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Theorem idsrngd 17055
Description: A commutative ring is a star ring when the conjugate operation is the identity. (Contributed by Thierry Arnoux, 19-Apr-2019.)
Hypotheses
Ref Expression
idsrngd.k  |-  B  =  ( Base `  R
)
idsrngd.c  |-  .*  =  ( *r `  R )
idsrngd.r  |-  ( ph  ->  R  e.  CRing )
idsrngd.i  |-  ( (
ph  /\  x  e.  B )  ->  (  .*  `  x )  =  x )
Assertion
Ref Expression
idsrngd  |-  ( ph  ->  R  e.  *Ring )
Distinct variable groups:    x,  .*    x, B    x, R    ph, x

Proof of Theorem idsrngd
Dummy variables  a 
b are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 idsrngd.k . . 3  |-  B  =  ( Base `  R
)
21a1i 11 . 2  |-  ( ph  ->  B  =  ( Base `  R ) )
3 eqidd 2452 . 2  |-  ( ph  ->  ( +g  `  R
)  =  ( +g  `  R ) )
4 eqidd 2452 . 2  |-  ( ph  ->  ( .r `  R
)  =  ( .r
`  R ) )
5 idsrngd.c . . 3  |-  .*  =  ( *r `  R )
65a1i 11 . 2  |-  ( ph  ->  .*  =  ( *r `  R ) )
7 idsrngd.r . . 3  |-  ( ph  ->  R  e.  CRing )
8 crngrng 16763 . . 3  |-  ( R  e.  CRing  ->  R  e.  Ring )
97, 8syl 16 . 2  |-  ( ph  ->  R  e.  Ring )
10 idsrngd.i . . . . . 6  |-  ( (
ph  /\  x  e.  B )  ->  (  .*  `  x )  =  x )
1110ralrimiva 2822 . . . . 5  |-  ( ph  ->  A. x  e.  B  (  .*  `  x )  =  x )
1211adantr 465 . . . 4  |-  ( (
ph  /\  a  e.  B )  ->  A. x  e.  B  (  .*  `  x )  =  x )
13 simpr 461 . . . . 5  |-  ( (
ph  /\  a  e.  B )  ->  a  e.  B )
14 simpr 461 . . . . . . 7  |-  ( ( ( ph  /\  a  e.  B )  /\  x  =  a )  ->  x  =  a )
1514fveq2d 5795 . . . . . 6  |-  ( ( ( ph  /\  a  e.  B )  /\  x  =  a )  -> 
(  .*  `  x
)  =  (  .* 
`  a ) )
1615, 14eqeq12d 2473 . . . . 5  |-  ( ( ( ph  /\  a  e.  B )  /\  x  =  a )  -> 
( (  .*  `  x )  =  x  <-> 
(  .*  `  a
)  =  a ) )
1713, 16rspcdv 3174 . . . 4  |-  ( (
ph  /\  a  e.  B )  ->  ( A. x  e.  B  (  .*  `  x )  =  x  ->  (  .*  `  a )  =  a ) )
1812, 17mpd 15 . . 3  |-  ( (
ph  /\  a  e.  B )  ->  (  .*  `  a )  =  a )
1918, 13eqeltrd 2539 . 2  |-  ( (
ph  /\  a  e.  B )  ->  (  .*  `  a )  e.  B )
2011adantr 465 . . . . 5  |-  ( (
ph  /\  b  e.  B )  ->  A. x  e.  B  (  .*  `  x )  =  x )
21203adant2 1007 . . . 4  |-  ( (
ph  /\  a  e.  B  /\  b  e.  B
)  ->  A. x  e.  B  (  .*  `  x )  =  x )
22 rnggrp 16758 . . . . . . 7  |-  ( R  e.  Ring  ->  R  e. 
