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Theorem iscrngd 15628
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 15627 . 2  |-  ( ph  ->  R  e.  Ring )
13 eqid 2389 . . . . 5  |-  (mulGrp `  R )  =  (mulGrp `  R )
14 eqid 2389 . . . . 5  |-  ( Base `  R )  =  (
Base `  R )
1513, 14mgpbas 15583 . . . 4  |-  ( Base `  R )  =  (
Base `  (mulGrp `  R
) )
161, 15syl6eq 2437 . . 3  |-  ( ph  ->  B  =  ( Base `  (mulGrp `  R )
) )
17 eqid 2389 . . . . 5  |-  ( .r
`  R )  =  ( .r `  R
)
1813, 17mgpplusg 15581 . . . 4  |-  ( .r
`  R )  =  ( +g  `  (mulGrp `  R ) )
193, 18syl6eq 2437 . . 3  |-  ( ph  ->  .x.  =  ( +g  `  (mulGrp `  R )
) )
2016, 19, 5, 6, 9, 10, 11ismndd 14648 . . 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 15353 . 2  |-  ( ph  ->  (mulGrp `  R )  e. CMnd )
2313iscrng 15600 . 2  |-  ( R  e.  CRing 
<->  ( R  e.  Ring  /\  (mulGrp `  R )  e. CMnd ) )
2412, 22, 23sylanbrc 646 1  |-  ( ph  ->  R  e.  CRing )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    /\ w3a 936    = wceq 1649    e. wcel 1717   ` cfv 5396  (class class class)co 6022   Basecbs 13398   +g cplusg 13458   .rcmulr 13459   Grpcgrp 14614  CMndccmn 15341  mulGrpcmgp 15577   Ringcrg 15589   CRingccrg 15590
This theorem is referenced by:  cncrng  16647
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2370  ax-sep 4273  ax-nul 4281  ax-pow 4320  ax-pr 4346  ax-un 4643  ax-cnex 8981  ax-resscn 8982  ax-1cn 8983  ax-icn 8984  ax-addcl 8985  ax-addrcl 8986  ax-mulcl 8987  ax-mulrcl 8988  ax-mulcom 8989  ax-addass 8990  ax-mulass 8991  ax-distr 8992  ax-i2m1 8993  ax-1ne0 8994  ax-1rid 8995  ax-rnegex 8996  ax-rrecex 8997  ax-cnre 8998  ax-pre-lttri 8999  ax-pre-lttrn 9000  ax-pre-ltadd 9001  ax-pre-mulgt0 9002
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2244  df-mo 2245  df-clab 2376  df-cleq 2382  df-clel 2385  df-nfc 2514  df-ne 2554  df-nel 2555  df-ral 2656  df-rex 2657  df-reu 2658  df-rab 2660  df-v 2903  df-sbc 3107  df-csb 3197  df-dif 3268  df-un 3270  df-in 3272  df-ss 3279  df-pss 3281  df-nul 3574  df-if 3685  df-pw 3746  df-sn 3765  df-pr 3766  df-tp 3767  df-op 3768  df-uni 3960  df-iun 4039  df-br 4156  df-opab 4210  df-mpt 4211  df-tr 4246  df-eprel 4437  df-id 4441  df-po 4446  df-so 4447  df-fr 4484  df-we 4486  df-ord 4527  df-on 4528  df-lim 4529  df-suc 4530  df-om 4788  df-xp 4826  df-rel 4827  df-cnv 4828  df-co 4829  df-dm 4830  df-rn 4831  df-res 4832  df-ima 4833  df-iota 5360  df-fun 5398  df-fn 5399  df-f 5400  df-f1 5401  df-fo 5402  df-f1o 5403  df-fv 5404  df-ov 6025  df-oprab 6026  df-mpt2 6027  df-riota 6487  df-recs 6571  df-rdg 6606  df-er 6843  df-en 7048  df-dom 7049  df-sdom 7050  df-pnf 9057  df-mnf 9058  df-xr 9059  df-ltxr 9060  df-le 9061  df-sub 9227  df-neg 9228  df-nn 9935  df-2 9992  df-ndx 13401  df-slot 13402  df-base 13403  df-sets 13404  df-plusg 13471  df-mnd 14619  df-cmn 15343  df-mgp 15578  df-rng 15592  df-cring 15593
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