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Mirrors > Home > MPE Home > Th. List > lmodvsubcl | Structured version Visualization version Unicode version |
Description: Closure of vector subtraction. (hvsubcl 27874 analog.) (Contributed by NM, 31-Mar-2014.) (Revised by Mario Carneiro, 19-Jun-2014.) |
Ref | Expression |
---|---|
lmodvsubcl.v | |
lmodvsubcl.m |
Ref | Expression |
---|---|
lmodvsubcl |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | lmodgrp 18870 | . 2 | |
2 | lmodvsubcl.v | . . 3 | |
3 | lmodvsubcl.m | . . 3 | |
4 | 2, 3 | grpsubcl 17495 | . 2 |
5 | 1, 4 | syl3an1 1359 | 1 |
Colors of variables: wff setvar class |
Syntax hints: wi 4 w3a 1037 wceq 1483 wcel 1990 cfv 5888 (class class class)co 6650 cbs 15857 cgrp 17422 csg 17424 clmod 18863 |
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-lmod 18865 |
This theorem is referenced by: lspsnsub 19007 lvecvscan 19111 ip2subdi 19989 ip2eq 19998 ipcau2 23033 nmparlem 23038 minveclem1 23195 minveclem2 23197 minveclem4 23203 minveclem6 23205 pjthlem1 23208 pjthlem2 23209 eqlkr 34386 lkrlsp 34389 mapdpglem1 36961 mapdpglem2 36962 mapdpglem5N 36966 mapdpglem8 36968 mapdpglem9 36969 mapdpglem13 36973 mapdpglem14 36974 mapdpglem27 36988 baerlem3lem2 36999 baerlem5alem2 37000 baerlem5blem2 37001 mapdheq4lem 37020 mapdh6lem1N 37022 mapdh6lem2N 37023 hdmap1l6lem1 37097 hdmap1l6lem2 37098 hdmap11 37140 hdmapinvlem4 37213 |
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