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Mirrors > Home > MPE Home > Th. List > rerpdivcl | Structured version Visualization version GIF version |
Description: Closure law for division of a real by a positive real. (Contributed by NM, 10-Nov-2008.) |
Ref | Expression |
---|---|
rerpdivcl | ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ+) → (𝐴 / 𝐵) ∈ ℝ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | rprene0 12400 | . 2 ⊢ (𝐵 ∈ ℝ+ → (𝐵 ∈ ℝ ∧ 𝐵 ≠ 0)) | |
2 | redivcl 11353 | . . 3 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐵 ≠ 0) → (𝐴 / 𝐵) ∈ ℝ) | |
3 | 2 | 3expb 1116 | . 2 ⊢ ((𝐴 ∈ ℝ ∧ (𝐵 ∈ ℝ ∧ 𝐵 ≠ 0)) → (𝐴 / 𝐵) ∈ ℝ) |
4 | 1, 3 | sylan2 594 | 1 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ+) → (𝐴 / 𝐵) ∈ ℝ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 ∈ wcel 2110 ≠ wne 3016 (class class class)co 7150 ℝcr 10530 0cc0 10531 / cdiv 11291 ℝ+crp 12383 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2156 ax-12 2172 ax-ext 2793 ax-sep 5196 ax-nul 5203 ax-pow 5259 ax-pr 5322 ax-un 7455 ax-resscn 10588 ax-1cn 10589 ax-icn 10590 ax-addcl 10591 ax-addrcl 10592 ax-mulcl 10593 ax-mulrcl 10594 ax-mulcom 10595 ax-addass 10596 ax-mulass 10597 ax-distr 10598 ax-i2m1 10599 ax-1ne0 10600 ax-1rid 10601 ax-rnegex 10602 ax-rrecex 10603 ax-cnre 10604 ax-pre-lttri 10605 ax-pre-lttrn 10606 ax-pre-ltadd 10607 ax-pre-mulgt0 10608 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3497 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4562 df-pr 4564 df-op 4568 df-uni 4833 df-br 5060 df-opab 5122 df-mpt 5140 df-id 5455 df-po 5469 df-so 5470 df-xp 5556 df-rel 5557 df-cnv 5558 df-co 5559 df-dm 5560 df-rn 5561 df-res 5562 df-ima 5563 df-iota 6309 df-fun 6352 df-fn 6353 df-f 6354 df-f1 6355 df-fo 6356 df-f1o 6357 df-fv 6358 df-riota 7108 df-ov 7153 df-oprab 7154 df-mpo 7155 df-er 8283 df-en 8504 df-dom 8505 df-sdom 8506 df-pnf 10671 df-mnf 10672 df-xr 10673 df-ltxr 10674 df-le 10675 df-sub 10866 df-neg 10867 df-div 11292 df-rp 12384 |
This theorem is referenced by: ledivge1le 12454 rerpdivcld 12456 icccntr 12872 refldivcl 13187 fldivle 13195 ltdifltdiv 13198 modvalr 13234 flpmodeq 13236 mod0 13238 negmod0 13240 modlt 13242 moddiffl 13244 moddifz 13245 modid 13258 modcyc 13268 modadd1 13270 modmul1 13286 moddi 13301 modsubdir 13302 modirr 13304 sqrtdiv 14619 divrcnv 15201 gexdvds 18703 aaliou3lem8 24928 logdivlt 25198 cxp2limlem 25547 harmonicbnd4 25582 logexprlim 25795 bposlem7 25860 bposlem9 25862 chebbnd1lem3 26041 chebbnd1 26042 chto1ub 26046 chpo1ub 26050 vmadivsum 26052 rplogsumlem1 26054 dchrvmasumlema 26070 dchrvmasumiflem1 26071 dchrisum0fno1 26081 mulogsumlem 26101 logdivsum 26103 mulog2sumlem1 26104 selberg2lem 26120 selberg3lem1 26127 pntrmax 26134 pntpbnd1a 26155 pntpbnd1 26156 pntpbnd2 26157 pntpbnd 26158 pntibndlem3 26162 pntlem3 26179 pntleml 26181 pnt2 26183 subfacval3 32431 heiborlem6 35088 fldivmod 44571 |
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