![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > odm1inv | Structured version Visualization version GIF version |
Description: The (order-1)th multiple of an element is its inverse. (Contributed by SN, 31-Jan-2025.) |
Ref | Expression |
---|---|
odm1inv.x | โข ๐ = (Baseโ๐บ) |
odm1inv.o | โข ๐ = (odโ๐บ) |
odm1inv.t | โข ยท = (.gโ๐บ) |
odm1inv.i | โข ๐ผ = (invgโ๐บ) |
odm1inv.g | โข (๐ โ ๐บ โ Grp) |
odm1inv.1 | โข (๐ โ ๐ด โ ๐) |
Ref | Expression |
---|---|
odm1inv | โข (๐ โ (((๐โ๐ด) โ 1) ยท ๐ด) = (๐ผโ๐ด)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | odm1inv.1 | . . . 4 โข (๐ โ ๐ด โ ๐) | |
2 | odm1inv.x | . . . . 5 โข ๐ = (Baseโ๐บ) | |
3 | odm1inv.o | . . . . 5 โข ๐ = (odโ๐บ) | |
4 | odm1inv.t | . . . . 5 โข ยท = (.gโ๐บ) | |
5 | eqid 2726 | . . . . 5 โข (0gโ๐บ) = (0gโ๐บ) | |
6 | 2, 3, 4, 5 | odid 19455 | . . . 4 โข (๐ด โ ๐ โ ((๐โ๐ด) ยท ๐ด) = (0gโ๐บ)) |
7 | 1, 6 | syl 17 | . . 3 โข (๐ โ ((๐โ๐ด) ยท ๐ด) = (0gโ๐บ)) |
8 | 2, 4 | mulg1 19005 | . . . 4 โข (๐ด โ ๐ โ (1 ยท ๐ด) = ๐ด) |
9 | 1, 8 | syl 17 | . . 3 โข (๐ โ (1 ยท ๐ด) = ๐ด) |
10 | 7, 9 | oveq12d 7422 | . 2 โข (๐ โ (((๐โ๐ด) ยท ๐ด)(-gโ๐บ)(1 ยท ๐ด)) = ((0gโ๐บ)(-gโ๐บ)๐ด)) |
11 | odm1inv.g | . . 3 โข (๐ โ ๐บ โ Grp) | |
12 | 2, 3, 1 | odcld 19469 | . . . 4 โข (๐ โ (๐โ๐ด) โ โ0) |
13 | 12 | nn0zd 12585 | . . 3 โข (๐ โ (๐โ๐ด) โ โค) |
14 | 1zzd 12594 | . . 3 โข (๐ โ 1 โ โค) | |
15 | eqid 2726 | . . . 4 โข (-gโ๐บ) = (-gโ๐บ) | |
16 | 2, 4, 15 | mulgsubdir 19038 | . . 3 โข ((๐บ โ Grp โง ((๐โ๐ด) โ โค โง 1 โ โค โง ๐ด โ ๐)) โ (((๐โ๐ด) โ 1) ยท ๐ด) = (((๐โ๐ด) ยท ๐ด)(-gโ๐บ)(1 ยท ๐ด))) |
17 | 11, 13, 14, 1, 16 | syl13anc 1369 | . 2 โข (๐ โ (((๐โ๐ด) โ 1) ยท ๐ด) = (((๐โ๐ด) ยท ๐ด)(-gโ๐บ)(1 ยท ๐ด))) |
18 | odm1inv.i | . . . 4 โข ๐ผ = (invgโ๐บ) | |
19 | 2, 15, 18, 5 | grpinvval2 18948 | . . 3 โข ((๐บ โ Grp โง ๐ด โ ๐) โ (๐ผโ๐ด) = ((0gโ๐บ)(-gโ๐บ)๐ด)) |
20 | 11, 1, 19 | syl2anc 583 | . 2 โข (๐ โ (๐ผโ๐ด) = ((0gโ๐บ)(-gโ๐บ)๐ด)) |
21 | 10, 17, 20 | 3eqtr4d 2776 | 1 โข (๐ โ (((๐โ๐ด) โ 1) ยท ๐ด) = (๐ผโ๐ด)) |
Colors of variables: wff setvar class |
Syntax hints: โ wi 4 = wceq 1533 โ wcel 2098 โcfv 6536 (class class class)co 7404 1c1 11110 โ cmin 11445 โคcz 12559 Basecbs 17150 0gc0g 17391 Grpcgrp 18860 invgcminusg 18861 -gcsg 18862 .gcmg 18992 odcod 19441 |
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 7721 ax-cnex 11165 ax-resscn 11166 ax-1cn 11167 ax-icn 11168 ax-addcl 11169 ax-addrcl 11170 ax-mulcl 11171 ax-mulrcl 11172 ax-mulcom 11173 ax-addass 11174 ax-mulass 11175 ax-distr 11176 ax-i2m1 11177 ax-1ne0 11178 ax-1rid 11179 ax-rnegex 11180 ax-rrecex 11181 ax-cnre 11182 ax-pre-lttri 11183 ax-pre-lttrn 11184 ax-pre-ltadd 11185 ax-pre-mulgt0 11186 |
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-rmo 3370 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-pss 3962 df-nul 4318 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4903 df-iun 4992 df-br 5142 df-opab 5204 df-mpt 5225 df-tr 5259 df-id 5567 df-eprel 5573 df-po 5581 df-so 5582 df-fr 5624 df-we 5626 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-pred 6293 df-ord 6360 df-on 6361 df-lim 6362 df-suc 6363 df-iota 6488 df-fun 6538 df-fn 6539 df-f 6540 df-f1 6541 df-fo 6542 df-f1o 6543 df-fv 6544 df-riota 7360 df-ov 7407 df-oprab 7408 df-mpo 7409 df-om 7852 df-1st 7971 df-2nd 7972 df-frecs 8264 df-wrecs 8295 df-recs 8369 df-rdg 8408 df-er 8702 df-en 8939 df-dom 8940 df-sdom 8941 df-sup 9436 df-inf 9437 df-pnf 11251 df-mnf 11252 df-xr 11253 df-ltxr 11254 df-le 11255 df-sub 11447 df-neg 11448 df-nn 12214 df-n0 12474 df-z 12560 df-uz 12824 df-fz 13488 df-seq 13970 df-0g 17393 df-mgm 18570 df-sgrp 18649 df-mnd 18665 df-grp 18863 df-minusg 18864 df-sbg 18865 df-mulg 18993 df-od 19445 |
This theorem is referenced by: finodsubmsubg 19484 |
Copyright terms: Public domain | W3C validator |