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Theorem pcmpt 15833
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, (𝑛𝐴), 1))
pcmpt.2 (𝜑 → ∀𝑛 ∈ ℙ 𝐴 ∈ ℕ0)
pcmpt.3 (𝜑𝑁 ∈ ℕ)
pcmpt.4 (𝜑𝑃 ∈ ℙ)
pcmpt.5 (𝑛 = 𝑃𝐴 = 𝐵)
Assertion
Ref Expression
pcmpt (𝜑 → (𝑃 pCnt (seq1( · , 𝐹)‘𝑁)) = if(𝑃𝑁, 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.
StepHypRef Expression
1 pcmpt.3 . 2 (𝜑𝑁 ∈ ℕ)
2 fveq2 6418 . . . . . 6 (𝑝 = 1 → (seq1( · , 𝐹)‘𝑝) = (seq1( · , 𝐹)‘1))
32oveq2d 6900 . . . . 5 (𝑝 = 1 → (𝑃 pCnt (seq1( · , 𝐹)‘𝑝)) = (𝑃 pCnt (seq1( · , 𝐹)‘1)))
4 breq2 4859 . . . . . 6 (𝑝 = 1 → (𝑃𝑝𝑃 ≤ 1))
54ifbid 4312 . . . . 5 (𝑝 = 1 → if(𝑃𝑝, 𝐵, 0) = if(𝑃 ≤ 1, 𝐵, 0))
63, 5eqeq12d 2832 . . . 4 (𝑝 = 1 → ((𝑃 pCnt (seq1( · , 𝐹)‘𝑝)) = if(𝑃𝑝, 𝐵, 0) ↔ (𝑃 pCnt (seq1( · , 𝐹)‘1)) = if(𝑃 ≤ 1, 𝐵, 0)))
76imbi2d 331 . . 3 (𝑝 = 1 → ((𝜑 → (𝑃 pCnt (seq1( · , 𝐹)‘𝑝)) = if(𝑃𝑝, 𝐵, 0)) ↔ (𝜑 → (𝑃 pCnt (seq1( · , 𝐹)‘1)) = if(𝑃 ≤ 1, 𝐵, 0))))
8 fveq2 6418 . . . . . 6 (𝑝 = 𝑘 → (seq1( · , 𝐹)‘𝑝) = (seq1( · , 𝐹)‘𝑘))
98oveq2d 6900 . . . . 5 (𝑝 = 𝑘 → (𝑃 pCnt (seq1( · , 𝐹)‘𝑝)) = (𝑃 pCnt (seq1( · , 𝐹)‘𝑘)))
10 breq2 4859 . . . . . 6 (𝑝 = 𝑘 → (𝑃𝑝𝑃𝑘))
1110ifbid 4312 . . . . 5 (𝑝 = 𝑘 → if(𝑃𝑝, 𝐵, 0) = if(𝑃𝑘, 𝐵, 0))
129, 11eqeq12d 2832 . . . 4 (𝑝 = 𝑘 → ((𝑃 pCnt (seq1( · , 𝐹)‘𝑝)) = if(𝑃𝑝, 𝐵, 0) ↔ (𝑃 pCnt (seq1( · , 𝐹)‘𝑘)) = if(𝑃𝑘, 𝐵, 0)))
1312imbi2d 331 . . 3 (𝑝 = 𝑘 → ((𝜑 → (𝑃 pCnt (seq1( · , 𝐹)‘𝑝)) = if(𝑃𝑝, 𝐵, 0)) ↔ (𝜑 → (𝑃 pCnt (seq1( · , 𝐹)‘𝑘)) = if(𝑃𝑘, 𝐵, 0))))
14 fveq2 6418 . . . . . 6 (𝑝 = (𝑘 + 1) → (seq1( · , 𝐹)‘𝑝) = (seq1( · , 𝐹)‘(𝑘 + 1)))
1514oveq2d 6900 . . . . 5 (𝑝 = (𝑘 + 1) → (𝑃 pCnt (seq1( · , 𝐹)‘𝑝)) = (𝑃 pCnt (seq1( · , 𝐹)‘(𝑘 + 1))))
16 breq2 4859 . . . . . 6 (𝑝 = (𝑘 + 1) → (𝑃𝑝𝑃 ≤ (𝑘 + 1)))
1716ifbid 4312 . . . . 5 (𝑝 = (𝑘 + 1) → if(𝑃𝑝, 𝐵, 0) = if(𝑃 ≤ (𝑘 + 1), 𝐵, 0))
1815, 17eqeq12d 2832 . . . 4 (𝑝 = (𝑘 + 1) → ((𝑃 pCnt (seq1( · , 𝐹)‘𝑝)) = if(𝑃𝑝, 𝐵, 0) ↔ (𝑃 pCnt (seq1( · , 𝐹)‘(𝑘 + 1))) = if(𝑃 ≤ (𝑘 + 1), 𝐵, 0)))
1918imbi2d 331 . . 3 (𝑝 = (𝑘 + 1) → ((𝜑 → (𝑃 pCnt (seq1( · , 𝐹)‘𝑝)) = if(𝑃𝑝, 𝐵, 0)) ↔ (𝜑 → (𝑃 pCnt (seq1( · , 𝐹)‘(𝑘 + 1))) = if(𝑃 ≤ (𝑘 + 1), 𝐵, 0))))
20 fveq2 6418 . . . . . 6 (𝑝 = 𝑁 → (seq1( · , 𝐹)‘𝑝) = (seq1( · , 𝐹)‘𝑁))
2120oveq2d 6900 . . . . 5 (𝑝 = 𝑁 → (𝑃 pCnt (seq1( · , 𝐹)‘𝑝)) = (𝑃 pCnt (seq1( · , 𝐹)‘𝑁)))
22 breq2 4859 . . . . . 6 (𝑝 = 𝑁 → (𝑃𝑝𝑃𝑁))
2322ifbid 4312 . . . . 5 (𝑝 = 𝑁 → if(𝑃𝑝, 𝐵, 0) = if(𝑃𝑁, 𝐵, 0))
2421, 23eqeq12d 2832 . . . 4 (𝑝 = 𝑁 → ((𝑃 pCnt (seq1( · , 𝐹)‘𝑝)) = if(𝑃𝑝, 𝐵, 0) ↔ (𝑃 pCnt (seq1( · , 𝐹)‘𝑁)) = if(𝑃𝑁, 𝐵, 0)))
2524imbi2d 331 . . 3 (𝑝 = 𝑁 → ((𝜑 → (𝑃 pCnt (seq1( · , 𝐹)‘𝑝)) = if(𝑃𝑝, 𝐵, 0)) ↔ (𝜑 → (𝑃 pCnt (seq1( · , 𝐹)‘𝑁)) = if(𝑃𝑁, 𝐵, 0))))
26 pcmpt.4 . . . 4 (𝜑𝑃 ∈ ℙ)
27 1z 11693 . . . . . . . . 9 1 ∈ ℤ
28 seq1 13057 . . . . . . . . 9 (1 ∈ ℤ → (seq1( · , 𝐹)‘1) = (𝐹‘1))
2927, 28ax-mp 5 . . . . . . . 8 (seq1( · , 𝐹)‘1) = (𝐹‘1)
30 1nn 11328 . . . . . . . . 9 1 ∈ ℕ
31 1nprm 15630 . . . . . . . . . . . 12 ¬ 1 ∈ ℙ
