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Theorem idsrngd 17643
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 2393 . 2  |-  ( ph  ->  ( +g  `  R
)  =  ( +g  `  R ) )
4 eqidd 2393 . 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 17341 . . 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 2806 . . . . 5  |-  ( ph  ->  A. x  e.  B  (  .*  `  x )  =  x )
1211adantr 463 . . . 4  |-  ( (
ph  /\  a  e.  B )  ->  A. x  e.  B  (  .*  `  x )  =  x )
13 simpr 459 . . . . 5  |-  ( (
ph  /\  a  e.  B )  ->  a  e.  B )
14 simpr 459 . . . . . . 7  |-  ( ( ( ph  /\  a  e.  B )  /\  x  =  a )  ->  x  =  a )
1514fveq2d 5791 . . . . . 6  |-  ( ( ( ph  /\  a  e.  B )  /\  x  =  a )  -> 
(  .*  `  x
)  =  (  .* 
`  a ) )
1615, 14eqeq12d 2414 . . . . 5  |-  ( ( ( ph  /\  a  e.  B )  /\  x  =  a )  -> 
( (  .*  `  x )  =  x  <-> 
(  .*  `  a
)  =  a ) )
1713, 16rspcdv 3151 . . . 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 2480 . 2  |-  ( (
ph  /\  a  e.  B )  ->  (  .*  `  a )  e.  B )
2011adantr 463 . . . . 5  |-  ( (
ph  /\  b  e.  B )  ->  A. x  e.  B  (  .*  `  x )  =  x )
21203adant2 1013 . . . 4  |-  ( (
ph  /\  a  e.  B  /\  b  e.  B
)  ->  A. x  e.  B  (  .*  `  x )  =  x )
22 ringgrp 17335 . . . . . . 7  |-  ( R  e.  Ring  ->  R  e. 
Grp )
239, 22syl 16 . . . . . 6  |-  ( ph  ->  R  e.  Grp )
24 eqid 2392 . . . . . . 7  |-  ( +g  `  R )  =  ( +g  `  R )
251, 24grpcl 16199 . . . . . 6  |-  ( ( R  e.  Grp  /\  a  e.  B  /\  b  e.  B )  ->  ( a ( +g  `  R ) b )  e.  B )
2623, 25syl3an1 1259 . . . . 5  |-  ( (
ph  /\  a  e.  B  /\  b  e.  B
)  ->  ( a
( +g  `  R ) b )  e.  B
)
27 simpr 459 . . . . . . 7  |-  ( ( ( ph  /\  a  e.  B  /\  b  e.  B )  /\  x  =  ( a ( +g  `  R ) b ) )  ->  x  =  ( a
( +g  `  R ) b ) )
2827fveq2d 5791 . . . . . 6  |-  ( ( ( ph  /\  a  e.  B  /\  b  e.  B )  /\  x  =  ( a ( +g  `  R ) b ) )  -> 
(  .*  `  x
)  =  (  .* 
`  ( a ( +g  `  R ) b ) ) )
2928, 27eqeq12d 2414 . . . . 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 3151 . . . 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 1014 . . . 4  |-  ( (
ph  /\  a  e.  B  /\  b  e.  B
)  ->  (  .*  `  a )  =  a )
33 simpr 459 . . . . . . 7  |-  ( (
ph  /\  b  e.  B )  ->  b  e.  B )
34 simpr 459 . . . . . . . . 9  |-  ( ( ( ph  /\  b  e.  B )  /\  x  =  b )  ->  x  =  b )
3534fveq2d 5791 . . . . . . . 8  |-  ( ( ( ph  /\  b  e.  B )  /\  x  =  b )  -> 
(  .*  `  x
)  =  (  .* 
`  b ) )
3635, 34eqeq12d 2414 . . . . . . 7  |-  ( ( ( ph  /\  b  e.  B )  /\  x  =  b )  -> 
( (  .*  `  x )  =  x  <-> 
(  .*  `  b
)  =  b ) )
3733, 36rspcdv 3151 . . . . . 6  |-  ( (
ph  /\  b  e.  B )  ->  ( A. x  e.  B  (  .*  `  x )  =  x  ->  (  .*  `  b )  =  b ) )
3820, 37mpd 15 . . . . 5  |-  ( (
ph  /\  b  e.  B )  ->  (  .*  `  b )  =  b )
39383adant2 1013 . . . 4  |-  ( (
ph  /\  a  e.  B  /\  b  e.  B
)  ->  (  .*  `  b )  =  b )
4032, 39oveq12d 6232 . . 3  |-  ( (
ph  /\  a  e.  B  /\  b  e.  B
)  ->  ( (  .*  `  a ) ( +g  `  R ) (  .*  `  b
) )  =  ( a ( +g  `  R
) b ) )
4131, 40eqtr4d 2436 . 2  |-  ( (
ph  /\  a  e.  B  /\  b  e.  B
)  ->  (  .*  `  ( a ( +g  `  R ) b ) )  =  ( (  .*  `  a ) ( +g  `  R
) (  .*  `  b ) ) )
42 eqid 2392 . . . . 5  |-  ( .r
`  R )  =  ( .r `  R
)
431, 42crngcom 17345 . . . 4  |-  ( ( R  e.  CRing  /\  a  e.  B  /\  b  e.  B )  ->  (
a ( .r `  R ) b )  =  ( b ( .r `  R ) a ) )
447, 43syl3an1 1259 . . 3  |-  ( (
ph  /\  a  e.  B  /\  b  e.  B
)  ->  ( a
( .r `  R
) b )  =  ( b ( .r
`  R ) a ) )
451, 42ringcl 17344 . . . . . 6  |-  ( ( R  e.  Ring  /\  a  e.  B  /\  b  e.  B )  ->  (
a ( .r `  R ) b )  e.  B )
