![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
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 11632 | . . 3 ⊢ (𝑁 ∈ ℕ → 𝑁 ∈ ℝ) | |
2 | nnne0 11659 | . . 3 ⊢ (𝑁 ∈ ℕ → 𝑁 ≠ 0) | |
3 | 1, 2 | jca 515 | . 2 ⊢ (𝑁 ∈ ℕ → (𝑁 ∈ ℝ ∧ 𝑁 ≠ 0)) |
4 | redivcl 11348 | . . 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 595 | 1 ⊢ ((𝐴 ∈ ℝ ∧ 𝑁 ∈ ℕ) → (𝐴 / 𝑁) ∈ ℝ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 ∈ wcel 2111 ≠ wne 2987 (class class class)co 7135 ℝcr 10525 0cc0 10526 / cdiv 11286 ℕcn 11625 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 ax-resscn 10583 ax-1cn 10584 ax-icn 10585 ax-addcl 10586 ax-addrcl 10587 ax-mulcl 10588 ax-mulrcl 10589 ax-mulcom 10590 ax-addass 10591 ax-mulass 10592 ax-distr 10593 ax-i2m1 10594 ax-1ne0 10595 ax-1rid 10596 ax-rnegex 10597 ax-rrecex 10598 ax-cnre 10599 ax-pre-lttri 10600 ax-pre-lttrn 10601 ax-pre-ltadd 10602 ax-pre-mulgt0 10603 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-nel 3092 df-ral 3111 df-rex 3112 df-reu 3113 df-rmo 3114 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-pss 3900 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-tp 4530 df-op 4532 df-uni 4801 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-tr 5137 df-id 5425 df-eprel 5430 df-po 5438 df-so 5439 df-fr 5478 df-we 5480 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-pred 6116 df-ord 6162 df-on 6163 df-lim 6164 df-suc 6165 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-riota 7093 df-ov 7138 df-oprab 7139 df-mpo 7140 df-om 7561 df-wrecs 7930 df-recs 7991 df-rdg 8029 df-er 8272 df-en 8493 df-dom 8494 df-sdom 8495 df-pnf 10666 df-mnf 10667 df-xr 10668 df-ltxr 10669 df-le 10670 df-sub 10861 df-neg 10862 df-div 11287 df-nn 11626 |
This theorem is referenced by: nnrecre 11667 nndivred 11679 fldiv2 13224 zmodcl 13254 iexpcyc 13565 sqrlem7 14600 expcnv 15211 ef01bndlem 15529 sin01bnd 15530 cos01bnd 15531 rpnnen2lem2 15560 rpnnen2lem3 15561 rpnnen2lem4 15562 rpnnen2lem9 15567 fldivp1 16223 ovoliunlem1 24106 dyadf 24195 dyadovol 24197 mbfi1fseqlem3 24321 mbfi1fseqlem4 24322 dveflem 24582 plyeq0lem 24807 tangtx 25098 tan4thpi 25107 root1id 25343 root1eq1 25344 root1cj 25345 cxpeq 25346 1cubrlem 25427 atan1 25514 log2tlbnd 25531 log2ublem1 25532 log2ublem2 25533 log2ub 25535 birthdaylem3 25539 birthday 25540 basellem5 25670 basellem8 25673 ppiub 25788 logfac2 25801 dchrptlem1 25848 dchrptlem2 25849 bposlem3 25870 bposlem4 25871 bposlem5 25872 bposlem6 25873 bposlem9 25876 vmadivsum 26066 dchrisum0lem1a 26070 dchrmusum2 26078 dchrvmasum2if 26081 dchrvmasumlem2 26082 dchrvmasumiflem1 26085 dchrvmasumiflem2 26086 dchrisum0re 26097 dchrisum0lem1b 26099 dchrisum0lem1 26100 dchrvmasumlem 26107 rplogsum 26111 mudivsum 26114 selberg2 26135 chpdifbndlem1 26137 selberg3lem1 26141 selbergr 26152 pntlemb 26181 pntlemg 26182 pntlemf 26189 snmlff 32689 sinccvglem 33028 circum 33030 poimirlem29 35086 poimirlem30 35087 poimirlem32 35089 |
Copyright terms: Public domain | W3C validator |