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Theorem pcmpt 16218

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 6645	. . . . . 6 ⊢ (𝑝 = 1 → (seq1( · , 𝐹)‘𝑝) = (seq1( · , 𝐹)‘1))
3	2	oveq2d 7151	. . . . 5 ⊢ (𝑝 = 1 → (𝑃 pCnt (seq1( · , 𝐹)‘𝑝)) = (𝑃 pCnt (seq1( · , 𝐹)‘1)))
4		breq2 5034	. . . . . 6 ⊢ (𝑝 = 1 → (𝑃 ≤ 𝑝 ↔ 𝑃 ≤ 1))
5	4	ifbid 4447	. . . . 5 ⊢ (𝑝 = 1 → if(𝑃 ≤ 𝑝, 𝐵, 0) = if(𝑃 ≤ 1, 𝐵, 0))
6	3, 5	eqeq12d 2814	. . . 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 344	. . 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 6645	. . . . . 6 ⊢ (𝑝 = 𝑘 → (seq1( · , 𝐹)‘𝑝) = (seq1( · , 𝐹)‘𝑘))
9	8	oveq2d 7151	. . . . 5 ⊢ (𝑝 = 𝑘 → (𝑃 pCnt (seq1( · , 𝐹)‘𝑝)) = (𝑃 pCnt (seq1( · , 𝐹)‘𝑘)))
10		breq2 5034	. . . . . 6 ⊢ (𝑝 = 𝑘 → (𝑃 ≤ 𝑝 ↔ 𝑃 ≤ 𝑘))
11	10	ifbid 4447	. . . . 5 ⊢ (𝑝 = 𝑘 → if(𝑃 ≤ 𝑝, 𝐵, 0) = if(𝑃 ≤ 𝑘, 𝐵, 0))
12	9, 11	eqeq12d 2814	. . . 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 344	. . 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 6645	. . . . . 6 ⊢ (𝑝 = (𝑘 + 1) → (seq1( · , 𝐹)‘𝑝) = (seq1( · , 𝐹)‘(𝑘 + 1)))
15	14	oveq2d 7151	. . . . 5 ⊢ (𝑝 = (𝑘 + 1) → (𝑃 pCnt (seq1( · , 𝐹)‘𝑝)) = (𝑃 pCnt (seq1( · , 𝐹)‘(𝑘 + 1))))
16		breq2 5034	. . . . . 6 ⊢ (𝑝 = (𝑘 + 1) → (𝑃 ≤ 𝑝 ↔ 𝑃 ≤ (𝑘 + 1)))
17	16	ifbid 4447	. . . . 5 ⊢ (𝑝 = (𝑘 + 1) → if(𝑃 ≤ 𝑝, 𝐵, 0) = if(𝑃 ≤ (𝑘 + 1), 𝐵, 0))
18	15, 17	eqeq12d 2814	. . . 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 344	. . 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 6645	. . . . . 6 ⊢ (𝑝 = 𝑁 → (seq1( · , 𝐹)‘𝑝) = (seq1( · , 𝐹)‘𝑁))
21	20	oveq2d 7151	. . . . 5 ⊢ (𝑝 = 𝑁 → (𝑃 pCnt (seq1( · , 𝐹)‘𝑝)) = (𝑃 pCnt (seq1( · , 𝐹)‘𝑁)))
22		breq2 5034	. . . . . 6 ⊢ (𝑝 = 𝑁 → (𝑃 ≤ 𝑝 ↔ 𝑃 ≤ 𝑁))
23	22	ifbid 4447	. . . . 5 ⊢ (𝑝 = 𝑁 → if(𝑃 ≤ 𝑝, 𝐵, 0) = if(𝑃 ≤ 𝑁, 𝐵, 0))
24	21, 23	eqeq12d 2814	. . . 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 344	. . 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 12000	. . . . . . . . 9 ⊢ 1 ∈ ℤ
28		seq1 13377	. . . . . . . . 9 ⊢ (1 ∈ ℤ → (seq1( · , 𝐹)‘1) = (𝐹‘1))
29	27, 28	ax-mp 5	. . . . . . . 8 ⊢ (seq1( · , 𝐹)‘1) = (𝐹‘1)
30		1nn 11636	. . . . . . . . 9 ⊢ 1 ∈ ℕ
31		1nprm 16013	. . . . . . . . . . . 12 ⊢ ¬ 1 ∈ ℙ
32		eleq1 2877	. . . . . . . . . . . 12 ⊢ (𝑛 = 1 → (𝑛 ∈ ℙ ↔ 1 ∈ ℙ))
33	31, 32	mtbiri 330	. . . . . . . . . . 11 ⊢ (𝑛 = 1 → ¬ 𝑛 ∈ ℙ)
34	33	iffalsed 4436	. . . . . . . . . 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 10626	. . . . . . . . . 10 ⊢ 1 ∈ V
37	34, 35, 36	fvmpt 6745	. . . . . . . . 9 ⊢ (1 ∈ ℕ → (𝐹‘1) = 1)
38	30, 37	ax-mp 5	. . . . . . . 8 ⊢ (𝐹‘1) = 1
39	29, 38	eqtri 2821	. . . . . . 7 ⊢ (seq1( · , 𝐹)‘1) = 1
