Theorem elrhmunit 13673

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

Assertion

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

Proof of Theorem elrhmunit

Step	Hyp	Ref	Expression
1		simpl 109	. . . 4 |- ( ( F e. ( R RingHom S ) /\ A e. (Unit ` R ) ) -> F e. ( R RingHom S ) )
2		eqidd 2194	. . . . 5 |- ( ( F e. ( R RingHom S ) /\ A e. (Unit ` R ) ) -> ( Base ` R ) = ( Base ` R ) )
3		eqidd 2194	. . . . 5 |- ( ( F e. ( R RingHom S ) /\ A e. (Unit ` R ) ) -> (Unit ` R ) = (Unit ` R ) )
4		rhmrcl1 13651	. . . . . . 7 |- ( F e. ( R RingHom S ) -> R e. Ring )
5	4	adantr 276	. . . . . 6 |- ( ( F e. ( R RingHom S ) /\ A e. (Unit ` R ) ) -> R e. Ring )
6		ringsrg 13543	. . . . . 6 |- ( R e. Ring -> R e. SRing )
7	5, 6	syl 14	. . . . 5 |- ( ( F e. ( R RingHom S ) /\ A e. (Unit ` R ) ) -> R e. SRing )
8		simpr 110	. . . . 5 |- ( ( F e. ( R RingHom S ) /\ A e. (Unit ` R ) ) -> A e. (Unit ` R ) )
9	2, 3, 7, 8	unitcld 13604	. . . 4 |- ( ( F e. ( R RingHom S ) /\ A e. (Unit ` R ) ) -> A e. ( Base ` R ) )
10		eqid 2193	. . . . . 6 |- ( Base ` R ) = ( Base ` R )
11		eqid 2193	. . . . . 6 |- ( 1r ` R ) = ( 1r ` R )
12	10, 11	ringidcl 13516	. . . . 5 |- ( R e. Ring -> ( 1r ` R ) e. ( Base ` R ) )
13	1, 4, 12	3syl 17	. . . 4 |- ( ( F e. ( R RingHom S ) /\ A e. (Unit ` R ) ) -> ( 1r ` R ) e. ( Base ` R ) )
14		eqidd 2194	. . . . . . 7 |- ( ( F e. ( R RingHom S ) /\ A e. (Unit ` R ) ) -> ( 1r ` R ) = ( 1r ` R ) )
15		eqidd 2194	. . . . . . 7 |- ( ( F e. ( R RingHom S ) /\ A e. (Unit ` R ) ) -> ( ||r ` R ) = ( ||r ` R ) )
16		eqidd 2194	. . . . . . 7 |- ( ( F e. ( R RingHom S ) /\ A e. (Unit ` R ) ) -> (oppr ` R ) = (oppr ` R ) )
17		eqidd 2194	. . . . . . 7 |- ( ( F e. ( R RingHom S ) /\ A e. (Unit ` R ) ) -> ( ||r ` (oppr ` R ) ) = ( ||r ` (oppr ` R ) ) )
18	3, 14, 15, 16, 17, 7	isunitd 13602	. . . . . 6 |- ( ( F e. ( R RingHom S ) /\ A e. (Unit ` R ) ) -> ( A e. (Unit ` R ) <-> ( A ( ||r ` R ) ( 1r ` R ) /\ A ( ||r ` (oppr ` R ) ) ( 1r ` R ) ) ) )
19	8, 18	mpbid 147	. . . . 5 |- ( ( F e. ( R RingHom S ) /\ A e. (Unit ` R ) ) -> ( A ( ||r ` R ) ( 1r ` R ) /\ A ( ||r ` (oppr ` R ) ) ( 1r ` R ) ) )
20	19	simpld 112	. . . 4 |- ( ( F e. ( R RingHom S ) /\ A e. (Unit ` R ) ) -> A ( ||r ` R ) ( 1r ` R ) )
21		eqid 2193	. . . . 5 |- ( ||r ` R ) = ( ||r ` R )
22		eqid 2193	. . . . 5 |- ( ||r ` S ) = ( ||r ` S )
23	10, 21, 22	rhmdvdsr 13671	. . . 4 |- ( ( ( F e. ( R RingHom S ) /\ A e. ( Base ` R ) /\ ( 1r ` R ) e. ( Base ` R ) ) /\ A ( ||r ` R ) ( 1r ` R ) ) -> ( F ` A ) ( ||r ` S ) ( F ` ( 1r ` R ) ) )
24	1, 9, 13, 20, 23	syl31anc 1252	. . 3 |- ( ( F e. ( R RingHom S ) /\ A e. (Unit ` R ) ) -> ( F ` A ) ( ||r ` S ) ( F ` ( 1r ` R ) ) )
25		eqid 2193	. . . . . 6 |- ( 1r ` S ) = ( 1r ` S )
26	11, 25	rhm1 13663	. . . . 5 |- ( F e. ( R RingHom S ) -> ( F ` ( 1r ` R ) ) = ( 1r ` S ) )
27	26	breq2d 4041	. . . 4 |- ( F e. ( R RingHom S ) -> ( ( F ` A ) ( ||r ` S ) ( F ` ( 1r ` R ) ) <-> ( F ` A ) ( ||r ` S ) ( 1r ` S ) ) )
28	27	adantr 276	. . 3 |- ( ( F e. ( R RingHom S ) /\ A e. (Unit ` R ) ) -> ( ( F ` A ) ( ||r ` S ) ( F ` ( 1r ` R ) ) <-> ( F ` A ) ( ||r ` S ) ( 1r ` S ) ) )
29	24, 28	mpbid 147	. 2 |- ( ( F e. ( R RingHom S ) /\ A e. (Unit ` R ) ) -> ( F ` A ) ( ||r ` S ) ( 1r ` S ) )
30		rhmopp 13672	. . . . 5 |- ( F e. ( R RingHom S ) -> F e. ( (oppr ` R ) RingHom (oppr ` S ) ) )
