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

Theorem isringd 17107
Description: Properties that determine a ring. (Contributed by NM, 2-Aug-2013.)
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 )
Assertion
Ref Expression
isringd  |-  ( ph  ->  R  e.  Ring )
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 isringd
StepHypRef Expression
1 isringd.g . 2  |-  ( ph  ->  R  e.  Grp )
2 isringd.b . . . 4  |-  ( ph  ->  B  =  ( Base `  R ) )
3 eqid 2443 . . . . 5  |-  (mulGrp `  R )  =  (mulGrp `  R )
4 eqid 2443 . . . . 5  |-  ( Base `  R )  =  (
Base `  R )
53, 4mgpbas 17021 . . . 4  |-  ( Base `  R )  =  (
Base `  (mulGrp `  R
) )
62, 5syl6eq 2500 . . 3  |-  ( ph  ->  B  =  ( Base `  (mulGrp `  R )
) )
7 isringd.t . . . 4  |-  ( ph  ->  .x.  =  ( .r
`  R ) )
8 eqid 2443 . . . . 5  |-  ( .r
`  R )  =  ( .r `  R
)
93, 8mgpplusg 17019 . . . 4  |-  ( .r
`  R )  =  ( +g  `  (mulGrp `  R ) )
107, 9syl6eq 2500 . . 3  |-  ( ph  ->  .x.  =  ( +g  `  (mulGrp `  R )
) )
11 isringd.c . . 3  |-  ( (
ph  /\  x  e.  B  /\  y  e.  B
)  ->  ( x  .x.  y )  e.  B
)
12 isringd.a . . 3  |-  ( (
ph  /\  ( x  e.  B  /\  y  e.  B  /\  z  e.  B ) )  -> 
( ( x  .x.  y )  .x.  z
)  =  ( x 
.x.  ( y  .x.  z ) ) )
13 isringd.u . . 3  |-  ( ph  ->  .1.  e.  B )
14 isringd.i . . 3  |-  ( (
ph  /\  x  e.  B )  ->  (  .1.  .x.  x )  =  x )
15 isringd.h . . 3  |-  ( (
ph  /\  x  e.  B )  ->  (
x  .x.  .1.  )  =  x )
166, 10, 11, 12, 13, 14, 15ismndd 15817 . 2  |-  ( ph  ->  (mulGrp `  R )  e.  Mnd )
172eleq2d 2513 . . . . . 6  |-  ( ph  ->  ( x  e.  B  <->  x  e.  ( Base `  R
) ) )
182eleq2d 2513 . . . . . 6  |-  ( ph  ->  ( y  e.  B  <->  y  e.  ( Base `  R
) ) )
192eleq2d 2513 . . . . . 6  |-  ( ph  ->  ( z  e.  B  <->  z  e.  ( Base `  R
) ) )
2017, 18, 193anbi123d 1300 . . . . 5  |-  ( ph  ->  ( ( x  e.  B  /\  y  e.  B  /\  z  e.  B )  <->  ( x  e.  ( Base `  R
)  /\  y  e.  ( Base `  R )  /\  z  e.  ( Base `  R ) ) ) )
2120biimpar 485 . . . 4  |-  ( (
ph  /\  ( x  e.  ( Base `  R
)  /\  y  e.  ( Base `  R )  /\  z  e.  ( Base `  R ) ) )  ->  ( x  e.  B  /\  y  e.  B  /\  z  e.  B ) )
22 isringd.d . . . . . 6  |-  ( (
ph  /\  ( x  e.  B  /\  y  e.  B  /\  z  e.  B ) )  -> 
( x  .x.  (
y  .+  z )
)  =  ( ( x  .x.  y ) 
.+  ( x  .x.  z ) ) )
237adantr 465 . . . . . . 7  |-  ( (
ph  /\  ( x  e.  B  /\  y  e.  B  /\  z  e.  B ) )  ->  .x.  =  ( .r `  R ) )
24 eqidd 2444 . . . . . . 7  |-  ( (
ph  /\  ( x  e.  B  /\  y  e.  B  /\  z  e.  B ) )  ->  x  =  x )
25 isringd.p . . . . . . . 8  |-  ( ph  ->  .+  =  ( +g  `  R ) )
2625oveqdr 6305 . . . . . . 7  |-  ( (
