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Theorem pcmpt 12288
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 5494 . . . . . 6 (𝑝 = 1 → (seq1( · , 𝐹)‘𝑝) = (seq1( · , 𝐹)‘1))
32oveq2d 5867 . . . . 5 (𝑝 = 1 → (𝑃 pCnt (seq1( · , 𝐹)‘𝑝)) = (𝑃 pCnt (seq1( · , 𝐹)‘1)))
4 breq2 3991 . . . . . 6 (𝑝 = 1 → (𝑃𝑝𝑃 ≤ 1))
54ifbid 3546 . . . . 5 (𝑝 = 1 → if(𝑃𝑝, 𝐵, 0) = if(𝑃 ≤ 1, 𝐵, 0))
63, 5eqeq12d 2185 . . . 4 (𝑝 = 1 → ((𝑃 pCnt (seq1( · , 𝐹)‘𝑝)) = if(𝑃𝑝, 𝐵, 0) ↔ (𝑃 pCnt (seq1( · , 𝐹)‘1)) = if(𝑃 ≤ 1, 𝐵, 0)))
76imbi2d 229 . . 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 5494 . . . . . 6 (𝑝 = 𝑘 → (seq1( · , 𝐹)‘𝑝) = (seq1( · , 𝐹)‘𝑘))
98oveq2d 5867 . . . . 5 (𝑝 = 𝑘 → (𝑃 pCnt (seq1( · , 𝐹)‘𝑝)) = (𝑃 pCnt (seq1( · , 𝐹)‘𝑘)))
10 breq2 3991 . . . . . 6 (𝑝 = 𝑘 → (𝑃𝑝𝑃𝑘))
1110ifbid 3546 . . . . 5 (𝑝 = 𝑘 → if(𝑃𝑝, 𝐵, 0) = if(𝑃𝑘, 𝐵, 0))
129, 11eqeq12d 2185 . . . 4 (𝑝 = 𝑘 → ((𝑃 pCnt (seq1( · , 𝐹)‘𝑝)) = if(𝑃𝑝, 𝐵, 0) ↔ (𝑃 pCnt (seq1( · , 𝐹)‘𝑘)) = if(𝑃𝑘, 𝐵, 0)))
1312imbi2d 229 . . 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 5494 . . . . . 6 (𝑝 = (𝑘 + 1) → (seq1( · , 𝐹)‘𝑝) = (seq1( · , 𝐹)‘(𝑘 + 1)))
1514oveq2d 5867 . . . . 5 (𝑝 = (𝑘 + 1) → (𝑃 pCnt (seq1( · , 𝐹)‘𝑝)) = (𝑃 pCnt (seq1( · , 𝐹)‘(𝑘 + 1))))
16 breq2 3991 . . . . . 6 (𝑝 = (𝑘 + 1) → (𝑃𝑝𝑃 ≤ (𝑘 + 1)))
1716ifbid 3546 . . . . 5 (𝑝 = (𝑘 + 1) → if(𝑃𝑝, 𝐵, 0) = if(𝑃 ≤ (𝑘 + 1), 𝐵, 0))
1815, 17eqeq12d 2185 . . . 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 229 . . 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 5494 . . . . . 6 (𝑝 = 𝑁 → (seq1( · , 𝐹)‘𝑝) = (seq1( · , 𝐹)‘𝑁))
2120oveq2d 5867 . . . . 5 (𝑝 = 𝑁 → (𝑃 pCnt (seq1( · , 𝐹)‘𝑝)) = (𝑃 pCnt (seq1( · , 𝐹)‘𝑁)))
22 breq2 3991 . . . . . 6 (𝑝 = 𝑁 → (𝑃𝑝𝑃𝑁))
2322ifbid 3546 . . . . 5 (𝑝 = 𝑁 → if(𝑃𝑝, 𝐵, 0) = if(𝑃𝑁, 𝐵, 0))
2421, 23eqeq12d 2185 . . . 4 (𝑝 = 𝑁 → ((𝑃 pCnt (seq1( · , 𝐹)‘𝑝)) = if(𝑃𝑝, 𝐵, 0) ↔ (𝑃 pCnt (seq1( · , 𝐹)‘𝑁)) = if(𝑃𝑁, 𝐵, 0)))
2524imbi2d 229 . . 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 12252 . . . . 5 (𝑃 ∈ ℙ → (𝑃 pCnt 1) = 0)
2826, 27syl 14 . . . 4 (𝜑 → (𝑃 pCnt 1) = 0)
29 1zzd 9232 . . . . . . 7 (𝜑 → 1 ∈ ℤ)
30 elnnuz 9516 . . . . . . . 8 (𝑖 ∈ ℕ ↔ 𝑖 ∈ (ℤ‘1))
31 simpr 109 . . . . . . . . . 10 ((𝜑𝑖 ∈ ℕ) → 𝑖 ∈ ℕ)
3231adantr 274 . . . . . . . . . . . 12 (((𝜑𝑖 ∈ ℕ) ∧ 𝑖 ∈ ℙ) → 𝑖 ∈ ℕ)
33 simpr 109 . . . . . . . . . . . . 13 (((𝜑𝑖 ∈ ℕ) ∧ 𝑖 ∈ ℙ) → 𝑖 ∈ ℙ)
34 pcmpt.2 . . . . . . . . . . . . . 14 (𝜑 → ∀𝑛 ∈ ℙ 𝐴 ∈ ℕ0)
3534ad2antrr 485 . . . . . . . . . . . . 13 (((𝜑𝑖 ∈ ℕ) ∧ 𝑖 ∈ ℙ) → ∀𝑛 ∈ ℙ 𝐴 ∈ ℕ0)
36 nfcsb1v 3082 . . . . . . . . . . . . . . 15 𝑛𝑖 / 𝑛𝐴
3736nfel1 2323 . . . . . . . . . . . . . 14 𝑛𝑖 / 𝑛𝐴 ∈ ℕ0
38 csbeq1a 3058 . . . . . . . . . . . . . . 15 (𝑛 = 𝑖𝐴 = 𝑖 / 𝑛𝐴)
3938eleq1d 2239 . . . . . . . . . . . . . 14 (𝑛 = 𝑖 → (𝐴 ∈ ℕ0𝑖 / 𝑛𝐴 ∈ ℕ0))
4037, 39rspc 2828 . . . . . . . . . . . . 13 (𝑖 ∈ ℙ → (∀𝑛 ∈ ℙ 𝐴 ∈ ℕ0𝑖 / 𝑛𝐴 ∈ ℕ0))
4133, 35, 40sylc 62 . . . . . . . . . . . 12 (((𝜑𝑖 ∈ ℕ) ∧ 𝑖 ∈ ℙ) → 𝑖 / 𝑛𝐴 ∈ ℕ0)