40	39	oveq2i 7146	. . . . . 6 ⊢ (𝑃 pCnt (seq1( · , 𝐹)‘1)) = (𝑃 pCnt 1)
41		pc1 16182	. . . . . 6 ⊢ (𝑃 ∈ ℙ → (𝑃 pCnt 1) = 0)
42	40, 41	syl5eq 2845	. . . . 5 ⊢ (𝑃 ∈ ℙ → (𝑃 pCnt (seq1( · , 𝐹)‘1)) = 0)
43		prmgt1 16031	. . . . . . 7 ⊢ (𝑃 ∈ ℙ → 1 < 𝑃)
44		1re 10630	. . . . . . . 8 ⊢ 1 ∈ ℝ
45		prmuz2 16030	. . . . . . . . 9 ⊢ (𝑃 ∈ ℙ → 𝑃 ∈ (ℤ_≥‘2))
46		eluzelre 12242	. . . . . . . . 9 ⊢ (𝑃 ∈ (ℤ_≥‘2) → 𝑃 ∈ ℝ)
47	45, 46	syl 17	. . . . . . . 8 ⊢ (𝑃 ∈ ℙ → 𝑃 ∈ ℝ)
48		ltnle 10709	. . . . . . . 8 ⊢ ((1 ∈ ℝ ∧ 𝑃 ∈ ℝ) → (1 < 𝑃 ↔ ¬ 𝑃 ≤ 1))
49	44, 47, 48	sylancr 590	. . . . . . 7 ⊢ (𝑃 ∈ ℙ → (1 < 𝑃 ↔ ¬ 𝑃 ≤ 1))
50	43, 49	mpbid 235	. . . . . 6 ⊢ (𝑃 ∈ ℙ → ¬ 𝑃 ≤ 1)
51	50	iffalsed 4436	. . . . 5 ⊢ (𝑃 ∈ ℙ → if(𝑃 ≤ 1, 𝐵, 0) = 0)
52	42, 51	eqtr4d 2836	. . . 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 484	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑘 ∈ ℕ ∧ (𝑘 + 1) = 𝑃)) → 𝑃 ∈ ℙ)
55		pcmpt.2	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → ∀𝑛 ∈ ℙ 𝐴 ∈ ℕ₀)
56	35, 55	pcmptcl 16217	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (𝐹:ℕ⟶ℕ ∧ seq1( · , 𝐹):ℕ⟶ℕ))
57	56	simpld 498	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝐹:ℕ⟶ℕ)
58		peano2nn 11637	. . . . . . . . . . . . . . 15 ⊢ (𝑘 ∈ ℕ → (𝑘 + 1) ∈ ℕ)
59		ffvelrn 6826	. . . . . . . . . . . . . . 15 ⊢ ((𝐹:ℕ⟶ℕ ∧ (𝑘 + 1) ∈ ℕ) → (𝐹‘(𝑘 + 1)) ∈ ℕ)
60	57, 58, 59	syl2an 598	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (𝐹‘(𝑘 + 1)) ∈ ℕ)
61	60	adantrr 716	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑘 ∈ ℕ ∧ (𝑘 + 1) = 𝑃)) → (𝐹‘(𝑘 + 1)) ∈ ℕ)
62	54, 61	pccld 16177	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑘 ∈ ℕ ∧ (𝑘 + 1) = 𝑃)) → (𝑃 pCnt (𝐹‘(𝑘 + 1))) ∈ ℕ₀)
63	62	nn0cnd 11945	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑘 ∈ ℕ ∧ (𝑘 + 1) = 𝑃)) → (𝑃 pCnt (𝐹‘(𝑘 + 1))) ∈ ℂ)
64	63	addid2d 10830	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑘 ∈ ℕ ∧ (𝑘 + 1) = 𝑃)) → (0 + (𝑃 pCnt (𝐹‘(𝑘 + 1)))) = (𝑃 pCnt (𝐹‘(𝑘 + 1))))
65	58	ad2antrl 727	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑘 ∈ ℕ ∧ (𝑘 + 1) = 𝑃)) → (𝑘 + 1) ∈ ℕ)
66		ovex 7168	. . . . . . . . . . . . . . 15 ⊢ (𝑛↑𝐴) ∈ V
67	66, 36	ifex 4473	. . . . . . . . . . . . . 14 ⊢ if(𝑛 ∈ ℙ, (𝑛↑𝐴), 1) ∈ V
68	67	csbex 5179	. . . . . . . . . . . . 13 ⊢ ⦋(𝑘 + 1) / 𝑛⦌if(𝑛 ∈ ℙ, (𝑛↑𝐴), 1) ∈ V
69	35	fvmpts 6748	. . . . . . . . . . . . . 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 7168	. . . . . . . . . . . . . . 15 ⊢ (𝑘 + 1) ∈ V
71		nfv 1915	. . . . . . . . . . . . . . . 16 ⊢ Ⅎ𝑛(𝑘 + 1) ∈ ℙ
72		nfcv 2955	. . . . . . . . . . . . . . . . 17 ⊢ Ⅎ𝑛(𝑘 + 1)
