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Theorem hlcom 21404
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.  CHil OLD  /\  A  e.  X  /\  B  e.  X )  ->  ( A G B )  =  ( B G A ) )

Proof of Theorem hlcom
StepHypRef Expression
1 hlnv 21395 . 2  |-  ( U  e.  CHil OLD  ->  U  e.  NrmCVec )
2 hladdf.1 . . 3  |-  X  =  ( BaseSet `  U )
3 hladdf.2 . . 3  |-  G  =  ( +v `  U
)
42, 3nvcom 21102 . 2  |-  ( ( U  e.  NrmCVec  /\  A  e.  X  /\  B  e.  X )  ->  ( A G B )  =  ( B G A ) )
51, 4syl3an1 1220 1  |-  ( ( U  e.  CHil OLD  /\  A  e.  X  /\  B  e.  X )  ->  ( A G B )  =  ( B G A ) )
Colors of variables: wff set class
Syntax hints:    -> wi 6    /\ w3a 939    = wceq 1619    e. wcel 1621   ` cfv 4638  (class class class)co 5757   NrmCVeccnv 21065   +vcpv 21066   BaseSetcba 21067   CHil
OLDchlo 21389
This theorem is referenced by:  axhvcom-zf  21488
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-13 1625  ax-14 1626  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1927  ax-ext 2237  ax-rep 4071  ax-sep 4081  ax-nul 4089  ax-pr 4152  ax-un 4449
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 941  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1884  df-eu 2121  df-mo 2122  df-clab 2243  df-cleq 2249  df-clel 2252  df-nfc 2381  df-ne 2421  df-ral 2520  df-rex 2521  df-reu 2522  df-rab 2523  df-v 2742  df-sbc 2936  df-csb 3024  df-dif 3097  df-un 3099  df-in 3101  df-ss 3108  df-nul 3398  df-if 3507  df-sn 3587  df-pr 3588  df-op 3590  df-uni 3769  df-iun 3848  df-br 3964  df-opab 4018  df-mpt 4019  df-id 4246  df-xp 4640  df-rel 4641  df-cnv 4642  df-co 4643  df-dm 4644  df-rn 4645  df-res 4646  df-ima 4647  df-fun 4648  df-fn 4649  df-f 4650  df-f1 4651  df-fo 4652  df-f1o 4653  df-fv 4654  df-ov 5760  df-oprab 5761  df-1st 6021  df-2nd 6022  df-ablo 20874  df-vc 21027  df-nv 21073  df-va 21076  df-ba 21077  df-sm 21078  df-0v 21079  df-nmcv 21081  df-cbn 21367  df-hlo 21390
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