4232, 41nnexpcld 10624 . . . . . . . . . . 11 (((𝜑𝑖 ∈ ℕ) ∧ 𝑖 ∈ ℙ) → (𝑖𝑖 / 𝑛𝐴) ∈ ℕ)
43 1nn 8882 . . . . . . . . . . . 12 1 ∈ ℕ
4443a1i 9 . . . . . . . . . . 11 (((𝜑𝑖 ∈ ℕ) ∧ ¬ 𝑖 ∈ ℙ) → 1 ∈ ℕ)
45 prmdc 12077 . . . . . . . . . . . 12 (𝑖 ∈ ℕ → DECID 𝑖 ∈ ℙ)
4645adantl 275 . . . . . . . . . . 11 ((𝜑𝑖 ∈ ℕ) → DECID 𝑖 ∈ ℙ)
4742, 44, 46ifcldadc 3554 . . . . . . . . . 10 ((𝜑𝑖 ∈ ℕ) → if(𝑖 ∈ ℙ, (𝑖𝑖 / 𝑛𝐴), 1) ∈ ℕ)
48 nfcv 2312 . . . . . . . . . . 11 𝑛𝑖
4948nfel1 2323 . . . . . . . . . . . 12 𝑛 𝑖 ∈ ℙ
50 nfcv 2312 . . . . . . . . . . . . 13 𝑛
5148, 50, 36nfov 5881 . . . . . . . . . . . 12 𝑛(𝑖𝑖 / 𝑛𝐴)
52 nfcv 2312 . . . . . . . . . . . 12 𝑛1
5349, 51, 52nfif 3553 . . . . . . . . . . 11 𝑛if(𝑖 ∈ ℙ, (𝑖𝑖 / 𝑛𝐴), 1)
54 eleq1 2233 . . . . . . . . . . . 12 (𝑛 = 𝑖 → (𝑛 ∈ ℙ ↔ 𝑖 ∈ ℙ))
55 id 19 . . . . . . . . . . . . 13 (𝑛 = 𝑖𝑛 = 𝑖)
5655, 38oveq12d 5869 . . . . . . . . . . . 12 (𝑛 = 𝑖 → (𝑛𝐴) = (𝑖𝑖 / 𝑛𝐴))
5754, 56ifbieq1d 3547 . . . . . . . . . . 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 5586 . . . . . . . . . 10 ((𝑖 ∈ ℕ ∧ if(𝑖 ∈ ℙ, (𝑖𝑖 / 𝑛𝐴), 1) ∈ ℕ) → (𝐹𝑖) = if(𝑖 ∈ ℙ, (𝑖𝑖 / 𝑛𝐴), 1))
6031, 47, 59syl2anc 409 . . . . . . . . 9 ((𝜑𝑖 ∈ ℕ) → (𝐹𝑖) = if(𝑖 ∈ ℙ, (𝑖𝑖 / 𝑛𝐴), 1))
6160, 47eqeltrd 2247 . . . . . . . 8 ((𝜑𝑖 ∈ ℕ) → (𝐹𝑖) ∈ ℕ)
6230, 61sylan2br 286 . . . . . . 7 ((𝜑𝑖 ∈ (ℤ‘1)) → (𝐹𝑖) ∈ ℕ)
63 nnmulcl 8892 . . . . . . . 8 ((𝑖 ∈ ℕ ∧ 𝑗 ∈ ℕ) → (𝑖 · 𝑗) ∈ ℕ)
6463adantl 275 . . . . . . 7 ((𝜑 ∧ (𝑖 ∈ ℕ ∧ 𝑗 ∈ ℕ)) → (𝑖 · 𝑗) ∈ ℕ)
6529, 62, 64seq3-1 10409 . . . . . 6 (𝜑 → (seq1( · , 𝐹)‘1) = (𝐹‘1))
66 1nprm 12061 . . . . . . . . . 10 ¬ 1 ∈ ℙ
67 eleq1 2233 . . . . . . . . . 10 (𝑛 = 1 → (𝑛 ∈ ℙ ↔ 1 ∈ ℙ))
6866, 67mtbiri 670 . . . . . . . . 9 (𝑛 = 1 → ¬ 𝑛 ∈ ℙ)
6968iffalsed 3535 . . . . . . . 8 (𝑛 = 1 → if(𝑛 ∈ ℙ, (𝑛𝐴), 1) = 1)
70 1ex 7908 . . . . . . . 8 1 ∈ V
7169, 58, 70fvmpt 5571 . . . . . . 7 (1 ∈ ℕ → (𝐹‘1) = 1)
7243, 71ax-mp 5 . . . . . 6 (𝐹‘1) = 1
7365, 72eqtrdi 2219 . . . . 5 (𝜑 → (seq1( · , 𝐹)‘1) = 1)
7473oveq2d 5867 . . . 4 (𝜑 → (𝑃 pCnt (seq1( · , 𝐹)‘1)) = (𝑃 pCnt 1))
75 prmgt1 12079 . . . . . . 7 (𝑃 ∈ ℙ → 1 < 𝑃)
76 1z 9231 . . . . . . . 8 1 ∈ ℤ
77 prmz 12058 . . . . . . . 8 (𝑃 ∈ ℙ → 𝑃 ∈ ℤ)
78 zltnle 9251 . . . . . . . 8 ((1 ∈ ℤ ∧ 𝑃 ∈ ℤ) → (1 < 𝑃 ↔ ¬ 𝑃 ≤ 1))
7976, 77, 78sylancr 412 . . . . . . 7 (𝑃 ∈ ℙ → (1 < 𝑃 ↔ ¬ 𝑃 ≤ 1))
8075, 79mpbid 146 . . . . . 6 (𝑃 ∈ ℙ → ¬ 𝑃 ≤ 1)
8180iffalsed 3535 . . . . 5 (𝑃 ∈ ℙ → if(𝑃 ≤ 1, 𝐵, 0) = 0)
8226, 81syl 14 . . . 4 (𝜑 → if(𝑃 ≤ 1, 𝐵, 0) = 0)
8328, 74, 823eqtr4d 2213 . . 3 (𝜑 → (𝑃 pCnt (seq1( · , 𝐹)‘1)) = if(𝑃 ≤ 1, 𝐵, 0))
8426adantr 274 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑘 ∈ ℕ ∧ (𝑘 + 1) = 𝑃)) → 𝑃 ∈ ℙ)
8558, 34pcmptcl 12287 . . . . . . . . . . . . . . . 16 (𝜑 → (𝐹:ℕ⟶ℕ ∧ seq1( · , 𝐹):ℕ⟶ℕ))
8685simpld 111 . . . . . . . . . . . . . . 15 (𝜑𝐹:ℕ⟶ℕ)
87 peano2nn 8883 . . . . . . . . . . . . . . 15 (𝑘 ∈ ℕ → (𝑘 + 1) ∈ ℕ)
88 ffvelrn 5627 . . . . . . . . . . . . . . 15 ((𝐹:ℕ⟶ℕ ∧ (𝑘 + 1) ∈ ℕ) → (𝐹‘(𝑘 + 1)) ∈ ℕ)
