Theorem hst1a 30101

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 30097	. 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 1144	1 ⊢ (𝑆 ∈ CHStates → (normℎ‘(𝑆‘ ℋ)) = 1)

Syntax hints:   → wi 4   ∧ wa 400   = wceq 1539   ∈ wcel 2112  ∀wral 3071   ⊆ wss 3859  ⟶wf 6332  ‘cfv 6336  (class class class)co 7151  0cc0 10576  1c1 10577   ℋchba 28802   +_ℎ cva 28803   ·_ih csp 28805  norm_ℎcno 28806   C_ℋ cch 28812  ⊥cort 28813   ∨_ℋ chj 28816  CHStateschst 28846

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2730  ax-sep 5170  ax-nul 5177  ax-pow 5235  ax-pr 5299  ax-un 7460  ax-hilex 28882

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 846  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2071  df-mo 2558  df-eu 2589  df-clab 2737  df-cleq 2751  df-clel 2831  df-nfc 2902  df-ral 3076  df-rex 3077  df-rab 3080  df-v 3412  df-sbc 3698  df-dif 3862  df-un 3864  df-in 3866  df-ss 3876  df-nul 4227  df-if 4422  df-pw 4497  df-sn 4524  df-pr 4526  df-op 4530  df-uni 4800  df-br 5034  df-opab 5096  df-id 5431  df-xp 5531  df-rel 5532  df-cnv 5533  df-co 5534  df-dm 5535  df-rn 5536  df-res 5537  df-ima 5538  df-iota 6295  df-fun 6338  df-fn 6339  df-f 6340  df-fv 6344  df-ov 7154  df-oprab 7155  df-mpo 7156  df-map 8419  df-sh 29090  df-ch 29104  df-hst 30095

This theorem is referenced by:  hstnmoc  30106  hst1h  30110