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Mirrors > Home > MPE Home > Th. List > srgcl | Structured version Visualization version GIF version |
Description: Closure of the multiplication operation of a semiring. (Contributed by NM, 26-Aug-2011.) (Revised by Mario Carneiro, 6-Jan-2015.) (Revised by Thierry Arnoux, 1-Apr-2018.) |
Ref | Expression |
---|---|
srgcl.b | โข ๐ต = (Baseโ๐ ) |
srgcl.t | โข ยท = (.rโ๐ ) |
Ref | Expression |
---|---|
srgcl | โข ((๐ โ SRing โง ๐ โ ๐ต โง ๐ โ ๐ต) โ (๐ ยท ๐) โ ๐ต) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2736 | . . 3 โข (mulGrpโ๐ ) = (mulGrpโ๐ ) | |
2 | 1 | srgmgp 19791 | . 2 โข (๐ โ SRing โ (mulGrpโ๐ ) โ Mnd) |
3 | srgcl.b | . . . 4 โข ๐ต = (Baseโ๐ ) | |
4 | 1, 3 | mgpbas 19771 | . . 3 โข ๐ต = (Baseโ(mulGrpโ๐ )) |
5 | srgcl.t | . . . 4 โข ยท = (.rโ๐ ) | |
6 | 1, 5 | mgpplusg 19769 | . . 3 โข ยท = (+gโ(mulGrpโ๐ )) |
7 | 4, 6 | mndcl 18438 | . 2 โข (((mulGrpโ๐ ) โ Mnd โง ๐ โ ๐ต โง ๐ โ ๐ต) โ (๐ ยท ๐) โ ๐ต) |
8 | 2, 7 | syl3an1 1163 | 1 โข ((๐ โ SRing โง ๐ โ ๐ต โง ๐ โ ๐ต) โ (๐ ยท ๐) โ ๐ต) |
Colors of variables: wff setvar class |
Syntax hints: โ wi 4 โง w3a 1087 = wceq 1539 โ wcel 2104 โcfv 6458 (class class class)co 7307 Basecbs 16957 .rcmulr 17008 Mndcmnd 18430 mulGrpcmgp 19765 SRingcsrg 19786 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1911 ax-6 1969 ax-7 2009 ax-8 2106 ax-9 2114 ax-10 2135 ax-11 2152 ax-12 2169 ax-ext 2707 ax-sep 5232 ax-nul 5239 ax-pow 5297 ax-pr 5361 ax-un 7620 ax-cnex 10973 ax-resscn 10974 ax-1cn 10975 ax-icn 10976 ax-addcl 10977 ax-addrcl 10978 ax-mulcl 10979 ax-mulrcl 10980 ax-mulcom 10981 ax-addass 10982 ax-mulass 10983 ax-distr 10984 ax-i2m1 10985 ax-1ne0 10986 ax-1rid 10987 ax-rnegex 10988 ax-rrecex 10989 ax-cnre 10990 ax-pre-lttri 10991 ax-pre-lttrn 10992 ax-pre-ltadd 10993 ax-pre-mulgt0 10994 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 846 df-3or 1088 df-3an 1089 df-tru 1542 df-fal 1552 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2887 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-reu 3286 df-rab 3287 df-v 3439 df-sbc 3722 df-csb 3838 df-dif 3895 df-un 3897 df-in 3899 df-ss 3909 df-pss 3911 df-nul 4263 df-if 4466 df-pw 4541 df-sn 4566 df-pr 4568 df-op 4572 df-uni 4845 df-iun 4933 df-br 5082 df-opab 5144 df-mpt 5165 df-tr 5199 df-id 5500 df-eprel 5506 df-po 5514 df-so 5515 df-fr 5555 df-we 5557 df-xp 5606 df-rel 5607 df-cnv 5608 df-co 5609 df-dm 5610 df-rn 5611 df-res 5612 df-ima 5613 df-pred 6217 df-ord 6284 df-on 6285 df-lim 6286 df-suc 6287 df-iota 6410 df-fun 6460 df-fn 6461 df-f 6462 df-f1 6463 df-fo 6464 df-f1o 6465 df-fv 6466 df-riota 7264 df-ov 7310 df-oprab 7311 df-mpo 7312 df-om 7745 df-2nd 7864 df-frecs 8128 df-wrecs 8159 df-recs 8233 df-rdg 8272 df-er 8529 df-en 8765 df-dom 8766 df-sdom 8767 df-pnf 11057 df-mnf 11058 df-xr 11059 df-ltxr 11060 df-le 11061 df-sub 11253 df-neg 11254 df-nn 12020 df-2 12082 df-sets 16910 df-slot 16928 df-ndx 16940 df-base 16958 df-plusg 17020 df-mgm 18371 df-sgrp 18420 df-mnd 18431 df-mgp 19766 df-srg 19787 |
This theorem is referenced by: srgfcl 19796 srgmulgass 19812 srgpcomppsc 19815 srglmhm 19816 srgrmhm 19817 srgbinomlem1 19821 srgbinomlem4 19824 srgbinomlem 19825 slmdmcl 31508 |
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