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Theorem isrngd 15375
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 2283 . . . . 5  |-  (mulGrp `  R )  =  (mulGrp `  R )
4 eqid 2283 . . . . 5  |-  ( Base `  R )  =  (
Base `  R )
53, 4mgpbas 15331 . . . 4  |-  ( Base `  R )  =  (
Base `  (mulGrp `  R
) )
62, 5syl6eq 2331 . . 3  |-  ( ph  ->  B  =  ( Base `  (mulGrp `  R )
) )
7 isrngd.t . . . 4  |-  ( ph  ->  .x.  =  ( .r
`  R ) )
8 eqid 2283 . . . . 5  |-  ( .r
`  R )  =  ( .r `  R
)
93, 8mgpplusg 15329 . . . 4  |-  ( .r
`  R )  =  ( +g  `  (mulGrp `  R ) )
107, 9syl6eq 2331 . . 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 14396 . 2  |-  ( ph  ->  (mulGrp `  R )  e.  Mnd )
172eleq2d 2350 . . . . . 6  |-  ( ph  ->  ( x  e.  B  <->  x  e.  ( Base `  R
) ) )
182eleq2d 2350 . . . . . 6  |-  ( ph  ->  ( y  e.  B  <->  y  e.  ( Base `  R
) ) )
192eleq2d 2350 . . . . . 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 2284 . . . . . . 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 5875 . . . . . . 7  |-  ( (
ph  /\  ( x  e.  B  /\  y  e.  B  /\  z  e.  B ) )  -> 
( y  .+  z
)  =  ( y ( +g  `  R
) z ) )
2823, 24, 27oveq123d 5879 . . . . . 6  |-  ( (
ph  /\  ( x  e.  B  /\  y  e.  B  /\  z  e.  B ) )  -> 
( x  .x.  (
y  .+  z )
)  =  ( x ( .r `  R
) ( y ( +g  `  R ) z ) ) )
2923oveqd 5875 . . . . . . 7  |-  ( (
ph  /\  ( x  e.  B  /\  y  e.  B  /\  z  e.  B ) )  -> 
( x  .x.  y
)  =  ( x ( .r `  R
) y ) )
3023oveqd 5875 . . . . . . 7  |-  ( (
ph  /\  ( x  e.  B  /\  y  e.  B  /\  z  e.  B ) )  -> 
( x  .x.  z
)  =  ( x ( .r `  R
) z ) )
3126, 29, 30oveq123d 5879 . . . . . 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 2323 . . . . 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 5875 . . . . . . 7  |-  ( (
ph  /\  ( x  e.  B  /\  y  e.  B  /\  z  e.  B ) )  -> 
( x  .+  y
)  =  ( x ( +g  `  R
) y ) )
35 eqidd 2284 . . . . . . 7  |-  ( (
ph  /\  ( x  e.  B  /\  y  e.  B  /\  z  e.  B ) )  -> 
z  =  z )
3623, 34, 35oveq123d 5879 . . . . . 6  |-  ( (
ph  /\  ( x  e.  B  /\  y  e.  B  /\  z  e.  B ) )  -> 
( ( x  .+  y )  .x.  z
)  =  ( ( x ( +g  `  R
) y ) ( .r `  R ) z ) )
3723oveqd 5875 . . . . . . 7  |-  ( (
ph  /\  ( x  e.  B  /\  y  e.  B  /\  z  e.  B ) )  -> 
( y  .x.  z
)  =  ( y ( .r `  R
) z ) )
3826, 30, 37oveq123d 5879 . . . . . 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 2323 . . . . 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 2636 . 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 2283 . . 3  |-  ( +g  `  R )  =  ( +g  `  R )
444, 3, 43, 8isrng 15345 . 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 1623    e. wcel 1684   A.wral 2543   ` cfv 5255  (class class class)co 5858   Basecbs 13148   +g cplusg 13208   .rcmulr 13209   Mndcmnd 14361   Grpcgrp 14362  mulGrpcmgp 15325   Ringcrg 15337
This theorem is referenced by:  iscrngd  15376  imasrng  15402  opprrng  15413  issubrg2  15565  psrrng  16155  matrng  27480  mendrng  27500  erngdvlem3  31179  erngdvlem3-rN  31187
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-cnex 8793  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814
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 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-riota 6304  df-recs 6388  df-rdg 6423  df-er 6660  df-en 6864  df-dom 6865  df-sdom 6866  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-nn 9747  df-2 9804  df-ndx 13151  df-slot 13152  df-base 13153  df-sets 13154  df-plusg 13221  df-mnd 14367  df-mgp 15326  df-rng 15340
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