32 eleq1 2884 . . . . . . . . . . . 12 (𝑛 = 1 → (𝑛 ∈ ℙ ↔ 1 ∈ ℙ))
3331, 32mtbiri 318 . . . . . . . . . . 11 (𝑛 = 1 → ¬ 𝑛 ∈ ℙ)
3433iffalsed 4301 . . . . . . . . . 10 (𝑛 = 1 → if(𝑛 ∈ ℙ, (𝑛𝐴), 1) = 1)
35 pcmpt.1 . . . . . . . . . 10 𝐹 = (𝑛 ∈ ℕ ↦ if(𝑛 ∈ ℙ, (𝑛𝐴), 1))
36 1ex 10331 . . . . . . . . . 10 1 ∈ V
3734, 35, 36fvmpt 6513 . . . . . . . . 9 (1 ∈ ℕ → (𝐹‘1) = 1)
3830, 37ax-mp 5 . . . . . . . 8 (𝐹‘1) = 1
3929, 38eqtri 2839 . . . . . . 7 (seq1( · , 𝐹)‘1) = 1
4039oveq2i 6895 . . . . . 6 (𝑃 pCnt (seq1( · , 𝐹)‘1)) = (𝑃 pCnt 1)
41 pc1 15797 . . . . . 6 (𝑃 ∈ ℙ → (𝑃 pCnt 1) = 0)
4240, 41syl5eq 2863 . . . . 5 (𝑃 ∈ ℙ → (𝑃 pCnt (seq1( · , 𝐹)‘1)) = 0)
43 prmgt1 15647 . . . . . . 7 (𝑃 ∈ ℙ → 1 < 𝑃)
44 1re 10335 . . . . . . . 8 1 ∈ ℝ
45 prmuz2 15646 . . . . . . . . 9 (𝑃 ∈ ℙ → 𝑃 ∈ (ℤ‘2))
46 eluzelre 11935 . . . . . . . . 9 (𝑃 ∈ (ℤ‘2) → 𝑃 ∈ ℝ)
4745, 46syl 17 . . . . . . . 8 (𝑃 ∈ ℙ → 𝑃 ∈ ℝ)
48 ltnle 10412 . . . . . . . 8 ((1 ∈ ℝ ∧ 𝑃 ∈ ℝ) → (1 < 𝑃 ↔ ¬ 𝑃 ≤ 1))
4944, 47, 48sylancr 577 . . . . . . 7 (𝑃 ∈ ℙ → (1 < 𝑃 ↔ ¬ 𝑃 ≤ 1))
5043, 49mpbid 223 . . . . . 6 (𝑃 ∈ ℙ → ¬ 𝑃 ≤ 1)
5150iffalsed 4301 . . . . 5 (𝑃 ∈ ℙ → if(𝑃 ≤ 1, 𝐵, 0) = 0)
5242, 51eqtr4d 2854 . . . 4 (𝑃 ∈ ℙ → (𝑃 pCnt (seq1( · , 𝐹)‘1)) = if(𝑃 ≤ 1, 𝐵, 0))
5326, 52syl 17 . . 3 (𝜑 → (𝑃 pCnt (seq1( · , 𝐹)‘1)) = if(𝑃 ≤ 1, 𝐵, 0))
5426adantr 468 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑘 ∈ ℕ ∧ (𝑘 + 1) = 𝑃)) → 𝑃 ∈ ℙ)
55 pcmpt.2 . . . . . . . . . . . . . . . . 17 (𝜑 → ∀𝑛 ∈ ℙ 𝐴 ∈ ℕ0)
5635, 55pcmptcl 15832 . . . . . . . . . . . . . . . 16 (𝜑 → (𝐹:ℕ⟶ℕ ∧ seq1( · , 𝐹):ℕ⟶ℕ))
5756simpld 484 . . . . . . . . . . . . . . 15 (𝜑𝐹:ℕ⟶ℕ)
58 peano2nn 11329 . . . . . . . . . . . . . . 15 (𝑘 ∈ ℕ → (𝑘 + 1) ∈ ℕ)
59 ffvelrn 6589 . . . . . . . . . . . . . . 15 ((𝐹:ℕ⟶ℕ ∧ (𝑘 + 1) ∈ ℕ) → (𝐹‘(𝑘 + 1)) ∈ ℕ)
6057, 58, 59syl2an 585 . . . . . . . . . . . . . 14 ((𝜑𝑘 ∈ ℕ) → (𝐹‘(𝑘 + 1)) ∈ ℕ)
6160adantrr 699 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑘 ∈ ℕ ∧ (𝑘 + 1) = 𝑃)) → (𝐹‘(𝑘 + 1)) ∈ ℕ)
6254, 61pccld 15792 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑘 ∈ ℕ ∧ (𝑘 + 1) = 𝑃)) → (𝑃 pCnt (𝐹‘(𝑘 + 1))) ∈ ℕ0)
6362nn0cnd 11639 . . . . . . . . . . 11 ((𝜑 ∧ (𝑘 ∈ ℕ ∧ (𝑘 + 1) = 𝑃)) → (𝑃 pCnt (𝐹‘(𝑘 + 1))) ∈ ℂ)
6463addid2d 10532 . . . . . . . . . 10 ((𝜑 ∧ (𝑘 ∈ ℕ ∧ (𝑘 + 1) = 𝑃)) → (0 + (𝑃 pCnt (𝐹‘(𝑘 + 1)))) = (𝑃 pCnt (𝐹‘(𝑘 + 1))))
6558ad2antrl 710 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑘 ∈ ℕ ∧ (𝑘 + 1) = 𝑃)) → (𝑘 + 1) ∈ ℕ)
66 ovex 6916 . . . . . . . . . . . . . . 15 (𝑛𝐴) ∈ V
6766, 36ifex 4338 . . . . . . . . . . . . . 14 if(𝑛 ∈ ℙ, (𝑛𝐴), 1) ∈ V
6867csbex 5001 . . . . . . . . . . . . 13 (𝑘 + 1) / 𝑛if(𝑛 ∈ ℙ, (𝑛𝐴), 1) ∈ V
6935fvmpts 6516 . . . . . . . . . . . . . 14 (((𝑘 + 1) ∈ ℕ ∧ (𝑘 + 1) / 𝑛if(𝑛 ∈ ℙ, (𝑛𝐴), 1) ∈ V) → (𝐹‘(𝑘 + 1)) = (𝑘 + 1) / 𝑛if(𝑛 ∈ ℙ, (𝑛𝐴), 1))
70 ovex 6916 . . . . . . . . . . . . . . 15 (𝑘 + 1) ∈ V
71 nfv 2005 . . . . . . . . . . . . . . . 16 𝑛(𝑘 + 1) ∈ ℙ
