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Theorem pcmpt 15802
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 6332 . . . . . 6 (𝑝 = 1 → (seq1( · , 𝐹)‘𝑝) = (seq1( · , 𝐹)‘1))
32oveq2d 6808 . . . . 5 (𝑝 = 1 → (𝑃 pCnt (seq1( · , 𝐹)‘𝑝)) = (𝑃 pCnt (seq1( · , 𝐹)‘1)))
4 breq2 4788 . . . . . 6 (𝑝 = 1 → (𝑃𝑝𝑃 ≤ 1))
54ifbid 4245 . . . . 5 (𝑝 = 1 → if(𝑃𝑝, 𝐵, 0) = if(𝑃 ≤ 1, 𝐵, 0))
63, 5eqeq12d 2785 . . . 4 (𝑝 = 1 → ((𝑃 pCnt (seq1( · , 𝐹)‘𝑝)) = if(𝑃𝑝, 𝐵, 0) ↔ (𝑃 pCnt (seq1( · , 𝐹)‘1)) = if(𝑃 ≤ 1, 𝐵, 0)))
76imbi2d 329 . . 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 6332 . . . . . 6 (𝑝 = 𝑘 → (seq1( · , 𝐹)‘𝑝) = (seq1( · , 𝐹)‘𝑘))
98oveq2d 6808 . . . . 5 (𝑝 = 𝑘 → (𝑃 pCnt (seq1( · , 𝐹)‘𝑝)) = (𝑃 pCnt (seq1( · , 𝐹)‘𝑘)))
10 breq2 4788 . . . . . 6 (𝑝 = 𝑘 → (𝑃𝑝𝑃𝑘))
1110ifbid 4245 . . . . 5 (𝑝 = 𝑘 → if(𝑃𝑝, 𝐵, 0) = if(𝑃𝑘, 𝐵, 0))
129, 11eqeq12d 2785 . . . 4 (𝑝 = 𝑘 → ((𝑃 pCnt (seq1( · , 𝐹)‘𝑝)) = if(𝑃𝑝, 𝐵, 0) ↔ (𝑃 pCnt (seq1( · , 𝐹)‘𝑘)) = if(𝑃𝑘, 𝐵, 0)))
1312imbi2d 329 . . 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 6332 . . . . . 6 (𝑝 = (𝑘 + 1) → (seq1( · , 𝐹)‘𝑝) = (seq1( · , 𝐹)‘(𝑘 + 1)))
1514oveq2d 6808 . . . . 5 (𝑝 = (𝑘 + 1) → (𝑃 pCnt (seq1( · , 𝐹)‘𝑝)) = (𝑃 pCnt (seq1( · , 𝐹)‘(𝑘 + 1))))
16 breq2 4788 . . . . . 6 (𝑝 = (𝑘 + 1) → (𝑃𝑝𝑃 ≤ (𝑘 + 1)))
1716ifbid 4245 . . . . 5 (𝑝 = (𝑘 + 1) → if(𝑃𝑝, 𝐵, 0) = if(𝑃 ≤ (𝑘 + 1), 𝐵, 0))
1815, 17eqeq12d 2785 . . . 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 329 . . 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 6332 . . . . . 6 (𝑝 = 𝑁 → (seq1( · , 𝐹)‘𝑝) = (seq1( · , 𝐹)‘𝑁))
2120oveq2d 6808 . . . . 5 (𝑝 = 𝑁 → (𝑃 pCnt (seq1( · , 𝐹)‘𝑝)) = (𝑃 pCnt (seq1( · , 𝐹)‘𝑁)))
22 breq2 4788 . . . . . 6 (𝑝 = 𝑁 → (𝑃𝑝𝑃𝑁))
2322ifbid 4245 . . . . 5 (𝑝 = 𝑁 → if(𝑃𝑝, 𝐵, 0) = if(𝑃𝑁, 𝐵, 0))
2421, 23eqeq12d 2785 . . . 4 (𝑝 = 𝑁 → ((𝑃 pCnt (seq1( · , 𝐹)‘𝑝)) = if(𝑃𝑝, 𝐵, 0) ↔ (𝑃 pCnt (seq1( · , 𝐹)‘𝑁)) = if(𝑃𝑁, 𝐵, 0)))
2524imbi2d 329 . . 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 11608 . . . . . . . . 9 1 ∈ ℤ
28 seq1 13020 . . . . . . . . 9 (1 ∈ ℤ → (seq1( · , 𝐹)‘1) = (𝐹‘1))
2927, 28ax-mp 5 . . . . . . . 8 (seq1( · , 𝐹)‘1) = (𝐹‘1)
30 1nn 11232 . . . . . . . . 9 1 ∈ ℕ
31 1nprm 15598 . . . . . . . . . . . 12 ¬ 1 ∈ ℙ
32 eleq1 2837 . . . . . . . . . . . 12 (𝑛 = 1 → (𝑛 ∈ ℙ ↔ 1 ∈ ℙ))
3331, 32mtbiri 316 . . . . . . . . . . 11 (𝑛 = 1 → ¬ 𝑛 ∈ ℙ)
3433iffalsed 4234 . . . . . . . . . 10 (𝑛 = 1 → if(𝑛 ∈ ℙ, (𝑛𝐴), 1) = 1)
35 pcmpt.1 . . . . . . . . . 10 𝐹 = (𝑛 ∈ ℕ ↦ if(𝑛 ∈ ℙ, (𝑛𝐴), 1))
36 1ex 10236 . . . . . . . . . 10 1 ∈ V
