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Mirrors > Home > ILE Home > Th. List > isgrpde | Unicode version |
Description: Deduce a group from its
properties. In this version of isgrpd 12789, we
don't assume there is an expression for the inverse of ![]() |
Ref | Expression |
---|---|
isgrpd.b |
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isgrpd.p |
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isgrpd.c |
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isgrpd.a |
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isgrpd.z |
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isgrpd.i |
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isgrpde.n |
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Ref | Expression |
---|---|
isgrpde |
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | isgrpd.b |
. 2
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2 | isgrpd.p |
. 2
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3 | isgrpd.z |
. . 3
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
4 | isgrpd.i |
. . 3
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
5 | isgrpd.c |
. . . 4
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
6 | isgrpd.a |
. . . 4
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7 | isgrpde.n |
. . . 4
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
8 | 5, 3, 4, 6, 7 | grpridd 12698 |
. . 3
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9 | 1, 2, 3, 4, 8 | grpidd 12694 |
. 2
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10 | 1, 2, 5, 6, 3, 4, 8 | ismndd 12730 |
. 2
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11 | 1, 2, 9, 10, 7 | isgrpd2e 12786 |
1
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Colors of variables: wff set class |
Syntax hints: ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-13 2150 ax-14 2151 ax-ext 2159 ax-sep 4118 ax-pow 4171 ax-pr 4206 ax-un 4430 ax-cnex 7893 ax-resscn 7894 ax-1re 7896 ax-addrcl 7899 |
This theorem depends on definitions: df-bi 117 df-3an 980 df-tru 1356 df-nf 1461 df-sb 1763 df-eu 2029 df-mo 2030 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-ral 2460 df-rex 2461 df-reu 2462 df-rmo 2463 df-rab 2464 df-v 2739 df-sbc 2963 df-csb 3058 df-un 3133 df-in 3135 df-ss 3142 df-pw 3576 df-sn 3597 df-pr 3598 df-op 3600 df-uni 3808 df-int 3843 df-br 4001 df-opab 4062 df-mpt 4063 df-id 4290 df-xp 4629 df-rel 4630 df-cnv 4631 df-co 4632 df-dm 4633 df-rn 4634 df-res 4635 df-iota 5174 df-fun 5214 df-fn 5215 df-fv 5220 df-riota 5825 df-ov 5872 df-inn 8909 df-2 8967 df-ndx 12448 df-slot 12449 df-base 12451 df-plusg 12531 df-0g 12655 df-mgm 12667 df-sgrp 12700 df-mnd 12710 df-grp 12770 |
This theorem is referenced by: isgrpd 12789 dfgrp2 12792 unitgrp 13110 |
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