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Theorem hoaddassi 31808
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 31772 . . . . . 6 ((𝑅: ℋ⟶ ℋ ∧ 𝑆: ℋ⟶ ℋ ∧ 𝑥 ∈ ℋ) → ((𝑅 +op 𝑆)‘𝑥) = ((𝑅𝑥) + (𝑆𝑥)))
41, 2, 3mp3an12 1451 . . . . 5 (𝑥 ∈ ℋ → ((𝑅 +op 𝑆)‘𝑥) = ((𝑅𝑥) + (𝑆𝑥)))
54oveq1d 7463 . . . 4 (𝑥 ∈ ℋ → (((𝑅 +op 𝑆)‘𝑥) + (𝑇𝑥)) = (((𝑅𝑥) + (𝑆𝑥)) + (𝑇𝑥)))
61, 2hoaddcli 31800 . . . . 5 (𝑅 +op 𝑆): ℋ⟶ ℋ
7 hods.3 . . . . 5 𝑇: ℋ⟶ ℋ
8 hosval 31772 . . . . 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 1451 . . . 4 (𝑥 ∈ ℋ → (((𝑅 +op 𝑆) +op 𝑇)‘𝑥) = (((𝑅 +op 𝑆)‘𝑥) + (𝑇𝑥)))
10 hosval 31772 . . . . . . 7 ((𝑆: ℋ⟶ ℋ ∧ 𝑇: ℋ⟶ ℋ ∧ 𝑥 ∈ ℋ) → ((𝑆 +op 𝑇)‘𝑥) = ((𝑆𝑥) + (𝑇𝑥)))
112, 7, 10mp3an12 1451 . . . . . 6 (𝑥 ∈ ℋ → ((𝑆 +op 𝑇)‘𝑥) = ((𝑆𝑥) + (𝑇𝑥)))
1211oveq2d 7464 . . . . 5 (𝑥 ∈ ℋ → ((𝑅𝑥) + ((𝑆 +op 𝑇)‘𝑥)) = ((𝑅𝑥) + ((𝑆𝑥) + (𝑇𝑥))))
132, 7hoaddcli 31800 . . . . . 6 (𝑆 +op 𝑇): ℋ⟶ ℋ
14 hosval 31772 . . . . . 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 1451 . . . . 5 (𝑥 ∈ ℋ → ((𝑅 +op (𝑆 +op 𝑇))‘𝑥) = ((𝑅𝑥) + ((𝑆 +op 𝑇)‘𝑥)))
161ffvelcdmi 7117 . . . . . 6 (𝑥 ∈ ℋ → (𝑅𝑥) ∈ ℋ)
172ffvelcdmi 7117 . . . . . 6 (𝑥 ∈ ℋ → (𝑆𝑥) ∈ ℋ)
187ffvelcdmi 7117 . . . . . 6 (𝑥 ∈ ℋ → (𝑇𝑥) ∈ ℋ)
19 ax-hvass 31034 . . . . . 6 (((𝑅𝑥) ∈ ℋ ∧ (𝑆𝑥) ∈ ℋ ∧ (𝑇𝑥) ∈ ℋ) → (((𝑅𝑥) + (𝑆𝑥)) + (𝑇𝑥)) = ((𝑅𝑥) + ((𝑆𝑥) + (𝑇𝑥))))
2016, 17, 18, 19syl3anc 1371 . . . . 5 (𝑥 ∈ ℋ → (((𝑅𝑥) + (𝑆𝑥)) + (𝑇𝑥)) = ((𝑅𝑥) + ((𝑆𝑥) + (𝑇𝑥))))
2112, 15, 203eqtr4d 2790 . . . 4 (𝑥 ∈ ℋ → ((𝑅 +op (𝑆 +op 𝑇))‘𝑥) = (((𝑅𝑥) + (𝑆𝑥)) + (𝑇𝑥)))
225, 9, 213eqtr4d 2790 . . 3 (𝑥 ∈ ℋ → (((𝑅 +op 𝑆) +op 𝑇)‘𝑥) = ((𝑅 +op (𝑆 +op 𝑇))‘𝑥))
2322rgen 3069 . 2 𝑥 ∈ ℋ (((𝑅 +op 𝑆) +op 𝑇)‘𝑥) = ((𝑅 +op (𝑆 +op 𝑇))‘𝑥)
246, 7hoaddcli 31800 . . 3 ((𝑅 +op 𝑆) +op 𝑇): ℋ⟶ ℋ
251, 13hoaddcli 31800 . . 3 (𝑅 +op (𝑆 +op 𝑇)): ℋ⟶ ℋ
2624, 25hoeqi 31793 . 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 230 1 ((𝑅 +op 𝑆) +op 𝑇) = (𝑅 +op (𝑆 +op 𝑇))
Colors of variables: wff setvar class
Syntax hints:   = wceq 1537  wcel 2108  wral 3067  wf 6569  cfv 6573  (class class class)co 7448  chba 30951   + cva 30952   +op chos 30970
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-rep 5303  ax-sep 5317  ax-nul 5324  ax-pow 5383  ax-pr 5447  ax-un 7770  ax-hilex 31031  ax-hfvadd 31032  ax-hvass 31034
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-ral 3068  df-rex 3077  df-reu 3389  df-rab 3444  df-v 3490  df-sbc 3805  df-csb 3922  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-nul 4353  df-if 4549  df-pw 4624  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-iun 5017  df-br 5167  df-opab 5229  df-mpt 5250  df-id 5593  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-rn 5711  df-res 5712  df-ima 5713  df-iota 6525  df-fun 6575  df-fn 6576  df-f 6577  df-f1 6578  df-fo 6579  df-f1o 6580  df-fv 6581  df-ov 7451  df-oprab 7452  df-mpo 7453  df-map 8886  df-hosum 31762
This theorem is referenced by:  hoadd12i  31809  hoadd32i  31810  hoaddass  31814  hosubeq0i  31858
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