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Theorem rhmopp 24210
Description: A ring homomorphism is also a ring homomorphism for the opposite rings. (Contributed by Thierry Arnoux, 27-Oct-2017.)
Assertion
Ref Expression
rhmopp  |-  ( F  e.  ( R RingHom  S
)  ->  F  e.  ( (oppr
`  R ) RingHom  (oppr `  S
) ) )

Proof of Theorem rhmopp
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqid 2404 . 2  |-  ( Base `  (oppr
`  R ) )  =  ( Base `  (oppr `  R
) )
2 eqid 2404 . 2  |-  ( 1r
`  (oppr
`  R ) )  =  ( 1r `  (oppr `  R ) )
3 eqid 2404 . 2  |-  ( 1r
`  (oppr
`  S ) )  =  ( 1r `  (oppr `  S ) )
4 eqid 2404 . 2  |-  ( .r
`  (oppr
`  R ) )  =  ( .r `  (oppr `  R ) )
5 eqid 2404 . 2  |-  ( .r
`  (oppr
`  S ) )  =  ( .r `  (oppr `  S ) )
6 rhmrcl1 15777 . . 3  |-  ( F  e.  ( R RingHom  S
)  ->  R  e.  Ring )
7 eqid 2404 . . . 4  |-  (oppr `  R
)  =  (oppr `  R
)
87opprrngb 15692 . . 3  |-  ( R  e.  Ring  <->  (oppr
`  R )  e. 
Ring )
96, 8sylib 189 . 2  |-  ( F  e.  ( R RingHom  S
)  ->  (oppr
`  R )  e. 
Ring )
10 rhmrcl2 15778 . . 3  |-  ( F  e.  ( R RingHom  S
)  ->  S  e.  Ring )
11 eqid 2404 . . . 4  |-  (oppr `  S
)  =  (oppr `  S
)
1211opprrngb 15692 . . 3  |-  ( S  e.  Ring  <->  (oppr
`  S )  e. 
Ring )
1310, 12sylib 189 . 2  |-  ( F  e.  ( R RingHom  S
)  ->  (oppr
`  S )  e. 
Ring )
14 eqid 2404 . . . . 5  |-  ( 1r
`  R )  =  ( 1r `  R
)
157, 14oppr1 15694 . . . 4  |-  ( 1r
`  R )  =  ( 1r `  (oppr `  R
) )
1615eqcomi 2408 . . 3  |-  ( 1r
`  (oppr
`  R ) )  =  ( 1r `  R )
17 eqid 2404 . . . . 5  |-  ( 1r
`  S )  =  ( 1r `  S
)
1811, 17oppr1 15694 . . . 4  |-  ( 1r
`  S )  =  ( 1r `  (oppr `  S
) )
1918eqcomi 2408 . . 3  |-  ( 1r
`  (oppr
`  S ) )  =  ( 1r `  S )
2016, 19rhm1 15786 . 2  |-  ( F  e.  ( R RingHom  S
)  ->  ( F `  ( 1r `  (oppr `  R
) ) )  =  ( 1r `  (oppr `  S
) ) )
21 simpl 444 . . . 4  |-  ( ( F  e.  ( R RingHom  S )  /\  (
x  e.  ( Base `  (oppr
`  R ) )  /\  y  e.  (
Base `  (oppr
`  R ) ) ) )  ->  F  e.  ( R RingHom  S )
)
22 simprr 734 . . . . 5  |-  ( ( F  e.  ( R RingHom  S )  /\  (
x  e.  ( Base `  (oppr
`  R ) )  /\  y  e.  (
Base `  (oppr
`  R ) ) ) )  ->  y  e.  ( Base `  (oppr `  R
) ) )
23 eqid 2404 . . . . . 6  |-  ( Base `  R )  =  (
Base `  R )
247, 23opprbas 15689 . . . . 5  |-  ( Base `  R )  =  (
Base `  (oppr
`  R ) )
2522, 24syl6eleqr 2495 . . . 4  |-  ( ( F  e.  ( R RingHom  S )  /\  (
x  e.  ( Base `  (oppr
`  R ) )  /\  y  e.  (
Base `  (oppr
`  R ) ) ) )  ->  y  e.  ( Base `  R
) )
26 simprl 733 . . . . 5  |-  ( ( F  e.  ( R RingHom  S )  /\  (
x  e.  ( Base `  (oppr
`  R ) )  /\  y  e.  (
Base `  (oppr
`  R ) ) ) )  ->  x  e.  ( Base `  (oppr `  R
) ) )
2726, 24syl6eleqr 2495 . . . 4  |-  ( ( F  e.  ( R RingHom  S )  /\  (
x  e.  ( Base `  (oppr
`  R ) )  /\  y  e.  (
Base `  (oppr
`  R ) ) ) )  ->  x  e.  ( Base `  R
) )
28 eqid 2404 . . . . 5  |-  ( .r
`  R )  =  ( .r `  R
)
29 eqid 2404 . . . . 5  |-  ( .r
`  S )  =  ( .r `  S
)
3023, 28, 29rhmmul 15783 . . . 4  |-  ( ( F  e.  ( R RingHom  S )  /\  y  e.  ( Base `  R
)  /\  x  e.  ( Base `  R )
)  ->  ( F `  ( y ( .r
`  R ) x ) )  =  ( ( F `  y
) ( .r `  S ) ( F `
