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Theorem kbop 23456
 Description: The outer product of two vectors, expressed as in Dirac notation, is an operator. (Contributed by NM, 30-May-2006.) (Revised by Mario Carneiro, 16-Nov-2013.) (New usage is discouraged.)
Assertion
Ref Expression
kbop

Proof of Theorem kbop
Dummy variable is distinct from all other variables.
StepHypRef Expression
1 hicl 22582 . . . . 5
2 hvmulcl 22516 . . . . 5
31, 2sylan 458 . . . 4
43an31s 782 . . 3
5 eqid 2436 . . 3
64, 5fmptd 5893 . 2
7 kbfval 23455 . . 3
87feq1d 5580 . 2
96, 8mpbird 224 1
 Colors of variables: wff set class Syntax hints:   wi 4   wa 359   wcel 1725   cmpt 4266  wf 5450  (class class class)co 6081  cc 8988  chil 22422   csm 22424   csp 22425   ck 22460 This theorem is referenced by:  kbpj  23459  kbass2  23620  kbass5  23623 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-rep 4320  ax-sep 4330  ax-nul 4338  ax-pr 4403  ax-hilex 22502  ax-hfvmul 22508  ax-hfi 22581 This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2710  df-rex 2711  df-reu 2712  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-sn 3820  df-pr 3821  df-op 3823  df-uni 4016  df-iun 4095  df-br 4213  df-opab 4267  df-mpt 4268  df-id 4498  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-f1 5459  df-fo 5460  df-f1o 5461  df-fv 5462  df-ov 6084  df-oprab 6085  df-mpt2 6086  df-kb 23354
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