Theorem hoadddir 31790

Description: Scalar product reverse distributive law for Hilbert space operators. (Contributed by NM, 25-Aug-2006.) (New usage is discouraged.)

Assertion

Ref	Expression
hoadddir	⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝑇: ℋ⟶ ℋ) → ((𝐴 + 𝐵) ·op 𝑇) = ((𝐴 ·op 𝑇) +op (𝐵 ·op 𝑇)))

Proof of Theorem hoadddir

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		addcl 11216	. . . . . . . 8 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (𝐴 + 𝐵) ∈ ℂ)
2	1	anim1i 615	. . . . . . 7 ⊢ (((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) ∧ 𝑇: ℋ⟶ ℋ) → ((𝐴 + 𝐵) ∈ ℂ ∧ 𝑇: ℋ⟶ ℋ))
3	2	3impa 1109	. . . . . 6 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝑇: ℋ⟶ ℋ) → ((𝐴 + 𝐵) ∈ ℂ ∧ 𝑇: ℋ⟶ ℋ))
4		homval 31727	. . . . . . 7 ⊢ (((𝐴 + 𝐵) ∈ ℂ ∧ 𝑇: ℋ⟶ ℋ ∧ 𝑥 ∈ ℋ) → (((𝐴 + 𝐵) ·op 𝑇)‘𝑥) = ((𝐴 + 𝐵) ·ℎ (𝑇‘𝑥)))
5	4	3expa 1118	. . . . . 6 ⊢ ((((𝐴 + 𝐵) ∈ ℂ ∧ 𝑇: ℋ⟶ ℋ) ∧ 𝑥 ∈ ℋ) → (((𝐴 + 𝐵) ·op 𝑇)‘𝑥) = ((𝐴 + 𝐵) ·ℎ (𝑇‘𝑥)))
6	3, 5	sylan 580	. . . . 5 ⊢ (((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝑇: ℋ⟶ ℋ) ∧ 𝑥 ∈ ℋ) → (((𝐴 + 𝐵) ·op 𝑇)‘𝑥) = ((𝐴 + 𝐵) ·ℎ (𝑇‘𝑥)))
7		homval 31727	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℂ ∧ 𝑇: ℋ⟶ ℋ ∧ 𝑥 ∈ ℋ) → ((𝐴 ·op 𝑇)‘𝑥) = (𝐴 ·ℎ (𝑇‘𝑥)))
8	7	3expa 1118	. . . . . . . 8 ⊢ (((𝐴 ∈ ℂ ∧ 𝑇: ℋ⟶ ℋ) ∧ 𝑥 ∈ ℋ) → ((𝐴 ·op 𝑇)‘𝑥) = (𝐴 ·ℎ (𝑇‘𝑥)))
9	8	3adantl2 1168	. . . . . . 7 ⊢ (((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝑇: ℋ⟶ ℋ) ∧ 𝑥 ∈ ℋ) → ((𝐴 ·op 𝑇)‘𝑥) = (𝐴 ·ℎ (𝑇‘𝑥)))
10		homval 31727	. . . . . . . . 9 ⊢ ((𝐵 ∈ ℂ ∧ 𝑇: ℋ⟶ ℋ ∧ 𝑥 ∈ ℋ) → ((𝐵 ·op 𝑇)‘𝑥) = (𝐵 ·ℎ (𝑇‘𝑥)))
11	10	3expa 1118	. . . . . . . 8 ⊢ (((𝐵 ∈ ℂ ∧ 𝑇: ℋ⟶ ℋ) ∧ 𝑥 ∈ ℋ) → ((𝐵 ·op 𝑇)‘𝑥) = (𝐵 ·ℎ (𝑇‘𝑥)))
12	11	3adantl1 1167	. . . . . . 7 ⊢ (((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝑇: ℋ⟶ ℋ) ∧ 𝑥 ∈ ℋ) → ((𝐵 ·op 𝑇)‘𝑥) = (𝐵 ·ℎ (𝑇‘𝑥)))
13	9, 12	oveq12d 7428	. . . . . 6 ⊢ (((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝑇: ℋ⟶ ℋ) ∧ 𝑥 ∈ ℋ) → (((𝐴 ·op 𝑇)‘𝑥) +ℎ ((𝐵 ·op 𝑇)‘𝑥)) = ((𝐴 ·ℎ (𝑇‘𝑥)) +ℎ (𝐵 ·ℎ (𝑇‘𝑥))))
14		ffvelcdm 7076	. . . . . . . . . 10 ⊢ ((𝑇: ℋ⟶ ℋ ∧ 𝑥 ∈ ℋ) → (𝑇‘𝑥) ∈ ℋ)
15		ax-hvdistr2 30995	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ (𝑇‘𝑥) ∈ ℋ) → ((𝐴 + 𝐵) ·ℎ (𝑇‘𝑥)) = ((𝐴 ·ℎ (𝑇‘𝑥)) +ℎ (𝐵 ·ℎ (𝑇‘𝑥))))
