![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > pcmptcl | Structured version Visualization version GIF version |
Description: Closure for the prime power map. (Contributed by Mario Carneiro, 12-Mar-2014.) |
Ref | Expression |
---|---|
pcmpt.1 | ⊢ 𝐹 = (𝑛 ∈ ℕ ↦ if(𝑛 ∈ ℙ, (𝑛↑𝐴), 1)) |
pcmpt.2 | ⊢ (𝜑 → ∀𝑛 ∈ ℙ 𝐴 ∈ ℕ0) |
Ref | Expression |
---|---|
pcmptcl | ⊢ (𝜑 → (𝐹:ℕ⟶ℕ ∧ seq1( · , 𝐹):ℕ⟶ℕ)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | pcmpt.2 | . . . 4 ⊢ (𝜑 → ∀𝑛 ∈ ℙ 𝐴 ∈ ℕ0) | |
2 | pm2.27 42 | . . . . . . . 8 ⊢ (𝑛 ∈ ℙ → ((𝑛 ∈ ℙ → 𝐴 ∈ ℕ0) → 𝐴 ∈ ℕ0)) | |
3 | iftrue 4490 | . . . . . . . . . . 11 ⊢ (𝑛 ∈ ℙ → if(𝑛 ∈ ℙ, (𝑛↑𝐴), 1) = (𝑛↑𝐴)) | |
4 | 3 | adantr 481 | . . . . . . . . . 10 ⊢ ((𝑛 ∈ ℙ ∧ 𝐴 ∈ ℕ0) → if(𝑛 ∈ ℙ, (𝑛↑𝐴), 1) = (𝑛↑𝐴)) |
5 | prmnn 16504 | . . . . . . . . . . 11 ⊢ (𝑛 ∈ ℙ → 𝑛 ∈ ℕ) | |
6 | nnexpcl 13934 | . . . . . . . . . . 11 ⊢ ((𝑛 ∈ ℕ ∧ 𝐴 ∈ ℕ0) → (𝑛↑𝐴) ∈ ℕ) | |
7 | 5, 6 | sylan 580 | . . . . . . . . . 10 ⊢ ((𝑛 ∈ ℙ ∧ 𝐴 ∈ ℕ0) → (𝑛↑𝐴) ∈ ℕ) |
8 | 4, 7 | eqeltrd 2838 | . . . . . . . . 9 ⊢ ((𝑛 ∈ ℙ ∧ 𝐴 ∈ ℕ0) → if(𝑛 ∈ ℙ, (𝑛↑𝐴), 1) ∈ ℕ) |
9 | 8 | ex 413 | . . . . . . . 8 ⊢ (𝑛 ∈ ℙ → (𝐴 ∈ ℕ0 → if(𝑛 ∈ ℙ, (𝑛↑𝐴), 1) ∈ ℕ)) |
10 | 2, 9 | syld 47 | . . . . . . 7 ⊢ (𝑛 ∈ ℙ → ((𝑛 ∈ ℙ → 𝐴 ∈ ℕ0) → if(𝑛 ∈ ℙ, (𝑛↑𝐴), 1) ∈ ℕ)) |
11 | iffalse 4493 | . . . . . . . . 9 ⊢ (¬ 𝑛 ∈ ℙ → if(𝑛 ∈ ℙ, (𝑛↑𝐴), 1) = 1) | |
12 | 1nn 12122 | . . . . . . . . 9 ⊢ 1 ∈ ℕ | |
13 | 11, 12 | eqeltrdi 2846 | . . . . . . . 8 ⊢ (¬ 𝑛 ∈ ℙ → if(𝑛 ∈ ℙ, (𝑛↑𝐴), 1) ∈ ℕ) |
14 | 13 | a1d 25 | . . . . . . 7 ⊢ (¬ 𝑛 ∈ ℙ → ((𝑛 ∈ ℙ → 𝐴 ∈ ℕ0) → if(𝑛 ∈ ℙ, (𝑛↑𝐴), 1) ∈ ℕ)) |
15 | 10, 14 | pm2.61i 182 | . . . . . 6 ⊢ ((𝑛 ∈ ℙ → 𝐴 ∈ ℕ0) → if(𝑛 ∈ ℙ, (𝑛↑𝐴), 1) ∈ ℕ) |
16 | 15 | a1d 25 | . . . . 5 ⊢ ((𝑛 ∈ ℙ → 𝐴 ∈ ℕ0) → (𝑛 ∈ ℕ → if(𝑛 ∈ ℙ, (𝑛↑𝐴), 1) ∈ ℕ)) |
17 | 16 | ralimi2 3079 | . . . 4 ⊢ (∀𝑛 ∈ ℙ 𝐴 ∈ ℕ0 → ∀𝑛 ∈ ℕ if(𝑛 ∈ ℙ, (𝑛↑𝐴), 1) ∈ ℕ) |
18 | 1, 17 | syl 17 | . . 3 ⊢ (𝜑 → ∀𝑛 ∈ ℕ if(𝑛 ∈ ℙ, (𝑛↑𝐴), 1) ∈ ℕ) |
19 | pcmpt.1 | . . . 4 ⊢ 𝐹 = (𝑛 ∈ ℕ ↦ if(𝑛 ∈ ℙ, (𝑛↑𝐴), 1)) | |
20 | 19 | fmpt 7054 | . . 3 ⊢ (∀𝑛 ∈ ℕ if(𝑛 ∈ ℙ, (𝑛↑𝐴), 1) ∈ ℕ ↔ 𝐹:ℕ⟶ℕ) |
21 | 18, 20 | sylib 217 | . 2 ⊢ (𝜑 → 𝐹:ℕ⟶ℕ) |
22 | nnuz 12760 | . . 3 ⊢ ℕ = (ℤ≥‘1) | |
23 | 1zzd 12492 | . . 3 ⊢ (𝜑 → 1 ∈ ℤ) | |
24 | 21 | ffvelcdmda 7031 | . . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (𝐹‘𝑘) ∈ ℕ) |
