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Theorem pcmpt 15596

Description: Construct a function with given prime count characteristics. (Contributed by Mario Carneiro, 12-Mar-2014.)

**Hypotheses**
Ref	Expression
pcmpt.1	⊢ 𝐹 = (𝑛 ∈ ℕ ↦ if(𝑛 ∈ ℙ, (𝑛↑𝐴), 1))
pcmpt.2	⊢ (𝜑 → ∀𝑛 ∈ ℙ 𝐴 ∈ ℕ₀)
pcmpt.3	⊢ (𝜑 → 𝑁 ∈ ℕ)
pcmpt.4	⊢ (𝜑 → 𝑃 ∈ ℙ)
pcmpt.5	⊢ (𝑛 = 𝑃 → 𝐴 = 𝐵)

**Assertion**
Ref	Expression
pcmpt	⊢ (𝜑 → (𝑃 pCnt (seq1( · , 𝐹)‘𝑁)) = if(𝑃 ≤ 𝑁, 𝐵, 0))

Distinct variable groups: 𝐵,𝑛 𝑃,𝑛 Allowed substitution hints: 𝜑(𝑛) 𝐴(𝑛) 𝐹(𝑛) 𝑁(𝑛)

Proof of Theorem pcmpt Dummy variables 𝑘 𝑝 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		pcmpt.3	. 2 ⊢ (𝜑 → 𝑁 ∈ ℕ)
2		fveq2 6191	. . . . . 6 ⊢ (𝑝 = 1 → (seq1( · , 𝐹)‘𝑝) = (seq1( · , 𝐹)‘1))
3	2	oveq2d 6666	. . . . 5 ⊢ (𝑝 = 1 → (𝑃 pCnt (seq1( · , 𝐹)‘𝑝)) = (𝑃 pCnt (seq1( · , 𝐹)‘1)))
4		breq2 4657	. . . . . 6 ⊢ (𝑝 = 1 → (𝑃 ≤ 𝑝 ↔ 𝑃 ≤ 1))
5	4	ifbid 4108	. . . . 5 ⊢ (𝑝 = 1 → if(𝑃 ≤ 𝑝, 𝐵, 0) = if(𝑃 ≤ 1, 𝐵, 0))
6	3, 5	eqeq12d 2637	. . . 4 ⊢ (𝑝 = 1 → ((𝑃 pCnt (seq1( · , 𝐹)‘𝑝)) = if(𝑃 ≤ 𝑝, 𝐵, 0) ↔ (𝑃 pCnt (seq1( · , 𝐹)‘1)) = if(𝑃 ≤ 1, 𝐵, 0)))
7	6	imbi2d 330	. . 3 ⊢ (𝑝 = 1 → ((𝜑 → (𝑃 pCnt (seq1( · , 𝐹)‘𝑝)) = if(𝑃 ≤ 𝑝, 𝐵, 0)) ↔ (𝜑 → (𝑃 pCnt (seq1( · , 𝐹)‘1)) = if(𝑃 ≤ 1, 𝐵, 0))))
8		fveq2 6191	. . . . . 6 ⊢ (𝑝 = 𝑘 → (seq1( · , 𝐹)‘𝑝) = (seq1( · , 𝐹)‘𝑘))
9	8	oveq2d 6666	. . . . 5 ⊢ (𝑝 = 𝑘 → (𝑃 pCnt (seq1( · , 𝐹)‘𝑝)) = (𝑃 pCnt (seq1( · , 𝐹)‘𝑘)))
10		breq2 4657	. . . . . 6 ⊢ (𝑝 = 𝑘 → (𝑃 ≤ 𝑝 ↔ 𝑃 ≤ 𝑘))
11	10	ifbid 4108	. . . . 5 ⊢ (𝑝 = 𝑘 → if(𝑃 ≤ 𝑝, 𝐵, 0) = if(𝑃 ≤ 𝑘, 𝐵, 0))
12	9, 11	eqeq12d 2637	. . . 4 ⊢ (𝑝 = 𝑘 → ((𝑃 pCnt (seq1( · , 𝐹)‘𝑝)) = if(𝑃 ≤ 𝑝, 𝐵, 0) ↔ (𝑃 pCnt (seq1( · , 𝐹)‘𝑘)) = if(𝑃 ≤ 𝑘, 𝐵, 0)))
13	12	imbi2d 330	. . 3 ⊢ (𝑝 = 𝑘 → ((𝜑 → (𝑃 pCnt (seq1( · , 𝐹)‘𝑝)) = if(𝑃 ≤ 𝑝, 𝐵, 0)) ↔ (𝜑 → (𝑃 pCnt (seq1( · , 𝐹)‘𝑘)) = if(𝑃 ≤ 𝑘, 𝐵, 0))))
14		fveq2 6191	. . . . . 6 ⊢ (𝑝 = (𝑘 + 1) → (seq1( · , 𝐹)‘𝑝) = (seq1( · , 𝐹)‘(𝑘 + 1)))
15	14	oveq2d 6666	. . . . 5 ⊢ (𝑝 = (𝑘 + 1) → (𝑃 pCnt (seq1( · , 𝐹)‘𝑝)) = (𝑃 pCnt (seq1( · , 𝐹)‘(𝑘 + 1))))
16		breq2 4657	. . . . . 6 ⊢ (𝑝 = (𝑘 + 1) → (𝑃 ≤ 𝑝 ↔ 𝑃 ≤ (𝑘 + 1)))
17	16	ifbid 4108	. . . . 5 ⊢ (𝑝 = (𝑘 + 1) → if(𝑃 ≤ 𝑝, 𝐵, 0) = if(𝑃 ≤ (𝑘 + 1), 𝐵, 0))
18	15, 17	eqeq12d 2637	. . . 4 ⊢ (𝑝 = (𝑘 + 1) → ((𝑃 pCnt (seq1( · , 𝐹)‘𝑝)) = if(𝑃 ≤ 𝑝, 𝐵, 0) ↔ (𝑃 pCnt (seq1( · , 𝐹)‘(𝑘 + 1))) = if(𝑃 ≤ (𝑘 + 1), 𝐵, 0)))
19	18	imbi2d 330	. . 3 ⊢ (𝑝 = (𝑘 + 1) → ((𝜑 → (𝑃 pCnt (seq1( · , 𝐹)‘𝑝)) = if(𝑃 ≤ 𝑝, 𝐵, 0)) ↔ (𝜑 → (𝑃 pCnt (seq1( · , 𝐹)‘(𝑘 + 1))) = if(𝑃 ≤ (𝑘 + 1), 𝐵, 0))))
20		fveq2 6191	. . . . . 6 ⊢ (𝑝 = 𝑁 → (seq1( · , 𝐹)‘𝑝) = (seq1( · , 𝐹)‘𝑁))
21	20	oveq2d 6666	. . . . 5 ⊢ (𝑝 = 𝑁 → (𝑃 pCnt (seq1( · , 𝐹)‘𝑝)) = (𝑃 pCnt (seq1( · , 𝐹)‘𝑁)))
22		breq2 4657	. . . . . 6 ⊢ (𝑝 = 𝑁 → (𝑃 ≤ 𝑝 ↔ 𝑃 ≤ 𝑁))
23	22	ifbid 4108	. . . . 5 ⊢ (𝑝 = 𝑁 → if(𝑃 ≤ 𝑝, 𝐵, 0) = if(𝑃 ≤ 𝑁, 𝐵, 0))
24	21, 23	eqeq12d 2637	. . . 4 ⊢ (𝑝 = 𝑁 → ((𝑃 pCnt (seq1( · , 𝐹)‘𝑝)) = if(𝑃 ≤ 𝑝, 𝐵, 0) ↔ (𝑃 pCnt (seq1( · , 𝐹)‘𝑁)) = if(𝑃 ≤ 𝑁, 𝐵, 0)))
25	24	imbi2d 330	. . 3 ⊢ (𝑝 = 𝑁 → ((𝜑 → (𝑃 pCnt (seq1( · , 𝐹)‘𝑝)) = if(𝑃 ≤ 𝑝, 𝐵, 0)) ↔ (𝜑 → (𝑃 pCnt (seq1( · , 𝐹)‘𝑁)) = if(𝑃 ≤ 𝑁, 𝐵, 0))))
26		pcmpt.4	. . . 4 ⊢ (𝜑 → 𝑃 ∈ ℙ)
27		1z 11407	. . . . . . . . 9 ⊢ 1 ∈ ℤ
28		seq1 12814	. . . . . . . . 9 ⊢ (1 ∈ ℤ → (seq1( · , 𝐹)‘1) = (𝐹‘1))