16	14, 15	syl3an3 1165	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ (𝑇: ℋ⟶ ℋ ∧ 𝑥 ∈ ℋ)) → ((𝐴 + 𝐵) ·ℎ (𝑇‘𝑥)) = ((𝐴 ·ℎ (𝑇‘𝑥)) +ℎ (𝐵 ·ℎ (𝑇‘𝑥))))
17	16	3exp 1119	. . . . . . . 8 ⊢ (𝐴 ∈ ℂ → (𝐵 ∈ ℂ → ((𝑇: ℋ⟶ ℋ ∧ 𝑥 ∈ ℋ) → ((𝐴 + 𝐵) ·ℎ (𝑇‘𝑥)) = ((𝐴 ·ℎ (𝑇‘𝑥)) +ℎ (𝐵 ·ℎ (𝑇‘𝑥))))))
18	17	exp4a 431	. . . . . . 7 ⊢ (𝐴 ∈ ℂ → (𝐵 ∈ ℂ → (𝑇: ℋ⟶ ℋ → (𝑥 ∈ ℋ → ((𝐴 + 𝐵) ·ℎ (𝑇‘𝑥)) = ((𝐴 ·ℎ (𝑇‘𝑥)) +ℎ (𝐵 ·ℎ (𝑇‘𝑥)))))))
19	18	3imp1 1348	. . . . . 6 ⊢ (((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝑇: ℋ⟶ ℋ) ∧ 𝑥 ∈ ℋ) → ((𝐴 + 𝐵) ·ℎ (𝑇‘𝑥)) = ((𝐴 ·ℎ (𝑇‘𝑥)) +ℎ (𝐵 ·ℎ (𝑇‘𝑥))))
20	13, 19	eqtr4d 2774	. . . . 5 ⊢ (((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝑇: ℋ⟶ ℋ) ∧ 𝑥 ∈ ℋ) → (((𝐴 ·op 𝑇)‘𝑥) +ℎ ((𝐵 ·op 𝑇)‘𝑥)) = ((𝐴 + 𝐵) ·ℎ (𝑇‘𝑥)))
21	6, 20	eqtr4d 2774	. . . 4 ⊢ (((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝑇: ℋ⟶ ℋ) ∧ 𝑥 ∈ ℋ) → (((𝐴 + 𝐵) ·op 𝑇)‘𝑥) = (((𝐴 ·op 𝑇)‘𝑥) +ℎ ((𝐵 ·op 𝑇)‘𝑥)))
22		homulcl 31745	. . . . . . 7 ⊢ ((𝐴 ∈ ℂ ∧ 𝑇: ℋ⟶ ℋ) → (𝐴 ·op 𝑇): ℋ⟶ ℋ)
23		homulcl 31745	. . . . . . 7 ⊢ ((𝐵 ∈ ℂ ∧ 𝑇: ℋ⟶ ℋ) → (𝐵 ·op 𝑇): ℋ⟶ ℋ)
24	22, 23	anim12i 613	. . . . . 6 ⊢ (((𝐴 ∈ ℂ ∧ 𝑇: ℋ⟶ ℋ) ∧ (𝐵 ∈ ℂ ∧ 𝑇: ℋ⟶ ℋ)) → ((𝐴 ·op 𝑇): ℋ⟶ ℋ ∧ (𝐵 ·op 𝑇): ℋ⟶ ℋ))
25	24	3impdir 1352	. . . . 5 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝑇: ℋ⟶ ℋ) → ((𝐴 ·op 𝑇): ℋ⟶ ℋ ∧ (𝐵 ·op 𝑇): ℋ⟶ ℋ))
26		hosval 31726	. . . . . 6 ⊢ (((𝐴 ·op 𝑇): ℋ⟶ ℋ ∧ (𝐵 ·op 𝑇): ℋ⟶ ℋ ∧ 𝑥 ∈ ℋ) → (((𝐴 ·op 𝑇) +op (𝐵 ·op 𝑇))‘𝑥) = (((𝐴 ·op 𝑇)‘𝑥) +ℎ ((𝐵 ·op 𝑇)‘𝑥)))
27	26	3expa 1118	. . . . 5 ⊢ ((((𝐴 ·op 𝑇): ℋ⟶ ℋ ∧ (𝐵 ·op 𝑇): ℋ⟶ ℋ) ∧ 𝑥 ∈ ℋ) → (((𝐴 ·op 𝑇) +op (𝐵 ·op 𝑇))‘𝑥) = (((𝐴 ·op 𝑇)‘𝑥) +ℎ ((𝐵 ·op 𝑇)‘𝑥)))
28	25, 27	sylan 580	. . . 4 ⊢ (((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝑇: ℋ⟶ ℋ) ∧ 𝑥 ∈ ℋ) → (((𝐴 ·op 𝑇) +op (𝐵 ·op 𝑇))‘𝑥) = (((𝐴 ·op 𝑇)‘𝑥) +ℎ ((𝐵 ·op 𝑇)‘𝑥)))
29	21, 28	eqtr4d 2774	. . 3 ⊢ (((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝑇: ℋ⟶ ℋ) ∧ 𝑥 ∈ ℋ) → (((𝐴 + 𝐵) ·op 𝑇)‘𝑥) = (((𝐴 ·op 𝑇) +op (𝐵 ·op 𝑇))‘𝑥))
30	29	ralrimiva 3133	. 2 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝑇: ℋ⟶ ℋ) → ∀𝑥 ∈ ℋ (((𝐴 + 𝐵) ·op 𝑇)‘𝑥) = (((𝐴 ·op 𝑇) +op (𝐵 ·op 𝑇))‘𝑥))
31		homulcl 31745	. . . 4 ⊢ (((𝐴 + 𝐵) ∈ ℂ ∧ 𝑇: ℋ⟶ ℋ) → ((𝐴 + 𝐵) ·op 𝑇): ℋ⟶ ℋ)
