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Theorem pcmpt 12324
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 𝑖 𝑗 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 pcmpt.3 . 2 (𝜑𝑁 ∈ ℕ)
2 fveq2 5511 . . . . . 6 (𝑝 = 1 → (seq1( · , 𝐹)‘𝑝) = (seq1( · , 𝐹)‘1))
32oveq2d 5885 . . . . 5 (𝑝 = 1 → (𝑃 pCnt (seq1( · , 𝐹)‘𝑝)) = (𝑃 pCnt (seq1( · , 𝐹)‘1)))
4 breq2 4004 . . . . . 6 (𝑝 = 1 → (𝑃𝑝𝑃 ≤ 1))
54ifbid 3555 . . . . 5 (𝑝 = 1 → if(𝑃𝑝, 𝐵, 0) = if(𝑃 ≤ 1, 𝐵, 0))
63, 5eqeq12d 2192 . . . 4 (𝑝 = 1 → ((𝑃 pCnt (seq1( · , 𝐹)‘𝑝)) = if(𝑃𝑝, 𝐵, 0) ↔ (𝑃 pCnt (seq1( · , 𝐹)‘1)) = if(𝑃 ≤ 1, 𝐵, 0)))
76imbi2d 230 . . 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 5511 . . . . . 6 (𝑝 = 𝑘 → (seq1( · , 𝐹)‘𝑝) = (seq1( · , 𝐹)‘𝑘))
98oveq2d 5885 . . . . 5 (𝑝 = 𝑘 → (𝑃 pCnt (seq1( · , 𝐹)‘𝑝)) = (𝑃 pCnt (seq1( · , 𝐹)‘𝑘)))
10 breq2 4004 . . . . . 6 (𝑝 = 𝑘 → (𝑃𝑝𝑃𝑘))
1110ifbid 3555 . . . . 5 (𝑝 = 𝑘 → if(𝑃𝑝, 𝐵, 0) = if(𝑃𝑘, 𝐵, 0))
129, 11eqeq12d 2192 . . . 4 (𝑝 = 𝑘 → ((𝑃 pCnt (seq1( · , 𝐹)‘𝑝)) = if(𝑃𝑝, 𝐵, 0) ↔ (𝑃 pCnt (seq1( · , 𝐹)‘𝑘)) = if(𝑃𝑘, 𝐵, 0)))
1312imbi2d 230 . . 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 5511 . . . . . 6 (𝑝 = (𝑘 + 1) → (seq1( · , 𝐹)‘𝑝) = (seq1( · , 𝐹)‘(𝑘 + 1)))
1514oveq2d 5885 . . . . 5 (𝑝 = (𝑘 + 1) → (𝑃 pCnt (seq1( · , 𝐹)‘𝑝)) = (𝑃 pCnt (seq1( · , 𝐹)‘(𝑘 + 1))))
16 breq2 4004 . . . . . 6 (𝑝 = (𝑘 + 1) → (𝑃𝑝𝑃 ≤ (𝑘 + 1)))
1716ifbid 3555 . . . . 5 (𝑝 = (𝑘 + 1) → if(𝑃𝑝, 𝐵, 0) = if(𝑃 ≤ (𝑘 + 1), 𝐵, 0))
1815, 17eqeq12d 2192 . . . 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 230 . . 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 5511 . . . . . 6 (𝑝 = 𝑁 → (seq1( · , 𝐹)‘𝑝) = (seq1( · , 𝐹)‘𝑁))
2120oveq2d 5885 . . . . 5 (𝑝 = 𝑁 → (𝑃 pCnt (seq1( · , 𝐹)‘𝑝)) = (𝑃 pCnt (seq1( · , 𝐹)‘𝑁)))
22 breq2 4004 . . . . . 6 (𝑝 = 𝑁 → (𝑃𝑝𝑃𝑁))
2322ifbid 3555 . . . . 5 (𝑝 = 𝑁 → if(𝑃𝑝, 𝐵, 0) = if(𝑃𝑁, 𝐵, 0))
2421, 23eqeq12d 2192 . . . 4 (𝑝 = 𝑁 → ((𝑃 pCnt (seq1( · , 𝐹)‘𝑝)) = if(𝑃𝑝, 𝐵, 0) ↔ (𝑃 pCnt (seq1( · , 𝐹)‘𝑁)) = if(𝑃𝑁, 𝐵, 0)))
2524imbi2d 230 . . 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 . . . . 5 (𝜑𝑃 ∈ ℙ)
27 pc1 12288 . . . . 5 (𝑃 ∈ ℙ → (𝑃 pCnt 1) = 0)
2826, 27syl 14 . . . 4 (𝜑 → (𝑃 pCnt 1) = 0)
29 1zzd 9269 . . . . . . 7 (𝜑 → 1 ∈ ℤ)
30 elnnuz 9553 . . . . . . . 8 (𝑖 ∈ ℕ ↔ 𝑖 ∈ (ℤ‘1))
31 simpr 110 . . . . . . . . . 10 ((𝜑𝑖 ∈ ℕ) → 𝑖 ∈ ℕ)
3231adantr 276 . . . . . . . . . . . 12 (((𝜑𝑖 ∈ ℕ) ∧ 𝑖 ∈ ℙ) → 𝑖 ∈ ℕ)
33 simpr 110 . . . . . . . . . . . . 13 (((𝜑𝑖 ∈ ℕ) ∧ 𝑖 ∈ ℙ) → 𝑖 ∈ ℙ)
34 pcmpt.2 . . . . . . . . . . . . . 14 (𝜑 → ∀𝑛 ∈ ℙ 𝐴 ∈ ℕ0)
3534ad2antrr 488 . . . . . . . . . . . . 13 (((𝜑𝑖 ∈ ℕ) ∧ 𝑖 ∈ ℙ) → ∀𝑛 ∈ ℙ 𝐴 ∈ ℕ0)
36 nfcsb1v 3090 . . . . . . . . . . . . . . 15 𝑛𝑖 / 𝑛𝐴
3736nfel1 2330 . . . . . . . . . . . . . 14 𝑛𝑖 / 𝑛𝐴 ∈ ℕ0
38 csbeq1a 3066 . . . . . . . . . . . . . . 15 (𝑛 = 𝑖𝐴 = 𝑖 / 𝑛𝐴)
3938eleq1d 2246 . . . . . . . . . . . . . 14 (𝑛 = 𝑖 → (𝐴 ∈ ℕ0𝑖 / 𝑛𝐴 ∈ ℕ0))
4037, 39rspc 2835 . . . . . . . . . . . . 13 (𝑖 ∈ ℙ → (∀𝑛 ∈ ℙ 𝐴 ∈ ℕ0𝑖 / 𝑛𝐴 ∈ ℕ0))
4133, 35, 40sylc 62 . . . . . . . . . . . 12 (((𝜑𝑖 ∈ ℕ) ∧ 𝑖 ∈ ℙ) → 𝑖 / 𝑛𝐴 ∈ ℕ0)
4232, 41nnexpcld 10661 . . . . . . . . . . 11 (((𝜑𝑖 ∈ ℕ) ∧ 𝑖 ∈ ℙ) → (𝑖𝑖 / 𝑛𝐴) ∈ ℕ)
43 1nn 8919 . . . . . . . . . . . 12 1 ∈ ℕ
