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Mirrors > Home > MPE Home > Th. List > ablcntzd | Structured version Visualization version Unicode version |
Description: All subgroups in an abelian group commute. (Contributed by Mario Carneiro, 19-Apr-2016.) |
Ref | Expression |
---|---|
ablcntzd.z | Cntz |
ablcntzd.a | |
ablcntzd.t | SubGrp |
ablcntzd.u | SubGrp |
Ref | Expression |
---|---|
ablcntzd |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ablcntzd.t | . . 3 SubGrp | |
2 | eqid 2622 | . . . 4 | |
3 | 2 | subgss 17595 | . . 3 SubGrp |
4 | 1, 3 | syl 17 | . 2 |
5 | ablcntzd.a | . . . 4 | |
6 | ablcmn 18199 | . . . 4 CMnd | |
7 | 5, 6 | syl 17 | . . 3 CMnd |
8 | ablcntzd.u | . . . 4 SubGrp | |
9 | 2 | subgss 17595 | . . . 4 SubGrp |
10 | 8, 9 | syl 17 | . . 3 |
11 | ablcntzd.z | . . . 4 Cntz | |
12 | 2, 11 | cntzcmn 18245 | . . 3 CMnd |
13 | 7, 10, 12 | syl2anc 693 | . 2 |
14 | 4, 13 | sseqtr4d 3642 | 1 |
Colors of variables: wff setvar class |
Syntax hints: wi 4 wceq 1483 wcel 1990 wss 3574 cfv 5888 cbs 15857 SubGrpcsubg 17588 Cntzccntz 17748 CMndccmn 18193 cabl 18194 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-rep 4771 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-ral 2917 df-rex 2918 df-reu 2919 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-op 4184 df-uni 4437 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-id 5024 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-ov 6653 df-subg 17591 df-cntz 17750 df-cmn 18195 df-abl 18196 |
This theorem is referenced by: lsmsubg2 18262 ablfacrp2 18466 ablfac1b 18469 pgpfaclem1 18480 pgpfaclem2 18481 pj1lmhm 19100 pj1lmhm2 19101 lvecindp 19138 lvecindp2 19139 pjdm2 20055 pjf2 20058 pjfo 20059 lshpsmreu 34396 lshpkrlem5 34401 |
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