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Theorem rhmopp 28044
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 2454 . 2  |-  ( Base `  (oppr
`  R ) )  =  ( Base `  (oppr `  R
) )
2 eqid 2454 . 2  |-  ( 1r
`  (oppr
`  R ) )  =  ( 1r `  (oppr `  R ) )
3 eqid 2454 . 2  |-  ( 1r
`  (oppr
`  S ) )  =  ( 1r `  (oppr `  S ) )
4 eqid 2454 . 2  |-  ( .r
`  (oppr
`  R ) )  =  ( .r `  (oppr `  R ) )
5 eqid 2454 . 2  |-  ( .r
`  (oppr
`  S ) )  =  ( .r `  (oppr `  S ) )
6 rhmrcl1 17563 . . 3  |-  ( F  e.  ( R RingHom  S
)  ->  R  e.  Ring )
7 eqid 2454 . . . 4  |-  (oppr `  R
)  =  (oppr `  R
)
87opprringb 17476 . . 3  |-  ( R  e.  Ring  <->  (oppr
`  R )  e. 
Ring )
96, 8sylib 196 . 2  |-  ( F  e.  ( R RingHom  S
)  ->  (oppr
`  R )  e. 
Ring )
10 rhmrcl2 17564 . . 3  |-  ( F  e.  ( R RingHom  S
)  ->  S  e.  Ring )
11 eqid 2454 . . . 4  |-  (oppr `  S
)  =  (oppr `  S
)
1211opprringb 17476 . . 3  |-  ( S  e.  Ring  <->  (oppr
`  S )  e. 
Ring )
1310, 12sylib 196 . 2  |-  ( F  e.  ( R RingHom  S
)  ->  (oppr
`  S )  e. 
Ring )
14 eqid 2454 . . . . 5  |-  ( 1r
`  R )  =  ( 1r `  R
)
157, 14oppr1 17478 . . . 4  |-  ( 1r
`  R )  =  ( 1r `  (oppr `  R
) )
1615eqcomi 2467 . . 3  |-  ( 1r
`  (oppr
`  R ) )  =  ( 1r `  R )
17 eqid 2454 . . . . 5  |-  ( 1r
`  S )  =  ( 1r `  S
)
1811, 17oppr1 17478 . . . 4  |-  ( 1r
`  S )  =  ( 1r `  (oppr `  S
) )
1918eqcomi 2467 . . 3  |-  ( 1r
`  (oppr
`  S ) )  =  ( 1r `  S )
2016, 19rhm1 17574 . 2  |-  ( F  e.  ( R RingHom  S
)  ->  ( F `  ( 1r `  (oppr `  R
) ) )  =  ( 1r `  (oppr `  S
) ) )
21 simpl 455 . . . 4  |-  ( ( F  e.  ( R RingHom  S )  /\  (
x  e.  ( Base `  (oppr
`  R ) )  /\  y  e.  (
Base `  (oppr
`  R ) ) ) )  ->  F  e.  ( R RingHom  S )
)
22 simprr 755 . . . . 5  |-  ( ( F  e.  ( R RingHom  S )  /\  (
x  e.  ( Base `  (oppr
`  R ) )  /\  y  e.  (
Base `  (oppr
`  R ) ) ) )  ->  y  e.  ( Base `  (oppr `  R
) ) )
23 eqid 2454 . . . . . 6  |-  ( Base `  R )  =  (
Base `  R )
247, 23opprbas 17473 . . . . 5  |-  ( Base `  R )  =  (
Base `  (oppr
`  R ) )
2522, 24syl6eleqr 2553 . . . 4  |-  ( ( F  e.  ( R RingHom  S )  /\  (
x  e.  ( Base `  (oppr
`  R ) )  /\  y  e.  (
Base `  (oppr
`  R ) ) ) )  ->  y  e.  ( Base `  R
) )
26 simprl 754 . . . . 5  |-  ( ( F  e.  ( R RingHom  S )  /\  (
x  e.  ( Base `  (oppr
`  R ) )  /\  y  e.  (
Base `  (oppr
`  R ) ) ) )  ->  x  e.  ( Base `  (oppr `  R
) ) )
2726, 24syl6eleqr 2553 . . . 4  |-  ( ( F  e.  ( R RingHom  S )  /\  (
x  e.  ( Base `  (oppr
`  R ) )  /\  y  e.  (
Base `  (oppr
`  R ) ) ) )  ->  x  e.  ( Base `  R
) )
28 eqid 2454 . . . . 5  |-  ( .r
`  R )  =  ( .r `  R
)
29 eqid 2454 . . . . 5  |-  ( .r
`  S )  =  ( .r `  S
)
3023, 28, 29rhmmul 17571 . . . 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 1226 . . 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 17470 . . . 4  |-  ( x ( .r `  (oppr `  R
) ) y )  =  ( y ( .r `  R ) x )
3332fveq2i 5851 . . 3  |-  ( F `
 ( x ( .r `  (oppr `  R
) ) y ) )  =  ( F `
 ( y ( .r `  R ) x ) )
34 eqid 2454 . . . 4  |-  ( Base `  S )  =  (
Base `  S )
3534, 29, 11, 5opprmul 17470 . . 3  |-  ( ( F `  x ) ( .r `  (oppr `  S
) ) ( F `
 y ) )  =  ( ( F `
 y ) ( .r `  S ) ( F `  x
) )
3631, 33, 353eqtr4g 2520 . 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 ringgrp 17398 . . . . 5  |-  ( (oppr `  R )  e.  Ring  -> 
(oppr `  R )  e.  Grp )
389, 37syl 16 . . . 4  |-  ( F  e.  ( R RingHom  S
)  ->  (oppr
`  R )  e. 
