Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
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 514 | . 2 ⊢ (𝜑 → (𝐶 ∈ ℝ ∧ 0 ≤ 𝐶)) |
6 | lemul1ad.5 | . 2 ⊢ (𝜑 → 𝐴 ≤ 𝐵) | |
7 | lemul1a 11488 | . 2 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ (𝐶 ∈ ℝ ∧ 0 ≤ 𝐶)) ∧ 𝐴 ≤ 𝐵) → (𝐴 · 𝐶) ≤ (𝐵 · 𝐶)) | |
8 | 1, 2, 5, 6, 7 | syl31anc 1369 | 1 ⊢ (𝜑 → (𝐴 · 𝐶) ≤ (𝐵 · 𝐶)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 ∈ wcel 2110 class class class wbr 5059 (class class class)co 7150 ℝcr 10530 0cc0 10531 · cmul 10536 ≤ cle 10670 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2156 ax-12 2172 ax-ext 2793 ax-sep 5196 ax-nul 5203 ax-pow 5259 ax-pr 5322 ax-un 7455 ax-resscn 10588 ax-1cn 10589 ax-icn 10590 ax-addcl 10591 ax-addrcl 10592 ax-mulcl 10593 ax-mulrcl 10594 ax-mulcom 10595 ax-addass 10596 ax-mulass 10597 ax-distr 10598 ax-i2m1 10599 ax-1ne0 10600 ax-1rid 10601 ax-rnegex 10602 ax-rrecex 10603 ax-cnre 10604 ax-pre-lttri 10605 ax-pre-lttrn 10606 ax-pre-ltadd 10607 ax-pre-mulgt0 10608 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rab 3147 df-v 3497 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4562 df-pr 4564 df-op 4568 df-uni 4833 df-br 5060 df-opab 5122 df-mpt 5140 df-id 5455 df-po 5469 df-so 5470 df-xp 5556 df-rel 5557 df-cnv 5558 df-co 5559 df-dm 5560 df-rn 5561 df-res 5562 df-ima 5563 df-iota 6309 df-fun 6352 df-fn 6353 df-f 6354 df-f1 6355 df-fo 6356 df-f1o 6357 df-fv 6358 df-riota 7108 df-ov 7153 df-oprab 7154 df-mpo 7155 df-er 8283 df-en 8504 df-dom 8505 df-sdom 8506 df-pnf 10671 df-mnf 10672 df-xr 10673 df-ltxr 10674 df-le 10675 df-sub 10866 df-neg 10867 |
This theorem is referenced by: bernneq 13584 o1fsum 15162 cvgrat 15233 prmreclem3 16248 nlmvscnlem2 23288 nghmcn 23348 ipcnlem2 23841 dvlip 24584 dvlipcn 24585 dvfsumlem4 24620 dvfsum2 24625 aalioulem3 24917 radcnvlem1 24995 radcnvlem2 24996 abelthlem5 25017 abelthlem7 25020 logtayllem 25236 abscxpbnd 25328 efrlim 25541 lgamgulmlem5 25604 chpub 25790 2sqlem8 25996 rplogsumlem1 26054 rpvmasumlem 26057 dchrisumlem3 26061 dchrvmasumlem3 26069 mulog2sumlem2 26105 selberglem2 26116 selberg2lem 26120 pntrlog2bndlem3 26149 pntrlog2bndlem5 26151 pntlemj 26173 ostth2lem2 26204 axpaschlem 26720 smcnlem 28468 htthlem 28688 lnconi 29804 cnlnadjlem7 29844 nnmulge 30468 nexple 31263 logdivsqrle 31916 hgt750lemf 31919 bfplem2 35095 fltnltalem 39267 jm2.24nn 39549 areaquad 39816 int-ineq2ndprincd 40539 fmul01lt1lem2 41858 dvbdfbdioolem1 42205 fourierdlem19 42404 fourierdlem39 42424 hsphoidmvle2 42860 hsphoidmvle 42861 hoidmvlelem2 42871 smfmullem1 43059 |
Copyright terms: Public domain | W3C validator |