469, 45syl3an1 1259 . . . . 5  |-  ( (
ph  /\  a  e.  B  /\  b  e.  B
)  ->  ( a
( .r `  R
) b )  e.  B )
47 simpr 459 . . . . . . 7  |-  ( ( ( ph  /\  a  e.  B  /\  b  e.  B )  /\  x  =  ( a ( .r `  R ) b ) )  ->  x  =  ( a
( .r `  R
) b ) )
4847fveq2d 5791 . . . . . 6  |-  ( ( ( ph  /\  a  e.  B  /\  b  e.  B )  /\  x  =  ( a ( .r `  R ) b ) )  -> 
(  .*  `  x
)  =  (  .* 
`  ( a ( .r `  R ) b ) ) )
4948, 47eqeq12d 2414 . . . . 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 3151 . . . 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 6232 . . 3  |-  ( (
ph  /\  a  e.  B  /\  b  e.  B
)  ->  ( (  .*  `  b ) ( .r `  R ) (  .*  `  a
) )  =  ( b ( .r `  R ) a ) )
5344, 51, 523eqtr4d 2443 . 2  |-  ( (
ph  /\  a  e.  B  /\  b  e.  B
)  ->  (  .*  `  ( a ( .r
`  R ) b ) )  =  ( (  .*  `  b
) ( .r `  R ) (  .* 
`  a ) ) )
5418fveq2d 5791 . . 3  |-  ( (
ph  /\  a  e.  B )  ->  (  .*  `  (  .*  `  a ) )  =  (  .*  `  a
) )
5554, 18eqtrd 2433 . 2  |-  ( (
ph  /\  a  e.  B )  ->  (  .*  `  (  .*  `  a ) )  =  a )
562, 3, 4, 6, 9, 19, 41, 53, 55issrngd 17642 1  |-  ( ph  ->  R  e.  *Ring )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 367    /\ w3a 971    = wceq 1399    e. wcel 1836   A.wral 2742   ` cfv 5509  (class class class)co 6214   Basecbs 14653   +g cplusg 14721   .rcmulr 14722   *rcstv 14723   Grpcgrp 16189   Ringcrg 17330   CRingccrg 17331   *Ringcsr 17625
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1633  ax-4 1646  ax-5 1719  ax-6 1765  ax-7 1808  ax-8 1838  ax-9 1840  ax-10 1855  ax-11 1860  ax-12 1872  ax-13 2016  ax-ext 2370  ax-rep 4491  ax-sep 4501  ax-nul 4509  ax-pow 4556  ax-pr 4614  ax-un 6509  ax-cnex 9477  ax-resscn 9478  ax-1cn 9479  ax-icn 9480  ax-addcl 9481  ax-addrcl 9482  ax-mulcl 9483  ax-mulrcl 9484  ax-mulcom 9485  ax-addass 9486  ax-mulass 9487  ax-distr 9488  ax-i2m1 9489  ax-1ne0 9490  ax-1rid 9491  ax-rnegex 9492  ax-rrecex 9493  ax-cnre 9494  ax-pre-lttri 9495  ax-pre-lttrn 9496  ax-pre-ltadd 9497  ax-pre-mulgt0 9498
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1402  df-ex 1628  df-nf 1632  df-sb 1758  df-eu 2232  df-mo 2233  df-clab 2378  df-cleq 2384  df-clel 2387  df-nfc 2542  df-ne 2589  df-nel 2590  df-ral 2747  df-rex 2748  df-reu 2749  df-rmo 2750  df-rab 2751  df-v 3049  df-sbc 3266  df-csb 3362  df-dif 3405  df-un 3407  df-in 3409  df-ss 3416  df-pss 3418  df-nul 3725  df-if 3871  df-pw 3942  df-sn 3958  df-pr 3960  df-tp 3962  df-op 3964  df-uni 4177  df-iun 4258  df-br 4381  df-opab 4439  df-mpt 4440  df-tr 4474  df-eprel 4718  df-id 4722  df-po 4727  df-so 4728  df-fr 4765  df-we 4767  df-ord 4808  df-on 4809  df-lim 4810  df-suc 4811  df-xp 4932  df-rel 4933  df-cnv 4934  df-co 4935  df-dm 4936  df-rn 4937  df-res 4938  df-ima 4939  df-iota 5473  df-fun 5511  df-fn 5512  df-f 5513  df-f1 5514  df-fo 5515  df-f1o 5516  df-fv 5517  df-riota 6176  df-ov 6217  df-oprab 6218  df-mpt2 6219  df-om 6618  df-tpos 6891  df-recs 6978  df-rdg 7012  df-er 7247  df-map 7358  df-en 7454  df-dom 7455  df-sdom 7456  df-pnf 9559  df-mnf 9560  df-xr 9561  df-ltxr 9562  df-le 9563  df-sub 9738  df-neg 9739  df-nn 10471  df-2 10529  df-3 10530  df-ndx 14656  df-slot 14657  df-base 14658  df-sets 14659  df-plusg 14734  df-mulr 14735  df-0g 14868  df-mgm 16008  df-sgrp 16047  df-mnd 16057  df-mhm 16102  df-grp 16193  df-ghm 16401  df-cmn 16936  df-mgp 17274  df-ur 17286  df-ring 17332  df-cring 17333  df-oppr 17404  df-rnghom 17496  df-staf 17626  df-srng 17627
This theorem is referenced by:  recrng  18767  frlmphl  18920
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