 x ) ) )
3121, 25, 27, 30syl3anc 1184 . . 3  |-  ( ( F  e.  ( R RingHom  S )  /\  (
x  e.  ( Base `  (oppr
`  R ) )  /\  y  e.  (
Base `  (oppr
`  R ) ) ) )  ->  ( F `  ( y
( .r `  R
) x ) )  =  ( ( F `
 y ) ( .r `  S ) ( F `  x
) ) )
3223, 28, 7, 4opprmul 15686 . . . 4  |-  ( x ( .r `  (oppr `  R
) ) y )  =  ( y ( .r `  R ) x )
3332fveq2i 5690 . . 3  |-  ( F `
 ( x ( .r `  (oppr `  R
) ) y ) )  =  ( F `
 ( y ( .r `  R ) x ) )
34 eqid 2404 . . . 4  |-  ( Base `  S )  =  (
Base `  S )
3534, 29, 11, 5opprmul 15686 . . 3  |-  ( ( F `  x ) ( .r `  (oppr `  S
) ) ( F `
 y ) )  =  ( ( F `
 y ) ( .r `  S ) ( F `  x
) )
3631, 33, 353eqtr4g 2461 . 2  |-  ( ( F  e.  ( R RingHom  S )  /\  (
x  e.  ( Base `  (oppr
`  R ) )  /\  y  e.  (
Base `  (oppr
`  R ) ) ) )  ->  ( F `  ( x
( .r `  (oppr `  R
) ) y ) )  =  ( ( F `  x ) ( .r `  (oppr `  S
) ) ( F `
 y ) ) )
37 rnggrp 15624 . . . . 5  |-  ( (oppr `  R )  e.  Ring  -> 
(oppr `  R )  e.  Grp )
389, 37syl 16 . . . 4  |-  ( F  e.  ( R RingHom  S
)  ->  (oppr
`  R )  e. 
Grp )
39 rnggrp 15624 . . . . 5  |-  ( (oppr `  S )  e.  Ring  -> 
(oppr `  S )  e.  Grp )
4013, 39syl 16 . . . 4  |-  ( F  e.  ( R RingHom  S
)  ->  (oppr
`  S )  e. 
Grp )
4123, 34rhmf 15782 . . . . 5  |-  ( F  e.  ( R RingHom  S
)  ->  F :
( Base `  R ) --> ( Base `  S )
)
42 rhmghm 15781 . . . . . . . . 9  |-  ( F  e.  ( R RingHom  S
)  ->  F  e.  ( R  GrpHom  S ) )
4342ad2antrr 707 . . . . . . . 8  |-  ( ( ( F  e.  ( R RingHom  S )  /\  x  e.  ( Base `  R
) )  /\  y  e.  ( Base `  R
) )  ->  F  e.  ( R  GrpHom  S ) )
44 simplr 732 . . . . . . . 8  |-  ( ( ( F  e.  ( R RingHom  S )  /\  x  e.  ( Base `  R
) )  /\  y  e.  ( Base `  R
) )  ->  x  e.  ( Base `  R
) )
45 simpr 448 . . . . . . . 8  |-  ( ( ( F  e.  ( R RingHom  S )  /\  x  e.  ( Base `  R
) )  /\  y  e.  ( Base `  R
) )  ->  y  e.  ( Base `  R
) )
46 eqid 2404 . . . . . . . . 9  |-  ( +g  `  R )  =  ( +g  `  R )
47 eqid 2404 . . . . . . . . 9  |-  ( +g  `  S )  =  ( +g  `  S )
4823, 46, 47ghmlin 14966 . . . . . . . 8  |-  ( ( F  e.  ( R 
GrpHom  S )  /\  x  e.  ( Base `  R
)  /\  y  e.  ( Base `  R )
)  ->  ( F `  ( x ( +g  `  R ) y ) )  =  ( ( F `  x ) ( +g  `  S
) ( F `  y ) ) )
4943, 44, 45, 48syl3anc 1184 . . . . . . 7  |-  ( ( ( F  e.  ( R RingHom  S )  /\  x  e.  ( Base `  R
) )  /\  y  e.  ( Base `  R
) )  ->  ( F `  ( x
( +g  `  R ) y ) )  =  ( ( F `  x ) ( +g  `  S ) ( F `
 y ) ) )
5049ralrimiva 2749 . . . . . 6  |-  ( ( F  e.  ( R RingHom  S )  /\  x  e.  ( Base `  R
) )  ->  A. y  e.  ( Base `  R
) ( F `  ( x ( +g  `  R ) y ) )  =  ( ( F `  x ) ( +g  `  S
) ( F `  y ) ) )
5150ralrimiva 2749 . . . . 5  |-  ( F  e.  ( R RingHom  S
)  ->  A. x  e.  ( Base `  R
) A. y  e.  ( Base `  R
) ( F `  ( x ( +g  `  R ) y ) )  =  ( ( F `  x ) ( +g  `  S
) ( F `  y ) ) )
5241, 51jca 519 . . . 4  |-  ( F  e.  ( R RingHom  S
)  ->  ( F : ( Base `  R
) --> ( Base `  S
)  /\  A. x  e.  ( Base `  R
) A. y  e.  ( Base `  R
) ( F `  ( x ( +g  `  R ) y ) )  =  ( ( F `  x ) ( +g  `  S
) ( F `  y ) ) ) )
5338, 40, 52jca31 521 . . 3  |-  ( F  e.  ( R RingHom  S
)  ->  ( (
(oppr `  R )  e.  Grp  /\  (oppr
`  S )  e. 
