Theorem hmopf 29653

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 ∈ 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 29652	. 2 ⊢ (𝑇 ∈ HrmOp ↔ (𝑇: ℋ⟶ ℋ ∧ ∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ (𝑥 ·ih (𝑇‘𝑦)) = ((𝑇‘𝑥) ·ih 𝑦)))
2	1	simplbi 500	1 ⊢ (𝑇 ∈ HrmOp → 𝑇: ℋ⟶ ℋ)

Syntax hints:   → wi 4   = wceq 1537   ∈ wcel 2114  ∀wral 3140  ⟶wf 6353  ‘cfv 6357  (class class class)co 7158   ℋchba 28698   ·_ih csp 28701  HrmOpcho 28729

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2795  ax-sep 5205  ax-nul 5212  ax-pow 5268  ax-pr 5332  ax-un 7463  ax-hilex 28778

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2654  df-clab 2802  df-cleq 2816  df-clel 2895  df-nfc 2965  df-ral 3145  df-rex 3146  df-rab 3149  df-v 3498  df-sbc 3775  df-dif 3941  df-un 3943  df-in 3945  df-ss 3954  df-nul 4294  df-if 4470  df-pw 4543  df-sn 4570  df-pr 4572  df-op 4576  df-uni 4841  df-br 5069  df-opab 5131  df-id 5462  df-xp 5563  df-rel 5564  df-cnv 5565  df-co 5566  df-dm 5567  df-rn 5568  df-iota 6316  df-fun 6359  df-fn 6360  df-f 6361  df-fv 6365  df-ov 7161  df-oprab 7162  df-mpo 7163  df-map 8410  df-hmop 29623

This theorem is referenced by:  hmopex  29654  hmopre  29702  hmopadj  29718  hmdmadj  29719  hmoplin  29721  eighmre  29742  eighmorth  29743  hmops  29799  hmopm  29800  hmopd  29801  hmopco  29802  leop2  29903  leoppos  29905  leoprf  29907  leopsq  29908  leopadd  29911  leopmuli  29912  leopmul  29913  leopmul2i  29914  leopnmid  29917  nmopleid  29918  opsqrlem1  29919  opsqrlem6  29924  elpjrn  29969