4443a1i 9 . . . . . . . . . . 11 (((𝜑𝑖 ∈ ℕ) ∧ ¬ 𝑖 ∈ ℙ) → 1 ∈ ℕ)
45 prmdc 12113 . . . . . . . . . . . 12 (𝑖 ∈ ℕ → DECID 𝑖 ∈ ℙ)
4645adantl 277 . . . . . . . . . . 11 ((𝜑𝑖 ∈ ℕ) → DECID 𝑖 ∈ ℙ)
4742, 44, 46ifcldadc 3563 . . . . . . . . . 10 ((𝜑𝑖 ∈ ℕ) → if(𝑖 ∈ ℙ, (𝑖𝑖 / 𝑛𝐴), 1) ∈ ℕ)
48 nfcv 2319 . . . . . . . . . . 11 𝑛𝑖
4948nfel1 2330 . . . . . . . . . . . 12 𝑛 𝑖 ∈ ℙ
50 nfcv 2319 . . . . . . . . . . . . 13 𝑛
5148, 50, 36nfov 5899 . . . . . . . . . . . 12 𝑛(𝑖𝑖 / 𝑛𝐴)
52 nfcv 2319 . . . . . . . . . . . 12 𝑛1
5349, 51, 52nfif 3562 . . . . . . . . . . 11 𝑛if(𝑖 ∈ ℙ, (𝑖𝑖 / 𝑛𝐴), 1)
54 eleq1 2240 . . . . . . . . . . . 12 (𝑛 = 𝑖 → (𝑛 ∈ ℙ ↔ 𝑖 ∈ ℙ))
55 id 19 . . . . . . . . . . . . 13 (𝑛 = 𝑖𝑛 = 𝑖)
5655, 38oveq12d 5887 . . . . . . . . . . . 12 (𝑛 = 𝑖 → (𝑛𝐴) = (𝑖𝑖 / 𝑛𝐴))
5754, 56ifbieq1d 3556 . . . . . . . . . . 11 (𝑛 = 𝑖 → if(𝑛 ∈ ℙ, (𝑛𝐴), 1) = if(𝑖 ∈ ℙ, (𝑖𝑖 / 𝑛𝐴), 1))
58 pcmpt.1 . . . . . . . . . . 11 𝐹 = (𝑛 ∈ ℕ ↦ if(𝑛 ∈ ℙ, (𝑛𝐴), 1))
5948, 53, 57, 58fvmptf 5604 . . . . . . . . . 10 ((𝑖 ∈ ℕ ∧ if(𝑖 ∈ ℙ, (𝑖𝑖 / 𝑛𝐴), 1) ∈ ℕ) → (𝐹𝑖) = if(𝑖 ∈ ℙ, (𝑖𝑖 / 𝑛𝐴), 1))
6031, 47, 59syl2anc 411 . . . . . . . . 9 ((𝜑𝑖 ∈ ℕ) → (𝐹𝑖) = if(𝑖 ∈ ℙ, (𝑖𝑖 / 𝑛𝐴), 1))
6160, 47eqeltrd 2254 . . . . . . . 8 ((𝜑𝑖 ∈ ℕ) → (𝐹𝑖) ∈ ℕ)
6230, 61sylan2br 288 . . . . . . 7 ((𝜑𝑖 ∈ (ℤ‘1)) → (𝐹𝑖) ∈ ℕ)
63 nnmulcl 8929 . . . . . . . 8 ((𝑖 ∈ ℕ ∧ 𝑗 ∈ ℕ) → (𝑖 · 𝑗) ∈ ℕ)
6463adantl 277 . . . . . . 7 ((𝜑 ∧ (𝑖 ∈ ℕ ∧ 𝑗 ∈ ℕ)) → (𝑖 · 𝑗) ∈ ℕ)
6529, 62, 64seq3-1 10446 . . . . . 6 (𝜑 → (seq1( · , 𝐹)‘1) = (𝐹‘1))
66 1nprm 12097 . . . . . . . . . 10 ¬ 1 ∈ ℙ
67 eleq1 2240 . . . . . . . . . 10 (𝑛 = 1 → (𝑛 ∈ ℙ ↔ 1 ∈ ℙ))
6866, 67mtbiri 675 . . . . . . . . 9 (𝑛 = 1 → ¬ 𝑛 ∈ ℙ)
6968iffalsed 3544 . . . . . . . 8 (𝑛 = 1 → if(𝑛 ∈ ℙ, (𝑛𝐴), 1) = 1)
70 1ex 7943 . . . . . . . 8 1 ∈ V
7169, 58, 70fvmpt 5589 . . . . . . 7 (1 ∈ ℕ → (𝐹‘1) = 1)
7243, 71ax-mp 5 . . . . . 6 (𝐹‘1) = 1
7365, 72eqtrdi 2226 . . . . 5 (𝜑 → (seq1( · , 𝐹)‘1) = 1)
7473oveq2d 5885 . . . 4 (𝜑 → (𝑃 pCnt (seq1( · , 𝐹)‘1)) = (𝑃 pCnt 1))
75 prmgt1 12115 . . . . . . 7 (𝑃 ∈ ℙ → 1 < 𝑃)
76 1z 9268 . . . . . . . 8 1 ∈ ℤ
77 prmz 12094 . . . . . . . 8 (𝑃 ∈ ℙ → 𝑃 ∈ ℤ)
78 zltnle 9288 . . . . . . . 8 ((1 ∈ ℤ ∧ 𝑃 ∈ ℤ) → (1 < 𝑃 ↔ ¬ 𝑃 ≤ 1))
7976, 77, 78sylancr 414 . . . . . . 7 (𝑃 ∈ ℙ → (1 < 𝑃 ↔ ¬ 𝑃 ≤ 1))
8075, 79mpbid 147 . . . . . 6 (𝑃 ∈ ℙ → ¬ 𝑃 ≤ 1)
8180iffalsed 3544 . . . . 5 (𝑃 ∈ ℙ → if(𝑃 ≤ 1, 𝐵, 0) = 0)
8226, 81syl 14 . . . 4 (𝜑 → if(𝑃 ≤ 1, 𝐵, 0) = 0)
8328, 74, 823eqtr4d 2220 . . 3 (𝜑 → (𝑃 pCnt (seq1( · , 𝐹)‘1)) = if(𝑃 ≤ 1, 𝐵, 0))
8426adantr 276 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑘 ∈ ℕ ∧ (𝑘 + 1) = 𝑃)) → 𝑃 ∈ ℙ)
8558, 34pcmptcl 12323 . . . . . . . . . . . . . . . 16 (𝜑 → (𝐹:ℕ⟶ℕ ∧ seq1( · , 𝐹):ℕ⟶ℕ))
8685simpld 112 . . . . . . . . . . . . . . 15 (𝜑𝐹:ℕ⟶ℕ)
87 peano2nn 8920 . . . . . . . . . . . . . . 15 (𝑘 ∈ ℕ → (𝑘 + 1) ∈ ℕ)
88 ffvelcdm 5645 . . . . . . . . . . . . . . 15 ((𝐹:ℕ⟶ℕ ∧ (𝑘 + 1) ∈ ℕ) → (𝐹‘(𝑘 + 1)) ∈ ℕ)
8986, 87, 88syl2an 289 . . . . . . . . . . . . . 14 ((𝜑𝑘 ∈ ℕ) → (𝐹‘(𝑘 + 1)) ∈ ℕ)
9089adantrr 479 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑘 ∈ ℕ ∧ (𝑘 + 1) = 𝑃)) → (𝐹‘(𝑘 + 1)) ∈ ℕ)
9184, 90pccld 12283 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑘 ∈ ℕ ∧ (𝑘 + 1) = 𝑃)) → (𝑃 pCnt (𝐹‘(𝑘 + 1))) ∈ ℕ0)
