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Theorem elrhmunit 28422
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 458 . . . 4  |-  ( ( F  e.  ( R RingHom  S )  /\  A  e.  (Unit `  R )
)  ->  F  e.  ( R RingHom  S ) )
2 eqid 2429 . . . . . 6  |-  ( Base `  R )  =  (
Base `  R )
3 eqid 2429 . . . . . 6  |-  (Unit `  R )  =  (Unit `  R )
42, 3unitss 17823 . . . . 5  |-  (Unit `  R )  C_  ( Base `  R )
5 simpr 462 . . . . 5  |-  ( ( F  e.  ( R RingHom  S )  /\  A  e.  (Unit `  R )
)  ->  A  e.  (Unit `  R ) )
64, 5sseldi 3468 . . . 4  |-  ( ( F  e.  ( R RingHom  S )  /\  A  e.  (Unit `  R )
)  ->  A  e.  ( Base `  R )
)
7 rhmrcl1 17882 . . . . 5  |-  ( F  e.  ( R RingHom  S
)  ->  R  e.  Ring )
8 eqid 2429 . . . . . 6  |-  ( 1r
`  R )  =  ( 1r `  R
)
92, 8ringidcl 17736 . . . . 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 2429 . . . . . . 7  |-  ( ||r `  R
)  =  ( ||r `  R
)
12 eqid 2429 . . . . . . 7  |-  (oppr `  R
)  =  (oppr `  R
)
13 eqid 2429 . . . . . . 7  |-  ( ||r `  (oppr `  R
) )  =  (
||r `  (oppr
`  R ) )
143, 8, 11, 12, 13isunit 17820 . . . . . 6  |-  ( A  e.  (Unit `  R
)  <->  ( A (
||r `  R ) ( 1r
`  R )  /\  A ( ||r `
 (oppr
`  R ) ) ( 1r `  R
) ) )
155, 14sylib 199 . . . . 5  |-  ( ( F  e.  ( R RingHom  S )  /\  A  e.  (Unit `  R )
)  ->  ( A
( ||r `
 R ) ( 1r `  R )  /\  A ( ||r `  (oppr `  R
) ) ( 1r
`  R ) ) )
1615simpld 460 . . . 4  |-  ( ( F  e.  ( R RingHom  S )  /\  A  e.  (Unit `  R )
)  ->  A ( ||r `  R ) ( 1r
`  R ) )
17 eqid 2429 . . . . 5  |-  ( ||r `  S
)  =  ( ||r `  S
)
182, 11, 17rhmdvdsr 28420 . . . 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 1267 . . 3  |-  ( ( F  e.  ( R RingHom  S )  /\  A  e.  (Unit `  R )
)  ->  ( F `  A ) ( ||r `  S
) ( F `  ( 1r `  R ) ) )
20 eqid 2429 . . . . . 6  |-  ( 1r
`  S )  =  ( 1r `  S
)
218, 20rhm1 17893 . . . . 5  |-  ( F  e.  ( R RingHom  S
)  ->  ( F `  ( 1r `  R
) )  =  ( 1r `  S ) )
2221breq2d 4438 . . . 4  |-  ( F  e.  ( R RingHom  S
)  ->  ( ( F `  A )
( ||r `
 S ) ( F `  ( 1r
`  R ) )  <-> 
( F `  A
) ( ||r `
 S ) ( 1r `  S ) ) )
2322adantr 466 . . 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 213 . 2  |-  ( ( F  e.  ( R RingHom  S )  /\  A  e.  (Unit `  R )
)  ->  ( F `  A ) ( ||r `  S
) ( 1r `  S ) )
25 rhmopp 28421 . . . . 5  |-  ( F  e.  ( R RingHom  S
)  ->  F  e.  ( (oppr
`  R ) RingHom  (oppr `  S
) ) )
2625adantr 466 . . . 4  |-  ( ( F  e.  ( R RingHom  S )  /\  A  e.  (Unit `  R )
)  ->  F  e.  ( (oppr
`  R ) RingHom  (oppr `  S
) ) )
2715simprd 464 . . . 4  |-  ( ( F  e.  ( R RingHom  S )  /\  A  e.  (Unit `  R )
)  ->  A ( ||r `  (oppr
`  R ) ) ( 1r `  R
) )
2812, 2opprbas 17792 . . . . 5  |-  ( Base `  R )  =  (
Base `  (oppr
`  R ) )
29 eqid 2429 . . . . 5  |-  ( ||r `  (oppr `  S
) )  =  (
||r `  (oppr
`  S ) )
3028, 13, 29rhmdvdsr 28420 . . . 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 1267 . . 3  |-  ( ( F  e.  ( R RingHom  S )  /\  A  e.  (Unit `  R )
)  ->  ( F `  A ) ( ||r `  (oppr `  S
) ) ( F `
 ( 1r `  R ) ) )
3221breq2d 4438 . . . 4  |-  ( F  e.  ( R RingHom  S
