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| Description: Multiplication of both sides of 'less than or equal to' by a nonnegative number. (Contributed by Mario Carneiro, 28-May-2016.) | 
| Ref | Expression | 
|---|---|
| ltp1d.1 | ⊢ (𝜑 → 𝐴 ∈ ℝ) | 
| divgt0d.2 | ⊢ (𝜑 → 𝐵 ∈ ℝ) | 
| lemul1ad.3 | ⊢ (𝜑 → 𝐶 ∈ ℝ) | 
| lemul1ad.4 | ⊢ (𝜑 → 0 ≤ 𝐶) | 
| lemul1ad.5 | ⊢ (𝜑 → 𝐴 ≤ 𝐵) | 
| Ref | Expression | 
|---|---|
| lemul1ad | ⊢ (𝜑 → (𝐴 · 𝐶) ≤ (𝐵 · 𝐶)) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | ltp1d.1 | . 2 ⊢ (𝜑 → 𝐴 ∈ ℝ) | |
| 2 | divgt0d.2 | . 2 ⊢ (𝜑 → 𝐵 ∈ ℝ) | |
| 3 | lemul1ad.3 | . . 3 ⊢ (𝜑 → 𝐶 ∈ ℝ) | |
| 4 | lemul1ad.4 | . . 3 ⊢ (𝜑 → 0 ≤ 𝐶) | |
| 5 | 3, 4 | jca 511 | . 2 ⊢ (𝜑 → (𝐶 ∈ ℝ ∧ 0 ≤ 𝐶)) | 
| 6 | lemul1ad.5 | . 2 ⊢ (𝜑 → 𝐴 ≤ 𝐵) | |
| 7 | lemul1a 12121 | . 2 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ (𝐶 ∈ ℝ ∧ 0 ≤ 𝐶)) ∧ 𝐴 ≤ 𝐵) → (𝐴 · 𝐶) ≤ (𝐵 · 𝐶)) | |
| 8 | 1, 2, 5, 6, 7 | syl31anc 1375 | 1 ⊢ (𝜑 → (𝐴 · 𝐶) ≤ (𝐵 · 𝐶)) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ∧ wa 395 ∈ wcel 2108 class class class wbr 5143 (class class class)co 7431 ℝcr 11154 0cc0 11155 · cmul 11160 ≤ cle 11296 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-resscn 11212 ax-1cn 11213 ax-icn 11214 ax-addcl 11215 ax-addrcl 11216 ax-mulcl 11217 ax-mulrcl 11218 ax-mulcom 11219 ax-addass 11220 ax-mulass 11221 ax-distr 11222 ax-i2m1 11223 ax-1ne0 11224 ax-1rid 11225 ax-rnegex 11226 ax-rrecex 11227 ax-cnre 11228 ax-pre-lttri 11229 ax-pre-lttrn 11230 ax-pre-ltadd 11231 ax-pre-mulgt0 11232 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-opab 5206 df-mpt 5226 df-id 5578 df-po 5592 df-so 5593 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-er 8745 df-en 8986 df-dom 8987 df-sdom 8988 df-pnf 11297 df-mnf 11298 df-xr 11299 df-ltxr 11300 df-le 11301 df-sub 11494 df-neg 11495 | 
| This theorem is referenced by: bernneq 14268 o1fsum 15849 cvgrat 15919 prmreclem3 16956 nlmvscnlem2 24706 nghmcn 24766 ipcnlem2 25278 dvlip 26032 dvlipcn 26033 dvfsumlem4 26070 dvfsum2 26075 aalioulem3 26376 radcnvlem1 26456 radcnvlem2 26457 abelthlem5 26479 abelthlem7 26482 logtayllem 26701 abscxpbnd 26796 efrlim 27012 efrlimOLD 27013 lgamgulmlem5 27076 chpub 27264 2sqlem8 27470 rplogsumlem1 27528 rpvmasumlem 27531 dchrisumlem3 27535 dchrvmasumlem3 27543 mulog2sumlem2 27579 selberglem2 27590 selberg2lem 27594 pntrlog2bndlem3 27623 pntrlog2bndlem5 27625 pntlemj 27647 ostth2lem2 27678 axpaschlem 28955 smcnlem 30716 htthlem 30936 lnconi 32052 cnlnadjlem7 32092 nnmulge 32749 nexple 32833 logdivsqrle 34665 hgt750lemf 34668 bfplem2 37830 aks4d1p1p7 42075 posbezout 42101 aks6d1c7lem1 42181 fltnltalem 42672 jm2.24nn 42971 areaquad 43228 int-ineq2ndprincd 44206 fmul01lt1lem2 45600 dvbdfbdioolem1 45943 fourierdlem19 46141 fourierdlem39 46161 hsphoidmvle2 46600 hsphoidmvle 46601 hoidmvlelem2 46611 smfmullem1 46806 | 
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