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Theorem nvnnncan2 21321
Description: Cancellation law for vector subtraction. (nnncan2 9174 analog.) (Contributed by NM, 15-Feb-2008.) (New usage is discouraged.)
Hypotheses
Ref Expression
nvmf.1  |-  X  =  ( BaseSet `  U )
nvmf.3  |-  M  =  ( -v `  U
)
Assertion
Ref Expression
nvnnncan2  |-  ( ( U  e.  NrmCVec  /\  ( A  e.  X  /\  B  e.  X  /\  C  e.  X )
)  ->  ( ( A M C ) M ( B M C ) )  =  ( A M B ) )

Proof of Theorem nvnnncan2
StepHypRef Expression
1 eqid 2358 . . 3  |-  ( +v
`  U )  =  ( +v `  U
)
21nvgrp 21287 . 2  |-  ( U  e.  NrmCVec  ->  ( +v `  U )  e.  GrpOp )
3 nvmf.1 . . . 4  |-  X  =  ( BaseSet `  U )
43, 1bafval 21274 . . 3  |-  X  =  ran  ( +v `  U )
5 nvmf.3 . . . 4  |-  M  =  ( -v `  U
)
61, 5vsfval 21305 . . 3  |-  M  =  (  /g  `  ( +v `  U ) )
74, 6grponnncan2 21033 . 2  |-  ( ( ( +v `  U
)  e.  GrpOp  /\  ( A  e.  X  /\  B  e.  X  /\  C  e.  X )
)  ->  ( ( A M C ) M ( B M C ) )  =  ( A M B ) )
82, 7sylan 457 1  |-  ( ( U  e.  NrmCVec  /\  ( A  e.  X  /\  B  e.  X  /\  C  e.  X )
)  ->  ( ( A M C ) M ( B M C ) )  =  ( A M B ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    /\ w3a 934    = wceq 1642    e. wcel 1710   ` cfv 5337  (class class class)co 5945   GrpOpcgr 20965   NrmCVeccnv 21254   +vcpv 21255   BaseSetcba 21256   -vcnsb 21259
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-13 1712  ax-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1930  ax-ext 2339  ax-rep 4212  ax-sep 4222  ax-nul 4230  ax-pow 4269  ax-pr 4295  ax-un 4594
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2213  df-mo 2214  df-clab 2345  df-cleq 2351  df-clel 2354  df-nfc 2483  df-ne 2523  df-ral 2624  df-rex 2625  df-reu 2626  df-rab 2628  df-v 2866  df-sbc 3068  df-csb 3158  df-dif 3231  df-un 3233  df-in 3235  df-ss 3242  df-nul 3532  df-if 3642  df-pw 3703  df-sn 3722  df-pr 3723  df-op 3725  df-uni 3909  df-iun 3988  df-br 4105  df-opab 4159  df-mpt 4160  df-id 4391  df-xp 4777  df-rel 4778  df-cnv 4779  df-co 4780  df-dm 4781  df-rn 4782  df-res 4783  df-ima 4784  df-iota 5301  df-fun 5339  df-fn 5340  df-f 5341  df-f1 5342  df-fo 5343  df-f1o 5344  df-fv 5345  df-ov 5948  df-oprab 5949  df-mpt2 5950  df-1st 6209  df-2nd 6210  df-riota 6391  df-grpo 20970  df-gid 20971  df-ginv 20972  df-gdiv 20973  df-ablo 21061  df-vc 21216  df-nv 21262  df-va 21265  df-ba 21266  df-sm 21267  df-0v 21268  df-vs 21269  df-nmcv 21270
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