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Theorem rngonegrmul 30327
Description: Negation of a product in a ring. (Contributed by Jeff Madsen, 19-Jun-2010.)
Hypotheses
Ref Expression
ringnegmul.1  |-  G  =  ( 1st `  R
)
ringnegmul.2  |-  H  =  ( 2nd `  R
)
ringnegmul.3  |-  X  =  ran  G
ringnegmul.4  |-  N  =  ( inv `  G
)
Assertion
Ref Expression
rngonegrmul  |-  ( ( R  e.  RingOps  /\  A  e.  X  /\  B  e.  X )  ->  ( N `  ( A H B ) )  =  ( A H ( N `  B ) ) )

Proof of Theorem rngonegrmul
StepHypRef Expression
1 ringnegmul.3 . . . . . . 7  |-  X  =  ran  G
2 ringnegmul.1 . . . . . . . 8  |-  G  =  ( 1st `  R
)
32rneqi 5216 . . . . . . 7  |-  ran  G  =  ran  ( 1st `  R
)
41, 3eqtri 2470 . . . . . 6  |-  X  =  ran  ( 1st `  R
)
5 ringnegmul.2 . . . . . 6  |-  H  =  ( 2nd `  R
)
6 eqid 2441 . . . . . 6  |-  (GId `  H )  =  (GId
`  H )
74, 5, 6rngo1cl 25300 . . . . 5  |-  ( R  e.  RingOps  ->  (GId `  H
)  e.  X )
8 ringnegmul.4 . . . . . 6  |-  N  =  ( inv `  G
)
92, 1, 8rngonegcl 30320 . . . . 5  |-  ( ( R  e.  RingOps  /\  (GId `  H )  e.  X
)  ->  ( N `  (GId `  H )
)  e.  X )
107, 9mpdan 668 . . . 4  |-  ( R  e.  RingOps  ->  ( N `  (GId `  H ) )  e.  X )
112, 5, 1rngoass 25258 . . . . . . 7  |-  ( ( R  e.  RingOps  /\  ( A  e.  X  /\  B  e.  X  /\  ( N `  (GId `  H ) )  e.  X ) )  -> 
( ( A H B ) H ( N `  (GId `  H ) ) )  =  ( A H ( B H ( N `  (GId `  H ) ) ) ) )
12113exp2 1213 . . . . . 6  |-  ( R  e.  RingOps  ->  ( A  e.  X  ->  ( B  e.  X  ->  ( ( N `  (GId `  H ) )  e.  X  ->  ( ( A H B ) H ( N `  (GId `  H ) ) )  =  ( A H ( B H ( N `  (GId `  H ) ) ) ) ) ) ) )
1312com24 87 . . . . 5  |-  ( R  e.  RingOps  ->  ( ( N `
 (GId `  H
) )  e.  X  ->  ( B  e.  X  ->  ( A  e.  X  ->  ( ( A H B ) H ( N `  (GId `  H ) ) )  =  ( A H ( B H ( N `  (GId `  H ) ) ) ) ) ) ) )
1413com34 83 . . . 4  |-  ( R  e.  RingOps  ->  ( ( N `
 (GId `  H
) )  e.  X  ->  ( A  e.  X  ->  ( B  e.  X  ->  ( ( A H B ) H ( N `  (GId `  H ) ) )  =  ( A H ( B H ( N `  (GId `  H ) ) ) ) ) ) ) )
1510, 14mpd 15 . . 3  |-  ( R  e.  RingOps  ->  ( A  e.  X  ->  ( B  e.  X  ->  ( ( A H B ) H ( N `  (GId `  H ) ) )  =  ( A H ( B H ( N `  (GId `  H ) ) ) ) ) ) )
16153imp 1189 . 2  |-  ( ( R  e.  RingOps  /\  A  e.  X  /\  B  e.  X )  ->  (
( A H B ) H ( N `
 (GId `  H
) ) )  =  ( A H ( B H ( N `
 (GId `  H
) ) ) ) )
172, 5, 1rngocl 25253 . . . . 5  |-  ( ( R  e.  RingOps  /\  A  e.  X  /\  B  e.  X )  ->  ( A H B )  e.  X )
18173expb 1196 . . . 4  |-  ( ( R  e.  RingOps  /\  ( A  e.  X  /\  B  e.  X )