Grp )
239, 22syl 16 . . . . . 6  |-  ( ph  ->  R  e.  Grp )
24 eqid 2451 . . . . . . 7  |-  ( +g  `  R )  =  ( +g  `  R )
251, 24grpcl 15655 . . . . . 6  |-  ( ( R  e.  Grp  /\  a  e.  B  /\  b  e.  B )  ->  ( a ( +g  `  R ) b )  e.  B )
2623, 25syl3an1 1252 . . . . 5  |-  ( (
ph  /\  a  e.  B  /\  b  e.  B
)  ->  ( a
( +g  `  R ) b )  e.  B
)
27 simpr 461 . . . . . . 7  |-  ( ( ( ph  /\  a  e.  B  /\  b  e.  B )  /\  x  =  ( a ( +g  `  R ) b ) )  ->  x  =  ( a
( +g  `  R ) b ) )
2827fveq2d 5795 . . . . . 6  |-  ( ( ( ph  /\  a  e.  B  /\  b  e.  B )  /\  x  =  ( a ( +g  `  R ) b ) )  -> 
(  .*  `  x
)  =  (  .* 
`  ( a ( +g  `  R ) b ) ) )
2928, 27eqeq12d 2473 . . . . 5  |-  ( ( ( ph  /\  a  e.  B  /\  b  e.  B )  /\  x  =  ( a ( +g  `  R ) b ) )  -> 
( (  .*  `  x )  =  x  <-> 
(  .*  `  (
a ( +g  `  R
) b ) )  =  ( a ( +g  `  R ) b ) ) )
3026, 29rspcdv 3174 . . . 4  |-  ( (
ph  /\  a  e.  B  /\  b  e.  B
)  ->  ( A. x  e.  B  (  .*  `  x )  =  x  ->  (  .*  `  ( a ( +g  `  R ) b ) )  =  ( a ( +g  `  R
) b ) ) )
3121, 30mpd 15 . . 3  |-  ( (
ph  /\  a  e.  B  /\  b  e.  B
)  ->  (  .*  `  ( a ( +g  `  R ) b ) )  =  ( a ( +g  `  R
) b ) )
32183adant3 1008 . . . 4  |-  ( (
ph  /\  a  e.  B  /\  b  e.  B
)  ->  (  .*  `  a )  =  a )
33 simpr 461 . . . . . . 7  |-  ( (
ph  /\  b  e.  B )  ->  b  e.  B )
34 simpr 461 . . . . . . . . 9  |-  ( ( ( ph  /\  b  e.  B )  /\  x  =  b )  ->  x  =  b )
3534fveq2d 5795 . . . . . . . 8  |-  ( ( ( ph  /\  b  e.  B )  /\  x  =  b )  -> 
(  .*  `  x
)  =  (  .* 
`  b ) )
3635, 34eqeq12d 2473 . . . . . . 7  |-  ( ( ( ph  /\  b  e.  B )  /\  x  =  b )  -> 
( (  .*  `  x )  =  x  <-> 
(  .*  `  b
)  =  b ) )
3733, 36rspcdv 3174 . . . . . 6  |-  ( (
ph  /\  b  e.  B )  ->  ( A. x  e.  B  (  .*  `  x )  =  x  ->  (  .*  `  b )  =  b ) )
3820, 37mpd 15 . . . . 5  |-  ( (
ph  /\  b  e.  B )  ->  (  .*  `  b )  =  b )
39383adant2 1007 . . . 4  |-  ( (
ph  /\  a  e.  B  /\  b  e.  B
)  ->  (  .*  `  b )  =  b )
4032, 39oveq12d 6210 . . 3  |-  ( (
ph  /\  a  e.  B  /\  b  e.  B
)  ->  ( (  .*  `  a ) ( +g  `  R ) (  .*  `  b
) )  =  ( a ( +g  `  R
) b ) )
4131, 40eqtr4d 2495 . 2  |-  ( (
ph  /\  a  e.  B  /\  b  e.  B
)  ->  (  .*  `  ( a ( +g  `  R ) b ) )  =  ( (  .*  `  a ) ( +g  `  R
) (  .*  `  b ) ) )
42 eqid 2451 . . . . 5  |-  ( .r
