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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 4745 | . . . . . 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 483 | . . . 4 β’ ((πΊ β Grp β§ π΄ β π) β ({π΄} β π¦ β π΄ β π¦)) |
4 | 3 | rabbidv 3416 | . . 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 4913 | . 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 18964 | . . . 4 β’ (πΊ β Grp β (SubGrpβπΊ) β (ACSβπ)) |
8 | 7 | acsmred 17537 | . . 3 β’ (πΊ β Grp β (SubGrpβπΊ) β (Mooreβπ)) |
9 | snssi 4769 | . . 3 β’ (π΄ β π β {π΄} β π) | |
10 | cycsubg2.k | . . . 4 β’ πΎ = (mrClsβ(SubGrpβπΊ)) | |
11 | 10 | mrcval 17491 | . . 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 597 | . 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 19002 | . 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 2787 | 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 397 = wceq 1542 β wcel 2107 {crab 3408 β wss 3911 {csn 4587 β© cint 4908 β¦ cmpt 5189 ran crn 5635 βcfv 6497 (class class class)co 7358 β€cz 12500 Basecbs 17084 Moorecmre 17463 mrClscmrc 17464 Grpcgrp 18749 .gcmg 18873 SubGrpcsubg 18923 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2708 ax-sep 5257 ax-nul 5264 ax-pow 5321 ax-pr 5385 ax-un 7673 ax-cnex 11108 ax-resscn 11109 ax-1cn 11110 ax-icn 11111 ax-addcl 11112 ax-addrcl 11113 ax-mulcl 11114 ax-mulrcl 11115 ax-mulcom 11116 ax-addass 11117 ax-mulass 11118 ax-distr 11119 ax-i2m1 11120 ax-1ne0 11121 ax-1rid 11122 ax-rnegex 11123 ax-rrecex 11124 ax-cnre 11125 ax-pre-lttri 11126 ax-pre-lttrn 11127 ax-pre-ltadd 11128 ax-pre-mulgt0 11129 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2815 df-nfc 2890 df-ne 2945 df-nel 3051 df-ral 3066 df-rex 3075 df-rmo 3354 df-reu 3355 df-rab 3409 df-v 3448 df-sbc 3741 df-csb 3857 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-pss 3930 df-nul 4284 df-if 4488 df-pw 4563 df-sn 4588 df-pr 4590 df-op 4594 df-uni 4867 df-int 4909 df-iun 4957 df-iin 4958 df-br 5107 df-opab 5169 df-mpt 5190 df-tr 5224 df-id 5532 df-eprel 5538 df-po 5546 df-so 5547 df-fr 5589 df-we 5591 df-xp 5640 df-rel 5641 df-cnv 5642 df-co 5643 df-dm 5644 df-rn 5645 df-res 5646 df-ima 5647 df-pred 6254 df-ord 6321 df-on 6322 df-lim 6323 df-suc 6324 df-iota 6449 df-fun 6499 df-fn 6500 df-f 6501 df-f1 6502 df-fo 6503 df-f1o 6504 df-fv 6505 df-riota 7314 df-ov 7361 df-oprab 7362 df-mpo 7363 df-om 7804 df-1st 7922 df-2nd 7923 df-frecs 8213 df-wrecs 8244 df-recs 8318 df-rdg 8357 df-1o 8413 df-er 8649 df-en 8885 df-dom 8886 df-sdom 8887 df-fin 8888 df-pnf 11192 df-mnf 11193 df-xr 11194 df-ltxr 11195 df-le 11196 df-sub 11388 df-neg 11389 df-nn 12155 df-2 12217 df-n0 12415 df-z 12501 df-uz 12765 df-fz 13426 df-seq 13908 df-sets 17037 df-slot 17055 df-ndx 17067 df-base 17085 df-ress 17114 df-plusg 17147 df-0g 17324 df-mre 17467 df-mrc 17468 df-acs 17470 df-mgm 18498 df-sgrp 18547 df-mnd 18558 df-submnd 18603 df-grp 18752 df-minusg 18753 df-mulg 18874 df-subg 18926 |
This theorem is referenced by: odf1o1 19355 odf1o2 19356 cycsubgcyg2 19680 pgpfac1lem2 19855 pgpfac1lem3 19857 pgpfac1lem4 19858 |
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