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Theorem rngosubdir 31926
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 2420 . . . . 5  |-  ( inv `  G )  =  ( inv `  G )
4 ringsubdi.4 . . . . 5  |-  D  =  (  /g  `  G
)
51, 2, 3, 4rngosub 31920 . . . 4  |-  ( ( R  e.  RingOps  /\  A  e.  X  /\  B  e.  X )  ->  ( A D B )  =  ( A G ( ( inv `  G
) `  B )
) )
653adant3r3 1216 . . 3  |-  ( ( R  e.  RingOps  /\  ( A  e.  X  /\  B  e.  X  /\  C  e.  X )
)  ->  ( A D B )  =  ( A G ( ( inv `  G ) `
 B ) ) )
76oveq1d 6311 . 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 25981 . . . . . 6  |-  ( ( R  e.  RingOps  /\  A  e.  X  /\  C  e.  X )  ->  ( A H C )  e.  X )
1093adant3r2 1215 . . . . 5  |-  ( ( R  e.  RingOps  /\  ( A  e.  X  /\  B  e.  X  /\  C  e.  X )
)  ->  ( A H C )  e.  X
)
111, 8, 2rngocl 25981 . . . . . 6  |-  ( ( R  e.  RingOps  /\  B  e.  X  /\  C  e.  X )  ->  ( B H C )  e.  X )
12113adant3r1 1214 . . . . 5  |-  ( ( R  e.  RingOps  /\  ( A  e.  X  /\  B  e.  X  /\  C  e.  X )
)  ->  ( B H C )  e.  X
)
1310, 12jca 534 . . . 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 31920 . . . . 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 1206 . . . 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 472 . . 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 25 . . . . . . 7  |-  ( R  e.  RingOps  ->  ( A  e.  X  ->  A  e.  X ) )
181, 2, 3rngonegcl 31917 . . . . . . . 8  |-  ( ( R  e.  RingOps  /\  B  e.  X )  ->  (
( inv `  G
) `  B )  e.  X )
1918ex 435 . . . . . . 7  |-  ( R  e.  RingOps  ->  ( B  e.  X  ->  ( ( inv `  G ) `  B )  e.  X
) )
20 idd 25 . . . . . . 7  |-  ( R  e.  RingOps  ->  ( C  e.  X  ->  C  e.  X ) )
2117, 19, 203anim123d 1342 . . . . . 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 430 . . . . 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 25985 . . . . 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 472 . . . 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 31923 . . . . . 6  |-  ( ( R  e.  RingOps  /\  B  e.  X  /\  C  e.  X )  ->  (
( inv `  G
) `  ( B H C ) )  =  ( ( ( inv `  G ) `  B
) H C ) )
26253adant3r1 1214 . . . . 5  |-  ( ( R  e.  RingOps  /\  ( A  e.  X  /\  B  e.  X  /\  C  e.  X )
)  ->  ( ( inv `  G ) `  ( B H C ) )  =  ( ( ( inv `  G
) `  B ) H C ) )
2726oveq2d 6312 . . . 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 2464 . . 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 2464 . 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 2464 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 370    /\ w3a 982    = wceq 1437    e. wcel 1867   ran crn 4846   ` cfv 5592  (class class class)co 6296   1stc1st 6796   2ndc2nd 6797   invcgn 25787    /g cgs 25788   RingOpscrngo 25974
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1748  ax-6 1794  ax-7 1838  ax-8 1869  ax-9 1871  ax-10 1886  ax-11 1891  ax-12 1904  ax-13 2052  ax-ext 2398  ax-rep 4529  ax-sep 4539  ax-nul 4547  ax-pow 4594  ax-pr 4652  ax-un 6588
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1787  df-eu 2267  df-mo 2268  df-clab 2406  df-cleq 2412  df-clel 2415  df-nfc 2570  df-ne 2618  df-ral 2778  df-rex 2779  df-reu 2780  df-rmo 2781  df-rab 2782  df-v 3080  df-sbc 3297  df-csb 3393  df-dif 3436  df-un 3438  df-in 3440  df-ss 3447  df-nul 3759  df-if 3907  df-pw 3978  df-sn 3994  df-pr 3996  df-op 4000  df-uni 4214  df-iun 4295  df-br 4418  df-opab 4476  df-mpt 4477  df-id 4760  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-iota 5556  df-fun 5594  df-fn 5595  df-f 5596  df-f1 5597  df-fo 5598  df-f1o 5599  df-fv 5600  df-riota 6258  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-1st 6798  df-2nd 6799  df-grpo 25790  df-gid 25791  df-ginv 25792  df-gdiv 25793  df-ablo 25881  df-ass 25912  df-exid 25914  df-mgmOLD 25918  df-sgrOLD 25930  df-mndo 25937  df-rngo 25975
This theorem is referenced by: (None)
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