9291nn0cnd 9220 . . . . . . . . . . 11 ((𝜑 ∧ (𝑘 ∈ ℕ ∧ (𝑘 + 1) = 𝑃)) → (𝑃 pCnt (𝐹‘(𝑘 + 1))) ∈ ℂ)
9392addid2d 8097 . . . . . . . . . 10 ((𝜑 ∧ (𝑘 ∈ ℕ ∧ (𝑘 + 1) = 𝑃)) → (0 + (𝑃 pCnt (𝐹‘(𝑘 + 1)))) = (𝑃 pCnt (𝐹‘(𝑘 + 1))))
9487ad2antrl 490 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑘 ∈ ℕ ∧ (𝑘 + 1) = 𝑃)) → (𝑘 + 1) ∈ ℕ)
9587ad2antlr 489 . . . . . . . . . . . . . . . 16 (((𝜑𝑘 ∈ ℕ) ∧ (𝑘 + 1) ∈ ℙ) → (𝑘 + 1) ∈ ℕ)
96 simpr 110 . . . . . . . . . . . . . . . . 17 (((𝜑𝑘 ∈ ℕ) ∧ (𝑘 + 1) ∈ ℙ) → (𝑘 + 1) ∈ ℙ)
9734ad2antrr 488 . . . . . . . . . . . . . . . . 17 (((𝜑𝑘 ∈ ℕ) ∧ (𝑘 + 1) ∈ ℙ) → ∀𝑛 ∈ ℙ 𝐴 ∈ ℕ0)
98 nfcsb1v 3090 . . . . . . . . . . . . . . . . . . 19 𝑛(𝑘 + 1) / 𝑛𝐴
9998nfel1 2330 . . . . . . . . . . . . . . . . . 18 𝑛(𝑘 + 1) / 𝑛𝐴 ∈ ℕ0
100 csbeq1a 3066 . . . . . . . . . . . . . . . . . . 19 (𝑛 = (𝑘 + 1) → 𝐴 = (𝑘 + 1) / 𝑛𝐴)
101100eleq1d 2246 . . . . . . . . . . . . . . . . . 18 (𝑛 = (𝑘 + 1) → (𝐴 ∈ ℕ0(𝑘 + 1) / 𝑛𝐴 ∈ ℕ0))
10299, 101rspc 2835 . . . . . . . . . . . . . . . . 17 ((𝑘 + 1) ∈ ℙ → (∀𝑛 ∈ ℙ 𝐴 ∈ ℕ0(𝑘 + 1) / 𝑛𝐴 ∈ ℕ0))
10396, 97, 102sylc 62 . . . . . . . . . . . . . . . 16 (((𝜑𝑘 ∈ ℕ) ∧ (𝑘 + 1) ∈ ℙ) → (𝑘 + 1) / 𝑛𝐴 ∈ ℕ0)
10495, 103nnexpcld 10661 . . . . . . . . . . . . . . 15 (((𝜑𝑘 ∈ ℕ) ∧ (𝑘 + 1) ∈ ℙ) → ((𝑘 + 1)↑(𝑘 + 1) / 𝑛𝐴) ∈ ℕ)
10543a1i 9 . . . . . . . . . . . . . . 15 (((𝜑𝑘 ∈ ℕ) ∧ ¬ (𝑘 + 1) ∈ ℙ) → 1 ∈ ℕ)
10687adantl 277 . . . . . . . . . . . . . . . 16 ((𝜑𝑘 ∈ ℕ) → (𝑘 + 1) ∈ ℕ)
107 prmdc 12113 . . . . . . . . . . . . . . . 16 ((𝑘 + 1) ∈ ℕ → DECID (𝑘 + 1) ∈ ℙ)
108106, 107syl 14 . . . . . . . . . . . . . . 15 ((𝜑𝑘 ∈ ℕ) → DECID (𝑘 + 1) ∈ ℙ)
109104, 105, 108ifcldadc 3563 . . . . . . . . . . . . . 14 ((𝜑𝑘 ∈ ℕ) → if((𝑘 + 1) ∈ ℙ, ((𝑘 + 1)↑(𝑘 + 1) / 𝑛𝐴), 1) ∈ ℕ)
110109adantrr 479 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑘 ∈ ℕ ∧ (𝑘 + 1) = 𝑃)) → if((𝑘 + 1) ∈ ℙ, ((𝑘 + 1)↑(𝑘 + 1) / 𝑛𝐴), 1) ∈ ℕ)
111 nfcv 2319 . . . . . . . . . . . . . 14 𝑛(𝑘 + 1)
112 nfv 1528 . . . . . . . . . . . . . . 15 𝑛(𝑘 + 1) ∈ ℙ
113111, 50, 98nfov 5899 . . . . . . . . . . . . . . 15 𝑛((𝑘 + 1)↑(𝑘 + 1) / 𝑛𝐴)
114112, 113, 52nfif 3562 . . . . . . . . . . . . . 14 𝑛if((𝑘 + 1) ∈ ℙ, ((𝑘 + 1)↑(𝑘 + 1) / 𝑛𝐴), 1)
115 eleq1 2240 . . . . . . . . . . . . . . 15 (𝑛 = (𝑘 + 1) → (𝑛 ∈ ℙ ↔ (𝑘 + 1) ∈ ℙ))
116 id 19 . . . . . . . . . . . . . . . 16 (𝑛 = (𝑘 + 1) → 𝑛 = (𝑘 + 1))
117116, 100oveq12d 5887 . . . . . . . . . . . . . . 15 (𝑛 = (𝑘 + 1) → (𝑛𝐴) = ((𝑘 + 1)↑(𝑘 + 1) / 𝑛𝐴))
118115, 117ifbieq1d 3556 . . . . . . . . . . . . . 14 (𝑛 = (𝑘 + 1) → if(𝑛 ∈ ℙ, (𝑛𝐴), 1) = if((𝑘 + 1) ∈ ℙ, ((𝑘 + 1)↑(𝑘 + 1) / 𝑛𝐴), 1))
119111, 114, 118, 58fvmptf 5604 . . . . . . . . . . . . 13 (((𝑘 + 1) ∈ ℕ ∧ if((𝑘 + 1) ∈ ℙ, ((𝑘 + 1)↑(𝑘 + 1) / 𝑛𝐴), 1) ∈ ℕ) → (𝐹‘(𝑘 + 1)) = if((𝑘 + 1) ∈ ℙ, ((𝑘 + 1)↑(𝑘 + 1) / 𝑛𝐴), 1))
12094, 110, 119syl2anc 411 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑘 ∈ ℕ ∧ (𝑘 + 1) = 𝑃)) → (𝐹‘(𝑘 + 1)) = if((𝑘 + 1) ∈ ℙ, ((𝑘 + 1)↑(𝑘 + 1) / 𝑛𝐴), 1))
121 simprr 531 . . . . . . . . . . . . . 14 ((𝜑 ∧ (𝑘 ∈ ℕ ∧ (𝑘 + 1) = 𝑃)) → (𝑘 + 1) = 𝑃)
122121, 84eqeltrd 2254 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑘 ∈ ℕ ∧ (𝑘 + 1) = 𝑃)) → (𝑘 + 1) ∈ ℙ)
