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.

Step	Hyp	Ref	Expression
1		idsrngd.k	. . 3 |- B = ( Base ` R )
2	1	a1i 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 )
6	5	a1i 11	. 2 |- ( ph -> .* = ( *r ` R ) )
7		idsrngd.r	. . 3 |- ( ph -> R e. CRing )
8		crngring 17341	. . 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 2806	. . . . 5 |- ( ph -> A. x e. B ( .* ` x ) = x )
12	11	adantr 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 )
15	14	fveq2d 5791	. . . . . 6 |- ( ( ( ph /\ a e. B ) /\ x = a ) -> ( .* ` x ) = ( .* ` a ) )
16	15, 14	eqeq12d 2414	. . . . 5 |- ( ( ( ph /\ a e. B ) /\ x = a ) -> ( ( .* ` x ) = x <-> ( .* ` a ) = a ) )
17	13, 16	rspcdv 3151	. . . 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 2480	. 2 |- ( ( ph /\ a e. B ) -> ( .* ` a ) e. B )
20	11	adantr 463	. . . . 5 |- ( ( ph /\ b e. B ) -> A. x e. B ( .* ` x ) = x )
21	20	3adant2 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 )
23	9, 22	syl 16	. . . . . 6 |- ( ph -> R e. Grp )
24		eqid 2392	. . . . . . 7 |- ( +g ` R ) = ( +g ` R )
25	1, 24	grpcl 16199	. . . . . 6 |- ( ( R e. Grp /\ a e. B /\ b e. B ) -> ( a ( +g ` R ) b ) e. B )
26	23, 25	syl3an1 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 ) )
28	27	fveq2d 5791	. . . . . 6 |- ( ( ( ph /\ a e. B /\ b e. B ) /\ x = ( a ( +g ` R ) b ) ) -> ( .* ` x ) = ( .* ` ( a ( +g ` R ) b ) ) )
29	28, 27	eqeq12d 2414	. . . . 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 3151	. . . 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 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 )
35	34	fveq2d 5791	. . . . . . . 8 |- ( ( ( ph /\ b e. B ) /\ x = b ) -> ( .* ` x ) = ( .* ` b ) )
36	35, 34	eqeq12d 2414	. . . . . . 7 |- ( ( ( ph /\ b e. B ) /\ x = b ) -> ( ( .* ` x ) = x <-> ( .* ` b ) = b ) )
37	33, 36	rspcdv 3151	. . . . . 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 1013	. . . 4 |- ( ( ph /\ a e. B /\ b e. B ) -> ( .* ` b ) = b )
40	32, 39	oveq12d 6232	. . 3 |- ( ( ph /\ a e. B /\ b e. B ) -> ( ( .* ` a ) ( +g ` R ) ( .* ` b ) ) = ( a ( +g ` R ) b ) )
41	31, 40	eqtr4d 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 )
43	1, 42	crngcom 17345	. . . 4 |- ( ( R e. CRing /\ a e. B /\ b e. B ) -> ( a ( .r ` R ) b ) = ( b ( .r ` R ) a ) )
44	7, 43	syl3an1 1259	. . 3 |- ( ( ph /\ a e. B /\ b e. B ) -> ( a ( .r ` R ) b ) = ( b ( .r ` R ) a ) )
45	1, 42	ringcl 17344	. . . . . 6 |- ( ( R e. Ring /\ a e. B /\ b e. B ) -> ( a ( .r ` R ) b ) e. B )
46	9, 45	syl3an1 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 ) )
48	47	fveq2d 5791	. . . . . 6 |- ( ( ( ph /\ a e. B /\ b e. B ) /\ x = ( a ( .r ` R ) b ) ) -> ( .* ` x ) = ( .* ` ( a ( .r ` R ) b ) ) )
49	48, 47	eqeq12d 2414	. . . . 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 3151	. . . 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 6232	. . 3 |- ( ( ph /\ a e. B /\ b e. B ) -> ( ( .* ` b ) ( .r ` R ) ( .* ` a ) ) = ( b ( .r ` R ) a ) )
53	44, 51, 52	3eqtr4d 2443	. 2 |- ( ( ph /\ a e. B /\ b e. B ) -> ( .* ` ( a ( .r ` R ) b ) ) = ( ( .* ` b ) ( .r ` R ) ( .* ` a ) ) )
54	18	fveq2d 5791	. . 3 |- ( ( ph /\ a e. B ) -> ( .* ` ( .* ` a ) ) = ( .* ` a ) )
55	54, 18	eqtrd 2433	. 2 |- ( ( ph /\ a e. B ) -> ( .* ` ( .* ` a ) ) = a )
56	2, 3, 4, 6, 9, 19, 41, 53, 55	issrngd 17642	1 |- ( ph -> R e. *Ring )

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