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Theorem isrngd 16669
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 2441 . . . . 5  |-  (mulGrp `  R )  =  (mulGrp `  R )
4 eqid 2441 . . . . 5  |-  ( Base `  R )  =  (
Base `  R )
53, 4mgpbas 16587 . . . 4  |-  ( Base `  R )  =  (
Base `  (mulGrp `  R
) )
62, 5syl6eq 2489 . . 3  |-  ( ph  ->  B  =  ( Base `  (mulGrp `  R )
) )
7 isrngd.t . . . 4  |-  ( ph  ->  .x.  =  ( .r
`  R ) )
8 eqid 2441 . . . . 5  |-  ( .r
`  R )  =  ( .r `  R
)
93, 8mgpplusg 16585 . . . 4  |-  ( .r
`  R )  =  ( +g  `  (mulGrp `  R ) )
107, 9syl6eq 2489 . . 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 15440 . 2  |-  ( ph  ->  (mulGrp `  R )  e.  Mnd )
172eleq2d 2508 . . . . . 6  |-  ( ph  ->  ( x  e.  B  <->  x  e.  ( Base `  R
) ) )
182eleq2d 2508 . . . . . 6  |-  ( ph  ->  ( y  e.  B  <->  y  e.  ( Base `  R
) ) )
192eleq2d 2508 . . . . . 6  |-  ( ph  ->  ( z  e.  B  <->  z  e.  ( Base `  R
) ) )
2017, 18, 193anbi123d 1284 . . . . 5  |-  ( ph  ->  ( ( x  e.  B  /\  y  e.  B  /\  z  e.  B )  <->  ( x  e.  ( Base `  R
)  /\  y  e.  ( Base `  R )  /\  z  e.  ( Base `  R ) ) ) )
2120biimpar 482 . . . 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 462 . . . . . . 7  |-  ( (
ph  /\  ( x  e.  B  /\  y  e.  B  /\  z  e.  B ) )  ->  .x.  =  ( .r `  R ) )
24 eqidd 2442 . . . . . . 7  |-  ( (
ph  /\  ( x  e.  B  /\  y  e.  B  /\  z  e.  B ) )  ->  x  =  x )
25 isrngd.p . . . . . . . 8  |-  ( ph  ->  .+  =  ( +g  `  R ) )
2625proplem3 14625 . . . . . . 7  |-  ( (
ph  /\  ( x  e.  B  /\  y  e.  B  /\  z  e.  B ) )  -> 
( y  .+  z
)  =  ( y ( +g  `  R
) z ) )
2723, 24, 26oveq123d 6111 . . . . . 6  |-  ( (
ph  /\  ( x  e.  B  /\  y  e.  B  /\  z  e.  B ) )  -> 
( x  .x.  (
y  .+  z )
)  =  ( x ( .r `  R
) ( y ( +g  `  R ) z ) ) )
2825adantr 462 . . . . . . 7  |-  ( (
ph  /\  ( x  e.  B  /\  y  e.  B  /\  z  e.  B ) )  ->  .+  =  ( +g  `  R ) )
297proplem3 14625 . . . . . . 7  |-  ( (
ph  /\  ( x  e.  B  /\  y  e.  B  /\  z  e.  B ) )  -> 
( x  .x.  y
)  =  ( x ( .r `  R
) y ) )
307proplem3 14625 . . . . . . 7  |-  ( (
ph  /\  ( x  e.  B  /\  y  e.  B  /\  z  e.  B ) )  -> 
( x  .x.  z
)  =  ( x ( .r `  R
) z ) )
3128, 29, 30oveq123d 6111 . . . . . 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 2481 . . . . 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 14625 . . . . . . 7  |-  ( (
ph  /\  ( x  e.  B  /\  y  e.  B  /\  z  e.  B ) )  -> 
( x  .+  y
)  =  ( x ( +g  `  R
) y ) )
35 eqidd 2442 . . . . . . 7  |-  ( (
ph  /\  ( x  e.  B  /\  y  e.  B  /\  z  e.  B ) )  -> 
z  =  z )
3623, 34, 35oveq123d 6111 . . . . . 6  |-  ( (
ph  /\  ( x  e.  B  /\  y  e.  B  /\  z  e.  B ) )  -> 
( ( x  .+  y )  .x.  z
)  =  ( ( x ( +g  `  R
) y ) ( .r `  R ) z ) )
377proplem3 14625 . . . . . . 7  |-  ( (
ph  /\  ( x  e.  B  /\  y  e.  B  /\  z  e.  B ) )  -> 