3734, 35, 36fvmpt 6424 . . . . . . . . 9 (1 ∈ ℕ → (𝐹‘1) = 1)
3830, 37ax-mp 5 . . . . . . . 8 (𝐹‘1) = 1
3929, 38eqtri 2792 . . . . . . 7 (seq1( · , 𝐹)‘1) = 1
4039oveq2i 6803 . . . . . 6 (𝑃 pCnt (seq1( · , 𝐹)‘1)) = (𝑃 pCnt 1)
41 pc1 15766 . . . . . 6 (𝑃 ∈ ℙ → (𝑃 pCnt 1) = 0)
4240, 41syl5eq 2816 . . . . 5 (𝑃 ∈ ℙ → (𝑃 pCnt (seq1( · , 𝐹)‘1)) = 0)
43 prmgt1 15615 . . . . . . 7 (𝑃 ∈ ℙ → 1 < 𝑃)
44 1re 10240 . . . . . . . 8 1 ∈ ℝ
45 prmuz2 15614 . . . . . . . . 9 (𝑃 ∈ ℙ → 𝑃 ∈ (ℤ‘2))
46 eluzelre 11898 . . . . . . . . 9 (𝑃 ∈ (ℤ‘2) → 𝑃 ∈ ℝ)
4745, 46syl 17 . . . . . . . 8 (𝑃 ∈ ℙ → 𝑃 ∈ ℝ)
48 ltnle 10318 . . . . . . . 8 ((1 ∈ ℝ ∧ 𝑃 ∈ ℝ) → (1 < 𝑃 ↔ ¬ 𝑃 ≤ 1))
4944, 47, 48sylancr 567 . . . . . . 7 (𝑃 ∈ ℙ → (1 < 𝑃 ↔ ¬ 𝑃 ≤ 1))
5043, 49mpbid 222 . . . . . 6 (𝑃 ∈ ℙ → ¬ 𝑃 ≤ 1)
5150iffalsed 4234 . . . . 5 (𝑃 ∈ ℙ → if(𝑃 ≤ 1, 𝐵, 0) = 0)
5242, 51eqtr4d 2807 . . . 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 466 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑘 ∈ ℕ ∧ (𝑘 + 1) = 𝑃)) → 𝑃 ∈ ℙ)
55 pcmpt.2 . . . . . . . . . . . . . . . . 17 (𝜑 → ∀𝑛 ∈ ℙ 𝐴 ∈ ℕ0)
5635, 55pcmptcl 15801 . . . . . . . . . . . . . . . 16 (𝜑 → (𝐹:ℕ⟶ℕ ∧ seq1( · , 𝐹):ℕ⟶ℕ))
5756simpld 476 . . . . . . . . . . . . . . 15 (𝜑𝐹:ℕ⟶ℕ)
58 peano2nn 11233 . . . . . . . . . . . . . . 15 (𝑘 ∈ ℕ → (𝑘 + 1) ∈ ℕ)
59 ffvelrn 6500 . . . . . . . . . . . . . . 15 ((𝐹:ℕ⟶ℕ ∧ (𝑘 + 1) ∈ ℕ) → (𝐹‘(𝑘 + 1)) ∈ ℕ)
6057, 58, 59syl2an 575 . . . . . . . . . . . . . 14 ((𝜑𝑘 ∈ ℕ) → (𝐹‘(𝑘 + 1)) ∈ ℕ)
6160adantrr 688 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑘 ∈ ℕ ∧ (𝑘 + 1) = 𝑃)) → (𝐹‘(𝑘 + 1)) ∈ ℕ)
6254, 61pccld 15761 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑘 ∈ ℕ ∧ (𝑘 + 1) = 𝑃)) → (𝑃 pCnt (𝐹‘(𝑘 + 1))) ∈ ℕ0)
6362nn0cnd 11554 . . . . . . . . . . 11 ((𝜑 ∧ (𝑘 ∈ ℕ ∧ (𝑘 + 1) = 𝑃)) → (𝑃 pCnt (𝐹‘(𝑘 + 1))) ∈ ℂ)
6463addid2d 10438 . . . . . . . . . 10 ((𝜑 ∧ (𝑘 ∈ ℕ ∧ (𝑘 + 1) = 𝑃)) → (0 + (𝑃 pCnt (𝐹‘(𝑘 + 1)))) = (𝑃 pCnt (𝐹‘(𝑘 + 1))))
6558ad2antrl 699 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑘 ∈ ℕ ∧ (𝑘 + 1) = 𝑃)) → (𝑘 + 1) ∈ ℕ)
66 ovex 6822 . . . . . . . . . . . . . . 15 (𝑛𝐴) ∈ V
6766, 36ifex 4293 . . . . . . . . . . . . . 14 if(𝑛 ∈ ℙ, (𝑛𝐴), 1) ∈ V
6867csbex 4924 . . . . . . . . . . . . 13 (𝑘 + 1) / 𝑛if(𝑛 ∈ ℙ, (𝑛𝐴), 1) ∈ V
6935fvmpts 6427 . . . . . . . . . . . . . 14 (((𝑘 + 1) ∈ ℕ ∧ (𝑘 + 1) / 𝑛if(𝑛 ∈ ℙ, (𝑛𝐴), 1) ∈ V) → (𝐹‘(𝑘 + 1)) = (𝑘 + 1) / 𝑛if(𝑛 ∈ ℙ, (𝑛𝐴), 1))
70 ovex 6822 . . . . . . . . . . . . . . 15 (𝑘 + 1) ∈ V
71 nfv 1994 . . . . . . . . . . . . . . . 16 𝑛(𝑘 + 1) ∈ ℙ
72 nfcv 2912 . . . . . . . . . . . . . . . . 17 𝑛(𝑘 + 1)
73 nfcv 2912 . . . . . . . . . . . . . . . . 17 𝑛
