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Theorem elrhmunit 29820
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
StepHypRef Expression
1 simpl 473 . . . 4 |- ( ( F e. ( R RingHom S ) /\ A e. (Unit ` R ) ) -> F e. ( R RingHom S ) )
2 eqid 2622 . . . . . 6 |- ( Base ` R ) = ( Base ` R )
3 eqid 2622 . . . . . 6 |- (Unit ` R ) = (Unit ` R )
42, 3unitss 18660 . . . . 5 |- (Unit ` R ) C_ ( Base ` R )
5 simpr 477 . . . . 5 |- ( ( F e. ( R RingHom S ) /\ A e. (Unit ` R ) ) -> A e. (Unit ` R ) )
64, 5sseldi 3601 . . . 4 |- ( ( F e. ( R RingHom S ) /\ A e. (Unit ` R ) ) -> A e. ( Base ` R ) )
7 rhmrcl1 18719 . . . . 5 |- ( F e. ( R RingHom S ) -> R e. Ring )
8 eqid 2622 . . . . . 6 |- ( 1r ` R ) = ( 1r ` R )
92, 8ringidcl 18568 . . . . 5 |- ( R e. Ring -> ( 1r ` R ) e. ( Base ` R ) )
101, 7, 93syl 18 . . . 4 |- ( ( F e. ( R RingHom S ) /\ A e. (Unit ` R ) ) -> ( 1r ` R ) e. ( Base ` R ) )
11 eqid 2622 . . . . . . 7 |- ( ||r ` R ) = ( ||r ` R )
12 eqid 2622 . . . . . . 7 |- (oppr ` R ) = (oppr ` R )
13 eqid 2622 . . . . . . 7 |- ( ||r ` (oppr ` R ) ) = ( ||r ` (oppr ` R ) )
143, 8, 11, 12, 13isunit 18657 . . . . . 6 |- ( A e. (Unit ` R ) <-> ( A ( ||r ` R ) ( 1r ` R ) /\ A ( ||r ` (oppr ` R ) ) ( 1r ` R ) ) )
155, 14sylib 208 . . . . 5 |- ( ( F e. ( R RingHom S ) /\ A e. (Unit ` R ) ) -> ( A ( ||r ` R ) ( 1r ` R ) /\ A ( ||r ` (oppr ` R ) ) ( 1r ` R ) ) )
1615simpld 475 . . . 4 |- ( ( F e. ( R RingHom S ) /\ A e. (Unit ` R ) ) -> A ( ||r ` R ) ( 1r ` R ) )
17 eqid 2622 . . . . 5 |- ( ||r ` S ) = ( ||r ` S )
182, 11, 17rhmdvdsr 29818 . . . 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 ) ) )
191, 6, 10, 16, 18syl31anc 1329 . . 3 |- ( ( F e. ( R RingHom S ) /\ A e. (Unit ` R ) ) -> ( F ` A ) ( ||r ` S ) ( F ` ( 1r ` R ) ) )
20 eqid 2622 . . . . . 6 |- ( 1r ` S ) = ( 1r ` S )
218, 20rhm1 18730 . . . . 5 |- ( F e. ( R RingHom S ) -> ( F ` ( 1r ` R ) ) = ( 1r ` S ) )
2221breq2d 4665 . . . 4 |- ( F e. ( R RingHom S ) -> ( ( F ` A ) ( ||r ` S ) ( F ` ( 1r ` R ) ) <-> ( F ` A ) ( ||r ` S ) ( 1r ` S ) ) )
2322adantr 481 . . 3 |- ( ( F e. ( R RingHom S ) /\ A e. (Unit ` R ) ) -> ( ( F ` A ) ( ||r ` S ) ( F ` ( 1r ` R ) ) <-> ( F ` A ) ( ||r ` S ) ( 1r ` S ) ) )
2419, 23mpbid 222 . 2 |- ( ( F e. ( R RingHom S ) /\ A e. (Unit ` R ) ) -> ( F ` A ) ( ||r ` S ) ( 1r ` S ) )
25 rhmopp 29819 . . . . 5 |- ( F e. ( R RingHom S ) -> F e. ( (oppr ` R ) RingHom (oppr ` S ) ) )
