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Mirrors > Home > MPE Home > Th. List > ringcom | Structured version Visualization version Unicode version |
Description: Commutativity of the additive group of a ring. (See also lmodcom 18909.) (Contributed by Gérard Lang, 4-Dec-2014.) |
Ref | Expression |
---|---|
ringacl.b | |
ringacl.p |
Ref | Expression |
---|---|
ringcom |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simp1 1061 | . . . . . . . 8 | |
2 | ringacl.b | . . . . . . . . . . 11 | |
3 | eqid 2622 | . . . . . . . . . . 11 | |
4 | 2, 3 | ringidcl 18568 | . . . . . . . . . 10 |
5 | 1, 4 | syl 17 | . . . . . . . . 9 |
6 | ringacl.p | . . . . . . . . . 10 | |
7 | 2, 6 | ringacl 18578 | . . . . . . . . 9 |
8 | 1, 5, 5, 7 | syl3anc 1326 | . . . . . . . 8 |
9 | simp2 1062 | . . . . . . . 8 | |
10 | simp3 1063 | . . . . . . . 8 | |
11 | eqid 2622 | . . . . . . . . 9 | |
12 | 2, 6, 11 | ringdi 18566 | . . . . . . . 8 |
13 | 1, 8, 9, 10, 12 | syl13anc 1328 | . . . . . . 7 |
14 | 2, 6 | ringacl 18578 | . . . . . . . 8 |
15 | 2, 6, 11 | ringdir 18567 | . . . . . . . 8 |
16 | 1, 5, 5, 14, 15 | syl13anc 1328 | . . . . . . 7 |
17 | 13, 16 | eqtr3d 2658 | . . . . . 6 |
18 | 2, 6, 11 | ringdir 18567 | . . . . . . . . 9 |
19 | 1, 5, 5, 9, 18 | syl13anc 1328 | . . . . . . . 8 |
20 | 2, 11, 3 | ringlidm 18571 | . . . . . . . . . 10 |
21 | 1, 9, 20 | syl2anc 693 | . . . . . . . . 9 |
22 | 21, 21 | oveq12d 6668 | . . . . . . . 8 |
23 | 19, 22 | eqtrd 2656 | . . . . . . 7 |
24 | 2, 6, 11 | ringdir 18567 | . . . . . . . . 9 |
25 | 1, 5, 5, 10, 24 | syl13anc 1328 | . . . . . . . 8 |
26 | 2, 11, 3 | ringlidm 18571 | . . . . . . . . . 10 |
27 | 1, 10, 26 | syl2anc 693 | . . . . . . . . 9 |
28 | 27, 27 | oveq12d 6668 | . . . . . . . 8 |
29 | 25, 28 | eqtrd 2656 | . . . . . . 7 |
30 | 23, 29 | oveq12d 6668 | . . . . . 6 |
31 | 2, 11, 3 | ringlidm 18571 | . . . . . . . 8 |
32 | 1, 14, 31 | syl2anc 693 | . . . . . . 7 |
33 | 32, 32 | oveq12d 6668 | . . . . . 6 |
34 | 17, 30, 33 | 3eqtr3d 2664 | . . . . 5 |
35 | ringgrp 18552 | . . . . . . 7 | |
36 | 1, 35 | syl 17 | . . . . . 6 |
37 | 2, 6 | ringacl 18578 | . . . . . . 7 |
38 | 1, 9, 9, 37 | syl3anc 1326 | . . . . . 6 |
39 | 2, 6 | grpass 17431 | . . . . . 6 |
40 | 36, 38, 10, 10, 39 | syl13anc 1328 | . . . . 5 |
41 | 2, 6 | grpass 17431 | . . . . . 6 |
42 | 36, 14, 9, 10, 41 | syl13anc 1328 | . . . . 5 |
43 | 34, 40, 42 | 3eqtr4d 2666 | . . . 4 |
44 | 2, 6 | ringacl 18578 | . . . . . 6 |
45 | 1, 38, 10, 44 | syl3anc 1326 | . . . . 5 |
46 | 2, 6 | ringacl 18578 | . . . . . 6 |
47 | 1, 14, 9, 46 | syl3anc 1326 | . . . . 5 |
48 | 2, 6 | grprcan 17455 | . . . . 5 |
49 | 36, 45, 47, 10, 48 | syl13anc 1328 | . . . 4 |
50 | 43, 49 | mpbid 222 | . . 3 |
51 | 2, 6 | grpass 17431 | . . . 4 |
52 | 36, 9, 9, 10, 51 | syl13anc 1328 | . . 3 |
53 | 2, 6 | grpass 17431 | . . . 4 |
54 | 36, 9, 10, 9, 53 | syl13anc 1328 | . . 3 |
55 | 50, 52, 54 | 3eqtr3d 2664 | . 2 |
56 | 2, 6 | ringacl 18578 | . . . 4 |
57 | 56 | 3com23 1271 | . . 3 |
58 | 2, 6 | grplcan 17477 | . . 3 |
59 | 36, 14, 57, 9, 58 | syl13anc 1328 | . 2 |
60 | 55, 59 | mpbid 222 | 1 |
Colors of variables: wff setvar class |
Syntax hints: wi 4 wb 196 w3a 1037 wceq 1483 wcel 1990 cfv 5888 (class class class)co 6650 cbs 15857 cplusg 15941 cmulr 15942 cgrp 17422 cur 18501 crg 18547 |
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 ax-cnex 9992 ax-resscn 9993 ax-1cn 9994 ax-icn 9995 ax-addcl 9996 ax-addrcl 9997 ax-mulcl 9998 ax-mulrcl 9999 ax-mulcom 10000 ax-addass 10001 ax-mulass 10002 ax-distr 10003 ax-i2m1 10004 ax-1ne0 10005 ax-1rid 10006 ax-rnegex 10007 ax-rrecex 10008 ax-cnre 10009 ax-pre-lttri 10010 ax-pre-lttrn 10011 ax-pre-ltadd 10012 ax-pre-mulgt0 10013 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 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-nel 2898 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-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-we 5075 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-pred 5680 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 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-om 7066 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-er 7742 df-en 7956 df-dom 7957 df-sdom 7958 df-pnf 10076 df-mnf 10077 df-xr 10078 df-ltxr 10079 df-le 10080 df-sub 10268 df-neg 10269 df-nn 11021 df-2 11079 df-ndx 15860 df-slot 15861 df-base 15863 df-sets 15864 df-plusg 15954 df-0g 16102 df-mgm 17242 df-sgrp 17284 df-mnd 17295 df-grp 17425 df-minusg 17426 df-mgp 18490 df-ur 18502 df-ring 18549 |
This theorem is referenced by: ringabl 18580 |
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