Theorem hocadddiri 31865

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 31854	. . . . 5 ⊢ (𝑅 +op 𝑆): ℋ⟶ ℋ
4		hods.3	. . . . 5 ⊢ 𝑇: ℋ⟶ ℋ
5	3, 4	hocoi 31850	. . . 4 ⊢ (𝑥 ∈ ℋ → (((𝑅 +op 𝑆) ∘ 𝑇)‘𝑥) = ((𝑅 +op 𝑆)‘(𝑇‘𝑥)))
6	1, 4	hocofi 31852	. . . . . 6 ⊢ (𝑅 ∘ 𝑇): ℋ⟶ ℋ
7	2, 4	hocofi 31852	. . . . . 6 ⊢ (𝑆 ∘ 𝑇): ℋ⟶ ℋ
8		hosval 31826	. . . . . 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 1454	. . . . 5 ⊢ (𝑥 ∈ ℋ → (((𝑅 ∘ 𝑇) +op (𝑆 ∘ 𝑇))‘𝑥) = (((𝑅 ∘ 𝑇)‘𝑥) +ℎ ((𝑆 ∘ 𝑇)‘𝑥)))
10	4	ffvelcdmi 7029	. . . . . . 7 ⊢ (𝑥 ∈ ℋ → (𝑇‘𝑥) ∈ ℋ)
11		hosval 31826	. . . . . . . 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 1454	. . . . . . 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 31850	. . . . . . 7 ⊢ (𝑥 ∈ ℋ → ((𝑅 ∘ 𝑇)‘𝑥) = (𝑅‘(𝑇‘𝑥)))
15	2, 4	hocoi 31850	. . . . . . 7 ⊢ (𝑥 ∈ ℋ → ((𝑆 ∘ 𝑇)‘𝑥) = (𝑆‘(𝑇‘𝑥)))
16	14, 15	oveq12d 7378	. . . . . 6 ⊢ (𝑥 ∈ ℋ → (((𝑅 ∘ 𝑇)‘𝑥) +ℎ ((𝑆 ∘ 𝑇)‘𝑥)) = ((𝑅‘(𝑇‘𝑥)) +ℎ (𝑆‘(𝑇‘𝑥))))
17	13, 16	eqtr4d 2775	. . . . 5 ⊢ (𝑥 ∈ ℋ → ((𝑅 +op 𝑆)‘(𝑇‘𝑥)) = (((𝑅 ∘ 𝑇)‘𝑥) +ℎ ((𝑆 ∘ 𝑇)‘𝑥)))
18	9, 17	eqtr4d 2775	. . . 4 ⊢ (𝑥 ∈ ℋ → (((𝑅 ∘ 𝑇) +op (𝑆 ∘ 𝑇))‘𝑥) = ((𝑅 +op 𝑆)‘(𝑇‘𝑥)))
19	5, 18	eqtr4d 2775	. . 3 ⊢ (𝑥 ∈ ℋ → (((𝑅 +op 𝑆) ∘ 𝑇)‘𝑥) = (((𝑅 ∘ 𝑇) +op (𝑆 ∘ 𝑇))‘𝑥))
20	19	rgen 3054	. 2 ⊢ ∀𝑥 ∈ ℋ (((𝑅 +op 𝑆) ∘ 𝑇)‘𝑥) = (((𝑅 ∘ 𝑇) +op (𝑆 ∘ 𝑇))‘𝑥)
21	3, 4	hocofi 31852	. . 3 ⊢ ((𝑅 +op 𝑆) ∘ 𝑇): ℋ⟶ ℋ
22	6, 7	hoaddcli 31854	. . 3 ⊢ ((𝑅 ∘ 𝑇) +op (𝑆 ∘ 𝑇)): ℋ⟶ ℋ
23	21, 22	hoeqi 31847	. 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 1542   ∈ wcel 2114  ∀wral 3052   ∘ ccom 5628  ⟶wf 6488  ‘cfv 6492  (class class class)co 7360   ℋchba 31005   +_ℎ cva 31006   +_op chos 31024

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-rep 5212  ax-sep 5231  ax-nul 5241  ax-pow 5302  ax-pr 5370  ax-un 7682  ax-hilex 31085  ax-hfvadd 31086

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3063  df-reu 3344  df-rab 3391  df-v 3432  df-sbc 3730  df-csb 3839  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-nul 4275  df-if 4468  df-pw 4544  df-sn 4569  df-pr 4571  df-op 4575  df-uni 4852  df-iun 4936  df-br 5087  df-opab 5149  df-mpt 5168  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 7363  df-oprab 7364  df-mpo 7365  df-map 8768  df-hosum 31816

This theorem is referenced by: (None)