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Mirrors > Home > HSE Home > Th. List > hst1a | Structured version Visualization version GIF version |
Description: Unit value of a Hilbert-space-valued state. (Contributed by NM, 25-Jun-2006.) (New usage is discouraged.) |
Ref | Expression |
---|---|
hst1a | ⊢ (𝑆 ∈ CHStates → (normℎ‘(𝑆‘ ℋ)) = 1) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ishst 32044 | . 2 ⊢ (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 | simp2bi 1143 | 1 ⊢ (𝑆 ∈ CHStates → (normℎ‘(𝑆‘ ℋ)) = 1) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 394 = wceq 1533 ∈ wcel 2098 ∀wral 3058 ⊆ wss 3949 ⟶wf 6549 ‘cfv 6553 (class class class)co 7426 0cc0 11146 1c1 11147 ℋchba 30749 +ℎ cva 30750 ·ih csp 30752 normℎcno 30753 Cℋ cch 30759 ⊥cort 30760 ∨ℋ chj 30763 CHStateschst 30793 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2699 ax-sep 5303 ax-nul 5310 ax-pow 5369 ax-pr 5433 ax-un 7746 ax-hilex 30829 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2529 df-eu 2558 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ral 3059 df-rex 3068 df-rab 3431 df-v 3475 df-sbc 3779 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-nul 4327 df-if 4533 df-pw 4608 df-sn 4633 df-pr 4635 df-op 4639 df-uni 4913 df-br 5153 df-opab 5215 df-id 5580 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-rn 5693 df-res 5694 df-ima 5695 df-iota 6505 df-fun 6555 df-fn 6556 df-f 6557 df-fv 6561 df-ov 7429 df-oprab 7430 df-mpo 7431 df-map 8853 df-sh 31037 df-ch 31051 df-hst 32042 |
This theorem is referenced by: hstnmoc 32053 hst1h 32057 |
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