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Mirrors > Home > MPE Home > Th. List > grppropd | Structured version Visualization version Unicode version |
Description: If two structures have the same group components (properties), one is a group iff the other one is. (Contributed by Stefan O'Rear, 27-Nov-2014.) (Revised by Mario Carneiro, 2-Oct-2015.) |
Ref | Expression |
---|---|
grppropd.1 | |
grppropd.2 | |
grppropd.3 |
Ref | Expression |
---|---|
grppropd |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | grppropd.1 | . . . 4 | |
2 | grppropd.2 | . . . 4 | |
3 | grppropd.3 | . . . 4 | |
4 | 1, 2, 3 | mndpropd 17316 | . . 3 |
5 | 1, 2, 3 | grpidpropd 17261 | . . . . . . . . 9 |
6 | 5 | adantr 481 | . . . . . . . 8 |
7 | 3, 6 | eqeq12d 2637 | . . . . . . 7 |
8 | 7 | anass1rs 849 | . . . . . 6 |
9 | 8 | rexbidva 3049 | . . . . 5 |
10 | 9 | ralbidva 2985 | . . . 4 |
11 | 1 | rexeqdv 3145 | . . . . 5 |
12 | 1, 11 | raleqbidv 3152 | . . . 4 |
13 | 2 | rexeqdv 3145 | . . . . 5 |
14 | 2, 13 | raleqbidv 3152 | . . . 4 |
15 | 10, 12, 14 | 3bitr3d 298 | . . 3 |
16 | 4, 15 | anbi12d 747 | . 2 |
17 | eqid 2622 | . . 3 | |
18 | eqid 2622 | . . 3 | |
19 | eqid 2622 | . . 3 | |
20 | 17, 18, 19 | isgrp 17428 | . 2 |
21 | eqid 2622 | . . 3 | |
22 | eqid 2622 | . . 3 | |
23 | eqid 2622 | . . 3 | |
24 | 21, 22, 23 | isgrp 17428 | . 2 |
25 | 16, 20, 24 | 3bitr4g 303 | 1 |
Colors of variables: wff setvar class |
Syntax hints: wi 4 wb 196 wa 384 wceq 1483 wcel 1990 wral 2912 wrex 2913 cfv 5888 (class class class)co 6650 cbs 15857 cplusg 15941 c0g 16100 cmnd 17294 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-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-ov 6653 df-0g 16102 df-mgm 17242 df-sgrp 17284 df-mnd 17295 df-grp 17425 |
This theorem is referenced by: grpprop 17438 ghmpropd 17698 oppggrpb 17788 ablpropd 18203 ringpropd 18582 lmodprop2d 18925 sralmod 19187 nmpropd2 22399 ngppropd 22441 tngngp2 22456 tnggrpr 22459 zhmnrg 30011 |
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