Grp )
39 ringgrp 17398 . . . . 5  |-  ( (oppr `  S )  e.  Ring  -> 
(oppr `  S )  e.  Grp )
4013, 39syl 16 . . . 4  |-  ( F  e.  ( R RingHom  S
)  ->  (oppr
`  S )  e. 
Grp )
4123, 34rhmf 17570 . . . . 5  |-  ( F  e.  ( R RingHom  S
)  ->  F :
( Base `  R ) --> ( Base `  S )
)
42 rhmghm 17569 . . . . . . . . 9  |-  ( F  e.  ( R RingHom  S
)  ->  F  e.  ( R  GrpHom  S ) )
4342ad2antrr 723 . . . . . . . 8  |-  ( ( ( F  e.  ( R RingHom  S )  /\  x  e.  ( Base `  R
) )  /\  y  e.  ( Base `  R
) )  ->  F  e.  ( R  GrpHom  S ) )
44 simplr 753 . . . . . . . 8  |-  ( ( ( F  e.  ( R RingHom  S )  /\  x  e.  ( Base `  R
) )  /\  y  e.  ( Base `  R
) )  ->  x  e.  ( Base `  R
) )
45 simpr 459 . . . . . . . 8  |-  ( ( ( F  e.  ( R RingHom  S )  /\  x  e.  ( Base `  R
) )  /\  y  e.  ( Base `  R
) )  ->  y  e.  ( Base `  R
) )
46 eqid 2454 . . . . . . . . 9  |-  ( +g  `  R )  =  ( +g  `  R )
47 eqid 2454 . . . . . . . . 9  |-  ( +g  `  S )  =  ( +g  `  S )
4823, 46, 47ghmlin 16471 . . . . . . . 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 1226 . . . . . . 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 2868 . . . . . 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 2868 . . . . 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 530 . . . 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 532 . . 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 17473 . . . 4  |-  ( Base `  S )  =  (
Base `  (oppr
`  S ) )
557, 46oppradd 17474 . . . 4  |-  ( +g  `  R )  =  ( +g  `  (oppr `  R
) )
5611, 47oppradd 17474 . . . 4  |-  ( +g  `  S )  =  ( +g  `  (oppr `  S
) )
5724, 54, 55, 56isghm 16466 . . 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 17572 1  |-  ( F  e.  ( R RingHom  S
)  ->  F  e.  ( (oppr
`  R ) RingHom  (oppr `  S
) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 367    = wceq 1398    e. wcel 1823   A.wral 2804   -->wf 5566   ` cfv 5570  (class class class)co 6270   Basecbs 14716   +g cplusg 14784   .rcmulr 14785   Grpcgrp 16252    GrpHom cghm 16463   1rcur 17348   Ringcrg 17393  opprcoppr 17466   RingHom crh 17556
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-rep 4550  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-un 6565  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 368  df-an 369  df-3or 972  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-nel 2652  df-ral 2809  df-rex 2810  df-reu 2811  df-rmo 2812  df-rab 2813  df-v 3108  df-sbc 3325  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-tp 4021  df-op 4023  df-uni 4236  df-iun 4317  df-br 4440  df-opab 4498  df-mpt 4499  df-tr 4533  df-eprel 4780  df-id 4784  df-po 4789  df-so 4790  df-fr 4827  df-we 4829  df-ord 4870  df-on 4871  df-lim 4872  df-suc 4873  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-f1 5575  df-fo 5576  df-f1o 5577  df-fv 5578  df-riota 6232  df-ov 6273  df-oprab 6274  df-mpt2 6275  df-om 6674  df-tpos 6947  df-recs 7034  df-rdg 7068  df-er 7303  df-map 7414  df-en 7510  df-dom 7511  df-sdom 7512  df-pnf 9619  df-mnf 9620  df-xr 9621  df-ltxr 9622  df-le 9623  df-sub 9798  df-neg 9799  df-nn 10532  df-2 10590  df-3 10591  df-ndx 14719  df-slot 14720  df-base 14721  df-sets 14722  df-plusg 14797  df-mulr 14798  df-0g 14931  df-mgm 16071  df-sgrp 16110  df-mnd 16120  df-mhm 16165  df-grp 16256  df-ghm 16464  df-mgp 17337  df-ur 17349  df-ring 17395  df-oppr 17467  df-rnghom 17559
This theorem is referenced by:  elrhmunit  28045
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