Theorem elrhmunit 13939

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 2206	. . . . 5 |- ( ( F e. ( R RingHom S ) /\ A e. (Unit ` R ) ) -> ( Base ` R ) = ( Base ` R ) )
3		eqidd 2206	. . . . 5 |- ( ( F e. ( R RingHom S ) /\ A e. (Unit ` R ) ) -> (Unit ` R ) = (Unit ` R ) )
4		rhmrcl1 13917	. . . . . . 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 13809	. . . . . 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 13870	. . . 4 |- ( ( F e. ( R RingHom S ) /\ A e. (Unit ` R ) ) -> A e. ( Base ` R ) )
10		eqid 2205	. . . . . 6 |- ( Base ` R ) = ( Base ` R )
11		eqid 2205	. . . . . 6 |- ( 1r ` R ) = ( 1r ` R )
12	10, 11	ringidcl 13782	. . . . 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 2206	. . . . . . 7 |- ( ( F e. ( R RingHom S ) /\ A e. (Unit ` R ) ) -> ( 1r ` R ) = ( 1r ` R ) )
15		eqidd 2206	. . . . . . 7 |- ( ( F e. ( R RingHom S ) /\ A e. (Unit ` R ) ) -> ( ||r ` R ) = ( ||r ` R ) )
16		eqidd 2206	. . . . . . 7 |- ( ( F e. ( R RingHom S ) /\ A e. (Unit ` R ) ) -> (oppr ` R ) = (oppr ` R ) )
17		eqidd 2206	. . . . . . 7 |- ( ( F e. ( R RingHom S ) /\ A e. (Unit ` R ) ) -> ( ||r ` (oppr ` R ) ) = ( ||r ` (oppr ` R ) ) )
18	3, 14, 15, 16, 17, 7	isunitd 13868	. . . . . 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 2205	. . . . 5 |- ( ||r ` R ) = ( ||r ` R )
22		eqid 2205	. . . . 5 |- ( ||r ` S ) = ( ||r ` S )
23	10, 21, 22	rhmdvdsr 13937	. . . 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 1253	. . 3 |- ( ( F e. ( R RingHom S ) /\ A e. (Unit ` R ) ) -> ( F ` A ) ( ||r ` S ) ( F ` ( 1r ` R ) ) )
25		eqid 2205	. . . . . 6 |- ( 1r ` S ) = ( 1r ` S )
26	11, 25	rhm1 13929	. . . . 5 |- ( F e. ( R RingHom S ) -> ( F ` ( 1r ` R ) ) = ( 1r ` S ) )
27	26	breq2d 4056	. . . 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 13938	. . . . 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 2205	. . . . . . 7 |- (oppr ` R ) = (oppr ` R )
33	32, 10	opprbasg 13837	. . . . . 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 2284	. . . 4 |- ( ( F e. ( R RingHom S ) /\ A e. (Unit ` R ) ) -> A e. ( Base ` (oppr ` R ) ) )
36	13, 34	eleqtrd 2284	. . . 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 2205	. . . . 5 |- ( Base ` (oppr ` R ) ) = ( Base ` (oppr ` R ) )
39		eqid 2205	. . . . 5 |- ( ||r ` (oppr ` R ) ) = ( ||r ` (oppr ` R ) )
40		eqid 2205	. . . . 5 |- ( ||r ` (oppr ` S ) ) = ( ||r ` (oppr ` S ) )
41	38, 39, 40	rhmdvdsr 13937	. . . 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 1253	. . 3 |- ( ( F e. ( R RingHom S ) /\ A e. (Unit ` R ) ) -> ( F ` A ) ( ||r ` (oppr ` S ) ) ( F ` ( 1r ` R ) ) )
43	26	breq2d 4056	. . . 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 2206	. . 3 |- ( ( F e. ( R RingHom S ) /\ A e. (Unit ` R ) ) -> (Unit ` S ) = (Unit ` S ) )
47		eqidd 2206	. . 3 |- ( ( F e. ( R RingHom S ) /\ A e. (Unit ` R ) ) -> ( 1r ` S ) = ( 1r ` S ) )
48		eqidd 2206	. . 3 |- ( ( F e. ( R RingHom S ) /\ A e. (Unit ` R ) ) -> ( ||r ` S ) = ( ||r ` S ) )
49		eqidd 2206	. . 3 |- ( ( F e. ( R RingHom S ) /\ A e. (Unit ` R ) ) -> (oppr ` S ) = (oppr ` S ) )
50		eqidd 2206	. . 3 |- ( ( F e. ( R RingHom S ) /\ A e. (Unit ` R ) ) -> ( ||r ` (oppr ` S ) ) = ( ||r ` (oppr ` S ) ) )
51		rhmrcl2 13918	. . . . 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 13809	. . . 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 13868	. 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 947	1 |- ( ( F e. ( R RingHom S ) /\ A e. (Unit ` R ) ) -> ( F ` A ) e. (Unit ` S ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1373   e. wcel 2176   class class class wbr 4044   ` cfv 5271  (class class class)co 5944   Basecbs 12832   1rcur 13721  SRingcsrg 13725   Ringcrg 13758  opp_rcoppr 13829   ||_rcdsr 13848  Unitcui 13849   RingHom crh 13912

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 711  ax-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-13 2178  ax-14 2179  ax-ext 2187  ax-coll 4159  ax-sep 4162  ax-nul 4170  ax-pow 4218  ax-pr 4253  ax-un 4480  ax-setind 4585  ax-cnex 8016  ax-resscn 8017  ax-1cn 8018  ax-1re 8019  ax-icn 8020  ax-addcl 8021  ax-addrcl 8022  ax-mulcl 8023  ax-addcom 8025  ax-addass 8027  ax-i2m1 8030  ax-0lt1 8031  ax-0id 8033  ax-rnegex 8034  ax-pre-ltirr 8037  ax-pre-lttrn 8039  ax-pre-ltadd 8041

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-fal 1379  df-nf 1484  df-sb 1786  df-eu 2057  df-mo 2058  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-ne 2377  df-nel 2472  df-ral 2489  df-rex 2490  df-reu 2491  df-rmo 2492  df-rab 2493  df-v 2774  df-sbc 2999  df-csb 3094  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3461  df-pw 3618  df-sn 3639  df-pr 3640  df-op 3642  df-uni 3851  df-int 3886  df-iun 3929  df-br 4045  df-opab 4106  df-mpt 4107  df-id 4340  df-xp 4681  df-rel 4682  df-cnv 4683  df-co 4684  df-dm 4685  df-rn 4686  df-res 4687  df-ima 4688  df-iota 5232  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-riota 5899  df-ov 5947  df-oprab 5948  df-mpo 5949  df-1st 6226  df-2nd 6227  df-tpos 6331  df-map 6737  df-pnf 8109  df-mnf 8110  df-ltxr 8112  df-inn 9037  df-2 9095  df-3 9096  df-ndx 12835  df-slot 12836  df-base 12838  df-sets 12839  df-plusg 12922  df-mulr 12923  df-0g 13090  df-mgm 13188  df-sgrp 13234  df-mnd 13249  df-mhm 13291  df-grp 13335  df-minusg 13336  df-ghm 13577  df-cmn 13622  df-abl 13623  df-mgp 13683  df-ur 13722  df-srg 13726  df-ring 13760  df-oppr 13830  df-dvdsr 13851  df-unit 13852  df-rhm 13914

This theorem is referenced by:  rhmunitinv  13940