MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  iscrngd Structured version   Unicode version

Theorem iscrngd 17429
Description: Properties that determine a commutative ring. (Contributed by Mario Carneiro, 7-Jan-2015.)
Hypotheses
Ref Expression
isringd.b  |-  ( ph  ->  B  =  ( Base `  R ) )
isringd.p  |-  ( ph  ->  .+  =  ( +g  `  R ) )
isringd.t  |-  ( ph  ->  .x.  =  ( .r
`  R ) )
isringd.g  |-  ( ph  ->  R  e.  Grp )
isringd.c  |-  ( (
ph  /\  x  e.  B  /\  y  e.  B
)  ->  ( x  .x.  y )  e.  B
)
isringd.a  |-  ( (
ph  /\  ( x  e.  B  /\  y  e.  B  /\  z  e.  B ) )  -> 
( ( x  .x.  y )  .x.  z
)  =  ( x 
.x.  ( y  .x.  z ) ) )
isringd.d  |-  ( (
ph  /\  ( x  e.  B  /\  y  e.  B  /\  z  e.  B ) )  -> 
( x  .x.  (
y  .+  z )
)  =  ( ( x  .x.  y ) 
.+  ( x  .x.  z ) ) )
isringd.e  |-  ( (
ph  /\  ( x  e.  B  /\  y  e.  B  /\  z  e.  B ) )  -> 
( ( x  .+  y )  .x.  z
)  =  ( ( x  .x.  z ) 
.+  ( y  .x.  z ) ) )
isringd.u  |-  ( ph  ->  .1.  e.  B )
isringd.i  |-  ( (
ph  /\  x  e.  B )  ->  (  .1.  .x.  x )  =  x )
isringd.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 isringd.b . . 3  |-  ( ph  ->  B  =  ( Base `  R ) )
2 isringd.p . . 3  |-  ( ph  ->  .+  =  ( +g  `  R ) )
3 isringd.t . . 3  |-  ( ph  ->  .x.  =  ( .r
`  R ) )
4 isringd.g . . 3  |-  ( ph  ->  R  e.  Grp )
5 isringd.c . . 3  |-  ( (
ph  /\  x  e.  B  /\  y  e.  B
)  ->  ( x  .x.  y )  e.  B
)
6 isringd.a . . 3  |-  ( (
ph  /\  ( x  e.  B  /\  y  e.  B  /\  z  e.  B ) )  -> 
( ( x  .x.  y )  .x.  z
)  =  ( x 
.x.  ( y  .x.  z ) ) )
7 isringd.d . . 3  |-  ( (
ph  /\  ( x  e.  B  /\  y  e.  B  /\  z  e.  B ) )  -> 
( x  .x.  (
y  .+  z )
)  =  ( ( x  .x.  y ) 
.+  ( x  .x.  z ) ) )
8 isringd.e . . 3  |-  ( (
ph  /\  ( x  e.  B  /\  y  e.  B  /\  z  e.  B ) )  -> 
( ( x  .+  y )  .x.  z
)  =  ( ( x  .x.  z ) 
.+  ( y  .x.  z ) ) )
9 isringd.u . . 3  |-  ( ph  ->  .1.  e.  B )
10 isringd.i . . 3  |-  ( (
ph  /\  x  e.  B )  ->  (  .1.  .x.  x )  =  x )
11 isringd.h . . 3  |-  ( (
ph  /\  x  e.  B )  ->  (
x  .x.  .1.  )  =  x )
121, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11isringd 17428 . 2  |-  ( ph  ->  R  e.  Ring )
13 eqid 2454 . . . . 5  |-  (mulGrp `  R )  =  (mulGrp `  R )
14 eqid 2454 . . . . 5  |-  ( Base `  R )  =  (
Base `  R )
1513, 14mgpbas 17342 . . . 4  |-  ( Base `  R )  =  (
Base `  (mulGrp `  R
) )
161, 15syl6eq 2511 . . 3  |-  ( ph  ->  B  =  ( Base `  (mulGrp `  R )
) )
17 eqid 2454 . . . . 5  |-  ( .r
`  R )  =  ( .r `  R
)
1813, 17mgpplusg 17340 . . . 4  |-  ( .r
`  R )  =  ( +g  `  (mulGrp `  R ) )
193, 18syl6eq 2511 . . 3  |-  ( ph  ->  .x.  =  ( +g  `  (mulGrp `  R )
) )
2016, 19, 5, 6, 9, 10, 11ismndd 16142 . . 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 17009 . 2  |-  ( ph  ->  (mulGrp `  R )  e. CMnd )
2313iscrng 17400 . 2  |-  ( R  e.  CRing 
<->  ( R  e.  Ring  /\  (mulGrp `  R )  e. CMnd ) )
2412, 22, 23sylanbrc 662 1  |-  ( ph  ->  R  e.  CRing )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 367    /\ w3a 971    = wceq 1398    e. wcel 1823   ` cfv 5570  (class class class)co 6270   Basecbs 14716   +g cplusg 14784   .rcmulr 14785   Grpcgrp 16252  CMndccmn 16997  mulGrpcmgp 17336   Ringcrg 17393   CRingccrg 17394
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-un 6565  ax-cnex 9537  ax-resscn 9538  ax-1cn 9539  ax-icn 9540  ax-addcl 9541  ax-addrcl 9542  ax-mulcl 9543  ax-mulrcl 9544  ax-mulcom 9545  ax-addass 9546  ax-mulass 9547  ax-distr 9548  ax-i2m1 9549  ax-1ne0 9550  ax-1rid 9551  ax-rnegex 9552  ax-rrecex 9553  ax-cnre 9554  ax-pre-lttri 9555  ax-pre-lttrn 9556  ax-pre-ltadd 9557  ax-pre-mulgt0 9558
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-nel 2652  df-ral 2809  df-rex 2810  df-reu 2811  df-rab 2813  df-v 3108  df-sbc 3325  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-tp 4021  df-op 4023  df-uni 4236  df-iun 4317  df-br 4440  df-opab 4498  df-mpt 4499  df-tr 4533  df-eprel 4780  df-id 4784  df-po 4789  df-so 4790  df-fr 4827  df-we 4829  df-ord 4870  df-on 4871  df-lim 4872  df-suc 4873  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-f1 5575  df-fo 5576  df-f1o 5577  df-fv 5578  df-riota 6232  df-ov 6273  df-oprab 6274  df-mpt2 6275  df-om 6674  df-recs 7034  df-rdg 7068  df-er 7303  df-en 7510  df-dom 7511  df-sdom 7512  df-pnf 9619  df-mnf 9620  df-xr 9621  df-ltxr 9622  df-le 9623  df-sub 9798  df-neg 9799  df-nn 10532  df-2 10590  df-ndx 14719  df-slot 14720  df-base 14721  df-sets 14722  df-plusg 14797  df-mgm 16071  df-sgrp 16110  df-mnd 16120  df-cmn 16999  df-mgp 17337  df-ring 17395  df-cring 17396
This theorem is referenced by:  cncrng  18634
  Copyright terms: Public domain W3C validator