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Theorem idsrngd 18168
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 2472 . 2  |-  ( ph  ->  ( +g  `  R
)  =  ( +g  `  R ) )
4 eqidd 2472 . 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 crngring 17869 . . 3  |-  ( R  e.  CRing  ->  R  e.  Ring )
97, 8syl 17 . 2  |-  ( ph  ->  R  e.  Ring )
10 idsrngd.i . . . . . 6  |-  ( (
ph  /\  x  e.  B )  ->  (  .*  `  x )  =  x )
1110ralrimiva 2809 . . . . 5  |-  ( ph  ->  A. x  e.  B  (  .*  `  x )  =  x )
1211adantr 472 . . . 4  |-  ( (
ph  /\  a  e.  B )  ->  A. x  e.  B  (  .*  `  x )  =  x )
13 simpr 468 . . . . 5  |-  ( (
ph  /\  a  e.  B )  ->  a  e.  B )
14 simpr 468 . . . . . . 7  |-  ( ( ( ph  /\  a  e.  B )  /\  x  =  a )  ->  x  =  a )
1514fveq2d 5883 . . . . . 6  |-  ( ( ( ph  /\  a  e.  B )  /\  x  =  a )  -> 
(  .*  `  x
)  =  (  .* 
`  a ) )
1615, 14eqeq12d 2486 . . . . 5  |-  ( ( ( ph  /\  a  e.  B )  /\  x  =  a )  -> 
( (  .*  `  x )  =  x  <-> 
(  .*  `  a
)  =  a ) )
1713, 16rspcdv 3139 . . . 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 2549 . 2  |-  ( (
ph  /\  a  e.  B )  ->  (  .*  `  a )  e.  B )
2011adantr 472 . . . . 5  |-  ( (
ph  /\  b  e.  B )  ->  A. x  e.  B  (  .*  `  x )  =  x )
21203adant2 1049 . . . 4  |-  ( (
ph  /\  a  e.  B  /\  b  e.  B
)  ->  A. x  e.  B  (  .*  `  x )  =  x )
22 ringgrp 17863 . . . . . . 7  |-  ( R  e.  Ring  ->  R  e. 
Grp )
239, 22syl 17 . . . . . 6  |-  ( ph  ->  R  e.  Grp )
24 eqid 2471 . . . . . . 7  |-  ( +g  `  R )  =  ( +g  `  R )
251, 24grpcl 16757 . . . . . 6  |-  ( ( R  e.  Grp  /\  a  e.  B  /\  b  e.  B )  ->  ( a ( +g  `  R ) b )  e.  B )
2623, 25syl3an1 1325 . . . . 5  |-  ( (
ph  /\  a  e.  B  /\  b  e.  B
)  ->  ( a
( +g  `  R ) b )  e.  B
)
27 simpr 468 . . . . . . 7  |-  ( ( ( ph  /\  a  e.  B  /\  b  e.  B )  /\  x  =  ( a ( +g  `  R ) b ) )  ->  x  =  ( a
( +g  `  R ) b ) )
2827fveq2d 5883 . . . . . 6  |-  ( ( ( ph  /\  a  e.  B  /\  b  e.  B )  /\  x  =  ( a ( +g  `  R ) b ) )  -> 
(  .*  `  x
)  =  (  .* 
`  ( a ( +g  `  R ) b ) ) )
2928, 27eqeq12d 2486 . . . . 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 3139 . . . 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 1050 . . . 4  |-  ( (
ph  /\  a  e.  B  /\  b  e.  B
)  ->  (  .*  `  a )  =  a )
33 simpr 468 . . . . . . 7  |-  ( (
ph  /\  b  e.  B )  ->  b  e.  B )
34 simpr 468 . . . . . . . . 9  |-  ( ( ( ph  /\  b  e.  B )  /\  x  =  b )  ->  x  =  b )
3534fveq2d 5883 . . . . . . . 8  |-  ( ( ( ph  /\  b  e.  B )  /\  x  =  b )  -> 
(  .*  `  x
)  =  (  .* 
`  b ) )
3635, 34eqeq12d 2486 . . . . . . 7  |-  ( ( ( ph  /\  b  e.  B )  /\  x  =  b )  -> 
( (  .*  `  x )  =  x  <-> 
(  .*  `  b
)  =  b ) )
3733, 36rspcdv 3139 . . . . . 6  |-  ( (
ph  /\  b  e.  B )  ->  ( A. x  e.  B  (  .*  `  x )  =  x  ->  (  .*  `  b )  =  b ) )
3820, 37mpd 15 . . . . 5  |-  ( (
ph  /\  b  e.  B )  ->  (  .*  `  b )  =  b )
39383adant2 1049 . . . 4  |-  ( (
ph  /\  a  e.  B  /\  b  e.  B
)  ->  (  .*  `  b )  =  b )
4032, 39oveq12d 6326 . . 3  |-  ( (
ph  /\  a  e.  B  /\  b  e.  B
)  ->  ( (  .*  `  a ) ( +g  `  R ) (  .*  `  b
) )  =  ( a ( +g  `  R
) b ) )
4131, 40eqtr4d 2508 . 2  |-  ( (
ph  /\  a  e.  B  /\  b  e.  B
)  ->  (  .*  `  ( a ( +g  `  R ) b ) )  =  ( (  .*  `  a ) ( +g  `  R
) (  .*  `  b ) ) )
42 eqid 2471 . . . . 5  |-  ( .r
`  R )  =  ( .r `  R
)
431, 42crngcom 17873 . . . 4  |-  ( ( R  e.  CRing  /\  a  e.  B  /\  b  e.  B )  ->  (
a ( .r `  R ) b )  =  ( b ( .r `  R ) a ) )
447, 43syl3an1 1325 . . 3  |-  ( (
ph  /\  a  e.  B  /\  b  e.  B
)  ->  ( a
( .r `  R
