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Mirrors > Home > MPE Home > Th. List > abladdsub4 | Structured version Visualization version Unicode version |
Description: Abelian group addition/subtraction law. (Contributed by NM, 31-Mar-2014.) |
Ref | Expression |
---|---|
ablsubadd.b | |
ablsubadd.p | |
ablsubadd.m |
Ref | Expression |
---|---|
abladdsub4 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ablgrp 18198 | . . . 4 | |
2 | 1 | 3ad2ant1 1082 | . . 3 |
3 | simp2l 1087 | . . . 4 | |
4 | simp2r 1088 | . . . 4 | |
5 | ablsubadd.b | . . . . 5 | |
6 | ablsubadd.p | . . . . 5 | |
7 | 5, 6 | grpcl 17430 | . . . 4 |
8 | 2, 3, 4, 7 | syl3anc 1326 | . . 3 |
9 | simp3l 1089 | . . . 4 | |
10 | simp3r 1090 | . . . 4 | |
11 | 5, 6 | grpcl 17430 | . . . 4 |
12 | 2, 9, 10, 11 | syl3anc 1326 | . . 3 |
13 | 5, 6 | grpcl 17430 | . . . 4 |
14 | 2, 9, 4, 13 | syl3anc 1326 | . . 3 |
15 | ablsubadd.m | . . . 4 | |
16 | 5, 15 | grpsubrcan 17496 | . . 3 |
17 | 2, 8, 12, 14, 16 | syl13anc 1328 | . 2 |
18 | simp1 1061 | . . . . 5 | |
19 | 5, 6, 15 | ablsub4 18218 | . . . . 5 |
20 | 18, 3, 4, 9, 4, 19 | syl122anc 1335 | . . . 4 |
21 | eqid 2622 | . . . . . . 7 | |
22 | 5, 21, 15 | grpsubid 17499 | . . . . . 6 |
23 | 2, 4, 22 | syl2anc 693 | . . . . 5 |
24 | 23 | oveq2d 6666 | . . . 4 |
25 | 5, 15 | grpsubcl 17495 | . . . . . 6 |
26 | 2, 3, 9, 25 | syl3anc 1326 | . . . . 5 |
27 | 5, 6, 21 | grprid 17453 | . . . . 5 |
28 | 2, 26, 27 | syl2anc 693 | . . . 4 |
29 | 20, 24, 28 | 3eqtrd 2660 | . . 3 |
30 | 5, 6, 15 | ablsub4 18218 | . . . . 5 |
31 | 18, 9, 10, 9, 4, 30 | syl122anc 1335 | . . . 4 |
32 | 5, 21, 15 | grpsubid 17499 | . . . . . 6 |
33 | 2, 9, 32 | syl2anc 693 | . . . . 5 |
34 | 33 | oveq1d 6665 | . . . 4 |
35 | 5, 15 | grpsubcl 17495 | . . . . . 6 |
36 | 2, 10, 4, 35 | syl3anc 1326 | . . . . 5 |
37 | 5, 6, 21 | grplid 17452 | . . . . 5 |
38 | 2, 36, 37 | syl2anc 693 | . . . 4 |
39 | 31, 34, 38 | 3eqtrd 2660 | . . 3 |
40 | 29, 39 | eqeq12d 2637 | . 2 |
41 | 17, 40 | bitr3d 270 | 1 |
Colors of variables: wff setvar class |
Syntax hints: wi 4 wb 196 wa 384 w3a 1037 wceq 1483 wcel 1990 cfv 5888 (class class class)co 6650 cbs 15857 cplusg 15941 c0g 16100 cgrp 17422 csg 17424 cabl 18194 |
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-rep 4771 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 |
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-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-op 4184 df-uni 4437 df-iun 4522 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-rn 5125 df-res 5126 df-ima 5127 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-riota 6611 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-1st 7168 df-2nd 7169 df-0g 16102 df-mgm 17242 df-sgrp 17284 df-mnd 17295 df-grp 17425 df-minusg 17426 df-sbg 17427 df-cmn 18195 df-abl 18196 |
This theorem is referenced by: lmodvaddsub4 18915 |
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