8986, 87, 88syl2an 287 . . . . . . . . . . . . . 14 ((𝜑𝑘 ∈ ℕ) → (𝐹‘(𝑘 + 1)) ∈ ℕ)
9089adantrr 476 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑘 ∈ ℕ ∧ (𝑘 + 1) = 𝑃)) → (𝐹‘(𝑘 + 1)) ∈ ℕ)
9184, 90pccld 12247 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑘 ∈ ℕ ∧ (𝑘 + 1) = 𝑃)) → (𝑃 pCnt (𝐹‘(𝑘 + 1))) ∈ ℕ0)
9291nn0cnd 9183 . . . . . . . . . . 11 ((𝜑 ∧ (𝑘 ∈ ℕ ∧ (𝑘 + 1) = 𝑃)) → (𝑃 pCnt (𝐹‘(𝑘 + 1))) ∈ ℂ)
9392addid2d 8062 . . . . . . . . . 10 ((𝜑 ∧ (𝑘 ∈ ℕ ∧ (𝑘 + 1) = 𝑃)) → (0 + (𝑃 pCnt (𝐹‘(𝑘 + 1)))) = (𝑃 pCnt (𝐹‘(𝑘 + 1))))
9487ad2antrl 487 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑘 ∈ ℕ ∧ (𝑘 + 1) = 𝑃)) → (𝑘 + 1) ∈ ℕ)
9587ad2antlr 486 . . . . . . . . . . . . . . . 16 (((𝜑𝑘 ∈ ℕ) ∧ (𝑘 + 1) ∈ ℙ) → (𝑘 + 1) ∈ ℕ)
96 simpr 109 . . . . . . . . . . . . . . . . 17 (((𝜑𝑘 ∈ ℕ) ∧ (𝑘 + 1) ∈ ℙ) → (𝑘 + 1) ∈ ℙ)
9734ad2antrr 485 . . . . . . . . . . . . . . . . 17 (((𝜑𝑘 ∈ ℕ) ∧ (𝑘 + 1) ∈ ℙ) → ∀𝑛 ∈ ℙ 𝐴 ∈ ℕ0)
98 nfcsb1v 3082 . . . . . . . . . . . . . . . . . . 19 𝑛(𝑘 + 1) / 𝑛𝐴
9998nfel1 2323 . . . . . . . . . . . . . . . . . 18 𝑛(𝑘 + 1) / 𝑛𝐴 ∈ ℕ0
100 csbeq1a 3058 . . . . . . . . . . . . . . . . . . 19 (𝑛 = (𝑘 + 1) → 𝐴 = (𝑘 + 1) / 𝑛𝐴)
101100eleq1d 2239 . . . . . . . . . . . . . . . . . 18 (𝑛 = (𝑘 + 1) → (𝐴 ∈ ℕ0(𝑘 + 1) / 𝑛𝐴 ∈ ℕ0))
10299, 101rspc 2828 . . . . . . . . . . . . . . . . 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 10624 . . . . . . . . . . . . . . 15 (((𝜑𝑘 ∈ ℕ) ∧ (𝑘 + 1) ∈ ℙ) → ((𝑘 + 1)↑(𝑘 + 1) / 𝑛𝐴) ∈ ℕ)
10543a1i 9 . . . . . . . . . . . . . . 15 (((𝜑𝑘 ∈ ℕ) ∧ ¬ (𝑘 + 1) ∈ ℙ) → 1 ∈ ℕ)
10687adantl 275 . . . . . . . . . . . . . . . 16 ((𝜑𝑘 ∈ ℕ) → (𝑘 + 1) ∈ ℕ)
107 prmdc 12077 . . . . . . . . . . . . . . . 16 ((𝑘 + 1) ∈ ℕ → DECID (𝑘 + 1) ∈ ℙ)
108106, 107syl 14 . . . . . . . . . . . . . . 15 ((𝜑𝑘 ∈ ℕ) → DECID (𝑘 + 1) ∈ ℙ)
109104, 105, 108ifcldadc 3554 . . . . . . . . . . . . . 14 ((𝜑𝑘 ∈ ℕ) → if((𝑘 + 1) ∈ ℙ, ((𝑘 + 1)↑(𝑘 + 1) / 𝑛𝐴), 1) ∈ ℕ)
110109adantrr 476 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑘 ∈ ℕ ∧ (𝑘 + 1) = 𝑃)) → if((𝑘 + 1) ∈ ℙ, ((𝑘 + 1)↑(𝑘 + 1) / 𝑛𝐴), 1) ∈ ℕ)
111 nfcv 2312 . . . . . . . . . . . . . 14 𝑛(𝑘 + 1)
112 nfv 1521 . . . . . . . . . . . . . . 15 𝑛(𝑘 + 1) ∈ ℙ
113111, 50, 98nfov 5881 . . . . . . . . . . . . . . 15 𝑛((𝑘 + 1)↑(𝑘 + 1) / 𝑛𝐴)
114112, 113, 52nfif 3553 . . . . . . . . . . . . . 14 𝑛if((𝑘 + 1) ∈ ℙ, ((𝑘 + 1)↑(𝑘 + 1) / 𝑛𝐴), 1)
115 eleq1 2233 . . . . . . . . . . . . . . 15 (𝑛 = (𝑘 + 1) → (𝑛 ∈ ℙ ↔ (𝑘 + 1) ∈ ℙ))
116 id 19 . . . . . . . . . . . . . . . 16 (𝑛 = (𝑘 + 1) → 𝑛 = (𝑘 + 1))
117116, 100oveq12d 5869 . . . . . . . . . . . . . . 15 (𝑛 = (𝑘 + 1) → (𝑛𝐴) = ((𝑘 + 1)↑(𝑘 + 1) / 𝑛𝐴))
118115, 117ifbieq1d 3547 . . . . . . . . . . . . . 14 (𝑛 = (𝑘 + 1) → if(𝑛 ∈ ℙ, (𝑛𝐴), 1) = if((𝑘 + 1) ∈ ℙ, ((𝑘 + 1)↑(𝑘 + 1) / 𝑛𝐴), 1))
119111, 114, 118, 58fvmptf 5586 . . . . . . . . . . . . 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 409 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑘 ∈ ℕ ∧ (𝑘 + 1) = 𝑃)) → (𝐹‘(𝑘 + 1)) = if((𝑘 + 1) ∈ ℙ, ((𝑘 + 1)↑(𝑘 + 1) / 𝑛𝐴), 1))
