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Theorem rhmunitinv 27678
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 17236 . . . . . 6  |-  ( F  e.  ( R RingHom  S
)  ->  R  e.  Ring )
2 eqid 2441 . . . . . . 7  |-  (Unit `  R )  =  (Unit `  R )
3 eqid 2441 . . . . . . 7  |-  ( invr `  R )  =  (
invr `  R )
4 eqid 2441 . . . . . . 7  |-  ( .r
`  R )  =  ( .r `  R
)
5 eqid 2441 . . . . . . 7  |-  ( 1r
`  R )  =  ( 1r `  R
)
62, 3, 4, 5unitlinv 17194 . . . . . 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 5856 . . . 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 2441 . . . . . . 7  |-  ( Base `  R )  =  (
Base `  R )
1110, 2unitss 17177 . . . . . 6  |-  (Unit `  R )  C_  ( Base `  R )
122, 3unitinvcl 17191 . . . . . . 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 3484 . . . . 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 3484 . . . . 5  |-  ( ( F  e.  ( R RingHom  S )  /\  A  e.  (Unit `  R )
)  ->  A  e.  ( Base `  R )
)
17 eqid 2441 . . . . . 6  |-  ( .r
`  S )  =  ( .r `  S
)
1810, 4, 17rhmmul 17244 . . . . 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 1227 . . . 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 2441 . . . . . 6  |-  ( 1r
`  S )  =  ( 1r `  S
)
215, 20rhm1 17247 . . . . 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 2490 . . 3  |-  ( ( F  e.  ( R RingHom  S )  /\  A  e.  (Unit `  R )
)  ->  ( ( F `  ( ( invr `  R ) `  A ) ) ( .r `  S ) ( F `  A
) )  =  ( 1r `  S ) )
24 rhmrcl2 17237 . . . . 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 27676 . . . 4  |-  ( ( F  e.  ( R RingHom  S )  /\  A  e.  (Unit `  R )
)  ->  ( F `  A )  e.  (Unit `  S ) )
27 eqid 2441 . . . . 5  |-  (Unit `  S )  =  (Unit `  S )
28 eqid 2441 . . . . 5  |-  ( invr `  S )  =  (
invr `  S )
2927, 28, 17, 20unitlinv 17194 . . . 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 2485 . 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 2441 . . . . . 6  |-  ( (mulGrp `  S )s  (Unit `  S )
)  =  ( (mulGrp `  S )s  (Unit `  S )
)
3327, 32unitgrp 17184 . . . . 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 27676 . . . 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 17191 . . . 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 17183 . . . 4  |-  (Unit `  S )  =  (
Base `  ( (mulGrp `  S )s  (Unit `  S )
) )
41 fvex 5862 . . . . 5  |-  (Unit `  S )  e.  _V
42 eqid 2441 . . . . . . 7  |-  (mulGrp `  S )  =  (mulGrp `  S )
4342, 17mgpplusg 17013 . . . . . 6  |-  ( .r
`  S )  =  ( +g  `  (mulGrp `  S ) )
4432, 43ressplusg 14611 . . . . 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 15952 . . 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 1229 . 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 1381    e. wcel 1802   _Vcvv 3093   ` cfv 5574  (class class class)co 6277   Basecbs 14504   ↾s cress 14505   +g cplusg 14569   .rcmulr 14570   Grpcgrp 15922  mulGrpcmgp 17009   1rcur 17021   Ringcrg 17066  Unitcui 17156   invrcinvr 17188   RingHom crh 17229
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1603  ax-4 1616  ax-5 1689  ax-6 1732  ax-7 1774  ax-8 1804  ax-9 1806  ax-10 1821  ax-11 1826  ax-12 1838  ax-13 1983  ax-ext 2419  ax-rep 4544  ax-sep 4554  ax-nul 4562  ax-pow 4611  ax-pr 4672  ax-un 6573  ax-cnex 9546  ax-resscn 9547  ax-1cn 9548  ax-icn 9549  ax-addcl 9550  ax-addrcl 9551  ax-mulcl 9552  ax-mulrcl 9553  ax-mulcom 9554  ax-addass 9555  ax-mulass 9556  ax-distr 9557  ax-i2m1 9558  ax-1ne0 9559  ax-1rid 9560  ax-rnegex 9561  ax-rrecex 9562  ax-cnre 9563  ax-pre-lttri 9564  ax-pre-lttrn 9565  ax-pre-ltadd 9566  ax-pre-mulgt0 9567
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 973  df-3an 974  df-tru 1384  df-ex 1598  df-nf 1602  df-sb 1725  df-eu 2270  df-mo 2271  df-clab 2427  df-cleq 2433  df-clel 2436  df-nfc 2591  df-ne 2638  df-nel 2639  df-ral 2796  df-rex 2797  df-reu 2798  df-rmo 2799  df-rab 2800  df-v 3095  df-sbc 3312  df-csb 3418  df-dif 3461  df-un 3463  df-in 3465  df-ss 3472  df-pss 3474  df-nul 3768  df-if 3923  df-pw 3995  df-sn 4011  df-pr 4013  df-tp 4015  df-op 4017  df-uni 4231  df-iun 4313  df-br 4434  df-opab 4492  df-mpt 4493  df-tr 4527  df-eprel 4777  df-id 4781  df-po 4786  df-so 4787  df-fr 4824  df-we 4826  df-ord 4867  df-on 4868  df-lim 4869  df-suc 4870  df-xp 4991  df-rel 4992  df-cnv 4993  df-co 4994  df-dm 4995  df-rn 4996  df-res 4997  df-ima 4998  df-iota 5537  df-fun 5576  df-fn 5577  df-f 5578  df-f1 5579  df-fo 5580  df-f1o 5581  df-fv 5582  df-riota 6238  df-ov 6280  df-oprab 6281  df-mpt2 6282  df-om 6682  df-tpos 6953  df-recs 7040  df-rdg 7074  df-er 7309  df-map 7420  df-en 7515  df-dom 7516  df-sdom 7517  df-pnf 9628  df-mnf 9629  df-xr 9630  df-ltxr 9631  df-le 9632  df-sub 9807  df-neg 9808  df-nn 10538  df-2 10595  df-3 10596  df-ndx 14507  df-slot 14508  df-base 14509  df-sets 14510  df-ress 14511  df-plusg 14582  df-mulr 14583  df-0g 14711  df-mgm 15741  df-sgrp 15780  df-mnd 15790  df-mhm 15835  df-grp 15926  df-minusg 15927  df-ghm 16134  df-mgp 17010  df-ur 17022  df-ring 17068  df-oppr 17140  df-dvdsr 17158  df-unit 17159  df-invr 17189  df-rnghom 17232
This theorem is referenced by: (None)
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