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Theorem rhmunitinv 27605
Description: Ring homomorphisms preserve the inverse of unit elements. (Contributed by Thierry Arnoux, 23-Oct-2017.)
Assertion
Ref Expression
rhmunitinv  |-  ( ( F  e.  ( R RingHom  S )  /\  A  e.  (Unit `  R )
)  ->  ( F `  ( ( invr `  R
) `  A )
)  =  ( (
invr `  S ) `  ( F `  A
) ) )

Proof of Theorem rhmunitinv
StepHypRef Expression
1 rhmrcl1 17217 . . . . . 6  |-  ( F  e.  ( R RingHom  S
)  ->  R  e.  Ring )
2 eqid 2467 . . . . . . 7  |-  (Unit `  R )  =  (Unit `  R )
3 eqid 2467 . . . . . . 7  |-  ( invr `  R )  =  (
invr `  R )
4 eqid 2467 . . . . . . 7  |-  ( .r
`  R )  =  ( .r `  R
)
5 eqid 2467 . . . . . . 7  |-  ( 1r
`  R )  =  ( 1r `  R
)
62, 3, 4, 5unitlinv 17175 . . . . . 6  |-  ( ( R  e.  Ring  /\  A  e.  (Unit `  R )
)  ->  ( (
( invr `  R ) `  A ) ( .r
`  R ) A )  =  ( 1r
`  R ) )
71, 6sylan 471 . . . . 5  |-  ( ( F  e.  ( R RingHom  S )  /\  A  e.  (Unit `  R )
)  ->  ( (
( invr `  R ) `  A ) ( .r
`  R ) A )  =  ( 1r
`  R ) )
87fveq2d 5875 . . . 4  |-  ( ( F  e.  ( R RingHom  S )  /\  A  e.  (Unit `  R )
)  ->  ( F `  ( ( ( invr `  R ) `  A
) ( .r `  R ) A ) )  =  ( F `
 ( 1r `  R ) ) )
9 simpl 457 . . . . 5  |-  ( ( F  e.  ( R RingHom  S )  /\  A  e.  (Unit `  R )
)  ->  F  e.  ( R RingHom  S ) )
10 eqid 2467 . . . . . . 7  |-  ( Base `  R )  =  (
Base `  R )
1110, 2unitss 17158 . . . . . 6  |-  (Unit `  R )  C_  ( Base `  R )
122, 3unitinvcl 17172 . . . . . . 7  |-  ( ( R  e.  Ring  /\  A  e.  (Unit `  R )
)  ->  ( ( invr `  R ) `  A )  e.  (Unit `  R ) )
131, 12sylan 471 . . . . . 6  |-  ( ( F  e.  ( R RingHom  S )  /\  A  e.  (Unit `  R )
)  ->  ( ( invr `  R ) `  A )  e.  (Unit `  R ) )
1411, 13sseldi 3507 . . . . 5  |-  ( ( F  e.  ( R RingHom  S )  /\  A  e.  (Unit `  R )
)  ->  ( ( invr `  R ) `  A )  e.  (
Base `  R )
)
15 simpr 461 . . . . . 6  |-  ( ( F  e.  ( R RingHom  S )  /\  A  e.  (Unit `  R )
)  ->  A  e.  (Unit `  R ) )
1611, 15sseldi 3507 . . . . 5  |-  ( ( F  e.  ( R RingHom  S )  /\  A  e.  (Unit `  R )
)  ->  A  e.  ( Base `  R )
)
17 eqid 2467 . . . . . 6  |-  ( .r
`  S )  =  ( .r `  S
)
1810, 4, 17rhmmul 17225 . . . . 5  |-  ( ( F  e.  ( R RingHom  S )  /\  (
( invr `  R ) `  A )  e.  (
Base `  R )  /\  A  e.  ( Base `  R ) )  ->  ( F `  ( ( ( invr `  R ) `  A
) ( .r `  R ) A ) )  =  ( ( F `  ( (
invr `  R ) `  A ) ) ( .r `  S ) ( F `  A
) ) )
199, 14, 16, 18syl3anc 1228 . . . 4  |-  ( ( F  e.  ( R RingHom  S )  /\  A  e.  (Unit `  R )
)  ->  ( F `  ( ( ( invr `  R ) `  A
) ( .r `  R ) A ) )  =  ( ( F `  ( (
invr `  R ) `  A ) ) ( .r `  S ) ( F `  A
) ) )
20 eqid 2467 . . . . . 6  |-  ( 1r
`  S )  =  ( 1r `  S
)
215, 20rhm1 17228 . . . . 5  |-  ( F  e.  ( R RingHom  S
)  ->  ( F `  ( 1r `  R
) )  =  ( 1r `  S ) )
2221adantr 465 . . . 4  |-  ( ( F  e.  ( R RingHom  S )  /\  A  e.  (Unit `  R )
)  ->  ( F `  ( 1r `  R
) )  =  ( 1r `  S ) )
