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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 13028 | . . 3 ⊢ (𝐴 ∈ ℝ+ → 𝐴 ∈ ℝ) | |
2 | rprene0 13037 | . . 3 ⊢ (𝐵 ∈ ℝ+ → (𝐵 ∈ ℝ ∧ 𝐵 ≠ 0)) | |
3 | redivcl 11976 | . . . 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 13022 | . . 3 ⊢ (𝐴 ∈ ℝ+ ↔ (𝐴 ∈ ℝ ∧ 0 < 𝐴)) | |
7 | elrp 13022 | . . 3 ⊢ (𝐵 ∈ ℝ+ ↔ (𝐵 ∈ ℝ ∧ 0 < 𝐵)) | |
8 | divgt0 12126 | . . 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 13022 | . 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 2099 ≠ wne 2930 class class class wbr 5144 (class class class)co 7414 ℝcr 11146 0cc0 11147 < clt 11287 / cdiv 11910 ℝ+crp 13020 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2697 ax-sep 5295 ax-nul 5302 ax-pow 5360 ax-pr 5424 ax-un 7736 ax-resscn 11204 ax-1cn 11205 ax-icn 11206 ax-addcl 11207 ax-addrcl 11208 ax-mulcl 11209 ax-mulrcl 11210 ax-mulcom 11211 ax-addass 11212 ax-mulass 11213 ax-distr 11214 ax-i2m1 11215 ax-1ne0 11216 ax-1rid 11217 ax-rnegex 11218 ax-rrecex 11219 ax-cnre 11220 ax-pre-lttri 11221 ax-pre-lttrn 11222 ax-pre-ltadd 11223 ax-pre-mulgt0 11224 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2704 df-cleq 2718 df-clel 2803 df-nfc 2878 df-ne 2931 df-nel 3037 df-ral 3052 df-rex 3061 df-rmo 3365 df-reu 3366 df-rab 3421 df-v 3465 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4324 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4907 df-br 5145 df-opab 5207 df-mpt 5228 df-id 5571 df-po 5585 df-so 5586 df-xp 5679 df-rel 5680 df-cnv 5681 df-co 5682 df-dm 5683 df-rn 5684 df-res 5685 df-ima 5686 df-iota 6496 df-fun 6546 df-fn 6547 df-f 6548 df-f1 6549 df-fo 6550 df-f1o 6551 df-fv 6552 df-riota 7370 df-ov 7417 df-oprab 7418 df-mpo 7419 df-er 8724 df-en 8965 df-dom 8966 df-sdom 8967 df-pnf 11289 df-mnf 11290 df-xr 11291 df-ltxr 11292 df-le 11293 df-sub 11485 df-neg 11486 df-div 11911 df-rp 13021 |
This theorem is referenced by: rpreccl 13046 rphalfcl 13047 rpdivcld 13079 bcrpcl 14318 01sqrexlem7 15246 caurcvgr 15671 isprm5 16701 4sqlem12 16951 sylow1lem1 19590 metss2lem 24506 metss2 24507 minveclem3 25443 ovoliunlem3 25519 vitalilem4 25626 aaliou3lem8 26368 abelthlem8 26464 pigt3 26540 pige3ALT 26542 advlogexp 26677 atan1 26951 log2cnv 26967 cxp2limlem 26999 harmonicbnd4 27034 basellem1 27104 logexprlim 27249 logfacrlim2 27250 bcmono 27301 bposlem1 27308 bposlem7 27314 bposlem9 27316 rplogsumlem1 27508 dchrisumlem3 27515 dchrvmasum2lem 27520 dchrvmasum2if 27521 dchrvmasumlem2 27522 dchrvmasumlem3 27523 dchrvmasumiflem2 27526 dchrisum0lem2a 27541 dchrisum0lem2 27542 mudivsum 27554 mulogsumlem 27555 mulogsum 27556 mulog2sumlem1 27558 mulog2sumlem2 27559 mulog2sumlem3 27560 selberglem1 27569 selberglem2 27570 selberg 27572 selberg3lem1 27581 selbergr 27592 pntpbnd1a 27609 pntibndlem1 27613 pntibndlem3 27616 pntlema 27620 pntlemb 27621 pntlemg 27622 pntlemr 27626 pntlemj 27627 pntlemf 27629 smcnlem 30625 blocnilem 30732 minvecolem3 30804 nmcexi 31954 rpdp2cl 32744 dp2ltc 32749 dpgti 32768 circum 35513 faclim 35579 taupilem1 37039 poimirlem29 37361 mblfinlem3 37371 itg2addnclem2 37384 itg2addnclem3 37385 ftc1anclem7 37411 ftc1anc 37413 heiborlem5 37527 heiborlem7 37529 proot1ex 42896 |
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