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Theorem rngosubdir 28928
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
rngosubdir  |-  ( ( R  e.  RingOps  /\  ( A  e.  X  /\  B  e.  X  /\  C  e.  X )
)  ->  ( ( A D B ) H C )  =  ( ( A H C ) D ( B H C ) ) )

Proof of Theorem rngosubdir
StepHypRef Expression
1 ringsubdi.1 . . . . 5  |-  G  =  ( 1st `  R
)
2 ringsubdi.3 . . . . 5  |-  X  =  ran  G
3 eqid 2454 . . . . 5  |-  ( inv `  G )  =  ( inv `  G )
4 ringsubdi.4 . . . . 5  |-  D  =  (  /g  `  G
)
51, 2, 3, 4rngosub 28922 . . . 4  |-  ( ( R  e.  RingOps  /\  A  e.  X  /\  B  e.  X )  ->  ( A D B )  =  ( A G ( ( inv `  G
) `  B )
) )
653adant3r3 1199 . . 3  |-  ( ( R  e.  RingOps  /\  ( A  e.  X  /\  B  e.  X  /\  C  e.  X )
)  ->  ( A D B )  =  ( A G ( ( inv `  G ) `
 B ) ) )
76oveq1d 6218 . 2  |-  ( ( R  e.  RingOps  /\  ( A  e.  X  /\  B  e.  X  /\  C  e.  X )
)  ->  ( ( A D B ) H C )  =  ( ( A G ( ( inv `  G
) `  B )
) H C ) )
8 ringsubdi.2 . . . . . . 7  |-  H  =  ( 2nd `  R
)
91, 8, 2rngocl 24041 . . . . . 6  |-  ( ( R  e.  RingOps  /\  A  e.  X  /\  C  e.  X )  ->  ( A H C )  e.  X )
1093adant3r2 1198 . . . . 5  |-  ( ( R  e.  RingOps  /\  ( A  e.  X  /\  B  e.  X  /\  C  e.  X )
)  ->  ( A H C )  e.  X
)
111, 8, 2rngocl 24041 . . . . . 6  |-  ( ( R  e.  RingOps  /\  B  e.  X  /\  C  e.  X )  ->  ( B H C )  e.  X )
12113adant3r1 1197 . . . . 5  |-  ( ( R  e.  RingOps  /\  ( A  e.  X  /\  B  e.  X  /\  C  e.  X )
)  ->  ( B H C )  e.  X
)
1310, 12jca 532 . . . 4  |-  ( ( R  e.  RingOps  /\  ( A  e.  X  /\  B  e.  X  /\  C  e.  X )
)  ->  ( ( A H C )  e.  X  /\  ( B H C )  e.  X ) )
141, 2, 3, 4rngosub 28922 . . . . 5  |-  ( ( R  e.  RingOps  /\  ( A H C )  e.  X  /\  ( B H C )  e.  X )  ->  (
( A H C ) D ( B H C ) )  =  ( ( A H C ) G ( ( inv `  G
) `  ( B H C ) ) ) )
15143expb 1189 . . . 4  |-  ( ( R  e.  RingOps  /\  (
( A H C )  e.  X  /\  ( B H C )  e.  X ) )  ->  ( ( A H C ) D ( B H C ) )  =  ( ( A H C ) G ( ( inv `  G ) `
 ( B H C ) ) ) )
1613, 15syldan 470 . . 3  |-  ( ( R  e.  RingOps  /\  ( A  e.  X  /\  B  e.  X  /\  C  e.  X )
)  ->  ( ( A H C ) D ( B H C ) )  =  ( ( A H C ) G ( ( inv `  G ) `
 ( B H C ) ) ) )
17 idd 24 . . . . . . 7  |-  ( R  e.  RingOps  ->  ( A  e.  X  ->  A  e.  X ) )
181, 2, 3rngonegcl 28919 . . . . . . . 8  |-  ( ( R  e.  RingOps  /\  B  e.  X )  ->  (
( inv `  G
) `  B )  e.  X )
1918ex 434 . . . . . . 7  |-  ( R  e.  RingOps  ->  ( B  e.  X  ->  ( ( inv `  G ) `  B )  e.  X
) )
20 idd 24 . . . . . . 7  |-  ( R  e.  RingOps  ->  ( C  e.  X  ->  C  e.  X ) )
2117, 19, 203anim123d 1297 . . . . . 6  |-  ( R  e.  RingOps  ->  ( ( A  e.  X  /\  B  e.  X  /\  C  e.  X )  ->  ( A  e.  X  /\  ( ( inv `  G
) `  B )  e.  X  /\  C  e.  X ) ) )
2221imp 429 . . . . 5  |-  ( ( R  e.  RingOps  /\  ( A  e.  X  /\  B  e.  X  /\  C  e.  X )