121 simprr 527 . . . . . . . . . . . . . 14 ((𝜑 ∧ (𝑘 ∈ ℕ ∧ (𝑘 + 1) = 𝑃)) → (𝑘 + 1) = 𝑃)
122121, 84eqeltrd 2247 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑘 ∈ ℕ ∧ (𝑘 + 1) = 𝑃)) → (𝑘 + 1) ∈ ℙ)
123122iftrued 3532 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑘 ∈ ℕ ∧ (𝑘 + 1) = 𝑃)) → if((𝑘 + 1) ∈ ℙ, ((𝑘 + 1)↑(𝑘 + 1) / 𝑛𝐴), 1) = ((𝑘 + 1)↑(𝑘 + 1) / 𝑛𝐴))
124121csbeq1d 3056 . . . . . . . . . . . . . 14 ((𝜑 ∧ (𝑘 ∈ ℕ ∧ (𝑘 + 1) = 𝑃)) → (𝑘 + 1) / 𝑛𝐴 = 𝑃 / 𝑛𝐴)
125 nfcvd 2313 . . . . . . . . . . . . . . . 16 (𝑃 ∈ ℙ → 𝑛𝐵)
126 pcmpt.5 . . . . . . . . . . . . . . . 16 (𝑛 = 𝑃𝐴 = 𝐵)
127125, 126csbiegf 3092 . . . . . . . . . . . . . . 15 (𝑃 ∈ ℙ → 𝑃 / 𝑛𝐴 = 𝐵)
12884, 127syl 14 . . . . . . . . . . . . . 14 ((𝜑 ∧ (𝑘 ∈ ℕ ∧ (𝑘 + 1) = 𝑃)) → 𝑃 / 𝑛𝐴 = 𝐵)
129124, 128eqtrd 2203 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑘 ∈ ℕ ∧ (𝑘 + 1) = 𝑃)) → (𝑘 + 1) / 𝑛𝐴 = 𝐵)
130121, 129oveq12d 5869 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑘 ∈ ℕ ∧ (𝑘 + 1) = 𝑃)) → ((𝑘 + 1)↑(𝑘 + 1) / 𝑛𝐴) = (𝑃𝐵))
131120, 123, 1303eqtrd 2207 . . . . . . . . . . 11 ((𝜑 ∧ (𝑘 ∈ ℕ ∧ (𝑘 + 1) = 𝑃)) → (𝐹‘(𝑘 + 1)) = (𝑃𝐵))
132131oveq2d 5867 . . . . . . . . . 10 ((𝜑 ∧ (𝑘 ∈ ℕ ∧ (𝑘 + 1) = 𝑃)) → (𝑃 pCnt (𝐹‘(𝑘 + 1))) = (𝑃 pCnt (𝑃𝐵)))
133126eleq1d 2239 . . . . . . . . . . . . . 14 (𝑛 = 𝑃 → (𝐴 ∈ ℕ0𝐵 ∈ ℕ0))
134133rspcv 2830 . . . . . . . . . . . . 13 (𝑃 ∈ ℙ → (∀𝑛 ∈ ℙ 𝐴 ∈ ℕ0𝐵 ∈ ℕ0))
13526, 34, 134sylc 62 . . . . . . . . . . . 12 (𝜑𝐵 ∈ ℕ0)
136135adantr 274 . . . . . . . . . . 11 ((𝜑 ∧ (𝑘 ∈ ℕ ∧ (𝑘 + 1) = 𝑃)) → 𝐵 ∈ ℕ0)
137 pcidlem 12269 . . . . . . . . . . 11 ((𝑃 ∈ ℙ ∧ 𝐵 ∈ ℕ0) → (𝑃 pCnt (𝑃𝐵)) = 𝐵)
13826, 136, 137syl2an2r 590 . . . . . . . . . 10 ((𝜑 ∧ (𝑘 ∈ ℕ ∧ (𝑘 + 1) = 𝑃)) → (𝑃 pCnt (𝑃𝐵)) = 𝐵)
13993, 132, 1383eqtrd 2207 . . . . . . . . 9 ((𝜑 ∧ (𝑘 ∈ ℕ ∧ (𝑘 + 1) = 𝑃)) → (0 + (𝑃 pCnt (𝐹‘(𝑘 + 1)))) = 𝐵)
140 oveq1 5858 . . . . . . . . . 10 ((𝑃 pCnt (seq1( · , 𝐹)‘𝑘)) = 0 → ((𝑃 pCnt (seq1( · , 𝐹)‘𝑘)) + (𝑃 pCnt (𝐹‘(𝑘 + 1)))) = (0 + (𝑃 pCnt (𝐹‘(𝑘 + 1)))))
141140eqeq1d 2179 . . . . . . . . 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 156 . . . . . . . 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 8878 . . . . . . . . . . . . . 14 (𝑘 ∈ ℕ → 𝑘 ∈ ℝ)
144143ltp1d 8839 . . . . . . . . . . . . 13 (𝑘 ∈ ℕ → 𝑘 < (𝑘 + 1))
145 nnz 9224 . . . . . . . . . . . . . 14 (𝑘 ∈ ℕ → 𝑘 ∈ ℤ)
14687nnzd 9326 . . . . . . . . . . . . . 14 (𝑘 ∈ ℕ → (𝑘 + 1) ∈ ℤ)
147 zltnle 9251 . . . . . . . . . . . . . 14 ((𝑘 ∈ ℤ ∧ (𝑘 + 1) ∈ ℤ) → (𝑘 < (𝑘 + 1) ↔ ¬ (𝑘 + 1) ≤ 𝑘))
148145, 146, 147syl2anc 409 . . . . . . . . . . . . 13 (𝑘 ∈ ℕ → (𝑘 < (𝑘 + 1) ↔ ¬ (𝑘 + 1) ≤ 𝑘))
149144, 148mpbid 146 . . . . . . . . . . . 12 (𝑘 ∈ ℕ → ¬ (𝑘 + 1) ≤ 𝑘)
150149ad2antrl 487 . . . . . . . . . . 11 ((𝜑 ∧ (𝑘 ∈ ℕ ∧ (𝑘 + 1) = 𝑃)) → ¬ (𝑘 + 1) ≤ 𝑘)
151121breq1d 3997 . . . . . . . . . . 11 ((𝜑 ∧ (𝑘 ∈ ℕ ∧ (𝑘 + 1) = 𝑃)) → ((𝑘 + 1) ≤ 𝑘𝑃𝑘))
