Theorem elrhmunit 14322

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 2233	. . . . 5 |- ( ( F e. ( R RingHom S ) /\ A e. (Unit ` R ) ) -> ( Base ` R ) = ( Base ` R ) )
3		eqidd 2233	. . . . 5 |- ( ( F e. ( R RingHom S ) /\ A e. (Unit ` R ) ) -> (Unit ` R ) = (Unit ` R ) )
4		rhmrcl1 14300	. . . . . . 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 14191	. . . . . 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 14253	. . . 4 |- ( ( F e. ( R RingHom S ) /\ A e. (Unit ` R ) ) -> A e. ( Base ` R ) )
10		eqid 2232	. . . . . 6 |- ( Base ` R ) = ( Base ` R )
11		eqid 2232	. . . . . 6 |- ( 1r ` R ) = ( 1r ` R )
12	10, 11	ringidcl 14164	. . . . 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 2233	. . . . . . 7 |- ( ( F e. ( R RingHom S ) /\ A e. (Unit ` R ) ) -> ( 1r ` R ) = ( 1r ` R ) )
15		eqidd 2233	. . . . . . 7 |- ( ( F e. ( R RingHom S ) /\ A e. (Unit ` R ) ) -> ( ||r ` R ) = ( ||r ` R ) )
16		eqidd 2233	. . . . . . 7 |- ( ( F e. ( R RingHom S ) /\ A e. (Unit ` R ) ) -> (oppr ` R ) = (oppr ` R ) )
17		eqidd 2233	. . . . . . 7 |- ( ( F e. ( R RingHom S ) /\ A e. (Unit ` R ) ) -> ( ||r ` (oppr ` R ) ) = ( ||r ` (oppr ` R ) ) )
18	3, 14, 15, 16, 17, 7	isunitd 14251	. . . . . 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 2232	. . . . 5 |- ( ||r ` R ) = ( ||r ` R )
22		eqid 2232	. . . . 5 |- ( ||r ` S ) = ( ||r ` S )
23	10, 21, 22	rhmdvdsr 14320	. . . 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 1277	. . 3 |- ( ( F e. ( R RingHom S ) /\ A e. (Unit ` R ) ) -> ( F ` A ) ( ||r ` S ) ( F ` ( 1r ` R ) ) )
25		eqid 2232	. . . . . 6 |- ( 1r ` S ) = ( 1r ` S )
26	11, 25	rhm1 14312	. . . . 5 |- ( F e. ( R RingHom S ) -> ( F ` ( 1r ` R ) ) = ( 1r ` S ) )
27	26	breq2d 4121	. . . 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 14321	. . . . 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 2232	. . . . . . 7 |- (oppr ` R ) = (oppr ` R )
33	32, 10	opprbasg 14219	. . . . . 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 2311	. . . 4 |- ( ( F e. ( R RingHom S ) /\ A e. (Unit ` R ) ) -> A e. ( Base ` (oppr ` R ) ) )
36	13, 34	eleqtrd 2311	. . . 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 2232	. . . . 5 |- ( Base ` (oppr ` R ) ) = ( Base ` (oppr ` R ) )
39		eqid 2232	. . . . 5 |- ( ||r ` (oppr ` R ) ) = ( ||r ` (oppr ` R ) )
40		eqid 2232	. . . . 5 |- ( ||r ` (oppr ` S ) ) = ( ||r ` (oppr ` S ) )
41	38, 39, 40	rhmdvdsr 14320	. . . 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 1277	. . 3 |- ( ( F e. ( R RingHom S ) /\ A e. (Unit ` R ) ) -> ( F ` A ) ( ||r ` (oppr ` S ) ) ( F ` ( 1r ` R ) ) )
43	26	breq2d 4121	. . . 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 2233	. . 3 |- ( ( F e. ( R RingHom S ) /\ A e. (Unit ` R ) ) -> (Unit ` S ) = (Unit ` S ) )
47		eqidd 2233	. . 3 |- ( ( F e. ( R RingHom S ) /\ A e. (Unit ` R ) ) -> ( 1r ` S ) = ( 1r ` S ) )
48		eqidd 2233	. . 3 |- ( ( F e. ( R RingHom S ) /\ A e. (Unit ` R ) ) -> ( ||r ` S ) = ( ||r ` S ) )
49		eqidd 2233	. . 3 |- ( ( F e. ( R RingHom S ) /\ A e. (Unit ` R ) ) -> (oppr ` S ) = (oppr ` S ) )
50		eqidd 2233	. . 3 |- ( ( F e. ( R RingHom S ) /\ A e. (Unit ` R ) ) -> ( ||r ` (oppr ` S ) ) = ( ||r ` (oppr ` S ) ) )
51		rhmrcl2 14301	. . . . 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 14191	. . . 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 14251	. 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 953	1 |- ( ( F e. ( R RingHom S ) /\ A e. (Unit ` R ) ) -> ( F ` A ) e. (Unit ` S ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1398   e. wcel 2203   class class class wbr 4109   ` cfv 5352  (class class class)co 6050   Basecbs 13212   1rcur 14103  SRingcsrg 14107   Ringcrg 14140  opp_rcoppr 14211   ||_rcdsr 14230  Unitcui 14231   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-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-rhm 14297

This theorem is referenced by:  rhmunitinv  14323