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.

Step	Hyp	Ref	Expression
1		idsrngd.k	. . 3 |- B = ( Base ` R )
2	1	a1i 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 )
6	5	a1i 11	. 2 |- ( ph -> .* = ( *r ` R ) )
7		idsrngd.r	. . 3 |- ( ph -> R e. CRing )
8		crngring 17869	. . 3 |- ( R e. CRing -> R e. Ring )
9	7, 8	syl 17	. 2 |- ( ph -> R e. Ring )
10		idsrngd.i	. . . . . 6 |- ( ( ph /\ x e. B ) -> ( .* ` x ) = x )
11	10	ralrimiva 2809	. . . . 5 |- ( ph -> A. x e. B ( .* ` x ) = x )
12	11	adantr 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 )
15	14	fveq2d 5883	. . . . . 6 |- ( ( ( ph /\ a e. B ) /\ x = a ) -> ( .* ` x ) = ( .* ` a ) )
16	15, 14	eqeq12d 2486	. . . . 5 |- ( ( ( ph /\ a e. B ) /\ x = a ) -> ( ( .* ` x ) = x <-> ( .* ` a ) = a ) )
17	13, 16	rspcdv 3139	. . . 4 |- ( ( ph /\ a e. B ) -> ( A. x e. B ( .* ` x ) = x -> ( .* ` a ) = a ) )
18	12, 17	mpd 15	. . 3 |- ( ( ph /\ a e. B ) -> ( .* ` a ) = a )
19	18, 13	eqeltrd 2549	. 2 |- ( ( ph /\ a e. B ) -> ( .* ` a ) e. B )
20	11	adantr 472	. . . . 5 |- ( ( ph /\ b e. B ) -> A. x e. B ( .* ` x ) = x )
21	20	3adant2 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 )
23	9, 22	syl 17	. . . . . 6 |- ( ph -> R e. Grp )
24		eqid 2471	. . . . . . 7 |- ( +g ` R ) = ( +g ` R )
25	1, 24	grpcl 16757	. . . . . 6 |- ( ( R e. Grp /\ a e. B /\ b e. B ) -> ( a ( +g ` R ) b ) e. B )
26	23, 25	syl3an1 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 ) )
28	27	fveq2d 5883	. . . . . 6 |- ( ( ( ph /\ a e. B /\ b e. B ) /\ x = ( a ( +g ` R ) b ) ) -> ( .* ` x ) = ( .* ` ( a ( +g ` R ) b ) ) )
29	28, 27	eqeq12d 2486	. . . . 5 |- ( ( ( ph /\ a e. B /\ b e. B ) /\ x = ( a ( +g ` R ) b ) ) -> ( ( .* ` x ) = x <-> ( .* ` ( a ( +g ` R ) b ) ) = ( a ( +g ` R ) b ) ) )
30	26, 29	rspcdv 3139	. . . 4 |- ( ( ph /\ a e. B /\ b e. B ) -> ( A. x e. B ( .* ` x ) = x -> ( .* ` ( a ( +g ` R ) b ) ) = ( a ( +g ` R ) b ) ) )
31	21, 30	mpd 15	. . 3 |- ( ( ph /\ a e. B /\ b e. B ) -> ( .* ` ( a ( +g ` R ) b ) ) = ( a ( +g ` R ) b ) )
32	18	3adant3 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 )
35	34	fveq2d 5883	. . . . . . . 8 |- ( ( ( ph /\ b e. B ) /\ x = b ) -> ( .* ` x ) = ( .* ` b ) )
36	35, 34	eqeq12d 2486	. . . . . . 7 |- ( ( ( ph /\ b e. B ) /\ x = b ) -> ( ( .* ` x ) = x <-> ( .* ` b ) = b ) )
37	33, 36	rspcdv 3139	. . . . . 6 |- ( ( ph /\ b e. B ) -> ( A. x e. B ( .* ` x ) = x -> ( .* ` b ) = b ) )
38	20, 37	mpd 15	. . . . 5 |- ( ( ph /\ b e. B ) -> ( .* ` b ) = b )
39	38	3adant2 1049	. . . 4 |- ( ( ph /\ a e. B /\ b e. B ) -> ( .* ` b ) = b )
40	32, 39	oveq12d 6326	. . 3 |- ( ( ph /\ a e. B /\ b e. B ) -> ( ( .* ` a ) ( +g ` R ) ( .* ` b ) ) = ( a ( +g ` R ) b ) )
41	31, 40	eqtr4d 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 )
43	1, 42	crngcom 17873	. . . 4 |- ( ( R e. CRing /\ a e. B /\ b e. B ) -> ( a ( .r ` R ) b ) = ( b ( .r ` R ) a ) )
44	7, 43	syl3an1 1325	. . 3 |- ( ( ph /\ a e. B /\ b e. B ) -> ( a ( .r ` R ) b ) = ( b ( .r ` R ) a ) )
45	1, 42	ringcl 17872	. . . . . 6 |- ( ( R e. Ring /\ a e. B /\ b e. B ) -> ( a ( .r ` R ) b ) e. B )
46	9, 45	syl3an1 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 ) )
48	47	fveq2d 5883	. . . . . 6 |- ( ( ( ph /\ a e. B /\ b e. B ) /\ x = ( a ( .r ` R ) b ) ) -> ( .* ` x ) = ( .* ` ( a ( .r ` R ) b ) ) )
49	48, 47	eqeq12d 2486	. . . . 5 |- ( ( ( ph /\ a e. B /\ b e. B ) /\ x = ( a ( .r ` R ) b ) ) -> ( ( .* ` x ) = x <-> ( .* ` ( a ( .r ` R ) b ) ) = ( a ( .r ` R ) b ) ) )
50	46, 49	rspcdv 3139	. . . 4 |- ( ( ph /\ a e. B /\ b e. B ) -> ( A. x e. B ( .* ` x ) = x -> ( .* ` ( a ( .r ` R ) b ) ) = ( a ( .r ` R ) b ) ) )
51	21, 50	mpd 15	. . 3 |- ( ( ph /\ a e. B /\ b e. B ) -> ( .* ` ( a ( .r ` R ) b ) ) = ( a ( .r ` R ) b ) )
52	39, 32	oveq12d 6326	. . 3 |- ( ( ph /\ a e. B /\ b e. B ) -> ( ( .* ` b ) ( .r ` R ) ( .* ` a ) ) = ( b ( .r ` R ) a ) )
53	44, 51, 52	3eqtr4d 2515	. 2 |- ( ( ph /\ a e. B /\ b e. B ) -> ( .* ` ( a ( .r ` R ) b ) ) = ( ( .* ` b ) ( .r ` R ) ( .* ` a ) ) )
54	18	fveq2d 5883	. . 3 |- ( ( ph /\ a e. B ) -> ( .* ` ( .* ` a ) ) = ( .* ` a ) )
55	54, 18	eqtrd 2505	. 2 |- ( ( ph /\ a e. B ) -> ( .* ` ( .* ` a ) ) = a )
56	2, 3, 4, 6, 9, 19, 41, 53, 55	issrngd 18167	1 |- ( ph -> R e. *Ring )

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