31	30	adantr 276	. . . 4 |- ( ( F e. ( R RingHom S ) /\ A e. (Unit ` R ) ) -> F e. ( (oppr ` R ) RingHom (oppr ` S ) ) )
32		eqid 2193	. . . . . . 7 |- (oppr ` R ) = (oppr ` R )
33	32, 10	opprbasg 13571	. . . . . 6 |- ( R e. Ring -> ( Base ` R ) = ( Base ` (oppr ` R ) ) )
34	5, 33	syl 14	. . . . 5 |- ( ( F e. ( R RingHom S ) /\ A e. (Unit ` R ) ) -> ( Base ` R ) = ( Base ` (oppr ` R ) ) )
35	9, 34	eleqtrd 2272	. . . 4 |- ( ( F e. ( R RingHom S ) /\ A e. (Unit ` R ) ) -> A e. ( Base ` (oppr ` R ) ) )
36	13, 34	eleqtrd 2272	. . . 4 |- ( ( F e. ( R RingHom S ) /\ A e. (Unit ` R ) ) -> ( 1r ` R ) e. ( Base ` (oppr ` R ) ) )
37	19	simprd 114	. . . 4 |- ( ( F e. ( R RingHom S ) /\ A e. (Unit ` R ) ) -> A ( ||r ` (oppr ` R ) ) ( 1r ` R ) )
38		eqid 2193	. . . . 5 |- ( Base ` (oppr ` R ) ) = ( Base ` (oppr ` R ) )
39		eqid 2193	. . . . 5 |- ( ||r ` (oppr ` R ) ) = ( ||r ` (oppr ` R ) )
40		eqid 2193	. . . . 5 |- ( ||r ` (oppr ` S ) ) = ( ||r ` (oppr ` S ) )
41	38, 39, 40	rhmdvdsr 13671	. . . 4 |- ( ( ( F e. ( (oppr ` R ) RingHom (oppr ` S ) ) /\ A e. ( Base ` (oppr ` R ) ) /\ ( 1r ` R ) e. ( Base ` (oppr ` R ) ) ) /\ A ( ||r ` (oppr ` R ) ) ( 1r ` R ) ) -> ( F ` A ) ( ||r ` (oppr ` S ) ) ( F ` ( 1r ` R ) ) )
42	31, 35, 36, 37, 41	syl31anc 1252	. . 3 |- ( ( F e. ( R RingHom S ) /\ A e. (Unit ` R ) ) -> ( F ` A ) ( ||r ` (oppr ` S ) ) ( F ` ( 1r ` R ) ) )
43	26	breq2d 4041	. . . 4 |- ( F e. ( R RingHom S ) -> ( ( F ` A ) ( ||r ` (oppr ` S ) ) ( F ` ( 1r ` R ) ) <-> ( F ` A ) ( ||r ` (oppr ` S ) ) ( 1r ` S ) ) )
44	43	adantr 276	. . 3 |- ( ( F e. ( R RingHom S ) /\ A e. (Unit ` R ) ) -> ( ( F ` A ) ( ||r ` (oppr ` S ) ) ( F ` ( 1r ` R ) ) <-> ( F ` A ) ( ||r ` (oppr ` S ) ) ( 1r ` S ) ) )
45	42, 44	mpbid 147	. 2 |- ( ( F e. ( R RingHom S ) /\ A e. (Unit ` R ) ) -> ( F ` A ) ( ||r ` (oppr ` S ) ) ( 1r ` S ) )
46		eqidd 2194	. . 3 |- ( ( F e. ( R RingHom S ) /\ A e. (Unit ` R ) ) -> (Unit ` S ) = (Unit ` S ) )
47		eqidd 2194	. . 3 |- ( ( F e. ( R RingHom S ) /\ A e. (Unit ` R ) ) -> ( 1r ` S ) = ( 1r ` S ) )
48		eqidd 2194	. . 3 |- ( ( F e. ( R RingHom S ) /\ A e. (Unit ` R ) ) -> ( ||r ` S ) = ( ||r ` S ) )
49		eqidd 2194	. . 3 |- ( ( F e. ( R RingHom S ) /\ A e. (Unit ` R ) ) -> (oppr ` S ) = (oppr ` S ) )
50		eqidd 2194	. . 3 |- ( ( F e. ( R RingHom S ) /\ A e. (Unit ` R ) ) -> ( ||r ` (oppr ` S ) ) = ( ||r ` (oppr ` S ) ) )
51		rhmrcl2 13652	. . . . 5 |- ( F e. ( R RingHom S ) -> S e. Ring )
52	51	adantr 276	. . . 4 |- ( ( F e. ( R RingHom S ) /\ A e. (Unit ` R ) ) -> S e. Ring )
53		ringsrg 13543	. . . 4 |- ( S e. Ring -> S e. SRing )
54	52, 53	syl 14	. . 3 |- ( ( F e. ( R RingHom S ) /\ A e. (Unit ` R ) ) -> S e. SRing )
55	46, 47, 48, 49, 50, 54	isunitd 13602	. 2 |- ( ( F e. ( R RingHom S ) /\ A e. (Unit ` R ) ) -> ( ( F ` A ) e. (Unit ` S ) <-> ( ( F ` A ) ( ||r ` S ) ( 1r ` S ) /\ ( F ` A ) ( ||r ` (oppr ` S ) ) ( 1r ` S ) ) ) )
56	29, 45, 55	mpbir2and 946	1 |- ( ( F e. ( R RingHom S ) /\ A e. (Unit ` R ) ) -> ( F ` A ) e. (Unit ` S ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1364   e. wcel 2164   class class class wbr 4029   ` cfv 5254  (class class class)co 5918   Basecbs 12618   1rcur 13455  SRingcsrg 13459   Ringcrg 13492  opp_rcoppr 13563   ||_rcdsr 13582  Unitcui 13583   RingHom crh 13646

