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Theorem iscrngd 16668
Description: Properties that determine a commutative ring. (Contributed by Mario Carneiro, 7-Jan-2015.)
Hypotheses
Ref Expression
isrngd.b  |-  ( ph  ->  B  =  ( Base `  R ) )
isrngd.p  |-  ( ph  ->  .+  =  ( +g  `  R ) )
isrngd.t  |-  ( ph  ->  .x.  =  ( .r
`  R ) )
isrngd.g  |-  ( ph  ->  R  e.  Grp )
isrngd.c  |-  ( (
ph  /\  x  e.  B  /\  y  e.  B
)  ->  ( x  .x.  y )  e.  B
)
isrngd.a  |-  ( (
ph  /\  ( x  e.  B  /\  y  e.  B  /\  z  e.  B ) )  -> 
( ( x  .x.  y )  .x.  z
)  =  ( x 
.x.  ( y  .x.  z ) ) )
isrngd.d  |-  ( (
ph  /\  ( x  e.  B  /\  y  e.  B  /\  z  e.  B ) )  -> 
( x  .x.  (
y  .+  z )
)  =  ( ( x  .x.  y ) 
.+  ( x  .x.  z ) ) )
isrngd.e  |-  ( (
ph  /\  ( x  e.  B  /\  y  e.  B  /\  z  e.  B ) )  -> 
( ( x  .+  y )  .x.  z
)  =  ( ( x  .x.  z ) 
.+  ( y  .x.  z ) ) )
isrngd.u  |-  ( ph  ->  .1.  e.  B )
isrngd.i  |-  ( (
ph  /\  x  e.  B )  ->  (  .1.  .x.  x )  =  x )
isrngd.h  |-  ( (
ph  /\  x  e.  B )  ->  (
x  .x.  .1.  )  =  x )
iscrngd.c  |-  ( (
ph  /\  x  e.  B  /\  y  e.  B
)  ->  ( x  .x.  y )  =  ( y  .x.  x ) )
Assertion
Ref Expression
iscrngd  |-  ( ph  ->  R  e.  CRing )
Distinct variable groups:    x,  .1.    x, y, z, B    ph, x, y, z    x, R, y, z
Allowed substitution hints:    .+ ( x, y, z)    .x. ( x, y, z)    .1. ( y, z)

Proof of Theorem iscrngd
StepHypRef Expression
1 isrngd.b . . 3  |-  ( ph  ->  B  =  ( Base `  R ) )
2 isrngd.p . . 3  |-  ( ph  ->  .+  =  ( +g  `  R ) )
3 isrngd.t . . 3  |-  ( ph  ->  .x.  =  ( .r
`  R ) )
4 isrngd.g . . 3  |-  ( ph  ->  R  e.  Grp )
5 isrngd.c . . 3  |-  ( (
ph  /\  x  e.  B  /\  y  e.  B
)  ->  ( x  .x.  y )  e.  B
)
6 isrngd.a . . 3  |-  ( (
ph  /\  ( x  e.  B  /\  y  e.  B  /\  z  e.  B ) )  -> 
( ( x  .x.  y )  .x.  z
)  =  ( x 
.x.  ( y  .x.  z ) ) )
7 isrngd.d . . 3  |-  ( (
ph  /\  ( x  e.  B  /\  y  e.  B  /\  z  e.  B ) )  -> 
( x  .x.  (
y  .+  z )
)  =  ( ( x  .x.  y ) 
.+  ( x  .x.  z ) ) )
8 isrngd.e . . 3  |-  ( (
ph  /\  ( x  e.  B  /\  y  e.  B  /\  z  e.  B ) )  -> 
( ( x  .+  y )  .x.  z
)  =  ( ( x  .x.  z ) 
.+  ( y  .x.  z ) ) )
9 isrngd.u . . 3  |-  ( ph  ->  .1.  e.  B )
10 isrngd.i . . 3  |-  ( (
ph  /\  x  e.  B )  ->  (  .1.  .x.  x )  =  x )
11 isrngd.h . . 3  |-  ( (
ph  /\  x  e.  B )  ->  (
x  .x.  .1.  )  =  x )
121, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11isrngd 16667 . 2  |-  ( ph  ->  R  e.  Ring )
13 eqid 2438 . . . . 5  |-  (mulGrp `  R )  =  (mulGrp `  R )
14 eqid 2438 . . . . 5  |-  ( Base `  R )  =  (
Base `  R )
1513, 14mgpbas 16585 . . . 4  |-  ( Base `  R )  =  (
Base `  (mulGrp `  R
) )