73		nfcv 2955	. . . . . . . . . . . . . . . . 17 ⊢ Ⅎ𝑛↑
74		nfcsb1v 3852	. . . . . . . . . . . . . . . . 17 ⊢ Ⅎ𝑛⦋(𝑘 + 1) / 𝑛⦌𝐴
75	72, 73, 74	nfov 7165	. . . . . . . . . . . . . . . 16 ⊢ Ⅎ𝑛((𝑘 + 1)↑⦋(𝑘 + 1) / 𝑛⦌𝐴)
76		nfcv 2955	. . . . . . . . . . . . . . . 16 ⊢ Ⅎ𝑛1
77	71, 75, 76	nfif 4454	. . . . . . . . . . . . . . 15 ⊢ Ⅎ𝑛if((𝑘 + 1) ∈ ℙ, ((𝑘 + 1)↑⦋(𝑘 + 1) / 𝑛⦌𝐴), 1)
78		eleq1 2877	. . . . . . . . . . . . . . . 16 ⊢ (𝑛 = (𝑘 + 1) → (𝑛 ∈ ℙ ↔ (𝑘 + 1) ∈ ℙ))
79		id 22	. . . . . . . . . . . . . . . . 17 ⊢ (𝑛 = (𝑘 + 1) → 𝑛 = (𝑘 + 1))
80		csbeq1a 3842	. . . . . . . . . . . . . . . . 17 ⊢ (𝑛 = (𝑘 + 1) → 𝐴 = ⦋(𝑘 + 1) / 𝑛⦌𝐴)
81	79, 80	oveq12d 7153	. . . . . . . . . . . . . . . 16 ⊢ (𝑛 = (𝑘 + 1) → (𝑛↑𝐴) = ((𝑘 + 1)↑⦋(𝑘 + 1) / 𝑛⦌𝐴))
82	78, 81	ifbieq1d 4448	. . . . . . . . . . . . . . 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 3862	. . . . . . . . . . . . . 14 ⊢ ⦋(𝑘 + 1) / 𝑛⦌if(𝑛 ∈ ℙ, (𝑛↑𝐴), 1) = if((𝑘 + 1) ∈ ℙ, ((𝑘 + 1)↑⦋(𝑘 + 1) / 𝑛⦌𝐴), 1)
84	69, 83	eqtrdi 2849	. . . . . . . . . . . . 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 589	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑘 ∈ ℕ ∧ (𝑘 + 1) = 𝑃)) → (𝐹‘(𝑘 + 1)) = if((𝑘 + 1) ∈ ℙ, ((𝑘 + 1)↑⦋(𝑘 + 1) / 𝑛⦌𝐴), 1))
86		simprr 772	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑘 ∈ ℕ ∧ (𝑘 + 1) = 𝑃)) → (𝑘 + 1) = 𝑃)
87	86, 54	eqeltrd 2890	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑘 ∈ ℕ ∧ (𝑘 + 1) = 𝑃)) → (𝑘 + 1) ∈ ℙ)
88	87	iftrued 4433	. . . . . . . . . . . 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 3832	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑘 ∈ ℕ ∧ (𝑘 + 1) = 𝑃)) → ⦋(𝑘 + 1) / 𝑛⦌𝐴 = ⦋𝑃 / 𝑛⦌𝐴)
90		nfcvd 2956	. . . . . . . . . . . . . . . 16 ⊢ (𝑃 ∈ ℙ → Ⅎ𝑛𝐵)
91		pcmpt.5	. . . . . . . . . . . . . . . 16 ⊢ (𝑛 = 𝑃 → 𝐴 = 𝐵)
92	90, 91	csbiegf 3861	. . . . . . . . . . . . . . 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 2833	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑘 ∈ ℕ ∧ (𝑘 + 1) = 𝑃)) → ⦋(𝑘 + 1) / 𝑛⦌𝐴 = 𝐵)
95	86, 94	oveq12d 7153	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑘 ∈ ℕ ∧ (𝑘 + 1) = 𝑃)) → ((𝑘 + 1)↑⦋(𝑘 + 1) / 𝑛⦌𝐴) = (𝑃↑𝐵))
96	85, 88, 95	3eqtrd 2837	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑘 ∈ ℕ ∧ (𝑘 + 1) = 𝑃)) → (𝐹‘(𝑘 + 1)) = (𝑃↑𝐵))
97	96	oveq2d 7151	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑘 ∈ ℕ ∧ (𝑘 + 1) = 𝑃)) → (𝑃 pCnt (𝐹‘(𝑘 + 1))) = (𝑃 pCnt (𝑃↑𝐵)))
98	91	eleq1d 2874	. . . . . . . . . . . . . 14 ⊢ (𝑛 = 𝑃 → (𝐴 ∈ ℕ₀ ↔ 𝐵 ∈ ℕ₀))
99	98	rspcv 3566	. . . . . . . . . . . . 13 ⊢ (𝑃 ∈ ℙ → (∀𝑛 ∈ ℙ 𝐴 ∈ ℕ₀ → 𝐵 ∈ ℕ₀))
100	26, 55, 99	sylc 65	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐵 ∈ ℕ₀)
