MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  pcmpt Structured version   Visualization version   GIF version

Theorem pcmpt 15577
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 6178 . . . . . 6 (𝑝 = 1 → (seq1( · , 𝐹)‘𝑝) = (seq1( · , 𝐹)‘1))
32oveq2d 6651 . . . . 5 (𝑝 = 1 → (𝑃 pCnt (seq1( · , 𝐹)‘𝑝)) = (𝑃 pCnt (seq1( · , 𝐹)‘1)))
4 breq2 4648 . . . . . 6 (𝑝 = 1 → (𝑃𝑝𝑃 ≤ 1))
54ifbid 4099 . . . . 5 (𝑝 = 1 → if(𝑃𝑝, 𝐵, 0) = if(𝑃 ≤ 1, 𝐵, 0))
63, 5eqeq12d 2635 . . . 4 (𝑝 = 1 → ((𝑃 pCnt (seq1( · , 𝐹)‘𝑝)) = if(𝑃𝑝, 𝐵, 0) ↔ (𝑃 pCnt (seq1( · , 𝐹)‘1)) = if(𝑃 ≤ 1, 𝐵, 0)))
76imbi2d 330 . . 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 6178 . . . . . 6 (𝑝 = 𝑘 → (seq1( · , 𝐹)‘𝑝) = (seq1( · , 𝐹)‘𝑘))
98oveq2d 6651 . . . . 5 (𝑝 = 𝑘 → (𝑃 pCnt (seq1( · , 𝐹)‘𝑝)) = (𝑃 pCnt (seq1( · , 𝐹)‘𝑘)))
10 breq2 4648 . . . . . 6 (𝑝 = 𝑘 → (𝑃𝑝𝑃𝑘))
1110ifbid 4099 . . . . 5 (𝑝 = 𝑘 → if(𝑃𝑝, 𝐵, 0) = if(𝑃𝑘, 𝐵, 0))
129, 11eqeq12d 2635 . . . 4 (𝑝 = 𝑘 → ((𝑃 pCnt (seq1( · , 𝐹)‘𝑝)) = if(𝑃𝑝, 𝐵, 0) ↔ (𝑃 pCnt (seq1( · , 𝐹)‘𝑘)) = if(𝑃𝑘, 𝐵, 0)))
1312imbi2d 330 . . 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 6178 . . . . . 6 (𝑝 = (𝑘 + 1) → (seq1( · , 𝐹)‘𝑝) = (seq1( · , 𝐹)‘(𝑘 + 1)))
1514oveq2d 6651 . . . . 5 (𝑝 = (𝑘 + 1) → (𝑃 pCnt (seq1( · , 𝐹)‘𝑝)) = (𝑃 pCnt (seq1( · , 𝐹)‘(𝑘 + 1))))
16 breq2 4648 . . . . . 6 (𝑝 = (𝑘 + 1) → (𝑃𝑝𝑃 ≤ (𝑘 + 1)))
1716ifbid 4099 . . . . 5 (𝑝 = (𝑘 + 1) → if(𝑃𝑝, 𝐵, 0) = if(𝑃 ≤ (𝑘 + 1), 𝐵, 0))
1815, 17eqeq12d 2635 . . . 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 330 . . 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 6178 . . . . . 6 (𝑝 = 𝑁 → (seq1( · , 𝐹)‘𝑝) = (seq1( · , 𝐹)‘𝑁))
2120oveq2d 6651 . . . . 5 (𝑝 = 𝑁 → (𝑃 pCnt (seq1( · , 𝐹)‘𝑝)) = (𝑃 pCnt (seq1( · , 𝐹)‘𝑁)))
22 breq2 4648 . . . . . 6 (𝑝 = 𝑁 → (𝑃𝑝𝑃𝑁))
2322ifbid 4099 . . . . 5 (𝑝 = 𝑁 → if(𝑃𝑝, 𝐵, 0) = if(𝑃𝑁, 𝐵, 0))
2421, 23eqeq12d 2635 . . . 4 (𝑝 = 𝑁 → ((𝑃 pCnt (seq1( · , 𝐹)‘𝑝)) = if(𝑃𝑝, 𝐵, 0) ↔ (𝑃 pCnt (seq1( · , 𝐹)‘𝑁)) = if(𝑃𝑁, 𝐵, 0)))
2524imbi2d 330 . . 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 11392 . . . . . . . . 9 1 ∈ ℤ
28 seq1 12797 . . . . . . . . 9 (1 ∈ ℤ → (seq1( · , 𝐹)‘1) = (𝐹‘1))
2927, 28ax-mp 5 . . . . . . . 8 (seq1( · , 𝐹)‘1) = (𝐹‘1)
30 1nn 11016 . . . . . . . . 9 1 ∈ ℕ
31 1nprm 15373 . . . . . . . . . . . 12 ¬ 1 ∈ ℙ
32 eleq1 2687 . . . . . . . . . . . 12 (𝑛 = 1 → (𝑛 ∈ ℙ ↔ 1 ∈ ℙ))
3331, 32mtbiri 317 . . . . . . . . . . 11 (𝑛 = 1 → ¬ 𝑛 ∈ ℙ)
3433iffalsed 4088 . . . . . . . . . 10 (𝑛 = 1 → if(𝑛 ∈ ℙ, (𝑛𝐴), 1) = 1)
