![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
Mirrors > Home > MPE Home > Th. List > lmodacl | Structured version Unicode version |
Description: Closure of ring addition for a left module. (Contributed by NM, 14-Jan-2014.) (Revised by Mario Carneiro, 19-Jun-2014.) |
Ref | Expression |
---|---|
lmodacl.f |
![]() ![]() ![]() ![]() ![]() ![]() ![]() |
lmodacl.k |
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
lmodacl.p |
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
Ref | Expression |
---|---|
lmodacl |
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | lmodacl.f |
. . 3
![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
2 | 1 | lmodfgrp 17075 |
. 2
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
3 | lmodacl.k |
. . 3
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
4 | lmodacl.p |
. . 3
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
5 | 3, 4 | grpcl 15665 |
. 2
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
6 | 2, 5 | syl3an1 1252 |
1
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
Colors of variables: wff setvar class |
Syntax hints: ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1592 ax-4 1603 ax-5 1671 ax-6 1710 ax-7 1730 ax-8 1760 ax-9 1762 ax-10 1777 ax-11 1782 ax-12 1794 ax-13 1954 ax-ext 2431 ax-nul 4524 ax-pow 4573 |
This theorem depends on definitions: df-bi 185 df-or 370 df-an 371 df-3an 967 df-tru 1373 df-ex 1588 df-nf 1591 df-sb 1703 df-eu 2265 df-mo 2266 df-clab 2438 df-cleq 2444 df-clel 2447 df-nfc 2602 df-ne 2647 df-ral 2801 df-rex 2802 df-rab 2805 df-v 3074 df-sbc 3289 df-dif 3434 df-un 3436 df-in 3438 df-ss 3445 df-nul 3741 df-if 3895 df-sn 3981 df-pr 3983 df-op 3987 df-uni 4195 df-br 4396 df-iota 5484 df-fv 5529 df-ov 6198 df-mnd 15529 df-grp 15659 df-rng 16765 df-lmod 17068 |
This theorem is referenced by: lmodcom 17109 lss1d 17162 lspsolvlem 17341 lfladdcl 33035 lshpkrlem5 33078 ldualvsdi2 33108 baerlem5blem1 35673 hgmapadd 35861 |
Copyright terms: Public domain | W3C validator |