101	100	adantr 484	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑘 ∈ ℕ ∧ (𝑘 + 1) = 𝑃)) → 𝐵 ∈ ℕ₀)
102		pcidlem 16198	. . . . . . . . . . 11 ⊢ ((𝑃 ∈ ℙ ∧ 𝐵 ∈ ℕ₀) → (𝑃 pCnt (𝑃↑𝐵)) = 𝐵)
103	54, 101, 102	syl2anc 587	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑘 ∈ ℕ ∧ (𝑘 + 1) = 𝑃)) → (𝑃 pCnt (𝑃↑𝐵)) = 𝐵)
104	64, 97, 103	3eqtrd 2837	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑘 ∈ ℕ ∧ (𝑘 + 1) = 𝑃)) → (0 + (𝑃 pCnt (𝐹‘(𝑘 + 1)))) = 𝐵)
105		oveq1 7142	. . . . . . . . . 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 2800	. . . . . . . . 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 250	. . . . . . . 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 11632	. . . . . . . . . . . . 13 ⊢ (𝑘 ∈ ℕ → 𝑘 ∈ ℝ)
109	108	ad2antrl 727	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑘 ∈ ℕ ∧ (𝑘 + 1) = 𝑃)) → 𝑘 ∈ ℝ)
110		ltp1 11469	. . . . . . . . . . . . 13 ⊢ (𝑘 ∈ ℝ → 𝑘 < (𝑘 + 1))
111		peano2re 10802	. . . . . . . . . . . . . 14 ⊢ (𝑘 ∈ ℝ → (𝑘 + 1) ∈ ℝ)
112		ltnle 10709	. . . . . . . . . . . . . 14 ⊢ ((𝑘 ∈ ℝ ∧ (𝑘 + 1) ∈ ℝ) → (𝑘 < (𝑘 + 1) ↔ ¬ (𝑘 + 1) ≤ 𝑘))
113	111, 112	mpdan 686	. . . . . . . . . . . . 13 ⊢ (𝑘 ∈ ℝ → (𝑘 < (𝑘 + 1) ↔ ¬ (𝑘 + 1) ≤ 𝑘))
114	110, 113	mpbid 235	. . . . . . . . . . . 12 ⊢ (𝑘 ∈ ℝ → ¬ (𝑘 + 1) ≤ 𝑘)
115	109, 114	syl 17	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑘 ∈ ℕ ∧ (𝑘 + 1) = 𝑃)) → ¬ (𝑘 + 1) ≤ 𝑘)
116	86	breq1d 5040	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑘 ∈ ℕ ∧ (𝑘 + 1) = 𝑃)) → ((𝑘 + 1) ≤ 𝑘 ↔ 𝑃 ≤ 𝑘))
117	115, 116	mtbid 327	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑘 ∈ ℕ ∧ (𝑘 + 1) = 𝑃)) → ¬ 𝑃 ≤ 𝑘)
118	117	iffalsed 4436	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑘 ∈ ℕ ∧ (𝑘 + 1) = 𝑃)) → if(𝑃 ≤ 𝑘, 𝐵, 0) = 0)
119	118	eqeq2d 2809	. . . . . . . 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 488	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → 𝑘 ∈ ℕ)
121		nnuz 12269	. . . . . . . . . . . . . 14 ⊢ ℕ = (ℤ_≥‘1)
122	120, 121	eleqtrdi 2900	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → 𝑘 ∈ (ℤ_≥‘1))
123		seqp1 13379	. . . . . . . . . . . . 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 7151	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (𝑃 pCnt (seq1( · , 𝐹)‘(𝑘 + 1))) = (𝑃 pCnt ((seq1( · , 𝐹)‘𝑘) · (𝐹‘(𝑘 + 1)))))
126	26	adantr 484	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → 𝑃 ∈ ℙ)
127	56	simprd 499	. . . . . . . . . . . . . 14 ⊢ (𝜑 → seq1( · , 𝐹):ℕ⟶ℕ)
128	127	ffvelrnda 6828	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (seq1( · , 𝐹)‘𝑘) ∈ ℕ)