35 pcmpt.1 . . . . . . . . . 10 𝐹 = (𝑛 ∈ ℕ ↦ if(𝑛 ∈ ℙ, (𝑛𝐴), 1))
36 1ex 10020 . . . . . . . . . 10 1 ∈ V
3734, 35, 36fvmpt 6269 . . . . . . . . 9 (1 ∈ ℕ → (𝐹‘1) = 1)
3830, 37ax-mp 5 . . . . . . . 8 (𝐹‘1) = 1
3929, 38eqtri 2642 . . . . . . 7 (seq1( · , 𝐹)‘1) = 1
4039oveq2i 6646 . . . . . 6 (𝑃 pCnt (seq1( · , 𝐹)‘1)) = (𝑃 pCnt 1)
41 pc1 15541 . . . . . 6 (𝑃 ∈ ℙ → (𝑃 pCnt 1) = 0)
4240, 41syl5eq 2666 . . . . 5 (𝑃 ∈ ℙ → (𝑃 pCnt (seq1( · , 𝐹)‘1)) = 0)
43 prmgt1 15390 . . . . . . 7 (𝑃 ∈ ℙ → 1 < 𝑃)
44 1re 10024 . . . . . . . 8 1 ∈ ℝ
45 prmuz2 15389 . . . . . . . . 9 (𝑃 ∈ ℙ → 𝑃 ∈ (ℤ‘2))
46 eluzelre 11683 . . . . . . . . 9 (𝑃 ∈ (ℤ‘2) → 𝑃 ∈ ℝ)
4745, 46syl 17 . . . . . . . 8 (𝑃 ∈ ℙ → 𝑃 ∈ ℝ)
48 ltnle 10102 . . . . . . . 8 ((1 ∈ ℝ ∧ 𝑃 ∈ ℝ) → (1 < 𝑃 ↔ ¬ 𝑃 ≤ 1))
4944, 47, 48sylancr 694 . . . . . . 7 (𝑃 ∈ ℙ → (1 < 𝑃 ↔ ¬ 𝑃 ≤ 1))
5043, 49mpbid 222 . . . . . 6 (𝑃 ∈ ℙ → ¬ 𝑃 ≤ 1)
5150iffalsed 4088 . . . . 5 (𝑃 ∈ ℙ → if(𝑃 ≤ 1, 𝐵, 0) = 0)
5242, 51eqtr4d 2657 . . . 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 481 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑘 ∈ ℕ ∧ (𝑘 + 1) = 𝑃)) → 𝑃 ∈ ℙ)
55 pcmpt.2 . . . . . . . . . . . . . . . . 17 (𝜑 → ∀𝑛 ∈ ℙ 𝐴 ∈ ℕ0)
5635, 55pcmptcl 15576 . . . . . . . . . . . . . . . 16 (𝜑 → (𝐹:ℕ⟶ℕ ∧ seq1( · , 𝐹):ℕ⟶ℕ))
5756simpld 475 . . . . . . . . . . . . . . 15 (𝜑𝐹:ℕ⟶ℕ)
58 peano2nn 11017 . . . . . . . . . . . . . . 15 (𝑘 ∈ ℕ → (𝑘 + 1) ∈ ℕ)
59 ffvelrn 6343 . . . . . . . . . . . . . . 15 ((𝐹:ℕ⟶ℕ ∧ (𝑘 + 1) ∈ ℕ) → (𝐹‘(𝑘 + 1)) ∈ ℕ)
6057, 58, 59syl2an 494 . . . . . . . . . . . . . 14 ((𝜑𝑘 ∈ ℕ) → (𝐹‘(𝑘 + 1)) ∈ ℕ)
6160adantrr 752 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑘 ∈ ℕ ∧ (𝑘 + 1) = 𝑃)) → (𝐹‘(𝑘 + 1)) ∈ ℕ)
6254, 61pccld 15536 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑘 ∈ ℕ ∧ (𝑘 + 1) = 𝑃)) → (𝑃 pCnt (𝐹‘(𝑘 + 1))) ∈ ℕ0)
6362nn0cnd 11338 . . . . . . . . . . 11 ((𝜑 ∧ (𝑘 ∈ ℕ ∧ (𝑘 + 1) = 𝑃)) → (𝑃 pCnt (𝐹‘(𝑘 + 1))) ∈ ℂ)
6463addid2d 10222 . . . . . . . . . 10 ((𝜑 ∧ (𝑘 ∈ ℕ ∧ (𝑘 + 1) = 𝑃)) → (0 + (𝑃 pCnt (𝐹‘(𝑘 + 1)))) = (𝑃 pCnt (𝐹‘(𝑘 + 1))))
6558ad2antrl 763 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑘 ∈ ℕ ∧ (𝑘 + 1) = 𝑃)) → (𝑘 + 1) ∈ ℕ)
66 ovex 6663 . . . . . . . . . . . . . . 15 (𝑛𝐴) ∈ V
6766, 36ifex 4147 . . . . . . . . . . . . . 14 if(𝑛 ∈ ℙ, (𝑛𝐴), 1) ∈ V
6867csbex 4784 . . . . . . . . . . . . 13 (𝑘 + 1) / 𝑛if(𝑛 ∈ ℙ, (𝑛𝐴), 1) ∈ V
6935fvmpts 6272 . . . . . . . . . . . . . 14 (((𝑘 + 1) ∈ ℕ ∧ (𝑘 + 1) / 𝑛if(𝑛 ∈ ℙ, (𝑛𝐴), 1) ∈ V) → (𝐹‘(𝑘 + 1)) = (𝑘 + 1) / 𝑛if(𝑛 ∈ ℙ, (𝑛𝐴), 1))
