Theorem rhmunitinv 14182

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 14159	. . . . . 6 |- ( F e. ( R RingHom S ) -> R e. Ring )
2		eqid 2229	. . . . . . 7 |- (Unit ` R ) = (Unit ` R )
3		eqid 2229	. . . . . . 7 |- ( invr ` R ) = ( invr ` R )
4		eqid 2229	. . . . . . 7 |- ( .r ` R ) = ( .r ` R )
5		eqid 2229	. . . . . . 7 |- ( 1r ` R ) = ( 1r ` R )
6	2, 3, 4, 5	unitlinv 14130	. . . . . 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 5639	. . . 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 2230	. . . . . . 7 |- ( ( F e. ( R RingHom S ) /\ A e. (Unit ` R ) ) -> ( Base ` R ) = ( Base ` R ) )
11		eqidd 2230	. . . . . . 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 14050	. . . . . . . 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 14113	. . . . . 6 |- ( ( F e. ( R RingHom S ) /\ A e. (Unit ` R ) ) -> (Unit ` R ) C_ ( Base ` R ) )
16	2, 3	unitinvcl 14127	. . . . . . 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 3226	. . . . 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 3226	. . . . 5 |- ( ( F e. ( R RingHom S ) /\ A e. (Unit ` R ) ) -> A e. ( Base ` R ) )
21		eqid 2229	. . . . . 6 |- ( Base ` R ) = ( Base ` R )
22		eqid 2229	. . . . . 6 |- ( .r ` S ) = ( .r ` S )
23	21, 4, 22	rhmmul 14168	. . . . 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 1271	. . . 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 2229	. . . . . 6 |- ( 1r ` S ) = ( 1r ` S )
26	5, 25	rhm1 14171	. . . . 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 2270	. . 3 |- ( ( F e. ( R RingHom S ) /\ A e. (Unit ` R ) ) -> ( ( F ` ( ( invr ` R ) ` A ) ) ( .r ` S ) ( F ` A ) ) = ( 1r ` S ) )
29		rhmrcl2 14160	. . . . 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 14181	. . . 4 |- ( ( F e. ( R RingHom S ) /\ A e. (Unit ` R ) ) -> ( F ` A ) e. (Unit ` S ) )
32		eqid 2229	. . . . 5 |- (Unit ` S ) = (Unit ` S )
33		eqid 2229	. . . . 5 |- ( invr ` S ) = ( invr ` S )
34	32, 33, 22, 25	unitlinv 14130	. . . 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 2265	. 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 2230	. . . . . 6 |- ( ( F e. ( R RingHom S ) /\ A e. (Unit ` R ) ) -> ( (mulGrp ` S ) ↾s (Unit ` S ) ) = ( (mulGrp ` S ) ↾s (Unit ` S ) ) )
38		eqid 2229	. . . . . . . 8 |- (mulGrp ` S ) = (mulGrp ` S )
39	38, 22	mgpplusgg 13927	. . . . . . 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 13131	. . . . . . . 8 |- Base Fn _V
42	30	elexd 2814	. . . . . . . 8 |- ( ( F e. ( R RingHom S ) /\ A e. (Unit ` R ) ) -> S e. _V )
43		funfvex 5652	. . . . . . . . 9 |- ( ( Fun Base /\ S e. dom Base ) -> ( Base ` S ) e. _V )
44	43	funfni 5429	. . . . . . . 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 2230	. . . . . . . 8 |- ( ( F e. ( R RingHom S ) /\ A e. (Unit ` R ) ) -> ( Base ` S ) = ( Base ` S ) )
47		eqidd 2230	. . . . . . . 8 |- ( ( F e. ( R RingHom S ) /\ A e. (Unit ` R ) ) -> (Unit ` S ) = (Unit ` S ) )
48		ringsrg 14050	. . . . . . . . 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 14113	. . . . . . 7 |- ( ( F e. ( R RingHom S ) /\ A e. (Unit ` R ) ) -> (Unit ` S ) C_ ( Base ` S ) )
51	45, 50	ssexd 4227	. . . . . 6 |- ( ( F e. ( R RingHom S ) /\ A e. (Unit ` R ) ) -> (Unit ` S ) e. _V )
52	38	mgpex 13928	. . . . . . 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 13202	. . . . 5 |- ( ( F e. ( R RingHom S ) /\ A e. (Unit ` R ) ) -> ( .r ` S ) = ( +g ` ( (mulGrp ` S ) ↾s (Unit ` S ) ) ) )
55	54	oveqd 6030	. . . 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 6030	. . . 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 2244	. . 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 2229	. . . . . . 7 |- ( (mulGrp ` S ) ↾s (Unit ` S ) ) = ( (mulGrp ` S ) ↾s (Unit ` S ) )
59	32, 58	unitgrp 14120	. . . . . 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 14181	. . . . . 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 14119	. . . . 5 |- ( ( F e. ( R RingHom S ) /\ A e. (Unit ` R ) ) -> (Unit ` S ) = ( Base ` ( (mulGrp ` S ) ↾s (Unit ` S ) ) ) )
65	63, 64	eleqtrd 2308	. . . 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 14127	. . . . . 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 2308	. . . 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 2308	. . . 4 |- ( ( F e. ( R RingHom S ) /\ A e. (Unit ` R ) ) -> ( F ` A ) e. ( Base ` ( (mulGrp ` S ) ↾s (Unit ` S ) ) ) )
70		eqid 2229	. . . . 5 |- ( Base ` ( (mulGrp ` S ) ↾s (Unit ` S ) ) ) = ( Base ` ( (mulGrp ` S ) ↾s (Unit ` S ) ) )
71		eqid 2229	. . . . 5 |- ( +g ` ( (mulGrp ` S ) ↾s (Unit ` S ) ) ) = ( +g ` ( (mulGrp ` S ) ↾s (Unit ` S ) ) )
72	70, 71	grprcan 13610	. . . 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 1273	. . 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 1395   e. wcel 2200   _Vcvv 2800   Fn wfn 5319   ` cfv 5324  (class class class)co 6013   Basecbs 13072   ↾_s cress 13073   +g cplusg 13150   .rcmulr 13151   Grpcgrp 13573  mulGrpcmgp 13923   1rcur 13962  SRingcsrg 13966   Ringcrg 13999  Unitcui 14090   invrcinvr 14124   RingHom crh 14154

