Theorem rhmunitinv 14323

Description: Ring homomorphisms preserve the inverse of unit elements. (Contributed by Thierry Arnoux, 23-Oct-2017.)

Assertion

Ref	Expression
rhmunitinv	|- ( ( F e. ( R RingHom S ) /\ A e. (Unit ` R ) ) -> ( F ` ( ( invr ` R ) ` A ) ) = ( ( invr ` S ) ` ( F ` A ) ) )

Proof of Theorem rhmunitinv

Step	Hyp	Ref	Expression
1		rhmrcl1 14300	. . . . . 6 |- ( F e. ( R RingHom S ) -> R e. Ring )
2		eqid 2232	. . . . . . 7 |- (Unit ` R ) = (Unit ` R )
3		eqid 2232	. . . . . . 7 |- ( invr ` R ) = ( invr ` R )
4		eqid 2232	. . . . . . 7 |- ( .r ` R ) = ( .r ` R )
5		eqid 2232	. . . . . . 7 |- ( 1r ` R ) = ( 1r ` R )
6	2, 3, 4, 5	unitlinv 14271	. . . . . 6 |- ( ( R e. Ring /\ A e. (Unit ` R ) ) -> ( ( ( invr ` R ) ` A ) ( .r ` R ) A ) = ( 1r ` R ) )
7	1, 6	sylan 283	. . . . 5 |- ( ( F e. ( R RingHom S ) /\ A e. (Unit ` R ) ) -> ( ( ( invr ` R ) ` A ) ( .r ` R ) A ) = ( 1r ` R ) )
8	7	fveq2d 5674	. . . 4 |- ( ( F e. ( R RingHom S ) /\ A e. (Unit ` R ) ) -> ( F ` ( ( ( invr ` R ) ` A ) ( .r ` R ) A ) ) = ( F ` ( 1r ` R ) ) )
9		simpl 109	. . . . 5 |- ( ( F e. ( R RingHom S ) /\ A e. (Unit ` R ) ) -> F e. ( R RingHom S ) )
10		eqidd 2233	. . . . . . 7 |- ( ( F e. ( R RingHom S ) /\ A e. (Unit ` R ) ) -> ( Base ` R ) = ( Base ` R ) )
11		eqidd 2233	. . . . . . 7 |- ( ( F e. ( R RingHom S ) /\ A e. (Unit ` R ) ) -> (Unit ` R ) = (Unit ` R ) )
12	1	adantr 276	. . . . . . . 8 |- ( ( F e. ( R RingHom S ) /\ A e. (Unit ` R ) ) -> R e. Ring )
13		ringsrg 14191	. . . . . . . 8 |- ( R e. Ring -> R e. SRing )
14	12, 13	syl 14	. . . . . . 7 |- ( ( F e. ( R RingHom S ) /\ A e. (Unit ` R ) ) -> R e. SRing )
15	10, 11, 14	unitssd 14254	. . . . . 6 |- ( ( F e. ( R RingHom S ) /\ A e. (Unit ` R ) ) -> (Unit ` R ) C_ ( Base ` R ) )
16	2, 3	unitinvcl 14268	. . . . . . 7 |- ( ( R e. Ring /\ A e. (Unit ` R ) ) -> ( ( invr ` R ) ` A ) e. (Unit ` R ) )
17	1, 16	sylan 283	. . . . . 6 |- ( ( F e. ( R RingHom S ) /\ A e. (Unit ` R ) ) -> ( ( invr ` R ) ` A ) e. (Unit ` R ) )
18	15, 17	sseldd 3239	. . . . 5 |- ( ( F e. ( R RingHom S ) /\ A e. (Unit ` R ) ) -> ( ( invr ` R ) ` A ) e. ( Base ` R ) )
19		simpr 110	. . . . . 6 |- ( ( F e. ( R RingHom S ) /\ A e. (Unit ` R ) ) -> A e. (Unit ` R ) )
20	15, 19	sseldd 3239	. . . . 5 |- ( ( F e. ( R RingHom S ) /\ A e. (Unit ` R ) ) -> A e. ( Base ` R ) )
21		eqid 2232	. . . . . 6 |- ( Base ` R ) = ( Base ` R )
