Theorem hoadddi 29741

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 1192	. . . . . 6 ⊢ (((𝐴 ∈ ℂ ∧ 𝑇: ℋ⟶ ℋ ∧ 𝑈: ℋ⟶ ℋ) ∧ 𝑥 ∈ ℋ) → 𝐴 ∈ ℂ)
2		ffvelrn 6862	. . . . . . 7 ⊢ ((𝑇: ℋ⟶ ℋ ∧ 𝑥 ∈ ℋ) → (𝑇‘𝑥) ∈ ℋ)
3	2	3ad2antl2 1187	. . . . . 6 ⊢ (((𝐴 ∈ ℂ ∧ 𝑇: ℋ⟶ ℋ ∧ 𝑈: ℋ⟶ ℋ) ∧ 𝑥 ∈ ℋ) → (𝑇‘𝑥) ∈ ℋ)
4		ffvelrn 6862	. . . . . . 7 ⊢ ((𝑈: ℋ⟶ ℋ ∧ 𝑥 ∈ ℋ) → (𝑈‘𝑥) ∈ ℋ)
5	4	3ad2antl3 1188	. . . . . 6 ⊢ (((𝐴 ∈ ℂ ∧ 𝑇: ℋ⟶ ℋ ∧ 𝑈: ℋ⟶ ℋ) ∧ 𝑥 ∈ ℋ) → (𝑈‘𝑥) ∈ ℋ)
6		ax-hvdistr1 28946	. . . . . 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 1372	. . . . 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 29678	. . . . . . . 8 ⊢ ((𝑇: ℋ⟶ ℋ ∧ 𝑈: ℋ⟶ ℋ ∧ 𝑥 ∈ ℋ) → ((𝑇 +op 𝑈)‘𝑥) = ((𝑇‘𝑥) +ℎ (𝑈‘𝑥)))
9	8	oveq2d 7189	. . . . . . 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 1119	. . . . . 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 1167	. . . . 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 29679	. . . . . . . 8 ⊢ ((𝐴 ∈ ℂ ∧ 𝑇: ℋ⟶ ℋ ∧ 𝑥 ∈ ℋ) → ((𝐴 ·op 𝑇)‘𝑥) = (𝐴 ·ℎ (𝑇‘𝑥)))
13	12	3expa 1119	. . . . . . 7 ⊢ (((𝐴 ∈ ℂ ∧ 𝑇: ℋ⟶ ℋ) ∧ 𝑥 ∈ ℋ) → ((𝐴 ·op 𝑇)‘𝑥) = (𝐴 ·ℎ (𝑇‘𝑥)))
14	13	3adantl3 1169	. . . . . 6 ⊢ (((𝐴 ∈ ℂ ∧ 𝑇: ℋ⟶ ℋ ∧ 𝑈: ℋ⟶ ℋ) ∧ 𝑥 ∈ ℋ) → ((𝐴 ·op 𝑇)‘𝑥) = (𝐴 ·ℎ (𝑇‘𝑥)))
15		homval 29679	. . . . . . . 8 ⊢ ((𝐴 ∈ ℂ ∧ 𝑈: ℋ⟶ ℋ ∧ 𝑥 ∈ ℋ) → ((𝐴 ·op 𝑈)‘𝑥) = (𝐴 ·ℎ (𝑈‘𝑥)))
16	15	3expa 1119	. . . . . . 7 ⊢ (((𝐴 ∈ ℂ ∧ 𝑈: ℋ⟶ ℋ) ∧ 𝑥 ∈ ℋ) → ((𝐴 ·op 𝑈)‘𝑥) = (𝐴 ·ℎ (𝑈‘𝑥)))
17	16	3adantl2 1168	. . . . . 6 ⊢ (((𝐴 ∈ ℂ ∧ 𝑇: ℋ⟶ ℋ ∧ 𝑈: ℋ⟶ ℋ) ∧ 𝑥 ∈ ℋ) → ((𝐴 ·op 𝑈)‘𝑥) = (𝐴 ·ℎ (𝑈‘𝑥)))
18	14, 17	oveq12d 7191	. . . . 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 2784	. . . 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 29696	. . . . . . 7 ⊢ ((𝑇: ℋ⟶ ℋ ∧ 𝑈: ℋ⟶ ℋ) → (𝑇 +op 𝑈): ℋ⟶ ℋ)
21	20	anim2i 620	. . . . . 6 ⊢ ((𝐴 ∈ ℂ ∧ (𝑇: ℋ⟶ ℋ ∧ 𝑈: ℋ⟶ ℋ)) → (𝐴 ∈ ℂ ∧ (𝑇 +op 𝑈): ℋ⟶ ℋ))
22	21	3impb 1116	. . . . 5 ⊢ ((𝐴 ∈ ℂ ∧ 𝑇: ℋ⟶ ℋ ∧ 𝑈: ℋ⟶ ℋ) → (𝐴 ∈ ℂ ∧ (𝑇 +op 𝑈): ℋ⟶ ℋ))
23		homval 29679	. . . . . 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 1119	. . . . 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 583	. . . 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 29697	. . . . . . 7 ⊢ ((𝐴 ∈ ℂ ∧ 𝑇: ℋ⟶ ℋ) → (𝐴 ·op 𝑇): ℋ⟶ ℋ)
27		homulcl 29697	. . . . . . 7 ⊢ ((𝐴 ∈ ℂ ∧ 𝑈: ℋ⟶ ℋ) → (𝐴 ·op 𝑈): ℋ⟶ ℋ)
28	26, 27	anim12i 616	. . . . . 6 ⊢ (((𝐴 ∈ ℂ ∧ 𝑇: ℋ⟶ ℋ) ∧ (𝐴 ∈ ℂ ∧ 𝑈: ℋ⟶ ℋ)) → ((𝐴 ·op 𝑇): ℋ⟶ ℋ ∧ (𝐴 ·op 𝑈): ℋ⟶ ℋ))
29	28	3impdi 1351	. . . . 5 ⊢ ((𝐴 ∈ ℂ ∧ 𝑇: ℋ⟶ ℋ ∧ 𝑈: ℋ⟶ ℋ) → ((𝐴 ·op 𝑇): ℋ⟶ ℋ ∧ (𝐴 ·op 𝑈): ℋ⟶ ℋ))
30		hosval 29678	. . . . . 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 1119	. . . . 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 583	. . . 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 2784	. . 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 3097	. 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 29697	. . . . 5 ⊢ ((𝐴 ∈ ℂ ∧ (𝑇 +op 𝑈): ℋ⟶ ℋ) → (𝐴 ·op (𝑇 +op 𝑈)): ℋ⟶ ℋ)
36	20, 35	sylan2 596	. . . 4 ⊢ ((𝐴 ∈ ℂ ∧ (𝑇: ℋ⟶ ℋ ∧ 𝑈: ℋ⟶ ℋ)) → (𝐴 ·op (𝑇 +op 𝑈)): ℋ⟶ ℋ)
37	36	3impb 1116	. . 3 ⊢ ((𝐴 ∈ ℂ ∧ 𝑇: ℋ⟶ ℋ ∧ 𝑈: ℋ⟶ ℋ) → (𝐴 ·op (𝑇 +op 𝑈)): ℋ⟶ ℋ)
38		hoaddcl 29696	. . . . 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 599	. . . 4 ⊢ (((𝐴 ∈ ℂ ∧ 𝑇: ℋ⟶ ℋ) ∧ (𝐴 ∈ ℂ ∧ 𝑈: ℋ⟶ ℋ)) → ((𝐴 ·op 𝑇) +op (𝐴 ·op 𝑈)): ℋ⟶ ℋ)
40	39	3impdi 1351	. . 3 ⊢ ((𝐴 ∈ ℂ ∧ 𝑇: ℋ⟶ ℋ ∧ 𝑈: ℋ⟶ ℋ) → ((𝐴 ·op 𝑇) +op (𝐴 ·op 𝑈)): ℋ⟶ ℋ)
41		hoeq 29698	. . 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 587	. 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 399   ∧ w3a 1088   = wceq 1542   ∈ wcel 2114  ∀wral 3054  ⟶wf 6336  ‘cfv 6340  (class class class)co 7173  ℂcc 10616   ℋchba 28857   +_ℎ cva 28858   ·_ℎ csm 28859   +_op chos 28876   ·_op chot 28877