( y  .x.  z
)  =  ( y ( .r `  R
) z ) )
3828, 30, 37oveq123d 6111 . . . . . 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 2481 . . . . 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 529 . . . 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 467 . . 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 2807 . 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 2441 . . 3  |-  ( +g  `  R )  =  ( +g  `  R )
444, 3, 43, 8isrng 16639 . 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 1167 1  |-  ( ph  ->  R  e.  Ring )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    /\ w3a 960    = wceq 1364    e. wcel 1761   A.wral 2713   ` cfv 5415  (class class class)co 6090   Basecbs 14170   +g cplusg 14234   .rcmulr 14235   Mndcmnd 15405   Grpcgrp 15406  mulGrpcmgp 16581   Ringcrg 16635
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1713  ax-7 1733  ax-8 1763  ax-9 1765  ax-10 1780  ax-11 1785  ax-12 1797  ax-13 1948  ax-ext 2422  ax-sep 4410  ax-nul 4418  ax-pow 4467  ax-pr 4528  ax-un 6371  ax-cnex 9334  ax-resscn 9335  ax-1cn 9336  ax-icn 9337  ax-addcl 9338  ax-addrcl 9339  ax-mulcl 9340  ax-mulrcl 9341  ax-mulcom 9342  ax-addass 9343  ax-mulass 9344  ax-distr 9345  ax-i2m1 9346  ax-1ne0 9347  ax-1rid 9348  ax-rnegex 9349  ax-rrecex 9350  ax-cnre 9351  ax-pre-lttri 9352  ax-pre-lttrn 9353  ax-pre-ltadd 9354  ax-pre-mulgt0 9355
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 961  df-3an 962  df-tru 1367  df-ex 1592  df-nf 1595  df-sb 1706  df-eu 2261  df-mo 2262  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-nel 2607  df-ral 2718  df-rex 2719  df-reu 2720  df-rab 2722  df-v 2972  df-sbc 3184  df-csb 3286  df-dif 3328  df-un 3330  df-in 3332  df-ss 3339  df-pss 3341  df-nul 3635  df-if 3789  df-pw 3859  df-sn 3875  df-pr 3877  df-tp 3879  df-op 3881  df-uni 4089  df-iun 4170  df-br 4290  df-opab 4348  df-mpt 4349  df-tr 4383  df-eprel 4628  df-id 4632  df-po 4637  df-so 4638  df-fr 4675  df-we 4677  df-ord 4718  df-on 4719  df-lim 4720  df-suc 4721  df-xp 4842  df-rel 4843  df-cnv 4844  df-co 4845  df-dm 4846  df-rn 4847  df-res 4848  df-ima 4849  df-iota 5378  df-fun 5417  df-fn 5418  df-f 5419  df-f1 5420  df-fo 5421  df-f1o 5422  df-fv 5423  df-riota 6049  df-ov 6093  df-oprab 6094  df-mpt2 6095  df-om 6476  df-recs 6828  df-rdg 6862  df-er 7097  df-en 7307  df-dom 7308  df-sdom 7309  df-pnf 9416  df-mnf 9417  df-xr 9418  df-ltxr 9419  df-le 9420  df-sub 9593  df-neg 9594  df-nn 10319  df-2 10376  df-ndx 14173  df-slot 14174  df-base 14175  df-sets 14176  df-plusg 14247  df-mnd 15411  df-mgp 16582  df-rng 16637
This theorem is referenced by:  iscrngd  16670  imasrng  16701  opprrng  16713  issubrg2  16865  psrrng  17461  matrng  18289  mendrng  29474  erngdvlem3  34356  erngdvlem3-rN  34364
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