`  R )  =  ( .r `  R
)
431, 42crngcom 16767 . . . 4  |-  ( ( R  e.  CRing  /\  a  e.  B  /\  b  e.  B )  ->  (
a ( .r `  R ) b )  =  ( b ( .r `  R ) a ) )
447, 43syl3an1 1252 . . 3  |-  ( (
ph  /\  a  e.  B  /\  b  e.  B
)  ->  ( a
( .r `  R
) b )  =  ( b ( .r
`  R ) a ) )
451, 42rngcl 16766 . . . . . 6  |-  ( ( R  e.  Ring  /\  a  e.  B  /\  b  e.  B )  ->  (
a ( .r `  R ) b )  e.  B )
469, 45syl3an1 1252 . . . . 5  |-  ( (
ph  /\  a  e.  B  /\  b  e.  B
)  ->  ( a
( .r `  R
) b )  e.  B )
47 simpr 461 . . . . . . 7  |-  ( ( ( ph  /\  a  e.  B  /\  b  e.  B )  /\  x  =  ( a ( .r `  R ) b ) )  ->  x  =  ( a
( .r `  R
) b ) )
4847fveq2d 5795 . . . . . 6  |-  ( ( ( ph  /\  a  e.  B  /\  b  e.  B )  /\  x  =  ( a ( .r `  R ) b ) )  -> 
(  .*  `  x
)  =  (  .* 
`  ( a ( .r `  R ) b ) ) )
4948, 47eqeq12d 2473 . . . . 5  |-  ( ( ( ph  /\  a  e.  B  /\  b  e.  B )  /\  x  =  ( a ( .r `  R ) b ) )  -> 
( (  .*  `  x )  =  x  <-> 
(  .*  `  (
a ( .r `  R ) b ) )  =  ( a ( .r `  R
) b ) ) )
5046, 49rspcdv 3174 . . . 4  |-  ( (
ph  /\  a  e.  B  /\  b  e.  B
)  ->  ( A. x  e.  B  (  .*  `  x )  =  x  ->  (  .*  `  ( a ( .r
`  R ) b ) )  =  ( a ( .r `  R ) b ) ) )
5121, 50mpd 15 . . 3  |-  ( (
ph  /\  a  e.  B  /\  b  e.  B
)  ->  (  .*  `  ( a ( .r
`  R ) b ) )  =  ( a ( .r `  R ) b ) )
5239, 32oveq12d 6210 . . 3  |-  ( (
ph  /\  a  e.  B  /\  b  e.  B
)  ->  ( (  .*  `  b ) ( .r `  R ) (  .*  `  a
) )  =  ( b ( .r `  R ) a ) )
5344, 51, 523eqtr4d 2502 . 2  |-  ( (
ph  /\  a  e.  B  /\  b  e.  B
)  ->  (  .*  `  ( a ( .r
`  R ) b ) )  =  ( (  .*  `  b
) ( .r `  R ) (  .* 
`  a ) ) )
54 simpr 461 . . . . . . 7  |-  ( ( ( ph  /\  a  e.  B )  /\  x  =  (  .*  `  a
) )  ->  x  =  (  .*  `  a
) )
5554fveq2d 5795 . . . . . 6  |-  ( ( ( ph  /\  a  e.  B )  /\  x  =  (  .*  `  a
) )  ->  (  .*  `  x )  =  (  .*  `  (  .*  `  a ) ) )
5655, 54eqeq12d 2473 . . . . 5  |-  ( ( ( ph  /\  a  e.  B )  /\  x  =  (  .*  `  a
) )  ->  (
(  .*  `  x
)  =  x  <->  (  .*  `  (  .*  `  a
) )  =  (  .*  `  a ) ) )
5719, 56rspcdv 3174 . . . 4  |-  ( (
ph  /\  a  e.  B )  ->  ( A. x  e.  B  (  .*  `  x )  =  x  ->  (  .*  `  (  .*  `  a ) )  =  (  .*  `  a
) ) )
5812, 57mpd 15 . . 3  |-  ( (
ph  /\  a  e.  B )  ->  (  .*  `  (  .*  `  a ) )  =  (  .*  `  a
) )
5958, 18eqtrd 2492 . 2  |-  ( (
