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Mirrors > Home > MPE Home > Th. List > mulg0 | Structured version Visualization version GIF version |
Description: Group multiple (exponentiation) operation at zero. (Contributed by Mario Carneiro, 11-Dec-2014.) |
Ref | Expression |
---|---|
mulg0.b | โข ๐ต = (Baseโ๐บ) |
mulg0.o | โข 0 = (0gโ๐บ) |
mulg0.t | โข ยท = (.gโ๐บ) |
Ref | Expression |
---|---|
mulg0 | โข (๐ โ ๐ต โ (0 ยท ๐) = 0 ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 0z 12568 | . 2 โข 0 โ โค | |
2 | mulg0.b | . . . 4 โข ๐ต = (Baseโ๐บ) | |
3 | eqid 2724 | . . . 4 โข (+gโ๐บ) = (+gโ๐บ) | |
4 | mulg0.o | . . . 4 โข 0 = (0gโ๐บ) | |
5 | eqid 2724 | . . . 4 โข (invgโ๐บ) = (invgโ๐บ) | |
6 | mulg0.t | . . . 4 โข ยท = (.gโ๐บ) | |
7 | eqid 2724 | . . . 4 โข seq1((+gโ๐บ), (โ ร {๐})) = seq1((+gโ๐บ), (โ ร {๐})) | |
8 | 2, 3, 4, 5, 6, 7 | mulgval 18995 | . . 3 โข ((0 โ โค โง ๐ โ ๐ต) โ (0 ยท ๐) = if(0 = 0, 0 , if(0 < 0, (seq1((+gโ๐บ), (โ ร {๐}))โ0), ((invgโ๐บ)โ(seq1((+gโ๐บ), (โ ร {๐}))โ-0))))) |
9 | eqid 2724 | . . . 4 โข 0 = 0 | |
10 | 9 | iftruei 4528 | . . 3 โข if(0 = 0, 0 , if(0 < 0, (seq1((+gโ๐บ), (โ ร {๐}))โ0), ((invgโ๐บ)โ(seq1((+gโ๐บ), (โ ร {๐}))โ-0)))) = 0 |
11 | 8, 10 | eqtrdi 2780 | . 2 โข ((0 โ โค โง ๐ โ ๐ต) โ (0 ยท ๐) = 0 ) |
12 | 1, 11 | mpan 687 | 1 โข (๐ โ ๐ต โ (0 ยท ๐) = 0 ) |
Colors of variables: wff setvar class |
Syntax hints: โ wi 4 โง wa 395 = wceq 1533 โ wcel 2098 ifcif 4521 {csn 4621 class class class wbr 5139 ร cxp 5665 โcfv 6534 (class class class)co 7402 0cc0 11107 1c1 11108 < clt 11247 -cneg 11444 โcn 12211 โคcz 12557 seqcseq 13967 Basecbs 17149 +gcplusg 17202 0gc0g 17390 invgcminusg 18860 .gcmg 18991 |
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 2695 ax-sep 5290 ax-nul 5297 ax-pow 5354 ax-pr 5418 ax-un 7719 ax-cnex 11163 ax-resscn 11164 ax-1cn 11165 ax-icn 11166 ax-addcl 11167 ax-addrcl 11168 ax-mulcl 11169 ax-mulrcl 11170 ax-mulcom 11171 ax-addass 11172 ax-mulass 11173 ax-distr 11174 ax-i2m1 11175 ax-1ne0 11176 ax-1rid 11177 ax-rnegex 11178 ax-rrecex 11179 ax-cnre 11180 ax-pre-lttri 11181 ax-pre-lttrn 11182 ax-pre-ltadd 11183 ax-pre-mulgt0 11184 |
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 2526 df-eu 2555 df-clab 2702 df-cleq 2716 df-clel 2802 df-nfc 2877 df-ne 2933 df-nel 3039 df-ral 3054 df-rex 3063 df-reu 3369 df-rab 3425 df-v 3468 df-sbc 3771 df-csb 3887 df-dif 3944 df-un 3946 df-in 3948 df-ss 3958 df-pss 3960 df-nul 4316 df-if 4522 df-pw 4597 df-sn 4622 df-pr 4624 df-op 4628 df-uni 4901 df-iun 4990 df-br 5140 df-opab 5202 df-mpt 5223 df-tr 5257 df-id 5565 df-eprel 5571 df-po 5579 df-so 5580 df-fr 5622 df-we 5624 df-xp 5673 df-rel 5674 df-cnv 5675 df-co 5676 df-dm 5677 df-rn 5678 df-res 5679 df-ima 5680 df-pred 6291 df-ord 6358 df-on 6359 df-lim 6360 df-suc 6361 df-iota 6486 df-fun 6536 df-fn 6537 df-f 6538 df-f1 6539 df-fo 6540 df-f1o 6541 df-fv 6542 df-riota 7358 df-ov 7405 df-oprab 7406 df-mpo 7407 df-om 7850 df-1st 7969 df-2nd 7970 df-frecs 8262 df-wrecs 8293 df-recs 8367 df-rdg 8406 df-er 8700 df-en 8937 df-dom 8938 df-sdom 8939 df-pnf 11249 df-mnf 11250 df-xr 11251 df-ltxr 11252 df-le 11253 df-sub 11445 df-neg 11446 df-nn 12212 df-n0 12472 df-z 12558 df-uz 12822 df-seq 13968 df-mulg 18992 |
This theorem is referenced by: ressmulgnn0 19001 mulgnn0gsum 19003 mulgnn0p1 19008 mulgnn0subcl 19010 mulgneg 19015 mulgaddcom 19021 mulginvcom 19022 mulgnn0z 19024 mulgnn0dir 19027 mulgneg2 19031 mulgnn0ass 19033 mhmmulg 19038 submmulg 19041 cycsubm 19124 odid 19454 oddvdsnn0 19460 oddvds 19463 odf1 19478 gexid 19497 mulgnn0di 19741 0cyg 19809 gsumconst 19850 srgmulgass 20118 srgpcomp 20119 srgbinomlem3 20129 srgbinomlem4 20130 srgbinom 20132 mulgass2 20204 lmodvsmmulgdi 20739 cnfldmulg 21282 cnfldexp 21283 freshmansdream 21458 assamulgscmlem1 21782 mplcoe3 21924 mplcoe5 21926 mplbas2 21928 psrbagev1 21969 psrbagev1OLD 21970 evlslem3 21974 evlslem1 21976 mhppwdeg 22022 ply1scltm 22144 chfacfscmulgsum 22706 chfacfpmmulgsum 22710 cpmadugsumlemF 22722 tmdmulg 23940 clmmulg 24972 dchrptlem2 27138 xrsmulgzz 32671 omndmul2 32723 omndmul 32725 archirng 32827 archirngz 32828 archiabllem1b 32831 archiabllem2c 32834 aks6d1c1p6 41481 evlsvvvallem 41662 evlsvvval 41664 selvvvval 41686 evlselv 41688 mhphflem 41697 mhphf 41698 lmodvsmdi 47307 |
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