Theorem hstel2 32294

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 32289	. . . 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 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 3959	. . . . . . 7 ⊢ (𝑥 = 𝐴 → (𝑥 ⊆ (⊥‘𝑦) ↔ 𝐴 ⊆ (⊥‘𝑦)))
5		fveq2 6834	. . . . . . . . . 10 ⊢ (𝑥 = 𝐴 → (𝑆‘𝑥) = (𝑆‘𝐴))
6	5	oveq1d 7373	. . . . . . . . 9 ⊢ (𝑥 = 𝐴 → ((𝑆‘𝑥) ·ih (𝑆‘𝑦)) = ((𝑆‘𝐴) ·ih (𝑆‘𝑦)))
7	6	eqeq1d 2738	. . . . . . . 8 ⊢ (𝑥 = 𝐴 → (((𝑆‘𝑥) ·ih (𝑆‘𝑦)) = 0 ↔ ((𝑆‘𝐴) ·ih (𝑆‘𝑦)) = 0))
8		fvoveq1 7381	. . . . . . . . 9 ⊢ (𝑥 = 𝐴 → (𝑆‘(𝑥 ∨ℋ 𝑦)) = (𝑆‘(𝐴 ∨ℋ 𝑦)))
9	5	oveq1d 7373	. . . . . . . . 9 ⊢ (𝑥 = 𝐴 → ((𝑆‘𝑥) +ℎ (𝑆‘𝑦)) = ((𝑆‘𝐴) +ℎ (𝑆‘𝑦)))
10	8, 9	eqeq12d 2752	. . . . . . . 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 6834	. . . . . . . 8 ⊢ (𝑦 = 𝐵 → (⊥‘𝑦) = (⊥‘𝐵))
14	13	sseq2d 3966	. . . . . . 7 ⊢ (𝑦 = 𝐵 → (𝐴 ⊆ (⊥‘𝑦) ↔ 𝐴 ⊆ (⊥‘𝐵)))
15		fveq2 6834	. . . . . . . . . 10 ⊢ (𝑦 = 𝐵 → (𝑆‘𝑦) = (𝑆‘𝐵))
16	15	oveq2d 7374	. . . . . . . . 9 ⊢ (𝑦 = 𝐵 → ((𝑆‘𝐴) ·ih (𝑆‘𝑦)) = ((𝑆‘𝐴) ·ih (𝑆‘𝐵)))
17	16	eqeq1d 2738	. . . . . . . 8 ⊢ (𝑦 = 𝐵 → (((𝑆‘𝐴) ·ih (𝑆‘𝑦)) = 0 ↔ ((𝑆‘𝐴) ·ih (𝑆‘𝐵)) = 0))
18		oveq2 7366	. . . . . . . . . 10 ⊢ (𝑦 = 𝐵 → (𝐴 ∨ℋ 𝑦) = (𝐴 ∨ℋ 𝐵))
19	18	fveq2d 6838	. . . . . . . . 9 ⊢ (𝑦 = 𝐵 → (𝑆‘(𝐴 ∨ℋ 𝑦)) = (𝑆‘(𝐴 ∨ℋ 𝐵)))
20	15	oveq2d 7374	. . . . . . . . 9 ⊢ (𝑦 = 𝐵 → ((𝑆‘𝐴) +ℎ (𝑆‘𝑦)) = ((𝑆‘𝐴) +ℎ (𝑆‘𝐵)))
21	19, 20	eqeq12d 2752	. . . . . . . 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 3587	. . . . 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 1541   ∈ wcel 2113  ∀wral 3051   ⊆ wss 3901  ⟶wf 6488  ‘cfv 6492  (class class class)co 7358  0cc0 11026  1c1 11027   ℋchba 30994   +_ℎ cva 30995   ·_ih csp 30997  norm_ℎcno 30998   C_ℋ cch 31004  ⊥cort 31005   ∨_ℋ chj 31008  CHStateschst 31038

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2184  ax-ext 2708  ax-sep 5241  ax-nul 5251  ax-pow 5310  ax-pr 5377  ax-un 7680  ax-hilex 31074

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ral 3052  df-rex 3061  df-rab 3400  df-v 3442  df-sbc 3741  df-dif 3904  df-un 3906  df-in 3908  df-ss 3918  df-nul 4286  df-if 4480  df-pw 4556  df-sn 4581  df-pr 4583  df-op 4587  df-uni 4864  df-br 5099  df-opab 5161  df-id 5519  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-rn 5635  df-res 5636  df-ima 5637  df-iota 6448  df-fun 6494  df-fn 6495  df-f 6496  df-fv 6500  df-ov 7361  df-oprab 7362  df-mpo 7363  df-map 8765  df-sh 31282  df-ch 31296  df-hst 32287

This theorem is referenced by:  hstorth  32295  hstosum  32296