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Theorem rngosubdi 26584
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 2283 . . . . 5  |-  ( inv `  G )  =  ( inv `  G )
4 ringsubdi.4 . . . . 5  |-  D  =  (  /g  `  G
)
51, 2, 3, 4rngosub 26579 . . . 4  |-  ( ( R  e.  RingOps  /\  B  e.  X  /\  C  e.  X )  ->  ( B D C )  =  ( B G ( ( inv `  G
) `  C )
) )
653adant3r1 1160 . . 3  |-  ( ( R  e.  RingOps  /\  ( A  e.  X  /\  B  e.  X  /\  C  e.  X )
)  ->  ( B D C )  =  ( B G ( ( inv `  G ) `
 C ) ) )
76oveq2d 5874 . 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 21049 . . . . . 6  |-  ( ( R  e.  RingOps  /\  A  e.  X  /\  B  e.  X )  ->  ( A H B )  e.  X )
1093adant3r3 1162 . . . . 5  |-  ( ( R  e.  RingOps  /\  ( A  e.  X  /\  B  e.  X  /\  C  e.  X )
)  ->  ( A H B )  e.  X
)
111, 8, 2rngocl 21049 . . . . . 6  |-  ( ( R  e.  RingOps  /\  A  e.  X  /\  C  e.  X )  ->  ( A H C )  e.  X )
12113adant3r2 1161 . . . . 5  |-  ( ( R  e.  RingOps  /\  ( A  e.  X  /\  B  e.  X  /\  C  e.  X )
)  ->  ( A H C )  e.  X
)
1310, 12jca 518 . . . 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 26579 . . . . 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 1152 . . . 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 456 . . 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 21 . . . . . . 7  |-  ( R  e.  RingOps  ->  ( A  e.  X  ->  A  e.  X ) )
18 idd 21 . . . . . . 7  |-  ( R  e.  RingOps  ->  ( B  e.  X  ->  B  e.  X ) )
191, 2, 3rngonegcl 26576 . . . . . . . 8  |-  ( ( R  e.  RingOps  /\  C  e.  X )  ->  (
( inv `  G
) `  C )  e.  X )
2019ex 423 . . . . . . 7  |-  ( R  e.  RingOps  ->  ( C  e.  X  ->  ( ( inv `  G ) `  C )  e.  X
) )
2117, 18, 203anim123d 1259 . . . . . 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 418 . . . . 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 21052 . . . . 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 456 . . . 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 26583 . . . . . 6  |-  ( ( R  e.  RingOps  /\  A  e.  X  /\  C  e.  X )  ->  (
( inv `  G
) `  ( A H C ) )  =  ( A H ( ( inv `  G
) `  C )
) )
26253adant3r2 1161 . . . . 5  |-  ( ( R  e.  RingOps  /\  ( A  e.  X  /\  B  e.  X  /\  C  e.  X )
)  ->  ( ( inv `  G ) `  ( A H C ) )  =  ( A H ( ( inv `  G ) `  C
) ) )
2726oveq2d 5874 . . . 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 2318 . . 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 2318 . 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 2318 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 358    /\ w3a 934    = wceq 1623    e. wcel 1684   ran crn 4690   ` cfv 5255  (class class class)co 5858   1stc1st 6120   2ndc2nd 6121   invcgn 20855    /g cgs 20856   RingOpscrngo 21042
This theorem is referenced by:  dmncan1  26701
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-riota 6304  df-grpo 20858  df-gid 20859  df-ginv 20860  df-gdiv 20861  df-ablo 20949  df-ass 20980  df-exid 20982  df-mgm 20986  df-sgr 20998  df-mndo 21005  df-rngo 21043
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