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Mirrors > Home > MPE Home > Th. List > lemul2 | Structured version Visualization version GIF version |
Description: Multiplication of both sides of 'less than or equal to' by a positive number. (Contributed by NM, 16-Mar-2005.) |
Ref | Expression |
---|---|
lemul2 | ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ (𝐶 ∈ ℝ ∧ 0 < 𝐶)) → (𝐴 ≤ 𝐵 ↔ (𝐶 · 𝐴) ≤ (𝐶 · 𝐵))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | lemul1 11206 | . 2 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ (𝐶 ∈ ℝ ∧ 0 < 𝐶)) → (𝐴 ≤ 𝐵 ↔ (𝐴 · 𝐶) ≤ (𝐵 · 𝐶))) | |
2 | recn 10343 | . . . . . 6 ⊢ (𝐴 ∈ ℝ → 𝐴 ∈ ℂ) | |
3 | recn 10343 | . . . . . 6 ⊢ (𝐶 ∈ ℝ → 𝐶 ∈ ℂ) | |
4 | mulcom 10339 | . . . . . 6 ⊢ ((𝐴 ∈ ℂ ∧ 𝐶 ∈ ℂ) → (𝐴 · 𝐶) = (𝐶 · 𝐴)) | |
5 | 2, 3, 4 | syl2an 591 | . . . . 5 ⊢ ((𝐴 ∈ ℝ ∧ 𝐶 ∈ ℝ) → (𝐴 · 𝐶) = (𝐶 · 𝐴)) |
6 | 5 | 3adant2 1167 | . . . 4 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → (𝐴 · 𝐶) = (𝐶 · 𝐴)) |
7 | recn 10343 | . . . . . 6 ⊢ (𝐵 ∈ ℝ → 𝐵 ∈ ℂ) | |
8 | mulcom 10339 | . . . . . 6 ⊢ ((𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) → (𝐵 · 𝐶) = (𝐶 · 𝐵)) | |
9 | 7, 3, 8 | syl2an 591 | . . . . 5 ⊢ ((𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → (𝐵 · 𝐶) = (𝐶 · 𝐵)) |
10 | 9 | 3adant1 1166 | . . . 4 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → (𝐵 · 𝐶) = (𝐶 · 𝐵)) |
11 | 6, 10 | breq12d 4887 | . . 3 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → ((𝐴 · 𝐶) ≤ (𝐵 · 𝐶) ↔ (𝐶 · 𝐴) ≤ (𝐶 · 𝐵))) |
12 | 11 | 3adant3r 1237 | . 2 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ (𝐶 ∈ ℝ ∧ 0 < 𝐶)) → ((𝐴 · 𝐶) ≤ (𝐵 · 𝐶) ↔ (𝐶 · 𝐴) ≤ (𝐶 · 𝐵))) |
13 | 1, 12 | bitrd 271 | 1 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ (𝐶 ∈ ℝ ∧ 0 < 𝐶)) → (𝐴 ≤ 𝐵 ↔ (𝐶 · 𝐴) ≤ (𝐶 · 𝐵))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 198 ∧ wa 386 ∧ w3a 1113 = wceq 1658 ∈ wcel 2166 class class class wbr 4874 (class class class)co 6906 ℂcc 10251 ℝcr 10252 0cc0 10253 · cmul 10258 < clt 10392 ≤ cle 10393 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1896 ax-4 1910 ax-5 2011 ax-6 2077 ax-7 2114 ax-8 2168 ax-9 2175 ax-10 2194 ax-11 2209 ax-12 2222 ax-13 2391 ax-ext 2804 ax-sep 5006 ax-nul 5014 ax-pow 5066 ax-pr 5128 ax-un 7210 ax-resscn 10310 ax-1cn 10311 ax-icn 10312 ax-addcl 10313 ax-addrcl 10314 ax-mulcl 10315 ax-mulrcl 10316 ax-mulcom 10317 ax-addass 10318 ax-mulass 10319 ax-distr 10320 ax-i2m1 10321 ax-1ne0 10322 ax-1rid 10323 ax-rnegex 10324 ax-rrecex 10325 ax-cnre 10326 ax-pre-lttri 10327 ax-pre-lttrn 10328 ax-pre-ltadd 10329 ax-pre-mulgt0 10330 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 881 df-3or 1114 df-3an 1115 df-tru 1662 df-ex 1881 df-nf 1885 df-sb 2070 df-mo 2606 df-eu 2641 df-clab 2813 df-cleq 2819 df-clel 2822 df-nfc 2959 df-ne 3001 df-nel 3104 df-ral 3123 df-rex 3124 df-reu 3125 df-rab 3127 df-v 3417 df-sbc 3664 df-csb 3759 df-dif 3802 df-un 3804 df-in 3806 df-ss 3813 df-nul 4146 df-if 4308 df-pw 4381 df-sn 4399 df-pr 4401 df-op 4405 df-uni 4660 df-br 4875 df-opab 4937 df-mpt 4954 df-id 5251 df-po 5264 df-so 5265 df-xp 5349 df-rel 5350 df-cnv 5351 df-co 5352 df-dm 5353 df-rn 5354 df-res 5355 df-ima 5356 df-iota 6087 df-fun 6126 df-fn 6127 df-f 6128 df-f1 6129 df-fo 6130 df-f1o 6131 df-fv 6132 df-riota 6867 df-ov 6909 df-oprab 6910 df-mpt2 6911 df-er 8010 df-en 8224 df-dom 8225 df-sdom 8226 df-pnf 10394 df-mnf 10395 df-xr 10396 df-ltxr 10397 df-le 10398 df-sub 10588 df-neg 10589 |
This theorem is referenced by: lediv2 11244 lemul2i 11278 lemul2d 12201 nnlesq 13263 sqrlem6 14366 qexpz 15977 vdwlem3 16059 vdwlem9 16065 iihalf2 23103 tcphcphlem1 23404 csbren 23568 trirn 23569 minveclem2 23595 itg2monolem1 23917 itg2monolem3 23919 itgabs 24001 abelthlem2 24586 pilem2 24606 logdivlti 24766 atans2 25072 leibpi 25083 log2tlbnd 25086 jensenlem2 25128 zetacvg 25155 basellem1 25221 basellem2 25222 basellem3 25223 chtub 25351 logfaclbnd 25361 bpos1lem 25421 bposlem2 25424 bposlem3 25425 bposlem4 25426 bposlem5 25427 bposlem6 25428 lgsquadlem1 25519 chebbnd1lem1 25572 chebbnd1lem3 25574 dchrisumlem1 25592 dchrisum0lem3 25622 mulog2sumlem1 25637 mulog2sumlem2 25638 chpdifbndlem1 25656 pntlemj 25706 pntlemo 25710 ostth2lem2 25737 ostth2lem3 25738 ostth3 25741 minvecolem2 28287 cdj3lem1 29849 subfaclim 31717 itgabsnc 34023 fzmul 34080 bfp 34166 irrapxlem1 38231 irrapxlem3 38233 pellfundex 38295 jm2.17b 38372 jm2.17c 38373 stoweidlem11 41023 stoweidlem26 41038 stoweidlem38 41050 lighneallem4a 42356 |
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