Theorem ishst 30477

Description: Property of a complex Hilbert-space-valued state. Definition of CH-states in [Mayet3] p. 9. (Contributed by NM, 25-Jun-2006.) (New usage is discouraged.)

Assertion

Ref	Expression
ishst	⊢ (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))))))

Distinct variable group:   𝑥,𝑦,𝑆

Proof of Theorem ishst

Dummy variable 𝑓 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		ax-hilex 29262	. . . 4 ⊢ ℋ ∈ V
2		chex 29489	. . . 4 ⊢ Cℋ ∈ V
3	1, 2	elmap 8617	. . 3 ⊢ (𝑆 ∈ ( ℋ ↑m Cℋ ) ↔ 𝑆: Cℋ ⟶ ℋ)
4	3	anbi1i 623	. 2 ⊢ ((𝑆 ∈ ( ℋ ↑m Cℋ ) ∧ ((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)))))) ↔ (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)))))))
5		fveq1 6755	. . . . 5 ⊢ (𝑓 = 𝑆 → (𝑓‘ ℋ) = (𝑆‘ ℋ))
6	5	fveqeq2d 6764	. . . 4 ⊢ (𝑓 = 𝑆 → ((normℎ‘(𝑓‘ ℋ)) = 1 ↔ (normℎ‘(𝑆‘ ℋ)) = 1))
7		fveq1 6755	. . . . . . . . 9 ⊢ (𝑓 = 𝑆 → (𝑓‘𝑥) = (𝑆‘𝑥))
8		fveq1 6755	. . . . . . . . 9 ⊢ (𝑓 = 𝑆 → (𝑓‘𝑦) = (𝑆‘𝑦))
9	7, 8	oveq12d 7273	. . . . . . . 8 ⊢ (𝑓 = 𝑆 → ((𝑓‘𝑥) ·ih (𝑓‘𝑦)) = ((𝑆‘𝑥) ·ih (𝑆‘𝑦)))
10	9	eqeq1d 2740	. . . . . . 7 ⊢ (𝑓 = 𝑆 → (((𝑓‘𝑥) ·ih (𝑓‘𝑦)) = 0 ↔ ((𝑆‘𝑥) ·ih (𝑆‘𝑦)) = 0))
11		fveq1 6755	. . . . . . . 8 ⊢ (𝑓 = 𝑆 → (𝑓‘(𝑥 ∨ℋ 𝑦)) = (𝑆‘(𝑥 ∨ℋ 𝑦)))
12	7, 8	oveq12d 7273	. . . . . . . 8 ⊢ (𝑓 = 𝑆 → ((𝑓‘𝑥) +ℎ (𝑓‘𝑦)) = ((𝑆‘𝑥) +ℎ (𝑆‘𝑦)))
13	11, 12	eqeq12d 2754	. . . . . . 7 ⊢ (𝑓 = 𝑆 → ((𝑓‘(𝑥 ∨ℋ 𝑦)) = ((𝑓‘𝑥) +ℎ (𝑓‘𝑦)) ↔ (𝑆‘(𝑥 ∨ℋ 𝑦)) = ((𝑆‘𝑥) +ℎ (𝑆‘𝑦))))
14	10, 13	anbi12d 630	. . . . . 6 ⊢ (𝑓 = 𝑆 → ((((𝑓‘𝑥) ·ih (𝑓‘𝑦)) = 0 ∧ (𝑓‘(𝑥 ∨ℋ 𝑦)) = ((𝑓‘𝑥) +ℎ (𝑓‘𝑦))) ↔ (((𝑆‘𝑥) ·ih (𝑆‘𝑦)) = 0 ∧ (𝑆‘(𝑥 ∨ℋ 𝑦)) = ((𝑆‘𝑥) +ℎ (𝑆‘𝑦)))))
15	14	imbi2d 340	. . . . 5 ⊢ (𝑓 = 𝑆 → ((𝑥 ⊆ (⊥‘𝑦) → (((𝑓‘𝑥) ·ih (𝑓‘𝑦)) = 0 ∧ (𝑓‘(𝑥 ∨ℋ 𝑦)) = ((𝑓‘𝑥) +ℎ (𝑓‘𝑦)))) ↔ (𝑥 ⊆ (⊥‘𝑦) → (((𝑆‘𝑥) ·ih (𝑆‘𝑦)) = 0 ∧ (𝑆‘(𝑥 ∨ℋ 𝑦)) = ((𝑆‘𝑥) +ℎ (𝑆‘𝑦))))))
16	15	2ralbidv 3122	. . . 4 ⊢ (𝑓 = 𝑆 → (∀𝑥 ∈ Cℋ ∀𝑦 ∈ Cℋ (𝑥 ⊆ (⊥‘𝑦) → (((𝑓‘𝑥) ·ih (𝑓‘𝑦)) = 0 ∧ (𝑓‘(𝑥 ∨ℋ 𝑦)) = ((𝑓‘𝑥) +ℎ (𝑓‘𝑦)))) ↔ ∀𝑥 ∈ Cℋ ∀𝑦 ∈ Cℋ (𝑥 ⊆ (⊥‘𝑦) → (((𝑆‘𝑥) ·ih (𝑆‘𝑦)) = 0 ∧ (𝑆‘(𝑥 ∨ℋ 𝑦)) = ((𝑆‘𝑥) +ℎ (𝑆‘𝑦))))))