129		nnz 11992	. . . . . . . . . . . . . 14 ⊢ ((seq1( · , 𝐹)‘𝑘) ∈ ℕ → (seq1( · , 𝐹)‘𝑘) ∈ ℤ)
130		nnne0 11659	. . . . . . . . . . . . . 14 ⊢ ((seq1( · , 𝐹)‘𝑘) ∈ ℕ → (seq1( · , 𝐹)‘𝑘) ≠ 0)
131	129, 130	jca 515	. . . . . . . . . . . . 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 11992	. . . . . . . . . . . . . 14 ⊢ ((𝐹‘(𝑘 + 1)) ∈ ℕ → (𝐹‘(𝑘 + 1)) ∈ ℤ)
134		nnne0 11659	. . . . . . . . . . . . . 14 ⊢ ((𝐹‘(𝑘 + 1)) ∈ ℕ → (𝐹‘(𝑘 + 1)) ≠ 0)
135	133, 134	jca 515	. . . . . . . . . . . . 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 16178	. . . . . . . . . . . 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 1368	. . . . . . . . . . 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 2833	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (𝑃 pCnt (seq1( · , 𝐹)‘(𝑘 + 1))) = ((𝑃 pCnt (seq1( · , 𝐹)‘𝑘)) + (𝑃 pCnt (𝐹‘(𝑘 + 1)))))
140	139	adantrr 716	. . . . . . . . 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 16008	. . . . . . . . . . . . . . 15 ⊢ (𝑃 ∈ ℙ → 𝑃 ∈ ℕ)
142	26, 141	syl 17	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝑃 ∈ ℕ)
143	142	nnred 11640	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝑃 ∈ ℝ)
144	143	adantr 484	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑘 ∈ ℕ ∧ (𝑘 + 1) = 𝑃)) → 𝑃 ∈ ℝ)
145	144	leidd 11195	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑘 ∈ ℕ ∧ (𝑘 + 1) = 𝑃)) → 𝑃 ≤ 𝑃)
146	145, 86	breqtrrd 5058	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑘 ∈ ℕ ∧ (𝑘 + 1) = 𝑃)) → 𝑃 ≤ (𝑘 + 1))
147	146	iftrued 4433	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑘 ∈ ℕ ∧ (𝑘 + 1) = 𝑃)) → if(𝑃 ≤ (𝑘 + 1), 𝐵, 0) = 𝐵)
148	140, 147	eqeq12d 2814	. . . . . . . 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 297	. . . . . . 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 460	. . . . . 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 716	. . . . . . . . . 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 777	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ (𝑘 ∈ ℕ ∧ (𝑘 + 1) ≠ 𝑃)) ∧ (𝑘 + 1) ∈ ℙ) → (𝑘 + 1) ≠ 𝑃)
153	152	necomd 3042	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ (𝑘 ∈ ℕ ∧ (𝑘 + 1) ≠ 𝑃)) ∧ (𝑘 + 1) ∈ ℙ) → 𝑃 ≠ (𝑘 + 1))
154	26	ad2antrr 725	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ (𝑘 ∈ ℕ ∧ (𝑘 + 1) ≠ 𝑃)) ∧ (𝑘 + 1) ∈ ℙ) → 𝑃 ∈ ℙ)
155		simpr 488	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ (𝑘 ∈ ℕ ∧ (𝑘 + 1) ≠ 𝑃)) ∧ (𝑘 + 1) ∈ ℙ) → (𝑘 + 1) ∈ ℙ)
156	55	ad2antrr 725	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ (𝑘 ∈ ℕ ∧ (𝑘 + 1) ≠ 𝑃)) ∧ (𝑘 + 1) ∈ ℙ) → ∀𝑛 ∈ ℙ 𝐴 ∈ ℕ₀)
157	74	nfel1 2971	. . . . . . . . . . . . . . . . . . 19 ⊢ Ⅎ𝑛⦋(𝑘 + 1) / 𝑛⦌𝐴 ∈ ℕ₀