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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-coll 4202  ax-sep 4205  ax-nul 4213  ax-pow 4262  ax-pr 4297  ax-un 4528  ax-setind 4633  ax-cnex 8113  ax-resscn 8114  ax-1cn 8115  ax-1re 8116  ax-icn 8117  ax-addcl 8118  ax-addrcl 8119  ax-mulcl 8120  ax-addcom 8122  ax-addass 8124  ax-i2m1 8127  ax-0lt1 8128  ax-0id 8130  ax-rnegex 8131  ax-pre-ltirr 8134  ax-pre-lttrn 8136  ax-pre-ltadd 8138

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-nel 2496  df-ral 2513  df-rex 2514  df-reu 2515  df-rmo 2516  df-rab 2517  df-v 2802  df-sbc 3030  df-csb 3126  df-dif 3200  df-un 3202  df-in 3204  df-ss 3211  df-nul 3493  df-pw 3652  df-sn 3673  df-pr 3674  df-op 3676  df-uni 3892  df-int 3927  df-iun 3970  df-br 4087  df-opab 4149  df-mpt 4150  df-id 4388  df-xp 4729  df-rel 4730  df-cnv 4731  df-co 4732  df-dm 4733  df-rn 4734  df-res 4735  df-ima 4736  df-iota 5284  df-fun 5326  df-fn 5327  df-f 5328  df-f1 5329  df-fo 5330  df-f1o 5331  df-fv 5332  df-riota 5966  df-ov 6016  df-oprab 6017  df-mpo 6018  df-1st 6298  df-2nd 6299  df-tpos 6406  df-map 6814  df-pnf 8206  df-mnf 8207  df-ltxr 8209  df-inn 9134  df-2 9192  df-3 9193  df-ndx 13075  df-slot 13076  df-base 13078  df-sets 13079  df-iress 13080  df-plusg 13163  df-mulr 13164  df-0g 13331  df-mgm 13429  df-sgrp 13475  df-mnd 13490  df-mhm 13532  df-grp 13576  df-minusg 13577  df-ghm 13818  df-cmn 13863  df-abl 13864  df-mgp 13924  df-ur 13963  df-srg 13967  df-ring 14001  df-oppr 14071  df-dvdsr 14092  df-unit 14093  df-invr 14125  df-rhm 14156

This theorem is referenced by: (None)