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Theorem pcadd 16923

Description: An inequality for the prime count of a sum. This is the source of the ultrametric inequality for the p-adic metric. (Contributed by Mario Carneiro, 9-Sep-2014.)

**Hypotheses**
Ref	Expression
pcadd.1	⊢ (𝜑 → 𝑃 ∈ ℙ)
pcadd.2	⊢ (𝜑 → 𝐴 ∈ ℚ)
pcadd.3	⊢ (𝜑 → 𝐵 ∈ ℚ)
pcadd.4	⊢ (𝜑 → (𝑃 pCnt 𝐴) ≤ (𝑃 pCnt 𝐵))

**Assertion**
Ref	Expression
pcadd	⊢ (𝜑 → (𝑃 pCnt 𝐴) ≤ (𝑃 pCnt (𝐴 + 𝐵)))

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

Step	Hyp	Ref	Expression
1		pcadd.2	. . 3 ⊢ (𝜑 → 𝐴 ∈ ℚ)
2		elq 12990	. . 3 ⊢ (𝐴 ∈ ℚ ↔ ∃𝑥 ∈ ℤ ∃𝑦 ∈ ℕ 𝐴 = (𝑥 / 𝑦))
3	1, 2	sylib 218	. 2 ⊢ (𝜑 → ∃𝑥 ∈ ℤ ∃𝑦 ∈ ℕ 𝐴 = (𝑥 / 𝑦))
4		pcadd.3	. . 3 ⊢ (𝜑 → 𝐵 ∈ ℚ)
5		elq 12990	. . 3 ⊢ (𝐵 ∈ ℚ ↔ ∃𝑧 ∈ ℤ ∃𝑤 ∈ ℕ 𝐵 = (𝑧 / 𝑤))
6	4, 5	sylib 218	. 2 ⊢ (𝜑 → ∃𝑧 ∈ ℤ ∃𝑤 ∈ ℕ 𝐵 = (𝑧 / 𝑤))
7		pcadd.1	. . . . . . . 8 ⊢ (𝜑 → 𝑃 ∈ ℙ)
8		pcxcl 16895	. . . . . . . 8 ⊢ ((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℚ) → (𝑃 pCnt 𝐴) ∈ ℝ^*)
9	7, 1, 8	syl2anc 584	. . . . . . 7 ⊢ (𝜑 → (𝑃 pCnt 𝐴) ∈ ℝ^*)
10	9	xrleidd 13191	. . . . . 6 ⊢ (𝜑 → (𝑃 pCnt 𝐴) ≤ (𝑃 pCnt 𝐴))
11	10	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ 𝐵 = 0) → (𝑃 pCnt 𝐴) ≤ (𝑃 pCnt 𝐴))
12		oveq2 7439	. . . . . . 7 ⊢ (𝐵 = 0 → (𝐴 + 𝐵) = (𝐴 + 0))
13		qcn 13003	. . . . . . . . 9 ⊢ (𝐴 ∈ ℚ → 𝐴 ∈ ℂ)
14	1, 13	syl 17	. . . . . . . 8 ⊢ (𝜑 → 𝐴 ∈ ℂ)
15	14	addridd 11459	. . . . . . 7 ⊢ (𝜑 → (𝐴 + 0) = 𝐴)
16	12, 15	sylan9eqr 2797	. . . . . 6 ⊢ ((𝜑 ∧ 𝐵 = 0) → (𝐴 + 𝐵) = 𝐴)
17	16	oveq2d 7447	. . . . 5 ⊢ ((𝜑 ∧ 𝐵 = 0) → (𝑃 pCnt (𝐴 + 𝐵)) = (𝑃 pCnt 𝐴))
18	11, 17	breqtrrd 5176	. . . 4 ⊢ ((𝜑 ∧ 𝐵 = 0) → (𝑃 pCnt 𝐴) ≤ (𝑃 pCnt (𝐴 + 𝐵)))
19	18	a1d 25	. . 3 ⊢ ((𝜑 ∧ 𝐵 = 0) → ((∃𝑥 ∈ ℤ ∃𝑦 ∈ ℕ 𝐴 = (𝑥 / 𝑦) ∧ ∃𝑧 ∈ ℤ ∃𝑤 ∈ ℕ 𝐵 = (𝑧 / 𝑤)) → (𝑃 pCnt 𝐴) ≤ (𝑃 pCnt (𝐴 + 𝐵))))
20		reeanv 3227	. . . 4 ⊢ (∃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)))
21		reeanv 3227	. . . . . 6 ⊢ (∃𝑦 ∈ ℕ ∃𝑤 ∈ ℕ (𝐴 = (𝑥 / 𝑦) ∧ 𝐵 = (𝑧 / 𝑤)) ↔ (∃𝑦 ∈ ℕ 𝐴 = (𝑥 / 𝑦) ∧ ∃𝑤 ∈ ℕ 𝐵 = (𝑧 / 𝑤)))
22	7	ad3antrrr 730	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝐵 ≠ 0) ∧ (𝑥 ∈ ℤ ∧ 𝑧 ∈ ℤ)) ∧ ((𝑦 ∈ ℕ ∧ 𝑤 ∈ ℕ) ∧ (𝐴 = (𝑥 / 𝑦) ∧ 𝐵 = (𝑧 / 𝑤)))) → 𝑃 ∈ ℙ)
23		prmnn 16708	. . . . . . . . . . . . . . . . 17 ⊢ (𝑃 ∈ ℙ → 𝑃 ∈ ℕ)
24	22, 23	syl 17	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ 𝐵 ≠ 0) ∧ (𝑥 ∈ ℤ ∧ 𝑧 ∈ ℤ)) ∧ ((𝑦 ∈ ℕ ∧ 𝑤 ∈ ℕ) ∧ (𝐴 = (𝑥 / 𝑦) ∧ 𝐵 = (𝑧 / 𝑤)))) → 𝑃 ∈ ℕ)
25		simplrl 777	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ 𝐵 ≠ 0) ∧ (𝑥 ∈ ℤ ∧ 𝑧 ∈ ℤ)) ∧ ((𝑦 ∈ ℕ ∧ 𝑤 ∈ ℕ) ∧ (𝐴 = (𝑥 / 𝑦) ∧ 𝐵 = (𝑧 / 𝑤)))) → 𝑥 ∈ ℤ)
26		simprrl 781	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝜑 ∧ 𝐵 ≠ 0) ∧ (𝑥 ∈ ℤ ∧ 𝑧 ∈ ℤ)) ∧ ((𝑦 ∈ ℕ ∧ 𝑤 ∈ ℕ) ∧ (𝐴 = (𝑥 / 𝑦) ∧ 𝐵 = (𝑧 / 𝑤)))) → 𝐴 = (𝑥 / 𝑦))
27		pc0 16888	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑃 ∈ ℙ → (𝑃 pCnt 0) = +∞)
28	22, 27	syl 17	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((((𝜑 ∧ 𝐵 ≠ 0) ∧ (𝑥 ∈ ℤ ∧ 𝑧 ∈ ℤ)) ∧ ((𝑦 ∈ ℕ ∧ 𝑤 ∈ ℕ) ∧ (𝐴 = (𝑥 / 𝑦) ∧ 𝐵 = (𝑧 / 𝑤)))) → (𝑃 pCnt 0) = +∞)
29	4	ad3antrrr 730	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((((𝜑 ∧ 𝐵 ≠ 0) ∧ (𝑥 ∈ ℤ ∧ 𝑧 ∈ ℤ)) ∧ ((𝑦 ∈ ℕ ∧ 𝑤 ∈ ℕ) ∧ (𝐴 = (𝑥 / 𝑦) ∧ 𝐵 = (𝑧 / 𝑤)))) → 𝐵 ∈ ℚ)
30		simpllr 776	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((((𝜑 ∧ 𝐵 ≠ 0) ∧ (𝑥 ∈ ℤ ∧ 𝑧 ∈ ℤ)) ∧ ((𝑦 ∈ ℕ ∧ 𝑤 ∈ ℕ) ∧ (𝐴 = (𝑥 / 𝑦) ∧ 𝐵 = (𝑧 / 𝑤)))) → 𝐵 ≠ 0)
31		pcqcl 16890	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝑃 ∈ ℙ ∧ (𝐵 ∈ ℚ ∧ 𝐵 ≠ 0)) → (𝑃 pCnt 𝐵) ∈ ℤ)
32	22, 29, 30, 31	syl12anc 837	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((((𝜑 ∧ 𝐵 ≠ 0) ∧ (𝑥 ∈ ℤ ∧ 𝑧 ∈ ℤ)) ∧ ((𝑦 ∈ ℕ ∧ 𝑤 ∈ ℕ) ∧ (𝐴 = (𝑥 / 𝑦) ∧ 𝐵 = (𝑧 / 𝑤)))) → (𝑃 pCnt 𝐵) ∈ ℤ)
33	32	zred 12720	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((((𝜑 ∧ 𝐵 ≠ 0) ∧ (𝑥 ∈ ℤ ∧ 𝑧 ∈ ℤ)) ∧ ((𝑦 ∈ ℕ ∧ 𝑤 ∈ ℕ) ∧ (𝐴 = (𝑥 / 𝑦) ∧ 𝐵 = (𝑧 / 𝑤)))) → (𝑃 pCnt 𝐵) ∈ ℝ)
34		ltpnf 13160	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝑃 pCnt 𝐵) ∈ ℝ → (𝑃 pCnt 𝐵) < +∞)
35		rexr 11305	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝑃 pCnt 𝐵) ∈ ℝ → (𝑃 pCnt 𝐵) ∈ ℝ^*)
36		pnfxr 11313	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ +∞ ∈ ℝ^*
37		xrltnle 11326	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((𝑃 pCnt 𝐵) ∈ ℝ^* ∧ +∞ ∈ ℝ^*) → ((𝑃 pCnt 𝐵) < +∞ ↔ ¬ +∞ ≤ (𝑃 pCnt 𝐵)))
38	35, 36, 37	sylancl 586	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝑃 pCnt 𝐵) ∈ ℝ → ((𝑃 pCnt 𝐵) < +∞ ↔ ¬ +∞ ≤ (𝑃 pCnt 𝐵)))
39	34, 38	mpbid 232	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑃 pCnt 𝐵) ∈ ℝ → ¬ +∞ ≤ (𝑃 pCnt 𝐵))