74 nfcsb1v 3696 . . . . . . . . . . . . . . . . 17 𝑛(𝑘 + 1) / 𝑛𝐴
7572, 73, 74nfov 6820 . . . . . . . . . . . . . . . 16 𝑛((𝑘 + 1)↑(𝑘 + 1) / 𝑛𝐴)
76 nfcv 2912 . . . . . . . . . . . . . . . 16 𝑛1
7771, 75, 76nfif 4252 . . . . . . . . . . . . . . 15 𝑛if((𝑘 + 1) ∈ ℙ, ((𝑘 + 1)↑(𝑘 + 1) / 𝑛𝐴), 1)
78 eleq1 2837 . . . . . . . . . . . . . . . 16 (𝑛 = (𝑘 + 1) → (𝑛 ∈ ℙ ↔ (𝑘 + 1) ∈ ℙ))
79 id 22 . . . . . . . . . . . . . . . . 17 (𝑛 = (𝑘 + 1) → 𝑛 = (𝑘 + 1))
80 csbeq1a 3689 . . . . . . . . . . . . . . . . 17 (𝑛 = (𝑘 + 1) → 𝐴 = (𝑘 + 1) / 𝑛𝐴)
8179, 80oveq12d 6810 . . . . . . . . . . . . . . . 16 (𝑛 = (𝑘 + 1) → (𝑛𝐴) = ((𝑘 + 1)↑(𝑘 + 1) / 𝑛𝐴))
8278, 81ifbieq1d 4246 . . . . . . . . . . . . . . 15 (𝑛 = (𝑘 + 1) → if(𝑛 ∈ ℙ, (𝑛𝐴), 1) = if((𝑘 + 1) ∈ ℙ, ((𝑘 + 1)↑(𝑘 + 1) / 𝑛𝐴), 1))
8370, 77, 82csbief 3705 . . . . . . . . . . . . . 14 (𝑘 + 1) / 𝑛if(𝑛 ∈ ℙ, (𝑛𝐴), 1) = if((𝑘 + 1) ∈ ℙ, ((𝑘 + 1)↑(𝑘 + 1) / 𝑛𝐴), 1)
8469, 83syl6eq 2820 . . . . . . . . . . . . 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 566 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑘 ∈ ℕ ∧ (𝑘 + 1) = 𝑃)) → (𝐹‘(𝑘 + 1)) = if((𝑘 + 1) ∈ ℙ, ((𝑘 + 1)↑(𝑘 + 1) / 𝑛𝐴), 1))
86 simprr 748 . . . . . . . . . . . . . 14 ((𝜑 ∧ (𝑘 ∈ ℕ ∧ (𝑘 + 1) = 𝑃)) → (𝑘 + 1) = 𝑃)
8786, 54eqeltrd 2849 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑘 ∈ ℕ ∧ (𝑘 + 1) = 𝑃)) → (𝑘 + 1) ∈ ℙ)
8887iftrued 4231 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑘 ∈ ℕ ∧ (𝑘 + 1) = 𝑃)) → if((𝑘 + 1) ∈ ℙ, ((𝑘 + 1)↑(𝑘 + 1) / 𝑛𝐴), 1) = ((𝑘 + 1)↑(𝑘 + 1) / 𝑛𝐴))
8986csbeq1d 3687 . . . . . . . . . . . . . 14 ((𝜑 ∧ (𝑘 ∈ ℕ ∧ (𝑘 + 1) = 𝑃)) → (𝑘 + 1) / 𝑛𝐴 = 𝑃 / 𝑛𝐴)
90 nfcvd 2913 . . . . . . . . . . . . . . . 16 (𝑃 ∈ ℙ → 𝑛𝐵)
91 pcmpt.5 . . . . . . . . . . . . . . . 16 (𝑛 = 𝑃𝐴 = 𝐵)
9290, 91csbiegf 3704 . . . . . . . . . . . . . . 15 (𝑃 ∈ ℙ → 𝑃 / 𝑛𝐴 = 𝐵)
9354, 92syl 17 . . . . . . . . . . . . . 14 ((𝜑 ∧ (𝑘 ∈ ℕ ∧ (𝑘 + 1) = 𝑃)) → 𝑃 / 𝑛𝐴 = 𝐵)
9489, 93eqtrd 2804 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑘 ∈ ℕ ∧ (𝑘 + 1) = 𝑃)) → (𝑘 + 1) / 𝑛𝐴 = 𝐵)
9586, 94oveq12d 6810 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑘 ∈ ℕ ∧ (𝑘 + 1) = 𝑃)) → ((𝑘 + 1)↑(𝑘 + 1) / 𝑛𝐴) = (𝑃𝐵))
9685, 88, 953eqtrd 2808 . . . . . . . . . . 11 ((𝜑 ∧ (𝑘 ∈ ℕ ∧ (𝑘 + 1) = 𝑃)) → (𝐹‘(𝑘 + 1)) = (𝑃𝐵))
9796oveq2d 6808 . . . . . . . . . 10 ((𝜑 ∧ (𝑘 ∈ ℕ ∧ (𝑘 + 1) = 𝑃)) → (𝑃 pCnt (𝐹‘(𝑘 + 1))) = (𝑃 pCnt (𝑃𝐵)))
9891eleq1d 2834 . . . . . . . . . . . . . 14 (𝑛 = 𝑃 → (𝐴 ∈ ℕ0𝐵 ∈ ℕ0))
9998rspcv 3454 . . . . . . . . . . . . 13 (𝑃 ∈ ℙ → (∀𝑛 ∈ ℙ 𝐴 ∈ ℕ0𝐵 ∈ ℕ0))
10026, 55, 99sylc 65 . . . . . . . . . . . 12 (𝜑𝐵 ∈ ℕ0)