ph  /\  a  e.  B )  ->  (  .*  `  (  .*  `  a ) )  =  a )
602, 3, 4, 6, 9, 19, 41, 53, 59issrngd 17054 1  |-  ( ph  ->  R  e.  *Ring )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    /\ w3a 965    = wceq 1370    e. wcel 1758   A.wral 2795   ` cfv 5518  (class class class)co 6192   Basecbs 14278   +g cplusg 14342   .rcmulr 14343   *rcstv 14344   Grpcgrp 15514   Ringcrg 16753   CRingccrg 16754   *Ringcsr 17037
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-rep 4503  ax-sep 4513  ax-nul 4521  ax-pow 4570  ax-pr 4631  ax-un 6474  ax-cnex 9441  ax-resscn 9442  ax-1cn 9443  ax-icn 9444  ax-addcl 9445  ax-addrcl 9446  ax-mulcl 9447  ax-mulrcl 9448  ax-mulcom 9449  ax-addass 9450  ax-mulass 9451  ax-distr 9452  ax-i2m1 9453  ax-1ne0 9454  ax-1rid 9455  ax-rnegex 9456  ax-rrecex 9457  ax-cnre 9458  ax-pre-lttri 9459  ax-pre-lttrn 9460  ax-pre-ltadd 9461  ax-pre-mulgt0 9462
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-nel 2647  df-ral 2800  df-rex 2801  df-reu 2802  df-rmo 2803  df-rab 2804  df-v 3072  df-sbc 3287  df-csb 3389  df-dif 3431  df-un 3433  df-in 3435  df-ss 3442  df-pss 3444  df-nul 3738  df-if 3892  df-pw 3962  df-sn 3978  df-pr 3980  df-tp 3982  df-op 3984  df-uni 4192  df-iun 4273  df-br 4393  df-opab 4451  df-mpt 4452  df-tr 4486  df-eprel 4732  df-id 4736  df-po 4741  df-so 4742  df-fr 4779  df-we 4781  df-ord 4822  df-on 4823  df-lim 4824  df-suc 4825  df-xp 4946  df-rel 4947  df-cnv 4948  df-co 4949  df-dm 4950  df-rn 4951  df-res 4952  df-ima 4953  df-iota 5481  df-fun 5520  df-fn 5521  df-f 5522  df-f1 5523  df-fo 5524  df-f1o 5525  df-fv 5526  df-riota 6153  df-ov 6195  df-oprab 6196  df-mpt2 6197  df-om 6579  df-tpos 6847  df-recs 6934  df-rdg 6968  df-er 7203  df-map 7318  df-en 7413  df-dom 7414  df-sdom 7415  df-pnf 9523  df-mnf 9524  df-xr 9525  df-ltxr 9526  df-le 9527  df-sub 9700  df-neg 9701  df-nn 10426  df-2 10483  df-3 10484  df-ndx 14281  df-slot 14282  df-base 14283  df-sets 14284  df-plusg 14355  df-mulr 14356  df-0g 14484  df-mnd 15519  df-mhm 15568  df-grp 15649  df-ghm 15849  df-cmn 16385  df-mgp 16699  df-ur 16711  df-rng 16755  df-cring 16756  df-oppr 16823  df-rnghom 16914  df-staf 17038  df-srng 17039
This theorem is referenced by:  recrng  18162  frlmphl  18317
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