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Theorem rhmopp 27458
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 2460 . 2  |-  ( Base `  (oppr
`  R ) )  =  ( Base `  (oppr `  R
) )
2 eqid 2460 . 2  |-  ( 1r
`  (oppr
`  R ) )  =  ( 1r `  (oppr `  R ) )
3 eqid 2460 . 2  |-  ( 1r
`  (oppr
`  S ) )  =  ( 1r `  (oppr `  S ) )
4 eqid 2460 . 2  |-  ( .r
`  (oppr
`  R ) )  =  ( .r `  (oppr `  R ) )
5 eqid 2460 . 2  |-  ( .r
`  (oppr
`  S ) )  =  ( .r `  (oppr `  S ) )
6 rhmrcl1 17145 . . 3  |-  ( F  e.  ( R RingHom  S
)  ->  R  e.  Ring )
7 eqid 2460 . . . 4  |-  (oppr `  R
)  =  (oppr `  R
)
87opprrngb 17058 . . 3  |-  ( R  e.  Ring  <->  (oppr
`  R )  e. 
Ring )
96, 8sylib 196 . 2  |-  ( F  e.  ( R RingHom  S
)  ->  (oppr
`  R )  e. 
Ring )
10 rhmrcl2 17146 . . 3  |-  ( F  e.  ( R RingHom  S
)  ->  S  e.  Ring )
11 eqid 2460 . . . 4  |-  (oppr `  S
)  =  (oppr `  S
)
1211opprrngb 17058 . . 3  |-  ( S  e.  Ring  <->  (oppr
`  S )  e. 
Ring )
1310, 12sylib 196 . 2  |-  ( F  e.  ( R RingHom  S
)  ->  (oppr
`  S )  e. 
Ring )
14 eqid 2460 . . . . 5  |-  ( 1r
`  R )  =  ( 1r `  R
)
157, 14oppr1 17060 . . . 4  |-  ( 1r
`  R )  =  ( 1r `  (oppr `  R
) )
1615eqcomi 2473 . . 3  |-  ( 1r
`  (oppr
`  R ) )  =  ( 1r `  R )
17 eqid 2460 . . . . 5  |-  ( 1r
`  S )  =  ( 1r `  S
)
1811, 17oppr1 17060 . . . 4  |-  ( 1r
`  S )  =  ( 1r `  (oppr `  S
) )
1918eqcomi 2473 . . 3  |-  ( 1r
`  (oppr
`  S ) )  =  ( 1r `  S )
2016, 19rhm1 17156 . 2  |-  ( F  e.  ( R RingHom  S
)  ->  ( F `  ( 1r `  (oppr `  R
) ) )  =  ( 1r `  (oppr `  S
) ) )
21 simpl 457 . . . 4  |-  ( ( F  e.  ( R RingHom  S )  /\  (
x  e.  ( Base `  (oppr
`  R ) )  /\  y  e.  (
Base `  (oppr
`  R ) ) ) )  ->  F  e.  ( R RingHom  S )
)
22 simprr 756 . . . . 5  |-  ( ( F  e.  ( R RingHom  S )  /\  (
x  e.  ( Base `  (oppr
`  R ) )  /\  y  e.  (
Base `  (oppr
`  R ) ) ) )  ->  y  e.  ( Base `  (oppr `  R
) ) )
23 eqid 2460 . . . . . 6  |-  ( Base `  R )  =  (
Base `  R )
247, 23opprbas 17055 . . . . 5  |-  ( Base `  R )  =  (
Base `  (oppr
`  R ) )
2522, 24syl6eleqr 2559 . . . 4  |-  ( ( F  e.  ( R RingHom  S )  /\  (
x  e.  ( Base `  (oppr
`  R ) )  /\  y  e.  (
Base `  (oppr
`  R ) ) ) )  ->  y  e.  ( Base `  R
) )
26 simprl 755 . . . . 5  |-  ( ( F  e.  ( R RingHom  S )  /\  (
x  e.  ( Base `  (oppr
`  R ) )  /\  y  e.  (
Base `  (oppr
`  R ) ) ) )  ->  x  e.  ( Base `  (oppr `  R
) ) )
2726, 24syl6eleqr 2559 . . . 4  |-  ( ( F  e.  ( R RingHom  S )  /\  (
x  e.  ( Base `  (oppr
`  R ) )  /\  y  e.  (
Base `  (oppr
`  R ) ) ) )  ->  x  e.  ( Base `  R
) )
28 eqid 2460 . . . . 5  |-  ( .r
`  R )  =  ( .r `  R
)
29 eqid 2460 . . . . 5  |-  ( .r
`  S )  =  ( .r `  S
)
3023, 28, 29rhmmul 17153 . . . 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 1223 . . 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 17052 . . . 4  |-  ( x ( .r `  (oppr `  R
) ) y )  =  ( y ( .r `  R ) x )
3332fveq2i 5860 . . 3  |-  ( F `
 ( x ( .r `  (oppr `  R
) ) y ) )  =  ( F `
 ( y ( .r `  R ) x ) )
34 eqid 2460 . . . 4  |-  ( Base `  S )  =  (
Base `  S )
3534, 29, 11, 5opprmul 17052 . . 3  |-  ( ( F `  x ) ( .r `  (oppr `  S
) ) ( F `
 y ) )  =  ( ( F `
 y ) ( .r `  S ) ( F `  x
) )
3631, 33, 353eqtr4g 2526 . 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 16984 . . . . 5  |-  ( (oppr `  R )  e.  Ring  -> 
(oppr `  R )  e.  Grp )
389, 37syl 16 . . . 4  |-  ( F  e.  ( R RingHom  S
)  ->  (oppr
`  R )  e. 
