df-gcd 16585 df-prm 16762 df-pc 16929

This theorem is used by: pcqdiv 16949 pcexp 16951 pcaddlem 16980 sylow1lem1 19725 padicabv 27866

W3C validator