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Mirrors > Home > MPE Home > Th. List > ltdiv1 | Structured version Visualization version GIF version |
Description: Division of both sides of 'less than' by a positive number. (Contributed by NM, 10-Oct-2004.) (Revised by Mario Carneiro, 27-May-2016.) |
Ref | Expression |
---|---|
ltdiv1 | ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ (𝐶 ∈ ℝ ∧ 0 < 𝐶)) → (𝐴 < 𝐵 ↔ (𝐴 / 𝐶) < (𝐵 / 𝐶))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simp1 1167 | . . 3 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ (𝐶 ∈ ℝ ∧ 0 < 𝐶)) → 𝐴 ∈ ℝ) | |
2 | simp2 1168 | . . 3 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ (𝐶 ∈ ℝ ∧ 0 < 𝐶)) → 𝐵 ∈ ℝ) | |
3 | simp3l 1259 | . . . 4 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ (𝐶 ∈ ℝ ∧ 0 < 𝐶)) → 𝐶 ∈ ℝ) | |
4 | simp3r 1260 | . . . . 5 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ (𝐶 ∈ ℝ ∧ 0 < 𝐶)) → 0 < 𝐶) | |
5 | 4 | gt0ne0d 10888 | . . . 4 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ (𝐶 ∈ ℝ ∧ 0 < 𝐶)) → 𝐶 ≠ 0) |
6 | 3, 5 | rereccld 11148 | . . 3 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ (𝐶 ∈ ℝ ∧ 0 < 𝐶)) → (1 / 𝐶) ∈ ℝ) |
7 | recgt0 11163 | . . . 4 ⊢ ((𝐶 ∈ ℝ ∧ 0 < 𝐶) → 0 < (1 / 𝐶)) | |
8 | 7 | 3ad2ant3 1166 | . . 3 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ (𝐶 ∈ ℝ ∧ 0 < 𝐶)) → 0 < (1 / 𝐶)) |
9 | ltmul1 11169 | . . 3 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ ((1 / 𝐶) ∈ ℝ ∧ 0 < (1 / 𝐶))) → (𝐴 < 𝐵 ↔ (𝐴 · (1 / 𝐶)) < (𝐵 · (1 / 𝐶)))) | |
10 | 1, 2, 6, 8, 9 | syl112anc 1494 | . 2 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ (𝐶 ∈ ℝ ∧ 0 < 𝐶)) → (𝐴 < 𝐵 ↔ (𝐴 · (1 / 𝐶)) < (𝐵 · (1 / 𝐶)))) |
11 | 1 | recnd 10361 | . . . 4 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ (𝐶 ∈ ℝ ∧ 0 < 𝐶)) → 𝐴 ∈ ℂ) |
12 | 3 | recnd 10361 | . . . 4 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ (𝐶 ∈ ℝ ∧ 0 < 𝐶)) → 𝐶 ∈ ℂ) |
13 | 11, 12, 5 | divrecd 11100 | . . 3 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ (𝐶 ∈ ℝ ∧ 0 < 𝐶)) → (𝐴 / 𝐶) = (𝐴 · (1 / 𝐶))) |
14 | 2 | recnd 10361 | . . . 4 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ (𝐶 ∈ ℝ ∧ 0 < 𝐶)) → 𝐵 ∈ ℂ) |
15 | 14, 12, 5 | divrecd 11100 | . . 3 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ (𝐶 ∈ ℝ ∧ 0 < 𝐶)) → (𝐵 / 𝐶) = (𝐵 · (1 / 𝐶))) |
16 | 13, 15 | breq12d 4860 | . 2 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ (𝐶 ∈ ℝ ∧ 0 < 𝐶)) → ((𝐴 / 𝐶) < (𝐵 / 𝐶) ↔ (𝐴 · (1 / 𝐶)) < (𝐵 · (1 / 𝐶)))) |