72 nfcv 2959 . . . . . . . . . . . . . . . . 17 𝑛(𝑘 + 1)
73 nfcv 2959 . . . . . . . . . . . . . . . . 17 𝑛
74 nfcsb1v 3755 . . . . . . . . . . . . . . . . 17 𝑛(𝑘 + 1) / 𝑛𝐴
7572, 73, 74nfov 6914 . . . . . . . . . . . . . . . 16 𝑛((𝑘 + 1)↑(𝑘 + 1) / 𝑛𝐴)
76 nfcv 2959 . . . . . . . . . . . . . . . 16 𝑛1
7771, 75, 76nfif 4319 . . . . . . . . . . . . . . 15 𝑛if((𝑘 + 1) ∈ ℙ, ((𝑘 + 1)↑(𝑘 + 1) / 𝑛𝐴), 1)
78 eleq1 2884 . . . . . . . . . . . . . . . 16 (𝑛 = (𝑘 + 1) → (𝑛 ∈ ℙ ↔ (𝑘 + 1) ∈ ℙ))
79 id 22 . . . . . . . . . . . . . . . . 17 (𝑛 = (𝑘 + 1) → 𝑛 = (𝑘 + 1))
80 csbeq1a 3748 . . . . . . . . . . . . . . . . 17 (𝑛 = (𝑘 + 1) → 𝐴 = (𝑘 + 1) / 𝑛𝐴)
8179, 80oveq12d 6902 . . . . . . . . . . . . . . . 16 (𝑛 = (𝑘 + 1) → (𝑛𝐴) = ((𝑘 + 1)↑(𝑘 + 1) / 𝑛𝐴))
8278, 81ifbieq1d 4313 . . . . . . . . . . . . . . 15 (𝑛 = (𝑘 + 1) → if(𝑛 ∈ ℙ, (𝑛𝐴), 1) = if((𝑘 + 1) ∈ ℙ, ((𝑘 + 1)↑(𝑘 + 1) / 𝑛𝐴), 1))
8370, 77, 82csbief 3764 . . . . . . . . . . . . . 14 (𝑘 + 1) / 𝑛if(𝑛 ∈ ℙ, (𝑛𝐴), 1) = if((𝑘 + 1) ∈ ℙ, ((𝑘 + 1)↑(𝑘 + 1) / 𝑛𝐴), 1)
8469, 83syl6eq 2867 . . . . . . . . . . . . 13 (((𝑘 + 1) ∈ ℕ ∧ (𝑘 + 1) / 𝑛if(𝑛 ∈ ℙ, (𝑛𝐴), 1) ∈ V) → (𝐹‘(𝑘 + 1)) = if((𝑘 + 1) ∈ ℙ, ((𝑘 + 1)↑(𝑘 + 1) / 𝑛𝐴), 1))
8565, 68, 84sylancl 576 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑘 ∈ ℕ ∧ (𝑘 + 1) = 𝑃)) → (𝐹‘(𝑘 + 1)) = if((𝑘 + 1) ∈ ℙ, ((𝑘 + 1)↑(𝑘 + 1) / 𝑛𝐴), 1))
86 simprr 780 . . . . . . . . . . . . . 14 ((𝜑 ∧ (𝑘 ∈ ℕ ∧ (𝑘 + 1) = 𝑃)) → (𝑘 + 1) = 𝑃)
8786, 54eqeltrd 2896 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑘 ∈ ℕ ∧ (𝑘 + 1) = 𝑃)) → (𝑘 + 1) ∈ ℙ)
8887iftrued 4298 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑘 ∈ ℕ ∧ (𝑘 + 1) = 𝑃)) → if((𝑘 + 1) ∈ ℙ, ((𝑘 + 1)↑(𝑘 + 1) / 𝑛𝐴), 1) = ((𝑘 + 1)↑(𝑘 + 1) / 𝑛𝐴))
8986csbeq1d 3746 . . . . . . . . . . . . . 14 ((𝜑 ∧ (𝑘 ∈ ℕ ∧ (𝑘 + 1) = 𝑃)) → (𝑘 + 1) / 𝑛𝐴 = 𝑃 / 𝑛𝐴)
90 nfcvd 2960 . . . . . . . . . . . . . . . 16 (𝑃 ∈ ℙ → 𝑛𝐵)
91 pcmpt.5 . . . . . . . . . . . . . . . 16 (𝑛 = 𝑃𝐴 = 𝐵)
9290, 91csbiegf 3763 . . . . . . . . . . . . . . 15 (𝑃 ∈ ℙ → 𝑃 / 𝑛𝐴 = 𝐵)
9354, 92syl 17 . . . . . . . . . . . . . 14 ((𝜑 ∧ (𝑘 ∈ ℕ ∧ (𝑘 + 1) = 𝑃)) → 𝑃 / 𝑛𝐴 = 𝐵)
9489, 93eqtrd 2851 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑘 ∈ ℕ ∧ (𝑘 + 1) = 𝑃)) → (𝑘 + 1) / 𝑛𝐴 = 𝐵)
9586, 94oveq12d 6902 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑘 ∈ ℕ ∧ (𝑘 + 1) = 𝑃)) → ((𝑘 + 1)↑(𝑘 + 1) / 𝑛𝐴) = (𝑃𝐵))
9685, 88, 953eqtrd 2855 . . . . . . . . . . 11 ((𝜑 ∧ (𝑘 ∈ ℕ ∧ (𝑘 + 1) = 𝑃)) → (𝐹‘(𝑘 + 1)) = (𝑃𝐵))
9796oveq2d 6900 . . . . . . . . . 10 ((𝜑 ∧ (𝑘 ∈ ℕ ∧ (𝑘 + 1) = 𝑃)) → (𝑃 pCnt (𝐹‘(𝑘 + 1))) = (𝑃 pCnt (𝑃𝐵)))
9891eleq1d 2881 . . . . . . . . . . . . . 14 (𝑛 = 𝑃 → (𝐴 ∈ ℕ0𝐵 ∈ ℕ0))
9998rspcv 3509 . . . . . . . . . . . . 13 (𝑃 ∈ ℙ → (∀𝑛 ∈ ℙ 𝐴 ∈ ℕ0𝐵 ∈ ℕ0))
10026, 55, 99sylc 65 . . . . . . . . . . . 12 (𝜑𝐵 ∈ ℕ0)
101100adantr 468 . . . . . . . . . . 11 ((𝜑 ∧ (𝑘 ∈ ℕ ∧ (𝑘 + 1) = 𝑃)) → 𝐵 ∈ ℕ0)
102 pcidlem 15813 . . . . . . . . . . 11 ((𝑃 ∈ ℙ ∧ 𝐵 ∈ ℕ0) → (𝑃 pCnt (𝑃𝐵)) = 𝐵)
