Hilbert Space Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  HSE Home  >  Th. List  >  hvmulcom Structured version   Unicode version

Theorem hvmulcom 25664
 Description: Scalar multiplication commutative law. (Contributed by NM, 19-May-2005.) (New usage is discouraged.)
Assertion
Ref Expression
hvmulcom

Proof of Theorem hvmulcom
StepHypRef Expression
1 mulcom 9578 . . . 4
21oveq1d 6299 . . 3
323adant3 1016 . 2
4 ax-hvmulass 25628 . 2
5 ax-hvmulass 25628 . . 3
653com12 1200 . 2
73, 4, 63eqtr3d 2516 1
 Colors of variables: wff setvar class Syntax hints:   wi 4   wa 369   w3a 973   wceq 1379   wcel 1767  (class class class)co 6284  cc 9490   cmul 9497  chil 25540   csm 25542 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-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-mulcom 9556  ax-hvmulass 25628 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-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-rex 2820  df-rab 2823  df-v 3115  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-br 4448  df-iota 5551  df-fv 5596  df-ov 6287 This theorem is referenced by:  hvmulcomi  25668  hvsubdistr1  25670  lnopmi  26623
 Copyright terms: Public domain W3C validator