Users' Mathboxes Mathbox for Jeff Madsen < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  rngosubdi Structured version   Unicode version

Theorem rngosubdi 31638
Description: Ring multiplication distributes over subtraction. (Contributed by Jeff Madsen, 19-Jun-2010.)
Hypotheses
Ref Expression
ringsubdi.1  |-  G  =  ( 1st `  R
)
ringsubdi.2  |-  H  =  ( 2nd `  R
)
ringsubdi.3  |-  X  =  ran  G
ringsubdi.4  |-  D  =  (  /g  `  G
)
Assertion
Ref Expression
rngosubdi  |-  ( ( R  e.  RingOps  /\  ( A  e.  X  /\  B  e.  X  /\  C  e.  X )
)  ->  ( A H ( B D C ) )  =  ( ( A H B ) D ( A H C ) ) )

Proof of Theorem rngosubdi
StepHypRef Expression
1 ringsubdi.1 . . . . 5  |-  G  =  ( 1st `  R
)
2 ringsubdi.3 . . . . 5  |-  X  =  ran  G
3 eqid 2402 . . . . 5  |-  ( inv `  G )  =  ( inv `  G )
4 ringsubdi.4 . . . . 5  |-  D  =  (  /g  `  G
)
51, 2, 3, 4rngosub 31633 . . . 4  |-  ( ( R  e.  RingOps  /\  B  e.  X  /\  C  e.  X )  ->  ( B D C )  =  ( B G ( ( inv `  G
) `  C )
) )
653adant3r1 1206 . . 3  |-  ( ( R  e.  RingOps  /\  ( A  e.  X  /\  B  e.  X  /\  C  e.  X )
)  ->  ( B D C )  =  ( B G ( ( inv `  G ) `
 C ) ) )
76oveq2d 6294 . 2  |-  ( ( R  e.  RingOps  /\  ( A  e.  X  /\  B  e.  X  /\  C  e.  X )
)  ->  ( A H ( B D C ) )  =  ( A H ( B G ( ( inv `  G ) `
 C ) ) ) )
8 ringsubdi.2 . . . . . . 7  |-  H  =  ( 2nd `  R
)
91, 8, 2rngocl 25798 . . . . . 6  |-  ( ( R  e.  RingOps  /\  A  e.  X  /\  B  e.  X )  ->  ( A H B )  e.  X )
1093adant3r3 1208 . . . . 5  |-  ( ( R  e.  RingOps  /\  ( A  e.  X  /\  B  e.  X  /\  C  e.  X )
)  ->  ( A H B )  e.  X
)
111, 8, 2rngocl 25798 . . . . . 6  |-  ( ( R  e.  RingOps  /\  A  e.  X  /\  C  e.  X )  ->  ( A H C )  e.  X )
12113adant3r2 1207 . . . . 5  |-  ( ( R  e.  RingOps  /\  ( A  e.  X  /\  B  e.  X  /\  C  e.  X )
)  ->  ( A H C )  e.  X
)
1310, 12jca 530 . . . 4  |-  ( ( R  e.  RingOps  /\  ( A  e.  X  /\  B  e.  X  /\  C  e.  X )
)  ->  ( ( A H B )  e.  X  /\  ( A H C )  e.  X ) )
141, 2, 3, 4rngosub 31633 . . . . 5  |-  ( ( R  e.  RingOps  /\  ( A H B )  e.  X  /\  ( A H C )  e.  X )  ->  (
( A H B ) D ( A H C ) )  =  ( ( A H B ) G ( ( inv `  G
) `  ( A H C ) ) ) )
15143expb 1198 . . . 4  |-  ( ( R  e.  RingOps  /\  (
( A H B )  e.  X  /\  ( A H C )  e.  X ) )  ->  ( ( A H B ) D ( A H C ) )  =  ( ( A H B ) G ( ( inv `  G ) `
 ( A H C ) ) ) )
1613, 15syldan 468 . . 3  |-  ( ( R  e.  RingOps  /\  ( A  e.  X  /\  B  e.  X  /\  C  e.  X )
)  ->  ( ( A H B ) D ( A H C ) )  =  ( ( A H B ) G ( ( inv `  G ) `
 ( A H C ) ) ) )
17 idd 24 . . . . . . 7  |-  ( R  e.  RingOps  ->  ( A  e.  X  ->  A  e.  X ) )
18 idd 24 . . . . . . 7  |-  ( R  e.  RingOps  ->  ( B  e.  X  ->  B  e.  X ) )
191, 2, 3rngonegcl 31630 . . . . . . . 8  |-  ( ( R  e.  RingOps  /\  C  e.  X )  ->  (
( inv `  G
) `  C )  e.  X )
2019ex 432 . . . . . . 7  |-  ( R  e.  RingOps  ->  ( C  e.  X  ->  ( ( inv `  G ) `  C )  e.  X
) )
2117, 18, 203anim123d 1308 . . . . . 6  |-  ( R  e.  RingOps  ->  ( ( A  e.  X  /\  B  e.  X  /\  C  e.  X )  ->  ( A  e.  X  /\  B  e.  X  /\  ( ( inv `  G
