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Mirrors > Home > MPE Home > Th. List > isgrpde | Structured version Visualization version Unicode version |
Description: Deduce a group from its properties. In this version of isgrpd 17444, we don't assume there is an expression for the inverse of . (Contributed by NM, 6-Jan-2015.) |
Ref | Expression |
---|---|
isgrpd.b | |
isgrpd.p | |
isgrpd.c | |
isgrpd.a | |
isgrpd.z | |
isgrpd.i | |
isgrpde.n |
Ref | Expression |
---|---|
isgrpde |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | isgrpd.b | . 2 | |
2 | isgrpd.p | . 2 | |
3 | isgrpd.z | . . 3 | |
4 | isgrpd.i | . . 3 | |
5 | isgrpd.c | . . . 4 | |
6 | isgrpd.a | . . . 4 | |
7 | isgrpde.n | . . . 4 | |
8 | 5, 3, 4, 6, 7 | grpridd 6874 | . . 3 |
9 | 1, 2, 3, 4, 8 | grpidd 17268 | . 2 |
10 | 1, 2, 5, 6, 3, 4, 8 | ismndd 17313 | . 2 |
11 | 1, 2, 9, 10, 7 | isgrpd2e 17441 | 1 |
Colors of variables: wff setvar class |
Syntax hints: wi 4 wa 384 w3a 1037 wceq 1483 wcel 1990 wrex 2913 cfv 5888 (class class class)co 6650 cbs 15857 cplusg 15941 cgrp 17422 |
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-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-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-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-iota 5851 df-fun 5890 df-fv 5896 df-riota 6611 df-ov 6653 df-0g 16102 df-mgm 17242 df-sgrp 17284 df-mnd 17295 df-grp 17425 |
This theorem is referenced by: isgrpd 17444 dfgrp2 17447 imasgrp2 17530 unitgrp 18667 |
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