Grp )
39 rnggrp 16984 . . . . 5  |-  ( (oppr `  S )  e.  Ring  -> 
(oppr `  S )  e.  Grp )
4013, 39syl 16 . . . 4  |-  ( F  e.  ( R RingHom  S
)  ->  (oppr
`  S )  e. 
Grp )
4123, 34rhmf 17152 . . . . 5  |-  ( F  e.  ( R RingHom  S
)  ->  F :
( Base `  R ) --> ( Base `  S )
)
42 rhmghm 17151 . . . . . . . . 9  |-  ( F  e.  ( R RingHom  S
)  ->  F  e.  ( R  GrpHom  S ) )
4342ad2antrr 725 . . . . . . . 8  |-  ( ( ( F  e.  ( R RingHom  S )  /\  x  e.  ( Base `  R
) )  /\  y  e.  ( Base `  R
) )  ->  F  e.  ( R  GrpHom  S ) )
44 simplr 754 . . . . . . . 8  |-  ( ( ( F  e.  ( R RingHom  S )  /\  x  e.  ( Base `  R
) )  /\  y  e.  ( Base `  R
) )  ->  x  e.  ( Base `  R
) )
45 simpr 461 . . . . . . . 8  |-  ( ( ( F  e.  ( R RingHom  S )  /\  x  e.  ( Base `  R
) )  /\  y  e.  ( Base `  R
) )  ->  y  e.  ( Base `  R
) )
46 eqid 2460 . . . . . . . . 9  |-  ( +g  `  R )  =  ( +g  `  R )
47 eqid 2460 . . . . . . . . 9  |-  ( +g  `  S )  =  ( +g  `  S )
4823, 46, 47ghmlin 16060 . . . . . . . 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 1223 . . . . . . 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 2871 . . . . . 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 2871 . . . . 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 532 . . . 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 534 . . 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 17055 . . . 4  |-  ( Base `  S )  =  (
Base `  (oppr
`  S ) )
557, 46oppradd 17056 . . . 4  |-  ( +g  `  R )  =  ( +g  `  (oppr `  R
) )
5611, 47oppradd 17056 . . . 4  |-  ( +g  `  S )  =  ( +g  `  (oppr `  S
) )
5724, 54, 55, 56isghm 16055 . . 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 212 . 2  |-  ( F  e.  ( R RingHom  S
)  ->  F  e.  ( (oppr
`  R )  GrpHom  (oppr `  S ) ) )
591, 2, 3, 4, 5, 9, 13, 20, 36, 58isrhm2d 17154 1  |-  ( F  e.  ( R RingHom  S
)  ->  F  e.  ( (oppr
`  R ) RingHom  (oppr `  S
) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1374    e. wcel 1762   A.wral 2807   -->wf 5575   ` cfv 5579  (class class class)co 6275   Basecbs 14479   +g cplusg 14544   .rcmulr 14545   Grpcgrp 15716    GrpHom cghm 16052   1rcur 16936   Ringcrg 16979  opprcoppr 17048   RingHom crh 17138
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-8 1764  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1961  ax-ext 2438  ax-rep 4551  ax-sep 4561  ax-nul 4569  ax-pow 4618  ax-pr 4679  ax-un 6567  ax-cnex 9537  ax-resscn 9538  ax-1cn 9539  ax-icn 9540  ax-addcl 9541  ax-addrcl 9542  ax-mulcl 9543  ax-mulrcl 9544  ax-mulcom 9545  ax-addass 9546  ax-mulass 9547  ax-distr 9548  ax-i2m1 9549  ax-1ne0 9550  ax-1rid 9551  ax-rnegex 9552  ax-rrecex 9553  ax-cnre 9554  ax-pre-lttri 9555  ax-pre-lttrn 9556  ax-pre-ltadd 9557  ax-pre-mulgt0 9558
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 969  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2272  df-mo 2273  df-clab 2446  df-cleq 2452  df-clel 2455  df-nfc 2610  df-ne 2657  df-nel 2658  df-ral 2812  df-rex 2813  df-reu 2814  df-rmo 2815  df-rab 2816  df-v 3108  df-sbc 3325  df-csb 3429  df-dif 3472  df-un 3474  df-in 3476  df-ss 3483  df-pss 3485  df-nul 3779  df-if 3933  df-pw 4005  df-sn 4021  df-pr 4023  df-tp 4025  df-op 4027  df-uni 4239  df-iun 4320  df-br 4441  df-opab 4499  df-mpt 4500  df-tr 4534  df-eprel 4784  df-id 4788  df-po 4793  df-so 4794  df-fr 4831  df-we 4833  df-ord 4874  df-on 4875  df-lim 4876  df-suc 4877  df-xp 4998  df-rel 4999  df-cnv 5000  df-co 5001  df-dm 5002  df-rn 5003  df-res 5004  df-ima 5005  df-iota 5542  df-fun 5581  df-fn 5582  df-f 5583  df-f1 5584  df-fo 5585  df-f1o 5586  df-fv 5587  df-riota 6236  df-ov 6278  df-oprab 6279  df-mpt2 6280  df-om 6672  df-tpos 6945  df-recs 7032  df-rdg 7066  df-er 7301  df-map 7412  df-en 7507  df-dom 7508  df-sdom 7509  df-pnf 9619  df-mnf 9620  df-xr 9621  df-ltxr 9622  df-le 9623  df-sub 9796  df-neg 9797  df-nn 10526  df-2 10583  df-3 10584  df-ndx 14482  df-slot 14483  df-base 14484  df-sets 14485  df-plusg 14557  df-mulr 14558  df-0g 14686  df-mnd 15721  df-mhm 15770  df-grp 15851  df-ghm 16053  df-mgp 16925  df-ur 16937  df-rng 16981  df-oppr 17049  df-rnghom 17141
This theorem is referenced by:  elrhmunit  27459
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