Theorem hmopf 28733

Description: A Hermitian operator is a Hilbert space operator (mapping). (Contributed by NM, 19-Mar-2006.) (New usage is discouraged.)

Assertion

Ref	Expression
hmopf	|- ( T e. HrmOp -> T : ~H --> ~H )

Proof of Theorem hmopf

Dummy variables x y are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		elhmop 28732	. 2 |- ( T e. HrmOp <-> ( T : ~H --> ~H /\ A. x e. ~H A. y e. ~H ( x .ih ( T ` y ) ) = ( ( T ` x ) .ih y ) ) )
2	1	simplbi 476	1 |- ( T e. HrmOp -> T : ~H --> ~H )

Syntax hints:   -> wi 4   = wceq 1483   e. wcel 1990   A.wral 2912   -->wf 5884   ` cfv 5888  (class class class)co 6650   ~Hchil 27776   .ih csp 27779   HrmOpcho 27807

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-hilex 27856

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-map 7859  df-hmop 28703

This theorem is referenced by:  hmopex  28734  hmopre  28782  hmopadj  28798  hmdmadj  28799  hmoplin  28801  eighmre  28822  eighmorth  28823  hmops  28879  hmopm  28880  hmopd  28881  hmopco  28882  leop2  28983  leoppos  28985  leoprf  28987  leopsq  28988  leopadd  28991  leopmuli  28992  leopmul  28993  leopmul2i  28994  leopnmid  28997  nmopleid  28998  opsqrlem1  28999  opsqrlem6  29004  elpjrn  29049