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 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2166  ax-14 2167  ax-ext 2175  ax-coll 4144  ax-sep 4147  ax-nul 4155  ax-pow 4203  ax-pr 4238  ax-un 4464  ax-setind 4569  ax-cnex 7963  ax-resscn 7964  ax-1cn 7965  ax-1re 7966  ax-icn 7967  ax-addcl 7968  ax-addrcl 7969  ax-mulcl 7970  ax-addcom 7972  ax-addass 7974  ax-i2m1 7977  ax-0lt1 7978  ax-0id 7980  ax-rnegex 7981  ax-pre-ltirr 7984  ax-pre-lttrn 7986  ax-pre-ltadd 7988

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ne 2365  df-nel 2460  df-ral 2477  df-rex 2478  df-reu 2479  df-rmo 2480  df-rab 2481  df-v 2762  df-sbc 2986  df-csb 3081  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3447  df-pw 3603  df-sn 3624  df-pr 3625  df-op 3627  df-uni 3836  df-int 3871  df-iun 3914  df-br 4030  df-opab 4091  df-mpt 4092  df-id 4324  df-xp 4665  df-rel 4666  df-cnv 4667  df-co 4668  df-dm 4669  df-rn 4670  df-res 4671  df-ima 4672  df-iota 5215  df-fun 5256  df-fn 5257  df-f 5258  df-f1 5259  df-fo 5260  df-f1o 5261  df-fv 5262  df-riota 5873  df-ov 5921  df-oprab 5922  df-mpo 5923  df-1st 6193  df-2nd 6194  df-tpos 6298  df-map 6704  df-pnf 8056  df-mnf 8057  df-ltxr 8059  df-inn 8983  df-2 9041  df-3 9042  df-ndx 12621  df-slot 12622  df-base 12624  df-sets 12625  df-plusg 12708  df-mulr 12709  df-0g 12869  df-mgm 12939  df-sgrp 12985  df-mnd 12998  df-mhm 13031  df-grp 13075  df-minusg 13076  df-ghm 13311  df-cmn 13356  df-abl 13357  df-mgp 13417  df-ur 13456  df-srg 13460  df-ring 13494  df-oppr 13564  df-dvdsr 13585  df-unit 13586  df-rhm 13648

This theorem is referenced by:  rhmunitinv  13674