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Mirrors > Home > MPE Home > Th. List > isgrpi | Structured version Visualization version Unicode version |
Description: Properties that determine a group. (negative) is normally dependent on i.e. read it as . (Contributed by NM, 3-Sep-2011.) |
Ref | Expression |
---|---|
isgrpi.b | |
isgrpi.p | |
isgrpi.c | |
isgrpi.a | |
isgrpi.z | |
isgrpi.i | |
isgrpi.n | |
isgrpi.j |
Ref | Expression |
---|---|
isgrpi |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | isgrpi.b | . . . 4 | |
2 | 1 | a1i 11 | . . 3 |
3 | isgrpi.p | . . . 4 | |
4 | 3 | a1i 11 | . . 3 |
5 | isgrpi.c | . . . 4 | |
6 | 5 | 3adant1 1079 | . . 3 |
7 | isgrpi.a | . . . 4 | |
8 | 7 | adantl 482 | . . 3 |
9 | isgrpi.z | . . . 4 | |
10 | 9 | a1i 11 | . . 3 |
11 | isgrpi.i | . . . 4 | |
12 | 11 | adantl 482 | . . 3 |
13 | isgrpi.n | . . . 4 | |
14 | 13 | adantl 482 | . . 3 |
15 | isgrpi.j | . . . 4 | |
16 | 15 | adantl 482 | . . 3 |
17 | 2, 4, 6, 8, 10, 12, 14, 16 | isgrpd 17444 | . 2 |
18 | 17 | trud 1493 | 1 |
Colors of variables: wff setvar class |
Syntax hints: wi 4 wa 384 w3a 1037 wceq 1483 wtru 1484 wcel 1990 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: isgrpix 17449 cnaddabl 18272 cncrng 19767 |
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