40	33, 39	syl 17	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((((𝜑 ∧ 𝐵 ≠ 0) ∧ (𝑥 ∈ ℤ ∧ 𝑧 ∈ ℤ)) ∧ ((𝑦 ∈ ℕ ∧ 𝑤 ∈ ℕ) ∧ (𝐴 = (𝑥 / 𝑦) ∧ 𝐵 = (𝑧 / 𝑤)))) → ¬ +∞ ≤ (𝑃 pCnt 𝐵))
41	28, 40	eqnbrtrd 5166	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((𝜑 ∧ 𝐵 ≠ 0) ∧ (𝑥 ∈ ℤ ∧ 𝑧 ∈ ℤ)) ∧ ((𝑦 ∈ ℕ ∧ 𝑤 ∈ ℕ) ∧ (𝐴 = (𝑥 / 𝑦) ∧ 𝐵 = (𝑧 / 𝑤)))) → ¬ (𝑃 pCnt 0) ≤ (𝑃 pCnt 𝐵))
42		pcadd.4	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝜑 → (𝑃 pCnt 𝐴) ≤ (𝑃 pCnt 𝐵))
43	42	ad3antrrr 730	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((((𝜑 ∧ 𝐵 ≠ 0) ∧ (𝑥 ∈ ℤ ∧ 𝑧 ∈ ℤ)) ∧ ((𝑦 ∈ ℕ ∧ 𝑤 ∈ ℕ) ∧ (𝐴 = (𝑥 / 𝑦) ∧ 𝐵 = (𝑧 / 𝑤)))) → (𝑃 pCnt 𝐴) ≤ (𝑃 pCnt 𝐵))
44		oveq2 7439	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝐴 = 0 → (𝑃 pCnt 𝐴) = (𝑃 pCnt 0))
45	44	breq1d 5158	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝐴 = 0 → ((𝑃 pCnt 𝐴) ≤ (𝑃 pCnt 𝐵) ↔ (𝑃 pCnt 0) ≤ (𝑃 pCnt 𝐵)))
46	43, 45	syl5ibcom 245	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((((𝜑 ∧ 𝐵 ≠ 0) ∧ (𝑥 ∈ ℤ ∧ 𝑧 ∈ ℤ)) ∧ ((𝑦 ∈ ℕ ∧ 𝑤 ∈ ℕ) ∧ (𝐴 = (𝑥 / 𝑦) ∧ 𝐵 = (𝑧 / 𝑤)))) → (𝐴 = 0 → (𝑃 pCnt 0) ≤ (𝑃 pCnt 𝐵)))
47	46	necon3bd 2952	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((𝜑 ∧ 𝐵 ≠ 0) ∧ (𝑥 ∈ ℤ ∧ 𝑧 ∈ ℤ)) ∧ ((𝑦 ∈ ℕ ∧ 𝑤 ∈ ℕ) ∧ (𝐴 = (𝑥 / 𝑦) ∧ 𝐵 = (𝑧 / 𝑤)))) → (¬ (𝑃 pCnt 0) ≤ (𝑃 pCnt 𝐵) → 𝐴 ≠ 0))
48	41, 47	mpd 15	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝜑 ∧ 𝐵 ≠ 0) ∧ (𝑥 ∈ ℤ ∧ 𝑧 ∈ ℤ)) ∧ ((𝑦 ∈ ℕ ∧ 𝑤 ∈ ℕ) ∧ (𝐴 = (𝑥 / 𝑦) ∧ 𝐵 = (𝑧 / 𝑤)))) → 𝐴 ≠ 0)
49	26, 48	eqnetrrd 3007	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝜑 ∧ 𝐵 ≠ 0) ∧ (𝑥 ∈ ℤ ∧ 𝑧 ∈ ℤ)) ∧ ((𝑦 ∈ ℕ ∧ 𝑤 ∈ ℕ) ∧ (𝐴 = (𝑥 / 𝑦) ∧ 𝐵 = (𝑧 / 𝑤)))) → (𝑥 / 𝑦) ≠ 0)
50		simprll 779	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((((𝜑 ∧ 𝐵 ≠ 0) ∧ (𝑥 ∈ ℤ ∧ 𝑧 ∈ ℤ)) ∧ ((𝑦 ∈ ℕ ∧ 𝑤 ∈ ℕ) ∧ (𝐴 = (𝑥 / 𝑦) ∧ 𝐵 = (𝑧 / 𝑤)))) → 𝑦 ∈ ℕ)
51	50	nncnd 12280	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((((𝜑 ∧ 𝐵 ≠ 0) ∧ (𝑥 ∈ ℤ ∧ 𝑧 ∈ ℤ)) ∧ ((𝑦 ∈ ℕ ∧ 𝑤 ∈ ℕ) ∧ (𝐴 = (𝑥 / 𝑦) ∧ 𝐵 = (𝑧 / 𝑤)))) → 𝑦 ∈ ℂ)
52	50	nnne0d 12314	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((((𝜑 ∧ 𝐵 ≠ 0) ∧ (𝑥 ∈ ℤ ∧ 𝑧 ∈ ℤ)) ∧ ((𝑦 ∈ ℕ ∧ 𝑤 ∈ ℕ) ∧ (𝐴 = (𝑥 / 𝑦) ∧ 𝐵 = (𝑧 / 𝑤)))) → 𝑦 ≠ 0)
53	51, 52	div0d 12040	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((𝜑 ∧ 𝐵 ≠ 0) ∧ (𝑥 ∈ ℤ ∧ 𝑧 ∈ ℤ)) ∧ ((𝑦 ∈ ℕ ∧ 𝑤 ∈ ℕ) ∧ (𝐴 = (𝑥 / 𝑦) ∧ 𝐵 = (𝑧 / 𝑤)))) → (0 / 𝑦) = 0)
54		oveq1 7438	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑥 = 0 → (𝑥 / 𝑦) = (0 / 𝑦))
55	54	eqeq1d 2737	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑥 = 0 → ((𝑥 / 𝑦) = 0 ↔ (0 / 𝑦) = 0))
56	53, 55	syl5ibrcom 247	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝜑 ∧ 𝐵 ≠ 0) ∧ (𝑥 ∈ ℤ ∧ 𝑧 ∈ ℤ)) ∧ ((𝑦 ∈ ℕ ∧ 𝑤 ∈ ℕ) ∧ (𝐴 = (𝑥 / 𝑦) ∧ 𝐵 = (𝑧 / 𝑤)))) → (𝑥 = 0 → (𝑥 / 𝑦) = 0))
57	56	necon3d 2959	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝜑 ∧ 𝐵 ≠ 0) ∧ (𝑥 ∈ ℤ ∧ 𝑧 ∈ ℤ)) ∧ ((𝑦 ∈ ℕ ∧ 𝑤 ∈ ℕ) ∧ (𝐴 = (𝑥 / 𝑦) ∧ 𝐵 = (𝑧 / 𝑤)))) → ((𝑥 / 𝑦) ≠ 0 → 𝑥 ≠ 0))
58	49, 57	mpd 15	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ 𝐵 ≠ 0) ∧ (𝑥 ∈ ℤ ∧ 𝑧 ∈ ℤ)) ∧ ((𝑦 ∈ ℕ ∧ 𝑤 ∈ ℕ) ∧ (𝐴 = (𝑥 / 𝑦) ∧ 𝐵 = (𝑧 / 𝑤)))) → 𝑥 ≠ 0)
59		pczcl 16882	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑃 ∈ ℙ ∧ (𝑥 ∈ ℤ ∧ 𝑥 ≠ 0)) → (𝑃 pCnt 𝑥) ∈ ℕ₀)
60	22, 25, 58, 59	syl12anc 837	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ 𝐵 ≠ 0) ∧ (𝑥 ∈ ℤ ∧ 𝑧 ∈ ℤ)) ∧ ((𝑦 ∈ ℕ ∧ 𝑤 ∈ ℕ) ∧ (𝐴 = (𝑥 / 𝑦) ∧ 𝐵 = (𝑧 / 𝑤)))) → (𝑃 pCnt 𝑥) ∈ ℕ₀)
61	24, 60	nnexpcld 14281	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ 𝐵 ≠ 0) ∧ (𝑥 ∈ ℤ ∧ 𝑧 ∈ ℤ)) ∧ ((𝑦 ∈ ℕ ∧ 𝑤 ∈ ℕ) ∧ (𝐴 = (𝑥 / 𝑦) ∧ 𝐵 = (𝑧 / 𝑤)))) → (𝑃↑(𝑃 pCnt 𝑥)) ∈ ℕ)
62	61	nncnd 12280	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝐵 ≠ 0) ∧ (𝑥 ∈ ℤ ∧ 𝑧 ∈ ℤ)) ∧ ((𝑦 ∈ ℕ ∧ 𝑤 ∈ ℕ) ∧ (𝐴 = (𝑥 / 𝑦) ∧ 𝐵 = (𝑧 / 𝑤)))) → (𝑃↑(𝑃 pCnt 𝑥)) ∈ ℂ)
63	62, 51	mulcomd 11280	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝐵 ≠ 0) ∧ (𝑥 ∈ ℤ ∧ 𝑧 ∈ ℤ)) ∧ ((𝑦 ∈ ℕ ∧ 𝑤 ∈ ℕ) ∧ (𝐴 = (𝑥 / 𝑦) ∧ 𝐵 = (𝑧 / 𝑤)))) → ((𝑃↑(𝑃 pCnt 𝑥)) · 𝑦) = (𝑦 · (𝑃↑(𝑃 pCnt 𝑥))))
64	63	oveq2d 7447	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝐵 ≠ 0) ∧ (𝑥 ∈ ℤ ∧ 𝑧 ∈ ℤ)) ∧ ((𝑦 ∈ ℕ ∧ 𝑤 ∈ ℕ) ∧ (𝐴 = (𝑥 / 𝑦) ∧ 𝐵 = (𝑧 / 𝑤)))) → ((𝑥 · (𝑃↑(𝑃 pCnt 𝑦))) / ((𝑃↑(𝑃 pCnt 𝑥)) · 𝑦)) = ((𝑥 · (𝑃↑(𝑃 pCnt 𝑦))) / (𝑦 · (𝑃↑(𝑃 pCnt 𝑥)))))
65	25	zcnd 12721	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝐵 ≠ 0) ∧ (𝑥 ∈ ℤ ∧ 𝑧 ∈ ℤ)) ∧ ((𝑦 ∈ ℕ ∧ 𝑤 ∈ ℕ) ∧ (𝐴 = (𝑥 / 𝑦) ∧ 𝐵 = (𝑧 / 𝑤)))) → 𝑥 ∈ ℂ)
66	22, 50	pccld 16884	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ 𝐵 ≠ 0) ∧ (𝑥 ∈ ℤ ∧ 𝑧 ∈ ℤ)) ∧ ((𝑦 ∈ ℕ ∧ 𝑤 ∈ ℕ) ∧ (𝐴 = (𝑥 / 𝑦) ∧ 𝐵 = (𝑧 / 𝑤)))) → (𝑃 pCnt 𝑦) ∈ ℕ₀)
