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Theorem nvaddsubass 21324
Description: Associative-type 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
nvaddsubass  |-  ( ( U  e.  NrmCVec  /\  ( A  e.  X  /\  B  e.  X  /\  C  e.  X )
)  ->  ( ( A G B ) M C )  =  ( A G ( B M C ) ) )

Proof of Theorem nvaddsubass
StepHypRef Expression
1 nvsubadd.2 . . 3  |-  G  =  ( +v `  U
)
21nvgrp 21281 . 2  |-  ( U  e.  NrmCVec  ->  G  e.  GrpOp )
3 nvsubadd.1 . . . 4  |-  X  =  ( BaseSet `  U )
43, 1bafval 21268 . . 3  |-  X  =  ran  G
5 nvsubadd.3 . . . 4  |-  M  =  ( -v `  U
)
61, 5vsfval 21299 . . 3  |-  M  =  (  /g  `  G
)
74, 6grpomuldivass 21022 . 2  |-  ( ( G  e.  GrpOp  /\  ( A  e.  X  /\  B  e.  X  /\  C  e.  X )
)  ->  ( ( A G B ) M C )  =  ( A G ( B M C ) ) )
82, 7sylan 457 1  |-  ( ( U  e.  NrmCVec  /\  ( A  e.  X  /\  B  e.  X  /\  C  e.  X )
)  ->  ( ( A G B ) M C )  =  ( A G ( B M C ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    /\ w3a 934    = wceq 1642    e. wcel 1710   ` cfv 5334  (class class class)co 5942   GrpOpcgr 20959   NrmCVeccnv 21248   +vcpv 21249   BaseSetcba 21250   -vcnsb 21253
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 4210  ax-sep 4220  ax-nul 4228  ax-pow 4267  ax-pr 4293  ax-un 4591
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 3907  df-iun 3986  df-br 4103  df-opab 4157  df-mpt 4158  df-id 4388  df-xp 4774  df-rel 4775  df-cnv 4776  df-co 4777  df-dm 4778  df-rn 4779  df-res 4780  df-ima 4781  df-iota 5298  df-fun 5336  df-fn 5337  df-f 5338  df-f1 5339  df-fo 5340  df-f1o 5341  df-fv 5342  df-ov 5945  df-oprab 5946  df-mpt2 5947  df-1st 6206  df-2nd 6207  df-riota 6388  df-grpo 20964  df-gid 20965  df-ginv 20966  df-gdiv 20967  df-ablo 21055  df-vc 21210  df-nv 21256  df-va 21259  df-ba 21260  df-sm 21261  df-0v 21262  df-vs 21263  df-nmcv 21264
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