)  ->  ( ( F `  A )
( ||r `
 (oppr
`  S ) ) ( F `  ( 1r `  R ) )  <-> 
( F `  A
) ( ||r `
 (oppr
`  S ) ) ( 1r `  S
) ) )
3332adantr 466 . . 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 213 . 2  |-  ( ( F  e.  ( R RingHom  S )  /\  A  e.  (Unit `  R )
)  ->  ( F `  A ) ( ||r `  (oppr `  S
) ) ( 1r
`  S ) )
35 eqid 2429 . . 3  |-  (Unit `  S )  =  (Unit `  S )
36 eqid 2429 . . 3  |-  (oppr `  S
)  =  (oppr `  S
)
3735, 20, 17, 36, 29isunit 17820 . 2  |-  ( ( F `  A )  e.  (Unit `  S
)  <->  ( ( F `
 A ) (
||r `  S ) ( 1r
`  S )  /\  ( F `  A ) ( ||r `
 (oppr
`  S ) ) ( 1r `  S
) ) )
3824, 34, 37sylanbrc 668 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 187    /\ wa 370    e. wcel 1870   class class class wbr 4426   ` cfv 5601  (class class class)co 6305   Basecbs 15084   1rcur 17670   Ringcrg 17715  opprcoppr 17785   ||rcdsr 17801  Unitcui 17802   RingHom crh 17875
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1751  ax-6 1797  ax-7 1841  ax-8 1872  ax-9 1874  ax-10 1889  ax-11 1894  ax-12 1907  ax-13 2055  ax-ext 2407  ax-rep 4538  ax-sep 4548  ax-nul 4556  ax-pow 4603  ax-pr 4661  ax-un 6597  ax-cnex 9594  ax-resscn 9595  ax-1cn 9596  ax-icn 9597  ax-addcl 9598  ax-addrcl 9599  ax-mulcl 9600  ax-mulrcl 9601  ax-mulcom 9602  ax-addass 9603  ax-mulass 9604  ax-distr 9605  ax-i2m1 9606  ax-1ne0 9607  ax-1rid 9608  ax-rnegex 9609  ax-rrecex 9610  ax-cnre 9611  ax-pre-lttri 9612  ax-pre-lttrn 9613  ax-pre-ltadd 9614  ax-pre-mulgt0 9615
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1790  df-eu 2270  df-mo 2271  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2579  df-ne 2627  df-nel 2628  df-ral 2787  df-rex 2788  df-reu 2789  df-rmo 2790  df-rab 2791  df-v 3089  df-sbc 3306  df-csb 3402  df-dif 3445  df-un 3447  df-in 3449  df-ss 3456  df-pss 3458  df-nul 3768  df-if 3916  df-pw 3987  df-sn 4003  df-pr 4005  df-tp 4007  df-op 4009  df-uni 4223  df-iun 4304  df-br 4427  df-opab 4485  df-mpt 4486  df-tr 4521  df-eprel 4765  df-id 4769  df-po 4775  df-so 4776  df-fr 4813  df-we 4815  df-xp 4860  df-rel 4861  df-cnv 4862  df-co 4863  df-dm 4864  df-rn 4865  df-res 4866  df-ima 4867  df-pred 5399  df-ord 5445  df-on 5446  df-lim 5447  df-suc 5448  df-iota 5565  df-fun 5603  df-fn 5604  df-f 5605  df-f1 5606  df-fo 5607  df-f1o 5608  df-fv 5609  df-riota 6267  df-ov 6308  df-oprab 6309  df-mpt2 6310  df-om 6707  df-tpos 6981  df-wrecs 7036  df-recs 7098  df-rdg 7136  df-er 7371  df-map 7482  df-en 7578  df-dom 7579  df-sdom 7580  df-pnf 9676  df-mnf 9677  df-xr 9678  df-ltxr 9679  df-le 9680  df-sub 9861  df-neg 9862  df-nn 10610  df-2 10668  df-3 10669  df-ndx 15087  df-slot 15088  df-base 15089  df-sets 15090  df-plusg 15165  df-mulr 15166  df-0g 15299  df-mgm 16439  df-sgrp 16478  df-mnd 16488  df-mhm 16533  df-grp 16624  df-ghm 16832  df-mgp 17659  df-ur 17671  df-ring 17717  df-oppr 17786  df-dvdsr 17804  df-unit 17805  df-rnghom 17878
This theorem is referenced by:  rhmunitinv  28424  qqhval2lem  28624
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