67	24, 66	nnexpcld 14281	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝐵 ≠ 0) ∧ (𝑥 ∈ ℤ ∧ 𝑧 ∈ ℤ)) ∧ ((𝑦 ∈ ℕ ∧ 𝑤 ∈ ℕ) ∧ (𝐴 = (𝑥 / 𝑦) ∧ 𝐵 = (𝑧 / 𝑤)))) → (𝑃↑(𝑃 pCnt 𝑦)) ∈ ℕ)
68	67	nncnd 12280	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝐵 ≠ 0) ∧ (𝑥 ∈ ℤ ∧ 𝑧 ∈ ℤ)) ∧ ((𝑦 ∈ ℕ ∧ 𝑤 ∈ ℕ) ∧ (𝐴 = (𝑥 / 𝑦) ∧ 𝐵 = (𝑧 / 𝑤)))) → (𝑃↑(𝑃 pCnt 𝑦)) ∈ ℂ)
69	61	nnne0d 12314	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝐵 ≠ 0) ∧ (𝑥 ∈ ℤ ∧ 𝑧 ∈ ℤ)) ∧ ((𝑦 ∈ ℕ ∧ 𝑤 ∈ ℕ) ∧ (𝐴 = (𝑥 / 𝑦) ∧ 𝐵 = (𝑧 / 𝑤)))) → (𝑃↑(𝑃 pCnt 𝑥)) ≠ 0)
70	67	nnne0d 12314	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝐵 ≠ 0) ∧ (𝑥 ∈ ℤ ∧ 𝑧 ∈ ℤ)) ∧ ((𝑦 ∈ ℕ ∧ 𝑤 ∈ ℕ) ∧ (𝐴 = (𝑥 / 𝑦) ∧ 𝐵 = (𝑧 / 𝑤)))) → (𝑃↑(𝑃 pCnt 𝑦)) ≠ 0)
71	65, 62, 51, 68, 69, 70, 52	divdivdivd 12088	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝐵 ≠ 0) ∧ (𝑥 ∈ ℤ ∧ 𝑧 ∈ ℤ)) ∧ ((𝑦 ∈ ℕ ∧ 𝑤 ∈ ℕ) ∧ (𝐴 = (𝑥 / 𝑦) ∧ 𝐵 = (𝑧 / 𝑤)))) → ((𝑥 / (𝑃↑(𝑃 pCnt 𝑥))) / (𝑦 / (𝑃↑(𝑃 pCnt 𝑦)))) = ((𝑥 · (𝑃↑(𝑃 pCnt 𝑦))) / ((𝑃↑(𝑃 pCnt 𝑥)) · 𝑦)))
72	26	oveq2d 7447	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ 𝐵 ≠ 0) ∧ (𝑥 ∈ ℤ ∧ 𝑧 ∈ ℤ)) ∧ ((𝑦 ∈ ℕ ∧ 𝑤 ∈ ℕ) ∧ (𝐴 = (𝑥 / 𝑦) ∧ 𝐵 = (𝑧 / 𝑤)))) → (𝑃 pCnt 𝐴) = (𝑃 pCnt (𝑥 / 𝑦)))
73		pcdiv 16886	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑃 ∈ ℙ ∧ (𝑥 ∈ ℤ ∧ 𝑥 ≠ 0) ∧ 𝑦 ∈ ℕ) → (𝑃 pCnt (𝑥 / 𝑦)) = ((𝑃 pCnt 𝑥) − (𝑃 pCnt 𝑦)))
74	22, 25, 58, 50, 73	syl121anc 1374	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ 𝐵 ≠ 0) ∧ (𝑥 ∈ ℤ ∧ 𝑧 ∈ ℤ)) ∧ ((𝑦 ∈ ℕ ∧ 𝑤 ∈ ℕ) ∧ (𝐴 = (𝑥 / 𝑦) ∧ 𝐵 = (𝑧 / 𝑤)))) → (𝑃 pCnt (𝑥 / 𝑦)) = ((𝑃 pCnt 𝑥) − (𝑃 pCnt 𝑦)))
75	72, 74	eqtrd 2775	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ 𝐵 ≠ 0) ∧ (𝑥 ∈ ℤ ∧ 𝑧 ∈ ℤ)) ∧ ((𝑦 ∈ ℕ ∧ 𝑤 ∈ ℕ) ∧ (𝐴 = (𝑥 / 𝑦) ∧ 𝐵 = (𝑧 / 𝑤)))) → (𝑃 pCnt 𝐴) = ((𝑃 pCnt 𝑥) − (𝑃 pCnt 𝑦)))
76	75	oveq2d 7447	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ 𝐵 ≠ 0) ∧ (𝑥 ∈ ℤ ∧ 𝑧 ∈ ℤ)) ∧ ((𝑦 ∈ ℕ ∧ 𝑤 ∈ ℕ) ∧ (𝐴 = (𝑥 / 𝑦) ∧ 𝐵 = (𝑧 / 𝑤)))) → (𝑃↑(𝑃 pCnt 𝐴)) = (𝑃↑((𝑃 pCnt 𝑥) − (𝑃 pCnt 𝑦))))
77	24	nncnd 12280	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ 𝐵 ≠ 0) ∧ (𝑥 ∈ ℤ ∧ 𝑧 ∈ ℤ)) ∧ ((𝑦 ∈ ℕ ∧ 𝑤 ∈ ℕ) ∧ (𝐴 = (𝑥 / 𝑦) ∧ 𝐵 = (𝑧 / 𝑤)))) → 𝑃 ∈ ℂ)
78	24	nnne0d 12314	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ 𝐵 ≠ 0) ∧ (𝑥 ∈ ℤ ∧ 𝑧 ∈ ℤ)) ∧ ((𝑦 ∈ ℕ ∧ 𝑤 ∈ ℕ) ∧ (𝐴 = (𝑥 / 𝑦) ∧ 𝐵 = (𝑧 / 𝑤)))) → 𝑃 ≠ 0)
79	66	nn0zd 12637	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ 𝐵 ≠ 0) ∧ (𝑥 ∈ ℤ ∧ 𝑧 ∈ ℤ)) ∧ ((𝑦 ∈ ℕ ∧ 𝑤 ∈ ℕ) ∧ (𝐴 = (𝑥 / 𝑦) ∧ 𝐵 = (𝑧 / 𝑤)))) → (𝑃 pCnt 𝑦) ∈ ℤ)
80	60	nn0zd 12637	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ 𝐵 ≠ 0) ∧ (𝑥 ∈ ℤ ∧ 𝑧 ∈ ℤ)) ∧ ((𝑦 ∈ ℕ ∧ 𝑤 ∈ ℕ) ∧ (𝐴 = (𝑥 / 𝑦) ∧ 𝐵 = (𝑧 / 𝑤)))) → (𝑃 pCnt 𝑥) ∈ ℤ)
81	77, 78, 79, 80	expsubd 14194	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ 𝐵 ≠ 0) ∧ (𝑥 ∈ ℤ ∧ 𝑧 ∈ ℤ)) ∧ ((𝑦 ∈ ℕ ∧ 𝑤 ∈ ℕ) ∧ (𝐴 = (𝑥 / 𝑦) ∧ 𝐵 = (𝑧 / 𝑤)))) → (𝑃↑((𝑃 pCnt 𝑥) − (𝑃 pCnt 𝑦))) = ((𝑃↑(𝑃 pCnt 𝑥)) / (𝑃↑(𝑃 pCnt 𝑦))))
82	76, 81	eqtrd 2775	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝐵 ≠ 0) ∧ (𝑥 ∈ ℤ ∧ 𝑧 ∈ ℤ)) ∧ ((𝑦 ∈ ℕ ∧ 𝑤 ∈ ℕ) ∧ (𝐴 = (𝑥 / 𝑦) ∧ 𝐵 = (𝑧 / 𝑤)))) → (𝑃↑(𝑃 pCnt 𝐴)) = ((𝑃↑(𝑃 pCnt 𝑥)) / (𝑃↑(𝑃 pCnt 𝑦))))
83	82	oveq2d 7447	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝐵 ≠ 0) ∧ (𝑥 ∈ ℤ ∧ 𝑧 ∈ ℤ)) ∧ ((𝑦 ∈ ℕ ∧ 𝑤 ∈ ℕ) ∧ (𝐴 = (𝑥 / 𝑦) ∧ 𝐵 = (𝑧 / 𝑤)))) → (𝐴 / (𝑃↑(𝑃 pCnt 𝐴))) = (𝐴 / ((𝑃↑(𝑃 pCnt 𝑥)) / (𝑃↑(𝑃 pCnt 𝑦)))))
84	26	oveq1d 7446	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝐵 ≠ 0) ∧ (𝑥 ∈ ℤ ∧ 𝑧 ∈ ℤ)) ∧ ((𝑦 ∈ ℕ ∧ 𝑤 ∈ ℕ) ∧ (𝐴 = (𝑥 / 𝑦) ∧ 𝐵 = (𝑧 / 𝑤)))) → (𝐴 / ((𝑃↑(𝑃 pCnt 𝑥)) / (𝑃↑(𝑃 pCnt 𝑦)))) = ((𝑥 / 𝑦) / ((𝑃↑(𝑃 pCnt 𝑥)) / (𝑃↑(𝑃 pCnt 𝑦)))))
85	65, 51, 62, 68, 52, 70, 69	divdivdivd 12088	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝐵 ≠ 0) ∧ (𝑥 ∈ ℤ ∧ 𝑧 ∈ ℤ)) ∧ ((𝑦 ∈ ℕ ∧ 𝑤 ∈ ℕ) ∧ (𝐴 = (𝑥 / 𝑦) ∧ 𝐵 = (𝑧 / 𝑤)))) → ((𝑥 / 𝑦) / ((𝑃↑(𝑃 pCnt 𝑥)) / (𝑃↑(𝑃 pCnt 𝑦)))) = ((𝑥 · (𝑃↑(𝑃 pCnt 𝑦))) / (𝑦 · (𝑃↑(𝑃 pCnt 𝑥)))))
86	83, 84, 85	3eqtrd 2779	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝐵 ≠ 0) ∧ (𝑥 ∈ ℤ ∧ 𝑧 ∈ ℤ)) ∧ ((𝑦 ∈ ℕ ∧ 𝑤 ∈ ℕ) ∧ (𝐴 = (𝑥 / 𝑦) ∧ 𝐵 = (𝑧 / 𝑤)))) → (𝐴 / (𝑃↑(𝑃 pCnt 𝐴))) = ((𝑥 · (𝑃↑(𝑃 pCnt 𝑦))) / (𝑦 · (𝑃↑(𝑃 pCnt 𝑥)))))
87	64, 71, 86	3eqtr4d 2785	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝐵 ≠ 0) ∧ (𝑥 ∈ ℤ ∧ 𝑧 ∈ ℤ)) ∧ ((𝑦 ∈ ℕ ∧ 𝑤 ∈ ℕ) ∧ (𝐴 = (𝑥 / 𝑦) ∧ 𝐵 = (𝑧 / 𝑤)))) → ((𝑥 / (𝑃↑(𝑃 pCnt 𝑥))) / (𝑦 / (𝑃↑(𝑃 pCnt 𝑦)))) = (𝐴 / (𝑃↑(𝑃 pCnt 𝐴))))
88	87	oveq2d 7447	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝐵 ≠ 0) ∧ (𝑥 ∈ ℤ ∧ 𝑧 ∈ ℤ)) ∧ ((𝑦 ∈ ℕ ∧ 𝑤 ∈ ℕ) ∧ (𝐴 = (𝑥 / 𝑦) ∧ 𝐵 = (𝑧 / 𝑤)))) → ((𝑃↑(𝑃 pCnt 𝐴)) · ((𝑥 / (𝑃↑(𝑃 pCnt 𝑥))) / (𝑦 / (𝑃↑(𝑃 pCnt 𝑦))))) = ((𝑃↑(𝑃 pCnt 𝐴)) · (𝐴 / (𝑃↑(𝑃 pCnt 𝐴)))))