25 | nnmulcl 12135 | . . . 4 ⊢ ((𝑘 ∈ ℕ ∧ 𝑝 ∈ ℕ) → (𝑘 · 𝑝) ∈ ℕ) | |
26 | 25 | adantl 482 | . . 3 ⊢ ((𝜑 ∧ (𝑘 ∈ ℕ ∧ 𝑝 ∈ ℕ)) → (𝑘 · 𝑝) ∈ ℕ) |
27 | 22, 23, 24, 26 | seqf 13883 | . 2 ⊢ (𝜑 → seq1( · , 𝐹):ℕ⟶ℕ) |
28 | 21, 27 | jca 512 | 1 ⊢ (𝜑 → (𝐹:ℕ⟶ℕ ∧ seq1( · , 𝐹):ℕ⟶ℕ)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 396 = wceq 1541 ∈ wcel 2106 ∀wral 3062 ifcif 4484 ↦ cmpt 5186 ⟶wf 6489 (class class class)co 7351 1c1 11010 · cmul 11014 ℕcn 12111 ℕ0cn0 12371 seqcseq 13860 ↑cexp 13921 ℙcprime 16501 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2707 ax-sep 5254 ax-nul 5261 ax-pow 5318 ax-pr 5382 ax-un 7664 ax-cnex 11065 ax-resscn 11066 ax-1cn 11067 ax-icn 11068 ax-addcl 11069 ax-addrcl 11070 ax-mulcl 11071 ax-mulrcl 11072 ax-mulcom 11073 ax-addass 11074 ax-mulass 11075 ax-distr 11076 ax-i2m1 11077 ax-1ne0 11078 ax-1rid 11079 ax-rnegex 11080 ax-rrecex 11081 ax-cnre 11082 ax-pre-lttri 11083 ax-pre-lttrn 11084 ax-pre-ltadd 11085 ax-pre-mulgt0 11086 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2887 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-reu 3352 df-rab 3406 df-v 3445 df-sbc 3738 df-csb 3854 df-dif 3911 df-un 3913 df-in 3915 df-ss 3925 df-pss 3927 df-nul 4281 df-if 4485 df-pw 4560 df-sn 4585 df-pr 4587 df-op 4591 df-uni 4864 df-iun 4954 df-br 5104 df-opab 5166 df-mpt 5187 df-tr 5221 df-id 5529 df-eprel 5535 df-po 5543 df-so 5544 df-fr 5586 df-we 5588 df-xp 5637 df-rel 5638 df-cnv 5639 df-co 5640 df-dm 5641 df-rn 5642 df-res 5643 df-ima 5644 df-pred 6251 df-ord 6318 df-on 6319 df-lim 6320 df-suc 6321 df-iota 6445 df-fun 6495 df-fn 6496 df-f 6497 df-f1 6498 df-fo 6499 df-f1o 6500 df-fv 6501 df-riota 7307 df-ov 7354 df-oprab 7355 df-mpo 7356 df-om 7795 df-1st 7913 df-2nd 7914 df-frecs 8204 df-wrecs 8235 df-recs 8309 df-rdg 8348 df-er 8606 df-en 8842 df-dom 8843 df-sdom 8844 df-pnf 11149 df-mnf 11150 df-xr 11151 df-ltxr 11152 df-le 11153 df-sub 11345 df-neg 11346 df-nn 12112 df-n0 12372 df-z 12458 df-uz 12722 df-fz 13379 df-seq 13861 df-exp 13922 df-prm 16502 |
This theorem is referenced by: pcmpt 16718 pcmpt2 16719 pcmptdvds 16720 pcprod 16721 1arithlem4 16752 bposlem3 26580 bposlem5 26582 bposlem6 26583 |
Copyright terms: Public domain | W3C validator |