HomeHome Metamath Proof Explorer < Previous   Next >
Related theorems
Unicode version
Theorem hlcom 8605
Description: Hilbert space vector addition is commutative.
Hypotheses
Ref Expression
hladdf.1 |- X = (Base` U)
hladdf.2 |- G = (+v` U)
Assertion
Ref Expression
hlcom |- ((U e. CHil /\ A e. X /\ B e. X) -> (AGB) = (BGA))
Proof of Theorem hlcom
StepHypRef Expression
1 hladdf.1 . . 3 |- X = (Base` U)
2 hladdf.2 . . 3 |- G = (+v` U)
31, 2nvcom 8243 . 2 |- ((U e. NrmCVec /\ A e. X /\ B e. X) -> (AGB) = (BGA))
4 hlnv 8598 . 2 |- (U e. CHil -> U e. NrmCVec)
53, 4syl3an1 861 1 |- ((U e. CHil /\ A e. X /\ B e. X) -> (AGB) = (BGA))
Colors of variables: wff set class
Syntax hints:   -> wi 3   /\ w3a 777   = wceq 958   e. wcel 960  ` cfv 3189  (class class class)co 3970  NrmCVeccnv 8206  +vcpv 8207  Basecba 8208  CHilchl 8592
This theorem is referenced by:  axhvcom 8855
This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 964  ax-gen 965  ax-8 966  ax-9 967  ax-10 968  ax-11 969  ax-12 970  ax-13 971  ax-14 972  ax-17 973  ax-4 975  ax-5o 977  ax-6o 980  ax-9o 1125  ax-10o 1142  ax-16 1212  ax-11o 1220  ax-ext 1462  ax-sep 2709  ax-nul 2716  ax-pow 2749  ax-pr 2786  ax-un 2873
This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3an 779  df-ex 983  df-sb 1174  df-eu 1384  df-mo 1385  df-clab 1467  df-cleq 1472  df-clel 1475  df-ne 1590  df-ral 1652  df-rex 1653  df-rab 1655  df-v 1815  df-sbc 1945  df-dif 2053  df-un 2054  df-in 2055  df-ss 2057  df-nul 2285  df-pw 2407  df-sn 2417  df-pr 2418  df-op 2421  df-uni 2509  df-br 2626  df-opab 2673  df-id 2842  df-xp 3191  df-rel 3192  df-cnv 3193  df-co 3194  df-dm 3195  df-rn 3196  df-res 3197  df-ima 3198  df-fun 3199  df-fn 3200  df-f 3201  df-fo 3203  df-fv 3205  df-opr 3972  df-oprab 3973  df-1st 4086  df-2nd 4087  df-grp 8041  df-gid 8042  df-abl 8103  df-vc 8168  df-nv 8214  df-va 8217  df-ba 8218  df-sm 8219  df-0v 8220  df-nm 8222  df-bn 8526  df-hl 8593
Copyright terms: Public domain