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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 12257 | . . 3 ⊢ (𝑁 ∈ ℕ → 𝑁 ∈ ℝ) | |
2 | nnne0 12284 | . . 3 ⊢ (𝑁 ∈ ℕ → 𝑁 ≠ 0) | |
3 | 1, 2 | jca 510 | . 2 ⊢ (𝑁 ∈ ℕ → (𝑁 ∈ ℝ ∧ 𝑁 ≠ 0)) |
4 | redivcl 11971 | . . 3 ⊢ ((𝐴 ∈ ℝ ∧ 𝑁 ∈ ℝ ∧ 𝑁 ≠ 0) → (𝐴 / 𝑁) ∈ ℝ) | |
5 | 4 | 3expb 1117 | . 2 ⊢ ((𝐴 ∈ ℝ ∧ (𝑁 ∈ ℝ ∧ 𝑁 ≠ 0)) → (𝐴 / 𝑁) ∈ ℝ) |
6 | 3, 5 | sylan2 591 | 1 ⊢ ((𝐴 ∈ ℝ ∧ 𝑁 ∈ ℕ) → (𝐴 / 𝑁) ∈ ℝ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 394 ∈ wcel 2098 ≠ wne 2929 (class class class)co 7419 ℝcr 11144 0cc0 11145 / cdiv 11908 ℕcn 12250 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2696 ax-sep 5300 ax-nul 5307 ax-pow 5365 ax-pr 5429 ax-un 7741 ax-resscn 11202 ax-1cn 11203 ax-icn 11204 ax-addcl 11205 ax-addrcl 11206 ax-mulcl 11207 ax-mulrcl 11208 ax-mulcom 11209 ax-addass 11210 ax-mulass 11211 ax-distr 11212 ax-i2m1 11213 ax-1ne0 11214 ax-1rid 11215 ax-rnegex 11216 ax-rrecex 11217 ax-cnre 11218 ax-pre-lttri 11219 ax-pre-lttrn 11220 ax-pre-ltadd 11221 ax-pre-mulgt0 11222 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2930 df-nel 3036 df-ral 3051 df-rex 3060 df-rmo 3363 df-reu 3364 df-rab 3419 df-v 3463 df-sbc 3774 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-pss 3964 df-nul 4323 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4910 df-iun 4999 df-br 5150 df-opab 5212 df-mpt 5233 df-tr 5267 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 6307 df-ord 6374 df-on 6375 df-lim 6376 df-suc 6377 df-iota 6501 df-fun 6551 df-fn 6552 df-f 6553 df-f1 6554 df-fo 6555 df-f1o 6556 df-fv 6557 df-riota 7375 df-ov 7422 df-oprab 7423 df-mpo 7424 df-om 7872 df-2nd 7995 df-frecs 8287 df-wrecs 8318 df-recs 8392 df-rdg 8431 df-er 8725 df-en 8965 df-dom 8966 df-sdom 8967 df-pnf 11287 df-mnf 11288 df-xr 11289 df-ltxr 11290 df-le 11291 df-sub 11483 df-neg 11484 df-div 11909 df-nn 12251 |
This theorem is referenced by: nnrecre 12292 nndivred 12304 fldiv2 13867 zmodcl 13897 iexpcyc 14211 01sqrexlem7 15236 expcnv 15851 ef01bndlem 16169 sin01bnd 16170 cos01bnd 16171 rpnnen2lem2 16200 rpnnen2lem3 16201 rpnnen2lem4 16202 rpnnen2lem9 16207 fldivp1 16874 ovoliunlem1 25480 dyadf 25569 dyadovol 25571 mbfi1fseqlem3 25696 mbfi1fseqlem4 25697 dveflem 25960 plyeq0lem 26194 tangtx 26490 tan4thpi 26499 root1id 26739 root1eq1 26740 root1cj 26741 cxpeq 26742 1cubrlem 26823 atan1 26910 log2tlbnd 26927 log2ublem1 26928 log2ublem2 26929 log2ub 26931 birthdaylem3 26935 birthday 26936 basellem5 27067 basellem8 27070 ppiub 27187 logfac2 27200 dchrptlem1 27247 dchrptlem2 27248 bposlem3 27269 bposlem4 27270 bposlem5 27271 bposlem6 27272 bposlem9 27275 vmadivsum 27465 dchrisum0lem1a 27469 dchrmusum2 27477 dchrvmasum2if 27480 dchrvmasumlem2 27481 dchrvmasumiflem1 27484 dchrvmasumiflem2 27485 dchrisum0re 27496 dchrisum0lem1b 27498 dchrisum0lem1 27499 dchrvmasumlem 27506 rplogsum 27510 mudivsum 27513 selberg2 27534 chpdifbndlem1 27536 selberg3lem1 27540 selbergr 27551 pntlemb 27580 pntlemg 27581 pntlemf 27588 snmlff 35072 sinccvglem 35409 circum 35411 poimirlem29 37255 poimirlem30 37256 poimirlem32 37258 |
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