22		eqid 2232	. . . . . 6 |- ( .r ` S ) = ( .r ` S )
23	21, 4, 22	rhmmul 14309	. . . . 5 |- ( ( F e. ( R RingHom S ) /\ ( ( invr ` R ) ` A ) e. ( Base ` R ) /\ A e. ( Base ` R ) ) -> ( F ` ( ( ( invr ` R ) ` A ) ( .r ` R ) A ) ) = ( ( F ` ( ( invr ` R ) ` A ) ) ( .r ` S ) ( F ` A ) ) )
24	9, 18, 20, 23	syl3anc 1274	. . . 4 |- ( ( F e. ( R RingHom S ) /\ A e. (Unit ` R ) ) -> ( F ` ( ( ( invr ` R ) ` A ) ( .r ` R ) A ) ) = ( ( F ` ( ( invr ` R ) ` A ) ) ( .r ` S ) ( F ` A ) ) )
25		eqid 2232	. . . . . 6 |- ( 1r ` S ) = ( 1r ` S )
26	5, 25	rhm1 14312	. . . . 5 |- ( F e. ( R RingHom S ) -> ( F ` ( 1r ` R ) ) = ( 1r ` S ) )
27	26	adantr 276	. . . 4 |- ( ( F e. ( R RingHom S ) /\ A e. (Unit ` R ) ) -> ( F ` ( 1r ` R ) ) = ( 1r ` S ) )
28	8, 24, 27	3eqtr3d 2273	. . 3 |- ( ( F e. ( R RingHom S ) /\ A e. (Unit ` R ) ) -> ( ( F ` ( ( invr ` R ) ` A ) ) ( .r ` S ) ( F ` A ) ) = ( 1r ` S ) )
29		rhmrcl2 14301	. . . . 5 |- ( F e. ( R RingHom S ) -> S e. Ring )
30	29	adantr 276	. . . 4 |- ( ( F e. ( R RingHom S ) /\ A e. (Unit ` R ) ) -> S e. Ring )
31		elrhmunit 14322	. . . 4 |- ( ( F e. ( R RingHom S ) /\ A e. (Unit ` R ) ) -> ( F ` A ) e. (Unit ` S ) )
32		eqid 2232	. . . . 5 |- (Unit ` S ) = (Unit ` S )
33		eqid 2232	. . . . 5 |- ( invr ` S ) = ( invr ` S )
34	32, 33, 22, 25	unitlinv 14271	. . . 4 |- ( ( S e. Ring /\ ( F ` A ) e. (Unit ` S ) ) -> ( ( ( invr ` S ) ` ( F ` A ) ) ( .r ` S ) ( F ` A ) ) = ( 1r ` S ) )
35	30, 31, 34	syl2anc 411	. . 3 |- ( ( F e. ( R RingHom S ) /\ A e. (Unit ` R ) ) -> ( ( ( invr ` S ) ` ( F ` A ) ) ( .r ` S ) ( F ` A ) ) = ( 1r ` S ) )
36	28, 35	eqtr4d 2268	. 2 |- ( ( F e. ( R RingHom S ) /\ A e. (Unit ` R ) ) -> ( ( F ` ( ( invr ` R ) ` A ) ) ( .r ` S ) ( F ` A ) ) = ( ( ( invr ` S ) ` ( F ` A ) ) ( .r ` S ) ( F ` A ) ) )
37		eqidd 2233	. . . . . 6 |- ( ( F e. ( R RingHom S ) /\ A e. (Unit ` R ) ) -> ( (mulGrp ` S ) ↾s (Unit ` S ) ) = ( (mulGrp ` S ) ↾s (Unit ` S ) ) )
38		eqid 2232	. . . . . . . 8 |- (mulGrp ` S ) = (mulGrp ` S )
39	38, 22	mgpplusgg 14068	. . . . . . 7 |- ( S e. Ring -> ( .r ` S ) = ( +g ` (mulGrp ` S ) ) )
40	30, 39	syl 14	. . . . . 6 |- ( ( F e. ( R RingHom S ) /\ A e. (Unit ` R ) ) -> ( .r ` S ) = ( +g ` (mulGrp ` S ) ) )
41		basfn 13271	. . . . . . . 8 |- Base Fn _V
42	30	elexd 2827	. . . . . . . 8 |- ( ( F e. ( R RingHom S ) /\ A e. (Unit ` R ) ) -> S e. _V )