17 | 10, 16 | bitr4d 274 | 1 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ (𝐶 ∈ ℝ ∧ 0 < 𝐶)) → (𝐴 < 𝐵 ↔ (𝐴 / 𝐶) < (𝐵 / 𝐶))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 198 ∧ wa 385 ∧ w3a 1108 ∈ wcel 2157 class class class wbr 4847 (class class class)co 6882 ℝcr 10227 0cc0 10228 1c1 10229 · cmul 10233 < clt 10367 / cdiv 10980 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1891 ax-4 1905 ax-5 2006 ax-6 2072 ax-7 2107 ax-8 2159 ax-9 2166 ax-10 2185 ax-11 2200 ax-12 2213 ax-13 2379 ax-ext 2781 ax-sep 4979 ax-nul 4987 ax-pow 5039 ax-pr 5101 ax-un 7187 ax-resscn 10285 ax-1cn 10286 ax-icn 10287 ax-addcl 10288 ax-addrcl 10289 ax-mulcl 10290 ax-mulrcl 10291 ax-mulcom 10292 ax-addass 10293 ax-mulass 10294 ax-distr 10295 ax-i2m1 10296 ax-1ne0 10297 ax-1rid 10298 ax-rnegex 10299 ax-rrecex 10300 ax-cnre 10301 ax-pre-lttri 10302 ax-pre-lttrn 10303 ax-pre-ltadd 10304 ax-pre-mulgt0 10305 |
This theorem depends on definitions: df-bi 199 df-an 386 df-or 875 df-3or 1109 df-3an 1110 df-tru 1657 df-ex 1876 df-nf 1880 df-sb 2065 df-mo 2593 df-eu 2611 df-clab 2790 df-cleq 2796 df-clel 2799 df-nfc 2934 df-ne 2976 df-nel 3079 df-ral 3098 df-rex 3099 df-reu 3100 df-rmo 3101 df-rab 3102 df-v 3391 df-sbc 3638 df-csb 3733 df-dif 3776 df-un 3778 df-in 3780 df-ss 3787 df-nul 4120 df-if 4282 df-pw 4355 df-sn 4373 df-pr 4375 df-op 4379 df-uni 4633 df-br 4848 df-opab 4910 df-mpt 4927 df-id 5224 df-po 5237 df-so 5238 df-xp 5322 df-rel 5323 df-cnv 5324 df-co 5325 df-dm 5326 df-rn 5327 df-res 5328 df-ima 5329 df-iota 6068 df-fun 6107 df-fn 6108 df-f 6109 df-f1 6110 df-fo 6111 df-f1o 6112 df-fv 6113 df-riota 6843 df-ov 6885 df-oprab 6886 df-mpt2 6887 df-er 7986 df-en 8200 df-dom 8201 df-sdom 8202 df-pnf 10369 df-mnf 10370 df-xr 10371 df-ltxr 10372 df-le 10373 df-sub 10562 df-neg 10563 df-div 10981 |
This theorem is referenced by: lediv1 11184 gt0div 11185 ltmuldiv 11192 ltdivmul 11194 ltdiv23 11210 ltdiv1i 11239 ltdiv1d 12166 flltdivnn0lt 12893 quoremz 12913 quoremnn0ALT 12915 fldiv 12918 hashdvds 15817 hashgcdlem 15830 dvcvx 24128 sinq12gt0 24605 tanord1 24629 atanlogsublem 24998 basellem4 25166 chtub 25293 bposlem7 25371 lgsquadlem1 25461 lgsquadlem2 25462 2lgslem1a2 25471 chebbnd1lem3 25516 dp2lt 30113 dpmul4 30142 cvmliftlem6 31793 cvmliftlem7 31794 cvmliftlem8 31795 cvmliftlem9 31796 cvmliftlem10 31797 nn0prpwlem 32833 nndivsub 32968 tan2h 33894 reglogltb 38245 stoweidlem14 40978 stoweidlem26 40990 |
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