123122iftrued 3541 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑘 ∈ ℕ ∧ (𝑘 + 1) = 𝑃)) → if((𝑘 + 1) ∈ ℙ, ((𝑘 + 1)↑(𝑘 + 1) / 𝑛𝐴), 1) = ((𝑘 + 1)↑(𝑘 + 1) / 𝑛𝐴))
124121csbeq1d 3064 . . . . . . . . . . . . . 14 ((𝜑 ∧ (𝑘 ∈ ℕ ∧ (𝑘 + 1) = 𝑃)) → (𝑘 + 1) / 𝑛𝐴 = 𝑃 / 𝑛𝐴)
125 nfcvd 2320 . . . . . . . . . . . . . . . 16 (𝑃 ∈ ℙ → 𝑛𝐵)
126 pcmpt.5 . . . . . . . . . . . . . . . 16 (𝑛 = 𝑃𝐴 = 𝐵)
127125, 126csbiegf 3100 . . . . . . . . . . . . . . 15 (𝑃 ∈ ℙ → 𝑃 / 𝑛𝐴 = 𝐵)
12884, 127syl 14 . . . . . . . . . . . . . 14 ((𝜑 ∧ (𝑘 ∈ ℕ ∧ (𝑘 + 1) = 𝑃)) → 𝑃 / 𝑛𝐴 = 𝐵)
129124, 128eqtrd 2210 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑘 ∈ ℕ ∧ (𝑘 + 1) = 𝑃)) → (𝑘 + 1) / 𝑛𝐴 = 𝐵)
130121, 129oveq12d 5887 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑘 ∈ ℕ ∧ (𝑘 + 1) = 𝑃)) → ((𝑘 + 1)↑(𝑘 + 1) / 𝑛𝐴) = (𝑃𝐵))
131120, 123, 1303eqtrd 2214 . . . . . . . . . . 11 ((𝜑 ∧ (𝑘 ∈ ℕ ∧ (𝑘 + 1) = 𝑃)) → (𝐹‘(𝑘 + 1)) = (𝑃𝐵))
132131oveq2d 5885 . . . . . . . . . 10 ((𝜑 ∧ (𝑘 ∈ ℕ ∧ (𝑘 + 1) = 𝑃)) → (𝑃 pCnt (𝐹‘(𝑘 + 1))) = (𝑃 pCnt (𝑃𝐵)))
133126eleq1d 2246 . . . . . . . . . . . . . 14 (𝑛 = 𝑃 → (𝐴 ∈ ℕ0𝐵 ∈ ℕ0))
134133rspcv 2837 . . . . . . . . . . . . 13 (𝑃 ∈ ℙ → (∀𝑛 ∈ ℙ 𝐴 ∈ ℕ0𝐵 ∈ ℕ0))
13526, 34, 134sylc 62 . . . . . . . . . . . 12 (𝜑𝐵 ∈ ℕ0)
136135adantr 276 . . . . . . . . . . 11 ((𝜑 ∧ (𝑘 ∈ ℕ ∧ (𝑘 + 1) = 𝑃)) → 𝐵 ∈ ℕ0)
137 pcidlem 12305 . . . . . . . . . . 11 ((𝑃 ∈ ℙ ∧ 𝐵 ∈ ℕ0) → (𝑃 pCnt (𝑃𝐵)) = 𝐵)
13826, 136, 137syl2an2r 595 . . . . . . . . . 10 ((𝜑 ∧ (𝑘 ∈ ℕ ∧ (𝑘 + 1) = 𝑃)) → (𝑃 pCnt (𝑃𝐵)) = 𝐵)
13993, 132, 1383eqtrd 2214 . . . . . . . . 9 ((𝜑 ∧ (𝑘 ∈ ℕ ∧ (𝑘 + 1) = 𝑃)) → (0 + (𝑃 pCnt (𝐹‘(𝑘 + 1)))) = 𝐵)
140 oveq1 5876 . . . . . . . . . 10 ((𝑃 pCnt (seq1( · , 𝐹)‘𝑘)) = 0 → ((𝑃 pCnt (seq1( · , 𝐹)‘𝑘)) + (𝑃 pCnt (𝐹‘(𝑘 + 1)))) = (0 + (𝑃 pCnt (𝐹‘(𝑘 + 1)))))
141140eqeq1d 2186 . . . . . . . . 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))
142139, 141syl5ibrcom 157 . . . . . . . 8 ((𝜑 ∧ (𝑘 ∈ ℕ ∧ (𝑘 + 1) = 𝑃)) → ((𝑃 pCnt (seq1( · , 𝐹)‘𝑘)) = 0 → ((𝑃 pCnt (seq1( · , 𝐹)‘𝑘)) + (𝑃 pCnt (𝐹‘(𝑘 + 1)))) = 𝐵))
143 nnre 8915 . . . . . . . . . . . . . 14 (𝑘 ∈ ℕ → 𝑘 ∈ ℝ)
144143ltp1d 8876 . . . . . . . . . . . . 13 (𝑘 ∈ ℕ → 𝑘 < (𝑘 + 1))
145 nnz 9261 . . . . . . . . . . . . . 14 (𝑘 ∈ ℕ → 𝑘 ∈ ℤ)
14687nnzd 9363 . . . . . . . . . . . . . 14 (𝑘 ∈ ℕ → (𝑘 + 1) ∈ ℤ)
147 zltnle 9288 . . . . . . . . . . . . . 14 ((𝑘 ∈ ℤ ∧ (𝑘 + 1) ∈ ℤ) → (𝑘 < (𝑘 + 1) ↔ ¬ (𝑘 + 1) ≤ 𝑘))
148145, 146, 147syl2anc 411 . . . . . . . . . . . . 13 (𝑘 ∈ ℕ → (𝑘 < (𝑘 + 1) ↔ ¬ (𝑘 + 1) ≤ 𝑘))
149144, 148mpbid 147 . . . . . . . . . . . 12 (𝑘 ∈ ℕ → ¬ (𝑘 + 1) ≤ 𝑘)
150149ad2antrl 490 . . . . . . . . . . 11 ((𝜑 ∧ (𝑘 ∈ ℕ ∧ (𝑘 + 1) = 𝑃)) → ¬ (𝑘 + 1) ≤ 𝑘)
151121breq1d 4010 . . . . . . . . . . 11 ((𝜑 ∧ (𝑘 ∈ ℕ ∧ (𝑘 + 1) = 𝑃)) → ((𝑘 + 1) ≤ 𝑘𝑃𝑘))
152150, 151mtbid 672 . . . . . . . . . 10 ((𝜑 ∧ (𝑘 ∈ ℕ ∧ (𝑘 + 1) = 𝑃)) → ¬ 𝑃𝑘)
153152iffalsed 3544 . . . . . . . . 9 ((𝜑 ∧ (𝑘 ∈ ℕ ∧ (𝑘 + 1) = 𝑃)) → if(𝑃𝑘, 𝐵, 0) = 0)
154153eqeq2d 2189 . . . . . . . 8 ((𝜑 ∧ (𝑘 ∈ ℕ ∧ (𝑘 + 1) = 𝑃)) → ((𝑃 pCnt (seq1( · , 𝐹)‘𝑘)) = if(𝑃𝑘, 𝐵, 0) ↔ (𝑃 pCnt (seq1( · , 𝐹)‘𝑘)) = 0))
155 simpr 110 . . . . . . . . . . . . . 14 ((𝜑𝑘 ∈ ℕ) → 𝑘 ∈ ℕ)
156 nnuz 9552 . . . . . . . . . . . . . 14 ℕ = (ℤ‘1)
