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Theorem hfsmval 31757
Description: Value of the sum of two Hilbert space functionals. (Contributed by NM, 23-May-2006.) (Revised by Mario Carneiro, 23-Aug-2014.) (New usage is discouraged.)
Assertion
Ref Expression
hfsmval ((𝑆: ℋ⟶ℂ ∧ 𝑇: ℋ⟶ℂ) → (𝑆 +fn 𝑇) = (𝑥 ∈ ℋ ↦ ((𝑆𝑥) + (𝑇𝑥))))
Distinct variable groups:   𝑥,𝑆   𝑥,𝑇

Proof of Theorem hfsmval
Dummy variables 𝑓 𝑔 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 cnex 11236 . . 3 ℂ ∈ V
2 ax-hilex 31018 . . 3 ℋ ∈ V
31, 2elmap 8911 . 2 (𝑆 ∈ (ℂ ↑m ℋ) ↔ 𝑆: ℋ⟶ℂ)
41, 2elmap 8911 . 2 (𝑇 ∈ (ℂ ↑m ℋ) ↔ 𝑇: ℋ⟶ℂ)
5 fveq1 6905 . . . . 5 (𝑓 = 𝑆 → (𝑓𝑥) = (𝑆𝑥))
65oveq1d 7446 . . . 4 (𝑓 = 𝑆 → ((𝑓𝑥) + (𝑔𝑥)) = ((𝑆𝑥) + (𝑔𝑥)))
76mpteq2dv 5244 . . 3 (𝑓 = 𝑆 → (𝑥 ∈ ℋ ↦ ((𝑓𝑥) + (𝑔𝑥))) = (𝑥 ∈ ℋ ↦ ((𝑆𝑥) + (𝑔𝑥))))
8 fveq1 6905 . . . . 5 (𝑔 = 𝑇 → (𝑔𝑥) = (𝑇𝑥))
98oveq2d 7447 . . . 4 (𝑔 = 𝑇 → ((𝑆𝑥) + (𝑔𝑥)) = ((𝑆𝑥) + (𝑇𝑥)))
109mpteq2dv 5244 . . 3 (𝑔 = 𝑇 → (𝑥 ∈ ℋ ↦ ((𝑆𝑥) + (𝑔𝑥))) = (𝑥 ∈ ℋ ↦ ((𝑆𝑥) + (𝑇𝑥))))
11 df-hfsum 31752 . . 3 +fn = (𝑓 ∈ (ℂ ↑m ℋ), 𝑔 ∈ (ℂ ↑m ℋ) ↦ (𝑥 ∈ ℋ ↦ ((𝑓𝑥) + (𝑔𝑥))))
122mptex 7243 . . 3 (𝑥 ∈ ℋ ↦ ((𝑆𝑥) + (𝑇𝑥))) ∈ V
137, 10, 11, 12ovmpo 7593 . 2 ((𝑆 ∈ (ℂ ↑m ℋ) ∧ 𝑇 ∈ (ℂ ↑m ℋ)) → (𝑆 +fn 𝑇) = (𝑥 ∈ ℋ ↦ ((𝑆𝑥) + (𝑇𝑥))))
143, 4, 13syl2anbr 599 1 ((𝑆: ℋ⟶ℂ ∧ 𝑇: ℋ⟶ℂ) → (𝑆 +fn 𝑇) = (𝑥 ∈ ℋ ↦ ((𝑆𝑥) + (𝑇𝑥))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1540  wcel 2108  cmpt 5225  wf 6557  cfv 6561  (class class class)co 7431  m cmap 8866  cc 11153   + caddc 11158  chba 30938   +fn chfs 30960
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755  ax-cnex 11211  ax-hilex 31018
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-ov 7434  df-oprab 7435  df-mpo 7436  df-map 8868  df-hfsum 31752
This theorem is referenced by:  hfsval  31762
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