Theorem hstel2 32251

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

Assertion

Ref	Expression
hstel2	⊢ (((𝑆 ∈ CHStates ∧ 𝐴 ∈ Cℋ ) ∧ (𝐵 ∈ Cℋ ∧ 𝐴 ⊆ (⊥‘𝐵))) → (((𝑆‘𝐴) ·ih (𝑆‘𝐵)) = 0 ∧ (𝑆‘(𝐴 ∨ℋ 𝐵)) = ((𝑆‘𝐴) +ℎ (𝑆‘𝐵))))

Proof of Theorem hstel2

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

Step	Hyp	Ref	Expression
1		ishst 32246	. . . 4 ⊢ (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	simp3bi 1147	. . 3 ⊢ (𝑆 ∈ CHStates → ∀𝑥 ∈ Cℋ ∀𝑦 ∈ Cℋ (𝑥 ⊆ (⊥‘𝑦) → (((𝑆‘𝑥) ·ih (𝑆‘𝑦)) = 0 ∧ (𝑆‘(𝑥 ∨ℋ 𝑦)) = ((𝑆‘𝑥) +ℎ (𝑆‘𝑦)))))
3	2	ad2antrr 725	. 2 ⊢ (((𝑆 ∈ CHStates ∧ 𝐴 ∈ Cℋ ) ∧ (𝐵 ∈ Cℋ ∧ 𝐴 ⊆ (⊥‘𝐵))) → ∀𝑥 ∈ Cℋ ∀𝑦 ∈ Cℋ (𝑥 ⊆ (⊥‘𝑦) → (((𝑆‘𝑥) ·ih (𝑆‘𝑦)) = 0 ∧ (𝑆‘(𝑥 ∨ℋ 𝑦)) = ((𝑆‘𝑥) +ℎ (𝑆‘𝑦)))))
4		sseq1 4034	. . . . . . 7 ⊢ (𝑥 = 𝐴 → (𝑥 ⊆ (⊥‘𝑦) ↔ 𝐴 ⊆ (⊥‘𝑦)))
5		fveq2 6920	. . . . . . . . . 10 ⊢ (𝑥 = 𝐴 → (𝑆‘𝑥) = (𝑆‘𝐴))
6	5	oveq1d 7463	. . . . . . . . 9 ⊢ (𝑥 = 𝐴 → ((𝑆‘𝑥) ·ih (𝑆‘𝑦)) = ((𝑆‘𝐴) ·ih (𝑆‘𝑦)))
7	6	eqeq1d 2742	. . . . . . . 8 ⊢ (𝑥 = 𝐴 → (((𝑆‘𝑥) ·ih (𝑆‘𝑦)) = 0 ↔ ((𝑆‘𝐴) ·ih (𝑆‘𝑦)) = 0))
8		fvoveq1 7471	. . . . . . . . 9 ⊢ (𝑥 = 𝐴 → (𝑆‘(𝑥 ∨ℋ 𝑦)) = (𝑆‘(𝐴 ∨ℋ 𝑦)))
9	5	oveq1d 7463	. . . . . . . . 9 ⊢ (𝑥 = 𝐴 → ((𝑆‘𝑥) +ℎ (𝑆‘𝑦)) = ((𝑆‘𝐴) +ℎ (𝑆‘𝑦)))
10	8, 9	eqeq12d 2756	. . . . . . . 8 ⊢ (𝑥 = 𝐴 → ((𝑆‘(𝑥 ∨ℋ 𝑦)) = ((𝑆‘𝑥) +ℎ (𝑆‘𝑦)) ↔ (𝑆‘(𝐴 ∨ℋ 𝑦)) = ((𝑆‘𝐴) +ℎ (𝑆‘𝑦))))
11	7, 10	anbi12d 631	. . . . . . 7 ⊢ (𝑥 = 𝐴 → ((((𝑆‘𝑥) ·ih (𝑆‘𝑦)) = 0 ∧ (𝑆‘(𝑥 ∨ℋ 𝑦)) = ((𝑆‘𝑥) +ℎ (𝑆‘𝑦))) ↔ (((𝑆‘𝐴) ·ih (𝑆‘𝑦)) = 0 ∧ (𝑆‘(𝐴 ∨ℋ 𝑦)) = ((𝑆‘𝐴) +ℎ (𝑆‘𝑦)))))
12	4, 11	imbi12d 344	. . . . . 6 ⊢ (𝑥 = 𝐴 → ((𝑥 ⊆ (⊥‘𝑦) → (((𝑆‘𝑥) ·ih (𝑆‘𝑦)) = 0 ∧ (𝑆‘(𝑥 ∨ℋ 𝑦)) = ((𝑆‘𝑥) +ℎ (𝑆‘𝑦)))) ↔ (𝐴 ⊆ (⊥‘𝑦) → (((𝑆‘𝐴) ·ih (𝑆‘𝑦)) = 0 ∧ (𝑆‘(𝐴 ∨ℋ 𝑦)) = ((𝑆‘𝐴) +ℎ (𝑆‘𝑦))))))
