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Mirrors > Home > MPE Home > Th. List > nndivre | Structured version Visualization version GIF version |
Description: The quotient of a real and a positive integer is real. (Contributed by NM, 28-Nov-2008.) |
Ref | Expression |
---|---|
nndivre | ⊢ ((𝐴 ∈ ℝ ∧ 𝑁 ∈ ℕ) → (𝐴 / 𝑁) ∈ ℝ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nnre 11320 | . . 3 ⊢ (𝑁 ∈ ℕ → 𝑁 ∈ ℝ) | |
2 | nnne0 11348 | . . 3 ⊢ (𝑁 ∈ ℕ → 𝑁 ≠ 0) | |
3 | 1, 2 | jca 508 | . 2 ⊢ (𝑁 ∈ ℕ → (𝑁 ∈ ℝ ∧ 𝑁 ≠ 0)) |
4 | redivcl 11036 | . . 3 ⊢ ((𝐴 ∈ ℝ ∧ 𝑁 ∈ ℝ ∧ 𝑁 ≠ 0) → (𝐴 / 𝑁) ∈ ℝ) | |
5 | 4 | 3expb 1150 | . 2 ⊢ ((𝐴 ∈ ℝ ∧ (𝑁 ∈ ℝ ∧ 𝑁 ≠ 0)) → (𝐴 / 𝑁) ∈ ℝ) |
6 | 3, 5 | sylan2 587 | 1 ⊢ ((𝐴 ∈ ℝ ∧ 𝑁 ∈ ℕ) → (𝐴 / 𝑁) ∈ ℝ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 385 ∈ wcel 2157 ≠ wne 2971 (class class class)co 6878 ℝcr 10223 0cc0 10224 / cdiv 10976 ℕcn 11312 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1891 ax-4 1905 ax-5 2006 ax-6 2072 ax-7 2107 ax-8 2159 ax-9 2166 ax-10 2185 ax-11 2200 ax-12 2213 ax-13 2377 ax-ext 2777 ax-sep 4975 ax-nul 4983 ax-pow 5035 ax-pr 5097 ax-un 7183 ax-resscn 10281 ax-1cn 10282 ax-icn 10283 ax-addcl 10284 ax-addrcl 10285 ax-mulcl 10286 ax-mulrcl 10287 ax-mulcom 10288 ax-addass 10289 ax-mulass 10290 ax-distr 10291 ax-i2m1 10292 ax-1ne0 10293 ax-1rid 10294 ax-rnegex 10295 ax-rrecex 10296 ax-cnre 10297 ax-pre-lttri 10298 ax-pre-lttrn 10299 ax-pre-ltadd 10300 ax-pre-mulgt0 10301 |
This theorem depends on definitions: df-bi 199 df-an 386 df-or 875 df-3or 1109 df-3an 1110 df-tru 1657 df-ex 1876 df-nf 1880 df-sb 2065 df-mo 2591 df-eu 2609 df-clab 2786 df-cleq 2792 df-clel 2795 df-nfc 2930 df-ne 2972 df-nel 3075 df-ral 3094 df-rex 3095 df-reu 3096 df-rmo 3097 df-rab 3098 df-v 3387 df-sbc 3634 df-csb 3729 df-dif 3772 df-un 3774 df-in 3776 df-ss 3783 df-pss 3785 df-nul 4116 df-if 4278 df-pw 4351 df-sn 4369 df-pr 4371 df-tp 4373 df-op 4375 df-uni 4629 df-iun 4712 df-br 4844 df-opab 4906 df-mpt 4923 df-tr 4946 df-id 5220 df-eprel 5225 df-po 5233 df-so 5234 df-fr 5271 df-we 5273 df-xp 5318 df-rel 5319 df-cnv 5320 df-co 5321 df-dm 5322 df-rn 5323 df-res 5324 df-ima 5325 df-pred 5898 df-ord 5944 df-on 5945 df-lim 5946 df-suc 5947 df-iota 6064 df-fun 6103 df-fn 6104 df-f 6105 df-f1 6106 df-fo 6107 df-f1o 6108 df-fv 6109 df-riota 6839 df-ov 6881 df-oprab 6882 df-mpt2 6883 df-om 7300 df-wrecs 7645 df-recs 7707 df-rdg 7745 df-er 7982 df-en 8196 df-dom 8197 df-sdom 8198 df-pnf 10365 df-mnf 10366 df-xr 10367 df-ltxr 10368 df-le 10369 df-sub 10558 df-neg 10559 df-div 10977 df-nn 11313 |
This theorem is referenced by: nnrecre 11355 nndivred 11367 fldiv2 12915 zmodcl 12945 iexpcyc 13223 sqrlem7 14330 expcnv 14934 ef01bndlem 15250 sin01bnd 15251 cos01bnd 15252 rpnnen2lem2 15280 rpnnen2lem3 15281 rpnnen2lem4 15282 rpnnen2lem9 15287 fldivp1 15934 ovoliunlem1 23610 dyadf 23699 dyadovol 23701 mbfi1fseqlem3 23825 mbfi1fseqlem4 23826 dveflem 24083 plyeq0lem 24307 tangtx 24599 tan4thpi 24608 root1id 24839 root1eq1 24840 root1cj 24841 cxpeq 24842 1cubrlem 24920 atan1 25007 log2tlbnd 25024 log2ublem1 25025 log2ublem2 25026 log2ub 25028 birthdaylem3 25032 birthday 25033 basellem5 25163 basellem8 25166 ppiub 25281 logfac2 25294 dchrptlem1 25341 dchrptlem2 25342 bposlem3 25363 bposlem4 25364 bposlem5 25365 bposlem6 25366 bposlem9 25369 vmadivsum 25523 dchrisum0lem1a 25527 dchrmusum2 25535 dchrvmasum2if 25538 dchrvmasumlem2 25539 dchrvmasumiflem1 25542 dchrvmasumiflem2 25543 dchrisum0re 25554 dchrisum0lem1b 25556 dchrisum0lem1 25557 dchrvmasumlem 25564 rplogsum 25568 mudivsum 25571 selberg2 25592 chpdifbndlem1 25594 selberg3lem1 25598 selbergr 25609 pntlemb 25638 pntlemg 25639 pntlemf 25646 snmlff 31828 sinccvglem 32081 circum 32083 poimirlem29 33927 poimirlem30 33928 poimirlem32 33930 |
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