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Theorem rhmopp 26255
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 2438 . 2  |-  ( Base `  (oppr
`  R ) )  =  ( Base `  (oppr `  R
) )
2 eqid 2438 . 2  |-  ( 1r
`  (oppr
`  R ) )  =  ( 1r `  (oppr `  R ) )
3 eqid 2438 . 2  |-  ( 1r
`  (oppr
`  S ) )  =  ( 1r `  (oppr `  S ) )
4 eqid 2438 . 2  |-  ( .r
`  (oppr
`  R ) )  =  ( .r `  (oppr `  R ) )
5 eqid 2438 . 2  |-  ( .r
`  (oppr
`  S ) )  =  ( .r `  (oppr `  S ) )
6 rhmrcl1 16799 . . 3  |-  ( F  e.  ( R RingHom  S
)  ->  R  e.  Ring )
7 eqid 2438 . . . 4  |-  (oppr `  R
)  =  (oppr `  R
)
87opprrngb 16714 . . 3  |-  ( R  e.  Ring  <->  (oppr
`  R )  e. 
Ring )
96, 8sylib 196 . 2  |-  ( F  e.  ( R RingHom  S
)  ->  (oppr
`  R )  e. 
Ring )
10 rhmrcl2 16800 . . 3  |-  ( F  e.  ( R RingHom  S
)  ->  S  e.  Ring )
11 eqid 2438 . . . 4  |-  (oppr `  S
)  =  (oppr `  S
)
1211opprrngb 16714 . . 3  |-  ( S  e.  Ring  <->  (oppr
`  S )  e. 
Ring )
1310, 12sylib 196 . 2  |-  ( F  e.  ( R RingHom  S
)  ->  (oppr
`  S )  e. 
Ring )
14 eqid 2438 . . . . 5  |-  ( 1r
`  R )  =  ( 1r `  R
)
157, 14oppr1 16716 . . . 4  |-  ( 1r
`  R )  =  ( 1r `  (oppr `  R
) )
1615eqcomi 2442 . . 3  |-  ( 1r
`  (oppr
`  R ) )  =  ( 1r `  R )
17 eqid 2438 . . . . 5  |-  ( 1r
`  S )  =  ( 1r `  S
)
1811, 17oppr1 16716 . . . 4  |-  ( 1r
`  S )  =  ( 1r `  (oppr `  S
) )
1918eqcomi 2442 . . 3  |-  ( 1r
`  (oppr
`  S ) )  =  ( 1r `  S )
2016, 19rhm1 16808 . 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 2438 . . . . . 6  |-  ( Base `  R )  =  (
Base `  R )
247, 23opprbas 16711 . . . . 5  |-  ( Base `  R )  =  (
Base `  (oppr
`  R ) )
2522, 24syl6eleqr 2529 . . . 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 2529 . . . 4  |-  ( ( F  e.  ( R RingHom  S )  /\  (
x  e.  ( Base `  (oppr
`  R ) )  /\  y  e.  (
Base `  (oppr
`  R ) ) ) )  ->  x  e.  ( Base `  R
) )
28 eqid 2438 . . . . 5  |-  ( .r
`  R )  =  ( .r `  R
)
29 eqid 2438 . . . . 5  |-  ( .r
`  S )  =  ( .r `  S
)
3023, 28, 29rhmmul 16805 . . . 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 1218 . . 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 16708 . . . 4  |-  ( x ( .r `  (oppr `  R
) ) y )  =  ( y ( .r `  R ) x )
3332fveq2i 5689 . . 3  |-  ( F `
 ( x ( .r `  (oppr `  R
) ) y ) )  =  ( F `
 ( y ( .r `  R ) x ) )
34 eqid 2438 . . . 4  |-  ( Base `  S )  =  (
Base `  S )
3534, 29, 11, 5opprmul 16708 . . 3  |-  ( ( F `  x ) ( .r `  (oppr `  S
) ) ( F `
 y ) )  =  ( ( F `
 y ) ( .r `  S ) ( F `  x
) )
3631, 33, 353eqtr4g 2495 . 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 16640 . . . . 5  |-  ( (oppr `  R )  e.  Ring  -> 
(oppr `  R )  e.  Grp )
389, 37syl 16 . . . 4  |-  ( F  e.  ( R RingHom  S
)  ->  (oppr
`  R )  e. 
