Theorem ringsrg 13016

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 13008	. 2 |- ( R e. Ring -> R e. CMnd )
2		eqid 2175	. . 3 |- (mulGrp ` R ) = (mulGrp ` R )
3	2	ringmgp 12978	. 2 |- ( R e. Ring -> (mulGrp ` R ) e. Mnd )
4		eqid 2175	. . . . 5 |- ( Base ` R ) = ( Base ` R )
5		eqid 2175	. . . . 5 |- ( +g ` R ) = ( +g ` R )
6		eqid 2175	. . . . 5 |- ( .r ` R ) = ( .r ` R )
7	4, 2, 5, 6	isring 12976	. . . 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 2175	. . . . . 6 |- ( 0g ` R ) = ( 0g ` R )
10	4, 6, 9	ringlz 13014	. . . . 5 |- ( ( R e. Ring /\ x e. ( Base ` R ) ) -> ( ( 0g ` R ) ( .r ` R ) x ) = ( 0g ` R ) )
11	4, 6, 9	ringrz 13015	. . . . 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 2548	. . 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 2601	. . 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 12941	. 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 2146   A.wral 2453   ` cfv 5208  (class class class)co 5865   Basecbs 12427   +g cplusg 12491   .rcmulr 12492   0gc0g 12625   Mndcmnd 12681   Grpcgrp 12737  CMndccmn 12884  mulGrpcmgp 12925  SRingcsrg 12939   Ringcrg 12972

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 1445  ax-7 1446  ax-gen 1447  ax-ie1 1491  ax-ie2 1492  ax-8 1502  ax-10 1503  ax-11 1504  ax-i12 1505  ax-bndl 1507  ax-4 1508  ax-17 1524  ax-i9 1528  ax-ial 1532  ax-i5r 1533  ax-13 2148  ax-14 2149  ax-ext 2157  ax-coll 4113  ax-sep 4116  ax-pow 4169  ax-pr 4203  ax-un 4427  ax-setind 4530  ax-cnex 7877  ax-resscn 7878  ax-1cn 7879  ax-1re 7880  ax-icn 7881  ax-addcl 7882  ax-addrcl 7883  ax-mulcl 7884  ax-addcom 7886  ax-addass 7888  ax-i2m1 7891  ax-0lt1 7892  ax-0id 7894  ax-rnegex 7895  ax-pre-ltirr 7898  ax-pre-ltadd 7902

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1459  df-sb 1761  df-eu 2027  df-mo 2028  df-clab 2162  df-cleq 2168  df-clel 2171  df-nfc 2306  df-ne 2346  df-nel 2441  df-ral 2458  df-rex 2459  df-reu 2460  df-rmo 2461  df-rab 2462  df-v 2737  df-sbc 2961  df-csb 3056  df-dif 3129  df-un 3131  df-in 3133  df-ss 3140  df-nul 3421  df-pw 3574  df-sn 3595  df-pr 3596  df-op 3598  df-uni 3806  df-int 3841  df-iun 3884  df-br 3999  df-opab 4060  df-mpt 4061  df-id 4287  df-xp 4626  df-rel 4627  df-cnv 4628  df-co 4629  df-dm 4630  df-rn 4631  df-res 4632  df-ima 4633  df-iota 5170  df-fun 5210  df-fn 5211  df-f 5212  df-f1 5213  df-fo 5214  df-f1o 5215  df-fv 5216  df-riota 5821  df-ov 5868  df-oprab 5869  df-mpo 5870  df-pnf 7968  df-mnf 7969  df-ltxr 7971  df-inn 8891  df-2 8949  df-3 8950  df-ndx 12430  df-slot 12431  df-base 12433  df-sets 12434  df-plusg 12504  df-mulr 12505  df-0g 12627  df-mgm 12639  df-sgrp 12672  df-mnd 12682  df-grp 12740  df-minusg 12741  df-cmn 12886  df-abl 12887  df-mgp 12926  df-ur 12936  df-srg 12940  df-ring 12974

This theorem is referenced by: (None)