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Theorem elrhmunit 27501
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 457 . . . 4  |-  ( ( F  e.  ( R RingHom  S )  /\  A  e.  (Unit `  R )
)  ->  F  e.  ( R RingHom  S ) )
2 eqid 2467 . . . . . 6  |-  ( Base `  R )  =  (
Base `  R )
3 eqid 2467 . . . . . 6  |-  (Unit `  R )  =  (Unit `  R )
42, 3unitss 17110 . . . . 5  |-  (Unit `  R )  C_  ( Base `  R )
5 simpr 461 . . . . 5  |-  ( ( F  e.  ( R RingHom  S )  /\  A  e.  (Unit `  R )
)  ->  A  e.  (Unit `  R ) )
64, 5sseldi 3502 . . . 4  |-  ( ( F  e.  ( R RingHom  S )  /\  A  e.  (Unit `  R )
)  ->  A  e.  ( Base `  R )
)
7 rhmrcl1 17169 . . . . 5  |-  ( F  e.  ( R RingHom  S
)  ->  R  e.  Ring )
8 eqid 2467 . . . . . 6  |-  ( 1r
`  R )  =  ( 1r `  R
)
92, 8rngidcl 17020 . . . . 5  |-  ( R  e.  Ring  ->  ( 1r
`  R )  e.  ( Base `  R
) )
101, 7, 93syl 20 . . . 4  |-  ( ( F  e.  ( R RingHom  S )  /\  A  e.  (Unit `  R )
)  ->  ( 1r `  R )  e.  (
Base `  R )
)
11 eqid 2467 . . . . . . 7  |-  ( ||r `  R
)  =  ( ||r `  R
)
12 eqid 2467 . . . . . . 7  |-  (oppr `  R
)  =  (oppr `  R
)
13 eqid 2467 . . . . . . 7  |-  ( ||r `  (oppr `  R
) )  =  (
||r `  (oppr
`  R ) )
143, 8, 11, 12, 13isunit 17107 . . . . . 6  |-  ( A  e.  (Unit `  R
)  <->  ( A (
||r `  R ) ( 1r
`  R )  /\  A ( ||r `
 (oppr
`  R ) ) ( 1r `  R
) ) )
155, 14sylib 196 . . . . 5  |-  ( ( F  e.  ( R RingHom  S )  /\  A  e.  (Unit `  R )
)  ->  ( A
( ||r `
 R ) ( 1r `  R )  /\  A ( ||r `  (oppr `  R
) ) ( 1r
`  R ) ) )
1615simpld 459 . . . 4  |-  ( ( F  e.  ( R RingHom  S )  /\  A  e.  (Unit `  R )
)  ->  A ( ||r `  R ) ( 1r
`  R ) )
17 eqid 2467 . . . . 5  |-  ( ||r `  S
)  =  ( ||r `  S
)
182, 11, 17rhmdvdsr 27499 . . . 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 1231 . . 3  |-  ( ( F  e.  ( R RingHom  S )  /\  A  e.  (Unit `  R )
)  ->  ( F `  A ) ( ||r `  S
) ( F `  ( 1r `  R ) ) )
20 eqid 2467 . . . . . 6  |-  ( 1r
`  S )  =  ( 1r `  S
)
218, 20rhm1 17180 . . . . 5  |-  ( F  e.  ( R RingHom  S
)  ->  ( F `  ( 1r `  R
) )  =  ( 1r `  S ) )
2221breq2d 4459 . . . 4  |-  ( F  e.  ( R RingHom  S
)  ->  ( ( F `  A )
( ||r `
 S ) ( F `  ( 1r
`  R ) )  <-> 
( F `  A
) ( ||r `
 S ) ( 1r `  S ) ) )
2322adantr 465 . . 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 210 . 2  |-  ( ( F  e.  ( R RingHom  S )  /\  A  e.  (Unit `  R )
)  ->  ( F `  A ) ( ||r `  S
) ( 1r `  S ) )
25 rhmopp 27500 . . . . 5  |-  ( F  e.  ( R RingHom  S
)  ->  F  e.  ( (oppr
`  R ) RingHom  (oppr `  S
) ) )
2625adantr 465 . . . 4  |-  ( ( F  e.  ( R RingHom  S )  /\  A  e.  (Unit `  R )
)  ->  F  e.  ( (oppr
`  R ) RingHom  (oppr `  S
) ) )
2715simprd 463 . . . 4  |-  ( ( F  e.  ( R RingHom  S )  /\  A  e.  (Unit `  R )
)  ->  A ( ||r `  (oppr
`  R ) ) ( 1r `  R
) )