238, 19, 223eqtr3d 2516 . . 3  |-  ( ( F  e.  ( R RingHom  S )  /\  A  e.  (Unit `  R )
)  ->  ( ( F `  ( ( invr `  R ) `  A ) ) ( .r `  S ) ( F `  A
) )  =  ( 1r `  S ) )
24 rhmrcl2 17218 . . . . 5  |-  ( F  e.  ( R RingHom  S
)  ->  S  e.  Ring )
2524adantr 465 . . . 4  |-  ( ( F  e.  ( R RingHom  S )  /\  A  e.  (Unit `  R )
)  ->  S  e.  Ring )
26 elrhmunit 27603 . . . 4  |-  ( ( F  e.  ( R RingHom  S )  /\  A  e.  (Unit `  R )
)  ->  ( F `  A )  e.  (Unit `  S ) )
27 eqid 2467 . . . . 5  |-  (Unit `  S )  =  (Unit `  S )
28 eqid 2467 . . . . 5  |-  ( invr `  S )  =  (
invr `  S )
2927, 28, 17, 20unitlinv 17175 . . . 4  |-  ( ( S  e.  Ring  /\  ( F `  A )  e.  (Unit `  S )
)  ->  ( (
( invr `  S ) `  ( F `  A
) ) ( .r
`  S ) ( F `  A ) )  =  ( 1r
`  S ) )
3025, 26, 29syl2anc 661 . . 3  |-  ( ( F  e.  ( R RingHom  S )  /\  A  e.  (Unit `  R )
)  ->  ( (
( invr `  S ) `  ( F `  A
) ) ( .r
`  S ) ( F `  A ) )  =  ( 1r
`  S ) )
3123, 30eqtr4d 2511 . 2  |-  ( ( F  e.  ( R RingHom  S )  /\  A  e.  (Unit `  R )
)  ->  ( ( F `  ( ( invr `  R ) `  A ) ) ( .r `  S ) ( F `  A
) )  =  ( ( ( invr `  S
) `  ( F `  A ) ) ( .r `  S ) ( F `  A
) ) )
32 eqid 2467 . . . . . 6  |-  ( (mulGrp `  S )s  (Unit `  S )
)  =  ( (mulGrp `  S )s  (Unit `  S )
)
3327, 32unitgrp 17165 . . . . 5  |-  ( S  e.  Ring  ->  ( (mulGrp `  S )s  (Unit `  S )
)  e.  Grp )
3424, 33syl 16 . . . 4  |-  ( F  e.  ( R RingHom  S
)  ->  ( (mulGrp `  S )s  (Unit `  S )
)  e.  Grp )
3534adantr 465 . . 3  |-  ( ( F  e.  ( R RingHom  S )  /\  A  e.  (Unit `  R )
)  ->  ( (mulGrp `  S )s  (Unit `  S )
)  e.  Grp )
36 elrhmunit 27603 . . . 4  |-  ( ( F  e.  ( R RingHom  S )  /\  (
( invr `  R ) `  A )  e.  (Unit `  R ) )  -> 
( F `  (
( invr `  R ) `  A ) )  e.  (Unit `  S )
)
3713, 36syldan 470 . . 3  |-  ( ( F  e.  ( R RingHom  S )  /\  A  e.  (Unit `  R )
)  ->  ( F `  ( ( invr `  R
) `  A )
)  e.  (Unit `  S ) )
3827, 28unitinvcl 17172 . . . 4  |-  ( ( S  e.  Ring  /\  ( F `  A )  e.  (Unit `  S )
)  ->  ( ( invr `  S ) `  ( F `  A ) )  e.  (Unit `  S ) )
3925, 26, 38syl2anc 661 . . 3  |-  ( ( F  e.  ( R RingHom  S )  /\  A  e.  (Unit `  R )
)  ->  ( ( invr `  S ) `  ( F `  A ) )  e.  (Unit `  S ) )
4027, 32unitgrpbas 17164 . . . 4  |-  (Unit `  S )  =  (
Base `  ( (mulGrp `  S )s  (Unit `  S )
) )
41 fvex 5881 . . . . 5  |-  (Unit `  S )  e.  _V
42 eqid 2467 . . . . . . 7  |-  (mulGrp `  S )  =  (mulGrp `  S )
4342, 17mgpplusg 16994 . . . . . 6  |-  ( .r
`  S )  =  ( +g  `  (mulGrp `  S ) )
4432, 43ressplusg 14609 . . . . 5  |-  ( (Unit `  S )  e.  _V  ->  ( .r `  S
)  =  ( +g  `  ( (mulGrp `  S
)s  (Unit `  S )
) ) )
4541, 44ax-mp 5 . . . 4  |-  ( .r
`  S )  =  ( +g  `  (
(mulGrp `  S )s  (Unit `  S ) ) )
4640, 45grprcan 15932 . . 3  |-  ( ( ( (mulGrp `  S
)s  (Unit `  S )
)  e.  Grp  /\  ( ( F `  ( ( invr `  R
) `  A )
)  e.  (Unit `  S )  /\  (
( invr `  S ) `  ( F `  A
