Theorem hocadddiri 32109

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 32098	. . . . 5 ⊢ (𝑅 +op 𝑆): ℋ⟶ ℋ
4		hods.3	. . . . 5 ⊢ 𝑇: ℋ⟶ ℋ
5	3, 4	hocoi 32094	. . . 4 ⊢ (𝑥 ∈ ℋ → (((𝑅 +op 𝑆) ∘ 𝑇)‘𝑥) = ((𝑅 +op 𝑆)‘(𝑇‘𝑥)))
6	1, 4	hocofi 32096	. . . . . 6 ⊢ (𝑅 ∘ 𝑇): ℋ⟶ ℋ
7	2, 4	hocofi 32096	. . . . . 6 ⊢ (𝑆 ∘ 𝑇): ℋ⟶ ℋ
8		hosval 32070	. . . . . 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 1480	. . . . 5 ⊢ (𝑥 ∈ ℋ → (((𝑅 ∘ 𝑇) +op (𝑆 ∘ 𝑇))‘𝑥) = (((𝑅 ∘ 𝑇)‘𝑥) +ℎ ((𝑆 ∘ 𝑇)‘𝑥)))
10	4	ffvelcdmi 7080	. . . . . . 7 ⊢ (𝑥 ∈ ℋ → (𝑇‘𝑥) ∈ ℋ)
11		hosval 32070	. . . . . . . 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 1480	. . . . . . 7 ⊢ ((𝑇‘𝑥) ∈ ℋ → ((𝑅 +op 𝑆)‘(𝑇‘𝑥)) = ((𝑅‘(𝑇‘𝑥)) +ℎ (𝑆‘(𝑇‘𝑥))))
13	10, 12	syl 18	. . . . . 6 ⊢ (𝑥 ∈ ℋ → ((𝑅 +op 𝑆)‘(𝑇‘𝑥)) = ((𝑅‘(𝑇‘𝑥)) +ℎ (𝑆‘(𝑇‘𝑥))))
14	1, 4	hocoi 32094	. . . . . . 7 ⊢ (𝑥 ∈ ℋ → ((𝑅 ∘ 𝑇)‘𝑥) = (𝑅‘(𝑇‘𝑥)))
15	2, 4	hocoi 32094	. . . . . . 7 ⊢ (𝑥 ∈ ℋ → ((𝑆 ∘ 𝑇)‘𝑥) = (𝑆‘(𝑇‘𝑥)))
16	14, 15	oveq12d 7430	. . . . . 6 ⊢ (𝑥 ∈ ℋ → (((𝑅 ∘ 𝑇)‘𝑥) +ℎ ((𝑆 ∘ 𝑇)‘𝑥)) = ((𝑅‘(𝑇‘𝑥)) +ℎ (𝑆‘(𝑇‘𝑥))))
17	13, 16	eqtr4d 2801	. . . . 5 ⊢ (𝑥 ∈ ℋ → ((𝑅 +op 𝑆)‘(𝑇‘𝑥)) = (((𝑅 ∘ 𝑇)‘𝑥) +ℎ ((𝑆 ∘ 𝑇)‘𝑥)))
18	9, 17	eqtr4d 2801	. . . 4 ⊢ (𝑥 ∈ ℋ → (((𝑅 ∘ 𝑇) +op (𝑆 ∘ 𝑇))‘𝑥) = ((𝑅 +op 𝑆)‘(𝑇‘𝑥)))
19	5, 18	eqtr4d 2801	. . 3 ⊢ (𝑥 ∈ ℋ → (((𝑅 +op 𝑆) ∘ 𝑇)‘𝑥) = (((𝑅 ∘ 𝑇) +op (𝑆 ∘ 𝑇))‘𝑥))
20	19	rgen 3081	. 2 ⊢ ∀𝑥 ∈ ℋ (((𝑅 +op 𝑆) ∘ 𝑇)‘𝑥) = (((𝑅 ∘ 𝑇) +op (𝑆 ∘ 𝑇))‘𝑥)
21	3, 4	hocofi 32096	. . 3 ⊢ ((𝑅 +op 𝑆) ∘ 𝑇): ℋ⟶ ℋ
22	6, 7	hoaddcli 32098	. . 3 ⊢ ((𝑅 ∘ 𝑇) +op (𝑆 ∘ 𝑇)): ℋ⟶ ℋ
23	21, 22	hoeqi 32091	. 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 233	1 ⊢ ((𝑅 +op 𝑆) ∘ 𝑇) = ((𝑅 ∘ 𝑇) +op (𝑆 ∘ 𝑇))

Syntax hints:   = wceq 1570   ∈ wcel 2143  ∀wral 3079   ∘ ccom 5667  ⟶wf 6534  ‘cfv 6538  (class class class)co 7412   ℋchba 31249   +_ℎ cva 31250   +_op chos 31268

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 31329  ax-hfvadd 31330

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 32060

This theorem is referenced by: (None)