89	1	ad3antrrr 730	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝐵 ≠ 0) ∧ (𝑥 ∈ ℤ ∧ 𝑧 ∈ ℤ)) ∧ ((𝑦 ∈ ℕ ∧ 𝑤 ∈ ℕ) ∧ (𝐴 = (𝑥 / 𝑦) ∧ 𝐵 = (𝑧 / 𝑤)))) → 𝐴 ∈ ℚ)
90	89, 13	syl 17	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝐵 ≠ 0) ∧ (𝑥 ∈ ℤ ∧ 𝑧 ∈ ℤ)) ∧ ((𝑦 ∈ ℕ ∧ 𝑤 ∈ ℕ) ∧ (𝐴 = (𝑥 / 𝑦) ∧ 𝐵 = (𝑧 / 𝑤)))) → 𝐴 ∈ ℂ)
91		pcqcl 16890	. . . . . . . . . . . . 13 ⊢ ((𝑃 ∈ ℙ ∧ (𝐴 ∈ ℚ ∧ 𝐴 ≠ 0)) → (𝑃 pCnt 𝐴) ∈ ℤ)
92	22, 89, 48, 91	syl12anc 837	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝐵 ≠ 0) ∧ (𝑥 ∈ ℤ ∧ 𝑧 ∈ ℤ)) ∧ ((𝑦 ∈ ℕ ∧ 𝑤 ∈ ℕ) ∧ (𝐴 = (𝑥 / 𝑦) ∧ 𝐵 = (𝑧 / 𝑤)))) → (𝑃 pCnt 𝐴) ∈ ℤ)
93	77, 78, 92	expclzd 14188	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝐵 ≠ 0) ∧ (𝑥 ∈ ℤ ∧ 𝑧 ∈ ℤ)) ∧ ((𝑦 ∈ ℕ ∧ 𝑤 ∈ ℕ) ∧ (𝐴 = (𝑥 / 𝑦) ∧ 𝐵 = (𝑧 / 𝑤)))) → (𝑃↑(𝑃 pCnt 𝐴)) ∈ ℂ)
94	77, 78, 92	expne0d 14189	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝐵 ≠ 0) ∧ (𝑥 ∈ ℤ ∧ 𝑧 ∈ ℤ)) ∧ ((𝑦 ∈ ℕ ∧ 𝑤 ∈ ℕ) ∧ (𝐴 = (𝑥 / 𝑦) ∧ 𝐵 = (𝑧 / 𝑤)))) → (𝑃↑(𝑃 pCnt 𝐴)) ≠ 0)
95	90, 93, 94	divcan2d 12043	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝐵 ≠ 0) ∧ (𝑥 ∈ ℤ ∧ 𝑧 ∈ ℤ)) ∧ ((𝑦 ∈ ℕ ∧ 𝑤 ∈ ℕ) ∧ (𝐴 = (𝑥 / 𝑦) ∧ 𝐵 = (𝑧 / 𝑤)))) → ((𝑃↑(𝑃 pCnt 𝐴)) · (𝐴 / (𝑃↑(𝑃 pCnt 𝐴)))) = 𝐴)
96	88, 95	eqtr2d 2776	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝐵 ≠ 0) ∧ (𝑥 ∈ ℤ ∧ 𝑧 ∈ ℤ)) ∧ ((𝑦 ∈ ℕ ∧ 𝑤 ∈ ℕ) ∧ (𝐴 = (𝑥 / 𝑦) ∧ 𝐵 = (𝑧 / 𝑤)))) → 𝐴 = ((𝑃↑(𝑃 pCnt 𝐴)) · ((𝑥 / (𝑃↑(𝑃 pCnt 𝑥))) / (𝑦 / (𝑃↑(𝑃 pCnt 𝑦))))))
97		simplrr 778	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ 𝐵 ≠ 0) ∧ (𝑥 ∈ ℤ ∧ 𝑧 ∈ ℤ)) ∧ ((𝑦 ∈ ℕ ∧ 𝑤 ∈ ℕ) ∧ (𝐴 = (𝑥 / 𝑦) ∧ 𝐵 = (𝑧 / 𝑤)))) → 𝑧 ∈ ℤ)
98		simprrr 782	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝜑 ∧ 𝐵 ≠ 0) ∧ (𝑥 ∈ ℤ ∧ 𝑧 ∈ ℤ)) ∧ ((𝑦 ∈ ℕ ∧ 𝑤 ∈ ℕ) ∧ (𝐴 = (𝑥 / 𝑦) ∧ 𝐵 = (𝑧 / 𝑤)))) → 𝐵 = (𝑧 / 𝑤))
99	98, 30	eqnetrrd 3007	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝜑 ∧ 𝐵 ≠ 0) ∧ (𝑥 ∈ ℤ ∧ 𝑧 ∈ ℤ)) ∧ ((𝑦 ∈ ℕ ∧ 𝑤 ∈ ℕ) ∧ (𝐴 = (𝑥 / 𝑦) ∧ 𝐵 = (𝑧 / 𝑤)))) → (𝑧 / 𝑤) ≠ 0)
100		simprlr 780	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((((𝜑 ∧ 𝐵 ≠ 0) ∧ (𝑥 ∈ ℤ ∧ 𝑧 ∈ ℤ)) ∧ ((𝑦 ∈ ℕ ∧ 𝑤 ∈ ℕ) ∧ (𝐴 = (𝑥 / 𝑦) ∧ 𝐵 = (𝑧 / 𝑤)))) → 𝑤 ∈ ℕ)
101	100	nncnd 12280	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((((𝜑 ∧ 𝐵 ≠ 0) ∧ (𝑥 ∈ ℤ ∧ 𝑧 ∈ ℤ)) ∧ ((𝑦 ∈ ℕ ∧ 𝑤 ∈ ℕ) ∧ (𝐴 = (𝑥 / 𝑦) ∧ 𝐵 = (𝑧 / 𝑤)))) → 𝑤 ∈ ℂ)
102	100	nnne0d 12314	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((((𝜑 ∧ 𝐵 ≠ 0) ∧ (𝑥 ∈ ℤ ∧ 𝑧 ∈ ℤ)) ∧ ((𝑦 ∈ ℕ ∧ 𝑤 ∈ ℕ) ∧ (𝐴 = (𝑥 / 𝑦) ∧ 𝐵 = (𝑧 / 𝑤)))) → 𝑤 ≠ 0)
103	101, 102	div0d 12040	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((𝜑 ∧ 𝐵 ≠ 0) ∧ (𝑥 ∈ ℤ ∧ 𝑧 ∈ ℤ)) ∧ ((𝑦 ∈ ℕ ∧ 𝑤 ∈ ℕ) ∧ (𝐴 = (𝑥 / 𝑦) ∧ 𝐵 = (𝑧 / 𝑤)))) → (0 / 𝑤) = 0)
104		oveq1 7438	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑧 = 0 → (𝑧 / 𝑤) = (0 / 𝑤))
105	104	eqeq1d 2737	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑧 = 0 → ((𝑧 / 𝑤) = 0 ↔ (0 / 𝑤) = 0))
106	103, 105	syl5ibrcom 247	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝜑 ∧ 𝐵 ≠ 0) ∧ (𝑥 ∈ ℤ ∧ 𝑧 ∈ ℤ)) ∧ ((𝑦 ∈ ℕ ∧ 𝑤 ∈ ℕ) ∧ (𝐴 = (𝑥 / 𝑦) ∧ 𝐵 = (𝑧 / 𝑤)))) → (𝑧 = 0 → (𝑧 / 𝑤) = 0))
107	106	necon3d 2959	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝜑 ∧ 𝐵 ≠ 0) ∧ (𝑥 ∈ ℤ ∧ 𝑧 ∈ ℤ)) ∧ ((𝑦 ∈ ℕ ∧ 𝑤 ∈ ℕ) ∧ (𝐴 = (𝑥 / 𝑦) ∧ 𝐵 = (𝑧 / 𝑤)))) → ((𝑧 / 𝑤) ≠ 0 → 𝑧 ≠ 0))
108	99, 107	mpd 15	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ 𝐵 ≠ 0) ∧ (𝑥 ∈ ℤ ∧ 𝑧 ∈ ℤ)) ∧ ((𝑦 ∈ ℕ ∧ 𝑤 ∈ ℕ) ∧ (𝐴 = (𝑥 / 𝑦) ∧ 𝐵 = (𝑧 / 𝑤)))) → 𝑧 ≠ 0)
109		pczcl 16882	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑃 ∈ ℙ ∧ (𝑧 ∈ ℤ ∧ 𝑧 ≠ 0)) → (𝑃 pCnt 𝑧) ∈ ℕ₀)
110	22, 97, 108, 109	syl12anc 837	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ 𝐵 ≠ 0) ∧ (𝑥 ∈ ℤ ∧ 𝑧 ∈ ℤ)) ∧ ((𝑦 ∈ ℕ ∧ 𝑤 ∈ ℕ) ∧ (𝐴 = (𝑥 / 𝑦) ∧ 𝐵 = (𝑧 / 𝑤)))) → (𝑃 pCnt 𝑧) ∈ ℕ₀)
111	24, 110	nnexpcld 14281	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ 𝐵 ≠ 0) ∧ (𝑥 ∈ ℤ ∧ 𝑧 ∈ ℤ)) ∧ ((𝑦 ∈ ℕ ∧ 𝑤 ∈ ℕ) ∧ (𝐴 = (𝑥 / 𝑦) ∧ 𝐵 = (𝑧 / 𝑤)))) → (𝑃↑(𝑃 pCnt 𝑧)) ∈ ℕ)
112	111	nncnd 12280	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝐵 ≠ 0) ∧ (𝑥 ∈ ℤ ∧ 𝑧 ∈ ℤ)) ∧ ((𝑦 ∈ ℕ ∧ 𝑤 ∈ ℕ) ∧ (𝐴 = (𝑥 / 𝑦) ∧ 𝐵 = (𝑧 / 𝑤)))) → (𝑃↑(𝑃 pCnt 𝑧)) ∈ ℂ)
113	112, 101	mulcomd 11280	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝐵 ≠ 0) ∧ (𝑥 ∈ ℤ ∧ 𝑧 ∈ ℤ)) ∧ ((𝑦 ∈ ℕ ∧ 𝑤 ∈ ℕ) ∧ (𝐴 = (𝑥 / 𝑦) ∧ 𝐵 = (𝑧 / 𝑤)))) → ((𝑃↑(𝑃 pCnt 𝑧)) · 𝑤) = (𝑤 · (𝑃↑(𝑃 pCnt 𝑧))))
114	113	oveq2d 7447	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝐵 ≠ 0) ∧ (𝑥 ∈ ℤ ∧ 𝑧 ∈ ℤ)) ∧ ((𝑦 ∈ ℕ ∧ 𝑤 ∈ ℕ) ∧ (𝐴 = (𝑥 / 𝑦) ∧ 𝐵 = (𝑧 / 𝑤)))) → ((𝑧 · (𝑃↑(𝑃 pCnt 𝑤))) / ((𝑃↑(𝑃 pCnt 𝑧)) · 𝑤)) = ((𝑧 · (𝑃↑(𝑃 pCnt 𝑤))) / (𝑤 · (𝑃↑(𝑃 pCnt 𝑧)))))
