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Theorem rngonegrmul 29958
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 5227 . . . . . . 7  |-  ran  G  =  ran  ( 1st `  R
)
41, 3eqtri 2496 . . . . . 6  |-  X  =  ran  ( 1st `  R
)
5 ringnegmul.2 . . . . . 6  |-  H  =  ( 2nd `  R
)
6 eqid 2467 . . . . . 6  |-  (GId `  H )  =  (GId
`  H )
74, 5, 6rngo1cl 25107 . . . . 5  |-  ( R  e.  RingOps  ->  (GId `  H
)  e.  X )
8 ringnegmul.4 . . . . . 6  |-  N  =  ( inv `  G
)
92, 1, 8rngonegcl 29951 . . . . 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 25065 . . . . . . 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 1214 . . . . . 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 1190 . 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 25060 . . . . 5  |-  ( ( R  e.  RingOps  /\  A  e.  X  /\  B  e.  X )  ->  ( A H B )  e.  X )
18173expb 1197 . . . 4  |-  ( ( R  e.  RingOps  /\  ( A  e.  X  /\  B  e.  X )
)  ->  ( A H B )  e.  X
)
192, 5, 1, 8, 6rngonegmn1r 29956 . . . 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 1192 . 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 29956 . . . 4  |-  ( ( R  e.  RingOps  /\  B  e.  X )  ->  ( N `  B )  =  ( B H ( N `  (GId `  H ) ) ) )
23223adant2 1015 . . 3  |-  ( ( R  e.  RingOps  /\  A  e.  X  /\  B  e.  X )  ->  ( N `  B )  =  ( B H ( N `  (GId `  H ) ) ) )
2423oveq2d 6298 . 2  |-  ( ( R  e.  RingOps  /\  A  e.  X  /\  B  e.  X )  ->  ( A H ( N `  B ) )  =  ( A H ( B H ( N `
 (GId `  H
) ) ) ) )
2516, 21, 243eqtr4d 2518 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 973    = wceq 1379    e. wcel 1767   ran crn 5000   ` cfv 5586  (class class class)co 6282   1stc1st 6779   2ndc2nd 6780  GIdcgi 24865   invcgn 24866   RingOpscrngo 25053
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6574
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-reu 2821  df-rmo 2822  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-id 4795  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5549  df-fun 5588  df-fn 5589  df-f 5590  df-f1 5591  df-fo 5592  df-f1o 5593  df-fv 5594  df-riota 6243  df-ov 6285  df-1st 6781  df-2nd 6782  df-grpo 24869  df-gid 24870  df-ginv 24871  df-ablo 24960  df-ass 24991  df-exid 24993  df-mgm 24997  df-sgr 25009  df-mndo 25016  df-rngo 25054
This theorem is referenced by:  rngosubdi  29959
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