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

Theorem kbfval 23486
 Description: The outer product of two vectors, expressed as in Dirac notation. See df-kb 23385. (Contributed by NM, 15-May-2006.) (Revised by Mario Carneiro, 23-Aug-2014.) (New usage is discouraged.)
Assertion
Ref Expression
kbfval
Distinct variable groups:   ,   ,

Proof of Theorem kbfval
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 oveq2 6118 . . 3
21mpteq2dv 4321 . 2
3 oveq2 6118 . . . 4
43oveq1d 6125 . . 3
54mpteq2dv 4321 . 2
6 df-kb 23385 . 2
7 ax-hilex 22533 . . 3
87mptex 5995 . 2
92, 5, 6, 8ovmpt2 6238 1
 Colors of variables: wff set class Syntax hints:   wi 4   wa 360   wceq 1653   wcel 1727   cmpt 4291  (class class class)co 6110  chil 22453   csm 22455   csp 22456   ck 22491 This theorem is referenced by:  kbop  23487  kbval  23488  kbmul  23489 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1668  ax-8 1689  ax-14 1731  ax-6 1746  ax-7 1751  ax-11 1763  ax-12 1953  ax-ext 2423  ax-rep 4345  ax-sep 4355  ax-nul 4363  ax-pr 4432  ax-hilex 22533 This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2291  df-mo 2292  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2567  df-ne 2607  df-ral 2716  df-rex 2717  df-reu 2718  df-rab 2720  df-v 2964  df-sbc 3168  df-csb 3268  df-dif 3309  df-un 3311  df-in 3313  df-ss 3320  df-nul 3614  df-if 3764  df-sn 3844  df-pr 3845  df-op 3847  df-uni 4040  df-iun 4119  df-br 4238  df-opab 4292  df-mpt 4293  df-id 4527  df-xp 4913  df-rel 4914  df-cnv 4915  df-co 4916  df-dm 4917  df-rn 4918  df-res 4919  df-ima 4920  df-iota 5447  df-fun 5485  df-fn 5486  df-f 5487  df-f1 5488  df-fo 5489  df-f1o 5490  df-fv 5491  df-ov 6113  df-oprab 6114  df-mpt2 6115  df-kb 23385
 Copyright terms: Public domain W3C validator