) `  C )  e.  X ) ) )
2221imp 427 . . . . 5  |-  ( ( R  e.  RingOps  /\  ( A  e.  X  /\  B  e.  X  /\  C  e.  X )
)  ->  ( A  e.  X  /\  B  e.  X  /\  ( ( inv `  G ) `
 C )  e.  X ) )
231, 8, 2rngodi 25801 . . . . 5  |-  ( ( R  e.  RingOps  /\  ( A  e.  X  /\  B  e.  X  /\  ( ( inv `  G
) `  C )  e.  X ) )  -> 
( A H ( B G ( ( inv `  G ) `
 C ) ) )  =  ( ( A H B ) G ( A H ( ( inv `  G
) `  C )
) ) )
2422, 23syldan 468 . . . 4  |-  ( ( R  e.  RingOps  /\  ( A  e.  X  /\  B  e.  X  /\  C  e.  X )
)  ->  ( A H ( B G ( ( inv `  G
) `  C )
) )  =  ( ( A H B ) G ( A H ( ( inv `  G ) `  C
) ) ) )
251, 8, 2, 3rngonegrmul 31637 . . . . . 6  |-  ( ( R  e.  RingOps  /\  A  e.  X  /\  C  e.  X )  ->  (
( inv `  G
) `  ( A H C ) )  =  ( A H ( ( inv `  G
) `  C )
) )
26253adant3r2 1207 . . . . 5  |-  ( ( R  e.  RingOps  /\  ( A  e.  X  /\  B  e.  X  /\  C  e.  X )
)  ->  ( ( inv `  G ) `  ( A H C ) )  =  ( A H ( ( inv `  G ) `  C
) ) )
2726oveq2d 6294 . . . 4  |-  ( ( R  e.  RingOps  /\  ( A  e.  X  /\  B  e.  X  /\  C  e.  X )
)  ->  ( ( A H B ) G ( ( inv `  G
) `  ( A H C ) ) )  =  ( ( A H B ) G ( A H ( ( inv `  G
) `  C )
) ) )
2824, 27eqtr4d 2446 . . 3  |-  ( ( R  e.  RingOps  /\  ( A  e.  X  /\  B  e.  X  /\  C  e.  X )
)  ->  ( A H ( B G ( ( inv `  G
) `  C )
) )  =  ( ( A H B ) G ( ( inv `  G ) `
 ( A H C ) ) ) )
2916, 28eqtr4d 2446 . 2  |-  ( ( R  e.  RingOps  /\  ( A  e.  X  /\  B  e.  X  /\  C  e.  X )
)  ->  ( ( A H B ) D ( A H C ) )  =  ( A H ( B G ( ( inv `  G ) `  C
) ) ) )
307, 29eqtr4d 2446 1  |-  ( ( R  e.  RingOps  /\  ( A  e.  X  /\  B  e.  X  /\  C  e.  X )
)  ->  ( A H ( B D C ) )  =  ( ( A H B ) D ( A H C ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 367    /\ w3a 974    = wceq 1405    e. wcel 1842   ran crn 4824   ` cfv 5569  (class class class)co 6278   1stc1st 6782   2ndc2nd 6783   invcgn 25604    /g cgs 25605   RingOpscrngo 25791
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-8 1844  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-rep 4507  ax-sep 4517  ax-nul 4525  ax-pow 4572  ax-pr 4630  ax-un 6574
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-ral 2759  df-rex 2760  df-reu 2761  df-rmo 2762  df-rab 2763  df-v 3061  df-sbc 3278  df-csb 3374  df-dif 3417  df-un 3419  df-in 3421  df-ss 3428  df-nul 3739  df-if 3886  df-pw 3957  df-sn 3973  df-pr 3975  df-op 3979  df-uni 4192  df-iun 4273  df-br 4396  df-opab 4454  df-mpt 4455  df-id 4738  df-xp 4829  df-rel 4830  df-cnv 4831  df-co 4832  df-dm 4833  df-rn 4834  df-res 4835  df-ima 4836  df-iota 5533  df-fun 5571  df-fn 5572  df-f 5573  df-f1 5574  df-fo 5575  df-f1o 5576  df-fv 5577  df-riota 6240  df-ov 6281  df-oprab 6282  df-mpt2 6283  df-1st 6784  df-2nd 6785  df-grpo 25607  df-gid 25608  df-ginv 25609  df-gdiv 25610  df-ablo 25698  df-ass 25729  df-exid 25731  df-mgmOLD 25735  df-sgrOLD 25747  df-mndo 25754  df-rngo 25792
This theorem is referenced by:  dmncan1  31755
  Copyright terms: Public domain W3C validator