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Theorem isrngd 15699
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 2437 . . . . 5  |-  (mulGrp `  R )  =  (mulGrp `  R )
4 eqid 2437 . . . . 5  |-  ( Base `  R )  =  (
Base `  R )
53, 4mgpbas 15655 . . . 4  |-  ( Base `  R )  =  (
Base `  (mulGrp `  R
) )
62, 5syl6eq 2485 . . 3  |-  ( ph  ->  B  =  ( Base `  (mulGrp `  R )
) )
7 isrngd.t . . . 4  |-  ( ph  ->  .x.  =  ( .r
`  R ) )
8 eqid 2437 . . . . 5  |-  ( .r
`  R )  =  ( .r `  R
)
93, 8mgpplusg 15653 . . . 4  |-  ( .r
`  R )  =  ( +g  `  (mulGrp `  R ) )
107, 9syl6eq 2485 . . 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 14720 . 2  |-  ( ph  ->  (mulGrp `  R )  e.  Mnd )
172eleq2d 2504 . . . . . 6  |-  ( ph  ->  ( x  e.  B  <->  x  e.  ( Base `  R
) ) )
182eleq2d 2504 . . . . . 6  |-  ( ph  ->  ( y  e.  B  <->  y  e.  ( Base `  R
) ) )
192eleq2d 2504 . . . . . 6  |-  ( ph  ->  ( z  e.  B  <->  z  e.  ( Base `  R
) ) )
2017, 18, 193anbi123d 1255 . . . . 5  |-  ( ph  ->  ( ( x  e.  B  /\  y  e.  B  /\  z  e.  B )  <->  ( x  e.  ( Base `  R
)  /\  y  e.  ( Base `  R )  /\  z  e.  ( Base `  R ) ) ) )
2120biimpar 473 . . . 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 453 . . . . . . 7  |-  ( (
ph  /\  ( x  e.  B  /\  y  e.  B  /\  z  e.  B ) )  ->  .x.  =  ( .r `  R ) )
24 eqidd 2438 . . . . . . 7  |-  ( (
ph  /\  ( x  e.  B  /\  y  e.  B  /\  z  e.  B ) )  ->  x  =  x )
25 isrngd.p . . . . . . . 8  |-  ( ph  ->  .+  =  ( +g  `  R ) )
2625proplem3 13917 . . . . . . 7  |-  ( (
ph  /\  ( x  e.  B  /\  y  e.  B  /\  z  e.  B ) )  -> 
( y  .+  z
)  =  ( y ( +g  `  R
) z ) )
2723, 24, 26oveq123d 6103 . . . . . 6  |-  ( (
ph  /\  ( x  e.  B  /\  y  e.  B  /\  z  e.  B ) )  -> 
( x  .x.  (
y  .+  z )
)  =  ( x ( .r `  R
) ( y ( +g  `  R ) z ) ) )
2825adantr 453 . . . . . . 7  |-  ( (
ph  /\  ( x  e.  B  /\  y  e.  B  /\  z  e.  B ) )  ->  .+  =  ( +g  `  R ) )
297proplem3 13917 . . . . . . 7  |-  ( (
ph  /\  ( x  e.  B  /\  y  e.  B  /\  z  e.  B ) )  -> 
( x  .x.  y
)  =  ( x ( .r `  R
) y ) )
307proplem3 13917 . . . . . . 7  |-  ( (
ph  /\  ( x  e.  B  /\  y  e.  B  /\  z  e.  B ) )  -> 
( x  .x.  z
)  =  ( x ( .r `  R
) z ) )
3128, 29, 30oveq123d 6103 . . . . . 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 2477 . . . . 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 ) ) )
3425proplem3 13917 . . . . . . 7  |-  ( (
ph  /\  ( x  e.  B  /\  y  e.  B  /\  z  e.  B ) )  -> 
( x  .+  y
)  =  ( x ( +g  `  R
) y ) )
35 eqidd 2438 . . . . . . 7  |-  ( (
ph  /\  ( x  e.  B  /\  y  e.  B  /\  z  e.  B ) )  -> 
z  =  z )
3623, 34, 35oveq123d 6103 . . . . . 6  |-  ( (
ph  /\  ( x  e.  B  /\  y  e.  B  /\  z  e.  B ) )  -> 
( ( x  .+  y )  .x.  z
)  =  ( ( x ( +g  `  R
) y ) ( .r `  R ) z ) )
