Theorem rhmopp 29819

Description: A ring homomorphism is also a ring homomorphism for the opposite rings. (Contributed by Thierry Arnoux, 27-Oct-2017.)

Assertion

Ref	Expression
rhmopp	|- ( F e. ( R RingHom S ) -> F e. ( (oppr ` R ) RingHom (oppr ` S ) ) )

Proof of Theorem rhmopp

Dummy variables x y are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqid 2622	. 2 |- ( Base ` (oppr ` R ) ) = ( Base ` (oppr ` R ) )
2		eqid 2622	. 2 |- ( 1r ` (oppr ` R ) ) = ( 1r ` (oppr ` R ) )
3		eqid 2622	. 2 |- ( 1r ` (oppr ` S ) ) = ( 1r ` (oppr ` S ) )
4		eqid 2622	. 2 |- ( .r ` (oppr ` R ) ) = ( .r ` (oppr ` R ) )
5		eqid 2622	. 2 |- ( .r ` (oppr ` S ) ) = ( .r ` (oppr ` S ) )
6		rhmrcl1 18719	. . 3 |- ( F e. ( R RingHom S ) -> R e. Ring )
7		eqid 2622	. . . 4 |- (oppr ` R ) = (oppr ` R )
8	7	opprringb 18632	. . 3 |- ( R e. Ring <-> (oppr ` R ) e. Ring )
9	6, 8	sylib 208	. 2 |- ( F e. ( R RingHom S ) -> (oppr ` R ) e. Ring )
10		rhmrcl2 18720	. . 3 |- ( F e. ( R RingHom S ) -> S e. Ring )
11		eqid 2622	. . . 4 |- (oppr ` S ) = (oppr ` S )
12	11	opprringb 18632	. . 3 |- ( S e. Ring <-> (oppr ` S ) e. Ring )
13	10, 12	sylib 208	. 2 |- ( F e. ( R RingHom S ) -> (oppr ` S ) e. Ring )
14		eqid 2622	. . . . 5 |- ( 1r ` R ) = ( 1r ` R )
15	7, 14	oppr1 18634	. . . 4 |- ( 1r ` R ) = ( 1r ` (oppr ` R ) )
16	15	eqcomi 2631	. . 3 |- ( 1r ` (oppr ` R ) ) = ( 1r ` R )
17		eqid 2622	. . . . 5 |- ( 1r ` S ) = ( 1r ` S )
18	11, 17	oppr1 18634	. . . 4 |- ( 1r ` S ) = ( 1r ` (oppr ` S ) )
19	18	eqcomi 2631	. . 3 |- ( 1r ` (oppr ` S ) ) = ( 1r ` S )
20	16, 19	rhm1 18730	. 2 |- ( F e. ( R RingHom S ) -> ( F ` ( 1r ` (oppr ` R ) ) ) = ( 1r ` (oppr ` S ) ) )
21		simpl 473	. . . 4 |- ( ( F e. ( R RingHom S ) /\ ( x e. ( Base ` (oppr ` R ) ) /\ y e. ( Base ` (oppr ` R ) ) ) ) -> F e. ( R RingHom S ) )
22		simprr 796	. . . . 5 |- ( ( F e. ( R RingHom S ) /\ ( x e. ( Base ` (oppr ` R ) ) /\ y e. ( Base ` (oppr ` R ) ) ) ) -> y e. ( Base ` (oppr ` R ) ) )
23		eqid 2622	. . . . . 6 |- ( Base ` R ) = ( Base ` R )
24	7, 23	opprbas 18629	. . . . 5 |- ( Base ` R ) = ( Base ` (oppr ` R ) )
25	22, 24	syl6eleqr 2712	. . . 4 |- ( ( F e. ( R RingHom S ) /\ ( x e. ( Base ` (oppr ` R ) ) /\ y e. ( Base ` (oppr ` R ) ) ) ) -> y e. ( Base ` R ) )
26		simprl 794	. . . . 5 |- ( ( F e. ( R RingHom S ) /\ ( x e. ( Base ` (oppr ` R ) ) /\ y e. ( Base ` (oppr ` R ) ) ) ) -> x e. ( Base ` (oppr ` R ) ) )
