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Theorem nvnnncan1 21978
Description: Cancellation law for vector subtraction. (nnncan1 9270 analog.) (Contributed by NM, 7-Mar-2008.) (New usage is discouraged.)
Hypotheses
Ref Expression
nvmf.1  |-  X  =  ( BaseSet `  U )
nvmf.3  |-  M  =  ( -v `  U
)
Assertion
Ref Expression
nvnnncan1  |-  ( ( U  e.  NrmCVec  /\  ( A  e.  X  /\  B  e.  X  /\  C  e.  X )
)  ->  ( ( A M B ) M ( A M C ) )  =  ( C M B ) )

Proof of Theorem nvnnncan1
StepHypRef Expression
1 eqid 2388 . . 3  |-  ( +v
`  U )  =  ( +v `  U
)
21nvablo 21944 . 2  |-  ( U  e.  NrmCVec  ->  ( +v `  U )  e.  AbelOp )
3 nvmf.1 . . . 4  |-  X  =  ( BaseSet `  U )
43, 1bafval 21932 . . 3  |-  X  =  ran  ( +v `  U )
5 nvmf.3 . . . 4  |-  M  =  ( -v `  U
)
61, 5vsfval 21963 . . 3  |-  M  =  (  /g  `  ( +v `  U ) )
74, 6ablonnncan1 21732 . 2  |-  ( ( ( +v `  U
)  e.  AbelOp  /\  ( A  e.  X  /\  B  e.  X  /\  C  e.  X )
)  ->  ( ( A M B ) M ( A M C ) )  =  ( C M B ) )
82, 7sylan 458 1  |-  ( ( U  e.  NrmCVec  /\  ( A  e.  X  /\  B  e.  X  /\  C  e.  X )
)  ->  ( ( A M B ) M ( A M C ) )  =  ( C M B ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    /\ w3a 936    = wceq 1649    e. wcel 1717   ` cfv 5395  (class class class)co 6021   AbelOpcablo 21718   NrmCVeccnv 21912   +vcpv 21913   BaseSetcba 21914   -vcnsb 21917
This theorem is referenced by:  minvecolem2  22226
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2369  ax-rep 4262  ax-sep 4272  ax-nul 4280  ax-pow 4319  ax-pr 4345  ax-un 4642
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2243  df-mo 2244  df-clab 2375  df-cleq 2381  df-clel 2384  df-nfc 2513  df-ne 2553  df-ral 2655  df-rex 2656  df-reu 2657  df-rab 2659  df-v 2902  df-sbc 3106  df-csb 3196  df-dif 3267  df-un 3269  df-in 3271  df-ss 3278  df-nul 3573  df-if 3684  df-pw 3745  df-sn 3764  df-pr 3765  df-op 3767  df-uni 3959  df-iun 4038  df-br 4155  df-opab 4209  df-mpt 4210  df-id 4440  df-xp 4825  df-rel 4826  df-cnv 4827  df-co 4828  df-dm 4829  df-rn 4830  df-res 4831  df-ima 4832  df-iota 5359  df-fun 5397  df-fn 5398  df-f 5399  df-f1 5400  df-fo 5401  df-f1o 5402  df-fv 5403  df-ov 6024  df-oprab 6025  df-mpt2 6026  df-1st 6289  df-2nd 6290  df-riota 6486  df-grpo 21628  df-gid 21629  df-ginv 21630  df-gdiv 21631  df-ablo 21719  df-vc 21874  df-nv 21920  df-va 21923  df-ba 21924  df-sm 21925  df-0v 21926  df-vs 21927  df-nmcv 21928
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