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Mirrors > Home > MPE Home > Th. List > rpdivcl | Structured version Visualization version GIF version |
Description: Closure law for division of positive reals. (Contributed by FL, 27-Dec-2007.) |
Ref | Expression |
---|---|
rpdivcl | ⊢ ((𝐴 ∈ ℝ+ ∧ 𝐵 ∈ ℝ+) → (𝐴 / 𝐵) ∈ ℝ+) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | rpre 13022 | . . 3 ⊢ (𝐴 ∈ ℝ+ → 𝐴 ∈ ℝ) | |
2 | rprene0 13031 | . . 3 ⊢ (𝐵 ∈ ℝ+ → (𝐵 ∈ ℝ ∧ 𝐵 ≠ 0)) | |
3 | redivcl 11971 | . . . 4 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐵 ≠ 0) → (𝐴 / 𝐵) ∈ ℝ) | |
4 | 3 | 3expb 1117 | . . 3 ⊢ ((𝐴 ∈ ℝ ∧ (𝐵 ∈ ℝ ∧ 𝐵 ≠ 0)) → (𝐴 / 𝐵) ∈ ℝ) |
5 | 1, 2, 4 | syl2an 594 | . 2 ⊢ ((𝐴 ∈ ℝ+ ∧ 𝐵 ∈ ℝ+) → (𝐴 / 𝐵) ∈ ℝ) |
6 | elrp 13016 | . . 3 ⊢ (𝐴 ∈ ℝ+ ↔ (𝐴 ∈ ℝ ∧ 0 < 𝐴)) | |
7 | elrp 13016 | . . 3 ⊢ (𝐵 ∈ ℝ+ ↔ (𝐵 ∈ ℝ ∧ 0 < 𝐵)) | |
8 | divgt0 12120 | . . 3 ⊢ (((𝐴 ∈ ℝ ∧ 0 < 𝐴) ∧ (𝐵 ∈ ℝ ∧ 0 < 𝐵)) → 0 < (𝐴 / 𝐵)) | |
9 | 6, 7, 8 | syl2anb 596 | . 2 ⊢ ((𝐴 ∈ ℝ+ ∧ 𝐵 ∈ ℝ+) → 0 < (𝐴 / 𝐵)) |
10 | elrp 13016 | . 2 ⊢ ((𝐴 / 𝐵) ∈ ℝ+ ↔ ((𝐴 / 𝐵) ∈ ℝ ∧ 0 < (𝐴 / 𝐵))) | |
11 | 5, 9, 10 | sylanbrc 581 | 1 ⊢ ((𝐴 ∈ ℝ+ ∧ 𝐵 ∈ ℝ+) → (𝐴 / 𝐵) ∈ ℝ+) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 394 ∈ wcel 2098 ≠ wne 2937 class class class wbr 5152 (class class class)co 7426 ℝcr 11145 0cc0 11146 < clt 11286 / cdiv 11909 ℝ+crp 13014 |
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 2699 ax-sep 5303 ax-nul 5310 ax-pow 5369 ax-pr 5433 ax-un 7746 ax-resscn 11203 ax-1cn 11204 ax-icn 11205 ax-addcl 11206 ax-addrcl 11207 ax-mulcl 11208 ax-mulrcl 11209 ax-mulcom 11210 ax-addass 11211 ax-mulass 11212 ax-distr 11213 ax-i2m1 11214 ax-1ne0 11215 ax-1rid 11216 ax-rnegex 11217 ax-rrecex 11218 ax-cnre 11219 ax-pre-lttri 11220 ax-pre-lttrn 11221 ax-pre-ltadd 11222 ax-pre-mulgt0 11223 |
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 2529 df-eu 2558 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-rmo 3374 df-reu 3375 df-rab 3431 df-v 3475 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-nul 4327 df-if 4533 df-pw 4608 df-sn 4633 df-pr 4635 df-op 4639 df-uni 4913 df-br 5153 df-opab 5215 df-mpt 5236 df-id 5580 df-po 5594 df-so 5595 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-rn 5693 df-res 5694 df-ima 5695 df-iota 6505 df-fun 6555 df-fn 6556 df-f 6557 df-f1 6558 df-fo 6559 df-f1o 6560 df-fv 6561 df-riota 7382 df-ov 7429 df-oprab 7430 df-mpo 7431 df-er 8731 df-en 8971 df-dom 8972 df-sdom 8973 df-pnf 11288 df-mnf 11289 df-xr 11290 df-ltxr 11291 df-le 11292 df-sub 11484 df-neg 11485 df-div 11910 df-rp 13015 |
This theorem is referenced by: rpreccl 13040 rphalfcl 13041 rpdivcld 13073 bcrpcl 14307 01sqrexlem7 15235 caurcvgr 15660 isprm5 16685 4sqlem12 16932 sylow1lem1 19560 metss2lem 24440 metss2 24441 minveclem3 25377 ovoliunlem3 25453 vitalilem4 25560 aaliou3lem8 26300 abelthlem8 26396 pigt3 26472 pige3ALT 26474 advlogexp 26609 atan1 26880 log2cnv 26896 cxp2limlem 26928 harmonicbnd4 26963 basellem1 27033 logexprlim 27178 logfacrlim2 27179 bcmono 27230 bposlem1 27237 bposlem7 27243 bposlem9 27245 rplogsumlem1 27437 dchrisumlem3 27444 dchrvmasum2lem 27449 dchrvmasum2if 27450 dchrvmasumlem2 27451 dchrvmasumlem3 27452 dchrvmasumiflem2 27455 dchrisum0lem2a 27470 dchrisum0lem2 27471 mudivsum 27483 mulogsumlem 27484 mulogsum 27485 mulog2sumlem1 27487 mulog2sumlem2 27488 mulog2sumlem3 27489 selberglem1 27498 selberglem2 27499 selberg 27501 selberg3lem1 27510 selbergr 27521 pntpbnd1a 27538 pntibndlem1 27542 pntibndlem3 27545 pntlema 27549 pntlemb 27550 pntlemg 27551 pntlemr 27555 pntlemj 27556 pntlemf 27558 smcnlem 30527 blocnilem 30634 minvecolem3 30706 nmcexi 31856 rpdp2cl 32626 dp2ltc 32631 dpgti 32650 circum 35311 faclim 35373 taupilem1 36833 poimirlem29 37155 mblfinlem3 37165 itg2addnclem2 37178 itg2addnclem3 37179 ftc1anclem7 37205 ftc1anc 37207 heiborlem5 37321 heiborlem7 37323 proot1ex 42655 |
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