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Theorem elrhmunit 26428
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 2452 . . . . . 6  |-  ( Base `  R )  =  (
Base `  R )
3 eqid 2452 . . . . . 6  |-  (Unit `  R )  =  (Unit `  R )
42, 3unitss 16870 . . . . 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 3457 . . . 4  |-  ( ( F  e.  ( R RingHom  S )  /\  A  e.  (Unit `  R )
)  ->  A  e.  ( Base `  R )
)
7 rhmrcl1 16927 . . . . 5  |-  ( F  e.  ( R RingHom  S
)  ->  R  e.  Ring )
8 eqid 2452 . . . . . 6  |-  ( 1r
`  R )  =  ( 1r `  R
)
92, 8rngidcl 16783 . . . . 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 2452 . . . . . . 7  |-  ( ||r `  R
)  =  ( ||r `  R
)
12 eqid 2452 . . . . . . 7  |-  (oppr `  R
)  =  (oppr `  R
)
13 eqid 2452 . . . . . . 7  |-  ( ||r `  (oppr `  R
) )  =  (
||r `  (oppr
`  R ) )
143, 8, 11, 12, 13isunit 16867 . . . . . 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 2452 . . . . 5  |-  ( ||r `  S
)  =  ( ||r `  S
)
182, 11, 17rhmdvdsr 26426 . . . 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 1222 . . 3  |-  ( ( F  e.  ( R RingHom  S )  /\  A  e.  (Unit `  R )
)  ->  ( F `  A ) ( ||r `  S
) ( F `  ( 1r `  R ) ) )
20 eqid 2452 . . . . . 6  |-  ( 1r
`  S )  =  ( 1r `  S
)
218, 20rhm1 16938 . . . . 5  |-  ( F  e.  ( R RingHom  S
)  ->  ( F `  ( 1r `  R
) )  =  ( 1r `  S ) )
2221breq2d 4407 . . . 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 26427 . . . . 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 16839 . . . . 5  |-  ( Base `  R )  =  (
Base `  (oppr
`  R ) )
29 eqid 2452 . . . . 5  |-  ( ||r `  (oppr `  S
) )  =  (
||r `  (oppr
`  S ) )
3028, 13, 29rhmdvdsr 26426 . . . 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 1222 . . 3  |-  ( ( F  e.  ( R RingHom  S )  /\  A  e.  (Unit `  R )
)  ->  ( F `  A ) ( ||r `  (oppr `  S
) ) ( F `
 ( 1r `  R ) ) )
3221breq2d 4407 . . . 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 2452 . . 3  |-  (Unit `  S )  =  (Unit `  S )
36 eqid 2452 . . 3  |-  (oppr `  S
)  =  (oppr `  S
)
3735, 20, 17, 36, 29isunit 16867 . 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 1758   class class class wbr 4395   ` cfv 5521  (class class class)co 6195   Basecbs 14287   1rcur 16720   Ringcrg 16763  opprcoppr 16832   ||rcdsr 16848  Unitcui 16849   RingHom crh 16922
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1954  ax-ext 2431  ax-rep 4506  ax-sep 4516  ax-nul 4524  ax-pow 4573  ax-pr 4634  ax-un 6477  ax-cnex 9444  ax-resscn 9445  ax-1cn 9446  ax-icn 9447  ax-addcl 9448  ax-addrcl 9449  ax-mulcl 9450  ax-mulrcl 9451  ax-mulcom 9452  ax-addass 9453  ax-mulass 9454  ax-distr 9455  ax-i2m1 9456  ax-1ne0 9457  ax-1rid 9458  ax-rnegex 9459  ax-rrecex 9460  ax-cnre 9461  ax-pre-lttri 9462  ax-pre-lttrn 9463  ax-pre-ltadd 9464  ax-pre-mulgt0 9465
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2265  df-mo 2266  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2602  df-ne 2647  df-nel 2648  df-ral 2801  df-rex 2802  df-reu 2803  df-rmo 2804  df-rab 2805  df-v 3074  df-sbc 3289  df-csb 3391  df-dif 3434  df-un 3436  df-in 3438  df-ss 3445  df-pss 3447  df-nul 3741  df-if 3895  df-pw 3965  df-sn 3981  df-pr 3983  df-tp 3985  df-op 3987  df-uni 4195  df-iun 4276  df-br 4396  df-opab 4454  df-mpt 4455  df-tr 4489  df-eprel 4735  df-id 4739  df-po 4744  df-so 4745  df-fr 4782  df-we 4784  df-ord 4825  df-on 4826  df-lim 4827  df-suc 4828  df-xp 4949  df-rel 4950  df-cnv 4951  df-co 4952  df-dm 4953  df-rn 4954  df-res 4955  df-ima 4956  df-iota 5484  df-fun 5523  df-fn 5524  df-f 5525  df-f1 5526  df-fo 5527  df-f1o 5528  df-fv 5529  df-riota 6156  df-ov 6198  df-oprab 6199  df-mpt2 6200  df-om 6582  df-tpos 6850  df-recs 6937  df-rdg 6971  df-er 7206  df-map 7321  df-en 7416  df-dom 7417  df-sdom 7418  df-pnf 9526  df-mnf 9527  df-xr 9528  df-ltxr 9529  df-le 9530  df-sub 9703  df-neg 9704  df-nn 10429  df-2 10486  df-3 10487  df-ndx 14290  df-slot 14291  df-base 14292  df-sets 14293  df-plusg 14365  df-mulr 14366  df-0g 14494  df-mnd 15529  df-mhm 15578  df-grp 15659  df-ghm 15859  df-mgp 16709  df-ur 16721  df-rng 16765  df-oppr 16833  df-dvdsr 16851  df-unit 16852  df-rnghom 16924
This theorem is referenced by:  rhmunitinv  26430  qqhval2lem  26550
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