)  ->  ( A  e.  X  /\  (
( inv `  G
) `  B )  e.  X  /\  C  e.  X ) )
231, 8, 2rngodir 24045 . . . . 5  |-  ( ( R  e.  RingOps  /\  ( A  e.  X  /\  ( ( inv `  G
) `  B )  e.  X  /\  C  e.  X ) )  -> 
( ( A G ( ( inv `  G
) `  B )
) H C )  =  ( ( A H C ) G ( ( ( inv `  G ) `  B
) H C ) ) )
2422, 23syldan 470 . . . 4  |-  ( ( R  e.  RingOps  /\  ( A  e.  X  /\  B  e.  X  /\  C  e.  X )
)  ->  ( ( A G ( ( inv `  G ) `  B
) ) H C )  =  ( ( A H C ) G ( ( ( inv `  G ) `
 B ) H C ) ) )
251, 8, 2, 3rngoneglmul 28925 . . . . . 6  |-  ( ( R  e.  RingOps  /\  B  e.  X  /\  C  e.  X )  ->  (
( inv `  G
) `  ( B H C ) )  =  ( ( ( inv `  G ) `  B
) H C ) )
26253adant3r1 1197 . . . . 5  |-  ( ( R  e.  RingOps  /\  ( A  e.  X  /\  B  e.  X  /\  C  e.  X )
)  ->  ( ( inv `  G ) `  ( B H C ) )  =  ( ( ( inv `  G
) `  B ) H C ) )
2726oveq2d 6219 . . . 4  |-  ( ( R  e.  RingOps  /\  ( A  e.  X  /\  B  e.  X  /\  C  e.  X )
)  ->  ( ( A H C ) G ( ( inv `  G
) `  ( B H C ) ) )  =  ( ( A H C ) G ( ( ( inv `  G ) `  B
) H C ) ) )
2824, 27eqtr4d 2498 . . 3  |-  ( ( R  e.  RingOps  /\  ( A  e.  X  /\  B  e.  X  /\  C  e.  X )
)  ->  ( ( A G ( ( inv `  G ) `  B
) ) H C )  =  ( ( A H C ) G ( ( inv `  G ) `  ( B H C ) ) ) )
2916, 28eqtr4d 2498 . 2  |-  ( ( R  e.  RingOps  /\  ( A  e.  X  /\  B  e.  X  /\  C  e.  X )
)  ->  ( ( A H C ) D ( B H C ) )  =  ( ( A G ( ( inv `  G
) `  B )
) H C ) )
307, 29eqtr4d 2498 1  |-  ( ( R  e.  RingOps  /\  ( A  e.  X  /\  B  e.  X  /\  C  e.  X )
)  ->  ( ( A D B ) H C )  =  ( ( A H C ) D ( B H C ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    /\ w3a 965    = wceq 1370    e. wcel 1758   ran crn 4952   ` cfv 5529  (class class class)co 6203   1stc1st 6688   2ndc2nd 6689   invcgn 23847    /g cgs 23848   RingOpscrngo 24034
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 1955  ax-ext 2432  ax-rep 4514  ax-sep 4524  ax-nul 4532  ax-pow 4581  ax-pr 4642  ax-un 6485
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 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-ral 2804  df-rex 2805  df-reu 2806  df-rmo 2807  df-rab 2808  df-v 3080  df-sbc 3295  df-csb 3399  df-dif 3442  df-un 3444  df-in 3446  df-ss 3453  df-nul 3749  df-if 3903  df-pw 3973  df-sn 3989  df-pr 3991  df-op 3995  df-uni 4203  df-iun 4284  df-br 4404  df-opab 4462  df-mpt 4463  df-id 4747  df-xp 4957  df-rel 4958  df-cnv 4959  df-co 4960  df-dm 4961  df-rn 4962  df-res 4963  df-ima 4964  df-iota 5492  df-fun 5531  df-fn 5532  df-f 5533  df-f1 5534  df-fo 5535  df-f1o 5536  df-fv 5537  df-riota 6164  df-ov 6206  df-oprab 6207  df-mpt2 6208  df-1st 6690  df-2nd 6691  df-grpo 23850  df-gid 23851  df-ginv 23852  df-gdiv 23853  df-ablo 23941  df-ass 23972  df-exid 23974  df-mgm 23978  df-sgr 23990  df-mndo 23997  df-rngo 24035
This theorem is referenced by: (None)
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