115	97	zcnd 12721	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝐵 ≠ 0) ∧ (𝑥 ∈ ℤ ∧ 𝑧 ∈ ℤ)) ∧ ((𝑦 ∈ ℕ ∧ 𝑤 ∈ ℕ) ∧ (𝐴 = (𝑥 / 𝑦) ∧ 𝐵 = (𝑧 / 𝑤)))) → 𝑧 ∈ ℂ)
116	22, 100	pccld 16884	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ 𝐵 ≠ 0) ∧ (𝑥 ∈ ℤ ∧ 𝑧 ∈ ℤ)) ∧ ((𝑦 ∈ ℕ ∧ 𝑤 ∈ ℕ) ∧ (𝐴 = (𝑥 / 𝑦) ∧ 𝐵 = (𝑧 / 𝑤)))) → (𝑃 pCnt 𝑤) ∈ ℕ₀)
117	24, 116	nnexpcld 14281	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝐵 ≠ 0) ∧ (𝑥 ∈ ℤ ∧ 𝑧 ∈ ℤ)) ∧ ((𝑦 ∈ ℕ ∧ 𝑤 ∈ ℕ) ∧ (𝐴 = (𝑥 / 𝑦) ∧ 𝐵 = (𝑧 / 𝑤)))) → (𝑃↑(𝑃 pCnt 𝑤)) ∈ ℕ)
118	117	nncnd 12280	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝐵 ≠ 0) ∧ (𝑥 ∈ ℤ ∧ 𝑧 ∈ ℤ)) ∧ ((𝑦 ∈ ℕ ∧ 𝑤 ∈ ℕ) ∧ (𝐴 = (𝑥 / 𝑦) ∧ 𝐵 = (𝑧 / 𝑤)))) → (𝑃↑(𝑃 pCnt 𝑤)) ∈ ℂ)
119	111	nnne0d 12314	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝐵 ≠ 0) ∧ (𝑥 ∈ ℤ ∧ 𝑧 ∈ ℤ)) ∧ ((𝑦 ∈ ℕ ∧ 𝑤 ∈ ℕ) ∧ (𝐴 = (𝑥 / 𝑦) ∧ 𝐵 = (𝑧 / 𝑤)))) → (𝑃↑(𝑃 pCnt 𝑧)) ≠ 0)
120	117	nnne0d 12314	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝐵 ≠ 0) ∧ (𝑥 ∈ ℤ ∧ 𝑧 ∈ ℤ)) ∧ ((𝑦 ∈ ℕ ∧ 𝑤 ∈ ℕ) ∧ (𝐴 = (𝑥 / 𝑦) ∧ 𝐵 = (𝑧 / 𝑤)))) → (𝑃↑(𝑃 pCnt 𝑤)) ≠ 0)
121	115, 112, 101, 118, 119, 120, 102	divdivdivd 12088	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝐵 ≠ 0) ∧ (𝑥 ∈ ℤ ∧ 𝑧 ∈ ℤ)) ∧ ((𝑦 ∈ ℕ ∧ 𝑤 ∈ ℕ) ∧ (𝐴 = (𝑥 / 𝑦) ∧ 𝐵 = (𝑧 / 𝑤)))) → ((𝑧 / (𝑃↑(𝑃 pCnt 𝑧))) / (𝑤 / (𝑃↑(𝑃 pCnt 𝑤)))) = ((𝑧 · (𝑃↑(𝑃 pCnt 𝑤))) / ((𝑃↑(𝑃 pCnt 𝑧)) · 𝑤)))
122	98	oveq2d 7447	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ 𝐵 ≠ 0) ∧ (𝑥 ∈ ℤ ∧ 𝑧 ∈ ℤ)) ∧ ((𝑦 ∈ ℕ ∧ 𝑤 ∈ ℕ) ∧ (𝐴 = (𝑥 / 𝑦) ∧ 𝐵 = (𝑧 / 𝑤)))) → (𝑃 pCnt 𝐵) = (𝑃 pCnt (𝑧 / 𝑤)))
123		pcdiv 16886	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑃 ∈ ℙ ∧ (𝑧 ∈ ℤ ∧ 𝑧 ≠ 0) ∧ 𝑤 ∈ ℕ) → (𝑃 pCnt (𝑧 / 𝑤)) = ((𝑃 pCnt 𝑧) − (𝑃 pCnt 𝑤)))
124	22, 97, 108, 100, 123	syl121anc 1374	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ 𝐵 ≠ 0) ∧ (𝑥 ∈ ℤ ∧ 𝑧 ∈ ℤ)) ∧ ((𝑦 ∈ ℕ ∧ 𝑤 ∈ ℕ) ∧ (𝐴 = (𝑥 / 𝑦) ∧ 𝐵 = (𝑧 / 𝑤)))) → (𝑃 pCnt (𝑧 / 𝑤)) = ((𝑃 pCnt 𝑧) − (𝑃 pCnt 𝑤)))
125	122, 124	eqtrd 2775	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ 𝐵 ≠ 0) ∧ (𝑥 ∈ ℤ ∧ 𝑧 ∈ ℤ)) ∧ ((𝑦 ∈ ℕ ∧ 𝑤 ∈ ℕ) ∧ (𝐴 = (𝑥 / 𝑦) ∧ 𝐵 = (𝑧 / 𝑤)))) → (𝑃 pCnt 𝐵) = ((𝑃 pCnt 𝑧) − (𝑃 pCnt 𝑤)))
126	125	oveq2d 7447	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ 𝐵 ≠ 0) ∧ (𝑥 ∈ ℤ ∧ 𝑧 ∈ ℤ)) ∧ ((𝑦 ∈ ℕ ∧ 𝑤 ∈ ℕ) ∧ (𝐴 = (𝑥 / 𝑦) ∧ 𝐵 = (𝑧 / 𝑤)))) → (𝑃↑(𝑃 pCnt 𝐵)) = (𝑃↑((𝑃 pCnt 𝑧) − (𝑃 pCnt 𝑤))))
127	116	nn0zd 12637	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ 𝐵 ≠ 0) ∧ (𝑥 ∈ ℤ ∧ 𝑧 ∈ ℤ)) ∧ ((𝑦 ∈ ℕ ∧ 𝑤 ∈ ℕ) ∧ (𝐴 = (𝑥 / 𝑦) ∧ 𝐵 = (𝑧 / 𝑤)))) → (𝑃 pCnt 𝑤) ∈ ℤ)
128	110	nn0zd 12637	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ 𝐵 ≠ 0) ∧ (𝑥 ∈ ℤ ∧ 𝑧 ∈ ℤ)) ∧ ((𝑦 ∈ ℕ ∧ 𝑤 ∈ ℕ) ∧ (𝐴 = (𝑥 / 𝑦) ∧ 𝐵 = (𝑧 / 𝑤)))) → (𝑃 pCnt 𝑧) ∈ ℤ)
129	77, 78, 127, 128	expsubd 14194	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ 𝐵 ≠ 0) ∧ (𝑥 ∈ ℤ ∧ 𝑧 ∈ ℤ)) ∧ ((𝑦 ∈ ℕ ∧ 𝑤 ∈ ℕ) ∧ (𝐴 = (𝑥 / 𝑦) ∧ 𝐵 = (𝑧 / 𝑤)))) → (𝑃↑((𝑃 pCnt 𝑧) − (𝑃 pCnt 𝑤))) = ((𝑃↑(𝑃 pCnt 𝑧)) / (𝑃↑(𝑃 pCnt 𝑤))))
130	126, 129	eqtrd 2775	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝐵 ≠ 0) ∧ (𝑥 ∈ ℤ ∧ 𝑧 ∈ ℤ)) ∧ ((𝑦 ∈ ℕ ∧ 𝑤 ∈ ℕ) ∧ (𝐴 = (𝑥 / 𝑦) ∧ 𝐵 = (𝑧 / 𝑤)))) → (𝑃↑(𝑃 pCnt 𝐵)) = ((𝑃↑(𝑃 pCnt 𝑧)) / (𝑃↑(𝑃 pCnt 𝑤))))
131	130	oveq2d 7447	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝐵 ≠ 0) ∧ (𝑥 ∈ ℤ ∧ 𝑧 ∈ ℤ)) ∧ ((𝑦 ∈ ℕ ∧ 𝑤 ∈ ℕ) ∧ (𝐴 = (𝑥 / 𝑦) ∧ 𝐵 = (𝑧 / 𝑤)))) → (𝐵 / (𝑃↑(𝑃 pCnt 𝐵))) = (𝐵 / ((𝑃↑(𝑃 pCnt 𝑧)) / (𝑃↑(𝑃 pCnt 𝑤)))))
132	98	oveq1d 7446	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝐵 ≠ 0) ∧ (𝑥 ∈ ℤ ∧ 𝑧 ∈ ℤ)) ∧ ((𝑦 ∈ ℕ ∧ 𝑤 ∈ ℕ) ∧ (𝐴 = (𝑥 / 𝑦) ∧ 𝐵 = (𝑧 / 𝑤)))) → (𝐵 / ((𝑃↑(𝑃 pCnt 𝑧)) / (𝑃↑(𝑃 pCnt 𝑤)))) = ((𝑧 / 𝑤) / ((𝑃↑(𝑃 pCnt 𝑧)) / (𝑃↑(𝑃 pCnt 𝑤)))))
133	115, 101, 112, 118, 102, 120, 119	divdivdivd 12088	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝐵 ≠ 0) ∧ (𝑥 ∈ ℤ ∧ 𝑧 ∈ ℤ)) ∧ ((𝑦 ∈ ℕ ∧ 𝑤 ∈ ℕ) ∧ (𝐴 = (𝑥 / 𝑦) ∧ 𝐵 = (𝑧 / 𝑤)))) → ((𝑧 / 𝑤) / ((𝑃↑(𝑃 pCnt 𝑧)) / (𝑃↑(𝑃 pCnt 𝑤)))) = ((𝑧 · (𝑃↑(𝑃 pCnt 𝑤))) / (𝑤 · (𝑃↑(𝑃 pCnt 𝑧)))))
134	131, 132, 133	3eqtrd 2779	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝐵 ≠ 0) ∧ (𝑥 ∈ ℤ ∧ 𝑧 ∈ ℤ)) ∧ ((𝑦 ∈ ℕ ∧ 𝑤 ∈ ℕ) ∧ (𝐴 = (𝑥 / 𝑦) ∧ 𝐵 = (𝑧 / 𝑤)))) → (𝐵 / (𝑃↑(𝑃 pCnt 𝐵))) = ((𝑧 · (𝑃↑(𝑃 pCnt 𝑤))) / (𝑤 · (𝑃↑(𝑃 pCnt 𝑧)))))
135	114, 121, 134	3eqtr4d 2785	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝐵 ≠ 0) ∧ (𝑥 ∈ ℤ ∧ 𝑧 ∈ ℤ)) ∧ ((𝑦 ∈ ℕ ∧ 𝑤 ∈ ℕ) ∧ (𝐴 = (𝑥 / 𝑦) ∧ 𝐵 = (𝑧 / 𝑤)))) → ((𝑧 / (𝑃↑(𝑃 pCnt 𝑧))) / (𝑤 / (𝑃↑(𝑃 pCnt 𝑤)))) = (𝐵 / (𝑃↑(𝑃 pCnt 𝐵))))
136	135	oveq2d 7447	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝐵 ≠ 0) ∧ (𝑥 ∈ ℤ ∧ 𝑧 ∈ ℤ)) ∧ ((𝑦 ∈ ℕ ∧ 𝑤 ∈ ℕ) ∧ (𝐴 = (𝑥 / 𝑦) ∧ 𝐵 = (𝑧 / 𝑤)))) → ((𝑃↑(𝑃 pCnt 𝐵)) · ((𝑧 / (𝑃↑(𝑃 pCnt 𝑧))) / (𝑤 / (𝑃↑(𝑃 pCnt 𝑤))))) = ((𝑃↑(𝑃 pCnt 𝐵)) · (𝐵 / (𝑃↑(𝑃 pCnt 𝐵)))))