157155, 156eleqtrdi 2270 . . . . . . . . . . . . 13 ((𝜑𝑘 ∈ ℕ) → 𝑘 ∈ (ℤ‘1))
15862adantlr 477 . . . . . . . . . . . . 13 (((𝜑𝑘 ∈ ℕ) ∧ 𝑖 ∈ (ℤ‘1)) → (𝐹𝑖) ∈ ℕ)
15963adantl 277 . . . . . . . . . . . . 13 (((𝜑𝑘 ∈ ℕ) ∧ (𝑖 ∈ ℕ ∧ 𝑗 ∈ ℕ)) → (𝑖 · 𝑗) ∈ ℕ)
160157, 158, 159seq3p1 10448 . . . . . . . . . . . 12 ((𝜑𝑘 ∈ ℕ) → (seq1( · , 𝐹)‘(𝑘 + 1)) = ((seq1( · , 𝐹)‘𝑘) · (𝐹‘(𝑘 + 1))))
161160oveq2d 5885 . . . . . . . . . . 11 ((𝜑𝑘 ∈ ℕ) → (𝑃 pCnt (seq1( · , 𝐹)‘(𝑘 + 1))) = (𝑃 pCnt ((seq1( · , 𝐹)‘𝑘) · (𝐹‘(𝑘 + 1)))))
16226adantr 276 . . . . . . . . . . . 12 ((𝜑𝑘 ∈ ℕ) → 𝑃 ∈ ℙ)
16385simprd 114 . . . . . . . . . . . . . 14 (𝜑 → seq1( · , 𝐹):ℕ⟶ℕ)
164163ffvelcdmda 5647 . . . . . . . . . . . . 13 ((𝜑𝑘 ∈ ℕ) → (seq1( · , 𝐹)‘𝑘) ∈ ℕ)
165 nnz 9261 . . . . . . . . . . . . . 14 ((seq1( · , 𝐹)‘𝑘) ∈ ℕ → (seq1( · , 𝐹)‘𝑘) ∈ ℤ)
166 nnne0 8936 . . . . . . . . . . . . . 14 ((seq1( · , 𝐹)‘𝑘) ∈ ℕ → (seq1( · , 𝐹)‘𝑘) ≠ 0)
167165, 166jca 306 . . . . . . . . . . . . 13 ((seq1( · , 𝐹)‘𝑘) ∈ ℕ → ((seq1( · , 𝐹)‘𝑘) ∈ ℤ ∧ (seq1( · , 𝐹)‘𝑘) ≠ 0))
168164, 167syl 14 . . . . . . . . . . . 12 ((𝜑𝑘 ∈ ℕ) → ((seq1( · , 𝐹)‘𝑘) ∈ ℤ ∧ (seq1( · , 𝐹)‘𝑘) ≠ 0))
169 nnz 9261 . . . . . . . . . . . . . 14 ((𝐹‘(𝑘 + 1)) ∈ ℕ → (𝐹‘(𝑘 + 1)) ∈ ℤ)
170 nnne0 8936 . . . . . . . . . . . . . 14 ((𝐹‘(𝑘 + 1)) ∈ ℕ → (𝐹‘(𝑘 + 1)) ≠ 0)
171169, 170jca 306 . . . . . . . . . . . . 13 ((𝐹‘(𝑘 + 1)) ∈ ℕ → ((𝐹‘(𝑘 + 1)) ∈ ℤ ∧ (𝐹‘(𝑘 + 1)) ≠ 0))
17289, 171syl 14 . . . . . . . . . . . 12 ((𝜑𝑘 ∈ ℕ) → ((𝐹‘(𝑘 + 1)) ∈ ℤ ∧ (𝐹‘(𝑘 + 1)) ≠ 0))
173 pcmul 12284 . . . . . . . . . . . 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)))))
174162, 168, 172, 173syl3anc 1238 . . . . . . . . . . 11 ((𝜑𝑘 ∈ ℕ) → (𝑃 pCnt ((seq1( · , 𝐹)‘𝑘) · (𝐹‘(𝑘 + 1)))) = ((𝑃 pCnt (seq1( · , 𝐹)‘𝑘)) + (𝑃 pCnt (𝐹‘(𝑘 + 1)))))
175161, 174eqtrd 2210 . . . . . . . . . 10 ((𝜑𝑘 ∈ ℕ) → (𝑃 pCnt (seq1( · , 𝐹)‘(𝑘 + 1))) = ((𝑃 pCnt (seq1( · , 𝐹)‘𝑘)) + (𝑃 pCnt (𝐹‘(𝑘 + 1)))))
176175adantrr 479 . . . . . . . . 9 ((𝜑 ∧ (𝑘 ∈ ℕ ∧ (𝑘 + 1) = 𝑃)) → (𝑃 pCnt (seq1( · , 𝐹)‘(𝑘 + 1))) = ((𝑃 pCnt (seq1( · , 𝐹)‘𝑘)) + (𝑃 pCnt (𝐹‘(𝑘 + 1)))))
177 prmnn 12093 . . . . . . . . . . . . . . 15 (𝑃 ∈ ℙ → 𝑃 ∈ ℕ)
17826, 177syl 14 . . . . . . . . . . . . . 14 (𝜑𝑃 ∈ ℕ)
179178nnred 8921 . . . . . . . . . . . . 13 (𝜑𝑃 ∈ ℝ)
180179adantr 276 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑘 ∈ ℕ ∧ (𝑘 + 1) = 𝑃)) → 𝑃 ∈ ℝ)
181180leidd 8461 . . . . . . . . . . 11 ((𝜑 ∧ (𝑘 ∈ ℕ ∧ (𝑘 + 1) = 𝑃)) → 𝑃𝑃)
182181, 121breqtrrd 4028 . . . . . . . . . 10 ((𝜑 ∧ (𝑘 ∈ ℕ ∧ (𝑘 + 1) = 𝑃)) → 𝑃 ≤ (𝑘 + 1))
183182iftrued 3541 . . . . . . . . 9 ((𝜑 ∧ (𝑘 ∈ ℕ ∧ (𝑘 + 1) = 𝑃)) → if(𝑃 ≤ (𝑘 + 1), 𝐵, 0) = 𝐵)
184176, 183eqeq12d 2192 . . . . . . . 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))
185142, 154, 1843imtr4d 203 . . . . . . 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)))
186185expr 375 . . . . . 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))))
187175adantrr 479 . . . . . . . . . 10 ((𝜑 ∧ (𝑘 ∈ ℕ ∧ (𝑘 + 1) ≠ 𝑃)) → (𝑃 pCnt (seq1( · , 𝐹)‘(𝑘 + 1))) = ((𝑃 pCnt (seq1( · , 𝐹)‘𝑘)) + (𝑃 pCnt (𝐹‘(𝑘 + 1)))))
188 simplrr 536 . . . . . . . . . . . . . . . 16 (((𝜑 ∧ (𝑘 ∈ ℕ ∧ (𝑘 + 1) ≠ 𝑃)) ∧ (𝑘 + 1) ∈ ℙ) → (𝑘 + 1) ≠ 𝑃)
189188necomd 2433 . . . . . . . . . . . . . . 15 (((𝜑 ∧ (𝑘 ∈ ℕ ∧ (𝑘 + 1) ≠ 𝑃)) ∧ (𝑘 + 1) ∈ ℙ) → 𝑃 ≠ (𝑘 + 1))