10354, 101, 102syl2anc 575 . . . . . . . . . 10 ((𝜑 ∧ (𝑘 ∈ ℕ ∧ (𝑘 + 1) = 𝑃)) → (𝑃 pCnt (𝑃𝐵)) = 𝐵)
10464, 97, 1033eqtrd 2855 . . . . . . . . 9 ((𝜑 ∧ (𝑘 ∈ ℕ ∧ (𝑘 + 1) = 𝑃)) → (0 + (𝑃 pCnt (𝐹‘(𝑘 + 1)))) = 𝐵)
105 oveq1 6891 . . . . . . . . . 10 ((𝑃 pCnt (seq1( · , 𝐹)‘𝑘)) = 0 → ((𝑃 pCnt (seq1( · , 𝐹)‘𝑘)) + (𝑃 pCnt (𝐹‘(𝑘 + 1)))) = (0 + (𝑃 pCnt (𝐹‘(𝑘 + 1)))))
106105eqeq1d 2819 . . . . . . . . 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))
107104, 106syl5ibrcom 238 . . . . . . . 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 11323 . . . . . . . . . . . . 13 (𝑘 ∈ ℕ → 𝑘 ∈ ℝ)
109108ad2antrl 710 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑘 ∈ ℕ ∧ (𝑘 + 1) = 𝑃)) → 𝑘 ∈ ℝ)
110 ltp1 11156 . . . . . . . . . . . . 13 (𝑘 ∈ ℝ → 𝑘 < (𝑘 + 1))
111 peano2re 10504 . . . . . . . . . . . . . 14 (𝑘 ∈ ℝ → (𝑘 + 1) ∈ ℝ)
112 ltnle 10412 . . . . . . . . . . . . . 14 ((𝑘 ∈ ℝ ∧ (𝑘 + 1) ∈ ℝ) → (𝑘 < (𝑘 + 1) ↔ ¬ (𝑘 + 1) ≤ 𝑘))
113111, 112mpdan 670 . . . . . . . . . . . . 13 (𝑘 ∈ ℝ → (𝑘 < (𝑘 + 1) ↔ ¬ (𝑘 + 1) ≤ 𝑘))
114110, 113mpbid 223 . . . . . . . . . . . 12 (𝑘 ∈ ℝ → ¬ (𝑘 + 1) ≤ 𝑘)
115109, 114syl 17 . . . . . . . . . . 11 ((𝜑 ∧ (𝑘 ∈ ℕ ∧ (𝑘 + 1) = 𝑃)) → ¬ (𝑘 + 1) ≤ 𝑘)
11686breq1d 4865 . . . . . . . . . . 11 ((𝜑 ∧ (𝑘 ∈ ℕ ∧ (𝑘 + 1) = 𝑃)) → ((𝑘 + 1) ≤ 𝑘𝑃𝑘))
117115, 116mtbid 315 . . . . . . . . . 10 ((𝜑 ∧ (𝑘 ∈ ℕ ∧ (𝑘 + 1) = 𝑃)) → ¬ 𝑃𝑘)
118117iffalsed 4301 . . . . . . . . 9 ((𝜑 ∧ (𝑘 ∈ ℕ ∧ (𝑘 + 1) = 𝑃)) → if(𝑃𝑘, 𝐵, 0) = 0)
119118eqeq2d 2827 . . . . . . . 8 ((𝜑 ∧ (𝑘 ∈ ℕ ∧ (𝑘 + 1) = 𝑃)) → ((𝑃 pCnt (seq1( · , 𝐹)‘𝑘)) = if(𝑃𝑘, 𝐵, 0) ↔ (𝑃 pCnt (seq1( · , 𝐹)‘𝑘)) = 0))
120 simpr 473 . . . . . . . . . . . . . 14 ((𝜑𝑘 ∈ ℕ) → 𝑘 ∈ ℕ)
121 nnuz 11961 . . . . . . . . . . . . . 14 ℕ = (ℤ‘1)
122120, 121syl6eleq 2906 . . . . . . . . . . . . 13 ((𝜑𝑘 ∈ ℕ) → 𝑘 ∈ (ℤ‘1))
123 seqp1 13059 . . . . . . . . . . . . 13 (𝑘 ∈ (ℤ‘1) → (seq1( · , 𝐹)‘(𝑘 + 1)) = ((seq1( · , 𝐹)‘𝑘) · (𝐹‘(𝑘 + 1))))
124122, 123syl 17 . . . . . . . . . . . 12 ((𝜑𝑘 ∈ ℕ) → (seq1( · , 𝐹)‘(𝑘 + 1)) = ((seq1( · , 𝐹)‘𝑘) · (𝐹‘(𝑘 + 1))))
125124oveq2d 6900 . . . . . . . . . . 11 ((𝜑𝑘 ∈ ℕ) → (𝑃 pCnt (seq1( · , 𝐹)‘(𝑘 + 1))) = (𝑃 pCnt ((seq1( · , 𝐹)‘𝑘) · (𝐹‘(𝑘 + 1)))))
12626adantr 468 . . . . . . . . . . . 12 ((𝜑𝑘 ∈ ℕ) → 𝑃 ∈ ℙ)
12756simprd 485 . . . . . . . . . . . . . 14 (𝜑 → seq1( · , 𝐹):ℕ⟶ℕ)
128127ffvelrnda 6591 . . . . . . . . . . . . 13 ((𝜑𝑘 ∈ ℕ) → (seq1( · , 𝐹)‘𝑘) ∈ ℕ)
129 nnz 11685 . . . . . . . . . . . . . 14 ((seq1( · , 𝐹)‘𝑘) ∈ ℕ → (seq1( · , 𝐹)‘𝑘) ∈ ℤ)
130 nnne0 11351 . . . . . . . . . . . . . 14 ((seq1( · , 𝐹)‘𝑘) ∈ ℕ → (seq1( · , 𝐹)‘𝑘) ≠ 0)
131129, 130jca 503 . . . . . . . . . . . . 13 ((seq1( · , 𝐹)‘𝑘) ∈ ℕ → ((seq1( · , 𝐹)‘𝑘) ∈ ℤ ∧ (seq1( · , 𝐹)‘𝑘) ≠ 0))
132128, 131syl 17 . . . . . . . . . . . 12 ((𝜑𝑘 ∈ ℕ) → ((seq1( · , 𝐹)‘𝑘) ∈ ℤ ∧ (seq1( · , 𝐹)‘𝑘) ≠ 0))
133 nnz 11685 . . . . . . . . . . . . . 14 ((𝐹‘(𝑘 + 1)) ∈ ℕ → (𝐹‘(𝑘 + 1)) ∈ ℤ)
