HSE Home Hilbert Space Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  HSE Home  >  Th. List  >  hon0 Structured version   Unicode version

Theorem hon0 25334
Description: A Hilbert space operator is not empty. (Contributed by NM, 24-Mar-2006.) (New usage is discouraged.)
Assertion
Ref Expression
hon0  |-  ( T : ~H --> ~H  ->  -.  T  =  (/) )

Proof of Theorem hon0
StepHypRef Expression
1 ax-hv0cl 24542 . . 3  |-  0h  e.  ~H
2 n0i 3742 . . 3  |-  ( 0h  e.  ~H  ->  -.  ~H  =  (/) )
31, 2ax-mp 5 . 2  |-  -.  ~H  =  (/)
4 fn0 5630 . . 3  |-  ( T  Fn  (/)  <->  T  =  (/) )
5 ffn 5659 . . . 4  |-  ( T : ~H --> ~H  ->  T  Fn  ~H )
6 fndmu 5612 . . . . 5  |-  ( ( T  Fn  ~H  /\  T  Fn  (/) )  ->  ~H  =  (/) )
76ex 434 . . . 4  |-  ( T  Fn  ~H  ->  ( T  Fn  (/)  ->  ~H  =  (/) ) )
85, 7syl 16 . . 3  |-  ( T : ~H --> ~H  ->  ( T  Fn  (/)  ->  ~H  =  (/) ) )
94, 8syl5bir 218 . 2  |-  ( T : ~H --> ~H  ->  ( T  =  (/)  ->  ~H  =  (/) ) )
103, 9mtoi 178 1  |-  ( T : ~H --> ~H  ->  -.  T  =  (/) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    = wceq 1370    e. wcel 1758   (/)c0 3737    Fn wfn 5513   -->wf 5514   ~Hchil 24458   0hc0v 24463
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-sep 4513  ax-nul 4521  ax-pr 4631  ax-hv0cl 24542
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-ral 2800  df-rex 2801  df-rab 2804  df-v 3072  df-dif 3431  df-un 3433  df-in 3435  df-ss 3442  df-nul 3738  df-if 3892  df-sn 3978  df-pr 3980  df-op 3984  df-br 4393  df-opab 4451  df-id 4736  df-xp 4946  df-rel 4947  df-cnv 4948  df-co 4949  df-dm 4950  df-fun 5520  df-fn 5521  df-f 5522
This theorem is referenced by:  hmdmadj  25481
  Copyright terms: Public domain W3C validator