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Theorem rngosubdi 28894
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 2451 . . . . 5  |-  ( inv `  G )  =  ( inv `  G )
4 ringsubdi.4 . . . . 5  |-  D  =  (  /g  `  G
)
51, 2, 3, 4rngosub 28889 . . . 4  |-  ( ( R  e.  RingOps  /\  B  e.  X  /\  C  e.  X )  ->  ( B D C )  =  ( B G ( ( inv `  G
) `  C )
) )
653adant3r1 1197 . . 3  |-  ( ( R  e.  RingOps  /\  ( A  e.  X  /\  B  e.  X  /\  C  e.  X )
)  ->  ( B D C )  =  ( B G ( ( inv `  G ) `
 C ) ) )
76oveq2d 6203 . 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 24001 . . . . . 6  |-  ( ( R  e.  RingOps  /\  A  e.  X  /\  B  e.  X )  ->  ( A H B )  e.  X )
1093adant3r3 1199 . . . . 5  |-  ( ( R  e.  RingOps  /\  ( A  e.  X  /\  B  e.  X  /\  C  e.  X )
)  ->  ( A H B )  e.  X
)
111, 8, 2rngocl 24001 . . . . . 6  |-  ( ( R  e.  RingOps  /\  A  e.  X  /\  C  e.  X )  ->  ( A H C )  e.  X )
12113adant3r2 1198 . . . . 5  |-  ( ( R  e.  RingOps  /\  ( A  e.  X  /\  B  e.  X  /\  C  e.  X )
)  ->  ( A H C )  e.  X
)
1310, 12jca 532 . . . 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 28889 . . . . 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 1189 . . . 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 470 . . 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 28886 . . . . . . . 8  |-  ( ( R  e.  RingOps  /\  C  e.  X )  ->  (
( inv `  G
) `  C )  e.  X )
2019ex 434 . . . . . . 7  |-  ( R  e.  RingOps  ->  ( C  e.  X  ->  ( ( inv `  G ) `  C )  e.  X
) )
2117, 18, 203anim123d 1297 . . . . . 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 429 . . . . 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 24004 . . . . 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 470 . . . 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 28893 . . . . . 6  |-  ( ( R  e.  RingOps  /\  A  e.  X  /\  C  e.  X )  ->  (
( inv `  G
) `  ( A H C ) )  =  ( A H ( ( inv `  G
) `  C )
) )
26253adant3r2 1198 . . . . 5  |-  ( ( R  e.  RingOps  /\  ( A  e.  X  /\  B  e.  X  /\  C  e.  X )
)  ->  ( ( inv `  G ) `  ( A H C ) )  =  ( A H ( ( inv `  G ) `  C
) ) )
2726oveq2d 6203 . . . 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 2494 . . 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 2494 . 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 2494 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 369    /\ w3a 965    = wceq 1370    e. wcel 1758   ran crn 4936   ` cfv 5513  (class class class)co 6187   1stc1st 6672   2ndc2nd 6673   invcgn 23807    /g cgs 23808   RingOpscrngo 23994
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-rep 4498  ax-sep 4508  ax-nul 4516  ax-pow 4565  ax-pr 4626  ax-un 6469
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2599  df-ne 2644  df-ral 2798  df-rex 2799  df-reu 2800  df-rmo 2801  df-rab 2802  df-v 3067  df-sbc 3282  df-csb 3384  df-dif 3426  df-un 3428  df-in 3430  df-ss 3437  df-nul 3733  df-if 3887  df-pw 3957  df-sn 3973  df-pr 3975  df-op 3979  df-uni 4187  df-iun 4268  df-br 4388  df-opab 4446  df-mpt 4447  df-id 4731  df-xp 4941  df-rel 4942  df-cnv 4943  df-co 4944  df-dm 4945  df-rn 4946  df-res 4947  df-ima 4948  df-iota 5476  df-fun 5515  df-fn 5516  df-f 5517  df-f1 5518  df-fo 5519  df-f1o 5520  df-fv 5521  df-riota 6148  df-ov 6190  df-oprab 6191  df-mpt2 6192  df-1st 6674  df-2nd 6675  df-grpo 23810  df-gid 23811  df-ginv 23812  df-gdiv 23813  df-ablo 23901  df-ass 23932  df-exid 23934  df-mgm 23938  df-sgr 23950  df-mndo 23957  df-rngo 23995
This theorem is referenced by:  dmncan1  29011
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