29	27, 28	ax-mp 5	. . . . . . . 8 ⊢ (seq1( · , 𝐹)‘1) = (𝐹‘1)
30		1nn 11031	. . . . . . . . 9 ⊢ 1 ∈ ℕ
31		1nprm 15392	. . . . . . . . . . . 12 ⊢ ¬ 1 ∈ ℙ
32		eleq1 2689	. . . . . . . . . . . 12 ⊢ (𝑛 = 1 → (𝑛 ∈ ℙ ↔ 1 ∈ ℙ))
33	31, 32	mtbiri 317	. . . . . . . . . . 11 ⊢ (𝑛 = 1 → ¬ 𝑛 ∈ ℙ)
34	33	iffalsed 4097	. . . . . . . . . 10 ⊢ (𝑛 = 1 → if(𝑛 ∈ ℙ, (𝑛↑𝐴), 1) = 1)
35		pcmpt.1	. . . . . . . . . 10 ⊢ 𝐹 = (𝑛 ∈ ℕ ↦ if(𝑛 ∈ ℙ, (𝑛↑𝐴), 1))
36		1ex 10035	. . . . . . . . . 10 ⊢ 1 ∈ V
37	34, 35, 36	fvmpt 6282	. . . . . . . . 9 ⊢ (1 ∈ ℕ → (𝐹‘1) = 1)
38	30, 37	ax-mp 5	. . . . . . . 8 ⊢ (𝐹‘1) = 1
39	29, 38	eqtri 2644	. . . . . . 7 ⊢ (seq1( · , 𝐹)‘1) = 1
40	39	oveq2i 6661	. . . . . 6 ⊢ (𝑃 pCnt (seq1( · , 𝐹)‘1)) = (𝑃 pCnt 1)
41		pc1 15560	. . . . . 6 ⊢ (𝑃 ∈ ℙ → (𝑃 pCnt 1) = 0)
42	40, 41	syl5eq 2668	. . . . 5 ⊢ (𝑃 ∈ ℙ → (𝑃 pCnt (seq1( · , 𝐹)‘1)) = 0)
43		prmgt1 15409	. . . . . . 7 ⊢ (𝑃 ∈ ℙ → 1 < 𝑃)
44		1re 10039	. . . . . . . 8 ⊢ 1 ∈ ℝ
45		prmuz2 15408	. . . . . . . . 9 ⊢ (𝑃 ∈ ℙ → 𝑃 ∈ (ℤ_≥‘2))
46		eluzelre 11698	. . . . . . . . 9 ⊢ (𝑃 ∈ (ℤ_≥‘2) → 𝑃 ∈ ℝ)
47	45, 46	syl 17	. . . . . . . 8 ⊢ (𝑃 ∈ ℙ → 𝑃 ∈ ℝ)
48		ltnle 10117	. . . . . . . 8 ⊢ ((1 ∈ ℝ ∧ 𝑃 ∈ ℝ) → (1 < 𝑃 ↔ ¬ 𝑃 ≤ 1))
49	44, 47, 48	sylancr 695	. . . . . . 7 ⊢ (𝑃 ∈ ℙ → (1 < 𝑃 ↔ ¬ 𝑃 ≤ 1))
50	43, 49	mpbid 222	. . . . . 6 ⊢ (𝑃 ∈ ℙ → ¬ 𝑃 ≤ 1)
51	50	iffalsed 4097	. . . . 5 ⊢ (𝑃 ∈ ℙ → if(𝑃 ≤ 1, 𝐵, 0) = 0)
52	42, 51	eqtr4d 2659	. . . 4 ⊢ (𝑃 ∈ ℙ → (𝑃 pCnt (seq1( · , 𝐹)‘1)) = if(𝑃 ≤ 1, 𝐵, 0))
53	26, 52	syl 17	. . 3 ⊢ (𝜑 → (𝑃 pCnt (seq1( · , 𝐹)‘1)) = if(𝑃 ≤ 1, 𝐵, 0))
54	26	adantr 481	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑘 ∈ ℕ ∧ (𝑘 + 1) = 𝑃)) → 𝑃 ∈ ℙ)
55		pcmpt.2	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → ∀𝑛 ∈ ℙ 𝐴 ∈ ℕ₀)
56	35, 55	pcmptcl 15595	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (𝐹:ℕ⟶ℕ ∧ seq1( · , 𝐹):ℕ⟶ℕ))
57	56	simpld 475	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝐹:ℕ⟶ℕ)
58		peano2nn 11032	. . . . . . . . . . . . . . 15 ⊢ (𝑘 ∈ ℕ → (𝑘 + 1) ∈ ℕ)
59		ffvelrn 6357	. . . . . . . . . . . . . . 15 ⊢ ((𝐹:ℕ⟶ℕ ∧ (𝑘 + 1) ∈ ℕ) → (𝐹‘(𝑘 + 1)) ∈ ℕ)
60	57, 58, 59	syl2an 494	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (𝐹‘(𝑘 + 1)) ∈ ℕ)
61	60	adantrr 753	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑘 ∈ ℕ ∧ (𝑘 + 1) = 𝑃)) → (𝐹‘(𝑘 + 1)) ∈ ℕ)
62	54, 61	pccld 15555	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑘 ∈ ℕ ∧ (𝑘 + 1) = 𝑃)) → (𝑃 pCnt (𝐹‘(𝑘 + 1))) ∈ ℕ₀)
63	62	nn0cnd 11353	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑘 ∈ ℕ ∧ (𝑘 + 1) = 𝑃)) → (𝑃 pCnt (𝐹‘(𝑘 + 1))) ∈ ℂ)
64	63	addid2d 10237	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑘 ∈ ℕ ∧ (𝑘 + 1) = 𝑃)) → (0 + (𝑃 pCnt (𝐹‘(𝑘 + 1)))) = (𝑃 pCnt (𝐹‘(𝑘 + 1))))
65	58	ad2antrl 764	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑘 ∈ ℕ ∧ (𝑘 + 1) = 𝑃)) → (𝑘 + 1) ∈ ℕ)
66		ovex 6678	. . . . . . . . . . . . . . 15 ⊢ (𝑛↑𝐴) ∈ V
67	66, 36	ifex 4156	. . . . . . . . . . . . . 14 ⊢ if(𝑛 ∈ ℙ, (𝑛↑𝐴), 1) ∈ V
68	67	csbex 4793	. . . . . . . . . . . . 13 ⊢ ⦋(𝑘 + 1) / 𝑛⦌if(𝑛 ∈ ℙ, (𝑛↑𝐴), 1) ∈ V
69	35	fvmpts 6285	. . . . . . . . . . . . . 14 ⊢ (((𝑘 + 1) ∈ ℕ ∧ ⦋(𝑘 + 1) / 𝑛⦌if(𝑛 ∈ ℙ, (𝑛↑𝐴), 1) ∈ V) → (𝐹‘(𝑘 + 1)) = ⦋(𝑘 + 1) / 𝑛⦌if(𝑛 ∈ ℙ, (𝑛↑𝐴), 1))
70		ovex 6678	. . . . . . . . . . . . . . 15 ⊢ (𝑘 + 1) ∈ V
