Theorem hstel2 30300

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 30295	. . . 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 1149	. . 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 726	. 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 3926	. . . . . . 7 ⊢ (𝑥 = 𝐴 → (𝑥 ⊆ (⊥‘𝑦) ↔ 𝐴 ⊆ (⊥‘𝑦)))
5		fveq2 6717	. . . . . . . . . 10 ⊢ (𝑥 = 𝐴 → (𝑆‘𝑥) = (𝑆‘𝐴))
6	5	oveq1d 7228	. . . . . . . . 9 ⊢ (𝑥 = 𝐴 → ((𝑆‘𝑥) ·ih (𝑆‘𝑦)) = ((𝑆‘𝐴) ·ih (𝑆‘𝑦)))
7	6	eqeq1d 2739	. . . . . . . 8 ⊢ (𝑥 = 𝐴 → (((𝑆‘𝑥) ·ih (𝑆‘𝑦)) = 0 ↔ ((𝑆‘𝐴) ·ih (𝑆‘𝑦)) = 0))
8		fvoveq1 7236	. . . . . . . . 9 ⊢ (𝑥 = 𝐴 → (𝑆‘(𝑥 ∨ℋ 𝑦)) = (𝑆‘(𝐴 ∨ℋ 𝑦)))
9	5	oveq1d 7228	. . . . . . . . 9 ⊢ (𝑥 = 𝐴 → ((𝑆‘𝑥) +ℎ (𝑆‘𝑦)) = ((𝑆‘𝐴) +ℎ (𝑆‘𝑦)))
10	8, 9	eqeq12d 2753	. . . . . . . 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 634	. . . . . . 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 348	. . . . . 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 6717	. . . . . . . 8 ⊢ (𝑦 = 𝐵 → (⊥‘𝑦) = (⊥‘𝐵))
14	13	sseq2d 3933	. . . . . . 7 ⊢ (𝑦 = 𝐵 → (𝐴 ⊆ (⊥‘𝑦) ↔ 𝐴 ⊆ (⊥‘𝐵)))
15		fveq2 6717	. . . . . . . . . 10 ⊢ (𝑦 = 𝐵 → (𝑆‘𝑦) = (𝑆‘𝐵))
16	15	oveq2d 7229	. . . . . . . . 9 ⊢ (𝑦 = 𝐵 → ((𝑆‘𝐴) ·ih (𝑆‘𝑦)) = ((𝑆‘𝐴) ·ih (𝑆‘𝐵)))
17	16	eqeq1d 2739	. . . . . . . 8 ⊢ (𝑦 = 𝐵 → (((𝑆‘𝐴) ·ih (𝑆‘𝑦)) = 0 ↔ ((𝑆‘𝐴) ·ih (𝑆‘𝐵)) = 0))
18		oveq2 7221	. . . . . . . . . 10 ⊢ (𝑦 = 𝐵 → (𝐴 ∨ℋ 𝑦) = (𝐴 ∨ℋ 𝐵))
19	18	fveq2d 6721	. . . . . . . . 9 ⊢ (𝑦 = 𝐵 → (𝑆‘(𝐴 ∨ℋ 𝑦)) = (𝑆‘(𝐴 ∨ℋ 𝐵)))
20	15	oveq2d 7229	. . . . . . . . 9 ⊢ (𝑦 = 𝐵 → ((𝑆‘𝐴) +ℎ (𝑆‘𝑦)) = ((𝑆‘𝐴) +ℎ (𝑆‘𝐵)))
21	19, 20	eqeq12d 2753	. . . . . . . 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 634	. . . . . . 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 348	. . . . . 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 3547	. . . . 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 458	. . 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 714	. 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 399   = wceq 1543   ∈ wcel 2110  ∀wral 3061   ⊆ wss 3866  ⟶wf 6376  ‘cfv 6380  (class class class)co 7213  0cc0 10729  1c1 10730   ℋchba 29000   +_ℎ cva 29001   ·_ih csp 29003  norm_ℎcno 29004   C_ℋ cch 29010  ⊥cort 29011   ∨_ℋ chj 29014  CHStateschst 29044

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2158  ax-12 2175  ax-ext 2708  ax-sep 5192  ax-nul 5199  ax-pow 5258  ax-pr 5322  ax-un 7523  ax-hilex 29080

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-nf 1792  df-sb 2071  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2886  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3410  df-sbc 3695  df-dif 3869  df-un 3871  df-in 3873  df-ss 3883  df-nul 4238  df-if 4440  df-pw 4515  df-sn 4542  df-pr 4544  df-op 4548  df-uni 4820  df-br 5054  df-opab 5116  df-id 5455  df-xp 5557  df-rel 5558  df-cnv 5559  df-co 5560  df-dm 5561  df-rn 5562  df-res 5563  df-ima 5564  df-iota 6338  df-fun 6382  df-fn 6383  df-f 6384  df-fv 6388  df-ov 7216  df-oprab 7217  df-mpo 7218  df-map 8510  df-sh 29288  df-ch 29302  df-hst 30293

This theorem is referenced by:  hstorth  30301  hstosum  30302