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Theorem hlcom 21425
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 21416 . 2  |-  ( U  e.  CHil OLD  ->  U  e.  NrmCVec )
2 hladdf.1 . . 3  |-  X  =  ( BaseSet `  U )
3 hladdf.2 . . 3  |-  G  =  ( +v `  U
)
42, 3nvcom 21123 . 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 4659  (class class class)co 5778   NrmCVeccnv 21086   +vcpv 21087   BaseSetcba 21088   CHil
OLDchlo 21410
This theorem is referenced by:  axhvcom-zf  21509
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 4091  ax-sep 4101  ax-nul 4109  ax-pr 4172  ax-un 4470
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 2521  df-rex 2522  df-reu 2523  df-rab 2525  df-v 2759  df-sbc 2953  df-csb 3043  df-dif 3116  df-un 3118  df-in 3120  df-ss 3127  df-nul 3417  df-if 3526  df-sn 3606  df-pr 3607  df-op 3609  df-uni 3788  df-iun 3867  df-br 3984  df-opab 4038  df-mpt 4039  df-id 4267  df-xp 4661  df-rel 4662  df-cnv 4663  df-co 4664  df-dm 4665  df-rn 4666  df-res 4667  df-ima 4668  df-fun 4669  df-fn 4670  df-f 4671  df-f1 4672  df-fo 4673  df-f1o 4674  df-fv 4675  df-ov 5781  df-oprab 5782  df-1st 6042  df-2nd 6043  df-ablo 20895  df-vc 21048  df-nv 21094  df-va 21097  df-ba 21098  df-sm 21099  df-0v 21100  df-nmcv 21102  df-cbn 21388  df-hlo 21411
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