Theorem hoadddi 32136

Description: Scalar product distributive law for Hilbert space operators. (Contributed by NM, 12-Aug-2006.) (New usage is discouraged.)

Assertion

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

Proof of Theorem hoadddi

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		simpl1 1210	. . . . . 6 ⊢ (((𝐴 ∈ ℂ ∧ 𝑇: ℋ⟶ ℋ ∧ 𝑈: ℋ⟶ ℋ) ∧ 𝑥 ∈ ℋ) → 𝐴 ∈ ℂ)
2		ffvelcdm 7078	. . . . . . 7 ⊢ ((𝑇: ℋ⟶ ℋ ∧ 𝑥 ∈ ℋ) → (𝑇‘𝑥) ∈ ℋ)
3	2	3ad2antl2 1205	. . . . . 6 ⊢ (((𝐴 ∈ ℂ ∧ 𝑇: ℋ⟶ ℋ ∧ 𝑈: ℋ⟶ ℋ) ∧ 𝑥 ∈ ℋ) → (𝑇‘𝑥) ∈ ℋ)
4		ffvelcdm 7078	. . . . . . 7 ⊢ ((𝑈: ℋ⟶ ℋ ∧ 𝑥 ∈ ℋ) → (𝑈‘𝑥) ∈ ℋ)
5	4	3ad2antl3 1206	. . . . . 6 ⊢ (((𝐴 ∈ ℂ ∧ 𝑇: ℋ⟶ ℋ ∧ 𝑈: ℋ⟶ ℋ) ∧ 𝑥 ∈ ℋ) → (𝑈‘𝑥) ∈ ℋ)
6		ax-hvdistr1 31341	. . . . . 6 ⊢ ((𝐴 ∈ ℂ ∧ (𝑇‘𝑥) ∈ ℋ ∧ (𝑈‘𝑥) ∈ ℋ) → (𝐴 ·ℎ ((𝑇‘𝑥) +ℎ (𝑈‘𝑥))) = ((𝐴 ·ℎ (𝑇‘𝑥)) +ℎ (𝐴 ·ℎ (𝑈‘𝑥))))
7	1, 3, 5, 6	syl3anc 1398	. . . . 5 ⊢ (((𝐴 ∈ ℂ ∧ 𝑇: ℋ⟶ ℋ ∧ 𝑈: ℋ⟶ ℋ) ∧ 𝑥 ∈ ℋ) → (𝐴 ·ℎ ((𝑇‘𝑥) +ℎ (𝑈‘𝑥))) = ((𝐴 ·ℎ (𝑇‘𝑥)) +ℎ (𝐴 ·ℎ (𝑈‘𝑥))))
8		hosval 32073	. . . . . . . 8 ⊢ ((𝑇: ℋ⟶ ℋ ∧ 𝑈: ℋ⟶ ℋ ∧ 𝑥 ∈ ℋ) → ((𝑇 +op 𝑈)‘𝑥) = ((𝑇‘𝑥) +ℎ (𝑈‘𝑥)))
9	8	oveq2d 7428	. . . . . . 7 ⊢ ((𝑇: ℋ⟶ ℋ ∧ 𝑈: ℋ⟶ ℋ ∧ 𝑥 ∈ ℋ) → (𝐴 ·ℎ ((𝑇 +op 𝑈)‘𝑥)) = (𝐴 ·ℎ ((𝑇‘𝑥) +ℎ (𝑈‘𝑥))))
10	9	3expa 1136	. . . . . 6 ⊢ (((𝑇: ℋ⟶ ℋ ∧ 𝑈: ℋ⟶ ℋ) ∧ 𝑥 ∈ ℋ) → (𝐴 ·ℎ ((𝑇 +op 𝑈)‘𝑥)) = (𝐴 ·ℎ ((𝑇‘𝑥) +ℎ (𝑈‘𝑥))))
11	10	3adantl1 1185	. . . . 5 ⊢ (((𝐴 ∈ ℂ ∧ 𝑇: ℋ⟶ ℋ ∧ 𝑈: ℋ⟶ ℋ) ∧ 𝑥 ∈ ℋ) → (𝐴 ·ℎ ((𝑇 +op 𝑈)‘𝑥)) = (𝐴 ·ℎ ((𝑇‘𝑥) +ℎ (𝑈‘𝑥))))
12		homval 32074	. . . . . . . 8 ⊢ ((𝐴 ∈ ℂ ∧ 𝑇: ℋ⟶ ℋ ∧ 𝑥 ∈ ℋ) → ((𝐴 ·op 𝑇)‘𝑥) = (𝐴 ·ℎ (𝑇‘𝑥)))
13	12	3expa 1136	. . . . . . 7 ⊢ (((𝐴 ∈ ℂ ∧ 𝑇: ℋ⟶ ℋ) ∧ 𝑥 ∈ ℋ) → ((𝐴 ·op 𝑇)‘𝑥) = (𝐴 ·ℎ (𝑇‘𝑥)))
14	13	3adantl3 1187	. . . . . 6 ⊢ (((𝐴 ∈ ℂ ∧ 𝑇: ℋ⟶ ℋ ∧ 𝑈: ℋ⟶ ℋ) ∧ 𝑥 ∈ ℋ) → ((𝐴 ·op 𝑇)‘𝑥) = (𝐴 ·ℎ (𝑇‘𝑥)))
