Theorem ringsrg 13155

Description: Any ring is also a semiring. (Contributed by Thierry Arnoux, 1-Apr-2018.)

Assertion

Ref	Expression
ringsrg	|- ( R e. Ring -> R e. SRing )

Proof of Theorem ringsrg

Dummy variables x y z are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		ringcmn 13147	. 2 |- ( R e. Ring -> R e. CMnd )
2		eqid 2177	. . 3 |- (mulGrp ` R ) = (mulGrp ` R )
3	2	ringmgp 13116	. 2 |- ( R e. Ring -> (mulGrp ` R ) e. Mnd )
4		eqid 2177	. . . . 5 |- ( Base ` R ) = ( Base ` R )
5		eqid 2177	. . . . 5 |- ( +g ` R ) = ( +g ` R )
6		eqid 2177	. . . . 5 |- ( .r ` R ) = ( .r ` R )
7	4, 2, 5, 6	isring 13114	. . . 4 |- ( R e. Ring <-> ( R e. Grp /\ (mulGrp ` R ) e. Mnd /\ A. x e. ( Base ` R ) A. y e. ( Base ` R ) A. z e. ( Base ` R ) ( ( x ( .r ` R ) ( y ( +g ` R ) z ) ) = ( ( x ( .r ` R ) y ) ( +g ` R ) ( x ( .r ` R ) z ) ) /\ ( ( x ( +g ` R ) y ) ( .r ` R ) z ) = ( ( x ( .r ` R ) z ) ( +g ` R ) ( y ( .r ` R ) z ) ) ) ) )
8	7	simp3bi 1014	. . 3 |- ( R e. Ring -> A. x e. ( Base ` R ) A. y e. ( Base ` R ) A. z e. ( Base ` R ) ( ( x ( .r ` R ) ( y ( +g ` R ) z ) ) = ( ( x ( .r ` R ) y ) ( +g ` R ) ( x ( .r ` R ) z ) ) /\ ( ( x ( +g ` R ) y ) ( .r ` R ) z ) = ( ( x ( .r ` R ) z ) ( +g ` R ) ( y ( .r ` R ) z ) ) ) )
9		eqid 2177	. . . . . 6 |- ( 0g ` R ) = ( 0g ` R )
10	4, 6, 9	ringlz 13153	. . . . 5 |- ( ( R e. Ring /\ x e. ( Base ` R ) ) -> ( ( 0g ` R ) ( .r ` R ) x ) = ( 0g ` R ) )
11	4, 6, 9	ringrz 13154	. . . . 5 |- ( ( R e. Ring /\ x e. ( Base ` R ) ) -> ( x ( .r ` R ) ( 0g ` R ) ) = ( 0g ` R ) )
12	10, 11	jca 306	. . . 4 |- ( ( R e. Ring /\ x e. ( Base ` R ) ) -> ( ( ( 0g ` R ) ( .r ` R ) x ) = ( 0g ` R ) /\ ( x ( .r ` R ) ( 0g ` R ) ) = ( 0g ` R ) ) )
13	12	ralrimiva 2550	. . 3 |- ( R e. Ring -> A. x e. ( Base ` R ) ( ( ( 0g ` R ) ( .r ` R ) x ) = ( 0g ` R ) /\ ( x ( .r ` R ) ( 0g ` R ) ) = ( 0g ` R ) ) )
14		r19.26 2603	. . 3 |- ( A. x e. ( Base ` R ) ( A. y e. ( Base ` R ) A. z e. ( Base ` R ) ( ( x ( .r ` R ) ( y ( +g ` R ) z ) ) = ( ( x ( .r ` R ) y ) ( +g ` R ) ( x ( .r ` R ) z ) ) /\ ( ( x ( +g ` R ) y ) ( .r ` R ) z ) = ( ( x ( .r ` R ) z ) ( +g ` R ) ( y ( .r ` R ) z ) ) ) /\ ( ( ( 0g ` R ) ( .r ` R ) x ) = ( 0g ` R ) /\ ( x ( .r ` R ) ( 0g ` R ) ) = ( 0g ` R ) ) ) <-> ( A. x e. ( Base ` R ) A. y e. ( Base ` R ) A. z e. ( Base ` R ) ( ( x ( .r ` R ) ( y ( +g ` R ) z ) ) = ( ( x ( .r ` R ) y ) ( +g ` R ) ( x ( .r ` R ) z ) ) /\ ( ( x ( +g ` R ) y ) ( .r ` R ) z ) = ( ( x ( .r ` R ) z ) ( +g ` R ) ( y ( .r ` R ) z ) ) ) /\ A. x e. ( Base ` R ) ( ( ( 0g ` R ) ( .r ` R ) x ) = ( 0g ` R ) /\ ( x ( .r ` R ) ( 0g ` R ) ) = ( 0g ` R ) ) ) )
15	8, 13, 14	sylanbrc 417	. 2 |- ( R e. Ring -> A. x e. ( Base ` R ) ( A. y e. ( Base ` R ) A. z e. ( Base ` R ) ( ( x ( .r ` R ) ( y ( +g ` R ) z ) ) = ( ( x ( .r ` R ) y ) ( +g ` R ) ( x ( .r ` R ) z ) ) /\ ( ( x ( +g ` R ) y ) ( .r ` R ) z ) = ( ( x ( .r ` R ) z ) ( +g ` R ) ( y ( .r ` R ) z ) ) ) /\ ( ( ( 0g ` R ) ( .r ` R ) x ) = ( 0g ` R ) /\ ( x ( .r ` R ) ( 0g ` R ) ) = ( 0g ` R ) ) ) )
16	4, 2, 5, 6, 9	issrg 13079	. 2 |- ( R e. SRing <-> ( R e. CMnd /\ (mulGrp ` R ) e. Mnd /\ A. x e. ( Base ` R ) ( A. y e. ( Base ` R ) A. z e. ( Base ` R ) ( ( x ( .r ` R ) ( y ( +g ` R ) z ) ) = ( ( x ( .r ` R ) y ) ( +g ` R ) ( x ( .r ` R ) z ) ) /\ ( ( x ( +g ` R ) y ) ( .r ` R ) z ) = ( ( x ( .r ` R ) z ) ( +g ` R ) ( y ( .r ` R ) z ) ) ) /\ ( ( ( 0g ` R ) ( .r ` R ) x ) = ( 0g ` R ) /\ ( x ( .r ` R ) ( 0g ` R ) ) = ( 0g ` R ) ) ) ) )
17	1, 3, 15, 16	syl3anbrc 1181	1 |- ( R e. Ring -> R e. SRing )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1353   e. wcel 2148   A.wral 2455   ` cfv 5215  (class class class)co 5872   Basecbs 12454   +g cplusg 12528   .rcmulr 12529   0gc0g 12693   Mndcmnd 12749   Grpcgrp 12809  CMndccmn 13019  mulGrpcmgp 13061  SRingcsrg 13077   Ringcrg 13110

