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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 12385 | . . 3 ⊢ (𝐴 ∈ ℝ+ → 𝐴 ∈ ℝ) | |
2 | rprene0 12394 | . . 3 ⊢ (𝐵 ∈ ℝ+ → (𝐵 ∈ ℝ ∧ 𝐵 ≠ 0)) | |
3 | redivcl 11348 | . . . 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 598 | . 2 ⊢ ((𝐴 ∈ ℝ+ ∧ 𝐵 ∈ ℝ+) → (𝐴 / 𝐵) ∈ ℝ) |
6 | elrp 12379 | . . 3 ⊢ (𝐴 ∈ ℝ+ ↔ (𝐴 ∈ ℝ ∧ 0 < 𝐴)) | |
7 | elrp 12379 | . . 3 ⊢ (𝐵 ∈ ℝ+ ↔ (𝐵 ∈ ℝ ∧ 0 < 𝐵)) | |
8 | divgt0 11497 | . . 3 ⊢ (((𝐴 ∈ ℝ ∧ 0 < 𝐴) ∧ (𝐵 ∈ ℝ ∧ 0 < 𝐵)) → 0 < (𝐴 / 𝐵)) | |
9 | 6, 7, 8 | syl2anb 600 | . 2 ⊢ ((𝐴 ∈ ℝ+ ∧ 𝐵 ∈ ℝ+) → 0 < (𝐴 / 𝐵)) |
10 | elrp 12379 | . 2 ⊢ ((𝐴 / 𝐵) ∈ ℝ+ ↔ ((𝐴 / 𝐵) ∈ ℝ ∧ 0 < (𝐴 / 𝐵))) | |
11 | 5, 9, 10 | sylanbrc 586 | 1 ⊢ ((𝐴 ∈ ℝ+ ∧ 𝐵 ∈ ℝ+) → (𝐴 / 𝐵) ∈ ℝ+) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 ∈ wcel 2111 ≠ wne 2987 class class class wbr 5030 (class class class)co 7135 ℝcr 10525 0cc0 10526 < clt 10664 / cdiv 11286 ℝ+crp 12377 |
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-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-op 4532 df-uni 4801 df-br 5031 df-opab 5093 df-mpt 5111 df-id 5425 df-po 5438 df-so 5439 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-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-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-rp 12378 |
This theorem is referenced by: rpreccl 12403 rphalfcl 12404 rpdivcld 12436 bcrpcl 13664 sqrlem7 14600 caurcvgr 15022 isprm5 16041 4sqlem12 16282 sylow1lem1 18715 metss2lem 23118 metss2 23119 minveclem3 24033 ovoliunlem3 24108 vitalilem4 24215 aaliou3lem8 24941 abelthlem8 25034 pigt3 25110 pige3ALT 25112 advlogexp 25246 atan1 25514 log2cnv 25530 cxp2limlem 25561 harmonicbnd4 25596 basellem1 25666 logexprlim 25809 logfacrlim2 25810 bcmono 25861 bposlem1 25868 bposlem7 25874 bposlem9 25876 rplogsumlem1 26068 dchrisumlem3 26075 dchrvmasum2lem 26080 dchrvmasum2if 26081 dchrvmasumlem2 26082 dchrvmasumlem3 26083 dchrvmasumiflem2 26086 dchrisum0lem2a 26101 dchrisum0lem2 26102 mudivsum 26114 mulogsumlem 26115 mulogsum 26116 mulog2sumlem1 26118 mulog2sumlem2 26119 mulog2sumlem3 26120 selberglem1 26129 selberglem2 26130 selberg 26132 selberg3lem1 26141 selbergr 26152 pntpbnd1a 26169 pntibndlem1 26173 pntibndlem3 26176 pntlema 26180 pntlemb 26181 pntlemg 26182 pntlemr 26186 pntlemj 26187 pntlemf 26189 smcnlem 28480 blocnilem 28587 minvecolem3 28659 nmcexi 29809 rpdp2cl 30584 dp2ltc 30589 dpgti 30608 circum 33030 faclim 33091 taupilem1 34735 poimirlem29 35086 mblfinlem3 35096 itg2addnclem2 35109 itg2addnclem3 35110 ftc1anclem7 35136 ftc1anc 35138 heiborlem5 35253 heiborlem7 35255 proot1ex 40145 |
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