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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 18896 | . . 3 β’ (πΊ β Grp β πΊ β Mnd) | |
2 | 0subg.z | . . . 4 β’ 0 = (0gβπΊ) | |
3 | 2 | 0subm 18768 | . . 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 2728 | . . . . 5 β’ (invgβπΊ) = (invgβπΊ) | |
6 | 2, 5 | grpinvid 18955 | . . . 4 β’ (πΊ β Grp β ((invgβπΊ)β 0 ) = 0 ) |
7 | fvex 6910 | . . . . 5 β’ ((invgβπΊ)β 0 ) β V | |
8 | 7 | elsn 4644 | . . . 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 6911 | . . . 4 β’ 0 β V |
11 | fveq2 6897 | . . . . 5 β’ (π = 0 β ((invgβπΊ)βπ) = ((invgβπΊ)β 0 )) | |
12 | 11 | eleq1d 2814 | . . . 4 β’ (π = 0 β (((invgβπΊ)βπ) β { 0 } β ((invgβπΊ)β 0 ) β { 0 })) |
13 | 10, 12 | ralsn 4686 | . . 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 19098 | . 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 712 | 1 β’ (πΊ β Grp β { 0 } β (SubGrpβπΊ)) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 = wceq 1534 β wcel 2099 βwral 3058 {csn 4629 βcfv 6548 0gc0g 17420 Mndcmnd 18693 SubMndcsubmnd 18738 Grpcgrp 18889 invgcminusg 18890 SubGrpcsubg 19074 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2699 ax-sep 5299 ax-nul 5306 ax-pow 5365 ax-pr 5429 ax-un 7740 ax-cnex 11194 ax-resscn 11195 ax-1cn 11196 ax-icn 11197 ax-addcl 11198 ax-addrcl 11199 ax-mulcl 11200 ax-mulrcl 11201 ax-mulcom 11202 ax-addass 11203 ax-mulass 11204 ax-distr 11205 ax-i2m1 11206 ax-1ne0 11207 ax-1rid 11208 ax-rnegex 11209 ax-rrecex 11210 ax-cnre 11211 ax-pre-lttri 11212 ax-pre-lttrn 11213 ax-pre-ltadd 11214 ax-pre-mulgt0 11215 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2530 df-eu 2559 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 3373 df-reu 3374 df-rab 3430 df-v 3473 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4909 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5576 df-eprel 5582 df-po 5590 df-so 5591 df-fr 5633 df-we 5635 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-pred 6305 df-ord 6372 df-on 6373 df-lim 6374 df-suc 6375 df-iota 6500 df-fun 6550 df-fn 6551 df-f 6552 df-f1 6553 df-fo 6554 df-f1o 6555 df-fv 6556 df-riota 7376 df-ov 7423 df-oprab 7424 df-mpo 7425 df-om 7871 df-2nd 7994 df-frecs 8286 df-wrecs 8317 df-recs 8391 df-rdg 8430 df-er 8724 df-en 8964 df-dom 8965 df-sdom 8966 df-pnf 11280 df-mnf 11281 df-xr 11282 df-ltxr 11283 df-le 11284 df-sub 11476 df-neg 11477 df-nn 12243 df-2 12305 df-sets 17132 df-slot 17150 df-ndx 17162 df-base 17180 df-ress 17209 df-plusg 17245 df-0g 17422 df-mgm 18599 df-sgrp 18678 df-mnd 18694 df-submnd 18740 df-grp 18892 df-minusg 18893 df-subg 19077 |
This theorem is referenced by: 0nsg 19123 eqg0subg 19150 qus0subgadd 19153 idressubgsymg 19364 pgp0 19550 slwn0 19569 lsm01 19625 lsm02 19626 dprdz 19986 dprdsn 19992 pgpfac1lem5 20035 tgptsmscls 24053 evpmsubg 32868 |
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