17	6, 16	anbi12d 630	. . 3 ⊢ (𝑓 = 𝑆 → (((normℎ‘(𝑓‘ ℋ)) = 1 ∧ ∀𝑥 ∈ Cℋ ∀𝑦 ∈ Cℋ (𝑥 ⊆ (⊥‘𝑦) → (((𝑓‘𝑥) ·ih (𝑓‘𝑦)) = 0 ∧ (𝑓‘(𝑥 ∨ℋ 𝑦)) = ((𝑓‘𝑥) +ℎ (𝑓‘𝑦))))) ↔ ((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)))))))
18		df-hst 30475	. . 3 ⊢ CHStates = {𝑓 ∈ ( ℋ ↑m Cℋ ) ∣ ((normℎ‘(𝑓‘ ℋ)) = 1 ∧ ∀𝑥 ∈ Cℋ ∀𝑦 ∈ Cℋ (𝑥 ⊆ (⊥‘𝑦) → (((𝑓‘𝑥) ·ih (𝑓‘𝑦)) = 0 ∧ (𝑓‘(𝑥 ∨ℋ 𝑦)) = ((𝑓‘𝑥) +ℎ (𝑓‘𝑦)))))}
19	17, 18	elrab2 3620	. 2 ⊢ (𝑆 ∈ CHStates ↔ (𝑆 ∈ ( ℋ ↑m Cℋ ) ∧ ((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)))))))
20		3anass 1093	. 2 ⊢ ((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))))) ↔ (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)))))))
21	4, 19, 20	3bitr4i 302	1 ⊢ (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))))))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 395   ∧ w3a 1085   = wceq 1539   ∈ wcel 2108  ∀wral 3063   ⊆ wss 3883  ⟶wf 6414  ‘cfv 6418  (class class class)co 7255   ↑_m cmap 8573  0cc0 10802  1c1 10803   ℋchba 29182   +_ℎ cva 29183   ·_ih csp 29185  norm_ℎcno 29186   C_ℋ cch 29192  ⊥cort 29193   ∨_ℋ chj 29196  CHStateschst 29226

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709  ax-sep 5218  ax-nul 5225  ax-pow 5283  ax-pr 5347  ax-un 7566  ax-hilex 29262

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2888  df-ral 3068  df-rex 3069  df-rab 3072  df-v 3424  df-sbc 3712  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4254  df-if 4457  df-pw 4532  df-sn 4559  df-pr 4561  df-op 4565  df-uni 4837  df-br 5071  df-opab 5133  df-id 5480  df-xp 5586  df-rel 5587  df-cnv 5588  df-co 5589  df-dm 5590  df-rn 5591  df-res 5592  df-ima 5593  df-iota 6376  df-fun 6420  df-fn 6421  df-f 6422  df-fv 6426  df-ov 7258  df-oprab 7259  df-mpo 7260  df-map 8575  df-sh 29470  df-ch 29484  df-hst 30475

This theorem is referenced by:  hstcl  30480  hst1a  30481  hstel2  30482  hstrlem3a  30523