Theorem elrhmunit 14178

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 2230	. . . . 5 |- ( ( F e. ( R RingHom S ) /\ A e. (Unit ` R ) ) -> ( Base ` R ) = ( Base ` R ) )
3		eqidd 2230	. . . . 5 |- ( ( F e. ( R RingHom S ) /\ A e. (Unit ` R ) ) -> (Unit ` R ) = (Unit ` R ) )
4		rhmrcl1 14156	. . . . . . 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 14047	. . . . . 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 14109	. . . 4 |- ( ( F e. ( R RingHom S ) /\ A e. (Unit ` R ) ) -> A e. ( Base ` R ) )
10		eqid 2229	. . . . . 6 |- ( Base ` R ) = ( Base ` R )
11		eqid 2229	. . . . . 6 |- ( 1r ` R ) = ( 1r ` R )
12	10, 11	ringidcl 14020	. . . . 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 2230	. . . . . . 7 |- ( ( F e. ( R RingHom S ) /\ A e. (Unit ` R ) ) -> ( 1r ` R ) = ( 1r ` R ) )
15		eqidd 2230	. . . . . . 7 |- ( ( F e. ( R RingHom S ) /\ A e. (Unit ` R ) ) -> ( ||r ` R ) = ( ||r ` R ) )
16		eqidd 2230	. . . . . . 7 |- ( ( F e. ( R RingHom S ) /\ A e. (Unit ` R ) ) -> (oppr ` R ) = (oppr ` R ) )
17		eqidd 2230	. . . . . . 7 |- ( ( F e. ( R RingHom S ) /\ A e. (Unit ` R ) ) -> ( ||r ` (oppr ` R ) ) = ( ||r ` (oppr ` R ) ) )
18	3, 14, 15, 16, 17, 7	isunitd 14107	. . . . . 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 2229	. . . . 5 |- ( ||r ` R ) = ( ||r ` R )
22		eqid 2229	. . . . 5 |- ( ||r ` S ) = ( ||r ` S )
23	10, 21, 22	rhmdvdsr 14176	. . . 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 1274	. . 3 |- ( ( F e. ( R RingHom S ) /\ A e. (Unit ` R ) ) -> ( F ` A ) ( ||r ` S ) ( F ` ( 1r ` R ) ) )
25		eqid 2229	. . . . . 6 |- ( 1r ` S ) = ( 1r ` S )
26	11, 25	rhm1 14168	. . . . 5 |- ( F e. ( R RingHom S ) -> ( F ` ( 1r ` R ) ) = ( 1r ` S ) )
27	26	breq2d 4096	. . . 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 14177	. . . . 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 2229	. . . . . . 7 |- (oppr ` R ) = (oppr ` R )
33	32, 10	opprbasg 14075	. . . . . 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 2308	. . . 4 |- ( ( F e. ( R RingHom S ) /\ A e. (Unit ` R ) ) -> A e. ( Base ` (oppr ` R ) ) )
36	13, 34	eleqtrd 2308	. . . 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 2229	. . . . 5 |- ( Base ` (oppr ` R ) ) = ( Base ` (oppr ` R ) )
39		eqid 2229	. . . . 5 |- ( ||r ` (oppr ` R ) ) = ( ||r ` (oppr ` R ) )
40		eqid 2229	. . . . 5 |- ( ||r ` (oppr ` S ) ) = ( ||r ` (oppr ` S ) )
41	38, 39, 40	rhmdvdsr 14176	. . . 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 1274	. . 3 |- ( ( F e. ( R RingHom S ) /\ A e. (Unit ` R ) ) -> ( F ` A ) ( ||r ` (oppr ` S ) ) ( F ` ( 1r ` R ) ) )
43	26	breq2d 4096	. . . 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 2230	. . 3 |- ( ( F e. ( R RingHom S ) /\ A e. (Unit ` R ) ) -> (Unit ` S ) = (Unit ` S ) )
47		eqidd 2230	. . 3 |- ( ( F e. ( R RingHom S ) /\ A e. (Unit ` R ) ) -> ( 1r ` S ) = ( 1r ` S ) )
48		eqidd 2230	. . 3 |- ( ( F e. ( R RingHom S ) /\ A e. (Unit ` R ) ) -> ( ||r ` S ) = ( ||r ` S ) )
49		eqidd 2230	. . 3 |- ( ( F e. ( R RingHom S ) /\ A e. (Unit ` R ) ) -> (oppr ` S ) = (oppr ` S ) )
50		eqidd 2230	. . 3 |- ( ( F e. ( R RingHom S ) /\ A e. (Unit ` R ) ) -> ( ||r ` (oppr ` S ) ) = ( ||r ` (oppr ` S ) ) )
51		rhmrcl2 14157	. . . . 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 14047	. . . 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 14107	. 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 950	1 |- ( ( F e. ( R RingHom S ) /\ A e. (Unit ` R ) ) -> ( F ` A ) e. (Unit ` S ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1395   e. wcel 2200   class class class wbr 4084   ` cfv 5322  (class class class)co 6011   Basecbs 13069   1rcur 13959  SRingcsrg 13963   Ringcrg 13996  opp_rcoppr 14067   ||_rcdsr 14086  Unitcui 14087   RingHom crh 14151

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 4200  ax-sep 4203  ax-nul 4211  ax-pow 4260  ax-pr 4295  ax-un 4526  ax-setind 4631  ax-cnex 8111  ax-resscn 8112  ax-1cn 8113  ax-1re 8114  ax-icn 8115  ax-addcl 8116  ax-addrcl 8117  ax-mulcl 8118  ax-addcom 8120  ax-addass 8122  ax-i2m1 8125  ax-0lt1 8126  ax-0id 8128  ax-rnegex 8129  ax-pre-ltirr 8132  ax-pre-lttrn 8134  ax-pre-ltadd 8136

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 3890  df-int 3925  df-iun 3968  df-br 4085  df-opab 4147  df-mpt 4148  df-id 4386  df-xp 4727  df-rel 4728  df-cnv 4729  df-co 4730  df-dm 4731  df-rn 4732  df-res 4733  df-ima 4734  df-iota 5282  df-fun 5324  df-fn 5325  df-f 5326  df-f1 5327  df-fo 5328  df-f1o 5329  df-fv 5330  df-riota 5964  df-ov 6014  df-oprab 6015  df-mpo 6016  df-1st 6296  df-2nd 6297  df-tpos 6404  df-map 6812  df-pnf 8204  df-mnf 8205  df-ltxr 8207  df-inn 9132  df-2 9190  df-3 9191  df-ndx 13072  df-slot 13073  df-base 13075  df-sets 13076  df-plusg 13160  df-mulr 13161  df-0g 13328  df-mgm 13426  df-sgrp 13472  df-mnd 13487  df-mhm 13529  df-grp 13573  df-minusg 13574  df-ghm 13815  df-cmn 13860  df-abl 13861  df-mgp 13921  df-ur 13960  df-srg 13964  df-ring 13998  df-oppr 14068  df-dvdsr 14089  df-unit 14090  df-rhm 14153

This theorem is referenced by:  rhmunitinv  14179