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Theorem nvnnncan2 25944
Description: Cancellation law for vector subtraction. (nnncan2 9891 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 2402 . . 3  |-  ( +v
`  U )  =  ( +v `  U
)
21nvgrp 25910 . 2  |-  ( U  e.  NrmCVec  ->  ( +v `  U )  e.  GrpOp )
3 nvmf.1 . . . 4  |-  X  =  ( BaseSet `  U )
43, 1bafval 25897 . . 3  |-  X  =  ran  ( +v `  U )
5 nvmf.3 . . . 4  |-  M  =  ( -v `  U
)
61, 5vsfval 25928 . . 3  |-  M  =  (  /g  `  ( +v `  U ) )
74, 6grponnncan2 25656 . 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 469 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 setvar class
Syntax hints:    -> wi 4    /\ wa 367    /\ w3a 974    = wceq 1405    e. wcel 1842   ` cfv 5568  (class class class)co 6277   GrpOpcgr 25588   NrmCVeccnv 25877   +vcpv 25878   BaseSetcba 25879   -vcnsb 25882
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-8 1844  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-rep 4506  ax-sep 4516  ax-nul 4524  ax-pow 4571  ax-pr 4629  ax-un 6573
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-ral 2758  df-rex 2759  df-reu 2760  df-rab 2762  df-v 3060  df-sbc 3277  df-csb 3373  df-dif 3416  df-un 3418  df-in 3420  df-ss 3427  df-nul 3738  df-if 3885  df-pw 3956  df-sn 3972  df-pr 3974  df-op 3978  df-uni 4191  df-iun 4272  df-br 4395  df-opab 4453  df-mpt 4454  df-id 4737  df-xp 4828  df-rel 4829  df-cnv 4830  df-co 4831  df-dm 4832  df-rn 4833  df-res 4834  df-ima 4835  df-iota 5532  df-fun 5570  df-fn 5571  df-f 5572  df-f1 5573  df-fo 5574  df-f1o 5575  df-fv 5576  df-riota 6239  df-ov 6280  df-oprab 6281  df-mpt2 6282  df-1st 6783  df-2nd 6784  df-grpo 25593  df-gid 25594  df-ginv 25595  df-gdiv 25596  df-ablo 25684  df-vc 25839  df-nv 25885  df-va 25888  df-ba 25889  df-sm 25890  df-0v 25891  df-vs 25892  df-nmcv 25893
This theorem is referenced by: (None)
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