152150, 151mtbid 667 . . . . . . . . . 10 ((𝜑 ∧ (𝑘 ∈ ℕ ∧ (𝑘 + 1) = 𝑃)) → ¬ 𝑃𝑘)
153152iffalsed 3535 . . . . . . . . 9 ((𝜑 ∧ (𝑘 ∈ ℕ ∧ (𝑘 + 1) = 𝑃)) → if(𝑃𝑘, 𝐵, 0) = 0)
154153eqeq2d 2182 . . . . . . . 8 ((𝜑 ∧ (𝑘 ∈ ℕ ∧ (𝑘 + 1) = 𝑃)) → ((𝑃 pCnt (seq1( · , 𝐹)‘𝑘)) = if(𝑃𝑘, 𝐵, 0) ↔ (𝑃 pCnt (seq1( · , 𝐹)‘𝑘)) = 0))
155 simpr 109 . . . . . . . . . . . . . 14 ((𝜑𝑘 ∈ ℕ) → 𝑘 ∈ ℕ)
156 nnuz 9515 . . . . . . . . . . . . . 14 ℕ = (ℤ‘1)
157155, 156eleqtrdi 2263 . . . . . . . . . . . . 13 ((𝜑𝑘 ∈ ℕ) → 𝑘 ∈ (ℤ‘1))
15862adantlr 474 . . . . . . . . . . . . 13 (((𝜑𝑘 ∈ ℕ) ∧ 𝑖 ∈ (ℤ‘1)) → (𝐹𝑖) ∈ ℕ)
15963adantl 275 . . . . . . . . . . . . 13 (((𝜑𝑘 ∈ ℕ) ∧ (𝑖 ∈ ℕ ∧ 𝑗 ∈ ℕ)) → (𝑖 · 𝑗) ∈ ℕ)
160157, 158, 159seq3p1 10411 . . . . . . . . . . . 12 ((𝜑𝑘 ∈ ℕ) → (seq1( · , 𝐹)‘(𝑘 + 1)) = ((seq1( · , 𝐹)‘𝑘) · (𝐹‘(𝑘 + 1))))
161160oveq2d 5867 . . . . . . . . . . 11 ((𝜑𝑘 ∈ ℕ) → (𝑃 pCnt (seq1( · , 𝐹)‘(𝑘 + 1))) = (𝑃 pCnt ((seq1( · , 𝐹)‘𝑘) · (𝐹‘(𝑘 + 1)))))
16226adantr 274 . . . . . . . . . . . 12 ((𝜑𝑘 ∈ ℕ) → 𝑃 ∈ ℙ)
16385simprd 113 . . . . . . . . . . . . . 14 (𝜑 → seq1( · , 𝐹):ℕ⟶ℕ)
164163ffvelrnda 5629 . . . . . . . . . . . . 13 ((𝜑𝑘 ∈ ℕ) → (seq1( · , 𝐹)‘𝑘) ∈ ℕ)
165 nnz 9224 . . . . . . . . . . . . . 14 ((seq1( · , 𝐹)‘𝑘) ∈ ℕ → (seq1( · , 𝐹)‘𝑘) ∈ ℤ)
166 nnne0 8899 . . . . . . . . . . . . . 14 ((seq1( · , 𝐹)‘𝑘) ∈ ℕ → (seq1( · , 𝐹)‘𝑘) ≠ 0)
167165, 166jca 304 . . . . . . . . . . . . 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 9224 . . . . . . . . . . . . . 14 ((𝐹‘(𝑘 + 1)) ∈ ℕ → (𝐹‘(𝑘 + 1)) ∈ ℤ)
170 nnne0 8899 . . . . . . . . . . . . . 14 ((𝐹‘(𝑘 + 1)) ∈ ℕ → (𝐹‘(𝑘 + 1)) ≠ 0)
171169, 170jca 304 . . . . . . . . . . . . 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 12248 . . . . . . . . . . . 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 1233 . . . . . . . . . . 11 ((𝜑𝑘 ∈ ℕ) → (𝑃 pCnt ((seq1( · , 𝐹)‘𝑘) · (𝐹‘(𝑘 + 1)))) = ((𝑃 pCnt (seq1( · , 𝐹)‘𝑘)) + (𝑃 pCnt (𝐹‘(𝑘 + 1)))))
175161, 174eqtrd 2203 . . . . . . . . . 10 ((𝜑𝑘 ∈ ℕ) → (𝑃 pCnt (seq1( · , 𝐹)‘(𝑘 + 1))) = ((𝑃 pCnt (seq1( · , 𝐹)‘𝑘)) + (𝑃 pCnt (𝐹‘(𝑘 + 1)))))
176175adantrr 476 . . . . . . . . 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 12057 . . . . . . . . . . . . . . 15 (𝑃 ∈ ℙ → 𝑃 ∈ ℕ)
17826, 177syl 14 . . . . . . . . . . . . . 14 (𝜑𝑃 ∈ ℕ)
179178nnred 8884 . . . . . . . . . . . . 13 (𝜑𝑃 ∈ ℝ)
180179adantr 274 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑘 ∈ ℕ ∧ (𝑘 + 1) = 𝑃)) → 𝑃 ∈ ℝ)
181180leidd 8426 . . . . . . . . . . 11 ((𝜑 ∧ (𝑘 ∈ ℕ ∧ (𝑘 + 1) = 𝑃)) → 𝑃𝑃)
182181, 121breqtrrd 4015 . . . . . . . . . 10 ((𝜑 ∧ (𝑘 ∈ ℕ ∧ (𝑘 + 1) = 𝑃)) → 𝑃 ≤ (𝑘 + 1))
183182iftrued 3532 . . . . . . . . 9 ((𝜑 ∧ (𝑘 ∈ ℕ ∧ (𝑘 + 1) = 𝑃)) → if(𝑃 ≤ (𝑘 + 1), 𝐵, 0) = 𝐵)
184176, 183eqeq12d 2185 . . . . . . . 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 202 . . . . . . 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 373 . . . . . 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 476 . . . . . . . . . 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 531 . . . . . . . . . . . . . . . 16 (((𝜑 ∧ (𝑘 ∈ ℕ ∧ (𝑘 + 1) ≠ 𝑃)) ∧ (𝑘 + 1) ∈ ℙ) → (𝑘 + 1) ≠ 𝑃)