32	1, 31	stoic3 1776	. . 3 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝑇: ℋ⟶ ℋ) → ((𝐴 + 𝐵) ·op 𝑇): ℋ⟶ ℋ)
33		hoaddcl 31744	. . . . 5 ⊢ (((𝐴 ·op 𝑇): ℋ⟶ ℋ ∧ (𝐵 ·op 𝑇): ℋ⟶ ℋ) → ((𝐴 ·op 𝑇) +op (𝐵 ·op 𝑇)): ℋ⟶ ℋ)
34	22, 23, 33	syl2an 596	. . . 4 ⊢ (((𝐴 ∈ ℂ ∧ 𝑇: ℋ⟶ ℋ) ∧ (𝐵 ∈ ℂ ∧ 𝑇: ℋ⟶ ℋ)) → ((𝐴 ·op 𝑇) +op (𝐵 ·op 𝑇)): ℋ⟶ ℋ)
35	34	3impdir 1352	. . 3 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝑇: ℋ⟶ ℋ) → ((𝐴 ·op 𝑇) +op (𝐵 ·op 𝑇)): ℋ⟶ ℋ)
36		hoeq 31746	. . 3 ⊢ ((((𝐴 + 𝐵) ·op 𝑇): ℋ⟶ ℋ ∧ ((𝐴 ·op 𝑇) +op (𝐵 ·op 𝑇)): ℋ⟶ ℋ) → (∀𝑥 ∈ ℋ (((𝐴 + 𝐵) ·op 𝑇)‘𝑥) = (((𝐴 ·op 𝑇) +op (𝐵 ·op 𝑇))‘𝑥) ↔ ((𝐴 + 𝐵) ·op 𝑇) = ((𝐴 ·op 𝑇) +op (𝐵 ·op 𝑇))))
37	32, 35, 36	syl2anc 584	. 2 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝑇: ℋ⟶ ℋ) → (∀𝑥 ∈ ℋ (((𝐴 + 𝐵) ·op 𝑇)‘𝑥) = (((𝐴 ·op 𝑇) +op (𝐵 ·op 𝑇))‘𝑥) ↔ ((𝐴 + 𝐵) ·op 𝑇) = ((𝐴 ·op 𝑇) +op (𝐵 ·op 𝑇))))
38	30, 37	mpbid 232	1 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝑇: ℋ⟶ ℋ) → ((𝐴 + 𝐵) ·op 𝑇) = ((𝐴 ·op 𝑇) +op (𝐵 ·op 𝑇)))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1086   = wceq 1540   ∈ wcel 2109  ∀wral 3052  ⟶wf 6532  ‘cfv 6536  (class class class)co 7410  ℂcc 11132   + caddc 11137   ℋchba 30905   +_ℎ cva 30906   ·_ℎ csm 30907   +_op chos 30924   ·_op chot 30925

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2708  ax-rep 5254  ax-sep 5271  ax-nul 5281  ax-pow 5340  ax-pr 5407  ax-un 7734  ax-addcl 11194  ax-hilex 30985  ax-hfvadd 30986  ax-hfvmul 30991  ax-hvdistr2 30995

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2810  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3062  df-reu 3365  df-rab 3421  df-v 3466  df-sbc 3771  df-csb 3880  df-dif 3934  df-un 3936  df-in 3938  df-ss 3948  df-nul 4314  df-if 4506  df-pw 4582  df-sn 4607  df-pr 4609  df-op 4613  df-uni 4889  df-iun 4974  df-br 5125  df-opab 5187  df-mpt 5207  df-id 5553  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-rn 5670  df-res 5671  df-ima 5672  df-iota 6489  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-ov 7413  df-oprab 7414  df-mpo 7415  df-map 8847  df-hosum 31716  df-homul 31717

This theorem is referenced by:  ho2times  31805