) b )  =  ( b ( .r
`  R ) a ) )
451, 42ringcl 17872 . . . . . 6  |-  ( ( R  e.  Ring  /\  a  e.  B  /\  b  e.  B )  ->  (
a ( .r `  R ) b )  e.  B )
469, 45syl3an1 1325 . . . . 5  |-  ( (
ph  /\  a  e.  B  /\  b  e.  B
)  ->  ( a
( .r `  R
) b )  e.  B )
47 simpr 468 . . . . . . 7  |-  ( ( ( ph  /\  a  e.  B  /\  b  e.  B )  /\  x  =  ( a ( .r `  R ) b ) )  ->  x  =  ( a
( .r `  R
) b ) )
4847fveq2d 5883 . . . . . 6  |-  ( ( ( ph  /\  a  e.  B  /\  b  e.  B )  /\  x  =  ( a ( .r `  R ) b ) )  -> 
(  .*  `  x
)  =  (  .* 
`  ( a ( .r `  R ) b ) ) )
4948, 47eqeq12d 2486 . . . . 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 3139 . . . 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 6326 . . 3  |-  ( (
ph  /\  a  e.  B  /\  b  e.  B
)  ->  ( (  .*  `  b ) ( .r `  R ) (  .*  `  a
) )  =  ( b ( .r `  R ) a ) )
5344, 51, 523eqtr4d 2515 . 2  |-  ( (
ph  /\  a  e.  B  /\  b  e.  B
)  ->  (  .*  `  ( a ( .r
`  R ) b ) )  =  ( (  .*  `  b
) ( .r `  R ) (  .* 
`  a ) ) )
5418fveq2d 5883 . . 3  |-  ( (
ph  /\  a  e.  B )  ->  (  .*  `  (  .*  `  a ) )  =  (  .*  `  a
) )
5554, 18eqtrd 2505 . 2  |-  ( (
ph  /\  a  e.  B )  ->  (  .*  `  (  .*  `  a ) )  =  a )
562, 3, 4, 6, 9, 19, 41, 53, 55issrngd 18167 1  |-  ( ph  ->  R  e.  *Ring )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 376    /\ w3a 1007    = wceq 1452    e. wcel 1904   A.wral 2756   ` cfv 5589  (class class class)co 6308   Basecbs 15199   +g cplusg 15268   .rcmulr 15269   *rcstv 15270   Grpcgrp 16747   Ringcrg 17858   CRingccrg 17859   *Ringcsr 18150
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-8 1906  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-rep 4508  ax-sep 4518  ax-nul 4527  ax-pow 4579  ax-pr 4639  ax-un 6602  ax-cnex 9613  ax-resscn 9614  ax-1cn 9615  ax-icn 9616  ax-addcl 9617  ax-addrcl 9618  ax-mulcl 9619  ax-mulrcl 9620  ax-mulcom 9621  ax-addass 9622  ax-mulass 9623  ax-distr 9624  ax-i2m1 9625  ax-1ne0 9626  ax-1rid 9627  ax-rnegex 9628  ax-rrecex 9629  ax-cnre 9630  ax-pre-lttri 9631  ax-pre-lttrn 9632  ax-pre-ltadd 9633  ax-pre-mulgt0 9634
This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3or 1008  df-3an 1009  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-nel 2644  df-ral 2761  df-rex 2762  df-reu 2763  df-rmo 2764  df-rab 2765  df-v 3033  df-sbc 3256  df-csb 3350  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-pss 3406  df-nul 3723  df-if 3873  df-pw 3944  df-sn 3960  df-pr 3962  df-tp 3964  df-op 3966  df-uni 4191  df-iun 4271  df-br 4396  df-opab 4455  df-mpt 4456  df-tr 4491  df-eprel 4750  df-id 4754  df-po 4760  df-so 4761  df-fr 4798  df-we 4800  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-pred 5387  df-ord 5433  df-on 5434  df-lim 5435  df-suc 5436  df-iota 5553  df-fun 5591  df-fn 5592  df-f 5593  df-f1 5594  df-fo 5595  df-f1o 5596  df-fv 5597  df-riota 6270  df-ov 6311  df-oprab 6312  df-mpt2 6313  df-om 6712  df-tpos 6991  df-wrecs 7046  df-recs 7108  df-rdg 7146  df-er 7381  df-map 7492  df-en 7588  df-dom 7589  df-sdom 7590  df-pnf 9695  df-mnf 9696  df-xr 9697  df-ltxr 9698  df-le 9699  df-sub 9882  df-neg 9883  df-nn 10632  df-2 10690  df-3 10691  df-ndx 15202  df-slot 15203  df-base 15204  df-sets 15205  df-plusg 15281  df-mulr 15282  df-0g 15418  df-mgm 16566  df-sgrp 16605  df-mnd 16615  df-mhm 16660  df-grp 16751  df-ghm 16959  df-cmn 17510  df-mgp 17802  df-ur 17814  df-ring 17860  df-cring 17861  df-oppr 17929  df-rnghom 18021  df-staf 18151  df-srng 18152
This theorem is referenced by:  recrng  19266  frlmphl  19416
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