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Theorem rngosubdir 26688
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 2296 . . . . 5  |-  ( inv `  G )  =  ( inv `  G )
4 ringsubdi.4 . . . . 5  |-  D  =  (  /g  `  G
)
51, 2, 3, 4rngosub 26682 . . . 4  |-  ( ( R  e.  RingOps  /\  A  e.  X  /\  B  e.  X )  ->  ( A D B )  =  ( A G ( ( inv `  G
) `  B )
) )
653adant3r3 1162 . . 3  |-  ( ( R  e.  RingOps  /\  ( A  e.  X  /\  B  e.  X  /\  C  e.  X )
)  ->  ( A D B )  =  ( A G ( ( inv `  G ) `
 B ) ) )
76oveq1d 5889 . 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 21065 . . . . . 6  |-  ( ( R  e.  RingOps  /\  A  e.  X  /\  C  e.  X )  ->  ( A H C )  e.  X )
1093adant3r2 1161 . . . . 5  |-  ( ( R  e.  RingOps  /\  ( A  e.  X  /\  B  e.  X  /\  C  e.  X )
)  ->  ( A H C )  e.  X
)
111, 8, 2rngocl 21065 . . . . . 6  |-  ( ( R  e.  RingOps  /\  B  e.  X  /\  C  e.  X )  ->  ( B H C )  e.  X )
12113adant3r1 1160 . . . . 5  |-  ( ( R  e.  RingOps  /\  ( A  e.  X  /\  B  e.  X  /\  C  e.  X )
)  ->  ( B H C )  e.  X
)
1310, 12jca 518 . . . 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 26682 . . . . 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 1152 . . . 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 456 . . 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 21 . . . . . . 7  |-  ( R  e.  RingOps  ->  ( A  e.  X  ->  A  e.  X ) )
181, 2, 3rngonegcl 26679 . . . . . . . 8  |-  ( ( R  e.  RingOps  /\  B  e.  X )  ->  (
( inv `  G
) `  B )  e.  X )
1918ex 423 . . . . . . 7  |-  ( R  e.  RingOps  ->  ( B  e.  X  ->  ( ( inv `  G ) `  B )  e.  X
) )
20 idd 21 . . . . . . 7  |-  ( R  e.  RingOps  ->  ( C  e.  X  ->  C  e.  X ) )
2117, 19, 203anim123d 1259 . . . . . 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 418 . . . . 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 21069 . . . . 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 456 . . . 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 26685 . . . . . 6  |-  ( ( R  e.  RingOps  /\  B  e.  X  /\  C  e.  X )  ->  (
( inv `  G
) `  ( B H C ) )  =  ( ( ( inv `  G ) `  B
) H C ) )
26253adant3r1 1160 . . . . 5  |-  ( ( R  e.  RingOps  /\  ( A  e.  X  /\  B  e.  X  /\  C  e.  X )
)  ->  ( ( inv `  G ) `  ( B H C ) )  =  ( ( ( inv `  G
) `  B ) H C ) )
2726oveq2d 5890 . . . 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 2331 . . 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 2331 . 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 2331 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 358    /\ w3a 934    = wceq 1632    e. wcel 1696   ran crn 4706   ` cfv 5271  (class class class)co 5874   1stc1st 6136   2ndc2nd 6137   invcgn 20871    /g cgs 20872   RingOpscrngo 21058
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-reu 2563  df-rmo 2564  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-1st 6138  df-2nd 6139  df-riota 6320  df-grpo 20874  df-gid 20875  df-ginv 20876  df-gdiv 20877  df-ablo 20965  df-ass 20996  df-exid 20998  df-mgm 21002  df-sgr 21014  df-mndo 21021  df-rngo 21059
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