2812, 2opprbas 17079 . . . . 5  |-  ( Base `  R )  =  (
Base `  (oppr
`  R ) )
29 eqid 2467 . . . . 5  |-  ( ||r `  (oppr `  S
) )  =  (
||r `  (oppr
`  S ) )
3028, 13, 29rhmdvdsr 27499 . . . 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 1231 . . 3  |-  ( ( F  e.  ( R RingHom  S )  /\  A  e.  (Unit `  R )
)  ->  ( F `  A ) ( ||r `  (oppr `  S
) ) ( F `
 ( 1r `  R ) ) )
3221breq2d 4459 . . . 4  |-  ( F  e.  ( R RingHom  S
)  ->  ( ( F `  A )
( ||r `
 (oppr
`  S ) ) ( F `  ( 1r `  R ) )  <-> 
( F `  A
) ( ||r `
 (oppr
`  S ) ) ( 1r `  S
) ) )
3332adantr 465 . . 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 210 . 2  |-  ( ( F  e.  ( R RingHom  S )  /\  A  e.  (Unit `  R )
)  ->  ( F `  A ) ( ||r `  (oppr `  S
) ) ( 1r
`  S ) )
35 eqid 2467 . . 3  |-  (Unit `  S )  =  (Unit `  S )
36 eqid 2467 . . 3  |-  (oppr `  S
)  =  (oppr `  S
)
3735, 20, 17, 36, 29isunit 17107 . 2  |-  ( ( F `  A )  e.  (Unit `  S
)  <->  ( ( F `
 A ) (
||r `  S ) ( 1r
`  S )  /\  ( F `  A ) ( ||r `
 (oppr
`  S ) ) ( 1r `  S
) ) )
3824, 34, 37sylanbrc 664 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 184    /\ wa 369    e. wcel 1767   class class class wbr 4447   ` cfv 5588  (class class class)co 6284   Basecbs 14490   1rcur 16955   Ringcrg 17000  opprcoppr 17072   ||rcdsr 17088  Unitcui 17089   RingHom crh 17162
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6576  ax-cnex 9548  ax-resscn 9549  ax-1cn 9550  ax-icn 9551  ax-addcl 9552  ax-addrcl 9553  ax-mulcl 9554  ax-mulrcl 9555  ax-mulcom 9556  ax-addass 9557  ax-mulass 9558  ax-distr 9559  ax-i2m1 9560  ax-1ne0 9561  ax-1rid 9562  ax-rnegex 9563  ax-rrecex 9564  ax-cnre 9565  ax-pre-lttri 9566  ax-pre-lttrn 9567  ax-pre-ltadd 9568  ax-pre-mulgt0 9569
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2819  df-rex 2820  df-reu 2821  df-rmo 2822  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-we 4840  df-ord 4881  df-on 4882  df-lim 4883  df-suc 4884  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5551  df-fun 5590  df-fn 5591  df-f 5592  df-f1 5593  df-fo 5594  df-f1o 5595  df-fv 5596  df-riota 6245  df-ov 6287  df-oprab 6288  df-mpt2 6289  df-om 6685  df-tpos 6955  df-recs 7042  df-rdg 7076  df-er 7311  df-map 7422  df-en 7517  df-dom 7518  df-sdom 7519  df-pnf 9630  df-mnf 9631  df-xr 9632  df-ltxr 9633  df-le 9634  df-sub 9807  df-neg 9808  df-nn 10537  df-2 10594  df-3 10595  df-ndx 14493  df-slot 14494  df-base 14495  df-sets 14496  df-plusg 14568  df-mulr 14569  df-0g 14697  df-mnd 15732  df-mhm 15786  df-grp 15867  df-ghm 16070  df-mgp 16944  df-ur 16956  df-rng 17002  df-oppr 17073  df-dvdsr 17091  df-unit 17092  df-rnghom 17165
This theorem is referenced by:  rhmunitinv  27503  qqhval2lem  27626
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