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Theorem rngsrg 16789
Description: Any ring is also a semiring. (Contributed by Thierry Arnoux, 1-Apr-2018.)
Assertion
Ref Expression
rngsrg  |-  ( R  e.  Ring  ->  R  e. SRing
)

Proof of Theorem rngsrg
Dummy variables  x  y  z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 rngcmn 16781 . . 3  |-  ( R  e.  Ring  ->  R  e. CMnd
)
2 eqid 2451 . . . . . 6  |-  ( Base `  R )  =  (
Base `  R )
3 eqid 2451 . . . . . 6  |-  (mulGrp `  R )  =  (mulGrp `  R )
4 eqid 2451 . . . . . 6  |-  ( +g  `  R )  =  ( +g  `  R )
5 eqid 2451 . . . . . 6  |-  ( .r
`  R )  =  ( .r `  R
)
62, 3, 4, 5isrng 16755 . . . . 5  |-  ( 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 ) ) ) ) )
76biimpi 194 . . . 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 ) ) ) ) )
87simp2d 1001 . . 3  |-  ( R  e.  Ring  ->  (mulGrp `  R )  e.  Mnd )
97simp3d 1002 . . . 4  |-  ( 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 ) ) ) )
10 eqid 2451 . . . . . . 7  |-  ( 0g
`  R )  =  ( 0g `  R
)
112, 5, 10rnglz 16787 . . . . . 6  |-  ( ( R  e.  Ring  /\  x  e.  ( Base `  R
) )  ->  (
( 0g `  R
) ( .r `  R ) x )  =  ( 0g `  R ) )
122, 5, 10rngrz 16788 . . . . . 6  |-  ( ( R  e.  Ring  /\  x  e.  ( Base `  R
) )  ->  (
x ( .r `  R ) ( 0g
`  R ) )  =  ( 0g `  R ) )
1311, 12jca 532 . . . . 5  |-  ( ( R  e.  Ring  /\  x  e.  ( Base `  R
) )  ->  (
( ( 0g `  R ) ( .r
`  R ) x )  =  ( 0g
`  R )  /\  ( x ( .r
`  R ) ( 0g `  R ) )  =  ( 0g
`  R ) ) )
1413ralrimiva 2822 . . . 4  |-  ( R  e.  Ring  ->  A. x  e.  ( Base `  R
) ( ( ( 0g `  R ) ( .r `  R
) x )  =  ( 0g `  R
)  /\  ( x
( .r `  R
) ( 0g `  R ) )  =  ( 0g `  R
) ) )
15 r19.26 2945 . . . 4  |-  ( 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
) ) ) )
169, 14, 15sylanbrc 664 . . 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 ) ) )  /\  ( ( ( 0g `  R ) ( .r `  R
) x )  =  ( 0g `  R
)  /\  ( x
( .r `  R
) ( 0g `  R ) )  =  ( 0g `  R
) ) ) )
171, 8, 163jca 1168 . 2  |-  ( R  e.  Ring  ->  ( 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
) ) ) ) )
182, 3, 4, 5, 10issrg 16714 . 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
) ) ) ) )
1917, 18sylibr 212 1  |-  ( R  e.  Ring  ->  R  e. SRing
)
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    /\ w3a 965    = wceq 1370    e. wcel 1758   A.wral 2795   ` cfv 5516  (class class class)co 6190   Basecbs 14276   +g cplusg 14340   .rcmulr 14341   0gc0g 14480   Mndcmnd 15511   Grpcgrp 15512  CMndccmn 16381  mulGrpcmgp 16696  SRingcsrg 16712   Ringcrg 16751
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-rep 4501  ax-sep 4511  ax-nul 4519  ax-pow 4568  ax-pr 4629  ax-un 6472  ax-cnex 9439  ax-resscn 9440  ax-1cn 9441  ax-icn 9442  ax-addcl 9443  ax-addrcl 9444  ax-mulcl 9445  ax-mulrcl 9446  ax-mulcom 9447  ax-addass 9448  ax-mulass 9449  ax-distr 9450  ax-i2m1 9451  ax-1ne0 9452  ax-1rid 9453  ax-rnegex 9454  ax-rrecex 9455  ax-cnre 9456  ax-pre-lttri 9457  ax-pre-lttrn 9458  ax-pre-ltadd 9459  ax-pre-mulgt0 9460
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-nel 2647  df-ral 2800  df-rex 2801  df-reu 2802  df-rmo 2803  df-rab 2804  df-v 3070  df-sbc 3285  df-csb 3387  df-dif 3429  df-un 3431  df-in 3433  df-ss 3440  df-pss 3442  df-nul 3736  df-if 3890  df-pw 3960  df-sn 3976  df-pr 3978  df-tp 3980  df-op 3982  df-uni 4190  df-iun 4271  df-br 4391  df-opab 4449  df-mpt 4450  df-tr 4484  df-eprel 4730  df-id 4734  df-po 4739  df-so 4740  df-fr 4777  df-we 4779  df-ord 4820  df-on 4821  df-lim 4822  df-suc 4823  df-xp 4944  df-rel 4945  df-cnv 4946  df-co 4947  df-dm 4948  df-rn 4949  df-res 4950  df-ima 4951  df-iota 5479  df-fun 5518  df-fn 5519  df-f 5520  df-f1 5521  df-fo 5522  df-f1o 5523  df-fv 5524  df-riota 6151  df-ov 6193  df-oprab 6194  df-mpt2 6195  df-om 6577  df-recs 6932  df-rdg 6966  df-er 7201  df-en 7411  df-dom 7412  df-sdom 7413  df-pnf 9521  df-mnf 9522  df-xr 9523  df-ltxr 9524  df-le 9525  df-sub 9698  df-neg 9699  df-nn 10424  df-2 10481  df-ndx 14279  df-slot 14280  df-base 14281  df-sets 14282  df-plusg 14353  df-0g 14482  df-mnd 15517  df-grp 15647  df-minusg 15648  df-cmn 16383  df-abl 16384  df-mgp 16697  df-ur 16709  df-srg 16713  df-rng 16753
This theorem is referenced by:  crngbinom  16819  mplcoe5lem  17654  mdet1  18523  lmodslmd  26354
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