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Mirrors > Home > HSE Home > Th. List > sthil | Structured version Visualization version GIF version |
Description: The value of a state at the full Hilbert space. (Contributed by NM, 23-Oct-1999.) (New usage is discouraged.) |
Ref | Expression |
---|---|
sthil | ⊢ (𝑆 ∈ States → (𝑆‘ ℋ) = 1) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | isst 29993 | . 2 ⊢ (𝑆 ∈ States ↔ (𝑆: Cℋ ⟶(0[,]1) ∧ (𝑆‘ ℋ) = 1 ∧ ∀𝑥 ∈ Cℋ ∀𝑦 ∈ Cℋ (𝑥 ⊆ (⊥‘𝑦) → (𝑆‘(𝑥 ∨ℋ 𝑦)) = ((𝑆‘𝑥) + (𝑆‘𝑦))))) | |
2 | 1 | simp2bi 1142 | 1 ⊢ (𝑆 ∈ States → (𝑆‘ ℋ) = 1) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1536 ∈ wcel 2113 ∀wral 3141 ⊆ wss 3939 ⟶wf 6354 ‘cfv 6358 (class class class)co 7159 0cc0 10540 1c1 10541 + caddc 10543 [,]cicc 12744 ℋchba 28699 Cℋ cch 28709 ⊥cort 28710 ∨ℋ chj 28713 Statescst 28742 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2796 ax-sep 5206 ax-nul 5213 ax-pow 5269 ax-pr 5333 ax-un 7464 ax-hilex 28779 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-mo 2621 df-eu 2653 df-clab 2803 df-cleq 2817 df-clel 2896 df-nfc 2966 df-ral 3146 df-rex 3147 df-rab 3150 df-v 3499 df-sbc 3776 df-dif 3942 df-un 3944 df-in 3946 df-ss 3955 df-nul 4295 df-if 4471 df-pw 4544 df-sn 4571 df-pr 4573 df-op 4577 df-uni 4842 df-br 5070 df-opab 5132 df-id 5463 df-xp 5564 df-rel 5565 df-cnv 5566 df-co 5567 df-dm 5568 df-rn 5569 df-res 5570 df-ima 5571 df-iota 6317 df-fun 6360 df-fn 6361 df-f 6362 df-fv 6366 df-ov 7162 df-oprab 7163 df-mpo 7164 df-map 8411 df-sh 28987 df-ch 29001 df-st 29991 |
This theorem is referenced by: sto1i 30016 st0 30029 |
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