189188necomd 2426 . . . . . . . . . . . . . . 15 (((𝜑 ∧ (𝑘 ∈ ℕ ∧ (𝑘 + 1) ≠ 𝑃)) ∧ (𝑘 + 1) ∈ ℙ) → 𝑃 ≠ (𝑘 + 1))
19026ad2antrr 485 . . . . . . . . . . . . . . . . 17 (((𝜑 ∧ (𝑘 ∈ ℕ ∧ (𝑘 + 1) ≠ 𝑃)) ∧ (𝑘 + 1) ∈ ℙ) → 𝑃 ∈ ℙ)
191 simpr 109 . . . . . . . . . . . . . . . . 17 (((𝜑 ∧ (𝑘 ∈ ℕ ∧ (𝑘 + 1) ≠ 𝑃)) ∧ (𝑘 + 1) ∈ ℙ) → (𝑘 + 1) ∈ ℙ)
19234ad2antrr 485 . . . . . . . . . . . . . . . . . 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 12097 . . . . . . . . . . . . . . . . 17 ((𝑃 ∈ ℙ ∧ (𝑘 + 1) ∈ ℙ ∧ (𝑘 + 1) / 𝑛𝐴 ∈ ℕ0) → (𝑃 ∥ ((𝑘 + 1)↑(𝑘 + 1) / 𝑛𝐴) → 𝑃 = (𝑘 + 1)))
195190, 191, 193, 194syl3anc 1233 . . . . . . . . . . . . . . . 16 (((𝜑 ∧ (𝑘 ∈ ℕ ∧ (𝑘 + 1) ≠ 𝑃)) ∧ (𝑘 + 1) ∈ ℙ) → (𝑃 ∥ ((𝑘 + 1)↑(𝑘 + 1) / 𝑛𝐴) → 𝑃 = (𝑘 + 1)))
196195necon3ad 2382 . . . . . . . . . . . . . . 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 487 . . . . . . . . . . . . . . . . 17 ((𝜑 ∧ (𝑘 ∈ ℕ ∧ (𝑘 + 1) ≠ 𝑃)) → (𝑘 + 1) ∈ ℕ)
199109adantrr 476 . . . . . . . . . . . . . . . . 17 ((𝜑 ∧ (𝑘 ∈ ℕ ∧ (𝑘 + 1) ≠ 𝑃)) → if((𝑘 + 1) ∈ ℙ, ((𝑘 + 1)↑(𝑘 + 1) / 𝑛𝐴), 1) ∈ ℕ)
200198, 199, 119syl2anc 409 . . . . . . . . . . . . . . . 16 ((𝜑 ∧ (𝑘 ∈ ℕ ∧ (𝑘 + 1) ≠ 𝑃)) → (𝐹‘(𝑘 + 1)) = if((𝑘 + 1) ∈ ℙ, ((𝑘 + 1)↑(𝑘 + 1) / 𝑛𝐴), 1))
201 iftrue 3530 . . . . . . . . . . . . . . . 16 ((𝑘 + 1) ∈ ℙ → if((𝑘 + 1) ∈ ℙ, ((𝑘 + 1)↑(𝑘 + 1) / 𝑛𝐴), 1) = ((𝑘 + 1)↑(𝑘 + 1) / 𝑛𝐴))
202200, 201sylan9eq 2223 . . . . . . . . . . . . . . 15 (((𝜑 ∧ (𝑘 ∈ ℕ ∧ (𝑘 + 1) ≠ 𝑃)) ∧ (𝑘 + 1) ∈ ℙ) → (𝐹‘(𝑘 + 1)) = ((𝑘 + 1)↑(𝑘 + 1) / 𝑛𝐴))
203202breq2d 3999 . . . . . . . . . . . . . 14 (((𝜑 ∧ (𝑘 ∈ ℕ ∧ (𝑘 + 1) ≠ 𝑃)) ∧ (𝑘 + 1) ∈ ℙ) → (𝑃 ∥ (𝐹‘(𝑘 + 1)) ↔ 𝑃 ∥ ((𝑘 + 1)↑(𝑘 + 1) / 𝑛𝐴)))
204197, 203mtbird 668 . . . . . . . . . . . . 13 (((𝜑 ∧ (𝑘 ∈ ℕ ∧ (𝑘 + 1) ≠ 𝑃)) ∧ (𝑘 + 1) ∈ ℙ) → ¬ 𝑃 ∥ (𝐹‘(𝑘 + 1)))
20586, 198, 88syl2an2r 590 . . . . . . . . . . . . . . 15 ((𝜑 ∧ (𝑘 ∈ ℕ ∧ (𝑘 + 1) ≠ 𝑃)) → (𝐹‘(𝑘 + 1)) ∈ ℕ)
206205adantr 274 . . . . . . . . . . . . . 14 (((𝜑 ∧ (𝑘 ∈ ℕ ∧ (𝑘 + 1) ≠ 𝑃)) ∧ (𝑘 + 1) ∈ ℙ) → (𝐹‘(𝑘 + 1)) ∈ ℕ)
207 pceq0 12268 . . . . . . . . . . . . . 14 ((𝑃 ∈ ℙ ∧ (𝐹‘(𝑘 + 1)) ∈ ℕ) → ((𝑃 pCnt (𝐹‘(𝑘 + 1))) = 0 ↔ ¬ 𝑃 ∥ (𝐹‘(𝑘 + 1))))
208190, 206, 207syl2anc 409 . . . . . . . . . . . . 13 (((𝜑 ∧ (𝑘 ∈ ℕ ∧ (𝑘 + 1) ≠ 𝑃)) ∧ (𝑘 + 1) ∈ ℙ) → ((𝑃 pCnt (𝐹‘(𝑘 + 1))) = 0 ↔ ¬ 𝑃 ∥ (𝐹‘(𝑘 + 1))))
209204, 208mpbird 166 . . . . . . . . . . . 12 (((𝜑 ∧ (𝑘 ∈ ℕ ∧ (𝑘 + 1) ≠ 𝑃)) ∧ (𝑘 + 1) ∈ ℙ) → (𝑃 pCnt (𝐹‘(𝑘 + 1))) = 0)
210 iffalse 3533 . . . . . . . . . . . . . . 15 (¬ (𝑘 + 1) ∈ ℙ → if((𝑘 + 1) ∈ ℙ, ((𝑘 + 1)↑(𝑘 + 1) / 𝑛𝐴), 1) = 1)
211200, 210sylan9eq 2223 . . . . . . . . . . . . . 14 (((𝜑 ∧ (𝑘 ∈ ℕ ∧ (𝑘 + 1) ≠ 𝑃)) ∧ ¬ (𝑘 + 1) ∈ ℙ) → (𝐹‘(𝑘 + 1)) = 1)
212211oveq2d 5867 . . . . . . . . . . . . 13 (((𝜑 ∧ (𝑘 ∈ ℕ ∧ (𝑘 + 1) ≠ 𝑃)) ∧ ¬ (𝑘 + 1) ∈ ℙ) → (𝑃 pCnt (𝐹‘(𝑘 + 1))) = (𝑃 pCnt 1))