27	26, 24	syl6eleqr 2712	. . . 4 |- ( ( F e. ( R RingHom S ) /\ ( x e. ( Base ` (oppr ` R ) ) /\ y e. ( Base ` (oppr ` R ) ) ) ) -> x e. ( Base ` R ) )
28		eqid 2622	. . . . 5 |- ( .r ` R ) = ( .r ` R )
29		eqid 2622	. . . . 5 |- ( .r ` S ) = ( .r ` S )
30	23, 28, 29	rhmmul 18727	. . . 4 |- ( ( F e. ( R RingHom S ) /\ y e. ( Base ` R ) /\ x e. ( Base ` R ) ) -> ( F ` ( y ( .r ` R ) x ) ) = ( ( F ` y ) ( .r ` S ) ( F ` x ) ) )
31	21, 25, 27, 30	syl3anc 1326	. . 3 |- ( ( F e. ( R RingHom S ) /\ ( x e. ( Base ` (oppr ` R ) ) /\ y e. ( Base ` (oppr ` R ) ) ) ) -> ( F ` ( y ( .r ` R ) x ) ) = ( ( F ` y ) ( .r ` S ) ( F ` x ) ) )
32	23, 28, 7, 4	opprmul 18626	. . . 4 |- ( x ( .r ` (oppr ` R ) ) y ) = ( y ( .r ` R ) x )
33	32	fveq2i 6194	. . 3 |- ( F ` ( x ( .r ` (oppr ` R ) ) y ) ) = ( F ` ( y ( .r ` R ) x ) )
34		eqid 2622	. . . 4 |- ( Base ` S ) = ( Base ` S )
35	34, 29, 11, 5	opprmul 18626	. . 3 |- ( ( F ` x ) ( .r ` (oppr ` S ) ) ( F ` y ) ) = ( ( F ` y ) ( .r ` S ) ( F ` x ) )
36	31, 33, 35	3eqtr4g 2681	. 2 |- ( ( F e. ( R RingHom S ) /\ ( x e. ( Base ` (oppr ` R ) ) /\ y e. ( Base ` (oppr ` R ) ) ) ) -> ( F ` ( x ( .r ` (oppr ` R ) ) y ) ) = ( ( F ` x ) ( .r ` (oppr ` S ) ) ( F ` y ) ) )
37		ringgrp 18552	. . . . 5 |- ( (oppr ` R ) e. Ring -> (oppr ` R ) e. Grp )
38	9, 37	syl 17	. . . 4 |- ( F e. ( R RingHom S ) -> (oppr ` R ) e. Grp )
39		ringgrp 18552	. . . . 5 |- ( (oppr ` S ) e. Ring -> (oppr ` S ) e. Grp )
40	13, 39	syl 17	. . . 4 |- ( F e. ( R RingHom S ) -> (oppr ` S ) e. Grp )
41	23, 34	rhmf 18726	. . . . 5 |- ( F e. ( R RingHom S ) -> F : ( Base ` R ) --> ( Base ` S ) )
42		rhmghm 18725	. . . . . . . . 9 |- ( F e. ( R RingHom S ) -> F e. ( R GrpHom S ) )
43	42	ad2antrr 762	. . . . . . . 8 |- ( ( ( F e. ( R RingHom S ) /\ x e. ( Base ` R ) ) /\ y e. ( Base ` R ) ) -> F e. ( R GrpHom S ) )
44		simplr 792	. . . . . . . 8 |- ( ( ( F e. ( R RingHom S ) /\ x e. ( Base ` R ) ) /\ y e. ( Base ` R ) ) -> x e. ( Base ` R ) )
45		simpr 477	. . . . . . . 8 |- ( ( ( F e. ( R RingHom S ) /\ x e. ( Base ` R ) ) /\ y e. ( Base ` R ) ) -> y e. ( Base ` R ) )
46		eqid 2622	. . . . . . . . 9 |- ( +g ` R ) = ( +g ` R )
47		eqid 2622	. . . . . . . . 9 |- ( +g ` S ) = ( +g ` S )
48	23, 46, 47	ghmlin 17665	. . . . . . . 8 |- ( ( F e. ( R GrpHom S ) /\ x e. ( Base ` R ) /\ y e. ( Base ` R ) ) -> ( F ` ( x ( +g ` R ) y ) ) = ( ( F ` x ) ( +g ` S ) ( F ` y ) ) )
49	43, 44, 45, 48	syl3anc 1326	. . . . . . 7 |- ( ( ( F e. ( R RingHom S ) /\ x e. ( Base ` R ) ) /\ y e. ( Base ` R ) ) -> ( F ` ( x ( +g ` R ) y ) ) = ( ( F ` x ) ( +g ` S ) ( F ` y ) ) )
