Mathbox for Thierry Arnoux |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > ogrpinv0lt | Structured version Visualization version Unicode version |
Description: In an ordered group, the ordering is compatible with group inverse. (Contributed by Thierry Arnoux, 3-Sep-2018.) |
Ref | Expression |
---|---|
ogrpinvlt.0 | |
ogrpinvlt.1 | |
ogrpinvlt.2 | |
ogrpinv0lt.3 |
Ref | Expression |
---|---|
ogrpinv0lt | oGrp |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simpll 790 | . . . 4 oGrp oGrp | |
2 | ogrpgrp 29703 | . . . . . 6 oGrp | |
3 | 1, 2 | syl 17 | . . . . 5 oGrp |
4 | ogrpinvlt.0 | . . . . . 6 | |
5 | ogrpinv0lt.3 | . . . . . 6 | |
6 | 4, 5 | grpidcl 17450 | . . . . 5 |
7 | 3, 6 | syl 17 | . . . 4 oGrp |
8 | simplr 792 | . . . 4 oGrp | |
9 | ogrpinvlt.2 | . . . . . 6 | |
10 | 4, 9 | grpinvcl 17467 | . . . . 5 |
11 | 3, 8, 10 | syl2anc 693 | . . . 4 oGrp |
12 | simpr 477 | . . . 4 oGrp | |
13 | ogrpinvlt.1 | . . . . 5 | |
14 | eqid 2622 | . . . . 5 | |
15 | 4, 13, 14 | ogrpaddlt 29718 | . . . 4 oGrp |
16 | 1, 7, 8, 11, 12, 15 | syl131anc 1339 | . . 3 oGrp |
17 | 4, 14, 5 | grplid 17452 | . . . 4 |
18 | 3, 11, 17 | syl2anc 693 | . . 3 oGrp |
19 | 4, 14, 5, 9 | grprinv 17469 | . . . 4 |
20 | 3, 8, 19 | syl2anc 693 | . . 3 oGrp |
21 | 16, 18, 20 | 3brtr3d 4684 | . 2 oGrp |
22 | simpll 790 | . . . 4 oGrp oGrp | |
23 | 22, 2 | syl 17 | . . . . 5 oGrp |
24 | simplr 792 | . . . . 5 oGrp | |
25 | 23, 24, 10 | syl2anc 693 | . . . 4 oGrp |
26 | 22, 2, 6 | 3syl 18 | . . . 4 oGrp |
27 | simpr 477 | . . . 4 oGrp | |
28 | 4, 13, 14 | ogrpaddlt 29718 | . . . 4 oGrp |
29 | 22, 25, 26, 24, 27, 28 | syl131anc 1339 | . . 3 oGrp |
30 | 4, 14, 5, 9 | grplinv 17468 | . . . 4 |
31 | 23, 24, 30 | syl2anc 693 | . . 3 oGrp |
32 | 4, 14, 5 | grplid 17452 | . . . 4 |
33 | 23, 24, 32 | syl2anc 693 | . . 3 oGrp |
34 | 29, 31, 33 | 3brtr3d 4684 | . 2 oGrp |
35 | 21, 34 | impbida 877 | 1 oGrp |
Colors of variables: wff setvar class |
Syntax hints: wi 4 wb 196 wa 384 wceq 1483 wcel 1990 class class class wbr 4653 cfv 5888 (class class class)co 6650 cbs 15857 cplusg 15941 c0g 16100 cplt 16941 cgrp 17422 cminusg 17423 oGrpcogrp 29698 |
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-rmo 2920 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-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-riota 6611 df-ov 6653 df-0g 16102 df-plt 16958 df-mgm 17242 df-sgrp 17284 df-mnd 17295 df-grp 17425 df-minusg 17426 df-omnd 29699 df-ogrp 29700 |
This theorem is referenced by: archirngz 29743 |
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