134 nnne0 11351 . . . . . . . . . . . . . 14 ((𝐹‘(𝑘 + 1)) ∈ ℕ → (𝐹‘(𝑘 + 1)) ≠ 0)
135133, 134jca 503 . . . . . . . . . . . . 13 ((𝐹‘(𝑘 + 1)) ∈ ℕ → ((𝐹‘(𝑘 + 1)) ∈ ℤ ∧ (𝐹‘(𝑘 + 1)) ≠ 0))
13660, 135syl 17 . . . . . . . . . . . 12 ((𝜑𝑘 ∈ ℕ) → ((𝐹‘(𝑘 + 1)) ∈ ℤ ∧ (𝐹‘(𝑘 + 1)) ≠ 0))
137 pcmul 15793 . . . . . . . . . . . 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)))))
138126, 132, 136, 137syl3anc 1483 . . . . . . . . . . 11 ((𝜑𝑘 ∈ ℕ) → (𝑃 pCnt ((seq1( · , 𝐹)‘𝑘) · (𝐹‘(𝑘 + 1)))) = ((𝑃 pCnt (seq1( · , 𝐹)‘𝑘)) + (𝑃 pCnt (𝐹‘(𝑘 + 1)))))
139125, 138eqtrd 2851 . . . . . . . . . 10 ((𝜑𝑘 ∈ ℕ) → (𝑃 pCnt (seq1( · , 𝐹)‘(𝑘 + 1))) = ((𝑃 pCnt (seq1( · , 𝐹)‘𝑘)) + (𝑃 pCnt (𝐹‘(𝑘 + 1)))))
140139adantrr 699 . . . . . . . . 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 15626 . . . . . . . . . . . . . . 15 (𝑃 ∈ ℙ → 𝑃 ∈ ℕ)
14226, 141syl 17 . . . . . . . . . . . . . 14 (𝜑𝑃 ∈ ℕ)
143142nnred 11332 . . . . . . . . . . . . 13 (𝜑𝑃 ∈ ℝ)
144143adantr 468 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑘 ∈ ℕ ∧ (𝑘 + 1) = 𝑃)) → 𝑃 ∈ ℝ)
145144leidd 10889 . . . . . . . . . . 11 ((𝜑 ∧ (𝑘 ∈ ℕ ∧ (𝑘 + 1) = 𝑃)) → 𝑃𝑃)
146145, 86breqtrrd 4883 . . . . . . . . . 10 ((𝜑 ∧ (𝑘 ∈ ℕ ∧ (𝑘 + 1) = 𝑃)) → 𝑃 ≤ (𝑘 + 1))
147146iftrued 4298 . . . . . . . . 9 ((𝜑 ∧ (𝑘 ∈ ℕ ∧ (𝑘 + 1) = 𝑃)) → if(𝑃 ≤ (𝑘 + 1), 𝐵, 0) = 𝐵)
148140, 147eqeq12d 2832 . . . . . . . 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))
149107, 119, 1483imtr4d 285 . . . . . . 7 ((𝜑 ∧ (𝑘 ∈ ℕ ∧ (𝑘 + 1) = 𝑃)) → ((𝑃 pCnt (seq1( · , 𝐹)‘𝑘)) = if(𝑃𝑘, 𝐵, 0) → (𝑃 pCnt (seq1( · , 𝐹)‘(𝑘 + 1))) = if(𝑃 ≤ (𝑘 + 1), 𝐵, 0)))
150149expr 446 . . . . . 6 ((𝜑𝑘 ∈ ℕ) → ((𝑘 + 1) = 𝑃 → ((𝑃 pCnt (seq1( · , 𝐹)‘𝑘)) = if(𝑃𝑘, 𝐵, 0) → (𝑃 pCnt (seq1( · , 𝐹)‘(𝑘 + 1))) = if(𝑃 ≤ (𝑘 + 1), 𝐵, 0))))
151139adantrr 699 . . . . . . . . . 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 787 . . . . . . . . . . . . . . . 16 (((𝜑 ∧ (𝑘 ∈ ℕ ∧ (𝑘 + 1) ≠ 𝑃)) ∧ (𝑘 + 1) ∈ ℙ) → (𝑘 + 1) ≠ 𝑃)
153152necomd 3044 . . . . . . . . . . . . . . 15 (((𝜑 ∧ (𝑘 ∈ ℕ ∧ (𝑘 + 1) ≠ 𝑃)) ∧ (𝑘 + 1) ∈ ℙ) → 𝑃 ≠ (𝑘 + 1))
15426ad2antrr 708 . . . . . . . . . . . . . . . . 17 (((𝜑 ∧ (𝑘 ∈ ℕ ∧ (𝑘 + 1) ≠ 𝑃)) ∧ (𝑘 + 1) ∈ ℙ) → 𝑃 ∈ ℙ)
155 simpr 473 . . . . . . . . . . . . . . . . 17 (((𝜑 ∧ (𝑘 ∈ ℕ ∧ (𝑘 + 1) ≠ 𝑃)) ∧ (𝑘 + 1) ∈ ℙ) → (𝑘 + 1) ∈ ℙ)
15655ad2antrr 708 . . . . . . . . . . . . . . . . . 18 (((𝜑 ∧ (𝑘 ∈ ℕ ∧ (𝑘 + 1) ≠ 𝑃)) ∧ (𝑘 + 1) ∈ ℙ) → ∀𝑛 ∈ ℙ 𝐴 ∈ ℕ0)
15774nfel1 2974 . . . . . . . . . . . . . . . . . . 19 𝑛(𝑘 + 1) / 𝑛𝐴 ∈ ℕ0
15880eleq1d 2881 . . . . . . . . . . . . . . . . . . 19 (𝑛 = (𝑘 + 1) → (𝐴 ∈ ℕ0(𝑘 + 1) / 𝑛𝐴 ∈ ℕ0))
159157, 158rspc 3507 . . . . . . . . . . . . . . . . . 18 ((𝑘 + 1) ∈ ℙ → (∀𝑛 ∈ ℙ 𝐴 ∈ ℕ0(𝑘 + 1) / 𝑛𝐴 ∈ ℕ0))
160155, 156, 159sylc 65 . . . . . . . . . . . . . . . . 17 (((𝜑 ∧ (𝑘 ∈ ℕ ∧ (𝑘 + 1) ≠ 𝑃)) ∧ (𝑘 + 1) ∈ ℙ) → (𝑘 + 1) / 𝑛𝐴 ∈ ℕ0)
161 prmdvdsexpr 15666 . . . . . . . . . . . . . . . . 17 ((𝑃 ∈ ℙ ∧ (𝑘 + 1) ∈ ℙ ∧ (𝑘 + 1) / 𝑛𝐴 ∈ ℕ0) → (𝑃 ∥ ((𝑘 + 1)↑(𝑘 + 1) / 𝑛𝐴) → 𝑃 = (𝑘 + 1)))