13		fveq2 6920	. . . . . . . 8 ⊢ (𝑦 = 𝐵 → (⊥‘𝑦) = (⊥‘𝐵))
14	13	sseq2d 4041	. . . . . . 7 ⊢ (𝑦 = 𝐵 → (𝐴 ⊆ (⊥‘𝑦) ↔ 𝐴 ⊆ (⊥‘𝐵)))
15		fveq2 6920	. . . . . . . . . 10 ⊢ (𝑦 = 𝐵 → (𝑆‘𝑦) = (𝑆‘𝐵))
16	15	oveq2d 7464	. . . . . . . . 9 ⊢ (𝑦 = 𝐵 → ((𝑆‘𝐴) ·ih (𝑆‘𝑦)) = ((𝑆‘𝐴) ·ih (𝑆‘𝐵)))
17	16	eqeq1d 2742	. . . . . . . 8 ⊢ (𝑦 = 𝐵 → (((𝑆‘𝐴) ·ih (𝑆‘𝑦)) = 0 ↔ ((𝑆‘𝐴) ·ih (𝑆‘𝐵)) = 0))
18		oveq2 7456	. . . . . . . . . 10 ⊢ (𝑦 = 𝐵 → (𝐴 ∨ℋ 𝑦) = (𝐴 ∨ℋ 𝐵))
19	18	fveq2d 6924	. . . . . . . . 9 ⊢ (𝑦 = 𝐵 → (𝑆‘(𝐴 ∨ℋ 𝑦)) = (𝑆‘(𝐴 ∨ℋ 𝐵)))
20	15	oveq2d 7464	. . . . . . . . 9 ⊢ (𝑦 = 𝐵 → ((𝑆‘𝐴) +ℎ (𝑆‘𝑦)) = ((𝑆‘𝐴) +ℎ (𝑆‘𝐵)))
21	19, 20	eqeq12d 2756	. . . . . . . 8 ⊢ (𝑦 = 𝐵 → ((𝑆‘(𝐴 ∨ℋ 𝑦)) = ((𝑆‘𝐴) +ℎ (𝑆‘𝑦)) ↔ (𝑆‘(𝐴 ∨ℋ 𝐵)) = ((𝑆‘𝐴) +ℎ (𝑆‘𝐵))))
22	17, 21	anbi12d 631	. . . . . . 7 ⊢ (𝑦 = 𝐵 → ((((𝑆‘𝐴) ·ih (𝑆‘𝑦)) = 0 ∧ (𝑆‘(𝐴 ∨ℋ 𝑦)) = ((𝑆‘𝐴) +ℎ (𝑆‘𝑦))) ↔ (((𝑆‘𝐴) ·ih (𝑆‘𝐵)) = 0 ∧ (𝑆‘(𝐴 ∨ℋ 𝐵)) = ((𝑆‘𝐴) +ℎ (𝑆‘𝐵)))))
23	14, 22	imbi12d 344	. . . . . 6 ⊢ (𝑦 = 𝐵 → ((𝐴 ⊆ (⊥‘𝑦) → (((𝑆‘𝐴) ·ih (𝑆‘𝑦)) = 0 ∧ (𝑆‘(𝐴 ∨ℋ 𝑦)) = ((𝑆‘𝐴) +ℎ (𝑆‘𝑦)))) ↔ (𝐴 ⊆ (⊥‘𝐵) → (((𝑆‘𝐴) ·ih (𝑆‘𝐵)) = 0 ∧ (𝑆‘(𝐴 ∨ℋ 𝐵)) = ((𝑆‘𝐴) +ℎ (𝑆‘𝐵))))))
24	12, 23	rspc2v 3646	. . . . 5 ⊢ ((𝐴 ∈ Cℋ ∧ 𝐵 ∈ Cℋ ) → (∀𝑥 ∈ Cℋ ∀𝑦 ∈ Cℋ (𝑥 ⊆ (⊥‘𝑦) → (((𝑆‘𝑥) ·ih (𝑆‘𝑦)) = 0 ∧ (𝑆‘(𝑥 ∨ℋ 𝑦)) = ((𝑆‘𝑥) +ℎ (𝑆‘𝑦)))) → (𝐴 ⊆ (⊥‘𝐵) → (((𝑆‘𝐴) ·ih (𝑆‘𝐵)) = 0 ∧ (𝑆‘(𝐴 ∨ℋ 𝐵)) = ((𝑆‘𝐴) +ℎ (𝑆‘𝐵))))))
25	24	com23 86	. . . 4 ⊢ ((𝐴 ∈ Cℋ ∧ 𝐵 ∈ Cℋ ) → (𝐴 ⊆ (⊥‘𝐵) → (∀𝑥 ∈ Cℋ ∀𝑦 ∈ Cℋ (𝑥 ⊆ (⊥‘𝑦) → (((𝑆‘𝑥) ·ih (𝑆‘𝑦)) = 0 ∧ (𝑆‘(𝑥 ∨ℋ 𝑦)) = ((𝑆‘𝑥) +ℎ (𝑆‘𝑦)))) → (((𝑆‘𝐴) ·ih (𝑆‘𝐵)) = 0 ∧ (𝑆‘(𝐴 ∨ℋ 𝐵)) = ((𝑆‘𝐴) +ℎ (𝑆‘𝐵))))))