101100adantr 466 . . . . . . . . . . 11 ((𝜑 ∧ (𝑘 ∈ ℕ ∧ (𝑘 + 1) = 𝑃)) → 𝐵 ∈ ℕ0)
102 pcidlem 15782 . . . . . . . . . . 11 ((𝑃 ∈ ℙ ∧ 𝐵 ∈ ℕ0) → (𝑃 pCnt (𝑃𝐵)) = 𝐵)
10354, 101, 102syl2anc 565 . . . . . . . . . 10 ((𝜑 ∧ (𝑘 ∈ ℕ ∧ (𝑘 + 1) = 𝑃)) → (𝑃 pCnt (𝑃𝐵)) = 𝐵)
10464, 97, 1033eqtrd 2808 . . . . . . . . 9 ((𝜑 ∧ (𝑘 ∈ ℕ ∧ (𝑘 + 1) = 𝑃)) → (0 + (𝑃 pCnt (𝐹‘(𝑘 + 1)))) = 𝐵)
105 oveq1 6799 . . . . . . . . . 10 ((𝑃 pCnt (seq1( · , 𝐹)‘𝑘)) = 0 → ((𝑃 pCnt (seq1( · , 𝐹)‘𝑘)) + (𝑃 pCnt (𝐹‘(𝑘 + 1)))) = (0 + (𝑃 pCnt (𝐹‘(𝑘 + 1)))))
106105eqeq1d 2772 . . . . . . . . 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 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 11228 . . . . . . . . . . . . 13 (𝑘 ∈ ℕ → 𝑘 ∈ ℝ)
109108ad2antrl 699 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑘 ∈ ℕ ∧ (𝑘 + 1) = 𝑃)) → 𝑘 ∈ ℝ)
110 ltp1 11062 . . . . . . . . . . . . 13 (𝑘 ∈ ℝ → 𝑘 < (𝑘 + 1))
111 peano2re 10410 . . . . . . . . . . . . . 14 (𝑘 ∈ ℝ → (𝑘 + 1) ∈ ℝ)
112 ltnle 10318 . . . . . . . . . . . . . 14 ((𝑘 ∈ ℝ ∧ (𝑘 + 1) ∈ ℝ) → (𝑘 < (𝑘 + 1) ↔ ¬ (𝑘 + 1) ≤ 𝑘))
113111, 112mpdan 659 . . . . . . . . . . . . 13 (𝑘 ∈ ℝ → (𝑘 < (𝑘 + 1) ↔ ¬ (𝑘 + 1) ≤ 𝑘))
114110, 113mpbid 222 . . . . . . . . . . . 12 (𝑘 ∈ ℝ → ¬ (𝑘 + 1) ≤ 𝑘)
115109, 114syl 17 . . . . . . . . . . 11 ((𝜑 ∧ (𝑘 ∈ ℕ ∧ (𝑘 + 1) = 𝑃)) → ¬ (𝑘 + 1) ≤ 𝑘)
11686breq1d 4794 . . . . . . . . . . 11 ((𝜑 ∧ (𝑘 ∈ ℕ ∧ (𝑘 + 1) = 𝑃)) → ((𝑘 + 1) ≤ 𝑘𝑃𝑘))
117115, 116mtbid 313 . . . . . . . . . 10 ((𝜑 ∧ (𝑘 ∈ ℕ ∧ (𝑘 + 1) = 𝑃)) → ¬ 𝑃𝑘)
118117iffalsed 4234 . . . . . . . . 9 ((𝜑 ∧ (𝑘 ∈ ℕ ∧ (𝑘 + 1) = 𝑃)) → if(𝑃𝑘, 𝐵, 0) = 0)
119118eqeq2d 2780 . . . . . . . 8 ((𝜑 ∧ (𝑘 ∈ ℕ ∧ (𝑘 + 1) = 𝑃)) → ((𝑃 pCnt (seq1( · , 𝐹)‘𝑘)) = if(𝑃𝑘, 𝐵, 0) ↔ (𝑃 pCnt (seq1( · , 𝐹)‘𝑘)) = 0))
120 simpr 471 . . . . . . . . . . . . . 14 ((𝜑𝑘 ∈ ℕ) → 𝑘 ∈ ℕ)
121 nnuz 11924 . . . . . . . . . . . . . 14 ℕ = (ℤ‘1)
122120, 121syl6eleq 2859 . . . . . . . . . . . . 13 ((𝜑𝑘 ∈ ℕ) → 𝑘 ∈ (ℤ‘1))
123 seqp1 13022 . . . . . . . . . . . . 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 6808 . . . . . . . . . . 11 ((𝜑𝑘 ∈ ℕ) → (𝑃 pCnt (seq1( · , 𝐹)‘(𝑘 + 1))) = (𝑃 pCnt ((seq1( · , 𝐹)‘𝑘) · (𝐹‘(𝑘 + 1)))))
12626adantr 466 . . . . . . . . . . . 12 ((𝜑𝑘 ∈ ℕ) → 𝑃 ∈ ℙ)
12756simprd 477 . . . . . . . . . . . . . 14 (𝜑 → seq1( · , 𝐹):ℕ⟶ℕ)
128127ffvelrnda 6502 . . . . . . . . . . . . 13 ((𝜑𝑘 ∈ ℕ) → (seq1( · , 𝐹)‘𝑘) ∈ ℕ)
129 nnz 11600 . . . . . . . . . . . . . 14 ((seq1( · , 𝐹)‘𝑘) ∈ ℕ → (seq1( · , 𝐹)‘𝑘) ∈ ℤ)
130 nnne0 11254 . . . . . . . . . . . . . 14 ((seq1( · , 𝐹)‘𝑘) ∈ ℕ → (seq1( · , 𝐹)‘𝑘) ≠ 0)
