Theorem ringsrg 14026

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 14012	. 2 |- ( R e. Ring -> R e. CMnd )
2		eqid 2229	. . 3 |- (mulGrp ` R ) = (mulGrp ` R )
3	2	ringmgp 13981	. 2 |- ( R e. Ring -> (mulGrp ` R ) e. Mnd )
4		eqid 2229	. . . . 5 |- ( Base ` R ) = ( Base ` R )
5		eqid 2229	. . . . 5 |- ( +g ` R ) = ( +g ` R )
6		eqid 2229	. . . . 5 |- ( .r ` R ) = ( .r ` R )
7	4, 2, 5, 6	isring 13979	. . . 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 1038	. . 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 2229	. . . . . 6 |- ( 0g ` R ) = ( 0g ` R )
10	4, 6, 9	ringlz 14022	. . . . 5 |- ( ( R e. Ring /\ x e. ( Base ` R ) ) -> ( ( 0g ` R ) ( .r ` R ) x ) = ( 0g ` R ) )
11	4, 6, 9	ringrz 14023	. . . . 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 2603	. . 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 2657	. . 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 13944	. 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 1205	1 |- ( R e. Ring -> R e. SRing )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1395   e. wcel 2200   A.wral 2508   ` cfv 5318  (class class class)co 6007   Basecbs 13048   +g cplusg 13126   .rcmulr 13127   0gc0g 13305   Mndcmnd 13465   Grpcgrp 13549  CMndccmn 13837  mulGrpcmgp 13899  SRingcsrg 13942   Ringcrg 13975

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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-coll 4199  ax-sep 4202  ax-pow 4258  ax-pr 4293  ax-un 4524  ax-setind 4629  ax-cnex 8101  ax-resscn 8102  ax-1cn 8103  ax-1re 8104  ax-icn 8105  ax-addcl 8106  ax-addrcl 8107  ax-mulcl 8108  ax-addcom 8110  ax-addass 8112  ax-i2m1 8115  ax-0lt1 8116  ax-0id 8118  ax-rnegex 8119  ax-pre-ltirr 8122  ax-pre-ltadd 8126

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-nel 2496  df-ral 2513  df-rex 2514  df-reu 2515  df-rmo 2516  df-rab 2517  df-v 2801  df-sbc 3029  df-csb 3125  df-dif 3199  df-un 3201  df-in 3203  df-ss 3210  df-nul 3492  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3889  df-int 3924  df-iun 3967  df-br 4084  df-opab 4146  df-mpt 4147  df-id 4384  df-xp 4725  df-rel 4726  df-cnv 4727  df-co 4728  df-dm 4729  df-rn 4730  df-res 4731  df-ima 4732  df-iota 5278  df-fun 5320  df-fn 5321  df-f 5322  df-f1 5323  df-fo 5324  df-f1o 5325  df-fv 5326  df-riota 5960  df-ov 6010  df-oprab 6011  df-mpo 6012  df-pnf 8194  df-mnf 8195  df-ltxr 8197  df-inn 9122  df-2 9180  df-3 9181  df-ndx 13051  df-slot 13052  df-base 13054  df-sets 13055  df-plusg 13139  df-mulr 13140  df-0g 13307  df-mgm 13405  df-sgrp 13451  df-mnd 13466  df-grp 13552  df-minusg 13553  df-cmn 13839  df-abl 13840  df-mgp 13900  df-ur 13939  df-srg 13943  df-ring 13977

This theorem is referenced by:  qusring2  14045  dvdsrcl2  14079  dvdsrid  14080  dvdsrtr  14081  dvdsrmul1  14082  dvdsrneg  14083  dvdsr01  14084  dvdsr02  14085  1unit  14087  opprunitd  14090  crngunit  14091  unitmulcl  14093  unitmulclb  14094  unitgrp  14096  unitabl  14097  unitgrpid  14098  unitsubm  14099  unitinvcl  14103  unitinvinv  14104  ringinvcl  14105  unitlinv  14106  unitrinv  14107  unitnegcl  14110  dvrvald  14114  unitdvcl  14116  dvrid  14117  dvrcan1  14120  dvrcan3  14121  dvreq1  14122  dvrdir  14123  rdivmuldivd  14124  unitpropdg  14128  invrpropdg  14129  rhmdvdsr  14155  elrhmunit  14157  rhmunitinv  14158  subrgdvds  14215  subrguss  14216  subrginv  14217  subrgunit  14219  subrgugrp  14220  subrgintm  14223  unitrrg  14247  rspsn  14514  cnfldui  14569  dvdsrzring  14583  znunit  14639