Theorem hstel2 32308

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 32303	. . . 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 1153	. . 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 732	. 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 3940	. . . . . . 7 ⊢ (𝑥 = 𝐴 → (𝑥 ⊆ (⊥‘𝑦) ↔ 𝐴 ⊆ (⊥‘𝑦)))
5		fveq2 6827	. . . . . . . . . 10 ⊢ (𝑥 = 𝐴 → (𝑆‘𝑥) = (𝑆‘𝐴))
6	5	oveq1d 7371	. . . . . . . . 9 ⊢ (𝑥 = 𝐴 → ((𝑆‘𝑥) ·ih (𝑆‘𝑦)) = ((𝑆‘𝐴) ·ih (𝑆‘𝑦)))
7	6	eqeq1d 2741	. . . . . . . 8 ⊢ (𝑥 = 𝐴 → (((𝑆‘𝑥) ·ih (𝑆‘𝑦)) = 0 ↔ ((𝑆‘𝐴) ·ih (𝑆‘𝑦)) = 0))
8		fvoveq1 7379	. . . . . . . . 9 ⊢ (𝑥 = 𝐴 → (𝑆‘(𝑥 ∨ℋ 𝑦)) = (𝑆‘(𝐴 ∨ℋ 𝑦)))
9	5	oveq1d 7371	. . . . . . . . 9 ⊢ (𝑥 = 𝐴 → ((𝑆‘𝑥) +ℎ (𝑆‘𝑦)) = ((𝑆‘𝐴) +ℎ (𝑆‘𝑦)))
10	8, 9	eqeq12d 2755	. . . . . . . 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 638	. . . . . . 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 345	. . . . . 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 6827	. . . . . . . 8 ⊢ (𝑦 = 𝐵 → (⊥‘𝑦) = (⊥‘𝐵))
14	13	sseq2d 3947	. . . . . . 7 ⊢ (𝑦 = 𝐵 → (𝐴 ⊆ (⊥‘𝑦) ↔ 𝐴 ⊆ (⊥‘𝐵)))
15		fveq2 6827	. . . . . . . . . 10 ⊢ (𝑦 = 𝐵 → (𝑆‘𝑦) = (𝑆‘𝐵))
16	15	oveq2d 7372	. . . . . . . . 9 ⊢ (𝑦 = 𝐵 → ((𝑆‘𝐴) ·ih (𝑆‘𝑦)) = ((𝑆‘𝐴) ·ih (𝑆‘𝐵)))
17	16	eqeq1d 2741	. . . . . . . 8 ⊢ (𝑦 = 𝐵 → (((𝑆‘𝐴) ·ih (𝑆‘𝑦)) = 0 ↔ ((𝑆‘𝐴) ·ih (𝑆‘𝐵)) = 0))
18		oveq2 7364	. . . . . . . . . 10 ⊢ (𝑦 = 𝐵 → (𝐴 ∨ℋ 𝑦) = (𝐴 ∨ℋ 𝐵))
19	18	fveq2d 6831	. . . . . . . . 9 ⊢ (𝑦 = 𝐵 → (𝑆‘(𝐴 ∨ℋ 𝑦)) = (𝑆‘(𝐴 ∨ℋ 𝐵)))
20	15	oveq2d 7372	. . . . . . . . 9 ⊢ (𝑦 = 𝐵 → ((𝑆‘𝐴) +ℎ (𝑆‘𝑦)) = ((𝑆‘𝐴) +ℎ (𝑆‘𝐵)))
21	19, 20	eqeq12d 2755	. . . . . . . 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 638	. . . . . . 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 345	. . . . . 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 3571	. . . . 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 455	. . 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 720	. 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 396   = wceq 1547   ∈ wcel 2119  ∀wral 3053   ⊆ wss 3883  ⟶wf 6481  ‘cfv 6485  (class class class)co 7356  0cc0 11029  1c1 11030   ℋchba 31008   +_ℎ cva 31009   ·_ih csp 31011  norm_ℎcno 31012   C_ℋ cch 31018  ⊥cort 31019   ∨_ℋ chj 31022  CHStateschst 31052

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-10 2152  ax-11 2168  ax-12 2189  ax-ext 2711  ax-sep 5218  ax-pow 5294  ax-pr 5362  ax-un 7678  ax-hilex 31088

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2543  df-eu 2573  df-clab 2718  df-cleq 2731  df-clel 2814  df-nfc 2888  df-ral 3054  df-rex 3064  df-rab 3392  df-v 3433  df-sbc 3724  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4262  df-if 4455  df-pw 4531  df-sn 4556  df-pr 4558  df-op 4562  df-uni 4839  df-br 5073  df-opab 5135  df-id 5513  df-xp 5624  df-rel 5625  df-cnv 5626  df-co 5627  df-dm 5628  df-rn 5629  df-res 5630  df-ima 5631  df-iota 6441  df-fun 6487  df-fn 6488  df-f 6489  df-fv 6493  df-ov 7359  df-oprab 7360  df-mpo 7361  df-map 8765  df-sh 31296  df-ch 31310  df-hst 32301

This theorem is referenced by:  hstorth  32309  hstosum  32310