131129, 130jca 495 . . . . . . . . . . . . 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 11600 . . . . . . . . . . . . . 14 ((𝐹‘(𝑘 + 1)) ∈ ℕ → (𝐹‘(𝑘 + 1)) ∈ ℤ)
134 nnne0 11254 . . . . . . . . . . . . . 14 ((𝐹‘(𝑘 + 1)) ∈ ℕ → (𝐹‘(𝑘 + 1)) ≠ 0)
135133, 134jca 495 . . . . . . . . . . . . 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 15762 . . . . . . . . . . . 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 1475 . . . . . . . . . . 11 ((𝜑𝑘 ∈ ℕ) → (𝑃 pCnt ((seq1( · , 𝐹)‘𝑘) · (𝐹‘(𝑘 + 1)))) = ((𝑃 pCnt (seq1( · , 𝐹)‘𝑘)) + (𝑃 pCnt (𝐹‘(𝑘 + 1)))))
139125, 138eqtrd 2804 . . . . . . . . . 10 ((𝜑𝑘 ∈ ℕ) → (𝑃 pCnt (seq1( · , 𝐹)‘(𝑘 + 1))) = ((𝑃 pCnt (seq1( · , 𝐹)‘𝑘)) + (𝑃 pCnt (𝐹‘(𝑘 + 1)))))
140139adantrr 688 . . . . . . . . 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 15594 . . . . . . . . . . . . . . 15 (𝑃 ∈ ℙ → 𝑃 ∈ ℕ)
14226, 141syl 17 . . . . . . . . . . . . . 14 (𝜑𝑃 ∈ ℕ)
143142nnred 11236 . . . . . . . . . . . . 13 (𝜑𝑃 ∈ ℝ)
144143adantr 466 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑘 ∈ ℕ ∧ (𝑘 + 1) = 𝑃)) → 𝑃 ∈ ℝ)
145144leidd 10795 . . . . . . . . . . 11 ((𝜑 ∧ (𝑘 ∈ ℕ ∧ (𝑘 + 1) = 𝑃)) → 𝑃𝑃)
146145, 86breqtrrd 4812 . . . . . . . . . 10 ((𝜑 ∧ (𝑘 ∈ ℕ ∧ (𝑘 + 1) = 𝑃)) → 𝑃 ≤ (𝑘 + 1))
147146iftrued 4231 . . . . . . . . 9 ((𝜑 ∧ (𝑘 ∈ ℕ ∧ (𝑘 + 1) = 𝑃)) → if(𝑃 ≤ (𝑘 + 1), 𝐵, 0) = 𝐵)
148140, 147eqeq12d 2785 . . . . . . . 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 283 . . . . . . 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 444 . . . . . 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 688 . . . . . . . . . 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 755 . . . . . . . . . . . . . . . 16 (((𝜑 ∧ (𝑘 ∈ ℕ ∧ (𝑘 + 1) ≠ 𝑃)) ∧ (𝑘 + 1) ∈ ℙ) → (𝑘 + 1) ≠ 𝑃)
153152necomd 2997 . . . . . . . . . . . . . . 15 (((𝜑 ∧ (𝑘 ∈ ℕ ∧ (𝑘 + 1) ≠ 𝑃)) ∧ (𝑘 + 1) ∈ ℙ) → 𝑃 ≠ (𝑘 + 1))
15426ad2antrr 697 . . . . . . . . . . . . . . . . 17 (((𝜑 ∧ (𝑘 ∈ ℕ ∧ (𝑘 + 1) ≠ 𝑃)) ∧ (𝑘 + 1) ∈ ℙ) → 𝑃 ∈ ℙ)
155 simpr 471 . . . . . . . . . . . . . . . . 17 (((𝜑 ∧ (𝑘 ∈ ℕ ∧ (𝑘 + 1) ≠ 𝑃)) ∧ (𝑘 + 1) ∈ ℙ) → (𝑘 + 1) ∈ ℙ)
15655ad2antrr 697 . . . . . . . . . . . . . . . . . 18 (((𝜑 ∧ (𝑘 ∈ ℕ ∧ (𝑘 + 1) ≠ 𝑃)) ∧ (𝑘 + 1) ∈ ℙ) → ∀𝑛 ∈ ℙ 𝐴 ∈ ℕ0)
15774nfel1 2927 . . . . . . . . . . . . . . . . . . 19 𝑛(𝑘 + 1) / 𝑛𝐴 ∈ ℕ0
15880eleq1d 2834 . . . . . . . . . . . . . . . . . . 19 (𝑛 = (𝑘 + 1) → (𝐴 ∈ ℕ0(𝑘 + 1) / 𝑛𝐴 ∈ ℕ0))
159157, 158rspc 3452 . . . . . . . . . . . . . . . . . 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 15635 . . . . . . . . . . . . . . . . 17 ((𝑃 ∈ ℙ ∧ (𝑘 + 1) ∈ ℙ ∧ (𝑘 + 1) / 𝑛𝐴 ∈ ℕ0) → (𝑃 ∥ ((𝑘 + 1)↑(𝑘 + 1) / 𝑛𝐴) → 𝑃 = (𝑘 + 1)))
