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.

Step	Hyp	Ref	Expression
1		idsrngd.k	. . 3 |- B = ( Base ` R )
2	1	a1i 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 )
6	5	a1i 11	. 2 |- ( ph -> .* = ( *r ` R ) )
7		idsrngd.r	. . 3 |- ( ph -> R e. CRing )
8		crngrng 16763	. . 3 |- ( R e. CRing -> R e. Ring )
9	7, 8	syl 16	. 2 |- ( ph -> R e. Ring )
10		idsrngd.i	. . . . . 6 |- ( ( ph /\ x e. B ) -> ( .* ` x ) = x )
11	10	ralrimiva 2822	. . . . 5 |- ( ph -> A. x e. B ( .* ` x ) = x )
12	11	adantr 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 )
15	14	fveq2d 5795	. . . . . 6 |- ( ( ( ph /\ a e. B ) /\ x = a ) -> ( .* ` x ) = ( .* ` a ) )
16	15, 14	eqeq12d 2473	. . . . 5 |- ( ( ( ph /\ a e. B ) /\ x = a ) -> ( ( .* ` x ) = x <-> ( .* ` a ) = a ) )
17	13, 16	rspcdv 3174	. . . 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 2539	. 2 |- ( ( ph /\ a e. B ) -> ( .* ` a ) e. B )
20	11	adantr 465	. . . . 5 |- ( ( ph /\ b e. B ) -> A. x e. B ( .* ` x ) = x )
21	20	3adant2 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 )
23	9, 22	syl 16	. . . . . 6 |- ( ph -> R e. Grp )
24		eqid 2451	. . . . . . 7 |- ( +g ` R ) = ( +g ` R )
25	1, 24	grpcl 15655	. . . . . 6 |- ( ( R e. Grp /\ a e. B /\ b e. B ) -> ( a ( +g ` R ) b ) e. B )
26	23, 25	syl3an1 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 ) )
28	27	fveq2d 5795	. . . . . 6 |- ( ( ( ph /\ a e. B /\ b e. B ) /\ x = ( a ( +g ` R ) b ) ) -> ( .* ` x ) = ( .* ` ( a ( +g ` R ) b ) ) )
29	28, 27	eqeq12d 2473	. . . . 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 3174	. . . 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 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 )
35	34	fveq2d 5795	. . . . . . . 8 |- ( ( ( ph /\ b e. B ) /\ x = b ) -> ( .* ` x ) = ( .* ` b ) )
36	35, 34	eqeq12d 2473	. . . . . . 7 |- ( ( ( ph /\ b e. B ) /\ x = b ) -> ( ( .* ` x ) = x <-> ( .* ` b ) = b ) )
37	33, 36	rspcdv 3174	. . . . . 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 1007	. . . 4 |- ( ( ph /\ a e. B /\ b e. B ) -> ( .* ` b ) = b )
40	32, 39	oveq12d 6210	. . 3 |- ( ( ph /\ a e. B /\ b e. B ) -> ( ( .* ` a ) ( +g ` R ) ( .* ` b ) ) = ( a ( +g ` R ) b ) )
41	31, 40	eqtr4d 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 )
43	1, 42	crngcom 16767	. . . 4 |- ( ( R e. CRing /\ a e. B /\ b e. B ) -> ( a ( .r ` R ) b ) = ( b ( .r ` R ) a ) )
44	7, 43	syl3an1 1252	. . 3 |- ( ( ph /\ a e. B /\ b e. B ) -> ( a ( .r ` R ) b ) = ( b ( .r ` R ) a ) )
45	1, 42	rngcl 16766	. . . . . 6 |- ( ( R e. Ring /\ a e. B /\ b e. B ) -> ( a ( .r ` R ) b ) e. B )
46	9, 45	syl3an1 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 ) )
48	47	fveq2d 5795	. . . . . 6 |- ( ( ( ph /\ a e. B /\ b e. B ) /\ x = ( a ( .r ` R ) b ) ) -> ( .* ` x ) = ( .* ` ( a ( .r ` R ) b ) ) )
49	48, 47	eqeq12d 2473	. . . . 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 3174	. . . 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 6210	. . 3 |- ( ( ph /\ a e. B /\ b e. B ) -> ( ( .* ` b ) ( .r ` R ) ( .* ` a ) ) = ( b ( .r ` R ) a ) )
53	44, 51, 52	3eqtr4d 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 ) )
55	54	fveq2d 5795	. . . . . 6 |- ( ( ( ph /\ a e. B ) /\ x = ( .* ` a ) ) -> ( .* ` x ) = ( .* ` ( .* ` a ) ) )
56	55, 54	eqeq12d 2473	. . . . 5 |- ( ( ( ph /\ a e. B ) /\ x = ( .* ` a ) ) -> ( ( .* ` x ) = x <-> ( .* ` ( .* ` a ) ) = ( .* ` a ) ) )
57	19, 56	rspcdv 3174	. . . 4 |- ( ( ph /\ a e. B ) -> ( A. x e. B ( .* ` x ) = x -> ( .* ` ( .* ` a ) ) = ( .* ` a ) ) )
58	12, 57	mpd 15	. . 3 |- ( ( ph /\ a e. B ) -> ( .* ` ( .* ` a ) ) = ( .* ` a ) )
59	58, 18	eqtrd 2492	. 2 |- ( ( ph /\ a e. B ) -> ( .* ` ( .* ` a ) ) = a )
60	2, 3, 4, 6, 9, 19, 41, 53, 59	issrngd 17054	1 |- ( ph -> R e. *Ring )

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