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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 4782 | . . . . . 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 481 | . . . 4 β’ ((πΊ β Grp β§ π΄ β π) β ({π΄} β π¦ β π΄ β π¦)) |
4 | 3 | rabbidv 3434 | . . 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 4948 | . 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 19088 | . . . 4 β’ (πΊ β Grp β (SubGrpβπΊ) β (ACSβπ)) |
8 | 7 | acsmred 17609 | . . 3 β’ (πΊ β Grp β (SubGrpβπΊ) β (Mooreβπ)) |
9 | snssi 4806 | . . 3 β’ (π΄ β π β {π΄} β π) | |
10 | cycsubg2.k | . . . 4 β’ πΎ = (mrClsβ(SubGrpβπΊ)) | |
11 | 10 | mrcval 17563 | . . 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 595 | . 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 19134 | . 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 2776 | 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 395 = wceq 1533 β wcel 2098 {crab 3426 β wss 3943 {csn 4623 β© cint 4943 β¦ cmpt 5224 ran crn 5670 βcfv 6537 (class class class)co 7405 β€cz 12562 Basecbs 17153 Moorecmre 17535 mrClscmrc 17536 Grpcgrp 18863 .gcmg 18995 SubGrpcsubg 19047 |
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 7722 ax-cnex 11168 ax-resscn 11169 ax-1cn 11170 ax-icn 11171 ax-addcl 11172 ax-addrcl 11173 ax-mulcl 11174 ax-mulrcl 11175 ax-mulcom 11176 ax-addass 11177 ax-mulass 11178 ax-distr 11179 ax-i2m1 11180 ax-1ne0 11181 ax-1rid 11182 ax-rnegex 11183 ax-rrecex 11184 ax-cnre 11185 ax-pre-lttri 11186 ax-pre-lttrn 11187 ax-pre-ltadd 11188 ax-pre-mulgt0 11189 |
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-int 4944 df-iun 4992 df-iin 4993 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 6294 df-ord 6361 df-on 6362 df-lim 6363 df-suc 6364 df-iota 6489 df-fun 6539 df-fn 6540 df-f 6541 df-f1 6542 df-fo 6543 df-f1o 6544 df-fv 6545 df-riota 7361 df-ov 7408 df-oprab 7409 df-mpo 7410 df-om 7853 df-1st 7974 df-2nd 7975 df-frecs 8267 df-wrecs 8298 df-recs 8372 df-rdg 8411 df-1o 8467 df-er 8705 df-en 8942 df-dom 8943 df-sdom 8944 df-fin 8945 df-pnf 11254 df-mnf 11255 df-xr 11256 df-ltxr 11257 df-le 11258 df-sub 11450 df-neg 11451 df-nn 12217 df-2 12279 df-n0 12477 df-z 12563 df-uz 12827 df-fz 13491 df-seq 13973 df-sets 17106 df-slot 17124 df-ndx 17136 df-base 17154 df-ress 17183 df-plusg 17219 df-0g 17396 df-mre 17539 df-mrc 17540 df-acs 17542 df-mgm 18573 df-sgrp 18652 df-mnd 18668 df-submnd 18714 df-grp 18866 df-minusg 18867 df-mulg 18996 df-subg 19050 |
This theorem is referenced by: odf1o1 19492 odf1o2 19493 cycsubgcyg2 19822 pgpfac1lem2 19997 pgpfac1lem3 19999 pgpfac1lem4 20000 |
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