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Mirrors > Home > MPE Home > Th. List > pcelnn | Structured version Visualization version GIF version |
Description: There are a positive number of powers of a prime 𝑃 in 𝑁 iff 𝑃 divides 𝑁. (Contributed by Mario Carneiro, 23-Feb-2014.) |
Ref | Expression |
---|---|
pcelnn | ⊢ ((𝑃 ∈ ℙ ∧ 𝑁 ∈ ℕ) → ((𝑃 pCnt 𝑁) ∈ ℕ ↔ 𝑃 ∥ 𝑁)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nnz 12610 | . . 3 ⊢ (𝑁 ∈ ℕ → 𝑁 ∈ ℤ) | |
2 | 1nn0 12519 | . . . 4 ⊢ 1 ∈ ℕ0 | |
3 | pcdvdsb 16838 | . . . 4 ⊢ ((𝑃 ∈ ℙ ∧ 𝑁 ∈ ℤ ∧ 1 ∈ ℕ0) → (1 ≤ (𝑃 pCnt 𝑁) ↔ (𝑃↑1) ∥ 𝑁)) | |
4 | 2, 3 | mp3an3 1447 | . . 3 ⊢ ((𝑃 ∈ ℙ ∧ 𝑁 ∈ ℤ) → (1 ≤ (𝑃 pCnt 𝑁) ↔ (𝑃↑1) ∥ 𝑁)) |
5 | 1, 4 | sylan2 592 | . 2 ⊢ ((𝑃 ∈ ℙ ∧ 𝑁 ∈ ℕ) → (1 ≤ (𝑃 pCnt 𝑁) ↔ (𝑃↑1) ∥ 𝑁)) |
6 | pccl 16818 | . . 3 ⊢ ((𝑃 ∈ ℙ ∧ 𝑁 ∈ ℕ) → (𝑃 pCnt 𝑁) ∈ ℕ0) | |
7 | elnnnn0c 12548 | . . . 4 ⊢ ((𝑃 pCnt 𝑁) ∈ ℕ ↔ ((𝑃 pCnt 𝑁) ∈ ℕ0 ∧ 1 ≤ (𝑃 pCnt 𝑁))) | |
8 | 7 | baibr 536 | . . 3 ⊢ ((𝑃 pCnt 𝑁) ∈ ℕ0 → (1 ≤ (𝑃 pCnt 𝑁) ↔ (𝑃 pCnt 𝑁) ∈ ℕ)) |
9 | 6, 8 | syl 17 | . 2 ⊢ ((𝑃 ∈ ℙ ∧ 𝑁 ∈ ℕ) → (1 ≤ (𝑃 pCnt 𝑁) ↔ (𝑃 pCnt 𝑁) ∈ ℕ)) |
10 | prmnn 16645 | . . . . . 6 ⊢ (𝑃 ∈ ℙ → 𝑃 ∈ ℕ) | |
11 | 10 | nncnd 12259 | . . . . 5 ⊢ (𝑃 ∈ ℙ → 𝑃 ∈ ℂ) |
12 | 11 | exp1d 14138 | . . . 4 ⊢ (𝑃 ∈ ℙ → (𝑃↑1) = 𝑃) |
13 | 12 | adantr 480 | . . 3 ⊢ ((𝑃 ∈ ℙ ∧ 𝑁 ∈ ℕ) → (𝑃↑1) = 𝑃) |
14 | 13 | breq1d 5158 | . 2 ⊢ ((𝑃 ∈ ℙ ∧ 𝑁 ∈ ℕ) → ((𝑃↑1) ∥ 𝑁 ↔ 𝑃 ∥ 𝑁)) |
15 | 5, 9, 14 | 3bitr3d 309 | 1 ⊢ ((𝑃 ∈ ℙ ∧ 𝑁 ∈ ℕ) → ((𝑃 pCnt 𝑁) ∈ ℕ ↔ 𝑃 ∥ 𝑁)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 395 = wceq 1534 ∈ wcel 2099 class class class wbr 5148 (class class class)co 7420 1c1 11140 ≤ cle 11280 ℕcn 12243 ℕ0cn0 12503 ℤcz 12589 ↑cexp 14059 ∥ cdvds 16231 ℙcprime 16642 pCnt cpc 16805 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2699 ax-sep 5299 ax-nul 5306 ax-pow 5365 ax-pr 5429 ax-un 7740 ax-cnex 11195 ax-resscn 11196 ax-1cn 11197 ax-icn 11198 ax-addcl 11199 ax-addrcl 11200 ax-mulcl 11201 ax-mulrcl 11202 ax-mulcom 11203 ax-addass 11204 ax-mulass 11205 ax-distr 11206 ax-i2m1 11207 ax-1ne0 11208 ax-1rid 11209 ax-rnegex 11210 ax-rrecex 11211 ax-cnre 11212 ax-pre-lttri 11213 ax-pre-lttrn 11214 ax-pre-ltadd 11215 ax-pre-mulgt0 11216 ax-pre-sup 11217 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2530 df-eu 2559 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-rmo 3373 df-reu 3374 df-rab 3430 df-v 3473 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4909 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5576 df-eprel 5582 df-po 5590 df-so 5591 df-fr 5633 df-we 5635 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-pred 6305 df-ord 6372 df-on 6373 df-lim 6374 df-suc 6375 df-iota 6500 df-fun 6550 df-fn 6551 df-f 6552 df-f1 6553 df-fo 6554 df-f1o 6555 df-fv 6556 df-riota 7376 df-ov 7423 df-oprab 7424 df-mpo 7425 df-om 7871 df-1st 7993 df-2nd 7994 df-frecs 8287 df-wrecs 8318 df-recs 8392 df-rdg 8431 df-1o 8487 df-2o 8488 df-er 8725 df-en 8965 df-dom 8966 df-sdom 8967 df-fin 8968 df-sup 9466 df-inf 9467 df-pnf 11281 df-mnf 11282 df-xr 11283 df-ltxr 11284 df-le 11285 df-sub 11477 df-neg 11478 df-div 11903 df-nn 12244 df-2 12306 df-3 12307 df-n0 12504 df-z 12590 df-uz 12854 df-q 12964 df-rp 13008 df-fl 13790 df-mod 13868 df-seq 14000 df-exp 14060 df-cj 15079 df-re 15080 df-im 15081 df-sqrt 15215 df-abs 15216 df-dvds 16232 df-gcd 16470 df-prm 16643 df-pc 16806 |
This theorem is referenced by: pceq0 16840 pc2dvds 16848 1arith 16896 isppw2 27060 sqf11 27084 sqff1o 27127 chtublem 27157 perfect 27177 lgsne0 27281 dchrisum0flblem2 27455 aks4d1p7d1 41553 aks4d1p8d2 41556 aks4d1p8d3 41557 aks4d1p8 41558 aks6d1c2p2 41590 aks6d1c7 41656 perfectALTV 47063 |
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