15		homval 32074	. . . . . . . 8 ⊢ ((𝐴 ∈ ℂ ∧ 𝑈: ℋ⟶ ℋ ∧ 𝑥 ∈ ℋ) → ((𝐴 ·op 𝑈)‘𝑥) = (𝐴 ·ℎ (𝑈‘𝑥)))
16	15	3expa 1136	. . . . . . 7 ⊢ (((𝐴 ∈ ℂ ∧ 𝑈: ℋ⟶ ℋ) ∧ 𝑥 ∈ ℋ) → ((𝐴 ·op 𝑈)‘𝑥) = (𝐴 ·ℎ (𝑈‘𝑥)))
17	16	3adantl2 1186	. . . . . 6 ⊢ (((𝐴 ∈ ℂ ∧ 𝑇: ℋ⟶ ℋ ∧ 𝑈: ℋ⟶ ℋ) ∧ 𝑥 ∈ ℋ) → ((𝐴 ·op 𝑈)‘𝑥) = (𝐴 ·ℎ (𝑈‘𝑥)))
18	14, 17	oveq12d 7430	. . . . 5 ⊢ (((𝐴 ∈ ℂ ∧ 𝑇: ℋ⟶ ℋ ∧ 𝑈: ℋ⟶ ℋ) ∧ 𝑥 ∈ ℋ) → (((𝐴 ·op 𝑇)‘𝑥) +ℎ ((𝐴 ·op 𝑈)‘𝑥)) = ((𝐴 ·ℎ (𝑇‘𝑥)) +ℎ (𝐴 ·ℎ (𝑈‘𝑥))))
19	7, 11, 18	3eqtr4d 2808	. . . 4 ⊢ (((𝐴 ∈ ℂ ∧ 𝑇: ℋ⟶ ℋ ∧ 𝑈: ℋ⟶ ℋ) ∧ 𝑥 ∈ ℋ) → (𝐴 ·ℎ ((𝑇 +op 𝑈)‘𝑥)) = (((𝐴 ·op 𝑇)‘𝑥) +ℎ ((𝐴 ·op 𝑈)‘𝑥)))
20		hoaddcl 32091	. . . . . . 7 ⊢ ((𝑇: ℋ⟶ ℋ ∧ 𝑈: ℋ⟶ ℋ) → (𝑇 +op 𝑈): ℋ⟶ ℋ)
21	20	anim2i 628	. . . . . 6 ⊢ ((𝐴 ∈ ℂ ∧ (𝑇: ℋ⟶ ℋ ∧ 𝑈: ℋ⟶ ℋ)) → (𝐴 ∈ ℂ ∧ (𝑇 +op 𝑈): ℋ⟶ ℋ))
22	21	3impb 1132	. . . . 5 ⊢ ((𝐴 ∈ ℂ ∧ 𝑇: ℋ⟶ ℋ ∧ 𝑈: ℋ⟶ ℋ) → (𝐴 ∈ ℂ ∧ (𝑇 +op 𝑈): ℋ⟶ ℋ))
23		homval 32074	. . . . . 6 ⊢ ((𝐴 ∈ ℂ ∧ (𝑇 +op 𝑈): ℋ⟶ ℋ ∧ 𝑥 ∈ ℋ) → ((𝐴 ·op (𝑇 +op 𝑈))‘𝑥) = (𝐴 ·ℎ ((𝑇 +op 𝑈)‘𝑥)))
24	23	3expa 1136	. . . . 5 ⊢ (((𝐴 ∈ ℂ ∧ (𝑇 +op 𝑈): ℋ⟶ ℋ) ∧ 𝑥 ∈ ℋ) → ((𝐴 ·op (𝑇 +op 𝑈))‘𝑥) = (𝐴 ·ℎ ((𝑇 +op 𝑈)‘𝑥)))
25	22, 24	sylan 591	. . . 4 ⊢ (((𝐴 ∈ ℂ ∧ 𝑇: ℋ⟶ ℋ ∧ 𝑈: ℋ⟶ ℋ) ∧ 𝑥 ∈ ℋ) → ((𝐴 ·op (𝑇 +op 𝑈))‘𝑥) = (𝐴 ·ℎ ((𝑇 +op 𝑈)‘𝑥)))
26		homulcl 32092	. . . . . . 7 ⊢ ((𝐴 ∈ ℂ ∧ 𝑇: ℋ⟶ ℋ) → (𝐴 ·op 𝑇): ℋ⟶ ℋ)
27		homulcl 32092	. . . . . . 7 ⊢ ((𝐴 ∈ ℂ ∧ 𝑈: ℋ⟶ ℋ) → (𝐴 ·op 𝑈): ℋ⟶ ℋ)
28	26, 27	anim12i 624	. . . . . 6 ⊢ (((𝐴 ∈ ℂ ∧ 𝑇: ℋ⟶ ℋ) ∧ (𝐴 ∈ ℂ ∧ 𝑈: ℋ⟶ ℋ)) → ((𝐴 ·op 𝑇): ℋ⟶ ℋ ∧ (𝐴 ·op 𝑈): ℋ⟶ ℋ))
29	28	3impdi 1369	. . . . 5 ⊢ ((𝐴 ∈ ℂ ∧ 𝑇: ℋ⟶ ℋ ∧ 𝑈: ℋ⟶ ℋ) → ((𝐴 ·op 𝑇): ℋ⟶ ℋ ∧ (𝐴 ·op 𝑈): ℋ⟶ ℋ))
