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Mirrors > Home > MPE Home > Th. List > mulm1d | Structured version Visualization version Unicode version |
Description: Product with minus one is negative. (Contributed by Mario Carneiro, 27-May-2016.) |
Ref | Expression |
---|---|
mulm1d.1 |
Ref | Expression |
---|---|
mulm1d |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | mulm1d.1 | . 2 | |
2 | mulm1 10471 | . 2 | |
3 | 1, 2 | syl 17 | 1 |
Colors of variables: wff setvar class |
Syntax hints: wi 4 wceq 1483 wcel 1990 (class class class)co 6650 cc 9934 c1 9937 cmul 9941 cneg 10267 |
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 |
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-ltxr 10079 df-sub 10268 df-neg 10269 |
This theorem is referenced by: recextlem1 10657 ofnegsub 11018 modnegd 12725 modsumfzodifsn 12743 m1expcl2 12882 remullem 13868 sqrtneglem 14007 iseraltlem2 14413 iseraltlem3 14414 fsumneg 14519 incexclem 14568 incexc 14569 risefallfac 14755 efi4p 14867 cosadd 14895 absefib 14928 efieq1re 14929 pwp1fsum 15114 bitsinv1lem 15163 bezoutlem1 15256 pythagtriplem4 15524 negcncf 22721 mbfneg 23417 itg1sub 23476 itgcnlem 23556 i1fibl 23574 itgitg1 23575 itgmulc2 23600 dvmptneg 23729 dvlipcn 23757 lhop2 23778 logneg 24334 lognegb 24336 tanarg 24365 logtayl 24406 logtayl2 24408 asinlem 24595 asinlem2 24596 asinsin 24619 efiatan2 24644 2efiatan 24645 atandmtan 24647 atantan 24650 atans2 24658 dvatan 24662 basellem5 24811 lgsdir2lem4 25053 gausslemma2dlem5a 25095 lgseisenlem1 25100 lgseisenlem2 25101 rpvmasum2 25201 ostth3 25327 smcnlem 27552 ipval2 27562 dipsubdir 27703 his2sub 27949 qqhval2lem 30025 fwddifnp1 32272 itgmulc2nc 33478 ftc1anclem5 33489 areacirclem1 33500 mzpsubmpt 37306 rmym1 37500 rngunsnply 37743 expgrowth 38534 isumneg 39834 climneg 39842 stoweidlem22 40239 stirlinglem5 40295 fourierdlem97 40420 sqwvfourb 40446 etransclem46 40497 smfneg 41010 sharhght 41054 sigaradd 41055 altgsumbcALT 42131 |
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