Theorem elrhmunit 13910

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 2205	. . . . 5 |- ( ( F e. ( R RingHom S ) /\ A e. (Unit ` R ) ) -> ( Base ` R ) = ( Base ` R ) )
3		eqidd 2205	. . . . 5 |- ( ( F e. ( R RingHom S ) /\ A e. (Unit ` R ) ) -> (Unit ` R ) = (Unit ` R ) )
4		rhmrcl1 13888	. . . . . . 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 13780	. . . . . 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 13841	. . . 4 |- ( ( F e. ( R RingHom S ) /\ A e. (Unit ` R ) ) -> A e. ( Base ` R ) )
10		eqid 2204	. . . . . 6 |- ( Base ` R ) = ( Base ` R )
11		eqid 2204	. . . . . 6 |- ( 1r ` R ) = ( 1r ` R )
12	10, 11	ringidcl 13753	. . . . 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 2205	. . . . . . 7 |- ( ( F e. ( R RingHom S ) /\ A e. (Unit ` R ) ) -> ( 1r ` R ) = ( 1r ` R ) )
15		eqidd 2205	. . . . . . 7 |- ( ( F e. ( R RingHom S ) /\ A e. (Unit ` R ) ) -> ( ||r ` R ) = ( ||r ` R ) )
16		eqidd 2205	. . . . . . 7 |- ( ( F e. ( R RingHom S ) /\ A e. (Unit ` R ) ) -> (oppr ` R ) = (oppr ` R ) )
17		eqidd 2205	. . . . . . 7 |- ( ( F e. ( R RingHom S ) /\ A e. (Unit ` R ) ) -> ( ||r ` (oppr ` R ) ) = ( ||r ` (oppr ` R ) ) )
18	3, 14, 15, 16, 17, 7	isunitd 13839	. . . . . 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 2204	. . . . 5 |- ( ||r ` R ) = ( ||r ` R )
22		eqid 2204	. . . . 5 |- ( ||r ` S ) = ( ||r ` S )
23	10, 21, 22	rhmdvdsr 13908	. . . 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 2204	. . . . . 6 |- ( 1r ` S ) = ( 1r ` S )
26	11, 25	rhm1 13900	. . . . 5 |- ( F e. ( R RingHom S ) -> ( F ` ( 1r ` R ) ) = ( 1r ` S ) )
27	26	breq2d 4055	. . . 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 13909	. . . . 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 2204	. . . . . . 7 |- (oppr ` R ) = (oppr ` R )
33	32, 10	opprbasg 13808	. . . . . 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 2283	. . . 4 |- ( ( F e. ( R RingHom S ) /\ A e. (Unit ` R ) ) -> A e. ( Base ` (oppr ` R ) ) )
36	13, 34	eleqtrd 2283	. . . 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 2204	. . . . 5 |- ( Base ` (oppr ` R ) ) = ( Base ` (oppr ` R ) )
39		eqid 2204	. . . . 5 |- ( ||r ` (oppr ` R ) ) = ( ||r ` (oppr ` R ) )
40		eqid 2204	. . . . 5 |- ( ||r ` (oppr ` S ) ) = ( ||r ` (oppr ` S ) )
41	38, 39, 40	rhmdvdsr 13908	. . . 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 4055	. . . 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 2205	. . 3 |- ( ( F e. ( R RingHom S ) /\ A e. (Unit ` R ) ) -> (Unit ` S ) = (Unit ` S ) )
47		eqidd 2205	. . 3 |- ( ( F e. ( R RingHom S ) /\ A e. (Unit ` R ) ) -> ( 1r ` S ) = ( 1r ` S ) )
48		eqidd 2205	. . 3 |- ( ( F e. ( R RingHom S ) /\ A e. (Unit ` R ) ) -> ( ||r ` S ) = ( ||r ` S ) )
49		eqidd 2205	. . 3 |- ( ( F e. ( R RingHom S ) /\ A e. (Unit ` R ) ) -> (oppr ` S ) = (oppr ` S ) )
50		eqidd 2205	. . 3 |- ( ( F e. ( R RingHom S ) /\ A e. (Unit ` R ) ) -> ( ||r ` (oppr ` S ) ) = ( ||r ` (oppr ` S ) ) )
51		rhmrcl2 13889	. . . . 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 13780	. . . 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 13839	. 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 1372   e. wcel 2175   class class class wbr 4043   ` cfv 5270  (class class class)co 5943   Basecbs 12803   1rcur 13692  SRingcsrg 13696   Ringcrg 13729  opp_rcoppr 13800   ||_rcdsr 13819  Unitcui 13820   RingHom crh 13883

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 1469  ax-7 1470  ax-gen 1471  ax-ie1 1515  ax-ie2 1516  ax-8 1526  ax-10 1527  ax-11 1528  ax-i12 1529  ax-bndl 1531  ax-4 1532  ax-17 1548  ax-i9 1552  ax-ial 1556  ax-i5r 1557  ax-13 2177  ax-14 2178  ax-ext 2186  ax-coll 4158  ax-sep 4161  ax-nul 4169  ax-pow 4217  ax-pr 4252  ax-un 4479  ax-setind 4584  ax-cnex 8015  ax-resscn 8016  ax-1cn 8017  ax-1re 8018  ax-icn 8019  ax-addcl 8020  ax-addrcl 8021  ax-mulcl 8022  ax-addcom 8024  ax-addass 8026  ax-i2m1 8029  ax-0lt1 8030  ax-0id 8032  ax-rnegex 8033  ax-pre-ltirr 8036  ax-pre-lttrn 8038  ax-pre-ltadd 8040

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1375  df-fal 1378  df-nf 1483  df-sb 1785  df-eu 2056  df-mo 2057  df-clab 2191  df-cleq 2197  df-clel 2200  df-nfc 2336  df-ne 2376  df-nel 2471  df-ral 2488  df-rex 2489  df-reu 2490  df-rmo 2491  df-rab 2492  df-v 2773  df-sbc 2998  df-csb 3093  df-dif 3167  df-un 3169  df-in 3171  df-ss 3178  df-nul 3460  df-pw 3617  df-sn 3638  df-pr 3639  df-op 3641  df-uni 3850  df-int 3885  df-iun 3928  df-br 4044  df-opab 4105  df-mpt 4106  df-id 4339  df-xp 4680  df-rel 4681  df-cnv 4682  df-co 4683  df-dm 4684  df-rn 4685  df-res 4686  df-ima 4687  df-iota 5231  df-fun 5272  df-fn 5273  df-f 5274  df-f1 5275  df-fo 5276  df-f1o 5277  df-fv 5278  df-riota 5898  df-ov 5946  df-oprab 5947  df-mpo 5948  df-1st 6225  df-2nd 6226  df-tpos 6330  df-map 6736  df-pnf 8108  df-mnf 8109  df-ltxr 8111  df-inn 9036  df-2 9094  df-3 9095  df-ndx 12806  df-slot 12807  df-base 12809  df-sets 12810  df-plusg 12893  df-mulr 12894  df-0g 13061  df-mgm 13159  df-sgrp 13205  df-mnd 13220  df-mhm 13262  df-grp 13306  df-minusg 13307  df-ghm 13548  df-cmn 13593  df-abl 13594  df-mgp 13654  df-ur 13693  df-srg 13697  df-ring 13731  df-oppr 13801  df-dvdsr 13822  df-unit 13823  df-rhm 13885

This theorem is referenced by:  rhmunitinv  13911