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Mirrors > Home > ILE Home > Th. List > grpidrcan | Unicode version |
Description: If right adding an element of a group to an arbitrary element of the group results in this element, the added element is the identity element and vice versa. (Contributed by AV, 15-Mar-2019.) |
Ref | Expression |
---|---|
grpidrcan.b | |
grpidrcan.p | |
grpidrcan.o |
Ref | Expression |
---|---|
grpidrcan |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | grpidrcan.b | . . . . 5 | |
2 | grpidrcan.p | . . . . 5 | |
3 | grpidrcan.o | . . . . 5 | |
4 | 1, 2, 3 | grprid 12767 | . . . 4 |
5 | 4 | 3adant3 1017 | . . 3 |
6 | 5 | eqeq2d 2187 | . 2 |
7 | simp1 997 | . . 3 | |
8 | simp3 999 | . . 3 | |
9 | 1, 3 | grpidcl 12764 | . . . 4 |
10 | 9 | 3ad2ant1 1018 | . . 3 |
11 | simp2 998 | . . 3 | |
12 | 1, 2 | grplcan 12791 | . . 3 |
13 | 7, 8, 10, 11, 12 | syl13anc 1240 | . 2 |
14 | 6, 13 | bitr3d 190 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wb 105 w3a 978 wceq 1353 wcel 2146 cfv 5208 (class class class)co 5865 cbs 12428 cplusg 12492 c0g 12626 cgrp 12738 |
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 1445 ax-7 1446 ax-gen 1447 ax-ie1 1491 ax-ie2 1492 ax-8 1502 ax-10 1503 ax-11 1504 ax-i12 1505 ax-bndl 1507 ax-4 1508 ax-17 1524 ax-i9 1528 ax-ial 1532 ax-i5r 1533 ax-13 2148 ax-14 2149 ax-ext 2157 ax-coll 4113 ax-sep 4116 ax-pow 4169 ax-pr 4203 ax-un 4427 ax-cnex 7877 ax-resscn 7878 ax-1re 7880 ax-addrcl 7883 |
This theorem depends on definitions: df-bi 117 df-3an 980 df-tru 1356 df-nf 1459 df-sb 1761 df-eu 2027 df-mo 2028 df-clab 2162 df-cleq 2168 df-clel 2171 df-nfc 2306 df-ral 2458 df-rex 2459 df-reu 2460 df-rmo 2461 df-rab 2462 df-v 2737 df-sbc 2961 df-csb 3056 df-un 3131 df-in 3133 df-ss 3140 df-pw 3574 df-sn 3595 df-pr 3596 df-op 3598 df-uni 3806 df-int 3841 df-iun 3884 df-br 3999 df-opab 4060 df-mpt 4061 df-id 4287 df-xp 4626 df-rel 4627 df-cnv 4628 df-co 4629 df-dm 4630 df-rn 4631 df-res 4632 df-ima 4633 df-iota 5170 df-fun 5210 df-fn 5211 df-f 5212 df-f1 5213 df-fo 5214 df-f1o 5215 df-fv 5216 df-riota 5821 df-ov 5868 df-inn 8891 df-2 8949 df-ndx 12431 df-slot 12432 df-base 12434 df-plusg 12505 df-0g 12628 df-mgm 12640 df-sgrp 12673 df-mnd 12683 df-grp 12741 df-minusg 12742 |
This theorem is referenced by: (None) |
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