ph  /\  ( x  e.  B  /\  y  e.  B  /\  z  e.  B ) )  -> 
( y  .+  z
)  =  ( y ( +g  `  R
) z ) )
2723, 24, 26oveq123d 6302 . . . . . 6  |-  ( (
ph  /\  ( x  e.  B  /\  y  e.  B  /\  z  e.  B ) )  -> 
( x  .x.  (
y  .+  z )
)  =  ( x ( .r `  R
) ( y ( +g  `  R ) z ) ) )
2825adantr 465 . . . . . . 7  |-  ( (
ph  /\  ( x  e.  B  /\  y  e.  B  /\  z  e.  B ) )  ->  .+  =  ( +g  `  R ) )
297oveqdr 6305 . . . . . . 7  |-  ( (
ph  /\  ( x  e.  B  /\  y  e.  B  /\  z  e.  B ) )  -> 
( x  .x.  y
)  =  ( x ( .r `  R
) y ) )
307oveqdr 6305 . . . . . . 7  |-  ( (
ph  /\  ( x  e.  B  /\  y  e.  B  /\  z  e.  B ) )  -> 
( x  .x.  z
)  =  ( x ( .r `  R
) z ) )
3128, 29, 30oveq123d 6302 . . . . . 6  |-  ( (
ph  /\  ( x  e.  B  /\  y  e.  B  /\  z  e.  B ) )  -> 
( ( x  .x.  y )  .+  (
x  .x.  z )
)  =  ( ( x ( .r `  R ) y ) ( +g  `  R
) ( x ( .r `  R ) z ) ) )
3222, 27, 313eqtr3d 2492 . . . . 5  |-  ( (
ph  /\  ( x  e.  B  /\  y  e.  B  /\  z  e.  B ) )  -> 
( x ( .r
`  R ) ( y ( +g  `  R
) z ) )  =  ( ( x ( .r `  R
) y ) ( +g  `  R ) ( x ( .r
`  R ) z ) ) )
33 isringd.e . . . . . 6  |-  ( (
ph  /\  ( x  e.  B  /\  y  e.  B  /\  z  e.  B ) )  -> 
( ( x  .+  y )  .x.  z
)  =  ( ( x  .x.  z ) 
.+  ( y  .x.  z ) ) )
3425oveqdr 6305 . . . . . . 7  |-  ( (
ph  /\  ( x  e.  B  /\  y  e.  B  /\  z  e.  B ) )  -> 
( x  .+  y
)  =  ( x ( +g  `  R
) y ) )
35 eqidd 2444 . . . . . . 7  |-  ( (
ph  /\  ( x  e.  B  /\  y  e.  B  /\  z  e.  B ) )  -> 
z  =  z )
3623, 34, 35oveq123d 6302 . . . . . 6  |-  ( (
ph  /\  ( x  e.  B  /\  y  e.  B  /\  z  e.  B ) )  -> 
( ( x  .+  y )  .x.  z
)  =  ( ( x ( +g  `  R
) y ) ( .r `  R ) z ) )
377oveqdr 6305 . . . . . . 7  |-  ( (
ph  /\  ( x  e.  B  /\  y  e.  B  /\  z  e.  B ) )  -> 
( y  .x.  z
)  =  ( y ( .r `  R
) z ) )
3828, 30, 37oveq123d 6302 . . . . . 6  |-  ( (
ph  /\  ( x  e.  B  /\  y  e.  B  /\  z  e.  B ) )  -> 
( ( x  .x.  z )  .+  (
y  .x.  z )
)  =  ( ( x ( .r `  R ) z ) ( +g  `  R
) ( y ( .r `  R ) z ) ) )
3933, 36, 383eqtr3d 2492 . . . . 5  |-  ( (
ph  /\  ( x  e.  B  /\  y  e.  B  /\  z  e.  B ) )  -> 
( ( x ( +g  `  R ) y ) ( .r
`  R ) z )  =  ( ( x ( .r `  R ) z ) ( +g  `  R
) ( y ( .r `  R ) z ) ) )
4032, 39jca 532 . . . 4  |-  ( (
ph  /\  ( x  e.  B  /\  y  e.  B  /\  z  e.  B ) )  -> 
( ( 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 ) ) ) )
4121, 40syldan 470 . . 3  |-  ( (
ph  /\  ( x  e.  ( Base `  R
)  /\  y  e.  ( Base `  R )  /\  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 ) ) ) )
4241ralrimivvva 2865 . 2  |-  ( ph  ->  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 ) ) ) )
