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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 29787 | . . 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 1126 | . 2 ⊢ (𝑆 ∈ CHStates → 𝑆: Cℋ ⟶ ℋ) |
3 | 2 | ffvelrnda 6674 | 1 ⊢ ((𝑆 ∈ CHStates ∧ 𝐴 ∈ Cℋ ) → (𝑆‘𝐴) ∈ ℋ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 387 = wceq 1508 ∈ wcel 2051 ∀wral 3081 ⊆ wss 3822 ⟶wf 6181 ‘cfv 6185 (class class class)co 6974 0cc0 10333 1c1 10334 ℋchba 28490 +ℎ cva 28491 ·ih csp 28493 normℎcno 28494 Cℋ cch 28500 ⊥cort 28501 ∨ℋ chj 28504 CHStateschst 28534 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1759 ax-4 1773 ax-5 1870 ax-6 1929 ax-7 1966 ax-8 2053 ax-9 2060 ax-10 2080 ax-11 2094 ax-12 2107 ax-13 2302 ax-ext 2743 ax-sep 5056 ax-nul 5063 ax-pow 5115 ax-pr 5182 ax-un 7277 ax-hilex 28570 |
This theorem depends on definitions: df-bi 199 df-an 388 df-or 835 df-3an 1071 df-tru 1511 df-ex 1744 df-nf 1748 df-sb 2017 df-mo 2548 df-eu 2585 df-clab 2752 df-cleq 2764 df-clel 2839 df-nfc 2911 df-ral 3086 df-rex 3087 df-rab 3090 df-v 3410 df-sbc 3675 df-dif 3825 df-un 3827 df-in 3829 df-ss 3836 df-nul 4173 df-if 4345 df-pw 4418 df-sn 4436 df-pr 4438 df-op 4442 df-uni 4709 df-br 4926 df-opab 4988 df-id 5308 df-xp 5409 df-rel 5410 df-cnv 5411 df-co 5412 df-dm 5413 df-rn 5414 df-res 5415 df-ima 5416 df-iota 6149 df-fun 6187 df-fn 6188 df-f 6189 df-fv 6193 df-ov 6977 df-oprab 6978 df-mpo 6979 df-map 8206 df-sh 28778 df-ch 28792 df-hst 29785 |
This theorem is referenced by: hstnmoc 29796 hstle1 29799 hst1h 29800 hst0h 29801 hstpyth 29802 hstle 29803 hstles 29804 hstoh 29805 hstrlem6 29837 |
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