Theorem hstel2 32247

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 32242	. . . 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 1146	. . 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 4020	. . . . . . 7 ⊢ (𝑥 = 𝐴 → (𝑥 ⊆ (⊥‘𝑦) ↔ 𝐴 ⊆ (⊥‘𝑦)))
5		fveq2 6906	. . . . . . . . . 10 ⊢ (𝑥 = 𝐴 → (𝑆‘𝑥) = (𝑆‘𝐴))
6	5	oveq1d 7445	. . . . . . . . 9 ⊢ (𝑥 = 𝐴 → ((𝑆‘𝑥) ·ih (𝑆‘𝑦)) = ((𝑆‘𝐴) ·ih (𝑆‘𝑦)))
7	6	eqeq1d 2736	. . . . . . . 8 ⊢ (𝑥 = 𝐴 → (((𝑆‘𝑥) ·ih (𝑆‘𝑦)) = 0 ↔ ((𝑆‘𝐴) ·ih (𝑆‘𝑦)) = 0))
8		fvoveq1 7453	. . . . . . . . 9 ⊢ (𝑥 = 𝐴 → (𝑆‘(𝑥 ∨ℋ 𝑦)) = (𝑆‘(𝐴 ∨ℋ 𝑦)))
9	5	oveq1d 7445	. . . . . . . . 9 ⊢ (𝑥 = 𝐴 → ((𝑆‘𝑥) +ℎ (𝑆‘𝑦)) = ((𝑆‘𝐴) +ℎ (𝑆‘𝑦)))
10	8, 9	eqeq12d 2750	. . . . . . . 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 632	. . . . . . 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 6906	. . . . . . . 8 ⊢ (𝑦 = 𝐵 → (⊥‘𝑦) = (⊥‘𝐵))
14	13	sseq2d 4027	. . . . . . 7 ⊢ (𝑦 = 𝐵 → (𝐴 ⊆ (⊥‘𝑦) ↔ 𝐴 ⊆ (⊥‘𝐵)))
15		fveq2 6906	. . . . . . . . . 10 ⊢ (𝑦 = 𝐵 → (𝑆‘𝑦) = (𝑆‘𝐵))
16	15	oveq2d 7446	. . . . . . . . 9 ⊢ (𝑦 = 𝐵 → ((𝑆‘𝐴) ·ih (𝑆‘𝑦)) = ((𝑆‘𝐴) ·ih (𝑆‘𝐵)))
17	16	eqeq1d 2736	. . . . . . . 8 ⊢ (𝑦 = 𝐵 → (((𝑆‘𝐴) ·ih (𝑆‘𝑦)) = 0 ↔ ((𝑆‘𝐴) ·ih (𝑆‘𝐵)) = 0))
18		oveq2 7438	. . . . . . . . . 10 ⊢ (𝑦 = 𝐵 → (𝐴 ∨ℋ 𝑦) = (𝐴 ∨ℋ 𝐵))
19	18	fveq2d 6910	. . . . . . . . 9 ⊢ (𝑦 = 𝐵 → (𝑆‘(𝐴 ∨ℋ 𝑦)) = (𝑆‘(𝐴 ∨ℋ 𝐵)))
20	15	oveq2d 7446	. . . . . . . . 9 ⊢ (𝑦 = 𝐵 → ((𝑆‘𝐴) +ℎ (𝑆‘𝑦)) = ((𝑆‘𝐴) +ℎ (𝑆‘𝐵)))
21	19, 20	eqeq12d 2750	. . . . . . . 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 632	. . . . . . 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 3632	. . . . 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 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 395   = wceq 1536   ∈ wcel 2105  ∀wral 3058   ⊆ wss 3962  ⟶wf 6558  ‘cfv 6562  (class class class)co 7430  0cc0 11152  1c1 11153   ℋchba 30947   +_ℎ cva 30948   ·_ih csp 30950  norm_ℎcno 30951   C_ℋ cch 30957  ⊥cort 30958   ∨_ℋ chj 30961  CHStateschst 30991

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-10 2138  ax-11 2154  ax-12 2174  ax-ext 2705  ax-sep 5301  ax-nul 5311  ax-pow 5370  ax-pr 5437  ax-un 7753  ax-hilex 31027

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1539  df-fal 1549  df-ex 1776  df-nf 1780  df-sb 2062  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2726  df-clel 2813  df-nfc 2889  df-ral 3059  df-rex 3068  df-rab 3433  df-v 3479  df-sbc 3791  df-dif 3965  df-un 3967  df-in 3969  df-ss 3979  df-nul 4339  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4912  df-br 5148  df-opab 5210  df-id 5582  df-xp 5694  df-rel 5695  df-cnv 5696  df-co 5697  df-dm 5698  df-rn 5699  df-res 5700  df-ima 5701  df-iota 6515  df-fun 6564  df-fn 6565  df-f 6566  df-fv 6570  df-ov 7433  df-oprab 7434  df-mpo 7435  df-map 8866  df-sh 31235  df-ch 31249  df-hst 32240

This theorem is referenced by:  hstorth  32248  hstosum  32249