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Theorem hocadddiri 29714
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 29703 . . . . 5 (𝑅 +op 𝑆): ℋ⟶ ℋ
4 hods.3 . . . . 5 𝑇: ℋ⟶ ℋ
53, 4hocoi 29699 . . . 4 (𝑥 ∈ ℋ → (((𝑅 +op 𝑆) ∘ 𝑇)‘𝑥) = ((𝑅 +op 𝑆)‘(𝑇𝑥)))
61, 4hocofi 29701 . . . . . 6 (𝑅𝑇): ℋ⟶ ℋ
72, 4hocofi 29701 . . . . . 6 (𝑆𝑇): ℋ⟶ ℋ
8 hosval 29675 . . . . . 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)))
104ffvelrni 6860 . . . . . . 7 (𝑥 ∈ ℋ → (𝑇𝑥) ∈ ℋ)
11 hosval 29675 . . . . . . . 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 29699 . . . . . . 7 (𝑥 ∈ ℋ → ((𝑅𝑇)‘𝑥) = (𝑅‘(𝑇𝑥)))
152, 4hocoi 29699 . . . . . . 7 (𝑥 ∈ ℋ → ((𝑆𝑇)‘𝑥) = (𝑆‘(𝑇𝑥)))
1614, 15oveq12d 7188 . . . . . 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 29701 . . 3 ((𝑅 +op 𝑆) ∘ 𝑇): ℋ⟶ ℋ
226, 7hoaddcli 29703 . . 3 ((𝑅𝑇) +op (𝑆𝑇)): ℋ⟶ ℋ
2321, 22hoeqi 29696 . 2 (∀𝑥 ∈ ℋ (((𝑅 +op 𝑆) ∘ 𝑇)‘𝑥) = (((𝑅𝑇) +op (𝑆𝑇))‘𝑥) ↔ ((𝑅 +op 𝑆) ∘ 𝑇) = ((𝑅𝑇) +op (𝑆𝑇)))
2420, 23mpbi 233 1 ((𝑅 +op 𝑆) ∘ 𝑇) = ((𝑅𝑇) +op (𝑆𝑇))
Colors of variables: wff setvar class
Syntax hints:   = wceq 1542  wcel 2114  wral 3053  ccom 5529  wf 6335  cfv 6339  (class class class)co 7170  chba 28854   + cva 28855   +op chos 28873
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2020  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2162  ax-12 2179  ax-ext 2710  ax-rep 5154  ax-sep 5167  ax-nul 5174  ax-pow 5232  ax-pr 5296  ax-un 7479  ax-hilex 28934  ax-hfvadd 28935
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1787  df-nf 1791  df-sb 2075  df-mo 2540  df-eu 2570  df-clab 2717  df-cleq 2730  df-clel 2811  df-nfc 2881  df-ne 2935  df-ral 3058  df-rex 3059  df-reu 3060  df-rab 3062  df-v 3400  df-sbc 3681  df-csb 3791  df-dif 3846  df-un 3848  df-in 3850  df-ss 3860  df-nul 4212  df-if 4415  df-pw 4490  df-sn 4517  df-pr 4519  df-op 4523  df-uni 4797  df-iun 4883  df-br 5031  df-opab 5093  df-mpt 5111  df-id 5429  df-xp 5531  df-rel 5532  df-cnv 5533  df-co 5534  df-dm 5535  df-rn 5536  df-res 5537  df-ima 5538  df-iota 6297  df-fun 6341  df-fn 6342  df-f 6343  df-f1 6344  df-fo 6345  df-f1o 6346  df-fv 6347  df-ov 7173  df-oprab 7174  df-mpo 7175  df-map 8439  df-hosum 29665
This theorem is referenced by: (None)
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