162154, 155, 160, 161syl3anc 1475 . . . . . . . . . . . . . . . 16 (((𝜑 ∧ (𝑘 ∈ ℕ ∧ (𝑘 + 1) ≠ 𝑃)) ∧ (𝑘 + 1) ∈ ℙ) → (𝑃 ∥ ((𝑘 + 1)↑(𝑘 + 1) / 𝑛𝐴) → 𝑃 = (𝑘 + 1)))
163162necon3ad 2955 . . . . . . . . . . . . . . 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 699 . . . . . . . . . . . . . . . . 17 ((𝜑 ∧ (𝑘 ∈ ℕ ∧ (𝑘 + 1) ≠ 𝑃)) → (𝑘 + 1) ∈ ℕ)
166165, 68, 84sylancl 566 . . . . . . . . . . . . . . . 16 ((𝜑 ∧ (𝑘 ∈ ℕ ∧ (𝑘 + 1) ≠ 𝑃)) → (𝐹‘(𝑘 + 1)) = if((𝑘 + 1) ∈ ℙ, ((𝑘 + 1)↑(𝑘 + 1) / 𝑛𝐴), 1))
167 iftrue 4229 . . . . . . . . . . . . . . . 16 ((𝑘 + 1) ∈ ℙ → if((𝑘 + 1) ∈ ℙ, ((𝑘 + 1)↑(𝑘 + 1) / 𝑛𝐴), 1) = ((𝑘 + 1)↑(𝑘 + 1) / 𝑛𝐴))
168166, 167sylan9eq 2824 . . . . . . . . . . . . . . 15 (((𝜑 ∧ (𝑘 ∈ ℕ ∧ (𝑘 + 1) ≠ 𝑃)) ∧ (𝑘 + 1) ∈ ℙ) → (𝐹‘(𝑘 + 1)) = ((𝑘 + 1)↑(𝑘 + 1) / 𝑛𝐴))
169168breq2d 4796 . . . . . . . . . . . . . 14 (((𝜑 ∧ (𝑘 ∈ ℕ ∧ (𝑘 + 1) ≠ 𝑃)) ∧ (𝑘 + 1) ∈ ℙ) → (𝑃 ∥ (𝐹‘(𝑘 + 1)) ↔ 𝑃 ∥ ((𝑘 + 1)↑(𝑘 + 1) / 𝑛𝐴)))
170164, 169mtbird 314 . . . . . . . . . . . . 13 (((𝜑 ∧ (𝑘 ∈ ℕ ∧ (𝑘 + 1) ≠ 𝑃)) ∧ (𝑘 + 1) ∈ ℙ) → ¬ 𝑃 ∥ (𝐹‘(𝑘 + 1)))
17157adantr 466 . . . . . . . . . . . . . . . 16 ((𝜑 ∧ (𝑘 ∈ ℕ ∧ (𝑘 + 1) ≠ 𝑃)) → 𝐹:ℕ⟶ℕ)
172171, 165, 59syl2anc 565 . . . . . . . . . . . . . . 15 ((𝜑 ∧ (𝑘 ∈ ℕ ∧ (𝑘 + 1) ≠ 𝑃)) → (𝐹‘(𝑘 + 1)) ∈ ℕ)
173172adantr 466 . . . . . . . . . . . . . 14 (((𝜑 ∧ (𝑘 ∈ ℕ ∧ (𝑘 + 1) ≠ 𝑃)) ∧ (𝑘 + 1) ∈ ℙ) → (𝐹‘(𝑘 + 1)) ∈ ℕ)
174 pceq0 15781 . . . . . . . . . . . . . 14 ((𝑃 ∈ ℙ ∧ (𝐹‘(𝑘 + 1)) ∈ ℕ) → ((𝑃 pCnt (𝐹‘(𝑘 + 1))) = 0 ↔ ¬ 𝑃 ∥ (𝐹‘(𝑘 + 1))))
175154, 173, 174syl2anc 565 . . . . . . . . . . . . 13 (((𝜑 ∧ (𝑘 ∈ ℕ ∧ (𝑘 + 1) ≠ 𝑃)) ∧ (𝑘 + 1) ∈ ℙ) → ((𝑃 pCnt (𝐹‘(𝑘 + 1))) = 0 ↔ ¬ 𝑃 ∥ (𝐹‘(𝑘 + 1))))
176170, 175mpbird 247 . . . . . . . . . . . 12 (((𝜑 ∧ (𝑘 ∈ ℕ ∧ (𝑘 + 1) ≠ 𝑃)) ∧ (𝑘 + 1) ∈ ℙ) → (𝑃 pCnt (𝐹‘(𝑘 + 1))) = 0)
177 iffalse 4232 . . . . . . . . . . . . . . 15 (¬ (𝑘 + 1) ∈ ℙ → if((𝑘 + 1) ∈ ℙ, ((𝑘 + 1)↑(𝑘 + 1) / 𝑛𝐴), 1) = 1)
178166, 177sylan9eq 2824 . . . . . . . . . . . . . 14 (((𝜑 ∧ (𝑘 ∈ ℕ ∧ (𝑘 + 1) ≠ 𝑃)) ∧ ¬ (𝑘 + 1) ∈ ℙ) → (𝐹‘(𝑘 + 1)) = 1)
179178oveq2d 6808 . . . . . . . . . . . . 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 697 . . . . . . . . . . . . 13 (((𝜑 ∧ (𝑘 ∈ ℕ ∧ (𝑘 + 1) ≠ 𝑃)) ∧ ¬ (𝑘 + 1) ∈ ℙ) → (𝑃 pCnt 1) = 0)
182179, 181eqtrd 2804 . . . . . . . . . . . 12 (((𝜑 ∧ (𝑘 ∈ ℕ ∧ (𝑘 + 1) ≠ 𝑃)) ∧ ¬ (𝑘 + 1) ∈ ℙ) → (𝑃 pCnt (𝐹‘(𝑘 + 1))) = 0)
183176, 182pm2.61dan 796 . . . . . . . . . . 11 ((𝜑 ∧ (𝑘 ∈ ℕ ∧ (𝑘 + 1) ≠ 𝑃)) → (𝑃 pCnt (𝐹‘(𝑘 + 1))) = 0)
184183oveq2d 6808 . . . . . . . . . 10 ((𝜑 ∧ (𝑘 ∈ ℕ ∧ (𝑘 + 1) ≠ 𝑃)) → ((𝑃 pCnt (seq1( · , 𝐹)‘𝑘)) + (𝑃 pCnt (𝐹‘(𝑘 + 1)))) = ((𝑃 pCnt (seq1( · , 𝐹)‘𝑘)) + 0))