71		nfv 1843	. . . . . . . . . . . . . . . 16 ⊢ Ⅎ𝑛(𝑘 + 1) ∈ ℙ
72		nfcv 2764	. . . . . . . . . . . . . . . . 17 ⊢ Ⅎ𝑛(𝑘 + 1)
73		nfcv 2764	. . . . . . . . . . . . . . . . 17 ⊢ Ⅎ𝑛↑
74		nfcsb1v 3549	. . . . . . . . . . . . . . . . 17 ⊢ Ⅎ𝑛⦋(𝑘 + 1) / 𝑛⦌𝐴
75	72, 73, 74	nfov 6676	. . . . . . . . . . . . . . . 16 ⊢ Ⅎ𝑛((𝑘 + 1)↑⦋(𝑘 + 1) / 𝑛⦌𝐴)
76		nfcv 2764	. . . . . . . . . . . . . . . 16 ⊢ Ⅎ𝑛1
77	71, 75, 76	nfif 4115	. . . . . . . . . . . . . . 15 ⊢ Ⅎ𝑛if((𝑘 + 1) ∈ ℙ, ((𝑘 + 1)↑⦋(𝑘 + 1) / 𝑛⦌𝐴), 1)
78		eleq1 2689	. . . . . . . . . . . . . . . 16 ⊢ (𝑛 = (𝑘 + 1) → (𝑛 ∈ ℙ ↔ (𝑘 + 1) ∈ ℙ))
79		id 22	. . . . . . . . . . . . . . . . 17 ⊢ (𝑛 = (𝑘 + 1) → 𝑛 = (𝑘 + 1))
80		csbeq1a 3542	. . . . . . . . . . . . . . . . 17 ⊢ (𝑛 = (𝑘 + 1) → 𝐴 = ⦋(𝑘 + 1) / 𝑛⦌𝐴)
81	79, 80	oveq12d 6668	. . . . . . . . . . . . . . . 16 ⊢ (𝑛 = (𝑘 + 1) → (𝑛↑𝐴) = ((𝑘 + 1)↑⦋(𝑘 + 1) / 𝑛⦌𝐴))
82	78, 81	ifbieq1d 4109	. . . . . . . . . . . . . . 15 ⊢ (𝑛 = (𝑘 + 1) → if(𝑛 ∈ ℙ, (𝑛↑𝐴), 1) = if((𝑘 + 1) ∈ ℙ, ((𝑘 + 1)↑⦋(𝑘 + 1) / 𝑛⦌𝐴), 1))
83	70, 77, 82	csbief 3558	. . . . . . . . . . . . . 14 ⊢ ⦋(𝑘 + 1) / 𝑛⦌if(𝑛 ∈ ℙ, (𝑛↑𝐴), 1) = if((𝑘 + 1) ∈ ℙ, ((𝑘 + 1)↑⦋(𝑘 + 1) / 𝑛⦌𝐴), 1)
84	69, 83	syl6eq 2672	. . . . . . . . . . . . 13 ⊢ (((𝑘 + 1) ∈ ℕ ∧ ⦋(𝑘 + 1) / 𝑛⦌if(𝑛 ∈ ℙ, (𝑛↑𝐴), 1) ∈ V) → (𝐹‘(𝑘 + 1)) = if((𝑘 + 1) ∈ ℙ, ((𝑘 + 1)↑⦋(𝑘 + 1) / 𝑛⦌𝐴), 1))
85	65, 68, 84	sylancl 694	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑘 ∈ ℕ ∧ (𝑘 + 1) = 𝑃)) → (𝐹‘(𝑘 + 1)) = if((𝑘 + 1) ∈ ℙ, ((𝑘 + 1)↑⦋(𝑘 + 1) / 𝑛⦌𝐴), 1))
86		simprr 796	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑘 ∈ ℕ ∧ (𝑘 + 1) = 𝑃)) → (𝑘 + 1) = 𝑃)
87	86, 54	eqeltrd 2701	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑘 ∈ ℕ ∧ (𝑘 + 1) = 𝑃)) → (𝑘 + 1) ∈ ℙ)
88	87	iftrued 4094	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑘 ∈ ℕ ∧ (𝑘 + 1) = 𝑃)) → if((𝑘 + 1) ∈ ℙ, ((𝑘 + 1)↑⦋(𝑘 + 1) / 𝑛⦌𝐴), 1) = ((𝑘 + 1)↑⦋(𝑘 + 1) / 𝑛⦌𝐴))
89	86	csbeq1d 3540	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑘 ∈ ℕ ∧ (𝑘 + 1) = 𝑃)) → ⦋(𝑘 + 1) / 𝑛⦌𝐴 = ⦋𝑃 / 𝑛⦌𝐴)
90		nfcvd 2765	. . . . . . . . . . . . . . . 16 ⊢ (𝑃 ∈ ℙ → Ⅎ𝑛𝐵)
91		pcmpt.5	. . . . . . . . . . . . . . . 16 ⊢ (𝑛 = 𝑃 → 𝐴 = 𝐵)
92	90, 91	csbiegf 3557	. . . . . . . . . . . . . . 15 ⊢ (𝑃 ∈ ℙ → ⦋𝑃 / 𝑛⦌𝐴 = 𝐵)
93	54, 92	syl 17	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑘 ∈ ℕ ∧ (𝑘 + 1) = 𝑃)) → ⦋𝑃 / 𝑛⦌𝐴 = 𝐵)
94	89, 93	eqtrd 2656	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑘 ∈ ℕ ∧ (𝑘 + 1) = 𝑃)) → ⦋(𝑘 + 1) / 𝑛⦌𝐴 = 𝐵)
95	86, 94	oveq12d 6668	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑘 ∈ ℕ ∧ (𝑘 + 1) = 𝑃)) → ((𝑘 + 1)↑⦋(𝑘 + 1) / 𝑛⦌𝐴) = (𝑃↑𝐵))
96	85, 88, 95	3eqtrd 2660	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑘 ∈ ℕ ∧ (𝑘 + 1) = 𝑃)) → (𝐹‘(𝑘 + 1)) = (𝑃↑𝐵))
97	96	oveq2d 6666	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑘 ∈ ℕ ∧ (𝑘 + 1) = 𝑃)) → (𝑃 pCnt (𝐹‘(𝑘 + 1))) = (𝑃 pCnt (𝑃↑𝐵)))
98	91	eleq1d 2686	. . . . . . . . . . . . . 14 ⊢ (𝑛 = 𝑃 → (𝐴 ∈ ℕ₀ ↔ 𝐵 ∈ ℕ₀))
99	98	rspcv 3305	. . . . . . . . . . . . 13 ⊢ (𝑃 ∈ ℙ → (∀𝑛 ∈ ℙ 𝐴 ∈ ℕ₀ → 𝐵 ∈ ℕ₀))
100	26, 55, 99	sylc 65	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐵 ∈ ℕ₀)
101	100	adantr 481	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑘 ∈ ℕ ∧ (𝑘 + 1) = 𝑃)) → 𝐵 ∈ ℕ₀)
