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Theorem isrngd 15391
Description: Properties that determine a ring. (Contributed by NM, 2-Aug-2013.)
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 )
Assertion
Ref Expression
isrngd  |-  ( 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 isrngd
StepHypRef Expression
1 isrngd.g . 2  |-  ( ph  ->  R  e.  Grp )
2 isrngd.b . . . 4  |-  ( ph  ->  B  =  ( Base `  R ) )
3 eqid 2296 . . . . 5  |-  (mulGrp `  R )  =  (mulGrp `  R )
4 eqid 2296 . . . . 5  |-  ( Base `  R )  =  (
Base `  R )
53, 4mgpbas 15347 . . . 4  |-  ( Base `  R )  =  (
Base `  (mulGrp `  R
) )
62, 5syl6eq 2344 . . 3  |-  ( ph  ->  B  =  ( Base `  (mulGrp `  R )
) )
7 isrngd.t . . . 4  |-  ( ph  ->  .x.  =  ( .r
`  R ) )
8 eqid 2296 . . . . 5  |-  ( .r
`  R )  =  ( .r `  R
)
93, 8mgpplusg 15345 . . . 4  |-  ( .r
`  R )  =  ( +g  `  (mulGrp `  R ) )
107, 9syl6eq 2344 . . 3  |-  ( ph  ->  .x.  =  ( +g  `  (mulGrp `  R )
) )
11 isrngd.c . . 3  |-  ( (
ph  /\  x  e.  B  /\  y  e.  B
)  ->  ( x  .x.  y )  e.  B
)
12 isrngd.a . . 3  |-  ( (
ph  /\  ( x  e.  B  /\  y  e.  B  /\  z  e.  B ) )  -> 
( ( x  .x.  y )  .x.  z
)  =  ( x 
.x.  ( y  .x.  z ) ) )
13 isrngd.u . . 3  |-  ( ph  ->  .1.  e.  B )
14 isrngd.i . . 3  |-  ( (
ph  /\  x  e.  B )  ->  (  .1.  .x.  x )  =  x )
15 isrngd.h . . 3  |-  ( (
ph  /\  x  e.  B )  ->  (
x  .x.  .1.  )  =  x )
166, 10, 11, 12, 13, 14, 15ismndd 14412 . 2  |-  ( ph  ->  (mulGrp `  R )  e.  Mnd )
172eleq2d 2363 . . . . . 6  |-  ( ph  ->  ( x  e.  B  <->  x  e.  ( Base `  R
) ) )
182eleq2d 2363 . . . . . 6  |-  ( ph  ->  ( y  e.  B  <->  y  e.  ( Base `  R
) ) )
192eleq2d 2363 . . . . . 6  |-  ( ph  ->  ( z  e.  B  <->  z  e.  ( Base `  R
) ) )
2017, 18, 193anbi123d 1252 . . . . 5  |-  ( ph  ->  ( ( x  e.  B  /\  y  e.  B  /\  z  e.  B )  <->  ( x  e.  ( Base `  R
)  /\  y  e.  ( Base `  R )  /\  z  e.  ( Base `  R ) ) ) )
2120biimpar 471 . . . 4  |-  ( (
ph  /\  ( x  e.  ( Base `  R
)  /\  y  e.  ( Base `  R )  /\  z  e.  ( Base `  R ) ) )  ->  ( x  e.  B  /\  y  e.  B  /\  z  e.  B ) )
22 isrngd.d . . . . . 6  |-  ( (
ph  /\  ( x  e.  B  /\  y  e.  B  /\  z  e.  B ) )  -> 
( x  .x.  (
y  .+  z )
)  =  ( ( x  .x.  y ) 
.+  ( x  .x.  z ) ) )
237adantr 451 . . . . . . 7  |-  ( (
ph  /\  ( x  e.  B  /\  y  e.  B  /\  z  e.  B ) )  ->  .x.  =  ( .r `  R ) )
24 eqidd 2297 . . . . . . 7  |-  ( (
ph  /\  ( x  e.  B  /\  y  e.  B  /\  z  e.  B ) )  ->  x  =  x )
25 isrngd.p . . . . . . . . 9  |-  ( ph  ->  .+  =  ( +g  `  R ) )
2625adantr 451 . . . . . . . 8  |-  ( (
ph  /\  ( x  e.  B  /\  y  e.  B  /\  z  e.  B ) )  ->  .+  =  ( +g  `  R ) )
2726oveqd 5891 . . . . . . 7  |-  ( (
ph  /\  ( x  e.  B  /\  y  e.  B  /\  z  e.  B ) )  -> 
( y  .+  z
)  =  ( y ( +g  `  R
) z ) )
2823, 24, 27oveq123d 5895 . . . . . 6  |-  ( (
ph  /\  ( x  e.  B  /\  y  e.  B  /\  z  e.  B ) )  -> 
( x  .x.  (
y  .+  z )
)  =  ( x ( .r `  R
) ( y ( +g  `  R ) z ) ) )