43 eqid 2443 . . 3  |-  ( +g  `  R )  =  ( +g  `  R )
444, 3, 43, 8isring 17076 . 2  |-  ( 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 ) ) ) ) )
451, 16, 42, 44syl3anbrc 1181 1  |-  ( ph  ->  R  e.  Ring )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    /\ w3a 974    = wceq 1383    e. wcel 1804   A.wral 2793   ` cfv 5578  (class class class)co 6281   Basecbs 14509   +g cplusg 14574   .rcmulr 14575   Mndcmnd 15793   Grpcgrp 15927  mulGrpcmgp 17015   Ringcrg 17072
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1605  ax-4 1618  ax-5 1691  ax-6 1734  ax-7 1776  ax-8 1806  ax-9 1808  ax-10 1823  ax-11 1828  ax-12 1840  ax-13 1985  ax-ext 2421  ax-sep 4558  ax-nul 4566  ax-pow 4615  ax-pr 4676  ax-un 6577  ax-cnex 9551  ax-resscn 9552  ax-1cn 9553  ax-icn 9554  ax-addcl 9555  ax-addrcl 9556  ax-mulcl 9557  ax-mulrcl 9558  ax-mulcom 9559  ax-addass 9560  ax-mulass 9561  ax-distr 9562  ax-i2m1 9563  ax-1ne0 9564  ax-1rid 9565  ax-rnegex 9566  ax-rrecex 9567  ax-cnre 9568  ax-pre-lttri 9569  ax-pre-lttrn 9570  ax-pre-ltadd 9571  ax-pre-mulgt0 9572
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 975  df-3an 976  df-tru 1386  df-ex 1600  df-nf 1604  df-sb 1727  df-eu 2272  df-mo 2273  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2593  df-ne 2640  df-nel 2641  df-ral 2798  df-rex 2799  df-reu 2800  df-rab 2802  df-v 3097  df-sbc 3314  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3771  df-if 3927  df-pw 3999  df-sn 4015  df-pr 4017  df-tp 4019  df-op 4021  df-uni 4235  df-iun 4317  df-br 4438  df-opab 4496  df-mpt 4497  df-tr 4531  df-eprel 4781  df-id 4785  df-po 4790  df-so 4791  df-fr 4828  df-we 4830  df-ord 4871  df-on 4872  df-lim 4873  df-suc 4874  df-xp 4995  df-rel 4996  df-cnv 4997  df-co 4998  df-dm 4999  df-rn 5000  df-res 5001  df-ima 5002  df-iota 5541  df-fun 5580  df-fn 5581  df-f 5582  df-f1 5583  df-fo 5584  df-f1o 5585  df-fv 5586  df-riota 6242  df-ov 6284  df-oprab 6285  df-mpt2 6286  df-om 6686  df-recs 7044  df-rdg 7078  df-er 7313  df-en 7519  df-dom 7520  df-sdom 7521  df-pnf 9633  df-mnf 9634  df-xr 9635  df-ltxr 9636  df-le 9637  df-sub 9812  df-neg 9813  df-nn 10543  df-2 10600  df-ndx 14512  df-slot 14513  df-base 14514  df-sets 14515  df-plusg 14587  df-mgm 15746  df-sgrp 15785  df-mnd 15795  df-mgp 17016  df-ring 17074
This theorem is referenced by:  iscrngd  17108  imasring  17142  opprring  17154  issubrg2  17323  psrring  17940  matring  18818  mendring  31117  erngdvlem3  36456  erngdvlem3-rN  36464
  Copyright terms: Public domain W3C validator