70 ovex 6663 . . . . . . . . . . . . . . 15 (𝑘 + 1) ∈ V
71 nfv 1841 . . . . . . . . . . . . . . . 16 𝑛(𝑘 + 1) ∈ ℙ
72 nfcv 2762 . . . . . . . . . . . . . . . . 17 𝑛(𝑘 + 1)
73 nfcv 2762 . . . . . . . . . . . . . . . . 17 𝑛
74 nfcsb1v 3542 . . . . . . . . . . . . . . . . 17 𝑛(𝑘 + 1) / 𝑛𝐴
7572, 73, 74nfov 6661 . . . . . . . . . . . . . . . 16 𝑛((𝑘 + 1)↑(𝑘 + 1) / 𝑛𝐴)
76 nfcv 2762 . . . . . . . . . . . . . . . 16 𝑛1
7771, 75, 76nfif 4106 . . . . . . . . . . . . . . 15 𝑛if((𝑘 + 1) ∈ ℙ, ((𝑘 + 1)↑(𝑘 + 1) / 𝑛𝐴), 1)
78 eleq1 2687 . . . . . . . . . . . . . . . 16 (𝑛 = (𝑘 + 1) → (𝑛 ∈ ℙ ↔ (𝑘 + 1) ∈ ℙ))
79 id 22 . . . . . . . . . . . . . . . . 17 (𝑛 = (𝑘 + 1) → 𝑛 = (𝑘 + 1))
80 csbeq1a 3535 . . . . . . . . . . . . . . . . 17 (𝑛 = (𝑘 + 1) → 𝐴 = (𝑘 + 1) / 𝑛𝐴)
8179, 80oveq12d 6653 . . . . . . . . . . . . . . . 16 (𝑛 = (𝑘 + 1) → (𝑛𝐴) = ((𝑘 + 1)↑(𝑘 + 1) / 𝑛𝐴))
8278, 81ifbieq1d 4100 . . . . . . . . . . . . . . 15 (𝑛 = (𝑘 + 1) → if(𝑛 ∈ ℙ, (𝑛𝐴), 1) = if((𝑘 + 1) ∈ ℙ, ((𝑘 + 1)↑(𝑘 + 1) / 𝑛𝐴), 1))
8370, 77, 82csbief 3551 . . . . . . . . . . . . . 14 (𝑘 + 1) / 𝑛if(𝑛 ∈ ℙ, (𝑛𝐴), 1) = if((𝑘 + 1) ∈ ℙ, ((𝑘 + 1)↑(𝑘 + 1) / 𝑛𝐴), 1)
8469, 83syl6eq 2670 . . . . . . . . . . . . 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 693 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑘 ∈ ℕ ∧ (𝑘 + 1) = 𝑃)) → (𝐹‘(𝑘 + 1)) = if((𝑘 + 1) ∈ ℙ, ((𝑘 + 1)↑(𝑘 + 1) / 𝑛𝐴), 1))
86 simprr 795 . . . . . . . . . . . . . 14 ((𝜑 ∧ (𝑘 ∈ ℕ ∧ (𝑘 + 1) = 𝑃)) → (𝑘 + 1) = 𝑃)
8786, 54eqeltrd 2699 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑘 ∈ ℕ ∧ (𝑘 + 1) = 𝑃)) → (𝑘 + 1) ∈ ℙ)
8887iftrued 4085 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑘 ∈ ℕ ∧ (𝑘 + 1) = 𝑃)) → if((𝑘 + 1) ∈ ℙ, ((𝑘 + 1)↑(𝑘 + 1) / 𝑛𝐴), 1) = ((𝑘 + 1)↑(𝑘 + 1) / 𝑛𝐴))
8986csbeq1d 3533 . . . . . . . . . . . . . 14 ((𝜑 ∧ (𝑘 ∈ ℕ ∧ (𝑘 + 1) = 𝑃)) → (𝑘 + 1) / 𝑛𝐴 = 𝑃 / 𝑛𝐴)
90 nfcvd 2763 . . . . . . . . . . . . . . . 16 (𝑃 ∈ ℙ → 𝑛𝐵)
91 pcmpt.5 . . . . . . . . . . . . . . . 16 (𝑛 = 𝑃𝐴 = 𝐵)
9290, 91csbiegf 3550 . . . . . . . . . . . . . . 15 (𝑃 ∈ ℙ → 𝑃 / 𝑛𝐴 = 𝐵)
9354, 92syl 17 . . . . . . . . . . . . . 14 ((𝜑 ∧ (𝑘 ∈ ℕ ∧ (𝑘 + 1) = 𝑃)) → 𝑃 / 𝑛𝐴 = 𝐵)
9489, 93eqtrd 2654 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑘 ∈ ℕ ∧ (𝑘 + 1) = 𝑃)) → (𝑘 + 1) / 𝑛𝐴 = 𝐵)
9586, 94oveq12d 6653 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑘 ∈ ℕ ∧ (𝑘 + 1) = 𝑃)) → ((𝑘 + 1)↑(𝑘 + 1) / 𝑛𝐴) = (𝑃𝐵))
9685, 88, 953eqtrd 2658 . . . . . . . . . . 11 ((𝜑 ∧ (𝑘 ∈ ℕ ∧ (𝑘 + 1) = 𝑃)) → (𝐹‘(𝑘 + 1)) = (𝑃𝐵))