Grp )
39 rnggrp 16640 . . . . 5  |-  ( (oppr `  S )  e.  Ring  -> 
(oppr `  S )  e.  Grp )
4013, 39syl 16 . . . 4  |-  ( F  e.  ( R RingHom  S
)  ->  (oppr
`  S )  e. 
Grp )
4123, 34rhmf 16804 . . . . 5  |-  ( F  e.  ( R RingHom  S
)  ->  F :
( Base `  R ) --> ( Base `  S )
)
42 rhmghm 16803 . . . . . . . . 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 2438 . . . . . . . . 9  |-  ( +g  `  R )  =  ( +g  `  R )
47 eqid 2438 . . . . . . . . 9  |-  ( +g  `  S )  =  ( +g  `  S )
4823, 46, 47ghmlin 15743 . . . . . . . 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 1218 . . . . . . 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 2794 . . . . . 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 2794 . . . . 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 16711 . . . 4  |-  ( Base `  S )  =  (
Base `  (oppr
`  S ) )
557, 46oppradd 16712 . . . 4  |-  ( +g  `  R )  =  ( +g  `  (oppr `  R
) )
5611, 47oppradd 16712 . . . 4  |-  ( +g  `  S )  =  ( +g  `  (oppr `  S
) )
5724, 54, 55, 56isghm 15738 . . 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 16806 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 1369    e. wcel 1756   A.wral 2710   -->wf 5409   ` cfv 5413  (class class class)co 6086   Basecbs 14166   +g cplusg 14230   .rcmulr 14231   Grpcgrp 15402    GrpHom cghm 15735   1rcur 16593   Ringcrg 16635  opprcoppr 16704   RingHom crh 16794
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 2419  ax-rep 4398  ax-sep 4408  ax-nul 4416  ax-pow 4465  ax-pr 4526  ax-un 6367  ax-cnex 9330  ax-resscn 9331  ax-1cn 9332  ax-icn 9333  ax-addcl 9334  ax-addrcl 9335  ax-mulcl 9336  ax-mulrcl 9337  ax-mulcom 9338  ax-addass 9339  ax-mulass 9340  ax-distr 9341  ax-i2m1 9342  ax-1ne0 9343  ax-1rid 9344  ax-rnegex 9345  ax-rrecex 9346  ax-cnre 9347  ax-pre-lttri 9348  ax-pre-lttrn 9349  ax-pre-ltadd 9350  ax-pre-mulgt0 9351
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 2256  df-mo 2257  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-nel 2604  df-ral 2715  df-rex 2716  df-reu 2717  df-rmo 2718  df-rab 2719  df-v 2969  df-sbc 3182  df-csb 3284  df-dif 3326  df-un 3328  df-in 3330  df-ss 3337  df-pss 3339  df-nul 3633  df-if 3787  df-pw 3857  df-sn 3873  df-pr 3875  df-tp 3877  df-op 3879  df-uni 4087  df-iun 4168  df-br 4288  df-opab 4346  df-mpt 4347  df-tr 4381  df-eprel 4627  df-id 4631  df-po 4636  df-so 4637  df-fr 4674  df-we 4676  df-ord 4717  df-on 4718  df-lim 4719  df-suc 4720  df-xp 4841  df-rel 4842  df-cnv 4843  df-co 4844  df-dm 4845  df-rn 4846  df-res 4847  df-ima 4848  df-iota 5376  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-riota 6047  df-ov 6089  df-oprab 6090  df-mpt2 6091  df-om 6472  df-tpos 6740  df-recs 6824  df-rdg 6858  df-er 7093  df-map 7208  df-en 7303  df-dom 7304  df-sdom 7305  df-pnf 9412  df-mnf 9413  df-xr 9414  df-ltxr 9415  df-le 9416  df-sub 9589  df-neg 9590  df-nn 10315  df-2 10372  df-3 10373  df-ndx 14169  df-slot 14170  df-base 14171  df-sets 14172  df-plusg 14243  df-mulr 14244  df-0g 14372  df-mnd 15407  df-mhm 15456  df-grp 15536  df-ghm 15736  df-mgp 16582  df-ur 16594  df-rng 16637  df-oppr 16705  df-rnghom 16796
This theorem is referenced by:  elrhmunit  26256
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