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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 29993 | . 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 1142 | 1 ⊢ (𝑆 ∈ CHStates → (normℎ‘(𝑆‘ ℋ)) = 1) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1537 ∈ wcel 2114 ∀wral 3140 ⊆ wss 3938 ⟶wf 6353 ‘cfv 6357 (class class class)co 7158 0cc0 10539 1c1 10540 ℋchba 28698 +ℎ cva 28699 ·ih csp 28701 normℎcno 28702 Cℋ cch 28708 ⊥cort 28709 ∨ℋ chj 28712 CHStateschst 28742 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 ax-hilex 28778 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ral 3145 df-rex 3146 df-rab 3149 df-v 3498 df-sbc 3775 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-op 4576 df-uni 4841 df-br 5069 df-opab 5131 df-id 5462 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-fv 6365 df-ov 7161 df-oprab 7162 df-mpo 7163 df-map 8410 df-sh 28986 df-ch 29000 df-hst 29991 |
This theorem is referenced by: hstnmoc 30002 hst1h 30006 |
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