9796oveq2d 6651 . . . . . . . . . 10 ((𝜑 ∧ (𝑘 ∈ ℕ ∧ (𝑘 + 1) = 𝑃)) → (𝑃 pCnt (𝐹‘(𝑘 + 1))) = (𝑃 pCnt (𝑃𝐵)))
9891eleq1d 2684 . . . . . . . . . . . . . 14 (𝑛 = 𝑃 → (𝐴 ∈ ℕ0𝐵 ∈ ℕ0))
9998rspcv 3300 . . . . . . . . . . . . 13 (𝑃 ∈ ℙ → (∀𝑛 ∈ ℙ 𝐴 ∈ ℕ0𝐵 ∈ ℕ0))
10026, 55, 99sylc 65 . . . . . . . . . . . 12 (𝜑𝐵 ∈ ℕ0)
101100adantr 481 . . . . . . . . . . 11 ((𝜑 ∧ (𝑘 ∈ ℕ ∧ (𝑘 + 1) = 𝑃)) → 𝐵 ∈ ℕ0)
102 pcidlem 15557 . . . . . . . . . . 11 ((𝑃 ∈ ℙ ∧ 𝐵 ∈ ℕ0) → (𝑃 pCnt (𝑃𝐵)) = 𝐵)
10354, 101, 102syl2anc 692 . . . . . . . . . 10 ((𝜑 ∧ (𝑘 ∈ ℕ ∧ (𝑘 + 1) = 𝑃)) → (𝑃 pCnt (𝑃𝐵)) = 𝐵)
10464, 97, 1033eqtrd 2658 . . . . . . . . 9 ((𝜑 ∧ (𝑘 ∈ ℕ ∧ (𝑘 + 1) = 𝑃)) → (0 + (𝑃 pCnt (𝐹‘(𝑘 + 1)))) = 𝐵)
105 oveq1 6642 . . . . . . . . . 10 ((𝑃 pCnt (seq1( · , 𝐹)‘𝑘)) = 0 → ((𝑃 pCnt (seq1( · , 𝐹)‘𝑘)) + (𝑃 pCnt (𝐹‘(𝑘 + 1)))) = (0 + (𝑃 pCnt (𝐹‘(𝑘 + 1)))))
106105eqeq1d 2622 . . . . . . . . 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 11012 . . . . . . . . . . . . 13 (𝑘 ∈ ℕ → 𝑘 ∈ ℝ)
109108ad2antrl 763 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑘 ∈ ℕ ∧ (𝑘 + 1) = 𝑃)) → 𝑘 ∈ ℝ)
110 ltp1 10846 . . . . . . . . . . . . 13 (𝑘 ∈ ℝ → 𝑘 < (𝑘 + 1))
111 peano2re 10194 . . . . . . . . . . . . . 14 (𝑘 ∈ ℝ → (𝑘 + 1) ∈ ℝ)
112 ltnle 10102 . . . . . . . . . . . . . 14 ((𝑘 ∈ ℝ ∧ (𝑘 + 1) ∈ ℝ) → (𝑘 < (𝑘 + 1) ↔ ¬ (𝑘 + 1) ≤ 𝑘))
113111, 112mpdan 701 . . . . . . . . . . . . 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 4654 . . . . . . . . . . 11 ((𝜑 ∧ (𝑘 ∈ ℕ ∧ (𝑘 + 1) = 𝑃)) → ((𝑘 + 1) ≤ 𝑘𝑃𝑘))
117115, 116mtbid 314 . . . . . . . . . 10 ((𝜑 ∧ (𝑘 ∈ ℕ ∧ (𝑘 + 1) = 𝑃)) → ¬ 𝑃𝑘)
118117iffalsed 4088 . . . . . . . . 9 ((𝜑 ∧ (𝑘 ∈ ℕ ∧ (𝑘 + 1) = 𝑃)) → if(𝑃𝑘, 𝐵, 0) = 0)
119118eqeq2d 2630 . . . . . . . 8 ((𝜑 ∧ (𝑘 ∈ ℕ ∧ (𝑘 + 1) = 𝑃)) → ((𝑃 pCnt (seq1( · , 𝐹)‘𝑘)) = if(𝑃𝑘, 𝐵, 0) ↔ (𝑃 pCnt (seq1( · , 𝐹)‘𝑘)) = 0))
120 simpr 477 . . . . . . . . . . . . . 14 ((𝜑𝑘 ∈ ℕ) → 𝑘 ∈ ℕ)
121 nnuz 11708 . . . . . . . . . . . . . 14 ℕ = (ℤ‘1)
122120, 121syl6eleq 2709 . . . . . . . . . . . . 13 ((𝜑𝑘 ∈ ℕ) → 𝑘 ∈ (ℤ‘1))
123 seqp1 12799 . . . . . . . . . . . . 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 6651 . . . . . . . . . . 11 ((𝜑𝑘 ∈ ℕ) → (𝑃 pCnt (seq1( · , 𝐹)‘(𝑘 + 1))) = (𝑃 pCnt ((seq1( · , 𝐹)‘𝑘) · (𝐹‘(𝑘 + 1)))))
12626adantr 481 . . . . . . . . . . . 12 ((𝜑𝑘 ∈ ℕ) → 𝑃 ∈ ℙ)
12756simprd 479 . . . . . . . . . . . . . 14 (𝜑 → seq1( · , 𝐹):ℕ⟶ℕ)
128127ffvelrnda 6345 . . . . . . . . . . . . 13 ((𝜑𝑘 ∈ ℕ) → (seq1( · , 𝐹)‘𝑘) ∈ ℕ)
