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Theorem rngosubdir 30187
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 2467 . . . . 5  |-  ( inv `  G )  =  ( inv `  G )
4 ringsubdi.4 . . . . 5  |-  D  =  (  /g  `  G
)
51, 2, 3, 4rngosub 30181 . . . 4  |-  ( ( R  e.  RingOps  /\  A  e.  X  /\  B  e.  X )  ->  ( A D B )  =  ( A G ( ( inv `  G
) `  B )
) )
653adant3r3 1207 . . 3  |-  ( ( R  e.  RingOps  /\  ( A  e.  X  /\  B  e.  X  /\  C  e.  X )
)  ->  ( A D B )  =  ( A G ( ( inv `  G ) `
 B ) ) )
76oveq1d 6300 . 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 25157 . . . . . 6  |-  ( ( R  e.  RingOps  /\  A  e.  X  /\  C  e.  X )  ->  ( A H C )  e.  X )
1093adant3r2 1206 . . . . 5  |-  ( ( R  e.  RingOps  /\  ( A  e.  X  /\  B  e.  X  /\  C  e.  X )
)  ->  ( A H C )  e.  X
)
111, 8, 2rngocl 25157 . . . . . 6  |-  ( ( R  e.  RingOps  /\  B  e.  X  /\  C  e.  X )  ->  ( B H C )  e.  X )
12113adant3r1 1205 . . . . 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 30181 . . . . 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 1197 . . . 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 30178 . . . . . . . 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 1306 . . . . . 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 25161 . . . . 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 30184 . . . . . 6  |-  ( ( R  e.  RingOps  /\  B  e.  X  /\  C  e.  X )  ->  (
( inv `  G
) `  ( B H C ) )  =  ( ( ( inv `  G ) `  B
) H C ) )
26253adant3r1 1205 . . . . 5  |-  ( ( R  e.  RingOps  /\  ( A  e.  X  /\  B  e.  X  /\  C  e.  X )
)  ->  ( ( inv `  G ) `  ( B H C ) )  =  ( ( ( inv `  G
) `  B ) H C ) )
2726oveq2d 6301 . . . 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 2511 . . 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 2511 . 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 2511 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 973    = wceq 1379    e. wcel 1767   ran crn 5000   ` cfv 5588  (class class class)co 6285   1stc1st 6783   2ndc2nd 6784   invcgn 24963    /g cgs 24964   RingOpscrngo 25150
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6577
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-reu 2821  df-rmo 2822  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-id 4795  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5551  df-fun 5590  df-fn 5591  df-f 5592  df-f1 5593  df-fo 5594  df-f1o 5595  df-fv 5596  df-riota 6246  df-ov 6288  df-oprab 6289  df-mpt2 6290  df-1st 6785  df-2nd 6786  df-grpo 24966  df-gid 24967  df-ginv 24968  df-gdiv 24969  df-ablo 25057  df-ass 25088  df-exid 25090  df-mgm 25094  df-sgr 25106  df-mndo 25113  df-rngo 25151
This theorem is referenced by: (None)
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