102		pcidlem 15576	. . . . . . . . . . 11 ⊢ ((𝑃 ∈ ℙ ∧ 𝐵 ∈ ℕ₀) → (𝑃 pCnt (𝑃↑𝐵)) = 𝐵)
103	54, 101, 102	syl2anc 693	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑘 ∈ ℕ ∧ (𝑘 + 1) = 𝑃)) → (𝑃 pCnt (𝑃↑𝐵)) = 𝐵)
104	64, 97, 103	3eqtrd 2660	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑘 ∈ ℕ ∧ (𝑘 + 1) = 𝑃)) → (0 + (𝑃 pCnt (𝐹‘(𝑘 + 1)))) = 𝐵)
105		oveq1 6657	. . . . . . . . . 10 ⊢ ((𝑃 pCnt (seq1( · , 𝐹)‘𝑘)) = 0 → ((𝑃 pCnt (seq1( · , 𝐹)‘𝑘)) + (𝑃 pCnt (𝐹‘(𝑘 + 1)))) = (0 + (𝑃 pCnt (𝐹‘(𝑘 + 1)))))
106	105	eqeq1d 2624	. . . . . . . . 9 ⊢ ((𝑃 pCnt (seq1( · , 𝐹)‘𝑘)) = 0 → (((𝑃 pCnt (seq1( · , 𝐹)‘𝑘)) + (𝑃 pCnt (𝐹‘(𝑘 + 1)))) = 𝐵 ↔ (0 + (𝑃 pCnt (𝐹‘(𝑘 + 1)))) = 𝐵))
107	104, 106	syl5ibrcom 237	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑘 ∈ ℕ ∧ (𝑘 + 1) = 𝑃)) → ((𝑃 pCnt (seq1( · , 𝐹)‘𝑘)) = 0 → ((𝑃 pCnt (seq1( · , 𝐹)‘𝑘)) + (𝑃 pCnt (𝐹‘(𝑘 + 1)))) = 𝐵))
108		nnre 11027	. . . . . . . . . . . . 13 ⊢ (𝑘 ∈ ℕ → 𝑘 ∈ ℝ)
109	108	ad2antrl 764	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑘 ∈ ℕ ∧ (𝑘 + 1) = 𝑃)) → 𝑘 ∈ ℝ)
110		ltp1 10861	. . . . . . . . . . . . 13 ⊢ (𝑘 ∈ ℝ → 𝑘 < (𝑘 + 1))
111		peano2re 10209	. . . . . . . . . . . . . 14 ⊢ (𝑘 ∈ ℝ → (𝑘 + 1) ∈ ℝ)
112		ltnle 10117	. . . . . . . . . . . . . 14 ⊢ ((𝑘 ∈ ℝ ∧ (𝑘 + 1) ∈ ℝ) → (𝑘 < (𝑘 + 1) ↔ ¬ (𝑘 + 1) ≤ 𝑘))
113	111, 112	mpdan 702	. . . . . . . . . . . . 13 ⊢ (𝑘 ∈ ℝ → (𝑘 < (𝑘 + 1) ↔ ¬ (𝑘 + 1) ≤ 𝑘))
114	110, 113	mpbid 222	. . . . . . . . . . . 12 ⊢ (𝑘 ∈ ℝ → ¬ (𝑘 + 1) ≤ 𝑘)
115	109, 114	syl 17	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑘 ∈ ℕ ∧ (𝑘 + 1) = 𝑃)) → ¬ (𝑘 + 1) ≤ 𝑘)
116	86	breq1d 4663	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑘 ∈ ℕ ∧ (𝑘 + 1) = 𝑃)) → ((𝑘 + 1) ≤ 𝑘 ↔ 𝑃 ≤ 𝑘))
117	115, 116	mtbid 314	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑘 ∈ ℕ ∧ (𝑘 + 1) = 𝑃)) → ¬ 𝑃 ≤ 𝑘)
118	117	iffalsed 4097	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑘 ∈ ℕ ∧ (𝑘 + 1) = 𝑃)) → if(𝑃 ≤ 𝑘, 𝐵, 0) = 0)
119	118	eqeq2d 2632	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑘 ∈ ℕ ∧ (𝑘 + 1) = 𝑃)) → ((𝑃 pCnt (seq1( · , 𝐹)‘𝑘)) = if(𝑃 ≤ 𝑘, 𝐵, 0) ↔ (𝑃 pCnt (seq1( · , 𝐹)‘𝑘)) = 0))
120		simpr 477	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → 𝑘 ∈ ℕ)
121		nnuz 11723	. . . . . . . . . . . . . 14 ⊢ ℕ = (ℤ_≥‘1)
122	120, 121	syl6eleq 2711	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → 𝑘 ∈ (ℤ_≥‘1))
123		seqp1 12816	. . . . . . . . . . . . 13 ⊢ (𝑘 ∈ (ℤ_≥‘1) → (seq1( · , 𝐹)‘(𝑘 + 1)) = ((seq1( · , 𝐹)‘𝑘) · (𝐹‘(𝑘 + 1))))
124	122, 123	syl 17	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (seq1( · , 𝐹)‘(𝑘 + 1)) = ((seq1( · , 𝐹)‘𝑘) · (𝐹‘(𝑘 + 1))))
125	124	oveq2d 6666	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (𝑃 pCnt (seq1( · , 𝐹)‘(𝑘 + 1))) = (𝑃 pCnt ((seq1( · , 𝐹)‘𝑘) · (𝐹‘(𝑘 + 1)))))
126	26	adantr 481	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → 𝑃 ∈ ℙ)
127	56	simprd 479	. . . . . . . . . . . . . 14 ⊢ (𝜑 → seq1( · , 𝐹):ℕ⟶ℕ)
128	127	ffvelrnda 6359	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (seq1( · , 𝐹)‘𝑘) ∈ ℕ)
129		nnz 11399	. . . . . . . . . . . . . 14 ⊢ ((seq1( · , 𝐹)‘𝑘) ∈ ℕ → (seq1( · , 𝐹)‘𝑘) ∈ ℤ)
130		nnne0 11053	. . . . . . . . . . . . . 14 ⊢ ((seq1( · , 𝐹)‘𝑘) ∈ ℕ → (seq1( · , 𝐹)‘𝑘) ≠ 0)