19026ad2antrr 488 . . . . . . . . . . . . . . . . 17 (((𝜑 ∧ (𝑘 ∈ ℕ ∧ (𝑘 + 1) ≠ 𝑃)) ∧ (𝑘 + 1) ∈ ℙ) → 𝑃 ∈ ℙ)
191 simpr 110 . . . . . . . . . . . . . . . . 17 (((𝜑 ∧ (𝑘 ∈ ℕ ∧ (𝑘 + 1) ≠ 𝑃)) ∧ (𝑘 + 1) ∈ ℙ) → (𝑘 + 1) ∈ ℙ)
19234ad2antrr 488 . . . . . . . . . . . . . . . . . 18 (((𝜑 ∧ (𝑘 ∈ ℕ ∧ (𝑘 + 1) ≠ 𝑃)) ∧ (𝑘 + 1) ∈ ℙ) → ∀𝑛 ∈ ℙ 𝐴 ∈ ℕ0)
193191, 192, 102sylc 62 . . . . . . . . . . . . . . . . 17 (((𝜑 ∧ (𝑘 ∈ ℕ ∧ (𝑘 + 1) ≠ 𝑃)) ∧ (𝑘 + 1) ∈ ℙ) → (𝑘 + 1) / 𝑛𝐴 ∈ ℕ0)
194 prmdvdsexpr 12133 . . . . . . . . . . . . . . . . 17 ((𝑃 ∈ ℙ ∧ (𝑘 + 1) ∈ ℙ ∧ (𝑘 + 1) / 𝑛𝐴 ∈ ℕ0) → (𝑃 ∥ ((𝑘 + 1)↑(𝑘 + 1) / 𝑛𝐴) → 𝑃 = (𝑘 + 1)))
195190, 191, 193, 194syl3anc 1238 . . . . . . . . . . . . . . . 16 (((𝜑 ∧ (𝑘 ∈ ℕ ∧ (𝑘 + 1) ≠ 𝑃)) ∧ (𝑘 + 1) ∈ ℙ) → (𝑃 ∥ ((𝑘 + 1)↑(𝑘 + 1) / 𝑛𝐴) → 𝑃 = (𝑘 + 1)))
196195necon3ad 2389 . . . . . . . . . . . . . . 15 (((𝜑 ∧ (𝑘 ∈ ℕ ∧ (𝑘 + 1) ≠ 𝑃)) ∧ (𝑘 + 1) ∈ ℙ) → (𝑃 ≠ (𝑘 + 1) → ¬ 𝑃 ∥ ((𝑘 + 1)↑(𝑘 + 1) / 𝑛𝐴)))
197189, 196mpd 13 . . . . . . . . . . . . . 14 (((𝜑 ∧ (𝑘 ∈ ℕ ∧ (𝑘 + 1) ≠ 𝑃)) ∧ (𝑘 + 1) ∈ ℙ) → ¬ 𝑃 ∥ ((𝑘 + 1)↑(𝑘 + 1) / 𝑛𝐴))
19887ad2antrl 490 . . . . . . . . . . . . . . . . 17 ((𝜑 ∧ (𝑘 ∈ ℕ ∧ (𝑘 + 1) ≠ 𝑃)) → (𝑘 + 1) ∈ ℕ)
199109adantrr 479 . . . . . . . . . . . . . . . . 17 ((𝜑 ∧ (𝑘 ∈ ℕ ∧ (𝑘 + 1) ≠ 𝑃)) → if((𝑘 + 1) ∈ ℙ, ((𝑘 + 1)↑(𝑘 + 1) / 𝑛𝐴), 1) ∈ ℕ)
200198, 199, 119syl2anc 411 . . . . . . . . . . . . . . . 16 ((𝜑 ∧ (𝑘 ∈ ℕ ∧ (𝑘 + 1) ≠ 𝑃)) → (𝐹‘(𝑘 + 1)) = if((𝑘 + 1) ∈ ℙ, ((𝑘 + 1)↑(𝑘 + 1) / 𝑛𝐴), 1))
201 iftrue 3539 . . . . . . . . . . . . . . . 16 ((𝑘 + 1) ∈ ℙ → if((𝑘 + 1) ∈ ℙ, ((𝑘 + 1)↑(𝑘 + 1) / 𝑛𝐴), 1) = ((𝑘 + 1)↑(𝑘 + 1) / 𝑛𝐴))
202200, 201sylan9eq 2230 . . . . . . . . . . . . . . 15 (((𝜑 ∧ (𝑘 ∈ ℕ ∧ (𝑘 + 1) ≠ 𝑃)) ∧ (𝑘 + 1) ∈ ℙ) → (𝐹‘(𝑘 + 1)) = ((𝑘 + 1)↑(𝑘 + 1) / 𝑛𝐴))
203202breq2d 4012 . . . . . . . . . . . . . 14 (((𝜑 ∧ (𝑘 ∈ ℕ ∧ (𝑘 + 1) ≠ 𝑃)) ∧ (𝑘 + 1) ∈ ℙ) → (𝑃 ∥ (𝐹‘(𝑘 + 1)) ↔ 𝑃 ∥ ((𝑘 + 1)↑(𝑘 + 1) / 𝑛𝐴)))
204197, 203mtbird 673 . . . . . . . . . . . . 13 (((𝜑 ∧ (𝑘 ∈ ℕ ∧ (𝑘 + 1) ≠ 𝑃)) ∧ (𝑘 + 1) ∈ ℙ) → ¬ 𝑃 ∥ (𝐹‘(𝑘 + 1)))
20586, 198, 88syl2an2r 595 . . . . . . . . . . . . . . 15 ((𝜑 ∧ (𝑘 ∈ ℕ ∧ (𝑘 + 1) ≠ 𝑃)) → (𝐹‘(𝑘 + 1)) ∈ ℕ)
206205adantr 276 . . . . . . . . . . . . . 14 (((𝜑 ∧ (𝑘 ∈ ℕ ∧ (𝑘 + 1) ≠ 𝑃)) ∧ (𝑘 + 1) ∈ ℙ) → (𝐹‘(𝑘 + 1)) ∈ ℕ)
207 pceq0 12304 . . . . . . . . . . . . . 14 ((𝑃 ∈ ℙ ∧ (𝐹‘(𝑘 + 1)) ∈ ℕ) → ((𝑃 pCnt (𝐹‘(𝑘 + 1))) = 0 ↔ ¬ 𝑃 ∥ (𝐹‘(𝑘 + 1))))
208190, 206, 207syl2anc 411 . . . . . . . . . . . . 13 (((𝜑 ∧ (𝑘 ∈ ℕ ∧ (𝑘 + 1) ≠ 𝑃)) ∧ (𝑘 + 1) ∈ ℙ) → ((𝑃 pCnt (𝐹‘(𝑘 + 1))) = 0 ↔ ¬ 𝑃 ∥ (𝐹‘(𝑘 + 1))))
209204, 208mpbird 167 . . . . . . . . . . . 12 (((𝜑 ∧ (𝑘 ∈ ℕ ∧ (𝑘 + 1) ≠ 𝑃)) ∧ (𝑘 + 1) ∈ ℙ) → (𝑃 pCnt (𝐹‘(𝑘 + 1))) = 0)
210 iffalse 3542 . . . . . . . . . . . . . . 15 (¬ (𝑘 + 1) ∈ ℙ → if((𝑘 + 1) ∈ ℙ, ((𝑘 + 1)↑(𝑘 + 1) / 𝑛𝐴), 1) = 1)
211200, 210sylan9eq 2230 . . . . . . . . . . . . . 14 (((𝜑 ∧ (𝑘 ∈ ℕ ∧ (𝑘 + 1) ≠ 𝑃)) ∧ ¬ (𝑘 + 1) ∈ ℙ) → (𝐹‘(𝑘 + 1)) = 1)
212211oveq2d 5885 . . . . . . . . . . . . 13 (((𝜑 ∧ (𝑘 ∈ ℕ ∧ (𝑘 + 1) ≠ 𝑃)) ∧ ¬ (𝑘 + 1) ∈ ℙ) → (𝑃 pCnt (𝐹‘(𝑘 + 1))) = (𝑃 pCnt 1))
21328ad2antrr 488 . . . . . . . . . . . . 13 (((𝜑 ∧ (𝑘 ∈ ℕ ∧ (𝑘 + 1) ≠ 𝑃)) ∧ ¬ (𝑘 + 1) ∈ ℙ) → (𝑃 pCnt 1) = 0)