43		funfvex 5687	. . . . . . . . 9 |- ( ( Fun Base /\ S e. dom Base ) -> ( Base ` S ) e. _V )
44	43	funfni 5458	. . . . . . . 8 |- ( ( Base Fn _V /\ S e. _V ) -> ( Base ` S ) e. _V )
45	41, 42, 44	sylancr 414	. . . . . . 7 |- ( ( F e. ( R RingHom S ) /\ A e. (Unit ` R ) ) -> ( Base ` S ) e. _V )
46		eqidd 2233	. . . . . . . 8 |- ( ( F e. ( R RingHom S ) /\ A e. (Unit ` R ) ) -> ( Base ` S ) = ( Base ` S ) )
47		eqidd 2233	. . . . . . . 8 |- ( ( F e. ( R RingHom S ) /\ A e. (Unit ` R ) ) -> (Unit ` S ) = (Unit ` S ) )
48		ringsrg 14191	. . . . . . . . 9 |- ( S e. Ring -> S e. SRing )
49	30, 48	syl 14	. . . . . . . 8 |- ( ( F e. ( R RingHom S ) /\ A e. (Unit ` R ) ) -> S e. SRing )
50	46, 47, 49	unitssd 14254	. . . . . . 7 |- ( ( F e. ( R RingHom S ) /\ A e. (Unit ` R ) ) -> (Unit ` S ) C_ ( Base ` S ) )
51	45, 50	ssexd 4250	. . . . . 6 |- ( ( F e. ( R RingHom S ) /\ A e. (Unit ` R ) ) -> (Unit ` S ) e. _V )
52	38	mgpex 14069	. . . . . . 7 |- ( S e. Ring -> (mulGrp ` S ) e. _V )
53	30, 52	syl 14	. . . . . 6 |- ( ( F e. ( R RingHom S ) /\ A e. (Unit ` R ) ) -> (mulGrp ` S ) e. _V )
54	37, 40, 51, 53	ressplusgd 13342	. . . . 5 |- ( ( F e. ( R RingHom S ) /\ A e. (Unit ` R ) ) -> ( .r ` S ) = ( +g ` ( (mulGrp ` S ) ↾s (Unit ` S ) ) ) )
55	54	oveqd 6067	. . . 4 |- ( ( F e. ( R RingHom S ) /\ A e. (Unit ` R ) ) -> ( ( F ` ( ( invr ` R ) ` A ) ) ( .r ` S ) ( F ` A ) ) = ( ( F ` ( ( invr ` R ) ` A ) ) ( +g ` ( (mulGrp ` S ) ↾s (Unit ` S ) ) ) ( F ` A ) ) )
56	54	oveqd 6067	. . . 4 |- ( ( F e. ( R RingHom S ) /\ A e. (Unit ` R ) ) -> ( ( ( invr ` S ) ` ( F ` A ) ) ( .r ` S ) ( F ` A ) ) = ( ( ( invr ` S ) ` ( F ` A ) ) ( +g ` ( (mulGrp ` S ) ↾s (Unit ` S ) ) ) ( F ` A ) ) )
57	55, 56	eqeq12d 2247	. . 3 |- ( ( F e. ( R RingHom S ) /\ A e. (Unit ` R ) ) -> ( ( ( F ` ( ( invr ` R ) ` A ) ) ( .r ` S ) ( F ` A ) ) = ( ( ( invr ` S ) ` ( F ` A ) ) ( .r ` S ) ( F ` A ) ) <-> ( ( F ` ( ( invr ` R ) ` A ) ) ( +g ` ( (mulGrp ` S ) ↾s (Unit ` S ) ) ) ( F ` A ) ) = ( ( ( invr ` S ) ` ( F ` A ) ) ( +g ` ( (mulGrp ` S ) ↾s (Unit ` S ) ) ) ( F ` A ) ) ) )
58		eqid 2232	. . . . . . 7 |- ( (mulGrp ` S ) ↾s (Unit ` S ) ) = ( (mulGrp ` S ) ↾s (Unit ` S ) )
59	32, 58	unitgrp 14261	. . . . . 6 |- ( S e. Ring -> ( (mulGrp ` S ) ↾s (Unit ` S ) ) e. Grp )
60	29, 59	syl 14	. . . . 5 |- ( F e. ( R RingHom S ) -> ( (mulGrp ` S ) ↾s (Unit ` S ) ) e. Grp )