21328ad2antrr 485 . . . . . . . . . . . . 13 (((𝜑 ∧ (𝑘 ∈ ℕ ∧ (𝑘 + 1) ≠ 𝑃)) ∧ ¬ (𝑘 + 1) ∈ ℙ) → (𝑃 pCnt 1) = 0)
214212, 213eqtrd 2203 . . . . . . . . . . . 12 (((𝜑 ∧ (𝑘 ∈ ℕ ∧ (𝑘 + 1) ≠ 𝑃)) ∧ ¬ (𝑘 + 1) ∈ ℙ) → (𝑃 pCnt (𝐹‘(𝑘 + 1))) = 0)
215 exmiddc 831 . . . . . . . . . . . . 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 793 . . . . . . . . . . 11 ((𝜑 ∧ (𝑘 ∈ ℕ ∧ (𝑘 + 1) ≠ 𝑃)) → (𝑃 pCnt (𝐹‘(𝑘 + 1))) = 0)
218217oveq2d 5867 . . . . . . . . . 10 ((𝜑 ∧ (𝑘 ∈ ℕ ∧ (𝑘 + 1) ≠ 𝑃)) → ((𝑃 pCnt (seq1( · , 𝐹)‘𝑘)) + (𝑃 pCnt (𝐹‘(𝑘 + 1)))) = ((𝑃 pCnt (seq1( · , 𝐹)‘𝑘)) + 0))
21926adantr 274 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑘 ∈ ℕ ∧ (𝑘 + 1) ≠ 𝑃)) → 𝑃 ∈ ℙ)
220164adantrr 476 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑘 ∈ ℕ ∧ (𝑘 + 1) ≠ 𝑃)) → (seq1( · , 𝐹)‘𝑘) ∈ ℕ)
221219, 220pccld 12247 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑘 ∈ ℕ ∧ (𝑘 + 1) ≠ 𝑃)) → (𝑃 pCnt (seq1( · , 𝐹)‘𝑘)) ∈ ℕ0)
222221nn0cnd 9183 . . . . . . . . . . 11 ((𝜑 ∧ (𝑘 ∈ ℕ ∧ (𝑘 + 1) ≠ 𝑃)) → (𝑃 pCnt (seq1( · , 𝐹)‘𝑘)) ∈ ℂ)
223222addid1d 8061 . . . . . . . . . 10 ((𝜑 ∧ (𝑘 ∈ ℕ ∧ (𝑘 + 1) ≠ 𝑃)) → ((𝑃 pCnt (seq1( · , 𝐹)‘𝑘)) + 0) = (𝑃 pCnt (seq1( · , 𝐹)‘𝑘)))
224187, 218, 2233eqtrd 2207 . . . . . . . . 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 487 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑘 ∈ ℕ ∧ (𝑘 + 1) ≠ 𝑃)) → (𝑘 + 1) ∈ ℤ)
227 zltlen 9283 . . . . . . . . . . . 12 ((𝑃 ∈ ℤ ∧ (𝑘 + 1) ∈ ℤ) → (𝑃 < (𝑘 + 1) ↔ (𝑃 ≤ (𝑘 + 1) ∧ (𝑘 + 1) ≠ 𝑃)))
228225, 226, 227syl2anc 409 . . . . . . . . . . 11 ((𝜑 ∧ (𝑘 ∈ ℕ ∧ (𝑘 + 1) ≠ 𝑃)) → (𝑃 < (𝑘 + 1) ↔ (𝑃 ≤ (𝑘 + 1) ∧ (𝑘 + 1) ≠ 𝑃)))
229 simprl 526 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑘 ∈ ℕ ∧ (𝑘 + 1) ≠ 𝑃)) → 𝑘 ∈ ℕ)
230 nnleltp1 9264 . . . . . . . . . . . 12 ((𝑃 ∈ ℕ ∧ 𝑘 ∈ ℕ) → (𝑃𝑘𝑃 < (𝑘 + 1)))
231178, 229, 230syl2an2r 590 . . . . . . . . . . 11 ((𝜑 ∧ (𝑘 ∈ ℕ ∧ (𝑘 + 1) ≠ 𝑃)) → (𝑃𝑘𝑃 < (𝑘 + 1)))
232 simprr 527 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑘 ∈ ℕ ∧ (𝑘 + 1) ≠ 𝑃)) → (𝑘 + 1) ≠ 𝑃)
233232biantrud 302 . . . . . . . . . . 11 ((𝜑 ∧ (𝑘 ∈ ℕ ∧ (𝑘 + 1) ≠ 𝑃)) → (𝑃 ≤ (𝑘 + 1) ↔ (𝑃 ≤ (𝑘 + 1) ∧ (𝑘 + 1) ≠ 𝑃)))
234228, 231, 2333bitr4rd 220 . . . . . . . . . 10 ((𝜑 ∧ (𝑘 ∈ ℕ ∧ (𝑘 + 1) ≠ 𝑃)) → (𝑃 ≤ (𝑘 + 1) ↔ 𝑃𝑘))
235234ifbid 3546 . . . . . . . . 9 ((𝜑 ∧ (𝑘 ∈ ℕ ∧ (𝑘 + 1) ≠ 𝑃)) → if(𝑃 ≤ (𝑘 + 1), 𝐵, 0) = if(𝑃𝑘, 𝐵, 0))
236224, 235eqeq12d 2185 . . . . . . . 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 157 . . . . . . 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 373 . . . . . 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 9326 . . . . . . . 8 ((𝜑𝑘 ∈ ℕ) → (𝑘 + 1) ∈ ℤ)
240162, 77syl 14 . . . . . . . 8 ((𝜑𝑘 ∈ ℕ) → 𝑃 ∈ ℤ)
241 zdceq 9280 . . . . . . . 8 (((𝑘 + 1) ∈ ℤ ∧ 𝑃 ∈ ℤ) → DECID (𝑘 + 1) = 𝑃)
242239, 240, 241syl2anc 409 . . . . . . 7 ((𝜑𝑘 ∈ ℕ) → DECID (𝑘 + 1) = 𝑃)
243 dcne 2351 . . . . . . 7 (DECID (𝑘 + 1) = 𝑃 ↔ ((𝑘 + 1) = 𝑃 ∨ (𝑘 + 1) ≠ 𝑃))
