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Mirrors > Home > HSE Home > Th. List > hstcl | Structured version Visualization version GIF version |
Description: Closure of the value of a Hilbert-space-valued state. (Contributed by NM, 25-Jun-2006.) (New usage is discouraged.) |
Ref | Expression |
---|---|
hstcl | ⊢ ((𝑆 ∈ CHStates ∧ 𝐴 ∈ Cℋ ) → (𝑆‘𝐴) ∈ ℋ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ishst 29073 | . . 3 ⊢ (𝑆 ∈ CHStates ↔ (𝑆: Cℋ ⟶ ℋ ∧ (normℎ‘(𝑆‘ ℋ)) = 1 ∧ ∀𝑥 ∈ Cℋ ∀𝑦 ∈ Cℋ (𝑥 ⊆ (⊥‘𝑦) → (((𝑆‘𝑥) ·ih (𝑆‘𝑦)) = 0 ∧ (𝑆‘(𝑥 ∨ℋ 𝑦)) = ((𝑆‘𝑥) +ℎ (𝑆‘𝑦)))))) | |
2 | 1 | simp1bi 1076 | . 2 ⊢ (𝑆 ∈ CHStates → 𝑆: Cℋ ⟶ ℋ) |
3 | 2 | ffvelrnda 6359 | 1 ⊢ ((𝑆 ∈ CHStates ∧ 𝐴 ∈ Cℋ ) → (𝑆‘𝐴) ∈ ℋ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 384 = wceq 1483 ∈ wcel 1990 ∀wral 2912 ⊆ wss 3574 ⟶wf 5884 ‘cfv 5888 (class class class)co 6650 0cc0 9936 1c1 9937 ℋchil 27776 +ℎ cva 27777 ·ih csp 27779 normℎcno 27780 Cℋ cch 27786 ⊥cort 27787 ∨ℋ chj 27790 CHStateschst 27820 |
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-res 5126 df-ima 5127 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-sh 28064 df-ch 28078 df-hst 29071 |
This theorem is referenced by: hstnmoc 29082 hstle1 29085 hst1h 29086 hst0h 29087 hstpyth 29088 hstle 29089 hstles 29090 hstoh 29091 hstrlem6 29123 |
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