129 nnz 11384 . . . . . . . . . . . . . 14 ((seq1( · , 𝐹)‘𝑘) ∈ ℕ → (seq1( · , 𝐹)‘𝑘) ∈ ℤ)
130 nnne0 11038 . . . . . . . . . . . . . 14 ((seq1( · , 𝐹)‘𝑘) ∈ ℕ → (seq1( · , 𝐹)‘𝑘) ≠ 0)
131129, 130jca 554 . . . . . . . . . . . . 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 11384 . . . . . . . . . . . . . 14 ((𝐹‘(𝑘 + 1)) ∈ ℕ → (𝐹‘(𝑘 + 1)) ∈ ℤ)
134 nnne0 11038 . . . . . . . . . . . . . 14 ((𝐹‘(𝑘 + 1)) ∈ ℕ → (𝐹‘(𝑘 + 1)) ≠ 0)
135133, 134jca 554 . . . . . . . . . . . . 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 15537 . . . . . . . . . . . 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 1324 . . . . . . . . . . 11 ((𝜑𝑘 ∈ ℕ) → (𝑃 pCnt ((seq1( · , 𝐹)‘𝑘) · (𝐹‘(𝑘 + 1)))) = ((𝑃 pCnt (seq1( · , 𝐹)‘𝑘)) + (𝑃 pCnt (𝐹‘(𝑘 + 1)))))
139125, 138eqtrd 2654 . . . . . . . . . 10 ((𝜑𝑘 ∈ ℕ) → (𝑃 pCnt (seq1( · , 𝐹)‘(𝑘 + 1))) = ((𝑃 pCnt (seq1( · , 𝐹)‘𝑘)) + (𝑃 pCnt (𝐹‘(𝑘 + 1)))))
140139adantrr 752 . . . . . . . . 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 15369 . . . . . . . . . . . . . . 15 (𝑃 ∈ ℙ → 𝑃 ∈ ℕ)
14226, 141syl 17 . . . . . . . . . . . . . 14 (𝜑𝑃 ∈ ℕ)
143142nnred 11020 . . . . . . . . . . . . 13 (𝜑𝑃 ∈ ℝ)
144143adantr 481 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑘 ∈ ℕ ∧ (𝑘 + 1) = 𝑃)) → 𝑃 ∈ ℝ)
145144leidd 10579 . . . . . . . . . . 11 ((𝜑 ∧ (𝑘 ∈ ℕ ∧ (𝑘 + 1) = 𝑃)) → 𝑃𝑃)
146145, 86breqtrrd 4672 . . . . . . . . . 10 ((𝜑 ∧ (𝑘 ∈ ℕ ∧ (𝑘 + 1) = 𝑃)) → 𝑃 ≤ (𝑘 + 1))
147146iftrued 4085 . . . . . . . . 9 ((𝜑 ∧ (𝑘 ∈ ℕ ∧ (𝑘 + 1) = 𝑃)) → if(𝑃 ≤ (𝑘 + 1), 𝐵, 0) = 𝐵)
148140, 147eqeq12d 2635 . . . . . . . 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 642 . . . . . 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 752 . . . . . . . . . 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 800 . . . . . . . . . . . . . . . 16 (((𝜑 ∧ (𝑘 ∈ ℕ ∧ (𝑘 + 1) ≠ 𝑃)) ∧ (𝑘 + 1) ∈ ℙ) → (𝑘 + 1) ≠ 𝑃)
153152necomd 2846 . . . . . . . . . . . . . . 15 (((𝜑 ∧ (𝑘 ∈ ℕ ∧ (𝑘 + 1) ≠ 𝑃)) ∧ (𝑘 + 1) ∈ ℙ) → 𝑃 ≠ (𝑘 + 1))
15426ad2antrr 761 . . . . . . . . . . . . . . . . 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)
15655ad2antrr 761 . . . . . . . . . . . . . . . . . 18 (((𝜑 ∧ (𝑘 ∈ ℕ ∧ (𝑘 + 1) ≠ 𝑃)) ∧ (𝑘 + 1) ∈ ℙ) → ∀𝑛 ∈ ℙ 𝐴 ∈ ℕ0)
15774nfel1 2776 . . . . . . . . . . . . . . . . . . 19 𝑛(𝑘 + 1) / 𝑛𝐴 ∈ ℕ0
15880eleq1d 2684 . . . . . . . . . . . . . . . . . . 19 (𝑛 = (𝑘 + 1) → (𝐴 ∈ ℕ0(𝑘 + 1) / 𝑛𝐴 ∈ ℕ0))
