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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 29821 | . 2 ⊢ (𝑇 ∈ HrmOp ↔ (𝑇: ℋ⟶ ℋ ∧ ∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ (𝑥 ·ih (𝑇‘𝑦)) = ((𝑇‘𝑥) ·ih 𝑦))) | |
2 | 1 | simplbi 501 | 1 ⊢ (𝑇 ∈ HrmOp → 𝑇: ℋ⟶ ℋ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1542 ∈ wcel 2114 ∀wral 3054 ⟶wf 6346 ‘cfv 6350 (class class class)co 7183 ℋchba 28867 ·ih csp 28870 HrmOpcho 28898 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2020 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2162 ax-12 2179 ax-ext 2711 ax-sep 5177 ax-nul 5184 ax-pow 5242 ax-pr 5306 ax-un 7492 ax-hilex 28947 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2075 df-mo 2541 df-eu 2571 df-clab 2718 df-cleq 2731 df-clel 2812 df-nfc 2882 df-ral 3059 df-rex 3060 df-rab 3063 df-v 3402 df-sbc 3686 df-dif 3856 df-un 3858 df-in 3860 df-ss 3870 df-nul 4222 df-if 4425 df-pw 4500 df-sn 4527 df-pr 4529 df-op 4533 df-uni 4807 df-br 5041 df-opab 5103 df-id 5439 df-xp 5541 df-rel 5542 df-cnv 5543 df-co 5544 df-dm 5545 df-rn 5546 df-iota 6308 df-fun 6352 df-fn 6353 df-f 6354 df-fv 6358 df-ov 7186 df-oprab 7187 df-mpo 7188 df-map 8452 df-hmop 29792 |
This theorem is referenced by: hmopex 29823 hmopre 29871 hmopadj 29887 hmdmadj 29888 hmoplin 29890 eighmre 29911 eighmorth 29912 hmops 29968 hmopm 29969 hmopd 29970 hmopco 29971 leop2 30072 leoppos 30074 leoprf 30076 leopsq 30077 leopadd 30080 leopmuli 30081 leopmul 30082 leopmul2i 30083 leopnmid 30086 nmopleid 30087 opsqrlem1 30088 opsqrlem6 30093 elpjrn 30138 |
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