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Mirrors > Home > HSE Home > Th. List > hmopf | Structured version Visualization version GIF version |
Description: A Hermitian operator is a Hilbert space operator (mapping). (Contributed by NM, 19-Mar-2006.) (New usage is discouraged.) |
Ref | Expression |
---|---|
hmopf | ⊢ (𝑇 ∈ HrmOp → 𝑇: ℋ⟶ ℋ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elhmop 29650 | . 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) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1537 ∈ wcel 2114 ∀wral 3138 ⟶wf 6351 ‘cfv 6355 (class class class)co 7156 ℋchba 28696 ·ih csp 28699 HrmOpcho 28727 |
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 2793 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 ax-hilex 28776 |
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 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3496 df-sbc 3773 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-op 4574 df-uni 4839 df-br 5067 df-opab 5129 df-id 5460 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-fv 6363 df-ov 7159 df-oprab 7160 df-mpo 7161 df-map 8408 df-hmop 29621 |
This theorem is referenced by: hmopex 29652 hmopre 29700 hmopadj 29716 hmdmadj 29717 hmoplin 29719 eighmre 29740 eighmorth 29741 hmops 29797 hmopm 29798 hmopd 29799 hmopco 29800 leop2 29901 leoppos 29903 leoprf 29905 leopsq 29906 leopadd 29909 leopmuli 29910 leopmul 29911 leopmul2i 29912 leopnmid 29915 nmopleid 29916 opsqrlem1 29917 opsqrlem6 29922 elpjrn 29967 |
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