158	80	eleq1d 2874	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑛 = (𝑘 + 1) → (𝐴 ∈ ℕ₀ ↔ ⦋(𝑘 + 1) / 𝑛⦌𝐴 ∈ ℕ₀))
159	157, 158	rspc 3559	. . . . . . . . . . . . . . . . . 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 16051	. . . . . . . . . . . . . . . . 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 1368	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ (𝑘 ∈ ℕ ∧ (𝑘 + 1) ≠ 𝑃)) ∧ (𝑘 + 1) ∈ ℙ) → (𝑃 ∥ ((𝑘 + 1)↑⦋(𝑘 + 1) / 𝑛⦌𝐴) → 𝑃 = (𝑘 + 1)))
163	162	necon3ad 3000	. . . . . . . . . . . . . . 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 727	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ (𝑘 ∈ ℕ ∧ (𝑘 + 1) ≠ 𝑃)) → (𝑘 + 1) ∈ ℕ)
166	165, 68, 84	sylancl 589	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ (𝑘 ∈ ℕ ∧ (𝑘 + 1) ≠ 𝑃)) → (𝐹‘(𝑘 + 1)) = if((𝑘 + 1) ∈ ℙ, ((𝑘 + 1)↑⦋(𝑘 + 1) / 𝑛⦌𝐴), 1))
167		iftrue 4431	. . . . . . . . . . . . . . . 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 2853	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ (𝑘 ∈ ℕ ∧ (𝑘 + 1) ≠ 𝑃)) ∧ (𝑘 + 1) ∈ ℙ) → (𝐹‘(𝑘 + 1)) = ((𝑘 + 1)↑⦋(𝑘 + 1) / 𝑛⦌𝐴))
169	168	breq2d 5042	. . . . . . . . . . . . . 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 328	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝑘 ∈ ℕ ∧ (𝑘 + 1) ≠ 𝑃)) ∧ (𝑘 + 1) ∈ ℙ) → ¬ 𝑃 ∥ (𝐹‘(𝑘 + 1)))
171	57	adantr 484	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ (𝑘 ∈ ℕ ∧ (𝑘 + 1) ≠ 𝑃)) → 𝐹:ℕ⟶ℕ)
172	171, 165, 59	syl2anc 587	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (𝑘 ∈ ℕ ∧ (𝑘 + 1) ≠ 𝑃)) → (𝐹‘(𝑘 + 1)) ∈ ℕ)
173	172	adantr 484	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ (𝑘 ∈ ℕ ∧ (𝑘 + 1) ≠ 𝑃)) ∧ (𝑘 + 1) ∈ ℙ) → (𝐹‘(𝑘 + 1)) ∈ ℕ)
174		pceq0 16197	. . . . . . . . . . . . . 14 ⊢ ((𝑃 ∈ ℙ ∧ (𝐹‘(𝑘 + 1)) ∈ ℕ) → ((𝑃 pCnt (𝐹‘(𝑘 + 1))) = 0 ↔ ¬ 𝑃 ∥ (𝐹‘(𝑘 + 1))))
175	154, 173, 174	syl2anc 587	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝑘 ∈ ℕ ∧ (𝑘 + 1) ≠ 𝑃)) ∧ (𝑘 + 1) ∈ ℙ) → ((𝑃 pCnt (𝐹‘(𝑘 + 1))) = 0 ↔ ¬ 𝑃 ∥ (𝐹‘(𝑘 + 1))))
176	170, 175	mpbird 260	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝑘 ∈ ℕ ∧ (𝑘 + 1) ≠ 𝑃)) ∧ (𝑘 + 1) ∈ ℙ) → (𝑃 pCnt (𝐹‘(𝑘 + 1))) = 0)
177		iffalse 4434	. . . . . . . . . . . . . . 15 ⊢ (¬ (𝑘 + 1) ∈ ℙ → if((𝑘 + 1) ∈ ℙ, ((𝑘 + 1)↑⦋(𝑘 + 1) / 𝑛⦌𝐴), 1) = 1)
178	166, 177	sylan9eq 2853	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ (𝑘 ∈ ℕ ∧ (𝑘 + 1) ≠ 𝑃)) ∧ ¬ (𝑘 + 1) ∈ ℙ) → (𝐹‘(𝑘 + 1)) = 1)
179	178	oveq2d 7151	. . . . . . . . . . . . 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 725	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝑘 ∈ ℕ ∧ (𝑘 + 1) ≠ 𝑃)) ∧ ¬ (𝑘 + 1) ∈ ℙ) → (𝑃 pCnt 1) = 0)
