![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > lemuldivd | Structured version Visualization version GIF version |
Description: 'Less than or equal to' relationship between division and multiplication. (Contributed by Mario Carneiro, 30-May-2016.) |
Ref | Expression |
---|---|
ltmul1d.1 | ⊢ (𝜑 → 𝐴 ∈ ℝ) |
ltmul1d.2 | ⊢ (𝜑 → 𝐵 ∈ ℝ) |
ltmul1d.3 | ⊢ (𝜑 → 𝐶 ∈ ℝ+) |
Ref | Expression |
---|---|
lemuldivd | ⊢ (𝜑 → ((𝐴 · 𝐶) ≤ 𝐵 ↔ 𝐴 ≤ (𝐵 / 𝐶))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ltmul1d.1 | . 2 ⊢ (𝜑 → 𝐴 ∈ ℝ) | |
2 | ltmul1d.2 | . 2 ⊢ (𝜑 → 𝐵 ∈ ℝ) | |
3 | ltmul1d.3 | . . 3 ⊢ (𝜑 → 𝐶 ∈ ℝ+) | |
4 | 3 | rpregt0d 12162 | . 2 ⊢ (𝜑 → (𝐶 ∈ ℝ ∧ 0 < 𝐶)) |
5 | lemuldiv 11233 | . 2 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ (𝐶 ∈ ℝ ∧ 0 < 𝐶)) → ((𝐴 · 𝐶) ≤ 𝐵 ↔ 𝐴 ≤ (𝐵 / 𝐶))) | |
6 | 1, 2, 4, 5 | syl3anc 1494 | 1 ⊢ (𝜑 → ((𝐴 · 𝐶) ≤ 𝐵 ↔ 𝐴 ≤ (𝐵 / 𝐶))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 198 ∧ wa 386 ∈ wcel 2164 class class class wbr 4873 (class class class)co 6905 ℝcr 10251 0cc0 10252 · cmul 10257 < clt 10391 ≤ cle 10392 / cdiv 11009 ℝ+crp 12112 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1894 ax-4 1908 ax-5 2009 ax-6 2075 ax-7 2112 ax-8 2166 ax-9 2173 ax-10 2192 ax-11 2207 ax-12 2220 ax-13 2389 ax-ext 2803 ax-sep 5005 ax-nul 5013 ax-pow 5065 ax-pr 5127 ax-un 7209 ax-resscn 10309 ax-1cn 10310 ax-icn 10311 ax-addcl 10312 ax-addrcl 10313 ax-mulcl 10314 ax-mulrcl 10315 ax-mulcom 10316 ax-addass 10317 ax-mulass 10318 ax-distr 10319 ax-i2m1 10320 ax-1ne0 10321 ax-1rid 10322 ax-rnegex 10323 ax-rrecex 10324 ax-cnre 10325 ax-pre-lttri 10326 ax-pre-lttrn 10327 ax-pre-ltadd 10328 ax-pre-mulgt0 10329 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 879 df-3or 1112 df-3an 1113 df-tru 1660 df-ex 1879 df-nf 1883 df-sb 2068 df-mo 2605 df-eu 2640 df-clab 2812 df-cleq 2818 df-clel 2821 df-nfc 2958 df-ne 3000 df-nel 3103 df-ral 3122 df-rex 3123 df-reu 3124 df-rmo 3125 df-rab 3126 df-v 3416 df-sbc 3663 df-csb 3758 df-dif 3801 df-un 3803 df-in 3805 df-ss 3812 df-nul 4145 df-if 4307 df-pw 4380 df-sn 4398 df-pr 4400 df-op 4404 df-uni 4659 df-br 4874 df-opab 4936 df-mpt 4953 df-id 5250 df-po 5263 df-so 5264 df-xp 5348 df-rel 5349 df-cnv 5350 df-co 5351 df-dm 5352 df-rn 5353 df-res 5354 df-ima 5355 df-iota 6086 df-fun 6125 df-fn 6126 df-f 6127 df-f1 6128 df-fo 6129 df-f1o 6130 df-fv 6131 df-riota 6866 df-ov 6908 df-oprab 6909 df-mpt2 6910 df-er 8009 df-en 8223 df-dom 8224 df-sdom 8225 df-pnf 10393 df-mnf 10394 df-xr 10395 df-ltxr 10396 df-le 10397 df-sub 10587 df-neg 10588 df-div 11010 df-rp 12113 |
This theorem is referenced by: leexp2a 13210 bitsfzolem 15529 bitsfzo 15530 bitscmp 15533 gexexlem 18608 ovolsca 23681 abelthlem7 24591 cxpaddle 24895 divsqrtsumo1 25123 fsumharmonic 25151 lgamgulmlem5 25172 basellem8 25227 fsumvma2 25352 chpchtsum 25357 chpub 25358 logexprlim 25363 efexple 25419 chpchtlim 25581 rplogsumlem2 25587 dchrisum0lem1a 25588 dchrmusum2 25596 dchrvmasumlem2 25600 dchrisum0lem1 25618 mulog2sumlem2 25637 vmalogdivsum2 25640 2vmadivsumlem 25642 selberglem2 25648 chpdifbndlem1 25655 selberg3lem1 25659 selberg4lem1 25662 pntrlog2bndlem5 25683 pntlemh 25701 pntlemn 25702 pntlemr 25704 pntlemj 25705 ttgcontlem1 26184 logdivsqrle 31266 unbdqndv2lem2 33022 itg2addnclem2 33998 fourierdlem64 41174 |
Copyright terms: Public domain | W3C validator |