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Theorem hocadddiri 31808
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 31797 . . . . 5 (𝑅 +op 𝑆): ℋ⟶ ℋ
4 hods.3 . . . . 5 𝑇: ℋ⟶ ℋ
53, 4hocoi 31793 . . . 4 (𝑥 ∈ ℋ → (((𝑅 +op 𝑆) ∘ 𝑇)‘𝑥) = ((𝑅 +op 𝑆)‘(𝑇𝑥)))
61, 4hocofi 31795 . . . . . 6 (𝑅𝑇): ℋ⟶ ℋ
72, 4hocofi 31795 . . . . . 6 (𝑆𝑇): ℋ⟶ ℋ
8 hosval 31769 . . . . . 6 (((𝑅𝑇): ℋ⟶ ℋ ∧ (𝑆𝑇): ℋ⟶ ℋ ∧ 𝑥 ∈ ℋ) → (((𝑅𝑇) +op (𝑆𝑇))‘𝑥) = (((𝑅𝑇)‘𝑥) + ((𝑆𝑇)‘𝑥)))
96, 7, 8mp3an12 1450 . . . . 5 (𝑥 ∈ ℋ → (((𝑅𝑇) +op (𝑆𝑇))‘𝑥) = (((𝑅𝑇)‘𝑥) + ((𝑆𝑇)‘𝑥)))
104ffvelcdmi 7103 . . . . . . 7 (𝑥 ∈ ℋ → (𝑇𝑥) ∈ ℋ)
11 hosval 31769 . . . . . . . 8 ((𝑅: ℋ⟶ ℋ ∧ 𝑆: ℋ⟶ ℋ ∧ (𝑇𝑥) ∈ ℋ) → ((𝑅 +op 𝑆)‘(𝑇𝑥)) = ((𝑅‘(𝑇𝑥)) + (𝑆‘(𝑇𝑥))))
121, 2, 11mp3an12 1450 . . . . . . 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 31793 . . . . . . 7 (𝑥 ∈ ℋ → ((𝑅𝑇)‘𝑥) = (𝑅‘(𝑇𝑥)))
152, 4hocoi 31793 . . . . . . 7 (𝑥 ∈ ℋ → ((𝑆𝑇)‘𝑥) = (𝑆‘(𝑇𝑥)))
1614, 15oveq12d 7449 . . . . . 6 (𝑥 ∈ ℋ → (((𝑅𝑇)‘𝑥) + ((𝑆𝑇)‘𝑥)) = ((𝑅‘(𝑇𝑥)) + (𝑆‘(𝑇𝑥))))
1713, 16eqtr4d 2778 . . . . 5 (𝑥 ∈ ℋ → ((𝑅 +op 𝑆)‘(𝑇𝑥)) = (((𝑅𝑇)‘𝑥) + ((𝑆𝑇)‘𝑥)))
189, 17eqtr4d 2778 . . . 4 (𝑥 ∈ ℋ → (((𝑅𝑇) +op (𝑆𝑇))‘𝑥) = ((𝑅 +op 𝑆)‘(𝑇𝑥)))
195, 18eqtr4d 2778 . . 3 (𝑥 ∈ ℋ → (((𝑅 +op 𝑆) ∘ 𝑇)‘𝑥) = (((𝑅𝑇) +op (𝑆𝑇))‘𝑥))
2019rgen 3061 . 2 𝑥 ∈ ℋ (((𝑅 +op 𝑆) ∘ 𝑇)‘𝑥) = (((𝑅𝑇) +op (𝑆𝑇))‘𝑥)
213, 4hocofi 31795 . . 3 ((𝑅 +op 𝑆) ∘ 𝑇): ℋ⟶ ℋ
226, 7hoaddcli 31797 . . 3 ((𝑅𝑇) +op (𝑆𝑇)): ℋ⟶ ℋ
2321, 22hoeqi 31790 . 2 (∀𝑥 ∈ ℋ (((𝑅 +op 𝑆) ∘ 𝑇)‘𝑥) = (((𝑅𝑇) +op (𝑆𝑇))‘𝑥) ↔ ((𝑅 +op 𝑆) ∘ 𝑇) = ((𝑅𝑇) +op (𝑆𝑇)))
2420, 23mpbi 230 1 ((𝑅 +op 𝑆) ∘ 𝑇) = ((𝑅𝑇) +op (𝑆𝑇))
Colors of variables: wff setvar class
Syntax hints:   = wceq 1537  wcel 2106  wral 3059  ccom 5693  wf 6559  cfv 6563  (class class class)co 7431  chba 30948   + cva 30949   +op chos 30967
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754  ax-hilex 31028  ax-hfvadd 31029
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-ov 7434  df-oprab 7435  df-mpo 7436  df-map 8867  df-hosum 31759
This theorem is referenced by: (None)
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