131	129, 130	jca 554	. . . . . . . . . . . . 13 ⊢ ((seq1( · , 𝐹)‘𝑘) ∈ ℕ → ((seq1( · , 𝐹)‘𝑘) ∈ ℤ ∧ (seq1( · , 𝐹)‘𝑘) ≠ 0))
132	128, 131	syl 17	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → ((seq1( · , 𝐹)‘𝑘) ∈ ℤ ∧ (seq1( · , 𝐹)‘𝑘) ≠ 0))
133		nnz 11399	. . . . . . . . . . . . . 14 ⊢ ((𝐹‘(𝑘 + 1)) ∈ ℕ → (𝐹‘(𝑘 + 1)) ∈ ℤ)
134		nnne0 11053	. . . . . . . . . . . . . 14 ⊢ ((𝐹‘(𝑘 + 1)) ∈ ℕ → (𝐹‘(𝑘 + 1)) ≠ 0)
135	133, 134	jca 554	. . . . . . . . . . . . 13 ⊢ ((𝐹‘(𝑘 + 1)) ∈ ℕ → ((𝐹‘(𝑘 + 1)) ∈ ℤ ∧ (𝐹‘(𝑘 + 1)) ≠ 0))
136	60, 135	syl 17	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → ((𝐹‘(𝑘 + 1)) ∈ ℤ ∧ (𝐹‘(𝑘 + 1)) ≠ 0))
137		pcmul 15556	. . . . . . . . . . . 12 ⊢ ((𝑃 ∈ ℙ ∧ ((seq1( · , 𝐹)‘𝑘) ∈ ℤ ∧ (seq1( · , 𝐹)‘𝑘) ≠ 0) ∧ ((𝐹‘(𝑘 + 1)) ∈ ℤ ∧ (𝐹‘(𝑘 + 1)) ≠ 0)) → (𝑃 pCnt ((seq1( · , 𝐹)‘𝑘) · (𝐹‘(𝑘 + 1)))) = ((𝑃 pCnt (seq1( · , 𝐹)‘𝑘)) + (𝑃 pCnt (𝐹‘(𝑘 + 1)))))
138	126, 132, 136, 137	syl3anc 1326	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (𝑃 pCnt ((seq1( · , 𝐹)‘𝑘) · (𝐹‘(𝑘 + 1)))) = ((𝑃 pCnt (seq1( · , 𝐹)‘𝑘)) + (𝑃 pCnt (𝐹‘(𝑘 + 1)))))
139	125, 138	eqtrd 2656	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (𝑃 pCnt (seq1( · , 𝐹)‘(𝑘 + 1))) = ((𝑃 pCnt (seq1( · , 𝐹)‘𝑘)) + (𝑃 pCnt (𝐹‘(𝑘 + 1)))))
140	139	adantrr 753	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑘 ∈ ℕ ∧ (𝑘 + 1) = 𝑃)) → (𝑃 pCnt (seq1( · , 𝐹)‘(𝑘 + 1))) = ((𝑃 pCnt (seq1( · , 𝐹)‘𝑘)) + (𝑃 pCnt (𝐹‘(𝑘 + 1)))))
141		prmnn 15388	. . . . . . . . . . . . . . 15 ⊢ (𝑃 ∈ ℙ → 𝑃 ∈ ℕ)
142	26, 141	syl 17	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝑃 ∈ ℕ)
143	142	nnred 11035	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝑃 ∈ ℝ)
144	143	adantr 481	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑘 ∈ ℕ ∧ (𝑘 + 1) = 𝑃)) → 𝑃 ∈ ℝ)
145	144	leidd 10594	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑘 ∈ ℕ ∧ (𝑘 + 1) = 𝑃)) → 𝑃 ≤ 𝑃)
146	145, 86	breqtrrd 4681	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑘 ∈ ℕ ∧ (𝑘 + 1) = 𝑃)) → 𝑃 ≤ (𝑘 + 1))
147	146	iftrued 4094	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑘 ∈ ℕ ∧ (𝑘 + 1) = 𝑃)) → if(𝑃 ≤ (𝑘 + 1), 𝐵, 0) = 𝐵)
148	140, 147	eqeq12d 2637	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑘 ∈ ℕ ∧ (𝑘 + 1) = 𝑃)) → ((𝑃 pCnt (seq1( · , 𝐹)‘(𝑘 + 1))) = if(𝑃 ≤ (𝑘 + 1), 𝐵, 0) ↔ ((𝑃 pCnt (seq1( · , 𝐹)‘𝑘)) + (𝑃 pCnt (𝐹‘(𝑘 + 1)))) = 𝐵))
149	107, 119, 148	3imtr4d 283	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑘 ∈ ℕ ∧ (𝑘 + 1) = 𝑃)) → ((𝑃 pCnt (seq1( · , 𝐹)‘𝑘)) = if(𝑃 ≤ 𝑘, 𝐵, 0) → (𝑃 pCnt (seq1( · , 𝐹)‘(𝑘 + 1))) = if(𝑃 ≤ (𝑘 + 1), 𝐵, 0)))
150	149	expr 643	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → ((𝑘 + 1) = 𝑃 → ((𝑃 pCnt (seq1( · , 𝐹)‘𝑘)) = if(𝑃 ≤ 𝑘, 𝐵, 0) → (𝑃 pCnt (seq1( · , 𝐹)‘(𝑘 + 1))) = if(𝑃 ≤ (𝑘 + 1), 𝐵, 0))))
151	139	adantrr 753	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑘 ∈ ℕ ∧ (𝑘 + 1) ≠ 𝑃)) → (𝑃 pCnt (seq1( · , 𝐹)‘(𝑘 + 1))) = ((𝑃 pCnt (seq1( · , 𝐹)‘𝑘)) + (𝑃 pCnt (𝐹‘(𝑘 + 1)))))
152		simplrr 801	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ (𝑘 ∈ ℕ ∧ (𝑘 + 1) ≠ 𝑃)) ∧ (𝑘 + 1) ∈ ℙ) → (𝑘 + 1) ≠ 𝑃)
153	152	necomd 2849	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ (𝑘 ∈ ℕ ∧ (𝑘 + 1) ≠ 𝑃)) ∧ (𝑘 + 1) ∈ ℙ) → 𝑃 ≠ (𝑘 + 1))
154	26	ad2antrr 762	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ (𝑘 ∈ ℕ ∧ (𝑘 + 1) ≠ 𝑃)) ∧ (𝑘 + 1) ∈ ℙ) → 𝑃 ∈ ℙ)
