Users' Mathboxes Mathbox for Thierry Arnoux < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  rhmunitinv Structured version   Unicode version

Theorem rhmunitinv 26295
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 16814 . . . . . 6  |-  ( F  e.  ( R RingHom  S
)  ->  R  e.  Ring )
2 eqid 2443 . . . . . . 7  |-  (Unit `  R )  =  (Unit `  R )
3 eqid 2443 . . . . . . 7  |-  ( invr `  R )  =  (
invr `  R )
4 eqid 2443 . . . . . . 7  |-  ( .r
`  R )  =  ( .r `  R
)
5 eqid 2443 . . . . . . 7  |-  ( 1r
`  R )  =  ( 1r `  R
)
62, 3, 4, 5unitlinv 16774 . . . . . 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 5700 . . . 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 2443 . . . . . . 7  |-  ( Base `  R )  =  (
Base `  R )
1110, 2unitss 16757 . . . . . 6  |-  (Unit `  R )  C_  ( Base `  R )
122, 3unitinvcl 16771 . . . . . . 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 3359 . . . . 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 3359 . . . . 5  |-  ( ( F  e.  ( R RingHom  S )  /\  A  e.  (Unit `  R )
)  ->  A  e.  ( Base `  R )
)
17 eqid 2443 . . . . . 6  |-  ( .r
`  S )  =  ( .r `  S
)
1810, 4, 17rhmmul 16822 . . . . 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 1218 . . . 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 2443 . . . . . 6  |-  ( 1r
`  S )  =  ( 1r `  S
)
215, 20rhm1 16825 . . . . 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 2483 . . 3  |-  ( ( F  e.  ( R RingHom  S )  /\  A  e.  (Unit `  R )
)  ->  ( ( F `  ( ( invr `  R ) `  A ) ) ( .r `  S ) ( F `  A
) )  =  ( 1r `  S ) )
24 rhmrcl2 16815 . . . . 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 26293 . . . 4  |-  ( ( F  e.  ( R RingHom  S )  /\  A  e.  (Unit `  R )
)  ->  ( F `  A )  e.  (Unit `  S ) )
27 eqid 2443 . . . . 5  |-  (Unit `  S )  =  (Unit `  S )
28 eqid 2443 . . . . 5  |-  ( invr `  S )  =  (
invr `  S )
2927, 28, 17, 20unitlinv 16774 . . . 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 2478 . 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 2443 . . . . . 6  |-  ( (mulGrp `  S )s  (Unit `  S )
)  =  ( (mulGrp `  S )s  (Unit `  S )
)
3327, 32unitgrp 16764 . . . . 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 26293 . . . 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 16771 . . . 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 16763 . . . 4  |-  (Unit `  S )  =  (
Base `  ( (mulGrp `  S )s  (Unit `  S )
) )
41 fvex 5706 . . . . 5  |-  (Unit `  S )  e.  _V
42 eqid 2443 . . . . . . 7  |-  (mulGrp `  S )  =  (mulGrp `  S )
4342, 17mgpplusg 16600 . . . . . 6  |-  ( .r
`  S )  =  ( +g  `  (mulGrp `  S ) )
4432, 43ressplusg 14285 . . . . 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 15576 . . 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 1220 . 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 1369    e. wcel 1756   _Vcvv 2977   ` cfv 5423  (class class class)co 6096   Basecbs 14179   ↾s cress 14180   +g cplusg 14243   .rcmulr 14244   Grpcgrp 15415  mulGrpcmgp 16596   1rcur 16608   Ringcrg 16650  Unitcui 16736   invrcinvr 16768   RingHom crh 16809
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4408  ax-sep 4418  ax-nul 4426  ax-pow 4475  ax-pr 4536  ax-un 6377  ax-cnex 9343  ax-resscn 9344  ax-1cn 9345  ax-icn 9346  ax-addcl 9347  ax-addrcl 9348  ax-mulcl 9349  ax-mulrcl 9350  ax-mulcom 9351  ax-addass 9352  ax-mulass 9353  ax-distr 9354  ax-i2m1 9355  ax-1ne0 9356  ax-1rid 9357  ax-rnegex 9358  ax-rrecex 9359  ax-cnre 9360  ax-pre-lttri 9361  ax-pre-lttrn 9362  ax-pre-ltadd 9363  ax-pre-mulgt0 9364
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2573  df-ne 2613  df-nel 2614  df-ral 2725  df-rex 2726  df-reu 2727  df-rmo 2728  df-rab 2729  df-v 2979  df-sbc 3192  df-csb 3294  df-dif 3336  df-un 3338  df-in 3340  df-ss 3347  df-pss 3349  df-nul 3643  df-if 3797  df-pw 3867  df-sn 3883  df-pr 3885  df-tp 3887  df-op 3889  df-uni 4097  df-iun 4178  df-br 4298  df-opab 4356  df-mpt 4357  df-tr 4391  df-eprel 4637  df-id 4641  df-po 4646  df-so 4647  df-fr 4684  df-we 4686  df-ord 4727  df-on 4728  df-lim 4729  df-suc 4730  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-iota 5386  df-fun 5425  df-fn 5426  df-f 5427  df-f1 5428  df-fo 5429  df-f1o 5430  df-fv 5431  df-riota 6057  df-ov 6099  df-oprab 6100  df-mpt2 6101  df-om 6482  df-tpos 6750  df-recs 6837  df-rdg 6871  df-er 7106  df-map 7221  df-en 7316  df-dom 7317  df-sdom 7318  df-pnf 9425  df-mnf 9426  df-xr 9427  df-ltxr 9428  df-le 9429  df-sub 9602  df-neg 9603  df-nn 10328  df-2 10385  df-3 10386  df-ndx 14182  df-slot 14183  df-base 14184  df-sets 14185  df-ress 14186  df-plusg 14256  df-mulr 14257  df-0g 14385  df-mnd 15420  df-mhm 15469  df-grp 15550  df-minusg 15551  df-ghm 15750  df-mgp 16597  df-ur 16609  df-rng 16652  df-oppr 16720  df-dvdsr 16738  df-unit 16739  df-invr 16769  df-rnghom 16811
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator