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Theorem hlcom 30827
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 = ( +𝑣𝑈)
Assertion
Ref Expression
hlcom ((𝑈 ∈ CHilOLD𝐴𝑋𝐵𝑋) → (𝐴𝐺𝐵) = (𝐵𝐺𝐴))

Proof of Theorem hlcom
StepHypRef Expression
1 hlnv 30818 . 2 (𝑈 ∈ CHilOLD𝑈 ∈ NrmCVec)
2 hladdf.1 . . 3 𝑋 = (BaseSet‘𝑈)
3 hladdf.2 . . 3 𝐺 = ( +𝑣𝑈)
42, 3nvcom 30548 . 2 ((𝑈 ∈ NrmCVec ∧ 𝐴𝑋𝐵𝑋) → (𝐴𝐺𝐵) = (𝐵𝐺𝐴))
51, 4syl3an1 1160 1 ((𝑈 ∈ CHilOLD𝐴𝑋𝐵𝑋) → (𝐴𝐺𝐵) = (𝐵𝐺𝐴))
Colors of variables: wff setvar class
Syntax hints:  wi 4  w3a 1084   = wceq 1534  wcel 2099  cfv 6543  (class class class)co 7413  NrmCVeccnv 30511   +𝑣 cpv 30512  BaseSetcba 30513  CHilOLDchlo 30812
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2697  ax-rep 5280  ax-sep 5294  ax-nul 5301  ax-pr 5423  ax-un 7735
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2704  df-cleq 2718  df-clel 2803  df-nfc 2878  df-ne 2931  df-ral 3052  df-rex 3061  df-reu 3365  df-rab 3420  df-v 3464  df-sbc 3776  df-csb 3892  df-dif 3949  df-un 3951  df-in 3953  df-ss 3963  df-nul 4323  df-if 4524  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4906  df-iun 4995  df-br 5144  df-opab 5206  df-mpt 5227  df-id 5570  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-rn 5683  df-res 5684  df-ima 5685  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-ov 7416  df-oprab 7417  df-1st 7992  df-2nd 7993  df-ablo 30472  df-vc 30486  df-nv 30519  df-va 30522  df-ba 30523  df-sm 30524  df-0v 30525  df-nmcv 30527  df-cbn 30790  df-hlo 30813
This theorem is referenced by:  axhvcom-zf  30910
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