2923oveqd 5891 . . . . . . 7  |-  ( (
ph  /\  ( x  e.  B  /\  y  e.  B  /\  z  e.  B ) )  -> 
( x  .x.  y
)  =  ( x ( .r `  R
) y ) )
3023oveqd 5891 . . . . . . 7  |-  ( (
ph  /\  ( x  e.  B  /\  y  e.  B  /\  z  e.  B ) )  -> 
( x  .x.  z
)  =  ( x ( .r `  R
) z ) )
3126, 29, 30oveq123d 5895 . . . . . 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, 28, 313eqtr3d 2336 . . . . 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 isrngd.e . . . . . 6  |-  ( (
ph  /\  ( x  e.  B  /\  y  e.  B  /\  z  e.  B ) )  -> 
( ( x  .+  y )  .x.  z
)  =  ( ( x  .x.  z ) 
.+  ( y  .x.  z ) ) )
3426oveqd 5891 . . . . . . 7  |-  ( (
ph  /\  ( x  e.  B  /\  y  e.  B  /\  z  e.  B ) )  -> 
( x  .+  y
)  =  ( x ( +g  `  R
) y ) )
35 eqidd 2297 . . . . . . 7  |-  ( (
ph  /\  ( x  e.  B  /\  y  e.  B  /\  z  e.  B ) )  -> 
z  =  z )
3623, 34, 35oveq123d 5895 . . . . . 6  |-  ( (
ph  /\  ( x  e.  B  /\  y  e.  B  /\  z  e.  B ) )  -> 
( ( x  .+  y )  .x.  z
)  =  ( ( x ( +g  `  R
) y ) ( .r `  R ) z ) )
3723oveqd 5891 . . . . . . 7  |-  ( (
ph  /\  ( x  e.  B  /\  y  e.  B  /\  z  e.  B ) )  -> 
( y  .x.  z
)  =  ( y ( .r `  R
) z ) )
3826, 30, 37oveq123d 5895 . . . . . 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 2336 . . . . 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 518 . . . 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 456 . . 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 2649 . 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 2296 . . 3  |-  ( +g  `  R )  =  ( +g  `  R )
444, 3, 43, 8isrng 15361 . 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 1136 1  |-  ( ph  ->  R  e.  Ring )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    /\ w3a 934    = wceq 1632    e. wcel 1696   A.wral 2556   ` cfv 5271  (class class class)co 5874   Basecbs 13164   +g cplusg 13224   .rcmulr 13225   Mndcmnd 14377   Grpcgrp 14378  mulGrpcmgp 15341   Ringcrg 15353
This theorem is referenced by:  iscrngd  15392  imasrng  15418  opprrng  15429  issubrg2  15581  psrrng  16171  matrng  27583  mendrng  27603  erngdvlem3  31801  erngdvlem3-rN  31809
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-cnex 8809  ax-resscn 8810  ax-1cn 8811  ax-icn 8812  ax-addcl 8813  ax-addrcl 8814  ax-mulcl 8815  ax-mulrcl 8816  ax-mulcom 8817  ax-addass 8818  ax-mulass 8819  ax-distr 8820  ax-i2m1 8821  ax-1ne0 8822  ax-1rid 8823  ax-rnegex 8824  ax-rrecex 8825  ax-cnre 8826  ax-pre-lttri 8827  ax-pre-lttrn 8828  ax-pre-ltadd 8829  ax-pre-mulgt0 8830
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-riota 6320  df-recs 6404  df-rdg 6439  df-er 6676  df-en 6880  df-dom 6881  df-sdom 6882  df-pnf 8885  df-mnf 8886  df-xr 8887  df-ltxr 8888  df-le 8889  df-sub 9055  df-neg 9056  df-nn 9763  df-2 9820  df-ndx 13167  df-slot 13168  df-base 13169  df-sets 13170  df-plusg 13237  df-mnd 14383  df-mgp 15342  df-rng 15356
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