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Mirrors > Home > MPE Home > Th. List > lemul1ad | Structured version Visualization version GIF version |
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 554 | . 2 ⊢ (𝜑 → (𝐶 ∈ ℝ ∧ 0 ≤ 𝐶)) |
6 | lemul1ad.5 | . 2 ⊢ (𝜑 → 𝐴 ≤ 𝐵) | |
7 | lemul1a 10877 | . 2 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ (𝐶 ∈ ℝ ∧ 0 ≤ 𝐶)) ∧ 𝐴 ≤ 𝐵) → (𝐴 · 𝐶) ≤ (𝐵 · 𝐶)) | |
8 | 1, 2, 5, 6, 7 | syl31anc 1329 | 1 ⊢ (𝜑 → (𝐴 · 𝐶) ≤ (𝐵 · 𝐶)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 384 ∈ wcel 1990 class class class wbr 4653 (class class class)co 6650 ℝcr 9935 0cc0 9936 · cmul 9941 ≤ cle 10075 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-resscn 9993 ax-1cn 9994 ax-icn 9995 ax-addcl 9996 ax-addrcl 9997 ax-mulcl 9998 ax-mulrcl 9999 ax-mulcom 10000 ax-addass 10001 ax-mulass 10002 ax-distr 10003 ax-i2m1 10004 ax-1ne0 10005 ax-1rid 10006 ax-rnegex 10007 ax-rrecex 10008 ax-cnre 10009 ax-pre-lttri 10010 ax-pre-lttrn 10011 ax-pre-ltadd 10012 ax-pre-mulgt0 10013 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-nel 2898 df-ral 2917 df-rex 2918 df-reu 2919 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-op 4184 df-uni 4437 df-br 4654 df-opab 4713 df-mpt 4730 df-id 5024 df-po 5035 df-so 5036 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-riota 6611 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-er 7742 df-en 7956 df-dom 7957 df-sdom 7958 df-pnf 10076 df-mnf 10077 df-xr 10078 df-ltxr 10079 df-le 10080 df-sub 10268 df-neg 10269 |
This theorem is referenced by: bernneq 12990 o1fsum 14545 cvgrat 14615 prmreclem3 15622 nlmvscnlem2 22489 nghmcn 22549 ipcnlem2 23043 dvlip 23756 dvlipcn 23757 dvfsumlem4 23792 dvfsum2 23797 aalioulem3 24089 radcnvlem1 24167 radcnvlem2 24168 abelthlem5 24189 abelthlem7 24192 logtayllem 24405 abscxpbnd 24494 efrlim 24696 lgamgulmlem5 24759 chpub 24945 2sqlem8 25151 rplogsumlem1 25173 rpvmasumlem 25176 dchrisumlem3 25180 dchrvmasumlem3 25188 mulog2sumlem2 25224 selberglem2 25235 selberg2lem 25239 pntrlog2bndlem3 25268 pntrlog2bndlem5 25270 pntlemj 25292 ostth2lem2 25323 axpaschlem 25820 smcnlem 27552 htthlem 27774 lnconi 28892 cnlnadjlem7 28932 nnmulge 29515 nexple 30071 logdivsqrle 30728 hgt750lemf 30731 bfplem2 33622 jm2.24nn 37526 areaquad 37802 int-ineq2ndprincd 38496 fmul01lt1lem2 39817 dvbdfbdioolem1 40143 fourierdlem19 40343 fourierdlem39 40363 hsphoidmvle2 40799 hsphoidmvle 40800 hoidmvlelem2 40810 smfmullem1 40998 |
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