![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > cycsubgcld | Structured version Visualization version GIF version |
Description: The cyclic subgroup generated by ๐ด is a subgroup. Deduction related to cycsubgcl 19000. (Contributed by Rohan Ridenour, 3-Aug-2023.) |
Ref | Expression |
---|---|
cycsubgcld.1 | โข ๐ต = (Baseโ๐บ) |
cycsubgcld.2 | โข ยท = (.gโ๐บ) |
cycsubgcld.3 | โข ๐น = (๐ โ โค โฆ (๐ ยท ๐ด)) |
cycsubgcld.4 | โข (๐ โ ๐บ โ Grp) |
cycsubgcld.5 | โข (๐ โ ๐ด โ ๐ต) |
Ref | Expression |
---|---|
cycsubgcld | โข (๐ โ ran ๐น โ (SubGrpโ๐บ)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cycsubgcld.4 | . . 3 โข (๐ โ ๐บ โ Grp) | |
2 | cycsubgcld.5 | . . 3 โข (๐ โ ๐ด โ ๐ต) | |
3 | cycsubgcld.1 | . . . 4 โข ๐ต = (Baseโ๐บ) | |
4 | cycsubgcld.2 | . . . 4 โข ยท = (.gโ๐บ) | |
5 | cycsubgcld.3 | . . . 4 โข ๐น = (๐ โ โค โฆ (๐ ยท ๐ด)) | |
6 | 3, 4, 5 | cycsubgcl 19000 | . . 3 โข ((๐บ โ Grp โง ๐ด โ ๐ต) โ (ran ๐น โ (SubGrpโ๐บ) โง ๐ด โ ran ๐น)) |
7 | 1, 2, 6 | syl2anc 585 | . 2 โข (๐ โ (ran ๐น โ (SubGrpโ๐บ) โง ๐ด โ ran ๐น)) |
8 | 7 | simpld 496 | 1 โข (๐ โ ran ๐น โ (SubGrpโ๐บ)) |
Colors of variables: wff setvar class |
Syntax hints: โ wi 4 โง wa 397 = wceq 1542 โ wcel 2107 โฆ cmpt 5189 ran crn 5635 โcfv 6497 (class class class)co 7358 โคcz 12500 Basecbs 17084 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-iun 4957 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-er 8649 df-en 8885 df-dom 8886 df-sdom 8887 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-mgm 18498 df-sgrp 18547 df-mnd 18558 df-grp 18752 df-minusg 18753 df-mulg 18874 df-subg 18926 |
This theorem is referenced by: ablsimpg1gend 19885 fincygsubgd 19891 |
Copyright terms: Public domain | W3C validator |