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Theorem hoaddassi 31630
Description: Associativity of sum of Hilbert space operators. (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
hoaddassi ((𝑅 +op 𝑆) +op 𝑇) = (𝑅 +op (𝑆 +op 𝑇))

Proof of Theorem hoaddassi
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
3 hosval 31594 . . . . . 6 ((𝑅: ℋ⟶ ℋ ∧ 𝑆: ℋ⟶ ℋ ∧ 𝑥 ∈ ℋ) → ((𝑅 +op 𝑆)‘𝑥) = ((𝑅𝑥) + (𝑆𝑥)))
41, 2, 3mp3an12 1447 . . . . 5 (𝑥 ∈ ℋ → ((𝑅 +op 𝑆)‘𝑥) = ((𝑅𝑥) + (𝑆𝑥)))
54oveq1d 7431 . . . 4 (𝑥 ∈ ℋ → (((𝑅 +op 𝑆)‘𝑥) + (𝑇𝑥)) = (((𝑅𝑥) + (𝑆𝑥)) + (𝑇𝑥)))
61, 2hoaddcli 31622 . . . . 5 (𝑅 +op 𝑆): ℋ⟶ ℋ
7 hods.3 . . . . 5 𝑇: ℋ⟶ ℋ
8 hosval 31594 . . . . 5 (((𝑅 +op 𝑆): ℋ⟶ ℋ ∧ 𝑇: ℋ⟶ ℋ ∧ 𝑥 ∈ ℋ) → (((𝑅 +op 𝑆) +op 𝑇)‘𝑥) = (((𝑅 +op 𝑆)‘𝑥) + (𝑇𝑥)))
96, 7, 8mp3an12 1447 . . . 4 (𝑥 ∈ ℋ → (((𝑅 +op 𝑆) +op 𝑇)‘𝑥) = (((𝑅 +op 𝑆)‘𝑥) + (𝑇𝑥)))
10 hosval 31594 . . . . . . 7 ((𝑆: ℋ⟶ ℋ ∧ 𝑇: ℋ⟶ ℋ ∧ 𝑥 ∈ ℋ) → ((𝑆 +op 𝑇)‘𝑥) = ((𝑆𝑥) + (𝑇𝑥)))
112, 7, 10mp3an12 1447 . . . . . 6 (𝑥 ∈ ℋ → ((𝑆 +op 𝑇)‘𝑥) = ((𝑆𝑥) + (𝑇𝑥)))
1211oveq2d 7432 . . . . 5 (𝑥 ∈ ℋ → ((𝑅𝑥) + ((𝑆 +op 𝑇)‘𝑥)) = ((𝑅𝑥) + ((𝑆𝑥) + (𝑇𝑥))))
132, 7hoaddcli 31622 . . . . . 6 (𝑆 +op 𝑇): ℋ⟶ ℋ
14 hosval 31594 . . . . . 6 ((𝑅: ℋ⟶ ℋ ∧ (𝑆 +op 𝑇): ℋ⟶ ℋ ∧ 𝑥 ∈ ℋ) → ((𝑅 +op (𝑆 +op 𝑇))‘𝑥) = ((𝑅𝑥) + ((𝑆 +op 𝑇)‘𝑥)))
151, 13, 14mp3an12 1447 . . . . 5 (𝑥 ∈ ℋ → ((𝑅 +op (𝑆 +op 𝑇))‘𝑥) = ((𝑅𝑥) + ((𝑆 +op 𝑇)‘𝑥)))
161ffvelcdmi 7088 . . . . . 6 (𝑥 ∈ ℋ → (𝑅𝑥) ∈ ℋ)
172ffvelcdmi 7088 . . . . . 6 (𝑥 ∈ ℋ → (𝑆𝑥) ∈ ℋ)
187ffvelcdmi 7088 . . . . . 6 (𝑥 ∈ ℋ → (𝑇𝑥) ∈ ℋ)
19 ax-hvass 30856 . . . . . 6 (((𝑅𝑥) ∈ ℋ ∧ (𝑆𝑥) ∈ ℋ ∧ (𝑇𝑥) ∈ ℋ) → (((𝑅𝑥) + (𝑆𝑥)) + (𝑇𝑥)) = ((𝑅𝑥) + ((𝑆𝑥) + (𝑇𝑥))))
2016, 17, 18, 19syl3anc 1368 . . . . 5 (𝑥 ∈ ℋ → (((𝑅𝑥) + (𝑆𝑥)) + (𝑇𝑥)) = ((𝑅𝑥) + ((𝑆𝑥) + (𝑇𝑥))))
2112, 15, 203eqtr4d 2775 . . . 4 (𝑥 ∈ ℋ → ((𝑅 +op (𝑆 +op 𝑇))‘𝑥) = (((𝑅𝑥) + (𝑆𝑥)) + (𝑇𝑥)))
225, 9, 213eqtr4d 2775 . . 3 (𝑥 ∈ ℋ → (((𝑅 +op 𝑆) +op 𝑇)‘𝑥) = ((𝑅 +op (𝑆 +op 𝑇))‘𝑥))
2322rgen 3053 . 2 𝑥 ∈ ℋ (((𝑅 +op 𝑆) +op 𝑇)‘𝑥) = ((𝑅 +op (𝑆 +op 𝑇))‘𝑥)
246, 7hoaddcli 31622 . . 3 ((𝑅 +op 𝑆) +op 𝑇): ℋ⟶ ℋ
251, 13hoaddcli 31622 . . 3 (𝑅 +op (𝑆 +op 𝑇)): ℋ⟶ ℋ
2624, 25hoeqi 31615 . 2 (∀𝑥 ∈ ℋ (((𝑅 +op 𝑆) +op 𝑇)‘𝑥) = ((𝑅 +op (𝑆 +op 𝑇))‘𝑥) ↔ ((𝑅 +op 𝑆) +op 𝑇) = (𝑅 +op (𝑆 +op 𝑇)))
2723, 26mpbi 229 1 ((𝑅 +op 𝑆) +op 𝑇) = (𝑅 +op (𝑆 +op 𝑇))
Colors of variables: wff setvar class
Syntax hints:   = wceq 1533  wcel 2098  wral 3051  wf 6539  cfv 6543  (class class class)co 7416  chba 30773   + cva 30774   +op chos 30792
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696  ax-rep 5280  ax-sep 5294  ax-nul 5301  ax-pow 5359  ax-pr 5423  ax-un 7738  ax-hilex 30853  ax-hfvadd 30854  ax-hvass 30856
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ne 2931  df-ral 3052  df-rex 3061  df-reu 3365  df-rab 3420  df-v 3465  df-sbc 3769  df-csb 3885  df-dif 3942  df-un 3944  df-in 3946  df-ss 3956  df-nul 4319  df-if 4525  df-pw 4600  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4904  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5227  df-id 5570  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-rn 5683  df-res 5684  df-ima 5685  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-ov 7419  df-oprab 7420  df-mpo 7421  df-map 8845  df-hosum 31584
This theorem is referenced by:  hoadd12i  31631  hoadd32i  31632  hoaddass  31636  hosubeq0i  31680
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