377proplem3 13917 . . . . . . 7  |-  ( (
ph  /\  ( x  e.  B  /\  y  e.  B  /\  z  e.  B ) )  -> 
( y  .x.  z
)  =  ( y ( .r `  R
) z ) )
3828, 30, 37oveq123d 6103 . . . . . 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 2477 . . . . 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 520 . . . 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 458 . . 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 2800 . 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 2437 . . 3  |-  ( +g  `  R )  =  ( +g  `  R )
444, 3, 43, 8isrng 15669 . 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 1139 1  |-  ( ph  ->  R  e.  Ring )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 360    /\ w3a 937    = wceq 1653    e. wcel 1726   A.wral 2706   ` cfv 5455  (class class class)co 6082   Basecbs 13470   +g cplusg 13530   .rcmulr 13531   Mndcmnd 14685   Grpcgrp 14686  mulGrpcmgp 15649   Ringcrg 15661
This theorem is referenced by:  iscrngd  15700  imasrng  15726  opprrng  15737  issubrg2  15889  psrrng  16475  matrng  27458  mendrng  27478  erngdvlem3  31788  erngdvlem3-rN  31796
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2418  ax-sep 4331  ax-nul 4339  ax-pow 4378  ax-pr 4404  ax-un 4702  ax-cnex 9047  ax-resscn 9048  ax-1cn 9049  ax-icn 9050  ax-addcl 9051  ax-addrcl 9052  ax-mulcl 9053  ax-mulrcl 9054  ax-mulcom 9055  ax-addass 9056  ax-mulass 9057  ax-distr 9058  ax-i2m1 9059  ax-1ne0 9060  ax-1rid 9061  ax-rnegex 9062  ax-rrecex 9063  ax-cnre 9064  ax-pre-lttri 9065  ax-pre-lttrn 9066  ax-pre-ltadd 9067  ax-pre-mulgt0 9068
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2286  df-mo 2287  df-clab 2424  df-cleq 2430  df-clel 2433  df-nfc 2562  df-ne 2602  df-nel 2603  df-ral 2711  df-rex 2712  df-reu 2713  df-rab 2715  df-v 2959  df-sbc 3163  df-csb 3253  df-dif 3324  df-un 3326  df-in 3328  df-ss 3335  df-pss 3337  df-nul 3630  df-if 3741  df-pw 3802  df-sn 3821  df-pr 3822  df-tp 3823  df-op 3824  df-uni 4017  df-iun 4096  df-br 4214  df-opab 4268  df-mpt 4269  df-tr 4304  df-eprel 4495  df-id 4499  df-po 4504  df-so 4505  df-fr 4542  df-we 4544  df-ord 4585  df-on 4586  df-lim 4587  df-suc 4588  df-om 4847  df-xp 4885  df-rel 4886  df-cnv 4887  df-co 4888  df-dm 4889  df-rn 4890  df-res 4891  df-ima 4892  df-iota 5419  df-fun 5457  df-fn 5458  df-f 5459  df-f1 5460  df-fo 5461  df-f1o 5462  df-fv 5463  df-ov 6085  df-oprab 6086  df-mpt2 6087  df-riota 6550  df-recs 6634  df-rdg 6669  df-er 6906  df-en 7111  df-dom 7112  df-sdom 7113  df-pnf 9123  df-mnf 9124  df-xr 9125  df-ltxr 9126  df-le 9127  df-sub 9294  df-neg 9295  df-nn 10002  df-2 10059  df-ndx 13473  df-slot 13474  df-base 13475  df-sets 13476  df-plusg 13543  df-mnd 14691  df-mgp 15650  df-rng 15664
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