Theorem hst1a 32246

Description: Unit value of a Hilbert-space-valued state. (Contributed by NM, 25-Jun-2006.) (New usage is discouraged.)

Assertion

Ref	Expression
hst1a	⊢ (𝑆 ∈ CHStates → (normℎ‘(𝑆‘ ℋ)) = 1)

Proof of Theorem hst1a

Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		ishst 32242	. 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 1145	1 ⊢ (𝑆 ∈ CHStates → (normℎ‘(𝑆‘ ℋ)) = 1)

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1536   ∈ wcel 2105  ∀wral 3058   ⊆ wss 3962  ⟶wf 6558  ‘cfv 6562  (class class class)co 7430  0cc0 11152  1c1 11153   ℋchba 30947   +_ℎ cva 30948   ·_ih csp 30950  norm_ℎcno 30951   C_ℋ cch 30957  ⊥cort 30958   ∨_ℋ chj 30961  CHStateschst 30991

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-10 2138  ax-11 2154  ax-12 2174  ax-ext 2705  ax-sep 5301  ax-nul 5311  ax-pow 5370  ax-pr 5437  ax-un 7753  ax-hilex 31027

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1539  df-fal 1549  df-ex 1776  df-nf 1780  df-sb 2062  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2726  df-clel 2813  df-nfc 2889  df-ral 3059  df-rex 3068  df-rab 3433  df-v 3479  df-sbc 3791  df-dif 3965  df-un 3967  df-in 3969  df-ss 3979  df-nul 4339  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4912  df-br 5148  df-opab 5210  df-id 5582  df-xp 5694  df-rel 5695  df-cnv 5696  df-co 5697  df-dm 5698  df-rn 5699  df-res 5700  df-ima 5701  df-iota 6515  df-fun 6564  df-fn 6565  df-f 6566  df-fv 6570  df-ov 7433  df-oprab 7434  df-mpo 7435  df-map 8866  df-sh 31235  df-ch 31249  df-hst 32240

This theorem is referenced by:  hstnmoc  32251  hst1h  32255