159157, 158rspc 3298 . . . . . . . . . . . . . . . . . 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 15410 . . . . . . . . . . . . . . . . 17 ((𝑃 ∈ ℙ ∧ (𝑘 + 1) ∈ ℙ ∧ (𝑘 + 1) / 𝑛𝐴 ∈ ℕ0) → (𝑃 ∥ ((𝑘 + 1)↑(𝑘 + 1) / 𝑛𝐴) → 𝑃 = (𝑘 + 1)))
162154, 155, 160, 161syl3anc 1324 . . . . . . . . . . . . . . . 16 (((𝜑 ∧ (𝑘 ∈ ℕ ∧ (𝑘 + 1) ≠ 𝑃)) ∧ (𝑘 + 1) ∈ ℙ) → (𝑃 ∥ ((𝑘 + 1)↑(𝑘 + 1) / 𝑛𝐴) → 𝑃 = (𝑘 + 1)))
163162necon3ad 2804 . . . . . . . . . . . . . . 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 763 . . . . . . . . . . . . . . . . 17 ((𝜑 ∧ (𝑘 ∈ ℕ ∧ (𝑘 + 1) ≠ 𝑃)) → (𝑘 + 1) ∈ ℕ)
166165, 68, 84sylancl 693 . . . . . . . . . . . . . . . 16 ((𝜑 ∧ (𝑘 ∈ ℕ ∧ (𝑘 + 1) ≠ 𝑃)) → (𝐹‘(𝑘 + 1)) = if((𝑘 + 1) ∈ ℙ, ((𝑘 + 1)↑(𝑘 + 1) / 𝑛𝐴), 1))
167 iftrue 4083 . . . . . . . . . . . . . . . 16 ((𝑘 + 1) ∈ ℙ → if((𝑘 + 1) ∈ ℙ, ((𝑘 + 1)↑(𝑘 + 1) / 𝑛𝐴), 1) = ((𝑘 + 1)↑(𝑘 + 1) / 𝑛𝐴))
168166, 167sylan9eq 2674 . . . . . . . . . . . . . . 15 (((𝜑 ∧ (𝑘 ∈ ℕ ∧ (𝑘 + 1) ≠ 𝑃)) ∧ (𝑘 + 1) ∈ ℙ) → (𝐹‘(𝑘 + 1)) = ((𝑘 + 1)↑(𝑘 + 1) / 𝑛𝐴))
169168breq2d 4656 . . . . . . . . . . . . . 14 (((𝜑 ∧ (𝑘 ∈ ℕ ∧ (𝑘 + 1) ≠ 𝑃)) ∧ (𝑘 + 1) ∈ ℙ) → (𝑃 ∥ (𝐹‘(𝑘 + 1)) ↔ 𝑃 ∥ ((𝑘 + 1)↑(𝑘 + 1) / 𝑛𝐴)))
170164, 169mtbird 315 . . . . . . . . . . . . 13 (((𝜑 ∧ (𝑘 ∈ ℕ ∧ (𝑘 + 1) ≠ 𝑃)) ∧ (𝑘 + 1) ∈ ℙ) → ¬ 𝑃 ∥ (𝐹‘(𝑘 + 1)))
17157adantr 481 . . . . . . . . . . . . . . . 16 ((𝜑 ∧ (𝑘 ∈ ℕ ∧ (𝑘 + 1) ≠ 𝑃)) → 𝐹:ℕ⟶ℕ)
172171, 165, 59syl2anc 692 . . . . . . . . . . . . . . 15 ((𝜑 ∧ (𝑘 ∈ ℕ ∧ (𝑘 + 1) ≠ 𝑃)) → (𝐹‘(𝑘 + 1)) ∈ ℕ)
173172adantr 481 . . . . . . . . . . . . . 14 (((𝜑 ∧ (𝑘 ∈ ℕ ∧ (𝑘 + 1) ≠ 𝑃)) ∧ (𝑘 + 1) ∈ ℙ) → (𝐹‘(𝑘 + 1)) ∈ ℕ)
174 pceq0 15556 . . . . . . . . . . . . . 14 ((𝑃 ∈ ℙ ∧ (𝐹‘(𝑘 + 1)) ∈ ℕ) → ((𝑃 pCnt (𝐹‘(𝑘 + 1))) = 0 ↔ ¬ 𝑃 ∥ (𝐹‘(𝑘 + 1))))
175154, 173, 174syl2anc 692 . . . . . . . . . . . . 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 4086 . . . . . . . . . . . . . . 15 (¬ (𝑘 + 1) ∈ ℙ → if((𝑘 + 1) ∈ ℙ, ((𝑘 + 1)↑(𝑘 + 1) / 𝑛𝐴), 1) = 1)
178166, 177sylan9eq 2674 . . . . . . . . . . . . . 14 (((𝜑 ∧ (𝑘 ∈ ℕ ∧ (𝑘 + 1) ≠ 𝑃)) ∧ ¬ (𝑘 + 1) ∈ ℙ) → (𝐹‘(𝑘 + 1)) = 1)
179178oveq2d 6651 . . . . . . . . . . . . 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 761 . . . . . . . . . . . . 13 (((𝜑 ∧ (𝑘 ∈ ℕ ∧ (𝑘 + 1) ≠ 𝑃)) ∧ ¬ (𝑘 + 1) ∈ ℙ) → (𝑃 pCnt 1) = 0)
182179, 181eqtrd 2654 . . . . . . . . . . . 12 (((𝜑 ∧ (𝑘 ∈ ℕ ∧ (𝑘 + 1) ≠ 𝑃)) ∧ ¬ (𝑘 + 1) ∈ ℙ) → (𝑃 pCnt (𝐹‘(𝑘 + 1))) = 0)
183176, 182pm2.61dan 831 . . . . . . . . . . 11 ((𝜑 ∧ (𝑘 ∈ ℕ ∧ (𝑘 + 1) ≠ 𝑃)) → (𝑃 pCnt (𝐹‘(𝑘 + 1))) = 0)
184183oveq2d 6651 . . . . . . . . . 10 ((𝜑 ∧ (𝑘 ∈ ℕ ∧ (𝑘 + 1) ≠ 𝑃)) → ((𝑃 pCnt (seq1( · , 𝐹)‘𝑘)) + (𝑃 pCnt (𝐹‘(𝑘 + 1)))) = ((𝑃 pCnt (seq1( · , 𝐹)‘𝑘)) + 0))