155		simpr 477	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ (𝑘 ∈ ℕ ∧ (𝑘 + 1) ≠ 𝑃)) ∧ (𝑘 + 1) ∈ ℙ) → (𝑘 + 1) ∈ ℙ)
156	55	ad2antrr 762	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ (𝑘 ∈ ℕ ∧ (𝑘 + 1) ≠ 𝑃)) ∧ (𝑘 + 1) ∈ ℙ) → ∀𝑛 ∈ ℙ 𝐴 ∈ ℕ₀)
157	74	nfel1 2779	. . . . . . . . . . . . . . . . . . 19 ⊢ Ⅎ𝑛⦋(𝑘 + 1) / 𝑛⦌𝐴 ∈ ℕ₀
158	80	eleq1d 2686	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑛 = (𝑘 + 1) → (𝐴 ∈ ℕ₀ ↔ ⦋(𝑘 + 1) / 𝑛⦌𝐴 ∈ ℕ₀))
159	157, 158	rspc 3303	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑘 + 1) ∈ ℙ → (∀𝑛 ∈ ℙ 𝐴 ∈ ℕ₀ → ⦋(𝑘 + 1) / 𝑛⦌𝐴 ∈ ℕ₀))
160	155, 156, 159	sylc 65	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ (𝑘 ∈ ℕ ∧ (𝑘 + 1) ≠ 𝑃)) ∧ (𝑘 + 1) ∈ ℙ) → ⦋(𝑘 + 1) / 𝑛⦌𝐴 ∈ ℕ₀)
161		prmdvdsexpr 15429	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑃 ∈ ℙ ∧ (𝑘 + 1) ∈ ℙ ∧ ⦋(𝑘 + 1) / 𝑛⦌𝐴 ∈ ℕ₀) → (𝑃 ∥ ((𝑘 + 1)↑⦋(𝑘 + 1) / 𝑛⦌𝐴) → 𝑃 = (𝑘 + 1)))
162	154, 155, 160, 161	syl3anc 1326	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ (𝑘 ∈ ℕ ∧ (𝑘 + 1) ≠ 𝑃)) ∧ (𝑘 + 1) ∈ ℙ) → (𝑃 ∥ ((𝑘 + 1)↑⦋(𝑘 + 1) / 𝑛⦌𝐴) → 𝑃 = (𝑘 + 1)))
163	162	necon3ad 2807	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ (𝑘 ∈ ℕ ∧ (𝑘 + 1) ≠ 𝑃)) ∧ (𝑘 + 1) ∈ ℙ) → (𝑃 ≠ (𝑘 + 1) → ¬ 𝑃 ∥ ((𝑘 + 1)↑⦋(𝑘 + 1) / 𝑛⦌𝐴)))
164	153, 163	mpd 15	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ (𝑘 ∈ ℕ ∧ (𝑘 + 1) ≠ 𝑃)) ∧ (𝑘 + 1) ∈ ℙ) → ¬ 𝑃 ∥ ((𝑘 + 1)↑⦋(𝑘 + 1) / 𝑛⦌𝐴))
165	58	ad2antrl 764	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ (𝑘 ∈ ℕ ∧ (𝑘 + 1) ≠ 𝑃)) → (𝑘 + 1) ∈ ℕ)
166	165, 68, 84	sylancl 694	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ (𝑘 ∈ ℕ ∧ (𝑘 + 1) ≠ 𝑃)) → (𝐹‘(𝑘 + 1)) = if((𝑘 + 1) ∈ ℙ, ((𝑘 + 1)↑⦋(𝑘 + 1) / 𝑛⦌𝐴), 1))
167		iftrue 4092	. . . . . . . . . . . . . . . 16 ⊢ ((𝑘 + 1) ∈ ℙ → if((𝑘 + 1) ∈ ℙ, ((𝑘 + 1)↑⦋(𝑘 + 1) / 𝑛⦌𝐴), 1) = ((𝑘 + 1)↑⦋(𝑘 + 1) / 𝑛⦌𝐴))
168	166, 167	sylan9eq 2676	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ (𝑘 ∈ ℕ ∧ (𝑘 + 1) ≠ 𝑃)) ∧ (𝑘 + 1) ∈ ℙ) → (𝐹‘(𝑘 + 1)) = ((𝑘 + 1)↑⦋(𝑘 + 1) / 𝑛⦌𝐴))
169	168	breq2d 4665	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ (𝑘 ∈ ℕ ∧ (𝑘 + 1) ≠ 𝑃)) ∧ (𝑘 + 1) ∈ ℙ) → (𝑃 ∥ (𝐹‘(𝑘 + 1)) ↔ 𝑃 ∥ ((𝑘 + 1)↑⦋(𝑘 + 1) / 𝑛⦌𝐴)))
170	164, 169	mtbird 315	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝑘 ∈ ℕ ∧ (𝑘 + 1) ≠ 𝑃)) ∧ (𝑘 + 1) ∈ ℙ) → ¬ 𝑃 ∥ (𝐹‘(𝑘 + 1)))
171	57	adantr 481	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ (𝑘 ∈ ℕ ∧ (𝑘 + 1) ≠ 𝑃)) → 𝐹:ℕ⟶ℕ)
172	171, 165, 59	syl2anc 693	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (𝑘 ∈ ℕ ∧ (𝑘 + 1) ≠ 𝑃)) → (𝐹‘(𝑘 + 1)) ∈ ℕ)
173	172	adantr 481	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ (𝑘 ∈ ℕ ∧ (𝑘 + 1) ≠ 𝑃)) ∧ (𝑘 + 1) ∈ ℙ) → (𝐹‘(𝑘 + 1)) ∈ ℕ)
174		pceq0 15575	. . . . . . . . . . . . . 14 ⊢ ((𝑃 ∈ ℙ ∧ (𝐹‘(𝑘 + 1)) ∈ ℕ) → ((𝑃 pCnt (𝐹‘(𝑘 + 1))) = 0 ↔ ¬ 𝑃 ∥ (𝐹‘(𝑘 + 1))))
175	154, 173, 174	syl2anc 693	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝑘 ∈ ℕ ∧ (𝑘 + 1) ≠ 𝑃)) ∧ (𝑘 + 1) ∈ ℙ) → ((𝑃 pCnt (𝐹‘(𝑘 + 1))) = 0 ↔ ¬ 𝑃 ∥ (𝐹‘(𝑘 + 1))))
176	170, 175	mpbird 247	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝑘 ∈ ℕ ∧ (𝑘 + 1) ≠ 𝑃)) ∧ (𝑘 + 1) ∈ ℙ) → (𝑃 pCnt (𝐹‘(𝑘 + 1))) = 0)
177		iffalse 4095	. . . . . . . . . . . . . . 15 ⊢ (¬ (𝑘 + 1) ∈ ℙ → if((𝑘 + 1) ∈ ℙ, ((𝑘 + 1)↑⦋(𝑘 + 1) / 𝑛⦌𝐴), 1) = 1)