162154, 155, 160, 161syl3anc 1483 . . . . . . . . . . . . . . . 16 (((𝜑 ∧ (𝑘 ∈ ℕ ∧ (𝑘 + 1) ≠ 𝑃)) ∧ (𝑘 + 1) ∈ ℙ) → (𝑃 ∥ ((𝑘 + 1)↑(𝑘 + 1) / 𝑛𝐴) → 𝑃 = (𝑘 + 1)))
163162necon3ad 3002 . . . . . . . . . . . . . . 15 (((𝜑 ∧ (𝑘 ∈ ℕ ∧ (𝑘 + 1) ≠ 𝑃)) ∧ (𝑘 + 1) ∈ ℙ) → (𝑃 ≠ (𝑘 + 1) → ¬ 𝑃 ∥ ((𝑘 + 1)↑(𝑘 + 1) / 𝑛𝐴)))
164153, 163mpd 15 . . . . . . . . . . . . . 14 (((𝜑 ∧ (𝑘 ∈ ℕ ∧ (𝑘 + 1) ≠ 𝑃)) ∧ (𝑘 + 1) ∈ ℙ) → ¬ 𝑃 ∥ ((𝑘 + 1)↑(𝑘 + 1) / 𝑛𝐴))
16558ad2antrl 710 . . . . . . . . . . . . . . . . 17 ((𝜑 ∧ (𝑘 ∈ ℕ ∧ (𝑘 + 1) ≠ 𝑃)) → (𝑘 + 1) ∈ ℕ)
166165, 68, 84sylancl 576 . . . . . . . . . . . . . . . 16 ((𝜑 ∧ (𝑘 ∈ ℕ ∧ (𝑘 + 1) ≠ 𝑃)) → (𝐹‘(𝑘 + 1)) = if((𝑘 + 1) ∈ ℙ, ((𝑘 + 1)↑(𝑘 + 1) / 𝑛𝐴), 1))
167 iftrue 4296 . . . . . . . . . . . . . . . 16 ((𝑘 + 1) ∈ ℙ → if((𝑘 + 1) ∈ ℙ, ((𝑘 + 1)↑(𝑘 + 1) / 𝑛𝐴), 1) = ((𝑘 + 1)↑(𝑘 + 1) / 𝑛𝐴))
168166, 167sylan9eq 2871 . . . . . . . . . . . . . . 15 (((𝜑 ∧ (𝑘 ∈ ℕ ∧ (𝑘 + 1) ≠ 𝑃)) ∧ (𝑘 + 1) ∈ ℙ) → (𝐹‘(𝑘 + 1)) = ((𝑘 + 1)↑(𝑘 + 1) / 𝑛𝐴))
169168breq2d 4867 . . . . . . . . . . . . . 14 (((𝜑 ∧ (𝑘 ∈ ℕ ∧ (𝑘 + 1) ≠ 𝑃)) ∧ (𝑘 + 1) ∈ ℙ) → (𝑃 ∥ (𝐹‘(𝑘 + 1)) ↔ 𝑃 ∥ ((𝑘 + 1)↑(𝑘 + 1) / 𝑛𝐴)))
170164, 169mtbird 316 . . . . . . . . . . . . 13 (((𝜑 ∧ (𝑘 ∈ ℕ ∧ (𝑘 + 1) ≠ 𝑃)) ∧ (𝑘 + 1) ∈ ℙ) → ¬ 𝑃 ∥ (𝐹‘(𝑘 + 1)))
17157adantr 468 . . . . . . . . . . . . . . . 16 ((𝜑 ∧ (𝑘 ∈ ℕ ∧ (𝑘 + 1) ≠ 𝑃)) → 𝐹:ℕ⟶ℕ)
172171, 165, 59syl2anc 575 . . . . . . . . . . . . . . 15 ((𝜑 ∧ (𝑘 ∈ ℕ ∧ (𝑘 + 1) ≠ 𝑃)) → (𝐹‘(𝑘 + 1)) ∈ ℕ)
173172adantr 468 . . . . . . . . . . . . . 14 (((𝜑 ∧ (𝑘 ∈ ℕ ∧ (𝑘 + 1) ≠ 𝑃)) ∧ (𝑘 + 1) ∈ ℙ) → (𝐹‘(𝑘 + 1)) ∈ ℕ)
174 pceq0 15812 . . . . . . . . . . . . . 14 ((𝑃 ∈ ℙ ∧ (𝐹‘(𝑘 + 1)) ∈ ℕ) → ((𝑃 pCnt (𝐹‘(𝑘 + 1))) = 0 ↔ ¬ 𝑃 ∥ (𝐹‘(𝑘 + 1))))
175154, 173, 174syl2anc 575 . . . . . . . . . . . . 13 (((𝜑 ∧ (𝑘 ∈ ℕ ∧ (𝑘 + 1) ≠ 𝑃)) ∧ (𝑘 + 1) ∈ ℙ) → ((𝑃 pCnt (𝐹‘(𝑘 + 1))) = 0 ↔ ¬ 𝑃 ∥ (𝐹‘(𝑘 + 1))))
176170, 175mpbird 248 . . . . . . . . . . . 12 (((𝜑 ∧ (𝑘 ∈ ℕ ∧ (𝑘 + 1) ≠ 𝑃)) ∧ (𝑘 + 1) ∈ ℙ) → (𝑃 pCnt (𝐹‘(𝑘 + 1))) = 0)
177 iffalse 4299 . . . . . . . . . . . . . . 15 (¬ (𝑘 + 1) ∈ ℙ → if((𝑘 + 1) ∈ ℙ, ((𝑘 + 1)↑(𝑘 + 1) / 𝑛𝐴), 1) = 1)
178166, 177sylan9eq 2871 . . . . . . . . . . . . . 14 (((𝜑 ∧ (𝑘 ∈ ℕ ∧ (𝑘 + 1) ≠ 𝑃)) ∧ ¬ (𝑘 + 1) ∈ ℙ) → (𝐹‘(𝑘 + 1)) = 1)
179178oveq2d 6900 . . . . . . . . . . . . 13 (((𝜑 ∧ (𝑘 ∈ ℕ ∧ (𝑘 + 1) ≠ 𝑃)) ∧ ¬ (𝑘 + 1) ∈ ℙ) → (𝑃 pCnt (𝐹‘(𝑘 + 1))) = (𝑃 pCnt 1))
18026, 41syl 17 . . . . . . . . . . . . . 14 (𝜑 → (𝑃 pCnt 1) = 0)
181180ad2antrr 708 . . . . . . . . . . . . 13 (((𝜑 ∧ (𝑘 ∈ ℕ ∧ (𝑘 + 1) ≠ 𝑃)) ∧ ¬ (𝑘 + 1) ∈ ℙ) → (𝑃 pCnt 1) = 0)
182179, 181eqtrd 2851 . . . . . . . . . . . 12 (((𝜑 ∧ (𝑘 ∈ ℕ ∧ (𝑘 + 1) ≠ 𝑃)) ∧ ¬ (𝑘 + 1) ∈ ℙ) → (𝑃 pCnt (𝐹‘(𝑘 + 1))) = 0)
183176, 182pm2.61dan 838 . . . . . . . . . . 11 ((𝜑 ∧ (𝑘 ∈ ℕ ∧ (𝑘 + 1) ≠ 𝑃)) → (𝑃 pCnt (𝐹‘(𝑘 + 1))) = 0)
184183oveq2d 6900 . . . . . . . . . 10 ((𝜑 ∧ (𝑘 ∈ ℕ ∧ (𝑘 + 1) ≠ 𝑃)) → ((𝑃 pCnt (seq1( · , 𝐹)‘𝑘)) + (𝑃 pCnt (𝐹‘(𝑘 + 1)))) = ((𝑃 pCnt (seq1( · , 𝐹)‘𝑘)) + 0))
