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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 2732 | . . . . 5 โข (0gโ๐บ) = (0gโ๐บ) | |
6 | 2, 3, 4, 5 | odid 19400 | . . . 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 18955 | . . . 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 7423 | . 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 19414 | . . . 4 โข (๐ โ (๐โ๐ด) โ โ0) |
13 | 12 | nn0zd 12580 | . . 3 โข (๐ โ (๐โ๐ด) โ โค) |
14 | 1zzd 12589 | . . 3 โข (๐ โ 1 โ โค) | |
15 | eqid 2732 | . . . 4 โข (-gโ๐บ) = (-gโ๐บ) | |
16 | 2, 4, 15 | mulgsubdir 18988 | . . 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 1372 | . 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 18902 | . . 3 โข ((๐บ โ Grp โง ๐ด โ ๐) โ (๐ผโ๐ด) = ((0gโ๐บ)(-gโ๐บ)๐ด)) |
20 | 11, 1, 19 | syl2anc 584 | . 2 โข (๐ โ (๐ผโ๐ด) = ((0gโ๐บ)(-gโ๐บ)๐ด)) |
21 | 10, 17, 20 | 3eqtr4d 2782 | 1 โข (๐ โ (((๐โ๐ด) โ 1) ยท ๐ด) = (๐ผโ๐ด)) |
Colors of variables: wff setvar class |
Syntax hints: โ wi 4 = wceq 1541 โ wcel 2106 โcfv 6540 (class class class)co 7405 1c1 11107 โ cmin 11440 โคcz 12554 Basecbs 17140 0gc0g 17381 Grpcgrp 18815 invgcminusg 18816 -gcsg 18817 .gcmg 18944 odcod 19386 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2703 ax-sep 5298 ax-nul 5305 ax-pow 5362 ax-pr 5426 ax-un 7721 ax-cnex 11162 ax-resscn 11163 ax-1cn 11164 ax-icn 11165 ax-addcl 11166 ax-addrcl 11167 ax-mulcl 11168 ax-mulrcl 11169 ax-mulcom 11170 ax-addass 11171 ax-mulass 11172 ax-distr 11173 ax-i2m1 11174 ax-1ne0 11175 ax-1rid 11176 ax-rnegex 11177 ax-rrecex 11178 ax-cnre 11179 ax-pre-lttri 11180 ax-pre-lttrn 11181 ax-pre-ltadd 11182 ax-pre-mulgt0 11183 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3376 df-reu 3377 df-rab 3433 df-v 3476 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4322 df-if 4528 df-pw 4603 df-sn 4628 df-pr 4630 df-op 4634 df-uni 4908 df-iun 4998 df-br 5148 df-opab 5210 df-mpt 5231 df-tr 5265 df-id 5573 df-eprel 5579 df-po 5587 df-so 5588 df-fr 5630 df-we 5632 df-xp 5681 df-rel 5682 df-cnv 5683 df-co 5684 df-dm 5685 df-rn 5686 df-res 5687 df-ima 5688 df-pred 6297 df-ord 6364 df-on 6365 df-lim 6366 df-suc 6367 df-iota 6492 df-fun 6542 df-fn 6543 df-f 6544 df-f1 6545 df-fo 6546 df-f1o 6547 df-fv 6548 df-riota 7361 df-ov 7408 df-oprab 7409 df-mpo 7410 df-om 7852 df-1st 7971 df-2nd 7972 df-frecs 8262 df-wrecs 8293 df-recs 8367 df-rdg 8406 df-er 8699 df-en 8936 df-dom 8937 df-sdom 8938 df-sup 9433 df-inf 9434 df-pnf 11246 df-mnf 11247 df-xr 11248 df-ltxr 11249 df-le 11250 df-sub 11442 df-neg 11443 df-nn 12209 df-n0 12469 df-z 12555 df-uz 12819 df-fz 13481 df-seq 13963 df-0g 17383 df-mgm 18557 df-sgrp 18606 df-mnd 18622 df-grp 18818 df-minusg 18819 df-sbg 18820 df-mulg 18945 df-od 19390 |
This theorem is referenced by: finodsubmsubg 19429 |
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