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-coll 4117  ax-sep 4120  ax-pow 4173  ax-pr 4208  ax-un 4432  ax-setind 4535  ax-cnex 7899  ax-resscn 7900  ax-1cn 7901  ax-1re 7902  ax-icn 7903  ax-addcl 7904  ax-addrcl 7905  ax-mulcl 7906  ax-addcom 7908  ax-addass 7910  ax-i2m1 7913  ax-0lt1 7914  ax-0id 7916  ax-rnegex 7917  ax-pre-ltirr 7920  ax-pre-ltadd 7924

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-nel 2443  df-ral 2460  df-rex 2461  df-reu 2462  df-rmo 2463  df-rab 2464  df-v 2739  df-sbc 2963  df-csb 3058  df-dif 3131  df-un 3133  df-in 3135  df-ss 3142  df-nul 3423  df-pw 3577  df-sn 3598  df-pr 3599  df-op 3601  df-uni 3810  df-int 3845  df-iun 3888  df-br 4003  df-opab 4064  df-mpt 4065  df-id 4292  df-xp 4631  df-rel 4632  df-cnv 4633  df-co 4634  df-dm 4635  df-rn 4636  df-res 4637  df-ima 4638  df-iota 5177  df-fun 5217  df-fn 5218  df-f 5219  df-f1 5220  df-fo 5221  df-f1o 5222  df-fv 5223  df-riota 5828  df-ov 5875  df-oprab 5876  df-mpo 5877  df-pnf 7990  df-mnf 7991  df-ltxr 7993  df-inn 8916  df-2 8974  df-3 8975  df-ndx 12457  df-slot 12458  df-base 12460  df-sets 12461  df-plusg 12541  df-mulr 12542  df-0g 12695  df-mgm 12707  df-sgrp 12740  df-mnd 12750  df-grp 12812  df-minusg 12813  df-cmn 13021  df-abl 13022  df-mgp 13062  df-ur 13074  df-srg 13078  df-ring 13112

This theorem is referenced by:  dvdsrcl2  13199  dvdsrid  13200  dvdsrtr  13201  dvdsrmul1  13202  dvdsrneg  13203  dvdsr01  13204  dvdsr02  13205  1unit  13207  opprunitd  13210  crngunit  13211  unitmulcl  13213  unitmulclb  13214  unitgrp  13216  unitabl  13217  unitgrpid  13218  unitsubm  13219  unitinvcl  13223  unitinvinv  13224  ringinvcl  13225  unitlinv  13226  unitrinv  13227  unitnegcl  13230  dvrvald  13234  unitdvcl  13236  dvrid  13237  dvrcan1  13240  dvrcan3  13241  dvreq1  13242  dvrdir  13243  rdivmuldivd  13244  unitpropdg  13248  invrpropdg  13249  subrgdvds  13294  subrguss  13295  subrginv  13296  subrgunit  13298  subrgugrp  13299  subrgintm  13302  dvdsrzring  13362