) )  e.  (Unit `  S )  /\  ( F `  A )  e.  (Unit `  S )
) )  ->  (
( ( F `  ( ( invr `  R
) `  A )
) ( .r `  S ) ( F `
 A ) )  =  ( ( (
invr `  S ) `  ( F `  A
) ) ( .r
`  S ) ( F `  A ) )  <->  ( F `  ( ( invr `  R
) `  A )
)  =  ( (
invr `  S ) `  ( F `  A
) ) ) )
4735, 37, 39, 26, 46syl13anc 1230 . 2  |-  ( ( F  e.  ( R RingHom  S )  /\  A  e.  (Unit `  R )
)  ->  ( (
( F `  (
( invr `  R ) `  A ) ) ( .r `  S ) ( F `  A
) )  =  ( ( ( invr `  S
) `  ( F `  A ) ) ( .r `  S ) ( F `  A
) )  <->  ( F `  ( ( invr `  R
) `  A )
)  =  ( (
invr `  S ) `  ( F `  A
) ) ) )
4831, 47mpbid 210 1  |-  ( ( F  e.  ( R RingHom  S )  /\  A  e.  (Unit `  R )
)  ->  ( F `  ( ( invr `  R
) `  A )
)  =  ( (
invr `  S ) `  ( F `  A
) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1379    e. wcel 1767   _Vcvv 3118   ` cfv 5593  (class class class)co 6294   Basecbs 14502   ↾s cress 14503   +g cplusg 14567   .rcmulr 14568   Grpcgrp 15902  mulGrpcmgp 16990   1rcur 17002   Ringcrg 17047  Unitcui 17137   invrcinvr 17169   RingHom crh 17210
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 4563  ax-sep 4573  ax-nul 4581  ax-pow 4630  ax-pr 4691  ax-un 6586  ax-cnex 9558  ax-resscn 9559  ax-1cn 9560  ax-icn 9561  ax-addcl 9562  ax-addrcl 9563  ax-mulcl 9564  ax-mulrcl 9565  ax-mulcom 9566  ax-addass 9567  ax-mulass 9568  ax-distr 9569  ax-i2m1 9570  ax-1ne0 9571  ax-1rid 9572  ax-rnegex 9573  ax-rrecex 9574  ax-cnre 9575  ax-pre-lttri 9576  ax-pre-lttrn 9577  ax-pre-ltadd 9578  ax-pre-mulgt0 9579
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 2822  df-rex 2823  df-reu 2824  df-rmo 2825  df-rab 2826  df-v 3120  df-sbc 3337  df-csb 3441  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-pss 3497  df-nul 3791  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-tp 4037  df-op 4039  df-uni 4251  df-iun 4332  df-br 4453  df-opab 4511  df-mpt 4512  df-tr 4546  df-eprel 4796  df-id 4800  df-po 4805  df-so 4806  df-fr 4843  df-we 4845  df-ord 4886  df-on 4887  df-lim 4888  df-suc 4889  df-xp 5010  df-rel 5011  df-cnv 5012  df-co 5013  df-dm 5014  df-rn 5015  df-res 5016  df-ima 5017  df-iota 5556  df-fun 5595  df-fn 5596  df-f 5597  df-f1 5598  df-fo 5599  df-f1o 5600  df-fv 5601  df-riota 6255  df-ov 6297  df-oprab 6298  df-mpt2 6299  df-om 6695  df-tpos 6965  df-recs 7052  df-rdg 7086  df-er 7321  df-map 7432  df-en 7527  df-dom 7528  df-sdom 7529  df-pnf 9640  df-mnf 9641  df-xr 9642  df-ltxr 9643  df-le 9644  df-sub 9817  df-neg 9818  df-nn 10547  df-2 10604  df-3 10605  df-ndx 14505  df-slot 14506  df-base 14507  df-sets 14508  df-ress 14509  df-plusg 14580  df-mulr 14581  df-0g 14709  df-mgm 15741  df-sgrp 15764  df-mnd 15774  df-mhm 15819  df-grp 15906  df-minusg 15907  df-ghm 16114  df-mgp 16991  df-ur 17003  df-ring 17049  df-oppr 17121  df-dvdsr 17139  df-unit 17140  df-invr 17170  df-rnghom 17213
This theorem is referenced by: (None)
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