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Theorem multinv 25525
Description: Multiplication by an additive inverse. (Contributed by FL, 2-Sep-2009.)
Hypotheses
Ref Expression
multinv.1  |-  X  =  ran  G
multinv.2  |-  G  =  ( 1st `  R
)
multinv.3  |-  H  =  ( 2nd `  R
)
Assertion
Ref Expression
multinv  |-  ( ( R  e.  RingOps  /\  A  e.  X  /\  B  e.  X )  ->  (
( ( inv `  G
) `  A ) H B )  =  ( ( inv `  G
) `  ( A H B ) ) )

Proof of Theorem multinv
StepHypRef Expression
1 simp1 955 . . . . 5  |-  ( ( R  e.  RingOps  /\  A  e.  X  /\  B  e.  X )  ->  R  e.  RingOps )
2 simp2 956 . . . . 5  |-  ( ( R  e.  RingOps  /\  A  e.  X  /\  B  e.  X )  ->  A  e.  X )
3 multinv.2 . . . . . . . . 9  |-  G  =  ( 1st `  R
)
43rngogrpo 21073 . . . . . . . 8  |-  ( R  e.  RingOps  ->  G  e.  GrpOp )
5 multinv.1 . . . . . . . . . 10  |-  X  =  ran  G
6 eqid 2296 . . . . . . . . . 10  |-  ( inv `  G )  =  ( inv `  G )
75, 6grpoinvcl 20909 . . . . . . . . 9  |-  ( ( G  e.  GrpOp  /\  A  e.  X )  ->  (
( inv `  G
) `  A )  e.  X )
87ex 423 . . . . . . . 8  |-  ( G  e.  GrpOp  ->  ( A  e.  X  ->  ( ( inv `  G ) `
 A )  e.  X ) )
94, 8syl 15 . . . . . . 7  |-  ( R  e.  RingOps  ->  ( A  e.  X  ->  ( ( inv `  G ) `  A )  e.  X
) )
109a1dd 42 . . . . . 6  |-  ( R  e.  RingOps  ->  ( A  e.  X  ->  ( B  e.  X  ->  ( ( inv `  G ) `
 A )  e.  X ) ) )
11103imp 1145 . . . . 5  |-  ( ( R  e.  RingOps  /\  A  e.  X  /\  B  e.  X )  ->  (
( inv `  G
) `  A )  e.  X )
12 simp3 957 . . . . 5  |-  ( ( R  e.  RingOps  /\  A  e.  X  /\  B  e.  X )  ->  B  e.  X )
13 multinv.3 . . . . . . 7  |-  H  =  ( 2nd `  R
)
143, 13, 5rngodir 21069 . . . . . 6  |-  ( ( R  e.  RingOps  /\  ( A  e.  X  /\  ( ( inv `  G
) `  A )  e.  X  /\  B  e.  X ) )  -> 
( ( A G ( ( inv `  G
) `  A )
) H B )  =  ( ( A H B ) G ( ( ( inv `  G ) `  A
) H B ) ) )
1514eqcomd 2301 . . . . 5  |-  ( ( R  e.  RingOps  /\  ( A  e.  X  /\  ( ( inv `  G
) `  A )  e.  X  /\  B  e.  X ) )  -> 
( ( A H B ) G ( ( ( inv `  G
) `  A ) H B ) )  =  ( ( A G ( ( inv `  G
) `  A )
) H B ) )
161, 2, 11, 12, 15syl13anc 1184 . . . 4  |-  ( ( R  e.  RingOps  /\  A  e.  X  /\  B  e.  X )  ->  (
( A H B ) G ( ( ( inv `  G
) `  A ) H B ) )  =  ( ( A G ( ( inv `  G
) `  A )
) H B ) )
1743ad2ant1 976 . . . . . 6  |-  ( ( R  e.  RingOps  /\  A  e.  X  /\  B  e.  X )  ->  G  e.  GrpOp )
18 eqid 2296 . . . . . . 7  |-  (GId `  G )  =  (GId
`  G )
195, 18, 6grporinv 20912 . . . . . 6  |-  ( ( G  e.  GrpOp  /\  A  e.  X )  ->  ( A G ( ( inv `  G ) `  A
