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Mirrors > Home > MPE Home > Th. List > ringcld | Structured version Visualization version GIF version |
Description: Closure of the multiplication operation of a ring. (Contributed by SN, 29-Jul-2024.) |
Ref | Expression |
---|---|
ringcld.b | โข ๐ต = (Baseโ๐ ) |
ringcld.t | โข ยท = (.rโ๐ ) |
ringcld.r | โข (๐ โ ๐ โ Ring) |
ringcld.x | โข (๐ โ ๐ โ ๐ต) |
ringcld.y | โข (๐ โ ๐ โ ๐ต) |
Ref | Expression |
---|---|
ringcld | โข (๐ โ (๐ ยท ๐) โ ๐ต) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ringcld.r | . 2 โข (๐ โ ๐ โ Ring) | |
2 | ringcld.x | . 2 โข (๐ โ ๐ โ ๐ต) | |
3 | ringcld.y | . 2 โข (๐ โ ๐ โ ๐ต) | |
4 | ringcld.b | . . 3 โข ๐ต = (Baseโ๐ ) | |
5 | ringcld.t | . . 3 โข ยท = (.rโ๐ ) | |
6 | 4, 5 | ringcl 20197 | . 2 โข ((๐ โ Ring โง ๐ โ ๐ต โง ๐ โ ๐ต) โ (๐ ยท ๐) โ ๐ต) |
7 | 1, 2, 3, 6 | syl3anc 1368 | 1 โข (๐ โ (๐ ยท ๐) โ ๐ต) |
Colors of variables: wff setvar class |
Syntax hints: โ wi 4 = wceq 1533 โ wcel 2098 โcfv 6553 (class class class)co 7426 Basecbs 17187 .rcmulr 17241 Ringcrg 20180 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2699 ax-sep 5303 ax-nul 5310 ax-pow 5369 ax-pr 5433 ax-un 7746 ax-cnex 11202 ax-resscn 11203 ax-1cn 11204 ax-icn 11205 ax-addcl 11206 ax-addrcl 11207 ax-mulcl 11208 ax-mulrcl 11209 ax-mulcom 11210 ax-addass 11211 ax-mulass 11212 ax-distr 11213 ax-i2m1 11214 ax-1ne0 11215 ax-1rid 11216 ax-rnegex 11217 ax-rrecex 11218 ax-cnre 11219 ax-pre-lttri 11220 ax-pre-lttrn 11221 ax-pre-ltadd 11222 ax-pre-mulgt0 11223 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2529 df-eu 2558 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-reu 3375 df-rab 3431 df-v 3475 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-pss 3968 df-nul 4327 df-if 4533 df-pw 4608 df-sn 4633 df-pr 4635 df-op 4639 df-uni 4913 df-iun 5002 df-br 5153 df-opab 5215 df-mpt 5236 df-tr 5270 df-id 5580 df-eprel 5586 df-po 5594 df-so 5595 df-fr 5637 df-we 5639 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-rn 5693 df-res 5694 df-ima 5695 df-pred 6310 df-ord 6377 df-on 6378 df-lim 6379 df-suc 6380 df-iota 6505 df-fun 6555 df-fn 6556 df-f 6557 df-f1 6558 df-fo 6559 df-f1o 6560 df-fv 6561 df-riota 7382 df-ov 7429 df-oprab 7430 df-mpo 7431 df-om 7877 df-2nd 8000 df-frecs 8293 df-wrecs 8324 df-recs 8398 df-rdg 8437 df-er 8731 df-en 8971 df-dom 8972 df-sdom 8973 df-pnf 11288 df-mnf 11289 df-xr 11290 df-ltxr 11291 df-le 11292 df-sub 11484 df-neg 11485 df-nn 12251 df-2 12313 df-sets 17140 df-slot 17158 df-ndx 17170 df-base 17188 df-plusg 17253 df-mgm 18607 df-sgrp 18686 df-mnd 18702 df-mgp 20082 df-ring 20182 |
This theorem is referenced by: gsumdixp 20262 xpsring1d 20276 rngqiprnglin 21199 frlmphl 21722 assapropd 21812 psrmulcllem 21895 psrass1 21914 psrass23l 21917 psrass23 21919 psdmul 22097 evls1fpws 22295 evls1muld 22298 evls1maprhm 22302 mamuass 22322 mamuvs1 22325 mamuvs2 22326 mavmulass 22471 mdetrsca 22525 rrgsubm 32976 erlbr2d 33003 erler 33004 rlocaddval 33007 rlocmulval 33008 rloccring 33009 rlocf1 33012 fracerl 33017 fracfld 33019 dvdsruasso 33114 rhmquskerlem 33165 rhmqusnsg 33168 elrspunsn 33170 mxidlirredi 33209 qsdrngilem 33230 rprmasso2 33268 rprmirredlem 33269 q1pdir 33306 q1pvsca 33307 r1pvsca 33308 r1pcyc 33310 r1padd1 33311 r1pid2 33312 irredminply 33417 aks6d1c1p4 41614 drnginvmuld 41794 rhmmpllem2 41814 rhmcomulmpl 41816 rhmmpl 41817 evlsvvval 41827 evlsbagval 41830 evlsmaprhm 41834 evlmulval 41840 selvvvval 41849 evlselv 41851 selvmul 41853 evlsmhpvvval 41859 mhphf 41861 prjspner1 42081 |
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