137		qcn 13003	. . . . . . . . . . . 12 ⊢ (𝐵 ∈ ℚ → 𝐵 ∈ ℂ)
138	29, 137	syl 17	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝐵 ≠ 0) ∧ (𝑥 ∈ ℤ ∧ 𝑧 ∈ ℤ)) ∧ ((𝑦 ∈ ℕ ∧ 𝑤 ∈ ℕ) ∧ (𝐴 = (𝑥 / 𝑦) ∧ 𝐵 = (𝑧 / 𝑤)))) → 𝐵 ∈ ℂ)
139	77, 78, 32	expclzd 14188	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝐵 ≠ 0) ∧ (𝑥 ∈ ℤ ∧ 𝑧 ∈ ℤ)) ∧ ((𝑦 ∈ ℕ ∧ 𝑤 ∈ ℕ) ∧ (𝐴 = (𝑥 / 𝑦) ∧ 𝐵 = (𝑧 / 𝑤)))) → (𝑃↑(𝑃 pCnt 𝐵)) ∈ ℂ)
140	77, 78, 32	expne0d 14189	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝐵 ≠ 0) ∧ (𝑥 ∈ ℤ ∧ 𝑧 ∈ ℤ)) ∧ ((𝑦 ∈ ℕ ∧ 𝑤 ∈ ℕ) ∧ (𝐴 = (𝑥 / 𝑦) ∧ 𝐵 = (𝑧 / 𝑤)))) → (𝑃↑(𝑃 pCnt 𝐵)) ≠ 0)
141	138, 139, 140	divcan2d 12043	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝐵 ≠ 0) ∧ (𝑥 ∈ ℤ ∧ 𝑧 ∈ ℤ)) ∧ ((𝑦 ∈ ℕ ∧ 𝑤 ∈ ℕ) ∧ (𝐴 = (𝑥 / 𝑦) ∧ 𝐵 = (𝑧 / 𝑤)))) → ((𝑃↑(𝑃 pCnt 𝐵)) · (𝐵 / (𝑃↑(𝑃 pCnt 𝐵)))) = 𝐵)
142	136, 141	eqtr2d 2776	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝐵 ≠ 0) ∧ (𝑥 ∈ ℤ ∧ 𝑧 ∈ ℤ)) ∧ ((𝑦 ∈ ℕ ∧ 𝑤 ∈ ℕ) ∧ (𝐴 = (𝑥 / 𝑦) ∧ 𝐵 = (𝑧 / 𝑤)))) → 𝐵 = ((𝑃↑(𝑃 pCnt 𝐵)) · ((𝑧 / (𝑃↑(𝑃 pCnt 𝑧))) / (𝑤 / (𝑃↑(𝑃 pCnt 𝑤))))))
143		eluz 12890	. . . . . . . . . . 11 ⊢ (((𝑃 pCnt 𝐴) ∈ ℤ ∧ (𝑃 pCnt 𝐵) ∈ ℤ) → ((𝑃 pCnt 𝐵) ∈ (ℤ_≥‘(𝑃 pCnt 𝐴)) ↔ (𝑃 pCnt 𝐴) ≤ (𝑃 pCnt 𝐵)))
144	92, 32, 143	syl2anc 584	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝐵 ≠ 0) ∧ (𝑥 ∈ ℤ ∧ 𝑧 ∈ ℤ)) ∧ ((𝑦 ∈ ℕ ∧ 𝑤 ∈ ℕ) ∧ (𝐴 = (𝑥 / 𝑦) ∧ 𝐵 = (𝑧 / 𝑤)))) → ((𝑃 pCnt 𝐵) ∈ (ℤ_≥‘(𝑃 pCnt 𝐴)) ↔ (𝑃 pCnt 𝐴) ≤ (𝑃 pCnt 𝐵)))
145	43, 144	mpbird 257	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝐵 ≠ 0) ∧ (𝑥 ∈ ℤ ∧ 𝑧 ∈ ℤ)) ∧ ((𝑦 ∈ ℕ ∧ 𝑤 ∈ ℕ) ∧ (𝐴 = (𝑥 / 𝑦) ∧ 𝐵 = (𝑧 / 𝑤)))) → (𝑃 pCnt 𝐵) ∈ (ℤ_≥‘(𝑃 pCnt 𝐴)))
146		pczdvds 16897	. . . . . . . . . . . 12 ⊢ ((𝑃 ∈ ℙ ∧ (𝑥 ∈ ℤ ∧ 𝑥 ≠ 0)) → (𝑃↑(𝑃 pCnt 𝑥)) ∥ 𝑥)
147	22, 25, 58, 146	syl12anc 837	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝐵 ≠ 0) ∧ (𝑥 ∈ ℤ ∧ 𝑧 ∈ ℤ)) ∧ ((𝑦 ∈ ℕ ∧ 𝑤 ∈ ℕ) ∧ (𝐴 = (𝑥 / 𝑦) ∧ 𝐵 = (𝑧 / 𝑤)))) → (𝑃↑(𝑃 pCnt 𝑥)) ∥ 𝑥)
148	61	nnzd 12638	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝐵 ≠ 0) ∧ (𝑥 ∈ ℤ ∧ 𝑧 ∈ ℤ)) ∧ ((𝑦 ∈ ℕ ∧ 𝑤 ∈ ℕ) ∧ (𝐴 = (𝑥 / 𝑦) ∧ 𝐵 = (𝑧 / 𝑤)))) → (𝑃↑(𝑃 pCnt 𝑥)) ∈ ℤ)
149		dvdsval2 16290	. . . . . . . . . . . 12 ⊢ (((𝑃↑(𝑃 pCnt 𝑥)) ∈ ℤ ∧ (𝑃↑(𝑃 pCnt 𝑥)) ≠ 0 ∧ 𝑥 ∈ ℤ) → ((𝑃↑(𝑃 pCnt 𝑥)) ∥ 𝑥 ↔ (𝑥 / (𝑃↑(𝑃 pCnt 𝑥))) ∈ ℤ))
150	148, 69, 25, 149	syl3anc 1370	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝐵 ≠ 0) ∧ (𝑥 ∈ ℤ ∧ 𝑧 ∈ ℤ)) ∧ ((𝑦 ∈ ℕ ∧ 𝑤 ∈ ℕ) ∧ (𝐴 = (𝑥 / 𝑦) ∧ 𝐵 = (𝑧 / 𝑤)))) → ((𝑃↑(𝑃 pCnt 𝑥)) ∥ 𝑥 ↔ (𝑥 / (𝑃↑(𝑃 pCnt 𝑥))) ∈ ℤ))
151	147, 150	mpbid 232	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝐵 ≠ 0) ∧ (𝑥 ∈ ℤ ∧ 𝑧 ∈ ℤ)) ∧ ((𝑦 ∈ ℕ ∧ 𝑤 ∈ ℕ) ∧ (𝐴 = (𝑥 / 𝑦) ∧ 𝐵 = (𝑧 / 𝑤)))) → (𝑥 / (𝑃↑(𝑃 pCnt 𝑥))) ∈ ℤ)
152		pczndvds2 16901	. . . . . . . . . . 11 ⊢ ((𝑃 ∈ ℙ ∧ (𝑥 ∈ ℤ ∧ 𝑥 ≠ 0)) → ¬ 𝑃 ∥ (𝑥 / (𝑃↑(𝑃 pCnt 𝑥))))
153	22, 25, 58, 152	syl12anc 837	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝐵 ≠ 0) ∧ (𝑥 ∈ ℤ ∧ 𝑧 ∈ ℤ)) ∧ ((𝑦 ∈ ℕ ∧ 𝑤 ∈ ℕ) ∧ (𝐴 = (𝑥 / 𝑦) ∧ 𝐵 = (𝑧 / 𝑤)))) → ¬ 𝑃 ∥ (𝑥 / (𝑃↑(𝑃 pCnt 𝑥))))
154	151, 153	jca 511	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝐵 ≠ 0) ∧ (𝑥 ∈ ℤ ∧ 𝑧 ∈ ℤ)) ∧ ((𝑦 ∈ ℕ ∧ 𝑤 ∈ ℕ) ∧ (𝐴 = (𝑥 / 𝑦) ∧ 𝐵 = (𝑧 / 𝑤)))) → ((𝑥 / (𝑃↑(𝑃 pCnt 𝑥))) ∈ ℤ ∧ ¬ 𝑃 ∥ (𝑥 / (𝑃↑(𝑃 pCnt 𝑥)))))
155		pcdvds 16898	. . . . . . . . . . . . 13 ⊢ ((𝑃 ∈ ℙ ∧ 𝑦 ∈ ℕ) → (𝑃↑(𝑃 pCnt 𝑦)) ∥ 𝑦)
156	22, 50, 155	syl2anc 584	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝐵 ≠ 0) ∧ (𝑥 ∈ ℤ ∧ 𝑧 ∈ ℤ)) ∧ ((𝑦 ∈ ℕ ∧ 𝑤 ∈ ℕ) ∧ (𝐴 = (𝑥 / 𝑦) ∧ 𝐵 = (𝑧 / 𝑤)))) → (𝑃↑(𝑃 pCnt 𝑦)) ∥ 𝑦)
157	67	nnzd 12638	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝐵 ≠ 0) ∧ (𝑥 ∈ ℤ ∧ 𝑧 ∈ ℤ)) ∧ ((𝑦 ∈ ℕ ∧ 𝑤 ∈ ℕ) ∧ (𝐴 = (𝑥 / 𝑦) ∧ 𝐵 = (𝑧 / 𝑤)))) → (𝑃↑(𝑃 pCnt 𝑦)) ∈ ℤ)
158	50	nnzd 12638	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝐵 ≠ 0) ∧ (𝑥 ∈ ℤ ∧ 𝑧 ∈ ℤ)) ∧ ((𝑦 ∈ ℕ ∧ 𝑤 ∈ ℕ) ∧ (𝐴 = (𝑥 / 𝑦) ∧ 𝐵 = (𝑧 / 𝑤)))) → 𝑦 ∈ ℤ)
159		dvdsval2 16290	. . . . . . . . . . . . 13 ⊢ (((𝑃↑(𝑃 pCnt 𝑦)) ∈ ℤ ∧ (𝑃↑(𝑃 pCnt 𝑦)) ≠ 0 ∧ 𝑦 ∈ ℤ) → ((𝑃↑(𝑃 pCnt 𝑦)) ∥ 𝑦 ↔ (𝑦 / (𝑃↑(𝑃 pCnt 𝑦))) ∈ ℤ))
160	157, 70, 158, 159	syl3anc 1370	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝐵 ≠ 0) ∧ (𝑥 ∈ ℤ ∧ 𝑧 ∈ ℤ)) ∧ ((𝑦 ∈ ℕ ∧ 𝑤 ∈ ℕ) ∧ (𝐴 = (𝑥 / 𝑦) ∧ 𝐵 = (𝑧 / 𝑤)))) → ((𝑃↑(𝑃 pCnt 𝑦)) ∥ 𝑦 ↔ (𝑦 / (𝑃↑(𝑃 pCnt 𝑦))) ∈ ℤ))
161	156, 160	mpbid 232	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝐵 ≠ 0) ∧ (𝑥 ∈ ℤ ∧ 𝑧 ∈ ℤ)) ∧ ((𝑦 ∈ ℕ ∧ 𝑤 ∈ ℕ) ∧ (𝐴 = (𝑥 / 𝑦) ∧ 𝐵 = (𝑧 / 𝑤)))) → (𝑦 / (𝑃↑(𝑃 pCnt 𝑦))) ∈ ℤ)
162	50	nnred 12279	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝐵 ≠ 0) ∧ (𝑥 ∈ ℤ ∧ 𝑧 ∈ ℤ)) ∧ ((𝑦 ∈ ℕ ∧ 𝑤 ∈ ℕ) ∧ (𝐴 = (𝑥 / 𝑦) ∧ 𝐵 = (𝑧 / 𝑤)))) → 𝑦 ∈ ℝ)