)  ->  ( A H B )  e.  X
)
192, 5, 1, 8, 6rngonegmn1r 30325 . . . 4  |-  ( ( R  e.  RingOps  /\  ( A H B )  e.  X )  ->  ( N `  ( A H B ) )  =  ( ( A H B ) H ( N `  (GId `  H ) ) ) )
2018, 19syldan 470 . . 3  |-  ( ( R  e.  RingOps  /\  ( A  e.  X  /\  B  e.  X )
)  ->  ( N `  ( A H B ) )  =  ( ( A H B ) H ( N `
 (GId `  H
) ) ) )
21203impb 1191 . 2  |-  ( ( R  e.  RingOps  /\  A  e.  X  /\  B  e.  X )  ->  ( N `  ( A H B ) )  =  ( ( A H B ) H ( N `  (GId `  H ) ) ) )
222, 5, 1, 8, 6rngonegmn1r 30325 . . . 4  |-  ( ( R  e.  RingOps  /\  B  e.  X )  ->  ( N `  B )  =  ( B H ( N `  (GId `  H ) ) ) )
23223adant2 1014 . . 3  |-  ( ( R  e.  RingOps  /\  A  e.  X  /\  B  e.  X )  ->  ( N `  B )  =  ( B H ( N `  (GId `  H ) ) ) )
2423oveq2d 6294 . 2  |-  ( ( R  e.  RingOps  /\  A  e.  X  /\  B  e.  X )  ->  ( A H ( N `  B ) )  =  ( A H ( B H ( N `
 (GId `  H
) ) ) ) )
2516, 21, 243eqtr4d 2492 1  |-  ( ( R  e.  RingOps  /\  A  e.  X  /\  B  e.  X )  ->  ( N `  ( A H B ) )  =  ( A H ( N `  B ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    /\ w3a 972    = wceq 1381    e. wcel 1802   ran crn 4987   ` cfv 5575  (class class class)co 6278   1stc1st 6780   2ndc2nd 6781  GIdcgi 25058   invcgn 25059   RingOpscrngo 25246
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1603  ax-4 1616  ax-5 1689  ax-6 1732  ax-7 1774  ax-8 1804  ax-9 1806  ax-10 1821  ax-11 1826  ax-12 1838  ax-13 1983  ax-ext 2419  ax-rep 4545  ax-sep 4555  ax-nul 4563  ax-pow 4612  ax-pr 4673  ax-un 6574
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 974  df-tru 1384  df-ex 1598  df-nf 1602  df-sb 1725  df-eu 2270  df-mo 2271  df-clab 2427  df-cleq 2433  df-clel 2436  df-nfc 2591  df-ne 2638  df-ral 2796  df-rex 2797  df-reu 2798  df-rmo 2799  df-rab 2800  df-v 3095  df-sbc 3312  df-csb 3419  df-dif 3462  df-un 3464  df-in 3466  df-ss 3473  df-nul 3769  df-if 3924  df-sn 4012  df-pr 4014  df-op 4018  df-uni 4232  df-iun 4314  df-br 4435  df-opab 4493  df-mpt 4494  df-id 4782  df-xp 4992  df-rel 4993  df-cnv 4994  df-co 4995  df-dm 4996  df-rn 4997  df-res 4998  df-ima 4999  df-iota 5538  df-fun 5577  df-fn 5578  df-f 5579  df-f1 5580  df-fo 5581  df-f1o 5582  df-fv 5583  df-riota 6239  df-ov 6281  df-1st 6782  df-2nd 6783  df-grpo 25062  df-gid 25063  df-ginv 25064  df-ablo 25153  df-ass 25184  df-exid 25186  df-mgmOLD 25190  df-sgrOLD 25202  df-mndo 25209  df-rngo 25247
This theorem is referenced by:  rngosubdi  30328
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