18526adantr 481 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑘 ∈ ℕ ∧ (𝑘 + 1) ≠ 𝑃)) → 𝑃 ∈ ℙ)
186128adantrr 752 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑘 ∈ ℕ ∧ (𝑘 + 1) ≠ 𝑃)) → (seq1( · , 𝐹)‘𝑘) ∈ ℕ)
187185, 186pccld 15536 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑘 ∈ ℕ ∧ (𝑘 + 1) ≠ 𝑃)) → (𝑃 pCnt (seq1( · , 𝐹)‘𝑘)) ∈ ℕ0)
188187nn0cnd 11338 . . . . . . . . . . 11 ((𝜑 ∧ (𝑘 ∈ ℕ ∧ (𝑘 + 1) ≠ 𝑃)) → (𝑃 pCnt (seq1( · , 𝐹)‘𝑘)) ∈ ℂ)
189188addid1d 10221 . . . . . . . . . 10 ((𝜑 ∧ (𝑘 ∈ ℕ ∧ (𝑘 + 1) ≠ 𝑃)) → ((𝑃 pCnt (seq1( · , 𝐹)‘𝑘)) + 0) = (𝑃 pCnt (seq1( · , 𝐹)‘𝑘)))
190151, 184, 1893eqtrd 2658 . . . . . . . . 9 ((𝜑 ∧ (𝑘 ∈ ℕ ∧ (𝑘 + 1) ≠ 𝑃)) → (𝑃 pCnt (seq1( · , 𝐹)‘(𝑘 + 1))) = (𝑃 pCnt (seq1( · , 𝐹)‘𝑘)))
191142adantr 481 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑘 ∈ ℕ ∧ (𝑘 + 1) ≠ 𝑃)) → 𝑃 ∈ ℕ)
192191nnred 11020 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑘 ∈ ℕ ∧ (𝑘 + 1) ≠ 𝑃)) → 𝑃 ∈ ℝ)
193165nnred 11020 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑘 ∈ ℕ ∧ (𝑘 + 1) ≠ 𝑃)) → (𝑘 + 1) ∈ ℝ)
194192, 193ltlend 10167 . . . . . . . . . . 11 ((𝜑 ∧ (𝑘 ∈ ℕ ∧ (𝑘 + 1) ≠ 𝑃)) → (𝑃 < (𝑘 + 1) ↔ (𝑃 ≤ (𝑘 + 1) ∧ (𝑘 + 1) ≠ 𝑃)))
195 simprl 793 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑘 ∈ ℕ ∧ (𝑘 + 1) ≠ 𝑃)) → 𝑘 ∈ ℕ)
196 nnleltp1 11417 . . . . . . . . . . . 12 ((𝑃 ∈ ℕ ∧ 𝑘 ∈ ℕ) → (𝑃𝑘𝑃 < (𝑘 + 1)))
197191, 195, 196syl2anc 692 . . . . . . . . . . 11 ((𝜑 ∧ (𝑘 ∈ ℕ ∧ (𝑘 + 1) ≠ 𝑃)) → (𝑃𝑘𝑃 < (𝑘 + 1)))
198 simprr 795 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑘 ∈ ℕ ∧ (𝑘 + 1) ≠ 𝑃)) → (𝑘 + 1) ≠ 𝑃)
199198biantrud 528 . . . . . . . . . . 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 4099 . . . . . . . . 9 ((𝜑 ∧ (𝑘 ∈ ℕ ∧ (𝑘 + 1) ≠ 𝑃)) → if(𝑃 ≤ (𝑘 + 1), 𝐵, 0) = if(𝑃𝑘, 𝐵, 0))
202190, 201eqeq12d 2635 . . . . . . . 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 642 . . . . . 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 2877 . . . . 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 451 . . . 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 11023 . 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 384   = wceq 1481  wcel 1988  wne 2791  wral 2909  Vcvv 3195  csb 3526  ifcif 4077   class class class wbr 4644  cmpt 4720  wf 5872  cfv 5876  (class class class)co 6635  cr 9920  0cc0 9921  1c1 9922   + caddc 9924   · cmul 9926   < clt 10059  cle 10060  cn 11005  2c2 11055  0cn0 11277  cz 11362  cuz 11672  seqcseq 12784  cexp 12843  cdvds 14964  cprime 15366   pCnt cpc 15522