182	179, 181	eqtrd 2833	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝑘 ∈ ℕ ∧ (𝑘 + 1) ≠ 𝑃)) ∧ ¬ (𝑘 + 1) ∈ ℙ) → (𝑃 pCnt (𝐹‘(𝑘 + 1))) = 0)
183	176, 182	pm2.61dan 812	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑘 ∈ ℕ ∧ (𝑘 + 1) ≠ 𝑃)) → (𝑃 pCnt (𝐹‘(𝑘 + 1))) = 0)
184	183	oveq2d 7151	. . . . . . . . . 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 484	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑘 ∈ ℕ ∧ (𝑘 + 1) ≠ 𝑃)) → 𝑃 ∈ ℙ)
186	128	adantrr 716	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑘 ∈ ℕ ∧ (𝑘 + 1) ≠ 𝑃)) → (seq1( · , 𝐹)‘𝑘) ∈ ℕ)
187	185, 186	pccld 16177	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑘 ∈ ℕ ∧ (𝑘 + 1) ≠ 𝑃)) → (𝑃 pCnt (seq1( · , 𝐹)‘𝑘)) ∈ ℕ₀)
188	187	nn0cnd 11945	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑘 ∈ ℕ ∧ (𝑘 + 1) ≠ 𝑃)) → (𝑃 pCnt (seq1( · , 𝐹)‘𝑘)) ∈ ℂ)
189	188	addid1d 10829	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑘 ∈ ℕ ∧ (𝑘 + 1) ≠ 𝑃)) → ((𝑃 pCnt (seq1( · , 𝐹)‘𝑘)) + 0) = (𝑃 pCnt (seq1( · , 𝐹)‘𝑘)))
190	151, 184, 189	3eqtrd 2837	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑘 ∈ ℕ ∧ (𝑘 + 1) ≠ 𝑃)) → (𝑃 pCnt (seq1( · , 𝐹)‘(𝑘 + 1))) = (𝑃 pCnt (seq1( · , 𝐹)‘𝑘)))
191	142	adantr 484	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑘 ∈ ℕ ∧ (𝑘 + 1) ≠ 𝑃)) → 𝑃 ∈ ℕ)
192	191	nnred 11640	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑘 ∈ ℕ ∧ (𝑘 + 1) ≠ 𝑃)) → 𝑃 ∈ ℝ)
193	165	nnred 11640	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑘 ∈ ℕ ∧ (𝑘 + 1) ≠ 𝑃)) → (𝑘 + 1) ∈ ℝ)
194	192, 193	ltlend 10774	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑘 ∈ ℕ ∧ (𝑘 + 1) ≠ 𝑃)) → (𝑃 < (𝑘 + 1) ↔ (𝑃 ≤ (𝑘 + 1) ∧ (𝑘 + 1) ≠ 𝑃)))
195		simprl 770	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑘 ∈ ℕ ∧ (𝑘 + 1) ≠ 𝑃)) → 𝑘 ∈ ℕ)
196		nnleltp1 12025	. . . . . . . . . . . 12 ⊢ ((𝑃 ∈ ℕ ∧ 𝑘 ∈ ℕ) → (𝑃 ≤ 𝑘 ↔ 𝑃 < (𝑘 + 1)))
197	191, 195, 196	syl2anc 587	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑘 ∈ ℕ ∧ (𝑘 + 1) ≠ 𝑃)) → (𝑃 ≤ 𝑘 ↔ 𝑃 < (𝑘 + 1)))
198		simprr 772	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑘 ∈ ℕ ∧ (𝑘 + 1) ≠ 𝑃)) → (𝑘 + 1) ≠ 𝑃)
199	198	biantrud 535	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑘 ∈ ℕ ∧ (𝑘 + 1) ≠ 𝑃)) → (𝑃 ≤ (𝑘 + 1) ↔ (𝑃 ≤ (𝑘 + 1) ∧ (𝑘 + 1) ≠ 𝑃)))
200	194, 197, 199	3bitr4rd 315	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑘 ∈ ℕ ∧ (𝑘 + 1) ≠ 𝑃)) → (𝑃 ≤ (𝑘 + 1) ↔ 𝑃 ≤ 𝑘))
201	200	ifbid 4447	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑘 ∈ ℕ ∧ (𝑘 + 1) ≠ 𝑃)) → if(𝑃 ≤ (𝑘 + 1), 𝐵, 0) = if(𝑃 ≤ 𝑘, 𝐵, 0))
