Mathbox for Alexander van der Vekens |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > lidlmsgrp | Structured version Visualization version Unicode version |
Description: The multiplicative group of a (left) ideal of a ring is a semigroup. (Contributed by AV, 17-Feb-2020.) |
Ref | Expression |
---|---|
lidlabl.l | LIdeal |
lidlabl.i | ↾s |
Ref | Expression |
---|---|
lidlmsgrp | mulGrp SGrp |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | lidlabl.l | . . 3 LIdeal | |
2 | lidlabl.i | . . 3 ↾s | |
3 | 1, 2 | lidlmmgm 41925 | . 2 mulGrp Mgm |
4 | eqid 2622 | . . . . . . 7 mulGrp mulGrp | |
5 | 4 | ringmgp 18553 | . . . . . 6 mulGrp |
6 | 5 | ad2antrr 762 | . . . . 5 mulGrp |
7 | 1, 2 | lidlssbas 41922 | . . . . . . . . 9 |
8 | 7 | sseld 3602 | . . . . . . . 8 |
9 | 7 | sseld 3602 | . . . . . . . 8 |
10 | 7 | sseld 3602 | . . . . . . . 8 |
11 | 8, 9, 10 | 3anim123d 1406 | . . . . . . 7 |
12 | 11 | adantl 482 | . . . . . 6 |
13 | 12 | imp 445 | . . . . 5 |
14 | eqid 2622 | . . . . . . 7 | |
15 | 4, 14 | mgpbas 18495 | . . . . . 6 mulGrp |
16 | eqid 2622 | . . . . . . 7 | |
17 | 4, 16 | mgpplusg 18493 | . . . . . 6 mulGrp |
18 | 15, 17 | mndass 17302 | . . . . 5 mulGrp |
19 | 6, 13, 18 | syl2anc 693 | . . . 4 |
20 | 19 | ralrimivvva 2972 | . . 3 |
21 | 2, 16 | ressmulr 16006 | . . . . . . . . 9 |
22 | 21 | eqcomd 2628 | . . . . . . . 8 |
23 | 22 | oveqd 6667 | . . . . . . . 8 |
24 | eqidd 2623 | . . . . . . . 8 | |
25 | 22, 23, 24 | oveq123d 6671 | . . . . . . 7 |
26 | eqidd 2623 | . . . . . . . 8 | |
27 | 22 | oveqd 6667 | . . . . . . . 8 |
28 | 22, 26, 27 | oveq123d 6671 | . . . . . . 7 |
29 | 25, 28 | eqeq12d 2637 | . . . . . 6 |
30 | 29 | adantl 482 | . . . . 5 |
31 | 30 | ralbidv 2986 | . . . 4 |
32 | 31 | 2ralbidv 2989 | . . 3 |
33 | 20, 32 | mpbird 247 | . 2 |
34 | eqid 2622 | . . . 4 mulGrp mulGrp | |
35 | eqid 2622 | . . . 4 | |
36 | 34, 35 | mgpbas 18495 | . . 3 mulGrp |
37 | eqid 2622 | . . . 4 | |
38 | 34, 37 | mgpplusg 18493 | . . 3 mulGrp |
39 | 36, 38 | issgrp 17285 | . 2 mulGrp SGrp mulGrp Mgm |
40 | 3, 33, 39 | sylanbrc 698 | 1 mulGrp SGrp |
Colors of variables: wff setvar class |
Syntax hints: wi 4 wb 196 wa 384 w3a 1037 wceq 1483 wcel 1990 wral 2912 cfv 5888 (class class class)co 6650 cbs 15857 ↾s cress 15858 cmulr 15942 Mgmcmgm 17240 SGrpcsgrp 17283 cmnd 17294 mulGrpcmgp 18489 crg 18547 LIdealclidl 19170 |
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-1st 7168 df-2nd 7169 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-3 11080 df-4 11081 df-5 11082 df-6 11083 df-7 11084 df-8 11085 df-ndx 15860 df-slot 15861 df-base 15863 df-sets 15864 df-ress 15865 df-plusg 15954 df-mulr 15955 df-sca 15957 df-vsca 15958 df-ip 15959 df-0g 16102 df-mgm 17242 df-sgrp 17284 df-mnd 17295 df-grp 17425 df-minusg 17426 df-sbg 17427 df-subg 17591 df-mgp 18490 df-ur 18502 df-ring 18549 df-subrg 18778 df-lmod 18865 df-lss 18933 df-sra 19172 df-rgmod 19173 df-lidl 19174 |
This theorem is referenced by: lidlrng 41927 |
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