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Theorem rngonegrmul 26259
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 5036 . . . . . . 7  |-  ran  G  =  ran  ( 1st `  R
)
41, 3eqtri 2407 . . . . . 6  |-  X  =  ran  ( 1st `  R
)
5 ringnegmul.2 . . . . . 6  |-  H  =  ( 2nd `  R
)
6 eqid 2387 . . . . . 6  |-  (GId `  H )  =  (GId
`  H )
74, 5, 6rngo1cl 21865 . . . . 5  |-  ( R  e.  RingOps  ->  (GId `  H
)  e.  X )
8 ringnegmul.4 . . . . . 6  |-  N  =  ( inv `  G
)
92, 1, 8rngonegcl 26252 . . . . 5  |-  ( ( R  e.  RingOps  /\  (GId `  H )  e.  X
)  ->  ( N `  (GId `  H )
)  e.  X )
107, 9mpdan 650 . . . 4  |-  ( R  e.  RingOps  ->  ( N `  (GId `  H ) )  e.  X )
112, 5, 1rngoass 21823 . . . . . . 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 1171 . . . . . 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 83 . . . . 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 79 . . . 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 1147 . 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 21818 . . . . 5  |-  ( ( R  e.  RingOps  /\  A  e.  X  /\  B  e.  X )  ->  ( A H B )  e.  X )
18173expb 1154 . . . 4  |-  ( ( R  e.  RingOps  /\  ( A  e.  X  /\  B  e.  X )
)  ->  ( A H B )  e.  X
)
192, 5, 1, 8, 6rngonegmn1r 26257 . . . 4  |-  ( ( R  e.  RingOps  /\  ( A H B )  e.  X )  ->  ( N `  ( A H B ) )  =  ( ( A H B ) H ( N `  (GId `  H ) ) ) )
2018, 19syldan 457 . . 3  |-  ( ( R  e.  RingOps  /\  ( A  e.  X  /\  B  e.  X )
)  ->  ( N `  ( A H B ) )  =  ( ( A H B ) H ( N `
 (GId `  H
) ) ) )
21203impb 1149 . 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 26257 . . . 4  |-  ( ( R  e.  RingOps  /\  B  e.  X )  ->  ( N `  B )  =  ( B H ( N `  (GId `  H ) ) ) )
23223adant2 976 . . 3  |-  ( ( R  e.  RingOps  /\  A  e.  X  /\  B  e.  X )  ->  ( N `  B )  =  ( B H ( N `  (GId `  H ) ) ) )
2423oveq2d 6036 . 2  |-  ( ( R  e.  RingOps  /\  A  e.  X  /\  B  e.  X )  ->  ( A H ( N `  B ) )  =  ( A H ( B H ( N `
 (GId `  H
) ) ) ) )
2516, 21, 243eqtr4d 2429 1  |-  ( ( R  e.  RingOps  /\  A  e.  X  /\  B  e.  X )  ->  ( N `  ( A H B ) )  =  ( A H ( N `  B ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    /\ w3a 936    = wceq 1649    e. wcel 1717   ran crn 4819   ` cfv 5394  (class class class)co 6020   1stc1st 6286   2ndc2nd 6287  GIdcgi 21623   invcgn 21624   RingOpscrngo 21811
This theorem is referenced by:  rngosubdi  26260
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 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2368  ax-rep 4261  ax-sep 4271  ax-nul 4279  ax-pow 4318  ax-pr 4344  ax-un 4641
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2242  df-mo 2243  df-clab 2374  df-cleq 2380  df-clel 2383  df-nfc 2512  df-ne 2552  df-ral 2654  df-rex 2655  df-reu 2656  df-rmo 2657  df-rab 2658  df-v 2901  df-sbc 3105  df-csb 3195  df-dif 3266  df-un 3268  df-in 3270  df-ss 3277  df-nul 3572  df-if 3683  df-sn 3763  df-pr 3764  df-op 3766  df-uni 3958  df-iun 4037  df-br 4154  df-opab 4208  df-mpt 4209  df-id 4439  df-xp 4824  df-rel 4825  df-cnv 4826  df-co 4827  df-dm 4828  df-rn 4829  df-res 4830  df-ima 4831  df-iota 5358  df-fun 5396  df-fn 5397  df-f 5398  df-f1 5399  df-fo 5400  df-f1o 5401  df-fv 5402  df-ov 6023  df-1st 6288  df-2nd 6289  df-riota 6485  df-grpo 21627  df-gid 21628  df-ginv 21629  df-ablo 21718  df-ass 21749  df-exid 21751  df-mgm 21755  df-sgr 21767  df-mndo 21774  df-rngo 21812
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