214212, 213eqtrd 2210 . . . . . . . . . . . 12 (((𝜑 ∧ (𝑘 ∈ ℕ ∧ (𝑘 + 1) ≠ 𝑃)) ∧ ¬ (𝑘 + 1) ∈ ℙ) → (𝑃 pCnt (𝐹‘(𝑘 + 1))) = 0)
215 exmiddc 836 . . . . . . . . . . . . 13 (DECID (𝑘 + 1) ∈ ℙ → ((𝑘 + 1) ∈ ℙ ∨ ¬ (𝑘 + 1) ∈ ℙ))
216198, 107, 2153syl 17 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑘 ∈ ℕ ∧ (𝑘 + 1) ≠ 𝑃)) → ((𝑘 + 1) ∈ ℙ ∨ ¬ (𝑘 + 1) ∈ ℙ))
217209, 214, 216mpjaodan 798 . . . . . . . . . . 11 ((𝜑 ∧ (𝑘 ∈ ℕ ∧ (𝑘 + 1) ≠ 𝑃)) → (𝑃 pCnt (𝐹‘(𝑘 + 1))) = 0)
218217oveq2d 5885 . . . . . . . . . 10 ((𝜑 ∧ (𝑘 ∈ ℕ ∧ (𝑘 + 1) ≠ 𝑃)) → ((𝑃 pCnt (seq1( · , 𝐹)‘𝑘)) + (𝑃 pCnt (𝐹‘(𝑘 + 1)))) = ((𝑃 pCnt (seq1( · , 𝐹)‘𝑘)) + 0))
21926adantr 276 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑘 ∈ ℕ ∧ (𝑘 + 1) ≠ 𝑃)) → 𝑃 ∈ ℙ)
220164adantrr 479 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑘 ∈ ℕ ∧ (𝑘 + 1) ≠ 𝑃)) → (seq1( · , 𝐹)‘𝑘) ∈ ℕ)
221219, 220pccld 12283 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑘 ∈ ℕ ∧ (𝑘 + 1) ≠ 𝑃)) → (𝑃 pCnt (seq1( · , 𝐹)‘𝑘)) ∈ ℕ0)
222221nn0cnd 9220 . . . . . . . . . . 11 ((𝜑 ∧ (𝑘 ∈ ℕ ∧ (𝑘 + 1) ≠ 𝑃)) → (𝑃 pCnt (seq1( · , 𝐹)‘𝑘)) ∈ ℂ)
223222addid1d 8096 . . . . . . . . . 10 ((𝜑 ∧ (𝑘 ∈ ℕ ∧ (𝑘 + 1) ≠ 𝑃)) → ((𝑃 pCnt (seq1( · , 𝐹)‘𝑘)) + 0) = (𝑃 pCnt (seq1( · , 𝐹)‘𝑘)))
224187, 218, 2233eqtrd 2214 . . . . . . . . 9 ((𝜑 ∧ (𝑘 ∈ ℕ ∧ (𝑘 + 1) ≠ 𝑃)) → (𝑃 pCnt (seq1( · , 𝐹)‘(𝑘 + 1))) = (𝑃 pCnt (seq1( · , 𝐹)‘𝑘)))
225219, 77syl 14 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑘 ∈ ℕ ∧ (𝑘 + 1) ≠ 𝑃)) → 𝑃 ∈ ℤ)
226146ad2antrl 490 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑘 ∈ ℕ ∧ (𝑘 + 1) ≠ 𝑃)) → (𝑘 + 1) ∈ ℤ)
227 zltlen 9320 . . . . . . . . . . . 12 ((𝑃 ∈ ℤ ∧ (𝑘 + 1) ∈ ℤ) → (𝑃 < (𝑘 + 1) ↔ (𝑃 ≤ (𝑘 + 1) ∧ (𝑘 + 1) ≠ 𝑃)))
228225, 226, 227syl2anc 411 . . . . . . . . . . 11 ((𝜑 ∧ (𝑘 ∈ ℕ ∧ (𝑘 + 1) ≠ 𝑃)) → (𝑃 < (𝑘 + 1) ↔ (𝑃 ≤ (𝑘 + 1) ∧ (𝑘 + 1) ≠ 𝑃)))
229 simprl 529 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑘 ∈ ℕ ∧ (𝑘 + 1) ≠ 𝑃)) → 𝑘 ∈ ℕ)
230 nnleltp1 9301 . . . . . . . . . . . 12 ((𝑃 ∈ ℕ ∧ 𝑘 ∈ ℕ) → (𝑃𝑘𝑃 < (𝑘 + 1)))
231178, 229, 230syl2an2r 595 . . . . . . . . . . 11 ((𝜑 ∧ (𝑘 ∈ ℕ ∧ (𝑘 + 1) ≠ 𝑃)) → (𝑃𝑘𝑃 < (𝑘 + 1)))
232 simprr 531 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑘 ∈ ℕ ∧ (𝑘 + 1) ≠ 𝑃)) → (𝑘 + 1) ≠ 𝑃)
233232biantrud 304 . . . . . . . . . . 11 ((𝜑 ∧ (𝑘 ∈ ℕ ∧ (𝑘 + 1) ≠ 𝑃)) → (𝑃 ≤ (𝑘 + 1) ↔ (𝑃 ≤ (𝑘 + 1) ∧ (𝑘 + 1) ≠ 𝑃)))
234228, 231, 2333bitr4rd 221 . . . . . . . . . 10 ((𝜑 ∧ (𝑘 ∈ ℕ ∧ (𝑘 + 1) ≠ 𝑃)) → (𝑃 ≤ (𝑘 + 1) ↔ 𝑃𝑘))
235234ifbid 3555 . . . . . . . . 9 ((𝜑 ∧ (𝑘 ∈ ℕ ∧ (𝑘 + 1) ≠ 𝑃)) → if(𝑃 ≤ (𝑘 + 1), 𝐵, 0) = if(𝑃𝑘, 𝐵, 0))
236224, 235eqeq12d 2192 . . . . . . . 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)))
237236biimprd 158 . . . . . . 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)))
238237expr 375 . . . . . 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))))
239106nnzd 9363 . . . . . . . 8 ((𝜑𝑘 ∈ ℕ) → (𝑘 + 1) ∈ ℤ)
240162, 77syl 14 . . . . . . . 8 ((𝜑𝑘 ∈ ℕ) → 𝑃 ∈ ℤ)
241 zdceq 9317 . . . . . . . 8 (((𝑘 + 1) ∈ ℤ ∧ 𝑃 ∈ ℤ) → DECID (𝑘 + 1) = 𝑃)
242239, 240, 241syl2anc 411 . . . . . . 7 ((𝜑𝑘 ∈ ℕ) → DECID (𝑘 + 1) = 𝑃)
243 dcne 2358 . . . . . . 7 (DECID (𝑘 + 1) = 𝑃 ↔ ((𝑘 + 1) = 𝑃 ∨ (𝑘 + 1) ≠ 𝑃))
244242, 243sylib 122 . . . . . 6 ((𝜑𝑘 ∈ ℕ) → ((𝑘 + 1) = 𝑃 ∨ (𝑘 + 1) ≠ 𝑃))
