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

Theorem rngosubdi 26253
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 2380 . . . . 5  |-  ( inv `  G )  =  ( inv `  G )
4 ringsubdi.4 . . . . 5  |-  D  =  (  /g  `  G
)
51, 2, 3, 4rngosub 26248 . . . 4  |-  ( ( R  e.  RingOps  /\  B  e.  X  /\  C  e.  X )  ->  ( B D C )  =  ( B G ( ( inv `  G
) `  C )
) )
653adant3r1 1162 . . 3  |-  ( ( R  e.  RingOps  /\  ( A  e.  X  /\  B  e.  X  /\  C  e.  X )
)  ->  ( B D C )  =  ( B G ( ( inv `  G ) `
 C ) ) )
76oveq2d 6029 . 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 21811 . . . . . 6  |-  ( ( R  e.  RingOps  /\  A  e.  X  /\  B  e.  X )  ->  ( A H B )  e.  X )
1093adant3r3 1164 . . . . 5  |-  ( ( R  e.  RingOps  /\  ( A  e.  X  /\  B  e.  X  /\  C  e.  X )
)  ->  ( A H B )  e.  X
)
111, 8, 2rngocl 21811 . . . . . 6  |-  ( ( R  e.  RingOps  /\  A  e.  X  /\  C  e.  X )  ->  ( A H C )  e.  X )
12113adant3r2 1163 . . . . 5  |-  ( ( R  e.  RingOps  /\  ( A  e.  X  /\  B  e.  X  /\  C  e.  X )
)  ->  ( A H C )  e.  X
)
1310, 12jca 519 . . . 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 26248 . . . . 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 1154 . . . 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 457 . . 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 22 . . . . . . 7  |-  ( R  e.  RingOps  ->  ( A  e.  X  ->  A  e.  X ) )
18 idd 22 . . . . . . 7  |-  ( R  e.  RingOps  ->  ( B  e.  X  ->  B  e.  X ) )
191, 2, 3rngonegcl 26245 . . . . . . . 8  |-  ( ( R  e.  RingOps  /\  C  e.  X )  ->  (
( inv `  G
) `  C )  e.  X )
2019ex 424 . . . . . . 7  |-  ( R  e.  RingOps  ->  ( C  e.  X  ->  ( ( inv `  G ) `  C )  e.  X
) )
2117, 18, 203anim123d 1261 . . . . . 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 419 . . . . 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 21814 . . . . 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 457 . . . 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 26252 . . . . . 6  |-  ( ( R  e.  RingOps  /\  A  e.  X  /\  C  e.  X )  ->  (
( inv `  G
) `  ( A H C ) )  =  ( A H ( ( inv `  G
) `  C )
) )
26253adant3r2 1163 . . . . 5  |-  ( ( R  e.  RingOps  /\  ( A  e.  X  /\  B  e.  X  /\  C  e.  X )
)  ->  ( ( inv `  G ) `  ( A H C ) )  =  ( A H ( ( inv `  G ) `  C
) ) )
2726oveq2d 6029 . . . 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 2415 . . 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 2415 . 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 2415 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 set class
Syntax hints:    -> wi 4    /\ wa 359    /\ w3a 936    = wceq 1649    e. wcel 1717   ran crn 4812   ` cfv 5387  (class class class)co 6013   1stc1st 6279   2ndc2nd 6280   invcgn 21617    /g cgs 21618   RingOpscrngo 21804
This theorem is referenced by:  dmncan1  26370
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 2361  ax-rep 4254  ax-sep 4264  ax-nul 4272  ax-pow 4311  ax-pr 4337  ax-un 4634
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 2235  df-mo 2236  df-clab 2367  df-cleq 2373  df-clel 2376  df-nfc 2505  df-ne 2545  df-ral 2647  df-rex 2648  df-reu 2649  df-rmo 2650  df-rab 2651  df-v 2894  df-sbc 3098  df-csb 3188  df-dif 3259  df-un 3261  df-in 3263  df-ss 3270  df-nul 3565  df-if 3676  df-pw 3737  df-sn 3756  df-pr 3757  df-op 3759  df-uni 3951  df-iun 4030  df-br 4147  df-opab 4201  df-mpt 4202  df-id 4432  df-xp 4817  df-rel 4818  df-cnv 4819  df-co 4820  df-dm 4821  df-rn 4822  df-res 4823  df-ima 4824  df-iota 5351  df-fun 5389  df-fn 5390  df-f 5391  df-f1 5392  df-fo 5393  df-f1o 5394  df-fv 5395  df-ov 6016  df-oprab 6017  df-mpt2 6018  df-1st 6281  df-2nd 6282  df-riota 6478  df-grpo 21620  df-gid 21621  df-ginv 21622  df-gdiv 21623  df-ablo 21711  df-ass 21742  df-exid 21744  df-mgm 21748  df-sgr 21760  df-mndo 21767  df-rngo 21805
  Copyright terms: Public domain W3C validator