26	25	impr 454	. . 3 ⊢ ((𝐴 ∈ Cℋ ∧ (𝐵 ∈ Cℋ ∧ 𝐴 ⊆ (⊥‘𝐵))) → (∀𝑥 ∈ Cℋ ∀𝑦 ∈ Cℋ (𝑥 ⊆ (⊥‘𝑦) → (((𝑆‘𝑥) ·ih (𝑆‘𝑦)) = 0 ∧ (𝑆‘(𝑥 ∨ℋ 𝑦)) = ((𝑆‘𝑥) +ℎ (𝑆‘𝑦)))) → (((𝑆‘𝐴) ·ih (𝑆‘𝐵)) = 0 ∧ (𝑆‘(𝐴 ∨ℋ 𝐵)) = ((𝑆‘𝐴) +ℎ (𝑆‘𝐵)))))
27	26	adantll 713	. 2 ⊢ (((𝑆 ∈ CHStates ∧ 𝐴 ∈ Cℋ ) ∧ (𝐵 ∈ Cℋ ∧ 𝐴 ⊆ (⊥‘𝐵))) → (∀𝑥 ∈ Cℋ ∀𝑦 ∈ Cℋ (𝑥 ⊆ (⊥‘𝑦) → (((𝑆‘𝑥) ·ih (𝑆‘𝑦)) = 0 ∧ (𝑆‘(𝑥 ∨ℋ 𝑦)) = ((𝑆‘𝑥) +ℎ (𝑆‘𝑦)))) → (((𝑆‘𝐴) ·ih (𝑆‘𝐵)) = 0 ∧ (𝑆‘(𝐴 ∨ℋ 𝐵)) = ((𝑆‘𝐴) +ℎ (𝑆‘𝐵)))))
28	3, 27	mpd 15	1 ⊢ (((𝑆 ∈ CHStates ∧ 𝐴 ∈ Cℋ ) ∧ (𝐵 ∈ Cℋ ∧ 𝐴 ⊆ (⊥‘𝐵))) → (((𝑆‘𝐴) ·ih (𝑆‘𝐵)) = 0 ∧ (𝑆‘(𝐴 ∨ℋ 𝐵)) = ((𝑆‘𝐴) +ℎ (𝑆‘𝐵))))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1537   ∈ wcel 2108  ∀wral 3067   ⊆ wss 3976  ⟶wf 6569  ‘cfv 6573  (class class class)co 7448  0cc0 11184  1c1 11185   ℋchba 30951   +_ℎ cva 30952   ·_ih csp 30954  norm_ℎcno 30955   C_ℋ cch 30961  ⊥cort 30962   ∨_ℋ chj 30965  CHStateschst 30995

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-sep 5317  ax-nul 5324  ax-pow 5383  ax-pr 5447  ax-un 7770  ax-hilex 31031

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ral 3068  df-rex 3077  df-rab 3444  df-v 3490  df-sbc 3805  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-nul 4353  df-if 4549  df-pw 4624  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-br 5167  df-opab 5229  df-id 5593  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-rn 5711  df-res 5712  df-ima 5713  df-iota 6525  df-fun 6575  df-fn 6576  df-f 6577  df-fv 6581  df-ov 7451  df-oprab 7452  df-mpo 7453  df-map 8886  df-sh 31239  df-ch 31253  df-hst 32244

This theorem is referenced by:  hstorth  32252  hstosum  32253