18526adantr 466 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑘 ∈ ℕ ∧ (𝑘 + 1) ≠ 𝑃)) → 𝑃 ∈ ℙ)
186128adantrr 688 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑘 ∈ ℕ ∧ (𝑘 + 1) ≠ 𝑃)) → (seq1( · , 𝐹)‘𝑘) ∈ ℕ)
187185, 186pccld 15761 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑘 ∈ ℕ ∧ (𝑘 + 1) ≠ 𝑃)) → (𝑃 pCnt (seq1( · , 𝐹)‘𝑘)) ∈ ℕ0)
188187nn0cnd 11554 . . . . . . . . . . 11 ((𝜑 ∧ (𝑘 ∈ ℕ ∧ (𝑘 + 1) ≠ 𝑃)) → (𝑃 pCnt (seq1( · , 𝐹)‘𝑘)) ∈ ℂ)
189188addid1d 10437 . . . . . . . . . 10 ((𝜑 ∧ (𝑘 ∈ ℕ ∧ (𝑘 + 1) ≠ 𝑃)) → ((𝑃 pCnt (seq1( · , 𝐹)‘𝑘)) + 0) = (𝑃 pCnt (seq1( · , 𝐹)‘𝑘)))
190151, 184, 1893eqtrd 2808 . . . . . . . . 9 ((𝜑 ∧ (𝑘 ∈ ℕ ∧ (𝑘 + 1) ≠ 𝑃)) → (𝑃 pCnt (seq1( · , 𝐹)‘(𝑘 + 1))) = (𝑃 pCnt (seq1( · , 𝐹)‘𝑘)))
191142adantr 466 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑘 ∈ ℕ ∧ (𝑘 + 1) ≠ 𝑃)) → 𝑃 ∈ ℕ)
192191nnred 11236 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑘 ∈ ℕ ∧ (𝑘 + 1) ≠ 𝑃)) → 𝑃 ∈ ℝ)
193165nnred 11236 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑘 ∈ ℕ ∧ (𝑘 + 1) ≠ 𝑃)) → (𝑘 + 1) ∈ ℝ)
194192, 193ltlend 10383 . . . . . . . . . . 11 ((𝜑 ∧ (𝑘 ∈ ℕ ∧ (𝑘 + 1) ≠ 𝑃)) → (𝑃 < (𝑘 + 1) ↔ (𝑃 ≤ (𝑘 + 1) ∧ (𝑘 + 1) ≠ 𝑃)))
195 simprl 746 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑘 ∈ ℕ ∧ (𝑘 + 1) ≠ 𝑃)) → 𝑘 ∈ ℕ)
196 nnleltp1 11633 . . . . . . . . . . . 12 ((𝑃 ∈ ℕ ∧ 𝑘 ∈ ℕ) → (𝑃𝑘𝑃 < (𝑘 + 1)))
197191, 195, 196syl2anc 565 . . . . . . . . . . 11 ((𝜑 ∧ (𝑘 ∈ ℕ ∧ (𝑘 + 1) ≠ 𝑃)) → (𝑃𝑘𝑃 < (𝑘 + 1)))
198 simprr 748 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑘 ∈ ℕ ∧ (𝑘 + 1) ≠ 𝑃)) → (𝑘 + 1) ≠ 𝑃)
199198biantrud 515 . . . . . . . . . . 11 ((𝜑 ∧ (𝑘 ∈ ℕ ∧ (𝑘 + 1) ≠ 𝑃)) → (𝑃 ≤ (𝑘 + 1) ↔ (𝑃 ≤ (𝑘 + 1) ∧ (𝑘 + 1) ≠ 𝑃)))
200194, 197, 1993bitr4rd 301 . . . . . . . . . 10 ((𝜑 ∧ (𝑘 ∈ ℕ ∧ (𝑘 + 1) ≠ 𝑃)) → (𝑃 ≤ (𝑘 + 1) ↔ 𝑃𝑘))
201200ifbid 4245 . . . . . . . . 9 ((𝜑 ∧ (𝑘 ∈ ℕ ∧ (𝑘 + 1) ≠ 𝑃)) → if(𝑃 ≤ (𝑘 + 1), 𝐵, 0) = if(𝑃𝑘, 𝐵, 0))
202190, 201eqeq12d 2785 . . . . . . . 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 238 . . . . . . 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 444 . . . . . 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 3028 . . . . 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 398 . . . 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 11239 . 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 196  wa 382   = wceq 1630  wcel 2144  wne 2942  wral 3060  Vcvv 3349  csb 3680  ifcif 4223   class class class wbr 4784  cmpt 4861  wf 6027  cfv 6031  (class class class)co 6792  cr 10136  0cc0 10137  1c1 10138   + caddc 10140   · cmul 10142   < clt 10275  cle 10276  cn 11221  2c2 11271  0cn0 11493  cz 11578  cuz 11887  seqcseq 13007  cexp 13066  cdvds 15188  cprime 15591   pCnt cpc 15747