178	166, 177	sylan9eq 2676	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ (𝑘 ∈ ℕ ∧ (𝑘 + 1) ≠ 𝑃)) ∧ ¬ (𝑘 + 1) ∈ ℙ) → (𝐹‘(𝑘 + 1)) = 1)
179	178	oveq2d 6666	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝑘 ∈ ℕ ∧ (𝑘 + 1) ≠ 𝑃)) ∧ ¬ (𝑘 + 1) ∈ ℙ) → (𝑃 pCnt (𝐹‘(𝑘 + 1))) = (𝑃 pCnt 1))
180	26, 41	syl 17	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (𝑃 pCnt 1) = 0)
181	180	ad2antrr 762	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝑘 ∈ ℕ ∧ (𝑘 + 1) ≠ 𝑃)) ∧ ¬ (𝑘 + 1) ∈ ℙ) → (𝑃 pCnt 1) = 0)
182	179, 181	eqtrd 2656	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝑘 ∈ ℕ ∧ (𝑘 + 1) ≠ 𝑃)) ∧ ¬ (𝑘 + 1) ∈ ℙ) → (𝑃 pCnt (𝐹‘(𝑘 + 1))) = 0)
183	176, 182	pm2.61dan 832	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑘 ∈ ℕ ∧ (𝑘 + 1) ≠ 𝑃)) → (𝑃 pCnt (𝐹‘(𝑘 + 1))) = 0)
184	183	oveq2d 6666	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑘 ∈ ℕ ∧ (𝑘 + 1) ≠ 𝑃)) → ((𝑃 pCnt (seq1( · , 𝐹)‘𝑘)) + (𝑃 pCnt (𝐹‘(𝑘 + 1)))) = ((𝑃 pCnt (seq1( · , 𝐹)‘𝑘)) + 0))
185	26	adantr 481	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑘 ∈ ℕ ∧ (𝑘 + 1) ≠ 𝑃)) → 𝑃 ∈ ℙ)
186	128	adantrr 753	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑘 ∈ ℕ ∧ (𝑘 + 1) ≠ 𝑃)) → (seq1( · , 𝐹)‘𝑘) ∈ ℕ)
187	185, 186	pccld 15555	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑘 ∈ ℕ ∧ (𝑘 + 1) ≠ 𝑃)) → (𝑃 pCnt (seq1( · , 𝐹)‘𝑘)) ∈ ℕ₀)
188	187	nn0cnd 11353	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑘 ∈ ℕ ∧ (𝑘 + 1) ≠ 𝑃)) → (𝑃 pCnt (seq1( · , 𝐹)‘𝑘)) ∈ ℂ)
189	188	addid1d 10236	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑘 ∈ ℕ ∧ (𝑘 + 1) ≠ 𝑃)) → ((𝑃 pCnt (seq1( · , 𝐹)‘𝑘)) + 0) = (𝑃 pCnt (seq1( · , 𝐹)‘𝑘)))
190	151, 184, 189	3eqtrd 2660	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑘 ∈ ℕ ∧ (𝑘 + 1) ≠ 𝑃)) → (𝑃 pCnt (seq1( · , 𝐹)‘(𝑘 + 1))) = (𝑃 pCnt (seq1( · , 𝐹)‘𝑘)))
191	142	adantr 481	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑘 ∈ ℕ ∧ (𝑘 + 1) ≠ 𝑃)) → 𝑃 ∈ ℕ)
192	191	nnred 11035	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑘 ∈ ℕ ∧ (𝑘 + 1) ≠ 𝑃)) → 𝑃 ∈ ℝ)
193	165	nnred 11035	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑘 ∈ ℕ ∧ (𝑘 + 1) ≠ 𝑃)) → (𝑘 + 1) ∈ ℝ)
194	192, 193	ltlend 10182	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑘 ∈ ℕ ∧ (𝑘 + 1) ≠ 𝑃)) → (𝑃 < (𝑘 + 1) ↔ (𝑃 ≤ (𝑘 + 1) ∧ (𝑘 + 1) ≠ 𝑃)))
195		simprl 794	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑘 ∈ ℕ ∧ (𝑘 + 1) ≠ 𝑃)) → 𝑘 ∈ ℕ)
196		nnleltp1 11432	. . . . . . . . . . . 12 ⊢ ((𝑃 ∈ ℕ ∧ 𝑘 ∈ ℕ) → (𝑃 ≤ 𝑘 ↔ 𝑃 < (𝑘 + 1)))
197	191, 195, 196	syl2anc 693	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑘 ∈ ℕ ∧ (𝑘 + 1) ≠ 𝑃)) → (𝑃 ≤ 𝑘 ↔ 𝑃 < (𝑘 + 1)))
198		simprr 796	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑘 ∈ ℕ ∧ (𝑘 + 1) ≠ 𝑃)) → (𝑘 + 1) ≠ 𝑃)
199	198	biantrud 528	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑘 ∈ ℕ ∧ (𝑘 + 1) ≠ 𝑃)) → (𝑃 ≤ (𝑘 + 1) ↔ (𝑃 ≤ (𝑘 + 1) ∧ (𝑘 + 1) ≠ 𝑃)))
200	194, 197, 199	3bitr4rd 301	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑘 ∈ ℕ ∧ (𝑘 + 1) ≠ 𝑃)) → (𝑃 ≤ (𝑘 + 1) ↔ 𝑃 ≤ 𝑘))
201	200	ifbid 4108	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑘 ∈ ℕ ∧ (𝑘 + 1) ≠ 𝑃)) → if(𝑃 ≤ (𝑘 + 1), 𝐵, 0) = if(𝑃 ≤ 𝑘, 𝐵, 0))