61	60	adantr 276	. . . 4 |- ( ( F e. ( R RingHom S ) /\ A e. (Unit ` R ) ) -> ( (mulGrp ` S ) ↾s (Unit ` S ) ) e. Grp )
62		elrhmunit 14322	. . . . . 6 |- ( ( F e. ( R RingHom S ) /\ ( ( invr ` R ) ` A ) e. (Unit ` R ) ) -> ( F ` ( ( invr ` R ) ` A ) ) e. (Unit ` S ) )
63	17, 62	syldan 282	. . . . 5 |- ( ( F e. ( R RingHom S ) /\ A e. (Unit ` R ) ) -> ( F ` ( ( invr ` R ) ` A ) ) e. (Unit ` S ) )
64	47, 37, 49	unitgrpbasd 14260	. . . . 5 |- ( ( F e. ( R RingHom S ) /\ A e. (Unit ` R ) ) -> (Unit ` S ) = ( Base ` ( (mulGrp ` S ) ↾s (Unit ` S ) ) ) )
65	63, 64	eleqtrd 2311	. . . 4 |- ( ( F e. ( R RingHom S ) /\ A e. (Unit ` R ) ) -> ( F ` ( ( invr ` R ) ` A ) ) e. ( Base ` ( (mulGrp ` S ) ↾s (Unit ` S ) ) ) )
66	32, 33	unitinvcl 14268	. . . . . 6 |- ( ( S e. Ring /\ ( F ` A ) e. (Unit ` S ) ) -> ( ( invr ` S ) ` ( F ` A ) ) e. (Unit ` S ) )
67	30, 31, 66	syl2anc 411	. . . . 5 |- ( ( F e. ( R RingHom S ) /\ A e. (Unit ` R ) ) -> ( ( invr ` S ) ` ( F ` A ) ) e. (Unit ` S ) )
68	67, 64	eleqtrd 2311	. . . 4 |- ( ( F e. ( R RingHom S ) /\ A e. (Unit ` R ) ) -> ( ( invr ` S ) ` ( F ` A ) ) e. ( Base ` ( (mulGrp ` S ) ↾s (Unit ` S ) ) ) )
69	31, 64	eleqtrd 2311	. . . 4 |- ( ( F e. ( R RingHom S ) /\ A e. (Unit ` R ) ) -> ( F ` A ) e. ( Base ` ( (mulGrp ` S ) ↾s (Unit ` S ) ) ) )
70		eqid 2232	. . . . 5 |- ( Base ` ( (mulGrp ` S ) ↾s (Unit ` S ) ) ) = ( Base ` ( (mulGrp ` S ) ↾s (Unit ` S ) ) )
71		eqid 2232	. . . . 5 |- ( +g ` ( (mulGrp ` S ) ↾s (Unit ` S ) ) ) = ( +g ` ( (mulGrp ` S ) ↾s (Unit ` S ) ) )
72	70, 71	grprcan 13750	. . . 4 |- ( ( ( (mulGrp ` S ) ↾s (Unit ` S ) ) e. Grp /\ ( ( F ` ( ( invr ` R ) ` A ) ) e. ( Base ` ( (mulGrp ` S ) ↾s (Unit ` S ) ) ) /\ ( ( invr ` S ) ` ( F ` A ) ) e. ( Base ` ( (mulGrp ` S ) ↾s (Unit ` S ) ) ) /\ ( F ` A ) e. ( Base ` ( (mulGrp ` S ) ↾s (Unit ` S ) ) ) ) ) -> ( ( ( F ` ( ( invr ` R ) ` A ) ) ( +g ` ( (mulGrp ` S ) ↾s (Unit ` S ) ) ) ( F ` A ) ) = ( ( ( invr ` S ) ` ( F ` A ) ) ( +g ` ( (mulGrp ` S ) ↾s (Unit ` S ) ) ) ( F ` A ) ) <-> ( F ` ( ( invr ` R ) ` A ) ) = ( ( invr ` S ) ` ( F ` A ) ) ) )
73	61, 65, 68, 69, 72	syl13anc 1276	. . 3 |- ( ( F e. ( R RingHom S ) /\ A e. (Unit ` R ) ) -> ( ( ( F ` ( ( invr ` R ) ` A ) ) ( +g ` ( (mulGrp ` S ) ↾s (Unit ` S ) ) ) ( F ` A ) ) = ( ( ( invr ` S ) ` ( F ` A ) ) ( +g ` ( (mulGrp ` S ) ↾s (Unit ` S ) ) ) ( F ` A ) ) <-> ( F ` ( ( invr ` R ) ` A ) ) = ( ( invr ` S ) ` ( F ` A ) ) ) )
