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Theorem hlcom 25492
Description: Hilbert space vector addition is commutative. (Contributed by NM, 7-Sep-2007.) (New usage is discouraged.)
Hypotheses
Ref Expression
hladdf.1  |-  X  =  ( BaseSet `  U )
hladdf.2  |-  G  =  ( +v `  U
)
Assertion
Ref Expression
hlcom  |-  ( ( U  e.  CHilOLD  /\  A  e.  X  /\  B  e.  X )  ->  ( A G B )  =  ( B G A ) )

Proof of Theorem hlcom
StepHypRef Expression
1 hlnv 25483 . 2  |-  ( U  e.  CHilOLD  ->  U  e.  NrmCVec )
2 hladdf.1 . . 3  |-  X  =  ( BaseSet `  U )
3 hladdf.2 . . 3  |-  G  =  ( +v `  U
)
42, 3nvcom 25190 . 2  |-  ( ( U  e.  NrmCVec  /\  A  e.  X  /\  B  e.  X )  ->  ( A G B )  =  ( B G A ) )
51, 4syl3an1 1261 1  |-  ( ( U  e.  CHilOLD  /\  A  e.  X  /\  B  e.  X )  ->  ( A G B )  =  ( B G A ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ w3a 973    = wceq 1379    e. wcel 1767   ` cfv 5586  (class class class)co 6282   NrmCVeccnv 25153   +vcpv 25154   BaseSetcba 25155   CHilOLDchlo 25477
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6574
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-reu 2821  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-id 4795  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5549  df-fun 5588  df-fn 5589  df-f 5590  df-f1 5591  df-fo 5592  df-f1o 5593  df-fv 5594  df-ov 6285  df-oprab 6286  df-1st 6781  df-2nd 6782  df-ablo 24960  df-vc 25115  df-nv 25161  df-va 25164  df-ba 25165  df-sm 25166  df-0v 25167  df-nmcv 25169  df-cbn 25455  df-hlo 25478
This theorem is referenced by:  axhvcom-zf  25576
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