163	67	nnred 12279	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝐵 ≠ 0) ∧ (𝑥 ∈ ℤ ∧ 𝑧 ∈ ℤ)) ∧ ((𝑦 ∈ ℕ ∧ 𝑤 ∈ ℕ) ∧ (𝐴 = (𝑥 / 𝑦) ∧ 𝐵 = (𝑧 / 𝑤)))) → (𝑃↑(𝑃 pCnt 𝑦)) ∈ ℝ)
164	50	nngt0d 12313	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝐵 ≠ 0) ∧ (𝑥 ∈ ℤ ∧ 𝑧 ∈ ℤ)) ∧ ((𝑦 ∈ ℕ ∧ 𝑤 ∈ ℕ) ∧ (𝐴 = (𝑥 / 𝑦) ∧ 𝐵 = (𝑧 / 𝑤)))) → 0 < 𝑦)
165	67	nngt0d 12313	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝐵 ≠ 0) ∧ (𝑥 ∈ ℤ ∧ 𝑧 ∈ ℤ)) ∧ ((𝑦 ∈ ℕ ∧ 𝑤 ∈ ℕ) ∧ (𝐴 = (𝑥 / 𝑦) ∧ 𝐵 = (𝑧 / 𝑤)))) → 0 < (𝑃↑(𝑃 pCnt 𝑦)))
166	162, 163, 164, 165	divgt0d 12201	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝐵 ≠ 0) ∧ (𝑥 ∈ ℤ ∧ 𝑧 ∈ ℤ)) ∧ ((𝑦 ∈ ℕ ∧ 𝑤 ∈ ℕ) ∧ (𝐴 = (𝑥 / 𝑦) ∧ 𝐵 = (𝑧 / 𝑤)))) → 0 < (𝑦 / (𝑃↑(𝑃 pCnt 𝑦))))
167		elnnz 12621	. . . . . . . . . . 11 ⊢ ((𝑦 / (𝑃↑(𝑃 pCnt 𝑦))) ∈ ℕ ↔ ((𝑦 / (𝑃↑(𝑃 pCnt 𝑦))) ∈ ℤ ∧ 0 < (𝑦 / (𝑃↑(𝑃 pCnt 𝑦)))))
168	161, 166, 167	sylanbrc 583	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝐵 ≠ 0) ∧ (𝑥 ∈ ℤ ∧ 𝑧 ∈ ℤ)) ∧ ((𝑦 ∈ ℕ ∧ 𝑤 ∈ ℕ) ∧ (𝐴 = (𝑥 / 𝑦) ∧ 𝐵 = (𝑧 / 𝑤)))) → (𝑦 / (𝑃↑(𝑃 pCnt 𝑦))) ∈ ℕ)
169		pcndvds2 16902	. . . . . . . . . . 11 ⊢ ((𝑃 ∈ ℙ ∧ 𝑦 ∈ ℕ) → ¬ 𝑃 ∥ (𝑦 / (𝑃↑(𝑃 pCnt 𝑦))))
170	22, 50, 169	syl2anc 584	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝐵 ≠ 0) ∧ (𝑥 ∈ ℤ ∧ 𝑧 ∈ ℤ)) ∧ ((𝑦 ∈ ℕ ∧ 𝑤 ∈ ℕ) ∧ (𝐴 = (𝑥 / 𝑦) ∧ 𝐵 = (𝑧 / 𝑤)))) → ¬ 𝑃 ∥ (𝑦 / (𝑃↑(𝑃 pCnt 𝑦))))
171	168, 170	jca 511	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝐵 ≠ 0) ∧ (𝑥 ∈ ℤ ∧ 𝑧 ∈ ℤ)) ∧ ((𝑦 ∈ ℕ ∧ 𝑤 ∈ ℕ) ∧ (𝐴 = (𝑥 / 𝑦) ∧ 𝐵 = (𝑧 / 𝑤)))) → ((𝑦 / (𝑃↑(𝑃 pCnt 𝑦))) ∈ ℕ ∧ ¬ 𝑃 ∥ (𝑦 / (𝑃↑(𝑃 pCnt 𝑦)))))
172		pczdvds 16897	. . . . . . . . . . . 12 ⊢ ((𝑃 ∈ ℙ ∧ (𝑧 ∈ ℤ ∧ 𝑧 ≠ 0)) → (𝑃↑(𝑃 pCnt 𝑧)) ∥ 𝑧)
173	22, 97, 108, 172	syl12anc 837	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝐵 ≠ 0) ∧ (𝑥 ∈ ℤ ∧ 𝑧 ∈ ℤ)) ∧ ((𝑦 ∈ ℕ ∧ 𝑤 ∈ ℕ) ∧ (𝐴 = (𝑥 / 𝑦) ∧ 𝐵 = (𝑧 / 𝑤)))) → (𝑃↑(𝑃 pCnt 𝑧)) ∥ 𝑧)
174	111	nnzd 12638	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝐵 ≠ 0) ∧ (𝑥 ∈ ℤ ∧ 𝑧 ∈ ℤ)) ∧ ((𝑦 ∈ ℕ ∧ 𝑤 ∈ ℕ) ∧ (𝐴 = (𝑥 / 𝑦) ∧ 𝐵 = (𝑧 / 𝑤)))) → (𝑃↑(𝑃 pCnt 𝑧)) ∈ ℤ)
175		dvdsval2 16290	. . . . . . . . . . . 12 ⊢ (((𝑃↑(𝑃 pCnt 𝑧)) ∈ ℤ ∧ (𝑃↑(𝑃 pCnt 𝑧)) ≠ 0 ∧ 𝑧 ∈ ℤ) → ((𝑃↑(𝑃 pCnt 𝑧)) ∥ 𝑧 ↔ (𝑧 / (𝑃↑(𝑃 pCnt 𝑧))) ∈ ℤ))
176	174, 119, 97, 175	syl3anc 1370	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝐵 ≠ 0) ∧ (𝑥 ∈ ℤ ∧ 𝑧 ∈ ℤ)) ∧ ((𝑦 ∈ ℕ ∧ 𝑤 ∈ ℕ) ∧ (𝐴 = (𝑥 / 𝑦) ∧ 𝐵 = (𝑧 / 𝑤)))) → ((𝑃↑(𝑃 pCnt 𝑧)) ∥ 𝑧 ↔ (𝑧 / (𝑃↑(𝑃 pCnt 𝑧))) ∈ ℤ))
177	173, 176	mpbid 232	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝐵 ≠ 0) ∧ (𝑥 ∈ ℤ ∧ 𝑧 ∈ ℤ)) ∧ ((𝑦 ∈ ℕ ∧ 𝑤 ∈ ℕ) ∧ (𝐴 = (𝑥 / 𝑦) ∧ 𝐵 = (𝑧 / 𝑤)))) → (𝑧 / (𝑃↑(𝑃 pCnt 𝑧))) ∈ ℤ)
178		pczndvds2 16901	. . . . . . . . . . 11 ⊢ ((𝑃 ∈ ℙ ∧ (𝑧 ∈ ℤ ∧ 𝑧 ≠ 0)) → ¬ 𝑃 ∥ (𝑧 / (𝑃↑(𝑃 pCnt 𝑧))))
179	22, 97, 108, 178	syl12anc 837	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝐵 ≠ 0) ∧ (𝑥 ∈ ℤ ∧ 𝑧 ∈ ℤ)) ∧ ((𝑦 ∈ ℕ ∧ 𝑤 ∈ ℕ) ∧ (𝐴 = (𝑥 / 𝑦) ∧ 𝐵 = (𝑧 / 𝑤)))) → ¬ 𝑃 ∥ (𝑧 / (𝑃↑(𝑃 pCnt 𝑧))))
180	177, 179	jca 511	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝐵 ≠ 0) ∧ (𝑥 ∈ ℤ ∧ 𝑧 ∈ ℤ)) ∧ ((𝑦 ∈ ℕ ∧ 𝑤 ∈ ℕ) ∧ (𝐴 = (𝑥 / 𝑦) ∧ 𝐵 = (𝑧 / 𝑤)))) → ((𝑧 / (𝑃↑(𝑃 pCnt 𝑧))) ∈ ℤ ∧ ¬ 𝑃 ∥ (𝑧 / (𝑃↑(𝑃 pCnt 𝑧)))))
181		pcdvds 16898	. . . . . . . . . . . . 13 ⊢ ((𝑃 ∈ ℙ ∧ 𝑤 ∈ ℕ) → (𝑃↑(𝑃 pCnt 𝑤)) ∥ 𝑤)
182	22, 100, 181	syl2anc 584	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝐵 ≠ 0) ∧ (𝑥 ∈ ℤ ∧ 𝑧 ∈ ℤ)) ∧ ((𝑦 ∈ ℕ ∧ 𝑤 ∈ ℕ) ∧ (𝐴 = (𝑥 / 𝑦) ∧ 𝐵 = (𝑧 / 𝑤)))) → (𝑃↑(𝑃 pCnt 𝑤)) ∥ 𝑤)
183	117	nnzd 12638	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝐵 ≠ 0) ∧ (𝑥 ∈ ℤ ∧ 𝑧 ∈ ℤ)) ∧ ((𝑦 ∈ ℕ ∧ 𝑤 ∈ ℕ) ∧ (𝐴 = (𝑥 / 𝑦) ∧ 𝐵 = (𝑧 / 𝑤)))) → (𝑃↑(𝑃 pCnt 𝑤)) ∈ ℤ)
184	100	nnzd 12638	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝐵 ≠ 0) ∧ (𝑥 ∈ ℤ ∧ 𝑧 ∈ ℤ)) ∧ ((𝑦 ∈ ℕ ∧ 𝑤 ∈ ℕ) ∧ (𝐴 = (𝑥 / 𝑦) ∧ 𝐵 = (𝑧 / 𝑤)))) → 𝑤 ∈ ℤ)
185		dvdsval2 16290	. . . . . . . . . . . . 13 ⊢ (((𝑃↑(𝑃 pCnt 𝑤)) ∈ ℤ ∧ (𝑃↑(𝑃 pCnt 𝑤)) ≠ 0 ∧ 𝑤 ∈ ℤ) → ((𝑃↑(𝑃 pCnt 𝑤)) ∥ 𝑤 ↔ (𝑤 / (𝑃↑(𝑃 pCnt 𝑤))) ∈ ℤ))
186	183, 120, 184, 185	syl3anc 1370	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝐵 ≠ 0) ∧ (𝑥 ∈ ℤ ∧ 𝑧 ∈ ℤ)) ∧ ((𝑦 ∈ ℕ ∧ 𝑤 ∈ ℕ) ∧ (𝐴 = (𝑥 / 𝑦) ∧ 𝐵 = (𝑧 / 𝑤)))) → ((𝑃↑(𝑃 pCnt 𝑤)) ∥ 𝑤 ↔ (𝑤 / (𝑃↑(𝑃 pCnt 𝑤))) ∈ ℤ))
187	182, 186	mpbid 232	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝐵 ≠ 0) ∧ (𝑥 ∈ ℤ ∧ 𝑧 ∈ ℤ)) ∧ ((𝑦 ∈ ℕ ∧ 𝑤 ∈ ℕ) ∧ (𝐴 = (𝑥 / 𝑦) ∧ 𝐵 = (𝑧 / 𝑤)))) → (𝑤 / (𝑃↑(𝑃 pCnt 𝑤))) ∈ ℤ)
188	100	nnred 12279	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝐵 ≠ 0) ∧ (𝑥 ∈ ℤ ∧ 𝑧 ∈ ℤ)) ∧ ((𝑦 ∈ ℕ ∧ 𝑤 ∈ ℕ) ∧ (𝐴 = (𝑥 / 𝑦) ∧ 𝐵 = (𝑧 / 𝑤)))) → 𝑤 ∈ ℝ)
189	117	nnred 12279	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝐵 ≠ 0) ∧ (𝑥 ∈ ℤ ∧ 𝑧 ∈ ℤ)) ∧ ((𝑦 ∈ ℕ ∧ 𝑤 ∈ ℕ) ∧ (𝐴 = (𝑥 / 𝑦) ∧ 𝐵 = (𝑧 / 𝑤)))) → (𝑃↑(𝑃 pCnt 𝑤)) ∈ ℝ)
