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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 11481 | . 2 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ (𝐶 ∈ ℝ ∧ 0 < 𝐶)) → (𝐴 ≤ 𝐵 ↔ (𝐴 · 𝐶) ≤ (𝐵 · 𝐶))) | |
2 | recn 10616 | . . . . . 6 ⊢ (𝐴 ∈ ℝ → 𝐴 ∈ ℂ) | |
3 | recn 10616 | . . . . . 6 ⊢ (𝐶 ∈ ℝ → 𝐶 ∈ ℂ) | |
4 | mulcom 10612 | . . . . . 6 ⊢ ((𝐴 ∈ ℂ ∧ 𝐶 ∈ ℂ) → (𝐴 · 𝐶) = (𝐶 · 𝐴)) | |
5 | 2, 3, 4 | syl2an 598 | . . . . 5 ⊢ ((𝐴 ∈ ℝ ∧ 𝐶 ∈ ℝ) → (𝐴 · 𝐶) = (𝐶 · 𝐴)) |
6 | 5 | 3adant2 1128 | . . . 4 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → (𝐴 · 𝐶) = (𝐶 · 𝐴)) |
7 | recn 10616 | . . . . . 6 ⊢ (𝐵 ∈ ℝ → 𝐵 ∈ ℂ) | |
8 | mulcom 10612 | . . . . . 6 ⊢ ((𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) → (𝐵 · 𝐶) = (𝐶 · 𝐵)) | |
9 | 7, 3, 8 | syl2an 598 | . . . . 5 ⊢ ((𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → (𝐵 · 𝐶) = (𝐶 · 𝐵)) |
10 | 9 | 3adant1 1127 | . . . 4 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → (𝐵 · 𝐶) = (𝐶 · 𝐵)) |
11 | 6, 10 | breq12d 5043 | . . 3 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → ((𝐴 · 𝐶) ≤ (𝐵 · 𝐶) ↔ (𝐶 · 𝐴) ≤ (𝐶 · 𝐵))) |
12 | 11 | 3adant3r 1178 | . 2 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ (𝐶 ∈ ℝ ∧ 0 < 𝐶)) → ((𝐴 · 𝐶) ≤ (𝐵 · 𝐶) ↔ (𝐶 · 𝐴) ≤ (𝐶 · 𝐵))) |
13 | 1, 12 | bitrd 282 | 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 209 ∧ wa 399 ∧ w3a 1084 = wceq 1538 ∈ wcel 2111 class class class wbr 5030 (class class class)co 7135 ℂcc 10524 ℝcr 10525 0cc0 10526 · cmul 10531 < clt 10664 ≤ cle 10665 |
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-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 |
This theorem is referenced by: lediv2 11519 lemul2i 11552 lemul2d 12463 nnlesq 13564 sqrlem6 14599 qexpz 16227 vdwlem3 16309 vdwlem9 16315 iihalf2 23538 tcphcphlem1 23839 csbren 24003 trirn 24004 minveclem2 24030 itg2monolem1 24354 itg2monolem3 24356 itgabs 24438 abelthlem2 25027 pilem2 25047 logdivlti 25211 atans2 25517 leibpi 25528 log2tlbnd 25531 jensenlem2 25573 zetacvg 25600 basellem1 25666 basellem2 25667 basellem3 25668 chtub 25796 logfaclbnd 25806 bpos1lem 25866 bposlem2 25869 bposlem3 25870 bposlem4 25871 bposlem5 25872 bposlem6 25873 lgsquadlem1 25964 chebbnd1lem1 26053 chebbnd1lem3 26055 dchrisumlem1 26073 dchrisum0lem3 26103 mulog2sumlem1 26118 mulog2sumlem2 26119 chpdifbndlem1 26137 pntlemj 26187 pntlemo 26191 ostth2lem2 26218 ostth2lem3 26219 ostth3 26222 minvecolem2 28658 cdj3lem1 30217 subfaclim 32548 itgabsnc 35126 fzmul 35179 bfp 35262 irrapxlem1 39763 irrapxlem3 39765 pellfundex 39827 jm2.17b 39902 jm2.17c 39903 stoweidlem11 42653 stoweidlem26 42668 stoweidlem38 42680 lighneallem4a 44126 |
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