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Mirrors > Home > MPE Home > Th. List > Mathboxes > mgpsumn | Structured version Visualization version Unicode version |
Description: If the group sum for the multiplicative group of a unital ring contains a summand/factor that is the one of the ring, this summand/ factor can be removed from the group sum. (Contributed by AV, 29-Dec-2018.) |
Ref | Expression |
---|---|
mgpsumunsn.m | mulGrp |
mgpsumunsn.t | |
mgpsumunsn.r | |
mgpsumunsn.n | |
mgpsumunsn.i | |
mgpsumunsn.a | |
mgpsumn.n | |
mgpsumn.1 |
Ref | Expression |
---|---|
mgpsumn | g g |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | mgpsumunsn.m | . . 3 mulGrp | |
2 | mgpsumunsn.t | . . 3 | |
3 | mgpsumunsn.r | . . 3 | |
4 | mgpsumunsn.n | . . 3 | |
5 | mgpsumunsn.i | . . 3 | |
6 | mgpsumunsn.a | . . 3 | |
7 | crngring 18558 | . . . . 5 | |
8 | 3, 7 | syl 17 | . . . 4 |
9 | eqid 2622 | . . . . 5 | |
10 | mgpsumn.n | . . . . 5 | |
11 | 9, 10 | ringidcl 18568 | . . . 4 |
12 | 8, 11 | syl 17 | . . 3 |
13 | mgpsumn.1 | . . 3 | |
14 | 1, 2, 3, 4, 5, 6, 12, 13 | mgpsumunsn 42140 | . 2 g g |
15 | 1, 9 | mgpbas 18495 | . . . 4 |
16 | 1 | crngmgp 18555 | . . . . 5 CMnd |
17 | 3, 16 | syl 17 | . . . 4 CMnd |
18 | diffi 8192 | . . . . 5 | |
19 | 4, 18 | syl 17 | . . . 4 |
20 | eldifi 3732 | . . . . . 6 | |
21 | 20, 6 | sylan2 491 | . . . . 5 |
22 | 21 | ralrimiva 2966 | . . . 4 |
23 | 15, 17, 19, 22 | gsummptcl 18366 | . . 3 g |
24 | 9, 2, 10 | ringridm 18572 | . . 3 g g g |
25 | 8, 23, 24 | syl2anc 693 | . 2 g g |
26 | 14, 25 | eqtrd 2656 | 1 g g |
Colors of variables: wff setvar class |
Syntax hints: wi 4 wa 384 wceq 1483 wcel 1990 cdif 3571 csn 4177 cmpt 4729 cfv 5888 (class class class)co 6650 cfn 7955 cbs 15857 cmulr 15942 g cgsu 16101 CMndccmn 18193 mulGrpcmgp 18489 cur 18501 crg 18547 ccrg 18548 |
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-inf2 8538 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-int 4476 df-iun 4522 df-iin 4523 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-se 5074 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-isom 5897 df-riota 6611 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-of 6897 df-om 7066 df-1st 7168 df-2nd 7169 df-supp 7296 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-1o 7560 df-oadd 7564 df-er 7742 df-en 7956 df-dom 7957 df-sdom 7958 df-fin 7959 df-fsupp 8276 df-oi 8415 df-card 8765 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-n0 11293 df-z 11378 df-uz 11688 df-fz 12327 df-fzo 12466 df-seq 12802 df-hash 13118 df-ndx 15860 df-slot 15861 df-base 15863 df-sets 15864 df-ress 15865 df-plusg 15954 df-0g 16102 df-gsum 16103 df-mre 16246 df-mrc 16247 df-acs 16249 df-mgm 17242 df-sgrp 17284 df-mnd 17295 df-submnd 17336 df-mulg 17541 df-cntz 17750 df-cmn 18195 df-mgp 18490 df-ur 18502 df-ring 18549 df-cring 18550 |
This theorem is referenced by: (None) |
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