50	49	ralrimiva 2966	. . . . . 6 |- ( ( F e. ( R RingHom S ) /\ x e. ( Base ` R ) ) -> A. y e. ( Base ` R ) ( F ` ( x ( +g ` R ) y ) ) = ( ( F ` x ) ( +g ` S ) ( F ` y ) ) )
51	50	ralrimiva 2966	. . . . 5 |- ( F e. ( R RingHom S ) -> A. x e. ( Base ` R ) A. y e. ( Base ` R ) ( F ` ( x ( +g ` R ) y ) ) = ( ( F ` x ) ( +g ` S ) ( F ` y ) ) )
52	41, 51	jca 554	. . . 4 |- ( F e. ( R RingHom S ) -> ( F : ( Base ` R ) --> ( Base ` S ) /\ A. x e. ( Base ` R ) A. y e. ( Base ` R ) ( F ` ( x ( +g ` R ) y ) ) = ( ( F ` x ) ( +g ` S ) ( F ` y ) ) ) )
53	38, 40, 52	jca31 557	. . 3 |- ( F e. ( R RingHom S ) -> ( ( (oppr ` R ) e. Grp /\ (oppr ` S ) e. Grp ) /\ ( F : ( Base ` R ) --> ( Base ` S ) /\ A. x e. ( Base ` R ) A. y e. ( Base ` R ) ( F ` ( x ( +g ` R ) y ) ) = ( ( F ` x ) ( +g ` S ) ( F ` y ) ) ) ) )
54	11, 34	opprbas 18629	. . . 4 |- ( Base ` S ) = ( Base ` (oppr ` S ) )
55	7, 46	oppradd 18630	. . . 4 |- ( +g ` R ) = ( +g ` (oppr ` R ) )
56	11, 47	oppradd 18630	. . . 4 |- ( +g ` S ) = ( +g ` (oppr ` S ) )
57	24, 54, 55, 56	isghm 17660	. . 3 |- ( F e. ( (oppr ` R ) GrpHom (oppr ` S ) ) <-> ( ( (oppr ` R ) e. Grp /\ (oppr ` S ) e. Grp ) /\ ( F : ( Base ` R ) --> ( Base ` S ) /\ A. x e. ( Base ` R ) A. y e. ( Base ` R ) ( F ` ( x ( +g ` R ) y ) ) = ( ( F ` x ) ( +g ` S ) ( F ` y ) ) ) ) )
58	53, 57	sylibr 224	. 2 |- ( F e. ( R RingHom S ) -> F e. ( (oppr ` R ) GrpHom (oppr ` S ) ) )
59	1, 2, 3, 4, 5, 9, 13, 20, 36, 58	isrhm2d 18728	1 |- ( F e. ( R RingHom S ) -> F e. ( (oppr ` R ) RingHom (oppr ` S ) ) )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   A.wral 2912   -->wf 5884   ` cfv 5888  (class class class)co 6650   Basecbs 15857   +g cplusg 15941   .rcmulr 15942   Grpcgrp 17422   GrpHom cghm 17657   1rcur 18501   Ringcrg 18547  opp_rcoppr 18622   RingHom crh 18712

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-tpos 7352  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-er 7742  df-map 7859  df-en 7956  df-dom 7957  df-sdom 7958  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-nn 11021  df-2 11079  df-3 11080  df-ndx 15860  df-slot 15861  df-base 15863  df-sets 15864  df-plusg 15954  df-mulr 15955  df-0g 16102  df-mgm 17242  df-sgrp 17284  df-mnd 17295  df-mhm 17335  df-grp 17425  df-ghm 17658  df-mgp 18490  df-ur 18502  df-ring 18549  df-oppr 18623  df-rnghom 18715

This theorem is referenced by:  elrhmunit  29820