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Mirrors > Home > MPE Home > Th. List > cycsubg2 | Structured version Visualization version GIF version |
Description: The subgroup generated by an element is exhausted by its multiples. (Contributed by Stefan O'Rear, 6-Sep-2015.) |
Ref | Expression |
---|---|
cycsubg2.x | β’ π = (BaseβπΊ) |
cycsubg2.t | β’ Β· = (.gβπΊ) |
cycsubg2.f | β’ πΉ = (π₯ β β€ β¦ (π₯ Β· π΄)) |
cycsubg2.k | β’ πΎ = (mrClsβ(SubGrpβπΊ)) |
Ref | Expression |
---|---|
cycsubg2 | β’ ((πΊ β Grp β§ π΄ β π) β (πΎβ{π΄}) = ran πΉ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | snssg 4792 | . . . . . 6 β’ (π΄ β π β (π΄ β π¦ β {π΄} β π¦)) | |
2 | 1 | bicomd 222 | . . . . 5 β’ (π΄ β π β ({π΄} β π¦ β π΄ β π¦)) |
3 | 2 | adantl 480 | . . . 4 β’ ((πΊ β Grp β§ π΄ β π) β ({π΄} β π¦ β π΄ β π¦)) |
4 | 3 | rabbidv 3438 | . . 3 β’ ((πΊ β Grp β§ π΄ β π) β {π¦ β (SubGrpβπΊ) β£ {π΄} β π¦} = {π¦ β (SubGrpβπΊ) β£ π΄ β π¦}) |
5 | 4 | inteqd 4958 | . 2 β’ ((πΊ β Grp β§ π΄ β π) β β© {π¦ β (SubGrpβπΊ) β£ {π΄} β π¦} = β© {π¦ β (SubGrpβπΊ) β£ π΄ β π¦}) |
6 | cycsubg2.x | . . . . 5 β’ π = (BaseβπΊ) | |
7 | 6 | subgacs 19130 | . . . 4 β’ (πΊ β Grp β (SubGrpβπΊ) β (ACSβπ)) |
8 | 7 | acsmred 17645 | . . 3 β’ (πΊ β Grp β (SubGrpβπΊ) β (Mooreβπ)) |
9 | snssi 4816 | . . 3 β’ (π΄ β π β {π΄} β π) | |
10 | cycsubg2.k | . . . 4 β’ πΎ = (mrClsβ(SubGrpβπΊ)) | |
11 | 10 | mrcval 17599 | . . 3 β’ (((SubGrpβπΊ) β (Mooreβπ) β§ {π΄} β π) β (πΎβ{π΄}) = β© {π¦ β (SubGrpβπΊ) β£ {π΄} β π¦}) |
12 | 8, 9, 11 | syl2an 594 | . 2 β’ ((πΊ β Grp β§ π΄ β π) β (πΎβ{π΄}) = β© {π¦ β (SubGrpβπΊ) β£ {π΄} β π¦}) |
13 | cycsubg2.t | . . 3 β’ Β· = (.gβπΊ) | |
14 | cycsubg2.f | . . 3 β’ πΉ = (π₯ β β€ β¦ (π₯ Β· π΄)) | |
15 | 6, 13, 14 | cycsubg 19177 | . 2 β’ ((πΊ β Grp β§ π΄ β π) β ran πΉ = β© {π¦ β (SubGrpβπΊ) β£ π΄ β π¦}) |
16 | 5, 12, 15 | 3eqtr4d 2778 | 1 β’ ((πΊ β Grp β§ π΄ β π) β (πΎβ{π΄}) = ran πΉ) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β wb 205 β§ wa 394 = wceq 1533 β wcel 2098 {crab 3430 β wss 3949 {csn 4632 β© cint 4953 β¦ cmpt 5235 ran crn 5683 βcfv 6553 (class class class)co 7426 β€cz 12598 Basecbs 17189 Moorecmre 17571 mrClscmrc 17572 Grpcgrp 18904 .gcmg 19037 SubGrpcsubg 19089 |
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 2699 ax-sep 5303 ax-nul 5310 ax-pow 5369 ax-pr 5433 ax-un 7748 ax-cnex 11204 ax-resscn 11205 ax-1cn 11206 ax-icn 11207 ax-addcl 11208 ax-addrcl 11209 ax-mulcl 11210 ax-mulrcl 11211 ax-mulcom 11212 ax-addass 11213 ax-mulass 11214 ax-distr 11215 ax-i2m1 11216 ax-1ne0 11217 ax-1rid 11218 ax-rnegex 11219 ax-rrecex 11220 ax-cnre 11221 ax-pre-lttri 11222 ax-pre-lttrn 11223 ax-pre-ltadd 11224 ax-pre-mulgt0 11225 |
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 2529 df-eu 2558 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-rmo 3374 df-reu 3375 df-rab 3431 df-v 3475 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-pss 3968 df-nul 4327 df-if 4533 df-pw 4608 df-sn 4633 df-pr 4635 df-op 4639 df-uni 4913 df-int 4954 df-iun 5002 df-iin 5003 df-br 5153 df-opab 5215 df-mpt 5236 df-tr 5270 df-id 5580 df-eprel 5586 df-po 5594 df-so 5595 df-fr 5637 df-we 5639 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-rn 5693 df-res 5694 df-ima 5695 df-pred 6310 df-ord 6377 df-on 6378 df-lim 6379 df-suc 6380 df-iota 6505 df-fun 6555 df-fn 6556 df-f 6557 df-f1 6558 df-fo 6559 df-f1o 6560 df-fv 6561 df-riota 7382 df-ov 7429 df-oprab 7430 df-mpo 7431 df-om 7879 df-1st 8001 df-2nd 8002 df-frecs 8295 df-wrecs 8326 df-recs 8400 df-rdg 8439 df-1o 8495 df-er 8733 df-en 8973 df-dom 8974 df-sdom 8975 df-fin 8976 df-pnf 11290 df-mnf 11291 df-xr 11292 df-ltxr 11293 df-le 11294 df-sub 11486 df-neg 11487 df-nn 12253 df-2 12315 df-n0 12513 df-z 12599 df-uz 12863 df-fz 13527 df-seq 14009 df-sets 17142 df-slot 17160 df-ndx 17172 df-base 17190 df-ress 17219 df-plusg 17255 df-0g 17432 df-mre 17575 df-mrc 17576 df-acs 17578 df-mgm 18609 df-sgrp 18688 df-mnd 18704 df-submnd 18750 df-grp 18907 df-minusg 18908 df-mulg 19038 df-subg 19092 |
This theorem is referenced by: odf1o1 19541 odf1o2 19542 cycsubgcyg2 19871 pgpfac1lem2 20046 pgpfac1lem3 20048 pgpfac1lem4 20049 |
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