190	100	nngt0d 12313	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝐵 ≠ 0) ∧ (𝑥 ∈ ℤ ∧ 𝑧 ∈ ℤ)) ∧ ((𝑦 ∈ ℕ ∧ 𝑤 ∈ ℕ) ∧ (𝐴 = (𝑥 / 𝑦) ∧ 𝐵 = (𝑧 / 𝑤)))) → 0 < 𝑤)
191	117	nngt0d 12313	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝐵 ≠ 0) ∧ (𝑥 ∈ ℤ ∧ 𝑧 ∈ ℤ)) ∧ ((𝑦 ∈ ℕ ∧ 𝑤 ∈ ℕ) ∧ (𝐴 = (𝑥 / 𝑦) ∧ 𝐵 = (𝑧 / 𝑤)))) → 0 < (𝑃↑(𝑃 pCnt 𝑤)))
192	188, 189, 190, 191	divgt0d 12201	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝐵 ≠ 0) ∧ (𝑥 ∈ ℤ ∧ 𝑧 ∈ ℤ)) ∧ ((𝑦 ∈ ℕ ∧ 𝑤 ∈ ℕ) ∧ (𝐴 = (𝑥 / 𝑦) ∧ 𝐵 = (𝑧 / 𝑤)))) → 0 < (𝑤 / (𝑃↑(𝑃 pCnt 𝑤))))
193		elnnz 12621	. . . . . . . . . . 11 ⊢ ((𝑤 / (𝑃↑(𝑃 pCnt 𝑤))) ∈ ℕ ↔ ((𝑤 / (𝑃↑(𝑃 pCnt 𝑤))) ∈ ℤ ∧ 0 < (𝑤 / (𝑃↑(𝑃 pCnt 𝑤)))))
194	187, 192, 193	sylanbrc 583	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝐵 ≠ 0) ∧ (𝑥 ∈ ℤ ∧ 𝑧 ∈ ℤ)) ∧ ((𝑦 ∈ ℕ ∧ 𝑤 ∈ ℕ) ∧ (𝐴 = (𝑥 / 𝑦) ∧ 𝐵 = (𝑧 / 𝑤)))) → (𝑤 / (𝑃↑(𝑃 pCnt 𝑤))) ∈ ℕ)
195		pcndvds2 16902	. . . . . . . . . . 11 ⊢ ((𝑃 ∈ ℙ ∧ 𝑤 ∈ ℕ) → ¬ 𝑃 ∥ (𝑤 / (𝑃↑(𝑃 pCnt 𝑤))))
196	22, 100, 195	syl2anc 584	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝐵 ≠ 0) ∧ (𝑥 ∈ ℤ ∧ 𝑧 ∈ ℤ)) ∧ ((𝑦 ∈ ℕ ∧ 𝑤 ∈ ℕ) ∧ (𝐴 = (𝑥 / 𝑦) ∧ 𝐵 = (𝑧 / 𝑤)))) → ¬ 𝑃 ∥ (𝑤 / (𝑃↑(𝑃 pCnt 𝑤))))
197	194, 196	jca 511	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝐵 ≠ 0) ∧ (𝑥 ∈ ℤ ∧ 𝑧 ∈ ℤ)) ∧ ((𝑦 ∈ ℕ ∧ 𝑤 ∈ ℕ) ∧ (𝐴 = (𝑥 / 𝑦) ∧ 𝐵 = (𝑧 / 𝑤)))) → ((𝑤 / (𝑃↑(𝑃 pCnt 𝑤))) ∈ ℕ ∧ ¬ 𝑃 ∥ (𝑤 / (𝑃↑(𝑃 pCnt 𝑤)))))
198	22, 96, 142, 145, 154, 171, 180, 197	pcaddlem 16922	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝐵 ≠ 0) ∧ (𝑥 ∈ ℤ ∧ 𝑧 ∈ ℤ)) ∧ ((𝑦 ∈ ℕ ∧ 𝑤 ∈ ℕ) ∧ (𝐴 = (𝑥 / 𝑦) ∧ 𝐵 = (𝑧 / 𝑤)))) → (𝑃 pCnt 𝐴) ≤ (𝑃 pCnt (𝐴 + 𝐵)))
199	198	expr 456	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝐵 ≠ 0) ∧ (𝑥 ∈ ℤ ∧ 𝑧 ∈ ℤ)) ∧ (𝑦 ∈ ℕ ∧ 𝑤 ∈ ℕ)) → ((𝐴 = (𝑥 / 𝑦) ∧ 𝐵 = (𝑧 / 𝑤)) → (𝑃 pCnt 𝐴) ≤ (𝑃 pCnt (𝐴 + 𝐵))))
200	199	rexlimdvva 3211	. . . . . 6 ⊢ (((𝜑 ∧ 𝐵 ≠ 0) ∧ (𝑥 ∈ ℤ ∧ 𝑧 ∈ ℤ)) → (∃𝑦 ∈ ℕ ∃𝑤 ∈ ℕ (𝐴 = (𝑥 / 𝑦) ∧ 𝐵 = (𝑧 / 𝑤)) → (𝑃 pCnt 𝐴) ≤ (𝑃 pCnt (𝐴 + 𝐵))))
201	21, 200	biimtrrid 243	. . . . 5 ⊢ (((𝜑 ∧ 𝐵 ≠ 0) ∧ (𝑥 ∈ ℤ ∧ 𝑧 ∈ ℤ)) → ((∃𝑦 ∈ ℕ 𝐴 = (𝑥 / 𝑦) ∧ ∃𝑤 ∈ ℕ 𝐵 = (𝑧 / 𝑤)) → (𝑃 pCnt 𝐴) ≤ (𝑃 pCnt (𝐴 + 𝐵))))
202	201	rexlimdvva 3211	. . . 4 ⊢ ((𝜑 ∧ 𝐵 ≠ 0) → (∃𝑥 ∈ ℤ ∃𝑧 ∈ ℤ (∃𝑦 ∈ ℕ 𝐴 = (𝑥 / 𝑦) ∧ ∃𝑤 ∈ ℕ 𝐵 = (𝑧 / 𝑤)) → (𝑃 pCnt 𝐴) ≤ (𝑃 pCnt (𝐴 + 𝐵))))
203	20, 202	biimtrrid 243	. . 3 ⊢ ((𝜑 ∧ 𝐵 ≠ 0) → ((∃𝑥 ∈ ℤ ∃𝑦 ∈ ℕ 𝐴 = (𝑥 / 𝑦) ∧ ∃𝑧 ∈ ℤ ∃𝑤 ∈ ℕ 𝐵 = (𝑧 / 𝑤)) → (𝑃 pCnt 𝐴) ≤ (𝑃 pCnt (𝐴 + 𝐵))))
204	19, 203	pm2.61dane 3027	. 2 ⊢ (𝜑 → ((∃𝑥 ∈ ℤ ∃𝑦 ∈ ℕ 𝐴 = (𝑥 / 𝑦) ∧ ∃𝑧 ∈ ℤ ∃𝑤 ∈ ℕ 𝐵 = (𝑧 / 𝑤)) → (𝑃 pCnt 𝐴) ≤ (𝑃 pCnt (𝐴 + 𝐵))))
205	3, 6, 204	mp2and 699	1 ⊢ (𝜑 → (𝑃 pCnt 𝐴) ≤ (𝑃 pCnt (𝐴 + 𝐵)))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1537 ∈ wcel 2106 ≠ wne 2938 ∃wrex 3068 class class class wbr 5148 ‘cfv 6563 (class class class)co 7431 ℂcc 11151 ℝcr 11152 0cc0 11153 + caddc 11156 · cmul 11158 +∞cpnf 11290 ℝ^*cxr 11292 < clt 11293 ≤ cle 11294 − cmin 11490 / cdiv 11918 ℕcn 12264 ℕ₀cn0 12524 ℤcz 12611 ℤ_≥cuz 12876 ℚcq 12988 ↑cexp 14099 ∥ cdvds 16287 ℙcprime 16705 pCnt cpc 16870

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 ax-cnex 11209 ax-resscn 11210 ax-1cn 11211 ax-icn 11212 ax-addcl 11213 ax-addrcl 11214 ax-mulcl 11215 ax-mulrcl 11216 ax-mulcom 11217 ax-addass 11218 ax-mulass 11219 ax-distr 11220 ax-i2m1 11221 ax-1ne0 11222 ax-1rid 11223 ax-rnegex 11224 ax-rrecex 11225 ax-cnre 11226 ax-pre-lttri 11227 ax-pre-lttrn 11228 ax-pre-ltadd 11229 ax-pre-mulgt0 11230 ax-pre-sup 11231

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-rmo 3378 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-pss 3983 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5583 df-eprel 5589 df-po 5597 df-so 5598 df-fr 5641 df-we 5643 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-pred 6323 df-ord 6389 df-on 6390 df-lim 6391 df-suc 6392 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-1st 8013 df-2nd 8014 df-frecs 8305 df-wrecs 8336 df-recs 8410 df-rdg 8449 df-1o 8505 df-2o 8506 df-er 8744 df-en 8985 df-dom 8986 df-sdom 8987 df-fin 8988 df-sup 9480 df-inf 9481 df-pnf 11295 df-mnf 11296 df-xr 11297 df-ltxr 11298 df-le 11299 df-sub 11492 df-neg 11493 df-div 11919 df-nn 12265 df-2 12327 df-3 12328 df-n0 12525 df-z 12612 df-uz 12877 df-q 12989 df-rp 13033 df-fl 13829 df-mod 13907 df-seq 14040 df-exp 14100 df-cj 15135 df-re 15136 df-im 15137 df-sqrt 15271 df-abs 15272 df-dvds 16288 df-gcd 16529 df-prm 16706 df-pc 16871

This theorem is referenced by: pcadd2 16924 padicabv 27689

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