161, 15syl6eq 2486 . . 3  |-  ( ph  ->  B  =  ( Base `  (mulGrp `  R )
) )
17 eqid 2438 . . . . 5  |-  ( .r
`  R )  =  ( .r `  R
)
1813, 17mgpplusg 16583 . . . 4  |-  ( .r
`  R )  =  ( +g  `  (mulGrp `  R ) )
193, 18syl6eq 2486 . . 3  |-  ( ph  ->  .x.  =  ( +g  `  (mulGrp `  R )
) )
2016, 19, 5, 6, 9, 10, 11ismndd 15436 . . 3  |-  ( ph  ->  (mulGrp `  R )  e.  Mnd )
21 iscrngd.c . . 3  |-  ( (
ph  /\  x  e.  B  /\  y  e.  B
)  ->  ( x  .x.  y )  =  ( y  .x.  x ) )
2216, 19, 20, 21iscmnd 16280 . 2  |-  ( ph  ->  (mulGrp `  R )  e. CMnd )
2313iscrng 16640 . 2  |-  ( R  e.  CRing 
<->  ( R  e.  Ring  /\  (mulGrp `  R )  e. CMnd ) )
2412, 22, 23sylanbrc 664 1  |-  ( ph  ->  R  e.  CRing )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    /\ w3a 965    = wceq 1369    e. wcel 1756   ` cfv 5413  (class class class)co 6086   Basecbs 14166   +g cplusg 14230   .rcmulr 14231   Grpcgrp 15402  CMndccmn 16268  mulGrpcmgp 16579   Ringcrg 16633   CRingccrg 16634
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2419  ax-sep 4408  ax-nul 4416  ax-pow 4465  ax-pr 4526  ax-un 6367  ax-cnex 9330  ax-resscn 9331  ax-1cn 9332  ax-icn 9333  ax-addcl 9334  ax-addrcl 9335  ax-mulcl 9336  ax-mulrcl 9337  ax-mulcom 9338  ax-addass 9339  ax-mulass 9340  ax-distr 9341  ax-i2m1 9342  ax-1ne0 9343  ax-1rid 9344  ax-rnegex 9345  ax-rrecex 9346  ax-cnre 9347  ax-pre-lttri 9348  ax-pre-lttrn 9349  ax-pre-ltadd 9350  ax-pre-mulgt0 9351
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2256  df-mo 2257  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-nel 2604  df-ral 2715  df-rex 2716  df-reu 2717  df-rab 2719  df-v 2969  df-sbc 3182  df-csb 3284  df-dif 3326  df-un 3328  df-in 3330  df-ss 3337  df-pss 3339  df-nul 3633  df-if 3787  df-pw 3857  df-sn 3873  df-pr 3875  df-tp 3877  df-op 3879  df-uni 4087  df-iun 4168  df-br 4288  df-opab 4346  df-mpt 4347  df-tr 4381  df-eprel 4627  df-id 4631  df-po 4636  df-so 4637  df-fr 4674  df-we 4676  df-ord 4717  df-on 4718  df-lim 4719  df-suc 4720  df-xp 4841  df-rel 4842  df-cnv 4843  df-co 4844  df-dm 4845  df-rn 4846  df-res 4847  df-ima 4848  df-iota 5376  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-riota 6047  df-ov 6089  df-oprab 6090  df-mpt2 6091  df-om 6472  df-recs 6824  df-rdg 6858  df-er 7093  df-en 7303  df-dom 7304  df-sdom 7305  df-pnf 9412  df-mnf 9413  df-xr 9414  df-ltxr 9415  df-le 9416  df-sub 9589  df-neg 9590  df-nn 10315  df-2 10372  df-ndx 14169  df-slot 14170  df-base 14171  df-sets 14172  df-plusg 14243  df-mnd 15407  df-cmn 16270  df-mgp 16580  df-rng 16635  df-cring 16636
This theorem is referenced by:  cncrng  17812
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