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Theorem hocadddiri 30763
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 30752 . . . . 5 (𝑅 +op 𝑆): ℋ⟶ ℋ
4 hods.3 . . . . 5 𝑇: ℋ⟶ ℋ
53, 4hocoi 30748 . . . 4 (𝑥 ∈ ℋ → (((𝑅 +op 𝑆) ∘ 𝑇)‘𝑥) = ((𝑅 +op 𝑆)‘(𝑇𝑥)))
61, 4hocofi 30750 . . . . . 6 (𝑅𝑇): ℋ⟶ ℋ
72, 4hocofi 30750 . . . . . 6 (𝑆𝑇): ℋ⟶ ℋ
8 hosval 30724 . . . . . 6 (((𝑅𝑇): ℋ⟶ ℋ ∧ (𝑆𝑇): ℋ⟶ ℋ ∧ 𝑥 ∈ ℋ) → (((𝑅𝑇) +op (𝑆𝑇))‘𝑥) = (((𝑅𝑇)‘𝑥) + ((𝑆𝑇)‘𝑥)))
96, 7, 8mp3an12 1452 . . . . 5 (𝑥 ∈ ℋ → (((𝑅𝑇) +op (𝑆𝑇))‘𝑥) = (((𝑅𝑇)‘𝑥) + ((𝑆𝑇)‘𝑥)))
104ffvelcdmi 7035 . . . . . . 7 (𝑥 ∈ ℋ → (𝑇𝑥) ∈ ℋ)
11 hosval 30724 . . . . . . . 8 ((𝑅: ℋ⟶ ℋ ∧ 𝑆: ℋ⟶ ℋ ∧ (𝑇𝑥) ∈ ℋ) → ((𝑅 +op 𝑆)‘(𝑇𝑥)) = ((𝑅‘(𝑇𝑥)) + (𝑆‘(𝑇𝑥))))
121, 2, 11mp3an12 1452 . . . . . . 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 30748 . . . . . . 7 (𝑥 ∈ ℋ → ((𝑅𝑇)‘𝑥) = (𝑅‘(𝑇𝑥)))
152, 4hocoi 30748 . . . . . . 7 (𝑥 ∈ ℋ → ((𝑆𝑇)‘𝑥) = (𝑆‘(𝑇𝑥)))
1614, 15oveq12d 7376 . . . . . 6 (𝑥 ∈ ℋ → (((𝑅𝑇)‘𝑥) + ((𝑆𝑇)‘𝑥)) = ((𝑅‘(𝑇𝑥)) + (𝑆‘(𝑇𝑥))))
1713, 16eqtr4d 2776 . . . . 5 (𝑥 ∈ ℋ → ((𝑅 +op 𝑆)‘(𝑇𝑥)) = (((𝑅𝑇)‘𝑥) + ((𝑆𝑇)‘𝑥)))
189, 17eqtr4d 2776 . . . 4 (𝑥 ∈ ℋ → (((𝑅𝑇) +op (𝑆𝑇))‘𝑥) = ((𝑅 +op 𝑆)‘(𝑇𝑥)))
195, 18eqtr4d 2776 . . 3 (𝑥 ∈ ℋ → (((𝑅 +op 𝑆) ∘ 𝑇)‘𝑥) = (((𝑅𝑇) +op (𝑆𝑇))‘𝑥))
2019rgen 3063 . 2 𝑥 ∈ ℋ (((𝑅 +op 𝑆) ∘ 𝑇)‘𝑥) = (((𝑅𝑇) +op (𝑆𝑇))‘𝑥)
213, 4hocofi 30750 . . 3 ((𝑅 +op 𝑆) ∘ 𝑇): ℋ⟶ ℋ
226, 7hoaddcli 30752 . . 3 ((𝑅𝑇) +op (𝑆𝑇)): ℋ⟶ ℋ
2321, 22hoeqi 30745 . 2 (∀𝑥 ∈ ℋ (((𝑅 +op 𝑆) ∘ 𝑇)‘𝑥) = (((𝑅𝑇) +op (𝑆𝑇))‘𝑥) ↔ ((𝑅 +op 𝑆) ∘ 𝑇) = ((𝑅𝑇) +op (𝑆𝑇)))
2420, 23mpbi 229 1 ((𝑅 +op 𝑆) ∘ 𝑇) = ((𝑅𝑇) +op (𝑆𝑇))
Colors of variables: wff setvar class
Syntax hints:   = wceq 1542  wcel 2107  wral 3061  ccom 5638  wf 6493  cfv 6497  (class class class)co 7358  chba 29903   + cva 29904   +op chos 29922
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-rep 5243  ax-sep 5257  ax-nul 5264  ax-pow 5321  ax-pr 5385  ax-un 7673  ax-hilex 29983  ax-hfvadd 29984
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3353  df-rab 3407  df-v 3446  df-sbc 3741  df-csb 3857  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4284  df-if 4488  df-pw 4563  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4867  df-iun 4957  df-br 5107  df-opab 5169  df-mpt 5190  df-id 5532  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-rn 5645  df-res 5646  df-ima 5647  df-iota 6449  df-fun 6499  df-fn 6500  df-f 6501  df-f1 6502  df-fo 6503  df-f1o 6504  df-fv 6505  df-ov 7361  df-oprab 7362  df-mpo 7363  df-map 8770  df-hosum 30714
This theorem is referenced by: (None)
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