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Theorem rngosubdi 26560
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 2435 . . . . 5  |-  ( inv `  G )  =  ( inv `  G )
4 ringsubdi.4 . . . . 5  |-  D  =  (  /g  `  G
)
51, 2, 3, 4rngosub 26555 . . . 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 6089 . 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 21962 . . . . . 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 21962 . . . . . 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 26555 . . . . 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 26552 . . . . . . . 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 21965 . . . . 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 26559 . . . . . 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 6089 . . . 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 2470 . . 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 2470 . 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 2470 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 1652    e. wcel 1725   ran crn 4871   ` cfv 5446  (class class class)co 6073   1stc1st 6339   2ndc2nd 6340   invcgn 21768    /g cgs 21769   RingOpscrngo 21955
This theorem is referenced by:  dmncan1  26677
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-rep 4312  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-reu 2704  df-rmo 2705  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-id 4490  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-1st 6341  df-2nd 6342  df-riota 6541  df-grpo 21771  df-gid 21772  df-ginv 21773  df-gdiv 21774  df-ablo 21862  df-ass 21893  df-exid 21895  df-mgm 21899  df-sgr 21911  df-mndo 21918  df-rngo 21956
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