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1869  ax-4 1884  ax-5 1990  ax-6 2056  ax-7 2092  ax-8 2146  ax-9 2153  ax-10 2173  ax-11 2189  ax-12 2202  ax-13 2407  ax-ext 2750  ax-sep 4912  ax-nul 4920  ax-pow 4971  ax-pr 5034  ax-un 7095  ax-cnex 10193  ax-resscn 10194  ax-1cn 10195  ax-icn 10196  ax-addcl 10197  ax-addrcl 10198  ax-mulcl 10199  ax-mulrcl 10200  ax-mulcom 10201  ax-addass 10202  ax-mulass 10203  ax-distr 10204  ax-i2m1 10205  ax-1ne0 10206  ax-1rid 10207  ax-rnegex 10208  ax-rrecex 10209  ax-cnre 10210  ax-pre-lttri 10211  ax-pre-lttrn 10212  ax-pre-ltadd 10213  ax-pre-mulgt0 10214  ax-pre-sup 10215
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 827  df-3or 1071  df-3an 1072  df-tru 1633  df-fal 1636  df-ex 1852  df-nf 1857  df-sb 2049  df-eu 2621  df-mo 2622  df-clab 2757  df-cleq 2763  df-clel 2766  df-nfc 2901  df-ne 2943  df-nel 3046  df-ral 3065  df-rex 3066  df-reu 3067  df-rmo 3068  df-rab 3069  df-v 3351  df-sbc 3586  df-csb 3681  df-dif 3724  df-un 3726  df-in 3728  df-ss 3735  df-pss 3737  df-nul 4062  df-if 4224  df-pw 4297  df-sn 4315  df-pr 4317  df-tp 4319  df-op 4321  df-uni 4573  df-iun 4654  df-br 4785  df-opab 4845  df-mpt 4862  df-tr 4885  df-id 5157  df-eprel 5162  df-po 5170  df-so 5171  df-fr 5208  df-we 5210  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262  df-pred 5823  df-ord 5869  df-on 5870  df-lim 5871  df-suc 5872  df-iota 5994  df-fun 6033  df-fn 6034  df-f 6035  df-f1 6036  df-fo 6037  df-f1o 6038  df-fv 6039  df-riota 6753  df-ov 6795  df-oprab 6796  df-mpt2 6797  df-om 7212  df-1st 7314  df-2nd 7315  df-wrecs 7558  df-recs 7620  df-rdg 7658  df-1o 7712  df-2o 7713  df-er 7895  df-en 8109  df-dom 8110  df-sdom 8111  df-fin 8112  df-sup 8503  df-inf 8504  df-pnf 10277  df-mnf 10278  df-xr 10279  df-ltxr 10280  df-le 10281  df-sub 10469  df-neg 10470  df-div 10886  df-nn 11222  df-2 11280  df-3 11281  df-n0 11494  df-z 11579  df-uz 11888  df-q 11991  df-rp 12035  df-fz 12533  df-fl 12800  df-mod 12876  df-seq 13008  df-exp 13067  df-cj 14046  df-re 14047  df-im 14048  df-sqrt 14182  df-abs 14183  df-dvds 15189  df-gcd 15424  df-prm 15592  df-pc 15748
This theorem is referenced by:  pcmpt2  15803  pcprod  15805  1arithlem4  15836  chtublem  25156  bposlem3  25231
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