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Theorem rngoneglmul 30594
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 5218 . . . . . . 7  |-  ran  G  =  ran  ( 1st `  R
)
41, 3eqtri 2483 . . . . . 6  |-  X  =  ran  ( 1st `  R
)
5 ringnegmul.2 . . . . . 6  |-  H  =  ( 2nd `  R
)
6 eqid 2454 . . . . . 6  |-  (GId `  H )  =  (GId
`  H )
74, 5, 6rngo1cl 25629 . . . . 5  |-  ( R  e.  RingOps  ->  (GId `  H
)  e.  X )
8 ringnegmul.4 . . . . . 6  |-  N  =  ( inv `  G
)
92, 1, 8rngonegcl 30588 . . . . 5  |-  ( ( R  e.  RingOps  /\  (GId `  H )  e.  X
)  ->  ( N `  (GId `  H )
)  e.  X )
107, 9mpdan 666 . . . 4  |-  ( R  e.  RingOps  ->  ( N `  (GId `  H ) )  e.  X )
112, 5, 1rngoass 25587 . . . . 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 1212 . . . 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 1188 . 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 30592 . . . 4  |-  ( ( R  e.  RingOps  /\  A  e.  X )  ->  ( N `  A )  =  ( ( N `
 (GId `  H
) ) H A ) )
16153adant3 1014 . . 3  |-  ( ( R  e.  RingOps  /\  A  e.  X  /\  B  e.  X )  ->  ( N `  A )  =  ( ( N `
 (GId `  H
) ) H A ) )
1716oveq1d 6285 . 2  |-  ( ( R  e.  RingOps  /\  A  e.  X  /\  B  e.  X )  ->  (
( N `  A
) H B )  =  ( ( ( N `  (GId `  H ) ) H A ) H B ) )
182, 5, 1rngocl 25582 . . . . 5  |-  ( ( R  e.  RingOps  /\  A  e.  X  /\  B  e.  X )  ->  ( A H B )  e.  X )
19183expb 1195 . . . 4  |-  ( ( R  e.  RingOps  /\  ( A  e.  X  /\  B  e.  X )
)  ->  ( A H B )  e.  X
)
202, 5, 1, 8, 6rngonegmn1l 30592 . . . 4  |-  ( ( R  e.  RingOps  /\  ( A H B )  e.  X )  ->  ( N `  ( A H B ) )  =  ( ( N `  (GId `  H ) ) H ( A H B ) ) )
2119, 20syldan 468 . . 3  |-  ( ( R  e.  RingOps  /\  ( A  e.  X  /\  B  e.  X )
)  ->  ( N `  ( A H B ) )  =  ( ( N `  (GId `  H ) ) H ( A H B ) ) )
22213impb 1190 . 2  |-  ( ( R  e.  RingOps  /\  A  e.  X  /\  B  e.  X )  ->  ( N `  ( A H B ) )  =  ( ( N `  (GId `  H ) ) H ( A H B ) ) )
2314, 17, 223eqtr4rd 2506 1  |-  ( ( R  e.  RingOps  /\  A  e.  X  /\  B  e.  X )  ->  ( N `  ( A H B ) )  =  ( ( N `  A ) H B ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 367    /\ w3a 971    = wceq 1398    e. wcel 1823   ran crn 4989   ` cfv 5570  (class class class)co 6270   1stc1st 6771   2ndc2nd 6772  GIdcgi 25387   invcgn 25388   RingOpscrngo 25575
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
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  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-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-nul 3784  df-if 3930  df-sn 4017  df-pr 4019  df-op 4023  df-uni 4236  df-iun 4317  df-br 4440  df-opab 4498  df-mpt 4499  df-id 4784  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-1st 6773  df-2nd 6774  df-grpo 25391  df-gid 25392  df-ginv 25393  df-ablo 25482  df-ass 25513  df-exid 25515  df-mgmOLD 25519  df-sgrOLD 25531  df-mndo 25538  df-rngo 25576
This theorem is referenced by:  rngosubdir  30597
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