) )  =  (GId
`  G ) )
2017, 2, 19syl2anc 642 . . . . 5  |-  ( ( R  e.  RingOps  /\  A  e.  X  /\  B  e.  X )  ->  ( A G ( ( inv `  G ) `  A
) )  =  (GId
`  G ) )
2120oveq1d 5889 . . . 4  |-  ( ( R  e.  RingOps  /\  A  e.  X  /\  B  e.  X )  ->  (
( A G ( ( inv `  G
) `  A )
) H B )  =  ( (GId `  G ) H B ) )
2218, 5, 3, 13rngolz 21084 . . . . 5  |-  ( ( R  e.  RingOps  /\  B  e.  X )  ->  (
(GId `  G ) H B )  =  (GId
`  G ) )
23223adant2 974 . . . 4  |-  ( ( R  e.  RingOps  /\  A  e.  X  /\  B  e.  X )  ->  (
(GId `  G ) H B )  =  (GId
`  G ) )
2416, 21, 233eqtrd 2332 . . 3  |-  ( ( R  e.  RingOps  /\  A  e.  X  /\  B  e.  X )  ->  (
( A H B ) G ( ( ( inv `  G
) `  A ) H B ) )  =  (GId `  G )
)
253, 13, 5rngocl 21065 . . . 4  |-  ( ( R  e.  RingOps  /\  A  e.  X  /\  B  e.  X )  ->  ( A H B )  e.  X )
263, 13, 5rngocl 21065 . . . . 5  |-  ( ( R  e.  RingOps  /\  (
( inv `  G
) `  A )  e.  X  /\  B  e.  X )  ->  (
( ( inv `  G
) `  A ) H B )  e.  X
)
2711, 26syld3an2 1229 . . . 4  |-  ( ( R  e.  RingOps  /\  A  e.  X  /\  B  e.  X )  ->  (
( ( inv `  G
) `  A ) H B )  e.  X
)
285, 18, 6grpoinvid1 20913 . . . 4  |-  ( ( G  e.  GrpOp  /\  ( A H B )  e.  X  /\  ( ( ( inv `  G
) `  A ) H B )  e.  X
)  ->  ( (
( inv `  G
) `  ( A H B ) )  =  ( ( ( inv `  G ) `  A
) H B )  <-> 
( ( A H B ) G ( ( ( inv `  G
) `  A ) H B ) )  =  (GId `  G )
) )
2917, 25, 27, 28syl3anc 1182 . . 3  |-  ( ( R  e.  RingOps  /\  A  e.  X  /\  B  e.  X )  ->  (
( ( inv `  G
) `  ( A H B ) )  =  ( ( ( inv `  G ) `  A
) H B )  <-> 
( ( A H B ) G ( ( ( inv `  G
) `  A ) H B ) )  =  (GId `  G )
) )
3024, 29mpbird 223 . 2  |-  ( ( R  e.  RingOps  /\  A  e.  X  /\  B  e.  X )  ->  (
( inv `  G
) `  ( A H B ) )  =  ( ( ( inv `  G ) `  A
) H B ) )
3130eqcomd 2301 1  |-  ( ( R  e.  RingOps  /\  A  e.  X  /\  B  e.  X )  ->  (
( ( inv `  G
) `  A ) H B )  =  ( ( inv `  G
) `  ( A H B ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    /\ w3a 934    = wceq 1632    e. wcel 1696   ran crn 4706   ` cfv 5271  (class class class)co 5874   1stc1st 6136   2ndc2nd 6137   GrpOpcgr 20869  GIdcgi 20870   invcgn 20871   RingOpscrngo 21058
This theorem is referenced by:  mult2inv  25527
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-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-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-1st 6138  df-2nd 6139  df-riota 6320  df-grpo 20874  df-gid 20875  df-ginv 20876  df-ablo 20965  df-rngo 21059
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