Theorem hocadddiri 31854

Description: Distributive law for Hilbert space operator sum. (Contributed by NM, 26-Nov-2000.) (New usage is discouraged.)

Hypotheses

Ref	Expression
hods.1	⊢ 𝑅: ℋ⟶ ℋ
hods.2	⊢ 𝑆: ℋ⟶ ℋ
hods.3	⊢ 𝑇: ℋ⟶ ℋ

Assertion

Ref	Expression
hocadddiri	⊢ ((𝑅 +op 𝑆) ∘ 𝑇) = ((𝑅 ∘ 𝑇) +op (𝑆 ∘ 𝑇))

Proof of Theorem hocadddiri

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		hods.1	. . . . . 6 ⊢ 𝑅: ℋ⟶ ℋ
2		hods.2	. . . . . 6 ⊢ 𝑆: ℋ⟶ ℋ
3	1, 2	hoaddcli 31843	. . . . 5 ⊢ (𝑅 +op 𝑆): ℋ⟶ ℋ
4		hods.3	. . . . 5 ⊢ 𝑇: ℋ⟶ ℋ
5	3, 4	hocoi 31839	. . . 4 ⊢ (𝑥 ∈ ℋ → (((𝑅 +op 𝑆) ∘ 𝑇)‘𝑥) = ((𝑅 +op 𝑆)‘(𝑇‘𝑥)))
6	1, 4	hocofi 31841	. . . . . 6 ⊢ (𝑅 ∘ 𝑇): ℋ⟶ ℋ
7	2, 4	hocofi 31841	. . . . . 6 ⊢ (𝑆 ∘ 𝑇): ℋ⟶ ℋ
8		hosval 31815	. . . . . 6 ⊢ (((𝑅 ∘ 𝑇): ℋ⟶ ℋ ∧ (𝑆 ∘ 𝑇): ℋ⟶ ℋ ∧ 𝑥 ∈ ℋ) → (((𝑅 ∘ 𝑇) +op (𝑆 ∘ 𝑇))‘𝑥) = (((𝑅 ∘ 𝑇)‘𝑥) +ℎ ((𝑆 ∘ 𝑇)‘𝑥)))
9	6, 7, 8	mp3an12 1453	. . . . 5 ⊢ (𝑥 ∈ ℋ → (((𝑅 ∘ 𝑇) +op (𝑆 ∘ 𝑇))‘𝑥) = (((𝑅 ∘ 𝑇)‘𝑥) +ℎ ((𝑆 ∘ 𝑇)‘𝑥)))
10	4	ffvelcdmi 7028	. . . . . . 7 ⊢ (𝑥 ∈ ℋ → (𝑇‘𝑥) ∈ ℋ)
11		hosval 31815	. . . . . . . 8 ⊢ ((𝑅: ℋ⟶ ℋ ∧ 𝑆: ℋ⟶ ℋ ∧ (𝑇‘𝑥) ∈ ℋ) → ((𝑅 +op 𝑆)‘(𝑇‘𝑥)) = ((𝑅‘(𝑇‘𝑥)) +ℎ (𝑆‘(𝑇‘𝑥))))
12	1, 2, 11	mp3an12 1453	. . . . . . 7 ⊢ ((𝑇‘𝑥) ∈ ℋ → ((𝑅 +op 𝑆)‘(𝑇‘𝑥)) = ((𝑅‘(𝑇‘𝑥)) +ℎ (𝑆‘(𝑇‘𝑥))))
13	10, 12	syl 17	. . . . . 6 ⊢ (𝑥 ∈ ℋ → ((𝑅 +op 𝑆)‘(𝑇‘𝑥)) = ((𝑅‘(𝑇‘𝑥)) +ℎ (𝑆‘(𝑇‘𝑥))))
14	1, 4	hocoi 31839	. . . . . . 7 ⊢ (𝑥 ∈ ℋ → ((𝑅 ∘ 𝑇)‘𝑥) = (𝑅‘(𝑇‘𝑥)))
15	2, 4	hocoi 31839	. . . . . . 7 ⊢ (𝑥 ∈ ℋ → ((𝑆 ∘ 𝑇)‘𝑥) = (𝑆‘(𝑇‘𝑥)))
16	14, 15	oveq12d 7376	. . . . . 6 ⊢ (𝑥 ∈ ℋ → (((𝑅 ∘ 𝑇)‘𝑥) +ℎ ((𝑆 ∘ 𝑇)‘𝑥)) = ((𝑅‘(𝑇‘𝑥)) +ℎ (𝑆‘(𝑇‘𝑥))))
17	13, 16	eqtr4d 2774	. . . . 5 ⊢ (𝑥 ∈ ℋ → ((𝑅 +op 𝑆)‘(𝑇‘𝑥)) = (((𝑅 ∘ 𝑇)‘𝑥) +ℎ ((𝑆 ∘ 𝑇)‘𝑥)))
18	9, 17	eqtr4d 2774	. . . 4 ⊢ (𝑥 ∈ ℋ → (((𝑅 ∘ 𝑇) +op (𝑆 ∘ 𝑇))‘𝑥) = ((𝑅 +op 𝑆)‘(𝑇‘𝑥)))
19	5, 18	eqtr4d 2774	. . 3 ⊢ (𝑥 ∈ ℋ → (((𝑅 +op 𝑆) ∘ 𝑇)‘𝑥) = (((𝑅 ∘ 𝑇) +op (𝑆 ∘ 𝑇))‘𝑥))
20	19	rgen 3053	. 2 ⊢ ∀𝑥 ∈ ℋ (((𝑅 +op 𝑆) ∘ 𝑇)‘𝑥) = (((𝑅 ∘ 𝑇) +op (𝑆 ∘ 𝑇))‘𝑥)
21	3, 4	hocofi 31841	. . 3 ⊢ ((𝑅 +op 𝑆) ∘ 𝑇): ℋ⟶ ℋ
22	6, 7	hoaddcli 31843	. . 3 ⊢ ((𝑅 ∘ 𝑇) +op (𝑆 ∘ 𝑇)): ℋ⟶ ℋ
23	21, 22	hoeqi 31836	. 2 ⊢ (∀𝑥 ∈ ℋ (((𝑅 +op 𝑆) ∘ 𝑇)‘𝑥) = (((𝑅 ∘ 𝑇) +op (𝑆 ∘ 𝑇))‘𝑥) ↔ ((𝑅 +op 𝑆) ∘ 𝑇) = ((𝑅 ∘ 𝑇) +op (𝑆 ∘ 𝑇)))
24	20, 23	mpbi 230	1 ⊢ ((𝑅 +op 𝑆) ∘ 𝑇) = ((𝑅 ∘ 𝑇) +op (𝑆 ∘ 𝑇))

Syntax hints:   = wceq 1541   ∈ wcel 2113  ∀wral 3051   ∘ ccom 5628  ⟶wf 6488  ‘cfv 6492  (class class class)co 7358   ℋchba 30994   +_ℎ cva 30995   +_op chos 31013

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-rep 5224  ax-sep 5241  ax-nul 5251  ax-pow 5310  ax-pr 5377  ax-un 7680  ax-hilex 31074  ax-hfvadd 31075

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-ne 2933  df-ral 3052  df-rex 3061  df-reu 3351  df-rab 3400  df-v 3442  df-sbc 3741  df-csb 3850  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-iun 4948  df-br 5099  df-opab 5161  df-mpt 5180  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-f1 6497  df-fo 6498  df-f1o 6499  df-fv 6500  df-ov 7361  df-oprab 7362  df-mpo 7363  df-map 8765  df-hosum 31805

This theorem is referenced by: (None)