2625adantr 481 . . . 4 |- ( ( F e. ( R RingHom S ) /\ A e. (Unit ` R ) ) -> F e. ( (oppr ` R ) RingHom (oppr ` S ) ) )
2715simprd 479 . . . 4 |- ( ( F e. ( R RingHom S ) /\ A e. (Unit ` R ) ) -> A ( ||r ` (oppr ` R ) ) ( 1r ` R ) )
2812, 2opprbas 18629 . . . . 5 |- ( Base ` R ) = ( Base ` (oppr ` R ) )
29 eqid 2622 . . . . 5 |- ( ||r ` (oppr ` S ) ) = ( ||r ` (oppr ` S ) )
3028, 13, 29rhmdvdsr 29818 . . . 4 |- ( ( ( F e. ( (oppr ` R ) RingHom (oppr ` S ) ) /\ A e. ( Base ` R ) /\ ( 1r ` R ) e. ( Base ` R ) ) /\ A ( ||r ` (oppr ` R ) ) ( 1r ` R ) ) -> ( F ` A ) ( ||r ` (oppr ` S ) ) ( F ` ( 1r ` R ) ) )
3126, 6, 10, 27, 30syl31anc 1329 . . 3 |- ( ( F e. ( R RingHom S ) /\ A e. (Unit ` R ) ) -> ( F ` A ) ( ||r ` (oppr ` S ) ) ( F ` ( 1r ` R ) ) )
3221breq2d 4665 . . . 4 |- ( F e. ( R RingHom S ) -> ( ( F ` A ) ( ||r ` (oppr ` S ) ) ( F ` ( 1r ` R ) ) <-> ( F ` A ) ( ||r ` (oppr ` S ) ) ( 1r ` S ) ) )
3332adantr 481 . . 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 ) ) )
3431, 33mpbid 222 . 2 |- ( ( F e. ( R RingHom S ) /\ A e. (Unit ` R ) ) -> ( F ` A ) ( ||r ` (oppr ` S ) ) ( 1r ` S ) )
35 eqid 2622 . . 3 |- (Unit ` S ) = (Unit ` S )
36 eqid 2622 . . 3 |- (oppr ` S ) = (oppr ` S )
3735, 20, 17, 36, 29isunit 18657 . 2 |- ( ( F ` A ) e. (Unit ` S ) <-> ( ( F ` A ) ( ||r ` S ) ( 1r ` S ) /\ ( F ` A ) ( ||r ` (oppr ` S ) ) ( 1r ` S ) ) )
3824, 34, 37sylanbrc 698 1 |- ( ( F e. ( R RingHom S ) /\ A e. (Unit ` R ) ) -> ( F ` A ) e. (Unit ` S ) )
Colors of variables: wff setvar class
Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   e. wcel 1990   class class class wbr 4653   ` cfv 5888  (class class class)co 6650   Basecbs 15857   1rcur 18501   Ringcrg 18547  opprcoppr 18622   ||rcdsr 18638  Unitcui 18639   RingHom crh 18712
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-tpos 7352  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-er 7742  df-map 7859  df-en 7956  df-dom 7957  df-sdom 7958  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-nn 11021  df-2 11079  df-3 11080  df-ndx 15860  df-slot 15861  df-base 15863  df-sets 15864  df-plusg 15954  df-mulr 15955  df-0g 16102  df-mgm 17242  df-sgrp 17284  df-mnd 17295  df-mhm 17335  df-grp 17425  df-ghm 17658  df-mgp 18490  df-ur 18502  df-ring 18549  df-oppr 18623  df-dvdsr 18641  df-unit 18642  df-rnghom 18715
This theorem is referenced by:  rhmunitinv  29822  qqhval2lem  30025
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