Grp )  /\  ( F : ( Base `  R
) --> ( Base `  S
)  /\  A. x  e.  ( Base `  R
) A. y  e.  ( Base `  R
) ( F `  ( x ( +g  `  R ) y ) )  =  ( ( F `  x ) ( +g  `  S
) ( F `  y ) ) ) ) )
5411, 34opprbas 15689 . . . 4  |-  ( Base `  S )  =  (
Base `  (oppr
`  S ) )
557, 46oppradd 15690 . . . 4  |-  ( +g  `  R )  =  ( +g  `  (oppr `  R
) )
5611, 47oppradd 15690 . . . 4  |-  ( +g  `  S )  =  ( +g  `  (oppr `  S
) )
5724, 54, 55, 56isghm 14961 . . 3  |-  ( F  e.  ( (oppr `  R
)  GrpHom  (oppr
`  S ) )  <-> 
( ( (oppr `  R
)  e.  Grp  /\  (oppr `  S )  e.  Grp )  /\  ( F :
( Base `  R ) --> ( Base `  S )  /\  A. x  e.  (
Base `  R ) A. y  e.  ( Base `  R ) ( F `  ( x ( +g  `  R
) y ) )  =  ( ( F `
 x ) ( +g  `  S ) ( F `  y
) ) ) ) )
5853, 57sylibr 204 . 2  |-  ( F  e.  ( R RingHom  S
)  ->  F  e.  ( (oppr
`  R )  GrpHom  (oppr `  S ) ) )
591, 2, 3, 4, 5, 9, 13, 20, 36, 58isrhm2d 15784 1  |-  ( F  e.  ( R RingHom  S
)  ->  F  e.  ( (oppr
`  R ) RingHom  (oppr `  S
) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1649    e. wcel 1721   A.wral 2666   -->wf 5409   ` cfv 5413  (class class class)co 6040   Basecbs 13424   +g cplusg 13484   .rcmulr 13485   Grpcgrp 14640    GrpHom cghm 14958   Ringcrg 15615   1rcur 15617  opprcoppr 15682   RingHom crh 15772
This theorem is referenced by:  elrhmunit  24211
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-rep 4280  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660  ax-cnex 9002  ax-resscn 9003  ax-1cn 9004  ax-icn 9005  ax-addcl 9006  ax-addrcl 9007  ax-mulcl 9008  ax-mulrcl 9009  ax-mulcom 9010  ax-addass 9011  ax-mulass 9012  ax-distr 9013  ax-i2m1 9014  ax-1ne0 9015  ax-1rid 9016  ax-rnegex 9017  ax-rrecex 9018  ax-cnre 9019  ax-pre-lttri 9020  ax-pre-lttrn 9021  ax-pre-ltadd 9022  ax-pre-mulgt0 9023
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-nel 2570  df-ral 2671  df-rex 2672  df-reu 2673  df-rmo 2674  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-pss 3296  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-tp 3782  df-op 3783  df-uni 3976  df-iun 4055  df-br 4173  df-opab 4227  df-mpt 4228  df-tr 4263  df-eprel 4454  df-id 4458  df-po 4463  df-so 4464  df-fr 4501  df-we 4503  df-ord 4544  df-on 4545  df-lim 4546  df-suc 4547  df-om 4805  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-ov 6043  df-oprab 6044  df-mpt2 6045  df-tpos 6438  df-riota 6508  df-recs 6592  df-rdg 6627  df-er 6864  df-map 6979  df-en 7069  df-dom 7070  df-sdom 7071  df-pnf 9078  df-mnf 9079  df-xr 9080  df-ltxr 9081  df-le 9082  df-sub 9249  df-neg 9250  df-nn 9957  df-2 10014  df-3 10015  df-ndx 13427  df-slot 13428  df-base 13429  df-sets 13430  df-plusg 13497  df-mulr 13498  df-0g 13682  df-mnd 14645  df-mhm 14693  df-grp 14767  df-ghm 14959  df-mgp 15604  df-rng 15618  df-ur 15620  df-oppr 15683  df-rnghom 15774
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