244242, 243sylib 121 . . . . . 6 ((𝜑𝑘 ∈ ℕ) → ((𝑘 + 1) = 𝑃 ∨ (𝑘 + 1) ≠ 𝑃))
245186, 238, 244mpjaod 713 . . . . 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 115 . . . 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 8887 . 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 103  wb 104  wo 703  DECID wdc 829   = wceq 1348  wcel 2141  wne 2340  wral 2448  csb 3049  ifcif 3525   class class class wbr 3987  cmpt 4048  wf 5192  cfv 5196  (class class class)co 5851  cr 7766  0cc0 7767  1c1 7768   + caddc 7770   · cmul 7772   < clt 7947  cle 7948  cn 8871  0cn0 9128  cz 9205  cuz 9480  seqcseq 10394  cexp 10468  cdvds 11742  cprime 12054   pCnt cpc 12231
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-coll 4102  ax-sep 4105  ax-nul 4113  ax-pow 4158  ax-pr 4192  ax-un 4416  ax-setind 4519  ax-iinf 4570  ax-cnex 7858  ax-resscn 7859  ax-1cn 7860  ax-1re 7861  ax-icn 7862  ax-addcl 7863  ax-addrcl 7864  ax-mulcl 7865  ax-mulrcl 7866  ax-addcom 7867  ax-mulcom 7868  ax-addass 7869  ax-mulass 7870  ax-distr 7871  ax-i2m1 7872  ax-0lt1 7873  ax-1rid 7874  ax-0id 7875  ax-rnegex 7876  ax-precex 7877  ax-cnre 7878  ax-pre-ltirr 7879  ax-pre-ltwlin 7880  ax-pre-lttrn 7881  ax-pre-apti 7882  ax-pre-ltadd 7883  ax-pre-mulgt0 7884  ax-pre-mulext 7885  ax-arch 7886  ax-caucvg 7887
This theorem depends on definitions:  df-bi 116  df-stab 826  df-dc 830  df-3or 974  df-3an 975  df-tru 1351  df-fal 1354  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ne 2341  df-nel 2436  df-ral 2453  df-rex 2454  df-reu 2455  df-rmo 2456  df-rab 2457  df-v 2732  df-sbc 2956  df-csb 3050  df-dif 3123  df-un 3125  df-in 3127  df-ss 3134  df-nul 3415  df-if 3526  df-pw 3566  df-sn 3587  df-pr 3588  df-op 3590  df-uni 3795  df-int 3830  df-iun 3873  df-br 3988  df-opab 4049  df-mpt 4050  df-tr 4086  df-id 4276  df-po 4279  df-iso 4280  df-iord 4349  df-on 4351  df-ilim 4352  df-suc 4354  df-iom 4573  df-xp 4615  df-rel 4616  df-cnv 4617  df-co 4618  df-dm 4619  df-rn 4620  df-res 4621  df-ima 4622  df-iota 5158  df-fun 5198  df-fn 5199  df-f 5200  df-f1 5201  df-fo 5202  df-f1o 5203  df-fv 5204  df-isom 5205  df-riota 5807  df-ov 5854  df-oprab 5855  df-mpo 5856  df-1st 6117  df-2nd 6118  df-recs 6282  df-frec 6368  df-1o 6393  df-2o 6394  df-er 6511  df-en 6717  df-fin 6719  df-sup 6959  df-inf 6960  df-pnf 7949  df-mnf 7950  df-xr 7951  df-ltxr 7952  df-le 7953  df-sub 8085  df-neg 8086  df-reap 8487  df-ap 8494  df-div 8583  df-inn 8872  df-2 8930  df-3 8931  df-4 8932  df-n0 9129  df-z 9206  df-uz 9481  df-q 9572  df-rp 9604  df-fz 9959  df-fzo 10092  df-fl 10219  df-mod 10272  df-seqfrec 10395  df-exp 10469  df-cj 10799  df-re 10800  df-im 10801  df-rsqrt 10955  df-abs 10956  df-dvds 11743  df-gcd 11891  df-prm 12055  df-pc 12232
This theorem is referenced by:  pcmpt2  12289  pcprod  12291  1arithlem4  12311
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