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Mirrors > Home > MPE Home > Th. List > pccld | Structured version Visualization version GIF version |
Description: Closure of the prime power function. (Contributed by Mario Carneiro, 29-May-2016.) |
Ref | Expression |
---|---|
pccld.1 | ⊢ (𝜑 → 𝑃 ∈ ℙ) |
pccld.2 | ⊢ (𝜑 → 𝑁 ∈ ℕ) |
Ref | Expression |
---|---|
pccld | ⊢ (𝜑 → (𝑃 pCnt 𝑁) ∈ ℕ0) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | pccld.1 | . 2 ⊢ (𝜑 → 𝑃 ∈ ℙ) | |
2 | pccld.2 | . 2 ⊢ (𝜑 → 𝑁 ∈ ℕ) | |
3 | pccl 16846 | . 2 ⊢ ((𝑃 ∈ ℙ ∧ 𝑁 ∈ ℕ) → (𝑃 pCnt 𝑁) ∈ ℕ0) | |
4 | 1, 2, 3 | syl2anc 582 | 1 ⊢ (𝜑 → (𝑃 pCnt 𝑁) ∈ ℕ0) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2099 (class class class)co 7416 ℕcn 12258 ℕ0cn0 12518 ℙcprime 16667 pCnt cpc 16833 |
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 2697 ax-sep 5296 ax-nul 5303 ax-pow 5361 ax-pr 5425 ax-un 7738 ax-cnex 11205 ax-resscn 11206 ax-1cn 11207 ax-icn 11208 ax-addcl 11209 ax-addrcl 11210 ax-mulcl 11211 ax-mulrcl 11212 ax-mulcom 11213 ax-addass 11214 ax-mulass 11215 ax-distr 11216 ax-i2m1 11217 ax-1ne0 11218 ax-1rid 11219 ax-rnegex 11220 ax-rrecex 11221 ax-cnre 11222 ax-pre-lttri 11223 ax-pre-lttrn 11224 ax-pre-ltadd 11225 ax-pre-mulgt0 11226 ax-pre-sup 11227 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2704 df-cleq 2718 df-clel 2803 df-nfc 2878 df-ne 2931 df-nel 3037 df-ral 3052 df-rex 3061 df-rmo 3364 df-reu 3365 df-rab 3420 df-v 3464 df-sbc 3776 df-csb 3892 df-dif 3949 df-un 3951 df-in 3953 df-ss 3963 df-pss 3966 df-nul 4323 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4906 df-iun 4995 df-br 5146 df-opab 5208 df-mpt 5229 df-tr 5263 df-id 5572 df-eprel 5578 df-po 5586 df-so 5587 df-fr 5629 df-we 5631 df-xp 5680 df-rel 5681 df-cnv 5682 df-co 5683 df-dm 5684 df-rn 5685 df-res 5686 df-ima 5687 df-pred 6304 df-ord 6371 df-on 6372 df-lim 6373 df-suc 6374 df-iota 6498 df-fun 6548 df-fn 6549 df-f 6550 df-f1 6551 df-fo 6552 df-f1o 6553 df-fv 6554 df-riota 7372 df-ov 7419 df-oprab 7420 df-mpo 7421 df-om 7869 df-1st 7995 df-2nd 7996 df-frecs 8288 df-wrecs 8319 df-recs 8393 df-rdg 8432 df-1o 8488 df-2o 8489 df-er 8726 df-en 8967 df-dom 8968 df-sdom 8969 df-fin 8970 df-sup 9478 df-inf 9479 df-pnf 11291 df-mnf 11292 df-xr 11293 df-ltxr 11294 df-le 11295 df-sub 11487 df-neg 11488 df-div 11913 df-nn 12259 df-2 12321 df-3 12322 df-n0 12519 df-z 12605 df-uz 12869 df-q 12979 df-rp 13023 df-fl 13806 df-mod 13884 df-seq 14016 df-exp 14076 df-cj 15099 df-re 15100 df-im 15101 df-sqrt 15235 df-abs 15236 df-dvds 16252 df-gcd 16490 df-prm 16668 df-pc 16834 |
This theorem is referenced by: pcqmul 16850 pcidlem 16869 pcgcd1 16874 pc2dvds 16876 pcz 16878 pcprmpw2 16879 dvdsprmpweq 16881 pcadd 16886 pcmpt 16889 pcfac 16896 oddprmdvds 16900 pockthg 16903 prmreclem2 16914 sylow1lem1 19592 sylow1lem3 19594 sylow1lem5 19596 pgpfi 19599 slwhash 19618 fislw 19619 gexexlem 19846 ablfac1lem 20064 ablfac1b 20066 ablfac1c 20067 ablfac1eu 20069 pgpfac1lem2 20071 pgpfac1lem3a 20072 ablfaclem3 20083 mumullem2 27205 chtublem 27237 pclogsum 27241 bposlem1 27310 bposlem3 27312 chebbnd1lem1 27495 dchrisum0flblem1 27534 dchrisum0flblem2 27535 aks4d1p6 41793 aks4d1p7d1 41794 aks4d1p8d2 41797 aks4d1p8d3 41798 aks4d1p8 41799 aks6d1c2p2 41831 aks6d1c7 41896 |
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