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Theorem rngosubdir 26570
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 2436 . . . . 5  |-  ( inv `  G )  =  ( inv `  G )
4 ringsubdi.4 . . . . 5  |-  D  =  (  /g  `  G
)
51, 2, 3, 4rngosub 26564 . . . 4  |-  ( ( R  e.  RingOps  /\  A  e.  X  /\  B  e.  X )  ->  ( A D B )  =  ( A G ( ( inv `  G
) `  B )
) )
653adant3r3 1164 . . 3  |-  ( ( R  e.  RingOps  /\  ( A  e.  X  /\  B  e.  X  /\  C  e.  X )
)  ->  ( A D B )  =  ( A G ( ( inv `  G ) `
 B ) ) )
76oveq1d 6096 . 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 21970 . . . . . 6  |-  ( ( R  e.  RingOps  /\  A  e.  X  /\  C  e.  X )  ->  ( A H C )  e.  X )
1093adant3r2 1163 . . . . 5  |-  ( ( R  e.  RingOps  /\  ( A  e.  X  /\  B  e.  X  /\  C  e.  X )
)  ->  ( A H C )  e.  X
)
111, 8, 2rngocl 21970 . . . . . 6  |-  ( ( R  e.  RingOps  /\  B  e.  X  /\  C  e.  X )  ->  ( B H C )  e.  X )
12113adant3r1 1162 . . . . 5  |-  ( ( R  e.  RingOps  /\  ( A  e.  X  /\  B  e.  X  /\  C  e.  X )
)  ->  ( B H C )  e.  X
)
1310, 12jca 519 . . . 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 26564 . . . . 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 1154 . . . 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 457 . . 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 22 . . . . . . 7  |-  ( R  e.  RingOps  ->  ( A  e.  X  ->  A  e.  X ) )
181, 2, 3rngonegcl 26561 . . . . . . . 8  |-  ( ( R  e.  RingOps  /\  B  e.  X )  ->  (
( inv `  G
) `  B )  e.  X )
1918ex 424 . . . . . . 7  |-  ( R  e.  RingOps  ->  ( B  e.  X  ->  ( ( inv `  G ) `  B )  e.  X
) )
20 idd 22 . . . . . . 7  |-  ( R  e.  RingOps  ->  ( C  e.  X  ->  C  e.  X ) )
2117, 19, 203anim123d 1261 . . . . . 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 419 . . . . 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 21974 . . . . 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 457 . . . 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 26567 . . . . . 6  |-  ( ( R  e.  RingOps  /\  B  e.  X  /\  C  e.  X )  ->  (
( inv `  G
) `  ( B H C ) )  =  ( ( ( inv `  G ) `  B
) H C ) )
26253adant3r1 1162 . . . . 5  |-  ( ( R  e.  RingOps  /\  ( A  e.  X  /\  B  e.  X  /\  C  e.  X )
)  ->  ( ( inv `  G ) `  ( B H C ) )  =  ( ( ( inv `  G
) `  B ) H C ) )
2726oveq2d 6097 . . . 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 2471 . . 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 2471 . 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 2471 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 set class
Syntax hints:    -> wi 4    /\ wa 359    /\ w3a 936    = wceq 1652    e. wcel 1725   ran crn 4879   ` cfv 5454  (class class class)co 6081   1stc1st 6347   2ndc2nd 6348   invcgn 21776    /g cgs 21777   RingOpscrngo 21963
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 2417  ax-rep 4320  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701
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 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2710  df-rex 2711  df-reu 2712  df-rmo 2713  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-op 3823  df-uni 4016  df-iun 4095  df-br 4213  df-opab 4267  df-mpt 4268  df-id 4498  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-f1 5459  df-fo 5460  df-f1o 5461  df-fv 5462  df-ov 6084  df-oprab 6085  df-mpt2 6086  df-1st 6349  df-2nd 6350  df-riota 6549  df-grpo 21779  df-gid 21780  df-ginv 21781  df-gdiv 21782  df-ablo 21870  df-ass 21901  df-exid 21903  df-mgm 21907  df-sgr 21919  df-mndo 21926  df-rngo 21964
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