![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > 0subg | Structured version Visualization version GIF version |
Description: The zero subgroup of an arbitrary group. (Contributed by Stefan O'Rear, 10-Dec-2014.) (Proof shortened by SN, 31-Jan-2025.) |
Ref | Expression |
---|---|
0subg.z | β’ 0 = (0gβπΊ) |
Ref | Expression |
---|---|
0subg | β’ (πΊ β Grp β { 0 } β (SubGrpβπΊ)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | grpmnd 18866 | . . 3 β’ (πΊ β Grp β πΊ β Mnd) | |
2 | 0subg.z | . . . 4 β’ 0 = (0gβπΊ) | |
3 | 2 | 0subm 18738 | . . 3 β’ (πΊ β Mnd β { 0 } β (SubMndβπΊ)) |
4 | 1, 3 | syl 17 | . 2 β’ (πΊ β Grp β { 0 } β (SubMndβπΊ)) |
5 | eqid 2724 | . . . . 5 β’ (invgβπΊ) = (invgβπΊ) | |
6 | 2, 5 | grpinvid 18925 | . . . 4 β’ (πΊ β Grp β ((invgβπΊ)β 0 ) = 0 ) |
7 | fvex 6895 | . . . . 5 β’ ((invgβπΊ)β 0 ) β V | |
8 | 7 | elsn 4636 | . . . 4 β’ (((invgβπΊ)β 0 ) β { 0 } β ((invgβπΊ)β 0 ) = 0 ) |
9 | 6, 8 | sylibr 233 | . . 3 β’ (πΊ β Grp β ((invgβπΊ)β 0 ) β { 0 }) |
10 | 2 | fvexi 6896 | . . . 4 β’ 0 β V |
11 | fveq2 6882 | . . . . 5 β’ (π = 0 β ((invgβπΊ)βπ) = ((invgβπΊ)β 0 )) | |
12 | 11 | eleq1d 2810 | . . . 4 β’ (π = 0 β (((invgβπΊ)βπ) β { 0 } β ((invgβπΊ)β 0 ) β { 0 })) |
13 | 10, 12 | ralsn 4678 | . . 3 β’ (βπ β { 0 } ((invgβπΊ)βπ) β { 0 } β ((invgβπΊ)β 0 ) β { 0 }) |
14 | 9, 13 | sylibr 233 | . 2 β’ (πΊ β Grp β βπ β { 0 } ((invgβπΊ)βπ) β { 0 }) |
15 | 5 | issubg3 19067 | . 2 β’ (πΊ β Grp β ({ 0 } β (SubGrpβπΊ) β ({ 0 } β (SubMndβπΊ) β§ βπ β { 0 } ((invgβπΊ)βπ) β { 0 }))) |
16 | 4, 14, 15 | mpbir2and 710 | 1 β’ (πΊ β Grp β { 0 } β (SubGrpβπΊ)) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 = wceq 1533 β wcel 2098 βwral 3053 {csn 4621 βcfv 6534 0gc0g 17390 Mndcmnd 18663 SubMndcsubmnd 18708 Grpcgrp 18859 invgcminusg 18860 SubGrpcsubg 19043 |
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 2695 ax-sep 5290 ax-nul 5297 ax-pow 5354 ax-pr 5418 ax-un 7719 ax-cnex 11163 ax-resscn 11164 ax-1cn 11165 ax-icn 11166 ax-addcl 11167 ax-addrcl 11168 ax-mulcl 11169 ax-mulrcl 11170 ax-mulcom 11171 ax-addass 11172 ax-mulass 11173 ax-distr 11174 ax-i2m1 11175 ax-1ne0 11176 ax-1rid 11177 ax-rnegex 11178 ax-rrecex 11179 ax-cnre 11180 ax-pre-lttri 11181 ax-pre-lttrn 11182 ax-pre-ltadd 11183 ax-pre-mulgt0 11184 |
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 2526 df-eu 2555 df-clab 2702 df-cleq 2716 df-clel 2802 df-nfc 2877 df-ne 2933 df-nel 3039 df-ral 3054 df-rex 3063 df-rmo 3368 df-reu 3369 df-rab 3425 df-v 3468 df-sbc 3771 df-csb 3887 df-dif 3944 df-un 3946 df-in 3948 df-ss 3958 df-pss 3960 df-nul 4316 df-if 4522 df-pw 4597 df-sn 4622 df-pr 4624 df-op 4628 df-uni 4901 df-iun 4990 df-br 5140 df-opab 5202 df-mpt 5223 df-tr 5257 df-id 5565 df-eprel 5571 df-po 5579 df-so 5580 df-fr 5622 df-we 5624 df-xp 5673 df-rel 5674 df-cnv 5675 df-co 5676 df-dm 5677 df-rn 5678 df-res 5679 df-ima 5680 df-pred 6291 df-ord 6358 df-on 6359 df-lim 6360 df-suc 6361 df-iota 6486 df-fun 6536 df-fn 6537 df-f 6538 df-f1 6539 df-fo 6540 df-f1o 6541 df-fv 6542 df-riota 7358 df-ov 7405 df-oprab 7406 df-mpo 7407 df-om 7850 df-2nd 7970 df-frecs 8262 df-wrecs 8293 df-recs 8367 df-rdg 8406 df-er 8700 df-en 8937 df-dom 8938 df-sdom 8939 df-pnf 11249 df-mnf 11250 df-xr 11251 df-ltxr 11252 df-le 11253 df-sub 11445 df-neg 11446 df-nn 12212 df-2 12274 df-sets 17102 df-slot 17120 df-ndx 17132 df-base 17150 df-ress 17179 df-plusg 17215 df-0g 17392 df-mgm 18569 df-sgrp 18648 df-mnd 18664 df-submnd 18710 df-grp 18862 df-minusg 18863 df-subg 19046 |
This theorem is referenced by: 0nsg 19092 eqg0subg 19118 qus0subgadd 19121 idressubgsymg 19326 pgp0 19512 slwn0 19531 lsm01 19587 lsm02 19588 dprdz 19948 dprdsn 19954 pgpfac1lem5 19997 tgptsmscls 23998 evpmsubg 32799 |
Copyright terms: Public domain | W3C validator |