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Theorem hoaddassi 30426
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 30390 . . . . . 6 ((𝑅: ℋ⟶ ℋ ∧ 𝑆: ℋ⟶ ℋ ∧ 𝑥 ∈ ℋ) → ((𝑅 +op 𝑆)‘𝑥) = ((𝑅𝑥) + (𝑆𝑥)))
41, 2, 3mp3an12 1450 . . . . 5 (𝑥 ∈ ℋ → ((𝑅 +op 𝑆)‘𝑥) = ((𝑅𝑥) + (𝑆𝑥)))
54oveq1d 7352 . . . 4 (𝑥 ∈ ℋ → (((𝑅 +op 𝑆)‘𝑥) + (𝑇𝑥)) = (((𝑅𝑥) + (𝑆𝑥)) + (𝑇𝑥)))
61, 2hoaddcli 30418 . . . . 5 (𝑅 +op 𝑆): ℋ⟶ ℋ
7 hods.3 . . . . 5 𝑇: ℋ⟶ ℋ
8 hosval 30390 . . . . 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 1450 . . . 4 (𝑥 ∈ ℋ → (((𝑅 +op 𝑆) +op 𝑇)‘𝑥) = (((𝑅 +op 𝑆)‘𝑥) + (𝑇𝑥)))
10 hosval 30390 . . . . . . 7 ((𝑆: ℋ⟶ ℋ ∧ 𝑇: ℋ⟶ ℋ ∧ 𝑥 ∈ ℋ) → ((𝑆 +op 𝑇)‘𝑥) = ((𝑆𝑥) + (𝑇𝑥)))
112, 7, 10mp3an12 1450 . . . . . 6 (𝑥 ∈ ℋ → ((𝑆 +op 𝑇)‘𝑥) = ((𝑆𝑥) + (𝑇𝑥)))
1211oveq2d 7353 . . . . 5 (𝑥 ∈ ℋ → ((𝑅𝑥) + ((𝑆 +op 𝑇)‘𝑥)) = ((𝑅𝑥) + ((𝑆𝑥) + (𝑇𝑥))))
132, 7hoaddcli 30418 . . . . . 6 (𝑆 +op 𝑇): ℋ⟶ ℋ
14 hosval 30390 . . . . . 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 1450 . . . . 5 (𝑥 ∈ ℋ → ((𝑅 +op (𝑆 +op 𝑇))‘𝑥) = ((𝑅𝑥) + ((𝑆 +op 𝑇)‘𝑥)))
161ffvelcdmi 7016 . . . . . 6 (𝑥 ∈ ℋ → (𝑅𝑥) ∈ ℋ)
172ffvelcdmi 7016 . . . . . 6 (𝑥 ∈ ℋ → (𝑆𝑥) ∈ ℋ)
187ffvelcdmi 7016 . . . . . 6 (𝑥 ∈ ℋ → (𝑇𝑥) ∈ ℋ)
19 ax-hvass 29652 . . . . . 6 (((𝑅𝑥) ∈ ℋ ∧ (𝑆𝑥) ∈ ℋ ∧ (𝑇𝑥) ∈ ℋ) → (((𝑅𝑥) + (𝑆𝑥)) + (𝑇𝑥)) = ((𝑅𝑥) + ((𝑆𝑥) + (𝑇𝑥))))
2016, 17, 18, 19syl3anc 1370 . . . . 5 (𝑥 ∈ ℋ → (((𝑅𝑥) + (𝑆𝑥)) + (𝑇𝑥)) = ((𝑅𝑥) + ((𝑆𝑥) + (𝑇𝑥))))
2112, 15, 203eqtr4d 2786 . . . 4 (𝑥 ∈ ℋ → ((𝑅 +op (𝑆 +op 𝑇))‘𝑥) = (((𝑅𝑥) + (𝑆𝑥)) + (𝑇𝑥)))
225, 9, 213eqtr4d 2786 . . 3 (𝑥 ∈ ℋ → (((𝑅 +op 𝑆) +op 𝑇)‘𝑥) = ((𝑅 +op (𝑆 +op 𝑇))‘𝑥))
2322rgen 3063 . 2 𝑥 ∈ ℋ (((𝑅 +op 𝑆) +op 𝑇)‘𝑥) = ((𝑅 +op (𝑆 +op 𝑇))‘𝑥)
246, 7hoaddcli 30418 . . 3 ((𝑅 +op 𝑆) +op 𝑇): ℋ⟶ ℋ
251, 13hoaddcli 30418 . . 3 (𝑅 +op (𝑆 +op 𝑇)): ℋ⟶ ℋ
2624, 25hoeqi 30411 . 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 1540  wcel 2105  wral 3061  wf 6475  cfv 6479  (class class class)co 7337  chba 29569   + cva 29570   +op chos 29588
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2707  ax-rep 5229  ax-sep 5243  ax-nul 5250  ax-pow 5308  ax-pr 5372  ax-un 7650  ax-hilex 29649  ax-hfvadd 29650  ax-hvass 29652
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2886  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3350  df-rab 3404  df-v 3443  df-sbc 3728  df-csb 3844  df-dif 3901  df-un 3903  df-in 3905  df-ss 3915  df-nul 4270  df-if 4474  df-pw 4549  df-sn 4574  df-pr 4576  df-op 4580  df-uni 4853  df-iun 4943  df-br 5093  df-opab 5155  df-mpt 5176  df-id 5518  df-xp 5626  df-rel 5627  df-cnv 5628  df-co 5629  df-dm 5630  df-rn 5631  df-res 5632  df-ima 5633  df-iota 6431  df-fun 6481  df-fn 6482  df-f 6483  df-f1 6484  df-fo 6485  df-f1o 6486  df-fv 6487  df-ov 7340  df-oprab 7341  df-mpo 7342  df-map 8688  df-hosum 30380
This theorem is referenced by:  hoadd12i  30427  hoadd32i  30428  hoaddass  30432  hosubeq0i  30476
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