202	190, 201	eqeq12d 2814	. . . . . . . 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 251	. . . . . . 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 460	. . . . . 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 3073	. . . . 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 417	. . . 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 11643	. 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 209 ∧ wa 399 = wceq 1538 ∈ wcel 2111 ≠ wne 2987 ∀wral 3106 Vcvv 3441 ⦋csb 3828 ifcif 4425 class class class wbr 5030 ↦ cmpt 5110 ⟶wf 6320 ‘cfv 6324 (class class class)co 7135 ℝcr 10525 0cc0 10526 1c1 10527 + caddc 10529 · cmul 10531 < clt 10664 ≤ cle 10665 ℕcn 11625 2c2 11680 ℕ₀cn0 11885 ℤcz 11969 ℤ_≥cuz 12231 seqcseq 13364 ↑cexp 13425 ∥ cdvds 15599 ℙcprime 16005 pCnt cpc 16163

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 ax-cnex 10582 ax-resscn 10583 ax-1cn 10584 ax-icn 10585 ax-addcl 10586 ax-addrcl 10587 ax-mulcl 10588 ax-mulrcl 10589 ax-mulcom 10590 ax-addass 10591 ax-mulass 10592 ax-distr 10593 ax-i2m1 10594 ax-1ne0 10595 ax-1rid 10596 ax-rnegex 10597 ax-rrecex 10598 ax-cnre 10599 ax-pre-lttri 10600 ax-pre-lttrn 10601 ax-pre-ltadd 10602 ax-pre-mulgt0 10603 ax-pre-sup 10604

This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-fal 1551 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-nel 3092 df-ral 3111 df-rex 3112 df-reu 3113 df-rmo 3114 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-pss 3900 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-tp 4530 df-op 4532 df-uni 4801 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-tr 5137 df-id 5425 df-eprel 5430 df-po 5438 df-so 5439 df-fr 5478 df-we 5480 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-pred 6116 df-ord 6162 df-on 6163 df-lim 6164 df-suc 6165 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-riota 7093 df-ov 7138 df-oprab 7139 df-mpo 7140 df-om 7561 df-1st 7671 df-2nd 7672 df-wrecs 7930 df-recs 7991 df-rdg 8029 df-1o 8085 df-2o 8086 df-er 8272 df-en 8493 df-dom 8494 df-sdom 8495 df-fin 8496 df-sup 8890 df-inf 8891 df-pnf 10666 df-mnf 10667 df-xr 10668 df-ltxr 10669 df-le 10670 df-sub 10861 df-neg 10862 df-div 11287 df-nn 11626 df-2 11688 df-3 11689 df-n0 11886 df-z 11970 df-uz 12232 df-q 12337 df-rp 12378 df-fz 12886 df-fl 13157 df-mod 13233 df-seq 13365 df-exp 13426 df-cj 14450 df-re 14451 df-im 14452 df-sqrt 14586 df-abs 14587 df-dvds 15600 df-gcd 15834 df-prm 16006 df-pc 16164

This theorem is referenced by: pcmpt2 16219 pcprod 16221 1arithlem4 16252 chtublem 25795 bposlem3 25870

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