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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 511 | . 2 ⊢ (𝜑 → (𝐶 ∈ ℝ ∧ 0 ≤ 𝐶)) |
6 | lemul1ad.5 | . 2 ⊢ (𝜑 → 𝐴 ≤ 𝐵) | |
7 | lemul1a 12119 | . 2 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ (𝐶 ∈ ℝ ∧ 0 ≤ 𝐶)) ∧ 𝐴 ≤ 𝐵) → (𝐴 · 𝐶) ≤ (𝐵 · 𝐶)) | |
8 | 1, 2, 5, 6, 7 | syl31anc 1372 | 1 ⊢ (𝜑 → (𝐴 · 𝐶) ≤ (𝐵 · 𝐶)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 ∈ wcel 2106 class class class wbr 5148 (class class class)co 7431 ℝcr 11152 0cc0 11153 · cmul 11158 ≤ cle 11294 |
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 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 ax-resscn 11210 ax-1cn 11211 ax-icn 11212 ax-addcl 11213 ax-addrcl 11214 ax-mulcl 11215 ax-mulrcl 11216 ax-mulcom 11217 ax-addass 11218 ax-mulass 11219 ax-distr 11220 ax-i2m1 11221 ax-1ne0 11222 ax-1rid 11223 ax-rnegex 11224 ax-rrecex 11225 ax-cnre 11226 ax-pre-lttri 11227 ax-pre-lttrn 11228 ax-pre-ltadd 11229 ax-pre-mulgt0 11230 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5583 df-po 5597 df-so 5598 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-er 8744 df-en 8985 df-dom 8986 df-sdom 8987 df-pnf 11295 df-mnf 11296 df-xr 11297 df-ltxr 11298 df-le 11299 df-sub 11492 df-neg 11493 |
This theorem is referenced by: bernneq 14265 o1fsum 15846 cvgrat 15916 prmreclem3 16952 nlmvscnlem2 24722 nghmcn 24782 ipcnlem2 25292 dvlip 26047 dvlipcn 26048 dvfsumlem4 26085 dvfsum2 26090 aalioulem3 26391 radcnvlem1 26471 radcnvlem2 26472 abelthlem5 26494 abelthlem7 26497 logtayllem 26716 abscxpbnd 26811 efrlim 27027 efrlimOLD 27028 lgamgulmlem5 27091 chpub 27279 2sqlem8 27485 rplogsumlem1 27543 rpvmasumlem 27546 dchrisumlem3 27550 dchrvmasumlem3 27558 mulog2sumlem2 27594 selberglem2 27605 selberg2lem 27609 pntrlog2bndlem3 27638 pntrlog2bndlem5 27640 pntlemj 27662 ostth2lem2 27693 axpaschlem 28970 smcnlem 30726 htthlem 30946 lnconi 32062 cnlnadjlem7 32102 nnmulge 32756 nexple 33990 logdivsqrle 34644 hgt750lemf 34647 bfplem2 37810 aks4d1p1p7 42056 posbezout 42082 aks6d1c7lem1 42162 fltnltalem 42649 jm2.24nn 42948 areaquad 43205 int-ineq2ndprincd 44183 fmul01lt1lem2 45541 dvbdfbdioolem1 45884 fourierdlem19 46082 fourierdlem39 46102 hsphoidmvle2 46541 hsphoidmvle 46542 hoidmvlelem2 46552 smfmullem1 46747 |
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