30		hosval 32073	. . . . . 6 ⊢ (((𝐴 ·op 𝑇): ℋ⟶ ℋ ∧ (𝐴 ·op 𝑈): ℋ⟶ ℋ ∧ 𝑥 ∈ ℋ) → (((𝐴 ·op 𝑇) +op (𝐴 ·op 𝑈))‘𝑥) = (((𝐴 ·op 𝑇)‘𝑥) +ℎ ((𝐴 ·op 𝑈)‘𝑥)))
31	30	3expa 1136	. . . . 5 ⊢ ((((𝐴 ·op 𝑇): ℋ⟶ ℋ ∧ (𝐴 ·op 𝑈): ℋ⟶ ℋ) ∧ 𝑥 ∈ ℋ) → (((𝐴 ·op 𝑇) +op (𝐴 ·op 𝑈))‘𝑥) = (((𝐴 ·op 𝑇)‘𝑥) +ℎ ((𝐴 ·op 𝑈)‘𝑥)))
32	29, 31	sylan 591	. . . 4 ⊢ (((𝐴 ∈ ℂ ∧ 𝑇: ℋ⟶ ℋ ∧ 𝑈: ℋ⟶ ℋ) ∧ 𝑥 ∈ ℋ) → (((𝐴 ·op 𝑇) +op (𝐴 ·op 𝑈))‘𝑥) = (((𝐴 ·op 𝑇)‘𝑥) +ℎ ((𝐴 ·op 𝑈)‘𝑥)))
33	19, 25, 32	3eqtr4d 2808	. . 3 ⊢ (((𝐴 ∈ ℂ ∧ 𝑇: ℋ⟶ ℋ ∧ 𝑈: ℋ⟶ ℋ) ∧ 𝑥 ∈ ℋ) → ((𝐴 ·op (𝑇 +op 𝑈))‘𝑥) = (((𝐴 ·op 𝑇) +op (𝐴 ·op 𝑈))‘𝑥))
34	33	ralrimiva 3157	. 2 ⊢ ((𝐴 ∈ ℂ ∧ 𝑇: ℋ⟶ ℋ ∧ 𝑈: ℋ⟶ ℋ) → ∀𝑥 ∈ ℋ ((𝐴 ·op (𝑇 +op 𝑈))‘𝑥) = (((𝐴 ·op 𝑇) +op (𝐴 ·op 𝑈))‘𝑥))
35		homulcl 32092	. . . . 5 ⊢ ((𝐴 ∈ ℂ ∧ (𝑇 +op 𝑈): ℋ⟶ ℋ) → (𝐴 ·op (𝑇 +op 𝑈)): ℋ⟶ ℋ)
36	20, 35	sylan2 604	. . . 4 ⊢ ((𝐴 ∈ ℂ ∧ (𝑇: ℋ⟶ ℋ ∧ 𝑈: ℋ⟶ ℋ)) → (𝐴 ·op (𝑇 +op 𝑈)): ℋ⟶ ℋ)
37	36	3impb 1132	. . 3 ⊢ ((𝐴 ∈ ℂ ∧ 𝑇: ℋ⟶ ℋ ∧ 𝑈: ℋ⟶ ℋ) → (𝐴 ·op (𝑇 +op 𝑈)): ℋ⟶ ℋ)
38		hoaddcl 32091	. . . . 5 ⊢ (((𝐴 ·op 𝑇): ℋ⟶ ℋ ∧ (𝐴 ·op 𝑈): ℋ⟶ ℋ) → ((𝐴 ·op 𝑇) +op (𝐴 ·op 𝑈)): ℋ⟶ ℋ)
39	26, 27, 38	syl2an 607	. . . 4 ⊢ (((𝐴 ∈ ℂ ∧ 𝑇: ℋ⟶ ℋ) ∧ (𝐴 ∈ ℂ ∧ 𝑈: ℋ⟶ ℋ)) → ((𝐴 ·op 𝑇) +op (𝐴 ·op 𝑈)): ℋ⟶ ℋ)
40	39	3impdi 1369	. . 3 ⊢ ((𝐴 ∈ ℂ ∧ 𝑇: ℋ⟶ ℋ ∧ 𝑈: ℋ⟶ ℋ) → ((𝐴 ·op 𝑇) +op (𝐴 ·op 𝑈)): ℋ⟶ ℋ)
41		hoeq 32093	. . 3 ⊢ (((𝐴 ·op (𝑇 +op 𝑈)): ℋ⟶ ℋ ∧ ((𝐴 ·op 𝑇) +op (𝐴 ·op 𝑈)): ℋ⟶ ℋ) → (∀𝑥 ∈ ℋ ((𝐴 ·op (𝑇 +op 𝑈))‘𝑥) = (((𝐴 ·op 𝑇) +op (𝐴 ·op 𝑈))‘𝑥) ↔ (𝐴 ·op (𝑇 +op 𝑈)) = ((𝐴 ·op 𝑇) +op (𝐴 ·op 𝑈))))
42	37, 40, 41	syl2anc 595	. 2 ⊢ ((𝐴 ∈ ℂ ∧ 𝑇: ℋ⟶ ℋ ∧ 𝑈: ℋ⟶ ℋ) → (∀𝑥 ∈ ℋ ((𝐴 ·op (𝑇 +op 𝑈))‘𝑥) = (((𝐴 ·op 𝑇) +op (𝐴 ·op 𝑈))‘𝑥) ↔ (𝐴 ·op (𝑇 +op 𝑈)) = ((𝐴 ·op 𝑇) +op (𝐴 ·op 𝑈))))
43	34, 42	mpbid 235	1 ⊢ ((𝐴 ∈ ℂ ∧ 𝑇: ℋ⟶ ℋ ∧ 𝑈: ℋ⟶ ℋ) → (𝐴 ·op (𝑇 +op 𝑈)) = ((𝐴 ·op 𝑇) +op (𝐴 ·op 𝑈)))

