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Theorem hocadddiri 28526
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) = ((𝑅𝑇) +op (𝑆𝑇))

Proof of Theorem hocadddiri
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 hods.1 . . . . . 6 𝑅: ℋ⟶ ℋ
2 hods.2 . . . . . 6 𝑆: ℋ⟶ ℋ
31, 2hoaddcli 28515 . . . . 5 (𝑅 +op 𝑆): ℋ⟶ ℋ
4 hods.3 . . . . 5 𝑇: ℋ⟶ ℋ
53, 4hocoi 28511 . . . 4 (𝑥 ∈ ℋ → (((𝑅 +op 𝑆) ∘ 𝑇)‘𝑥) = ((𝑅 +op 𝑆)‘(𝑇𝑥)))
61, 4hocofi 28513 . . . . . 6 (𝑅𝑇): ℋ⟶ ℋ
72, 4hocofi 28513 . . . . . 6 (𝑆𝑇): ℋ⟶ ℋ
8 hosval 28487 . . . . . 6 (((𝑅𝑇): ℋ⟶ ℋ ∧ (𝑆𝑇): ℋ⟶ ℋ ∧ 𝑥 ∈ ℋ) → (((𝑅𝑇) +op (𝑆𝑇))‘𝑥) = (((𝑅𝑇)‘𝑥) + ((𝑆𝑇)‘𝑥)))
96, 7, 8mp3an12 1411 . . . . 5 (𝑥 ∈ ℋ → (((𝑅𝑇) +op (𝑆𝑇))‘𝑥) = (((𝑅𝑇)‘𝑥) + ((𝑆𝑇)‘𝑥)))
104ffvelrni 6324 . . . . . . 7 (𝑥 ∈ ℋ → (𝑇𝑥) ∈ ℋ)
11 hosval 28487 . . . . . . . 8 ((𝑅: ℋ⟶ ℋ ∧ 𝑆: ℋ⟶ ℋ ∧ (𝑇𝑥) ∈ ℋ) → ((𝑅 +op 𝑆)‘(𝑇𝑥)) = ((𝑅‘(𝑇𝑥)) + (𝑆‘(𝑇𝑥))))
121, 2, 11mp3an12 1411 . . . . . . 7 ((𝑇𝑥) ∈ ℋ → ((𝑅 +op 𝑆)‘(𝑇𝑥)) = ((𝑅‘(𝑇𝑥)) + (𝑆‘(𝑇𝑥))))
1310, 12syl 17 . . . . . 6 (𝑥 ∈ ℋ → ((𝑅 +op 𝑆)‘(𝑇𝑥)) = ((𝑅‘(𝑇𝑥)) + (𝑆‘(𝑇𝑥))))
141, 4hocoi 28511 . . . . . . 7 (𝑥 ∈ ℋ → ((𝑅𝑇)‘𝑥) = (𝑅‘(𝑇𝑥)))
152, 4hocoi 28511 . . . . . . 7 (𝑥 ∈ ℋ → ((𝑆𝑇)‘𝑥) = (𝑆‘(𝑇𝑥)))
1614, 15oveq12d 6633 . . . . . 6 (𝑥 ∈ ℋ → (((𝑅𝑇)‘𝑥) + ((𝑆𝑇)‘𝑥)) = ((𝑅‘(𝑇𝑥)) + (𝑆‘(𝑇𝑥))))
1713, 16eqtr4d 2658 . . . . 5 (𝑥 ∈ ℋ → ((𝑅 +op 𝑆)‘(𝑇𝑥)) = (((𝑅𝑇)‘𝑥) + ((𝑆𝑇)‘𝑥)))
189, 17eqtr4d 2658 . . . 4 (𝑥 ∈ ℋ → (((𝑅𝑇) +op (𝑆𝑇))‘𝑥) = ((𝑅 +op 𝑆)‘(𝑇𝑥)))
195, 18eqtr4d 2658 . . 3 (𝑥 ∈ ℋ → (((𝑅 +op 𝑆) ∘ 𝑇)‘𝑥) = (((𝑅𝑇) +op (𝑆𝑇))‘𝑥))
2019rgen 2918 . 2 𝑥 ∈ ℋ (((𝑅 +op 𝑆) ∘ 𝑇)‘𝑥) = (((𝑅𝑇) +op (𝑆𝑇))‘𝑥)
213, 4hocofi 28513 . . 3 ((𝑅 +op 𝑆) ∘ 𝑇): ℋ⟶ ℋ
226, 7hoaddcli 28515 . . 3 ((𝑅𝑇) +op (𝑆𝑇)): ℋ⟶ ℋ
2321, 22hoeqi 28508 . 2 (∀𝑥 ∈ ℋ (((𝑅 +op 𝑆) ∘ 𝑇)‘𝑥) = (((𝑅𝑇) +op (𝑆𝑇))‘𝑥) ↔ ((𝑅 +op 𝑆) ∘ 𝑇) = ((𝑅𝑇) +op (𝑆𝑇)))
2420, 23mpbi 220 1 ((𝑅 +op 𝑆) ∘ 𝑇) = ((𝑅𝑇) +op (𝑆𝑇))
Colors of variables: wff setvar class
Syntax hints:   = wceq 1480  wcel 1987  wral 2908  ccom 5088  wf 5853  cfv 5857  (class class class)co 6615  chil 27664   + cva 27665   +op chos 27683
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4741  ax-sep 4751  ax-nul 4759  ax-pow 4813  ax-pr 4877  ax-un 6914  ax-hilex 27744  ax-hfvadd 27745
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2913  df-rex 2914  df-reu 2915  df-rab 2917  df-v 3192  df-sbc 3423  df-csb 3520  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-nul 3898  df-if 4065  df-pw 4138  df-sn 4156  df-pr 4158  df-op 4162  df-uni 4410  df-iun 4494  df-br 4624  df-opab 4684  df-mpt 4685  df-id 4999  df-xp 5090  df-rel 5091  df-cnv 5092  df-co 5093  df-dm 5094  df-rn 5095  df-res 5096  df-ima 5097  df-iota 5820  df-fun 5859  df-fn 5860  df-f 5861  df-f1 5862  df-fo 5863  df-f1o 5864  df-fv 5865  df-ov 6618  df-oprab 6619  df-mpt2 6620  df-map 7819  df-hosum 28477
This theorem is referenced by: (None)
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