245186, 238, 244mpjaod 718 . . . . 5 ((𝜑𝑘 ∈ ℕ) → ((𝑃 pCnt (seq1( · , 𝐹)‘𝑘)) = if(𝑃𝑘, 𝐵, 0) → (𝑃 pCnt (seq1( · , 𝐹)‘(𝑘 + 1))) = if(𝑃 ≤ (𝑘 + 1), 𝐵, 0)))
246245expcom 116 . . . 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))))
247246a2d 26 . . 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))))
2487, 13, 19, 25, 83, 247nnind 8924 . 2 (𝑁 ∈ ℕ → (𝜑 → (𝑃 pCnt (seq1( · , 𝐹)‘𝑁)) = if(𝑃𝑁, 𝐵, 0)))
2491, 248mpcom 36 1 (𝜑 → (𝑃 pCnt (seq1( · , 𝐹)‘𝑁)) = if(𝑃𝑁, 𝐵, 0))
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wa 104  wb 105  wo 708  DECID wdc 834   = wceq 1353  wcel 2148  wne 2347  wral 2455  csb 3057  ifcif 3534   class class class wbr 4000  cmpt 4061  wf 5208  cfv 5212  (class class class)co 5869  cr 7801  0cc0 7802  1c1 7803   + caddc 7805   · cmul 7807   < clt 7982  cle 7983  cn 8908  0cn0 9165  cz 9242  cuz 9517  seqcseq 10431  cexp 10505  cdvds 11778  cprime 12090   pCnt cpc 12267
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-coll 4115  ax-sep 4118  ax-nul 4126  ax-pow 4171  ax-pr 4206  ax-un 4430  ax-setind 4533  ax-iinf 4584  ax-cnex 7893  ax-resscn 7894  ax-1cn 7895  ax-1re 7896  ax-icn 7897  ax-addcl 7898  ax-addrcl 7899  ax-mulcl 7900  ax-mulrcl 7901  ax-addcom 7902  ax-mulcom 7903  ax-addass 7904  ax-mulass 7905  ax-distr 7906  ax-i2m1 7907  ax-0lt1 7908  ax-1rid 7909  ax-0id 7910  ax-rnegex 7911  ax-precex 7912  ax-cnre 7913  ax-pre-ltirr 7914  ax-pre-ltwlin 7915  ax-pre-lttrn 7916  ax-pre-apti 7917  ax-pre-ltadd 7918  ax-pre-mulgt0 7919  ax-pre-mulext 7920  ax-arch 7921  ax-caucvg 7922
This theorem depends on definitions:  df-bi 117  df-stab 831  df-dc 835  df-3or 979  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-nel 2443  df-ral 2460  df-rex 2461  df-reu 2462  df-rmo 2463  df-rab 2464  df-v 2739  df-sbc 2963  df-csb 3058  df-dif 3131  df-un 3133  df-in 3135  df-ss 3142  df-nul 3423  df-if 3535  df-pw 3576  df-sn 3597  df-pr 3598  df-op 3600  df-uni 3808  df-int 3843  df-iun 3886  df-br 4001  df-opab 4062  df-mpt 4063  df-tr 4099  df-id 4290  df-po 4293  df-iso 4294  df-iord 4363  df-on 4365  df-ilim 4366  df-suc 4368  df-iom 4587  df-xp 4629  df-rel 4630  df-cnv 4631  df-co 4632  df-dm 4633  df-rn 4634  df-res 4635  df-ima 4636  df-iota 5174  df-fun 5214  df-fn 5215  df-f 5216  df-f1 5217  df-fo 5218  df-f1o 5219  df-fv 5220  df-isom 5221  df-riota 5825  df-ov 5872  df-oprab 5873  df-mpo 5874  df-1st 6135  df-2nd 6136  df-recs 6300  df-frec 6386  df-1o 6411  df-2o 6412  df-er 6529  df-en 6735  df-fin 6737  df-sup 6977  df-inf 6978  df-pnf 7984  df-mnf 7985  df-xr 7986  df-ltxr 7987  df-le 7988  df-sub 8120  df-neg 8121  df-reap 8522  df-ap 8529  df-div 8619  df-inn 8909  df-2 8967  df-3 8968  df-4 8969  df-n0 9166  df-z 9243  df-uz 9518  df-q 9609  df-rp 9641  df-fz 9996  df-fzo 10129  df-fl 10256  df-mod 10309  df-seqfrec 10432  df-exp 10506  df-cj 10835  df-re 10836  df-im 10837  df-rsqrt 10991  df-abs 10992  df-dvds 11779  df-gcd 11927  df-prm 12091  df-pc 12268
This theorem is referenced by:  pcmpt2  12325  pcprod  12327  1arithlem4  12347
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