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 400   ∧ w3a 1103   = wceq 1570   ∈ wcel 2143  ∀wral 3079  ⟶wf 6534  ‘cfv 6538  (class class class)co 7412  ℂcc 11099   ℋchba 31252   +_ℎ cva 31253   ·_ℎ csm 31254   +_op chos 31271   ·_op chot 31272

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-rep 5239  ax-sep 5258  ax-nul 5270  ax-pow 5338  ax-pr 5406  ax-un 7734  ax-hilex 31332  ax-hfvadd 31333  ax-hfvmul 31338  ax-hvdistr1 31341

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-nf 1814  df-sb 2097  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-ral 3080  df-rex 3090  df-reu 3370  df-rab 3417  df-v 3457  df-sbc 3746  df-csb 3855  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4288  df-if 4489  df-pw 4565  df-sn 4591  df-pr 4593  df-op 4597  df-uni 4874  df-iun 4959  df-br 5111  df-opab 5175  df-mpt 5194  df-id 5558  df-xp 5669  df-rel 5670  df-cnv 5671  df-co 5672  df-dm 5673  df-rn 5674  df-res 5675  df-ima 5676  df-iota 6494  df-fun 6540  df-fn 6541  df-f 6542  df-f1 6543  df-fo 6544  df-f1o 6545  df-fv 6546  df-ov 7415  df-oprab 7416  df-mpo 7417  df-map 8827  df-hosum 32063  df-homul 32064

This theorem is referenced by:  hosubdi  32141  honegdi  32142  ho2times  32152  opsqrlem6  32478