74	57, 73	bitrd 188	. 2 |- ( ( F e. ( R RingHom S ) /\ A e. (Unit ` R ) ) -> ( ( ( F ` ( ( invr ` R ) ` A ) ) ( .r ` S ) ( F ` A ) ) = ( ( ( invr ` S ) ` ( F ` A ) ) ( .r ` S ) ( F ` A ) ) <-> ( F ` ( ( invr ` R ) ` A ) ) = ( ( invr ` S ) ` ( F ` A ) ) ) )
75	36, 74	mpbid 147	1 |- ( ( F e. ( R RingHom S ) /\ A e. (Unit ` R ) ) -> ( F ` ( ( invr ` R ) ` A ) ) = ( ( invr ` S ) ` ( F ` A ) ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1398   e. wcel 2203   _Vcvv 2813   Fn wfn 5347   ` cfv 5352  (class class class)co 6050   Basecbs 13212   ↾_s cress 13213   +g cplusg 13290   .rcmulr 13291   Grpcgrp 13713  mulGrpcmgp 14064   1rcur 14103  SRingcsrg 14107   Ringcrg 14140  Unitcui 14231   invrcinvr 14265   RingHom crh 14295

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2205  ax-14 2206  ax-ext 2214  ax-coll 4225  ax-sep 4228  ax-nul 4236  ax-pow 4287  ax-pr 4322  ax-un 4554  ax-setind 4659  ax-cnex 8218  ax-resscn 8219  ax-1cn 8220  ax-1re 8221  ax-icn 8222  ax-addcl 8223  ax-addrcl 8224  ax-mulcl 8225  ax-addcom 8227  ax-addass 8229  ax-i2m1 8232  ax-0lt1 8233  ax-0id 8235  ax-rnegex 8236  ax-pre-ltirr 8239  ax-pre-lttrn 8241  ax-pre-ltadd 8243

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2083  df-mo 2084  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ne 2413  df-nel 2508  df-ral 2525  df-rex 2526  df-reu 2527  df-rmo 2528  df-rab 2529  df-v 2815  df-sbc 3043  df-csb 3139  df-dif 3213  df-un 3215  df-in 3217  df-ss 3224  df-nul 3509  df-pw 3671  df-sn 3695  df-pr 3696  df-op 3698  df-uni 3915  df-int 3950  df-iun 3993  df-br 4110  df-opab 4172  df-mpt 4173  df-id 4414  df-xp 4755  df-rel 4756  df-cnv 4757  df-co 4758  df-dm 4759  df-rn 4760  df-res 4761  df-ima 4762  df-iota 5312  df-fun 5354  df-fn 5355  df-f 5356  df-f1 5357  df-fo 5358  df-f1o 5359  df-fv 5360  df-riota 6003  df-ov 6053  df-oprab 6054  df-mpo 6055  df-1st 6334  df-2nd 6335  df-tpos 6476  df-map 6884  df-pnf 8310  df-mnf 8311  df-ltxr 8313  df-inn 9238  df-2 9296  df-3 9297  df-ndx 13215  df-slot 13216  df-base 13218  df-sets 13219  df-iress 13220  df-plusg 13303  df-mulr 13304  df-0g 13471  df-mgm 13569  df-sgrp 13615  df-mnd 13630  df-mhm 13672  df-grp 13716  df-minusg 13717  df-ghm 13958  df-cmn 14003  df-abl 14004  df-mgp 14065  df-ur 14104  df-srg 14108  df-ring 14142  df-oppr 14212  df-dvdsr 14233  df-unit 14234  df-invr 14266  df-rhm 14297

This theorem is referenced by: (None)