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Theorem multinv 25422
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 21057 . . . . . . . 8  |-  ( R  e.  RingOps  ->  G  e.  GrpOp )
5 multinv.1 . . . . . . . . . 10  |-  X  =  ran  G
6 eqid 2283 . . . . . . . . . 10  |-  ( inv `  G )  =  ( inv `  G )
75, 6grpoinvcl 20893 . . . . . . . . 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 21053 . . . . . 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 2288 . . . . 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 2283 . . . . . . 7  |-  (GId `  G )  =  (GId
`  G )
195, 18, 6grporinv 20896 . . . . . 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 5873 . . . 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 21068 . . . . 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 2319 . . 3  |-  ( ( R  e.  RingOps  /\  A  e.  X  /\  B  e.  X )  ->  (
( A H B ) G ( ( ( inv `  G
) `  A ) H B ) )  =  (GId `  G )
)
253, 13, 5rngocl 21049 . . . 4  |-  ( ( R  e.  RingOps  /\  A  e.  X  /\  B  e.  X )  ->  ( A H B )  e.  X )
263, 13, 5rngocl 21049 . . . . 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 20897 . . . 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 2288 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 1623    e. wcel 1684   ran crn 4690   ` cfv 5255  (class class class)co 5858   1stc1st 6120   2ndc2nd 6121   GrpOpcgr 20853  GIdcgi 20854   invcgn 20855   RingOpscrngo 21042
This theorem is referenced by:  mult2inv  25424
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-1st 6122  df-2nd 6123  df-riota 6304  df-grpo 20858  df-gid 20859  df-ginv 20860  df-ablo 20949  df-rngo 21043
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