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Mirrors > Home > MPE Home > Th. List > nnexpcld | Structured version Visualization version GIF version |
Description: Closure of exponentiation of nonnegative integers. (Contributed by Mario Carneiro, 28-May-2016.) |
Ref | Expression |
---|---|
nnexpcld.1 | ⊢ (𝜑 → 𝐴 ∈ ℕ) |
nnexpcld.2 | ⊢ (𝜑 → 𝑁 ∈ ℕ0) |
Ref | Expression |
---|---|
nnexpcld | ⊢ (𝜑 → (𝐴↑𝑁) ∈ ℕ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nnexpcld.1 | . 2 ⊢ (𝜑 → 𝐴 ∈ ℕ) | |
2 | nnexpcld.2 | . 2 ⊢ (𝜑 → 𝑁 ∈ ℕ0) | |
3 | nnexpcl 12873 | . 2 ⊢ ((𝐴 ∈ ℕ ∧ 𝑁 ∈ ℕ0) → (𝐴↑𝑁) ∈ ℕ) | |
4 | 1, 2, 3 | syl2anc 693 | 1 ⊢ (𝜑 → (𝐴↑𝑁) ∈ ℕ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 1990 (class class class)co 6650 ℕcn 11020 ℕ0cn0 11292 ↑cexp 12860 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-cnex 9992 ax-resscn 9993 ax-1cn 9994 ax-icn 9995 ax-addcl 9996 ax-addrcl 9997 ax-mulcl 9998 ax-mulrcl 9999 ax-mulcom 10000 ax-addass 10001 ax-mulass 10002 ax-distr 10003 ax-i2m1 10004 ax-1ne0 10005 ax-1rid 10006 ax-rnegex 10007 ax-rrecex 10008 ax-cnre 10009 ax-pre-lttri 10010 ax-pre-lttrn 10011 ax-pre-ltadd 10012 ax-pre-mulgt0 10013 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-nel 2898 df-ral 2917 df-rex 2918 df-reu 2919 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-pred 5680 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-riota 6611 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-om 7066 df-2nd 7169 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-er 7742 df-en 7956 df-dom 7957 df-sdom 7958 df-pnf 10076 df-mnf 10077 df-xr 10078 df-ltxr 10079 df-le 10080 df-sub 10268 df-neg 10269 df-nn 11021 df-n0 11293 df-z 11378 df-uz 11688 df-seq 12802 df-exp 12861 |
This theorem is referenced by: bitsp1 15153 bitsfzolem 15156 bitsfzo 15157 bitsmod 15158 bitsfi 15159 bitscmp 15160 bitsinv1lem 15163 bitsinv1 15164 2ebits 15169 bitsinvp1 15171 sadcaddlem 15179 sadadd3 15183 sadaddlem 15188 sadasslem 15192 bitsres 15195 bitsuz 15196 bitsshft 15197 smumullem 15214 smumul 15215 rplpwr 15276 rppwr 15277 pclem 15543 pcprendvds2 15546 pcpre1 15547 pcpremul 15548 pcdvdsb 15573 pcidlem 15576 pcid 15577 pcdvdstr 15580 pcgcd1 15581 pcprmpw2 15586 pcaddlem 15592 pcadd 15593 pcfaclem 15602 pcfac 15603 pcbc 15604 oddprmdvds 15607 prmpwdvds 15608 pockthlem 15609 2expltfac 15799 pgpfi1 18010 sylow1lem1 18013 sylow1lem3 18015 sylow1lem4 18016 sylow1lem5 18017 pgpfi 18020 gexexlem 18255 ablfac1lem 18467 ablfac1b 18469 ablfac1eu 18472 aalioulem2 24088 aalioulem5 24091 aaliou3lem9 24105 isppw2 24841 sgmppw 24922 fsumvma2 24939 pclogsum 24940 chpchtsum 24944 logfacubnd 24946 bposlem1 25009 bposlem5 25013 gausslemma2d 25099 lgseisen 25104 chebbnd1lem1 25158 rpvmasumlem 25176 dchrisum0flblem1 25197 dchrisum0flblem2 25198 ostth2lem2 25323 ostth2lem3 25324 oddpwdc 30416 eulerpartlemgh 30440 jm3.1lem3 37586 inductionexd 38453 stoweidlem25 40242 stoweidlem45 40262 wallispi2lem1 40288 ovnsubaddlem1 40784 ovolval5lem2 40867 fmtnoodd 41445 fmtnof1 41447 fmtnosqrt 41451 fmtnorec4 41461 odz2prm2pw 41475 fmtnoprmfac1lem 41476 fmtnoprmfac1 41477 fmtnoprmfac2lem1 41478 fmtnoprmfac2 41479 2pwp1prm 41503 lighneallem1 41522 proththdlem 41530 proththd 41531 pw2m1lepw2m1 42310 nnpw2even 42323 logbpw2m1 42361 nnpw2pmod 42377 nnpw2p 42380 nnolog2flm1 42384 dignn0flhalflem1 42409 |
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