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2020  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2162  ax-12 2179  ax-ext 2711  ax-rep 5155  ax-sep 5168  ax-nul 5175  ax-pow 5233  ax-pr 5297  ax-un 7482  ax-hilex 28937  ax-hfvadd 28938  ax-hfvmul 28943  ax-hvdistr1 28946

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1787  df-nf 1791  df-sb 2075  df-mo 2541  df-eu 2571  df-clab 2718  df-cleq 2731  df-clel 2812  df-nfc 2882  df-ne 2936  df-ral 3059  df-rex 3060  df-reu 3061  df-rab 3063  df-v 3401  df-sbc 3682  df-csb 3792  df-dif 3847  df-un 3849  df-in 3851  df-ss 3861  df-nul 4213  df-if 4416  df-pw 4491  df-sn 4518  df-pr 4520  df-op 4524  df-uni 4798  df-iun 4884  df-br 5032  df-opab 5094  df-mpt 5112  df-id 5430  df-xp 5532  df-rel 5533  df-cnv 5534  df-co 5535  df-dm 5536  df-rn 5537  df-res 5538  df-ima 5539  df-iota 6298  df-fun 6342  df-fn 6343  df-f 6344  df-f1 6345  df-fo 6346  df-f1o 6347  df-fv 6348  df-ov 7176  df-oprab 7177  df-mpo 7178  df-map 8442  df-hosum 29668  df-homul 29669

This theorem is referenced by:  hosubdi  29746  honegdi  29747  ho2times  29757  opsqrlem6  30083