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1720  ax-4 1735  ax-5 1837  ax-6 1886  ax-7 1933  ax-8 1990  ax-9 1997  ax-10 2017  ax-11 2032  ax-12 2045  ax-13 2244  ax-ext 2600  ax-sep 4772  ax-nul 4780  ax-pow 4834  ax-pr 4897  ax-un 6934  ax-cnex 9977  ax-resscn 9978  ax-1cn 9979  ax-icn 9980  ax-addcl 9981  ax-addrcl 9982  ax-mulcl 9983  ax-mulrcl 9984  ax-mulcom 9985  ax-addass 9986  ax-mulass 9987  ax-distr 9988  ax-i2m1 9989  ax-1ne0 9990  ax-1rid 9991  ax-rnegex 9992  ax-rrecex 9993  ax-cnre 9994  ax-pre-lttri 9995  ax-pre-lttrn 9996  ax-pre-ltadd 9997  ax-pre-mulgt0 9998  ax-pre-sup 9999
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1484  df-fal 1487  df-ex 1703  df-nf 1708  df-sb 1879  df-eu 2472  df-mo 2473  df-clab 2607  df-cleq 2613  df-clel 2616  df-nfc 2751  df-ne 2792  df-nel 2895  df-ral 2914  df-rex 2915  df-reu 2916  df-rmo 2917  df-rab 2918  df-v 3197  df-sbc 3430  df-csb 3527  df-dif 3570  df-un 3572  df-in 3574  df-ss 3581  df-pss 3583  df-nul 3908  df-if 4078  df-pw 4151  df-sn 4169  df-pr 4171  df-tp 4173  df-op 4175  df-uni 4428  df-int 4467  df-iun 4513  df-br 4645  df-opab 4704  df-mpt 4721  df-tr 4744  df-id 5014  df-eprel 5019  df-po 5025  df-so 5026  df-fr 5063  df-we 5065  df-xp 5110  df-rel 5111  df-cnv 5112  df-co 5113  df-dm 5114  df-rn 5115  df-res 5116  df-ima 5117  df-pred 5668  df-ord 5714  df-on 5715  df-lim 5716  df-suc 5717  df-iota 5839  df-fun 5878  df-fn 5879  df-f 5880  df-f1 5881  df-fo 5882  df-f1o 5883  df-fv 5884  df-riota 6596  df-ov 6638  df-oprab 6639  df-mpt2 6640  df-om 7051  df-1st 7153  df-2nd 7154  df-wrecs 7392  df-recs 7453  df-rdg 7491  df-1o 7545  df-2o 7546  df-oadd 7549  df-er 7727  df-en 7941  df-dom 7942  df-sdom 7943  df-fin 7944  df-sup 8333  df-inf 8334  df-pnf 10061  df-mnf 10062  df-xr 10063  df-ltxr 10064  df-le 10065  df-sub 10253  df-neg 10254  df-div 10670  df-nn 11006  df-2 11064  df-3 11065  df-n0 11278  df-z 11363  df-uz 11673  df-q 11774  df-rp 11818  df-fz 12312  df-fl 12576  df-mod 12652  df-seq 12785  df-exp 12844  df-cj 13820  df-re 13821  df-im 13822  df-sqrt 13956  df-abs 13957  df-dvds 14965  df-gcd 15198  df-prm 15367  df-pc 15523
This theorem is referenced by:  pcmpt2  15578  pcprod  15580  1arithlem4  15611  chtublem  24917  bposlem3  24992
  Copyright terms: Public domain W3C validator