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Theorem rngoneglmul 26569
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
rngoneglmul  |-  ( ( R  e.  RingOps  /\  A  e.  X  /\  B  e.  X )  ->  ( N `  ( A H B ) )  =  ( ( N `  A ) H B ) )

Proof of Theorem rngoneglmul
StepHypRef Expression
1 ringnegmul.3 . . . . . . 7  |-  X  =  ran  G
2 ringnegmul.1 . . . . . . . 8  |-  G  =  ( 1st `  R
)
32rneqi 5098 . . . . . . 7  |-  ran  G  =  ran  ( 1st `  R
)
41, 3eqtri 2458 . . . . . 6  |-  X  =  ran  ( 1st `  R
)
5 ringnegmul.2 . . . . . 6  |-  H  =  ( 2nd `  R
)
6 eqid 2438 . . . . . 6  |-  (GId `  H )  =  (GId
`  H )
74, 5, 6rngo1cl 22019 . . . . 5  |-  ( R  e.  RingOps  ->  (GId `  H
)  e.  X )
8 ringnegmul.4 . . . . . 6  |-  N  =  ( inv `  G
)
92, 1, 8rngonegcl 26563 . . . . 5  |-  ( ( R  e.  RingOps  /\  (GId `  H )  e.  X
)  ->  ( N `  (GId `  H )
)  e.  X )
107, 9mpdan 651 . . . 4  |-  ( R  e.  RingOps  ->  ( N `  (GId `  H ) )  e.  X )
112, 5, 1rngoass 21977 . . . . 5  |-  ( ( R  e.  RingOps  /\  (
( N `  (GId `  H ) )  e.  X  /\  A  e.  X  /\  B  e.  X ) )  -> 
( ( ( N `
 (GId `  H
) ) H A ) H B )  =  ( ( N `
 (GId `  H
) ) H ( A H B ) ) )
12113exp2 1172 . . . 4  |-  ( R  e.  RingOps  ->  ( ( N `
 (GId `  H
) )  e.  X  ->  ( A  e.  X  ->  ( B  e.  X  ->  ( ( ( N `
 (GId `  H
) ) H A ) H B )  =  ( ( N `
 (GId `  H
) ) H ( A H B ) ) ) ) ) )
1310, 12mpd 15 . . 3  |-  ( R  e.  RingOps  ->  ( A  e.  X  ->  ( B  e.  X  ->  ( ( ( N `  (GId `  H ) ) H A ) H B )  =  ( ( N `  (GId `  H ) ) H ( A H B ) ) ) ) )
14133imp 1148 . 2  |-  ( ( R  e.  RingOps  /\  A  e.  X  /\  B  e.  X )  ->  (
( ( N `  (GId `  H ) ) H A ) H B )  =  ( ( N `  (GId `  H ) ) H ( A H B ) ) )
152, 5, 1, 8, 6rngonegmn1l 26567 . . . 4  |-  ( ( R  e.  RingOps  /\  A  e.  X )  ->  ( N `  A )  =  ( ( N `
 (GId `  H
) ) H A ) )
16153adant3 978 . . 3  |-  ( ( R  e.  RingOps  /\  A  e.  X  /\  B  e.  X )  ->  ( N `  A )  =  ( ( N `
 (GId `  H
) ) H A ) )
1716oveq1d 6098 . 2  |-  ( ( R  e.  RingOps  /\  A  e.  X  /\  B  e.  X )  ->  (
( N `  A
) H B )  =  ( ( ( N `  (GId `  H ) ) H A ) H B ) )
182, 5, 1rngocl 21972 . . . . 5  |-  ( ( R  e.  RingOps  /\  A  e.  X  /\  B  e.  X )  ->  ( A H B )  e.  X )
19183expb 1155 . . . 4  |-  ( ( R  e.  RingOps  /\  ( A  e.  X  /\  B  e.  X )
)  ->  ( A H B )  e.  X
)
202, 5, 1, 8, 6rngonegmn1l 26567 . . . 4  |-  ( ( R  e.  RingOps  /\  ( A H B )  e.  X )  ->  ( N `  ( A H B ) )  =  ( ( N `  (GId `  H ) ) H ( A H B ) ) )
2119, 20syldan 458 . . 3  |-  ( ( R  e.  RingOps  /\  ( A  e.  X  /\  B  e.  X )
)  ->  ( N `  ( A H B ) )  =  ( ( N `  (GId `  H ) ) H ( A H B ) ) )
22213impb 1150 . 2  |-  ( ( R  e.  RingOps  /\  A  e.  X  /\  B  e.  X )  ->  ( N `  ( A H B ) )  =  ( ( N `  (GId `  H ) ) H ( A H B ) ) )
2314, 17, 223eqtr4rd 2481 1  |-  ( ( R  e.  RingOps  /\  A  e.  X  /\  B  e.  X )  ->  ( N `  ( A H B ) )  =  ( ( N `  A ) H B ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 360    /\ w3a 937    = wceq 1653    e. wcel 1726   ran crn 4881   ` cfv 5456  (class class class)co 6083   1stc1st 6349   2ndc2nd 6350  GIdcgi 21777   invcgn 21778   RingOpscrngo 21965
This theorem is referenced by:  rngosubdir  26572
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-rep 4322  ax-sep 4332  ax-nul 4340  ax-pow 4379  ax-pr 4405  ax-un 4703
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-reu 2714  df-rmo 2715  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-iun 4097  df-br 4215  df-opab 4269  df-mpt 4270  df-id 4500  df-xp 4886  df-rel 4887  df-cnv 4888  df-co 4889  df-dm 4890  df-rn 4891  df-res 4892  df-ima 4893  df-iota 5420  df-fun 5458  df-fn 5459  df-f 5460  df-f1 5461  df-fo 5462  df-f1o 5463  df-fv 5464  df-ov 6086  df-1st 6351  df-2nd 6352  df-riota 6551  df-grpo 21781  df-gid 21782  df-ginv 21783  df-ablo 21872  df-ass 21903  df-exid 21905  df-mgm 21909  df-sgr 21921  df-mndo 21928  df-rngo 21966
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