18526adantr 468 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑘 ∈ ℕ ∧ (𝑘 + 1) ≠ 𝑃)) → 𝑃 ∈ ℙ)
186128adantrr 699 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑘 ∈ ℕ ∧ (𝑘 + 1) ≠ 𝑃)) → (seq1( · , 𝐹)‘𝑘) ∈ ℕ)
187185, 186pccld 15792 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑘 ∈ ℕ ∧ (𝑘 + 1) ≠ 𝑃)) → (𝑃 pCnt (seq1( · , 𝐹)‘𝑘)) ∈ ℕ0)
188187nn0cnd 11639 . . . . . . . . . . 11 ((𝜑 ∧ (𝑘 ∈ ℕ ∧ (𝑘 + 1) ≠ 𝑃)) → (𝑃 pCnt (seq1( · , 𝐹)‘𝑘)) ∈ ℂ)
189188addid1d 10531 . . . . . . . . . 10 ((𝜑 ∧ (𝑘 ∈ ℕ ∧ (𝑘 + 1) ≠ 𝑃)) → ((𝑃 pCnt (seq1( · , 𝐹)‘𝑘)) + 0) = (𝑃 pCnt (seq1( · , 𝐹)‘𝑘)))
190151, 184, 1893eqtrd 2855 . . . . . . . . 9 ((𝜑 ∧ (𝑘 ∈ ℕ ∧ (𝑘 + 1) ≠ 𝑃)) → (𝑃 pCnt (seq1( · , 𝐹)‘(𝑘 + 1))) = (𝑃 pCnt (seq1( · , 𝐹)‘𝑘)))
191142adantr 468 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑘 ∈ ℕ ∧ (𝑘 + 1) ≠ 𝑃)) → 𝑃 ∈ ℕ)
192191nnred 11332 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑘 ∈ ℕ ∧ (𝑘 + 1) ≠ 𝑃)) → 𝑃 ∈ ℝ)
193165nnred 11332 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑘 ∈ ℕ ∧ (𝑘 + 1) ≠ 𝑃)) → (𝑘 + 1) ∈ ℝ)
194192, 193ltlend 10477 . . . . . . . . . . 11 ((𝜑 ∧ (𝑘 ∈ ℕ ∧ (𝑘 + 1) ≠ 𝑃)) → (𝑃 < (𝑘 + 1) ↔ (𝑃 ≤ (𝑘 + 1) ∧ (𝑘 + 1) ≠ 𝑃)))
195 simprl 778 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑘 ∈ ℕ ∧ (𝑘 + 1) ≠ 𝑃)) → 𝑘 ∈ ℕ)
196 nnleltp1 11718 . . . . . . . . . . . 12 ((𝑃 ∈ ℕ ∧ 𝑘 ∈ ℕ) → (𝑃𝑘𝑃 < (𝑘 + 1)))
197191, 195, 196syl2anc 575 . . . . . . . . . . 11 ((𝜑 ∧ (𝑘 ∈ ℕ ∧ (𝑘 + 1) ≠ 𝑃)) → (𝑃𝑘𝑃 < (𝑘 + 1)))
198 simprr 780 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑘 ∈ ℕ ∧ (𝑘 + 1) ≠ 𝑃)) → (𝑘 + 1) ≠ 𝑃)
199198biantrud 523 . . . . . . . . . . 11 ((𝜑 ∧ (𝑘 ∈ ℕ ∧ (𝑘 + 1) ≠ 𝑃)) → (𝑃 ≤ (𝑘 + 1) ↔ (𝑃 ≤ (𝑘 + 1) ∧ (𝑘 + 1) ≠ 𝑃)))
200194, 197, 1993bitr4rd 303 . . . . . . . . . 10 ((𝜑 ∧ (𝑘 ∈ ℕ ∧ (𝑘 + 1) ≠ 𝑃)) → (𝑃 ≤ (𝑘 + 1) ↔ 𝑃𝑘))
201200ifbid 4312 . . . . . . . . 9 ((𝜑 ∧ (𝑘 ∈ ℕ ∧ (𝑘 + 1) ≠ 𝑃)) → if(𝑃 ≤ (𝑘 + 1), 𝐵, 0) = if(𝑃𝑘, 𝐵, 0))
202190, 201eqeq12d 2832 . . . . . . . 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(𝑃𝑘, 𝐵, 0)))
203202biimprd 239 . . . . . . 7 ((𝜑 ∧ (𝑘 ∈ ℕ ∧ (𝑘 + 1) ≠ 𝑃)) → ((𝑃 pCnt (seq1( · , 𝐹)‘𝑘)) = if(𝑃𝑘, 𝐵, 0) → (𝑃 pCnt (seq1( · , 𝐹)‘(𝑘 + 1))) = if(𝑃 ≤ (𝑘 + 1), 𝐵, 0)))
204203expr 446 . . . . . 6 ((𝜑𝑘 ∈ ℕ) → ((𝑘 + 1) ≠ 𝑃 → ((𝑃 pCnt (seq1( · , 𝐹)‘𝑘)) = if(𝑃𝑘, 𝐵, 0) → (𝑃 pCnt (seq1( · , 𝐹)‘(𝑘 + 1))) = if(𝑃 ≤ (𝑘 + 1), 𝐵, 0))))
205150, 204pm2.61dne 3075 . . . . 5 ((𝜑𝑘 ∈ ℕ) → ((𝑃 pCnt (seq1( · , 𝐹)‘𝑘)) = if(𝑃𝑘, 𝐵, 0) → (𝑃 pCnt (seq1( · , 𝐹)‘(𝑘 + 1))) = if(𝑃 ≤ (𝑘 + 1), 𝐵, 0)))
206205expcom 400 . . . 4 (𝑘 ∈ ℕ → (𝜑 → ((𝑃 pCnt (seq1( · , 𝐹)‘𝑘)) = if(𝑃𝑘, 𝐵, 0) → (𝑃 pCnt (seq1( · , 𝐹)‘(𝑘 + 1))) = if(𝑃 ≤ (𝑘 + 1), 𝐵, 0))))
207206a2d 29 . . 3 (𝑘 ∈ ℕ → ((𝜑 → (𝑃 pCnt (seq1( · , 𝐹)‘𝑘)) = if(𝑃𝑘, 𝐵, 0)) → (𝜑 → (𝑃 pCnt (seq1( · , 𝐹)‘(𝑘 + 1))) = if(𝑃 ≤ (𝑘 + 1), 𝐵, 0))))
2087, 13, 19, 25, 53, 207nnind 11335 . 2 (𝑁 ∈ ℕ → (𝜑 → (𝑃 pCnt (seq1( · , 𝐹)‘𝑁)) = if(𝑃𝑁, 𝐵, 0)))
