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Theorem nvaddsub 26166
Description: Commutative/associative law for vector addition and subtraction. (Contributed by NM, 24-Jan-2008.) (New usage is discouraged.)
Hypotheses
Ref Expression
nvsubadd.1  |-  X  =  ( BaseSet `  U )
nvsubadd.2  |-  G  =  ( +v `  U
)
nvsubadd.3  |-  M  =  ( -v `  U
)
Assertion
Ref Expression
nvaddsub  |-  ( ( U  e.  NrmCVec  /\  ( A  e.  X  /\  B  e.  X  /\  C  e.  X )
)  ->  ( ( A G B ) M C )  =  ( ( A M C ) G B ) )

Proof of Theorem nvaddsub
StepHypRef Expression
1 nvsubadd.2 . . 3  |-  G  =  ( +v `  U
)
21nvablo 26121 . 2  |-  ( U  e.  NrmCVec  ->  G  e.  AbelOp )
3 nvsubadd.1 . . . 4  |-  X  =  ( BaseSet `  U )
43, 1bafval 26109 . . 3  |-  X  =  ran  G
5 nvsubadd.3 . . . 4  |-  M  =  ( -v `  U
)
61, 5vsfval 26140 . . 3  |-  M  =  (  /g  `  G
)
74, 6ablomuldiv 25903 . 2  |-  ( ( G  e.  AbelOp  /\  ( A  e.  X  /\  B  e.  X  /\  C  e.  X )
)  ->  ( ( A G B ) M C )  =  ( ( A M C ) G B ) )
82, 7sylan 473 1  |-  ( ( U  e.  NrmCVec  /\  ( A  e.  X  /\  B  e.  X  /\  C  e.  X )
)  ->  ( ( A G B ) M C )  =  ( ( A M C ) G B ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 370    /\ w3a 982    = wceq 1437    e. wcel 1867   ` cfv 5592  (class class class)co 6296   AbelOpcablo 25895   NrmCVeccnv 26089   +vcpv 26090   BaseSetcba 26091   -vcnsb 26094
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1748  ax-6 1794  ax-7 1838  ax-8 1869  ax-9 1871  ax-10 1886  ax-11 1891  ax-12 1904  ax-13 2052  ax-ext 2398  ax-rep 4529  ax-sep 4539  ax-nul 4547  ax-pow 4594  ax-pr 4652  ax-un 6588
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1787  df-eu 2267  df-mo 2268  df-clab 2406  df-cleq 2412  df-clel 2415  df-nfc 2570  df-ne 2618  df-ral 2778  df-rex 2779  df-reu 2780  df-rab 2782  df-v 3080  df-sbc 3297  df-csb 3393  df-dif 3436  df-un 3438  df-in 3440  df-ss 3447  df-nul 3759  df-if 3907  df-pw 3978  df-sn 3994  df-pr 3996  df-op 4000  df-uni 4214  df-iun 4295  df-br 4418  df-opab 4476  df-mpt 4477  df-id 4760  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-iota 5556  df-fun 5594  df-fn 5595  df-f 5596  df-f1 5597  df-fo 5598  df-f1o 5599  df-fv 5600  df-riota 6258  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-1st 6798  df-2nd 6799  df-grpo 25805  df-gid 25806  df-ginv 25807  df-gdiv 25808  df-ablo 25896  df-vc 26051  df-nv 26097  df-va 26100  df-ba 26101  df-sm 26102  df-0v 26103  df-vs 26104  df-nmcv 26105
This theorem is referenced by:  nvnpcan  26167
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