![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > ltmulgt11d | Structured version Visualization version GIF version |
Description: Multiplication by a number greater than 1. (Contributed by Mario Carneiro, 28-May-2016.) |
Ref | Expression |
---|---|
rpgecld.1 | โข (๐ โ ๐ด โ โ) |
rpgecld.2 | โข (๐ โ ๐ต โ โ+) |
Ref | Expression |
---|---|
ltmulgt11d | โข (๐ โ (1 < ๐ด โ ๐ต < (๐ต ยท ๐ด))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | rpgecld.2 | . . 3 โข (๐ โ ๐ต โ โ+) | |
2 | 1 | rpred 13048 | . 2 โข (๐ โ ๐ต โ โ) |
3 | rpgecld.1 | . 2 โข (๐ โ ๐ด โ โ) | |
4 | 1 | rpgt0d 13051 | . 2 โข (๐ โ 0 < ๐ต) |
5 | ltmulgt11 12104 | . 2 โข ((๐ต โ โ โง ๐ด โ โ โง 0 < ๐ต) โ (1 < ๐ด โ ๐ต < (๐ต ยท ๐ด))) | |
6 | 2, 3, 4, 5 | syl3anc 1368 | 1 โข (๐ โ (1 < ๐ด โ ๐ต < (๐ต ยท ๐ด))) |
Colors of variables: wff setvar class |
Syntax hints: โ wi 4 โ wb 205 โ wcel 2098 class class class wbr 5143 (class class class)co 7416 โcr 11137 0cc0 11138 1c1 11139 ยท cmul 11143 < clt 11278 โ+crp 13006 |
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 2166 ax-ext 2696 ax-sep 5294 ax-nul 5301 ax-pow 5359 ax-pr 5423 ax-un 7738 ax-resscn 11195 ax-1cn 11196 ax-icn 11197 ax-addcl 11198 ax-addrcl 11199 ax-mulcl 11200 ax-mulrcl 11201 ax-mulcom 11202 ax-addass 11203 ax-mulass 11204 ax-distr 11205 ax-i2m1 11206 ax-1ne0 11207 ax-1rid 11208 ax-rnegex 11209 ax-rrecex 11210 ax-cnre 11211 ax-pre-lttri 11212 ax-pre-lttrn 11213 ax-pre-ltadd 11214 ax-pre-mulgt0 11215 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 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 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2931 df-nel 3037 df-ral 3052 df-rex 3061 df-reu 3365 df-rab 3420 df-v 3465 df-sbc 3769 df-csb 3885 df-dif 3942 df-un 3944 df-in 3946 df-ss 3956 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4904 df-br 5144 df-opab 5206 df-mpt 5227 df-id 5570 df-po 5584 df-so 5585 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-riota 7372 df-ov 7419 df-oprab 7420 df-mpo 7421 df-er 8723 df-en 8963 df-dom 8964 df-sdom 8965 df-pnf 11280 df-mnf 11281 df-ltxr 11283 df-sub 11476 df-neg 11477 df-rp 13007 |
This theorem is referenced by: aks4d1p8 41614 |
Copyright terms: Public domain | W3C validator |