2091, 208mpcom 38 1 (𝜑 → (𝑃 pCnt (seq1( · , 𝐹)‘𝑁)) = if(𝑃𝑁, 𝐵, 0))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 197  wa 384   = wceq 1637  wcel 2157  wne 2989  wral 3107  Vcvv 3402  csb 3739  ifcif 4290   class class class wbr 4855  cmpt 4934  wf 6107  cfv 6111  (class class class)co 6884  cr 10230  0cc0 10231  1c1 10232   + caddc 10234   · cmul 10236   < clt 10369  cle 10370  cn 11315  2c2 11368  0cn0 11579  cz 11663  cuz 11924  seqcseq 13044  cexp 13103  cdvds 15223  cprime 15623   pCnt cpc 15778
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1877  ax-4 1894  ax-5 2001  ax-6 2069  ax-7 2105  ax-8 2159  ax-9 2166  ax-10 2186  ax-11 2202  ax-12 2215  ax-13 2422  ax-ext 2795  ax-sep 4988  ax-nul 4996  ax-pow 5048  ax-pr 5109  ax-un 7189  ax-cnex 10287  ax-resscn 10288  ax-1cn 10289  ax-icn 10290  ax-addcl 10291  ax-addrcl 10292  ax-mulcl 10293  ax-mulrcl 10294  ax-mulcom 10295  ax-addass 10296  ax-mulass 10297  ax-distr 10298  ax-i2m1 10299  ax-1ne0 10300  ax-1rid 10301  ax-rnegex 10302  ax-rrecex 10303  ax-cnre 10304  ax-pre-lttri 10305  ax-pre-lttrn 10306  ax-pre-ltadd 10307  ax-pre-mulgt0 10308  ax-pre-sup 10309
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 866  df-3or 1101  df-3an 1102  df-tru 1641  df-fal 1651  df-ex 1860  df-nf 1864  df-sb 2062  df-mo 2635  df-eu 2642  df-clab 2804  df-cleq 2810  df-clel 2813  df-nfc 2948  df-ne 2990  df-nel 3093  df-ral 3112  df-rex 3113  df-reu 3114  df-rmo 3115  df-rab 3116  df-v 3404  df-sbc 3645  df-csb 3740  df-dif 3783  df-un 3785  df-in 3787  df-ss 3794  df-pss 3796  df-nul 4128  df-if 4291  df-pw 4364  df-sn 4382  df-pr 4384  df-tp 4386  df-op 4388  df-uni 4642  df-iun 4725  df-br 4856  df-opab 4918  df-mpt 4935  df-tr 4958  df-id 5232  df-eprel 5237  df-po 5245  df-so 5246  df-fr 5283  df-we 5285  df-xp 5330  df-rel 5331  df-cnv 5332  df-co 5333  df-dm 5334  df-rn 5335  df-res 5336  df-ima 5337  df-pred 5907  df-ord 5953  df-on 5954  df-lim 5955  df-suc 5956  df-iota 6074  df-fun 6113  df-fn 6114  df-f 6115  df-f1 6116  df-fo 6117  df-f1o 6118  df-fv 6119  df-riota 6845  df-ov 6887  df-oprab 6888  df-mpt2 6889  df-om 7306  df-1st 7408  df-2nd 7409  df-wrecs 7652  df-recs 7714  df-rdg 7752  df-1o 7806  df-2o 7807  df-er 7989  df-en 8203  df-dom 8204  df-sdom 8205  df-fin 8206  df-sup 8597  df-inf 8598  df-pnf 10371  df-mnf 10372  df-xr 10373  df-ltxr 10374  df-le 10375  df-sub 10563  df-neg 10564  df-div 10980  df-nn 11316  df-2 11376  df-3 11377  df-n0 11580  df-z 11664  df-uz 11925  df-q 12028  df-rp 12067  df-fz 12570  df-fl 12837  df-mod 12913  df-seq 13045  df-exp 13104  df-cj 14082  df-re 14083  df-im 14084  df-sqrt 14218  df-abs 14219  df-dvds 15224  df-gcd 15456  df-prm 15624  df-pc 15779
This theorem is referenced by:  pcmpt2  15834  pcprod  15836  1arithlem4  15867  chtublem  25173  bposlem3  25248
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