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Mirrors > Home > MPE Home > Th. List > ltmul1dd | Structured version Visualization version GIF version |
Description: The ratio of nonnegative and positive numbers is nonnegative. (Contributed by Mario Carneiro, 30-May-2016.) |
Ref | Expression |
---|---|
ltmul1d.1 | โข (๐ โ ๐ด โ โ) |
ltmul1d.2 | โข (๐ โ ๐ต โ โ) |
ltmul1d.3 | โข (๐ โ ๐ถ โ โ+) |
ltdiv1dd.4 | โข (๐ โ ๐ด < ๐ต) |
Ref | Expression |
---|---|
ltmul1dd | โข (๐ โ (๐ด ยท ๐ถ) < (๐ต ยท ๐ถ)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ltdiv1dd.4 | . 2 โข (๐ โ ๐ด < ๐ต) | |
2 | ltmul1d.1 | . . 3 โข (๐ โ ๐ด โ โ) | |
3 | ltmul1d.2 | . . 3 โข (๐ โ ๐ต โ โ) | |
4 | ltmul1d.3 | . . 3 โข (๐ โ ๐ถ โ โ+) | |
5 | 2, 3, 4 | ltmul1d 13063 | . 2 โข (๐ โ (๐ด < ๐ต โ (๐ด ยท ๐ถ) < (๐ต ยท ๐ถ))) |
6 | 1, 5 | mpbid 231 | 1 โข (๐ โ (๐ด ยท ๐ถ) < (๐ต ยท ๐ถ)) |
Colors of variables: wff setvar class |
Syntax hints: โ wi 4 โ wcel 2098 class class class wbr 5141 (class class class)co 7405 โcr 11111 ยท cmul 11117 < clt 11252 โ+crp 12980 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2697 ax-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 ax-un 7722 ax-resscn 11169 ax-1cn 11170 ax-icn 11171 ax-addcl 11172 ax-addrcl 11173 ax-mulcl 11174 ax-mulrcl 11175 ax-mulcom 11176 ax-addass 11177 ax-mulass 11178 ax-distr 11179 ax-i2m1 11180 ax-1ne0 11181 ax-1rid 11182 ax-rnegex 11183 ax-rrecex 11184 ax-cnre 11185 ax-pre-lttri 11186 ax-pre-lttrn 11187 ax-pre-ltadd 11188 ax-pre-mulgt0 11189 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ne 2935 df-nel 3041 df-ral 3056 df-rex 3065 df-reu 3371 df-rab 3427 df-v 3470 df-sbc 3773 df-csb 3889 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-nul 4318 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4903 df-br 5142 df-opab 5204 df-mpt 5225 df-id 5567 df-po 5581 df-so 5582 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-iota 6489 df-fun 6539 df-fn 6540 df-f 6541 df-f1 6542 df-fo 6543 df-f1o 6544 df-fv 6545 df-riota 7361 df-ov 7408 df-oprab 7409 df-mpo 7410 df-er 8705 df-en 8942 df-dom 8943 df-sdom 8944 df-pnf 11254 df-mnf 11255 df-ltxr 11257 df-sub 11450 df-neg 11451 df-rp 12981 |
This theorem is referenced by: mul2lt0bi 13086 o1rlimmul 15569 2expltfac 17035 lhop1lem 25901 ftalem5 26964 chtppilimlem1 27361 pntibndlem2 27479 pntlemb 27485 pntlemr 27490 ostth2lem1 27506 ostth2lem3 27523 tgoldbachgtde 34201 fltnltalem 41982 ioodvbdlimc1lem1 45219 stoweidlem11 45299 stoweidlem13 45301 stoweidlem26 45314 wallispi 45358 stirlinglem1 45362 dirkercncflem4 45394 fourierdlem4 45399 |
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