202	190, 201	eqeq12d 2637	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑘 ∈ ℕ ∧ (𝑘 + 1) ≠ 𝑃)) → ((𝑃 pCnt (seq1( · , 𝐹)‘(𝑘 + 1))) = if(𝑃 ≤ (𝑘 + 1), 𝐵, 0) ↔ (𝑃 pCnt (seq1( · , 𝐹)‘𝑘)) = if(𝑃 ≤ 𝑘, 𝐵, 0)))
203	202	biimprd 238	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑘 ∈ ℕ ∧ (𝑘 + 1) ≠ 𝑃)) → ((𝑃 pCnt (seq1( · , 𝐹)‘𝑘)) = if(𝑃 ≤ 𝑘, 𝐵, 0) → (𝑃 pCnt (seq1( · , 𝐹)‘(𝑘 + 1))) = if(𝑃 ≤ (𝑘 + 1), 𝐵, 0)))
204	203	expr 643	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → ((𝑘 + 1) ≠ 𝑃 → ((𝑃 pCnt (seq1( · , 𝐹)‘𝑘)) = if(𝑃 ≤ 𝑘, 𝐵, 0) → (𝑃 pCnt (seq1( · , 𝐹)‘(𝑘 + 1))) = if(𝑃 ≤ (𝑘 + 1), 𝐵, 0))))
205	150, 204	pm2.61dne 2880	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → ((𝑃 pCnt (seq1( · , 𝐹)‘𝑘)) = if(𝑃 ≤ 𝑘, 𝐵, 0) → (𝑃 pCnt (seq1( · , 𝐹)‘(𝑘 + 1))) = if(𝑃 ≤ (𝑘 + 1), 𝐵, 0)))
206	205	expcom 451	. . . 4 ⊢ (𝑘 ∈ ℕ → (𝜑 → ((𝑃 pCnt (seq1( · , 𝐹)‘𝑘)) = if(𝑃 ≤ 𝑘, 𝐵, 0) → (𝑃 pCnt (seq1( · , 𝐹)‘(𝑘 + 1))) = if(𝑃 ≤ (𝑘 + 1), 𝐵, 0))))
207	206	a2d 29	. . 3 ⊢ (𝑘 ∈ ℕ → ((𝜑 → (𝑃 pCnt (seq1( · , 𝐹)‘𝑘)) = if(𝑃 ≤ 𝑘, 𝐵, 0)) → (𝜑 → (𝑃 pCnt (seq1( · , 𝐹)‘(𝑘 + 1))) = if(𝑃 ≤ (𝑘 + 1), 𝐵, 0))))
208	7, 13, 19, 25, 53, 207	nnind 11038	. 2 ⊢ (𝑁 ∈ ℕ → (𝜑 → (𝑃 pCnt (seq1( · , 𝐹)‘𝑁)) = if(𝑃 ≤ 𝑁, 𝐵, 0)))
209	1, 208	mpcom 38	1 ⊢ (𝜑 → (𝑃 pCnt (seq1( · , 𝐹)‘𝑁)) = if(𝑃 ≤ 𝑁, 𝐵, 0))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 196 ∧ wa 384 = wceq 1483 ∈ wcel 1990 ≠ wne 2794 ∀wral 2912 Vcvv 3200 ⦋csb 3533 ifcif 4086 class class class wbr 4653 ↦ cmpt 4729 ⟶wf 5884 ‘cfv 5888 (class class class)co 6650 ℝcr 9935 0cc0 9936 1c1 9937 + caddc 9939 · cmul 9941 < clt 10074 ≤ cle 10075 ℕcn 11020 2c2 11070 ℕ₀cn0 11292 ℤcz 11377 ℤ_≥cuz 11687 seqcseq 12801 ↑cexp 12860 ∥ cdvds 14983 ℙcprime 15385 pCnt cpc 15541

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-cnex 9992 ax-resscn 9993 ax-1cn 9994 ax-icn 9995 ax-addcl 9996 ax-addrcl 9997 ax-mulcl 9998 ax-mulrcl 9999 ax-mulcom 10000 ax-addass 10001 ax-mulass 10002 ax-distr 10003 ax-i2m1 10004 ax-1ne0 10005 ax-1rid 10006 ax-rnegex 10007 ax-rrecex 10008 ax-cnre 10009 ax-pre-lttri 10010 ax-pre-lttrn 10011 ax-pre-ltadd 10012 ax-pre-mulgt0 10013 ax-pre-sup 10014

This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-fal 1489 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-nel 2898 df-ral 2917 df-rex 2918 df-reu 2919 df-rmo 2920 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-pred 5680 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-riota 6611 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-om 7066 df-1st 7168 df-2nd 7169 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-1o 7560 df-2o 7561 df-er 7742 df-en 7956 df-dom 7957 df-sdom 7958 df-fin 7959 df-sup 8348 df-inf 8349 df-pnf 10076 df-mnf 10077 df-xr 10078 df-ltxr 10079 df-le 10080 df-sub 10268 df-neg 10269 df-div 10685 df-nn 11021 df-2 11079 df-3 11080 df-n0 11293 df-z 11378 df-uz 11688 df-q 11789 df-rp 11833 df-fz 12327 df-fl 12593 df-mod 12669 df-seq 12802 df-exp 12861 df-cj 13839 df-re 13840 df-im 13841 df-sqrt 13975 df-abs 13976 df-dvds 14984 df-gcd 15217 df-prm 15386 df-pc 15542

This theorem is referenced by: pcmpt2 15597 pcprod 15599 1arithlem4 15630 chtublem 24936 bposlem3 25011

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