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Theorem hfsval 28909
Description: Value of the sum of two Hilbert space functionals. (Contributed by NM, 23-May-2006.) (Revised by Mario Carneiro, 16-Nov-2013.) (New usage is discouraged.)
Assertion
Ref Expression
hfsval ((𝑆: ℋ⟶ℂ ∧ 𝑇: ℋ⟶ℂ ∧ 𝐴 ∈ ℋ) → ((𝑆 +fn 𝑇)‘𝐴) = ((𝑆𝐴) + (𝑇𝐴)))

Proof of Theorem hfsval
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 hfsmval 28904 . . . 4 ((𝑆: ℋ⟶ℂ ∧ 𝑇: ℋ⟶ℂ) → (𝑆 +fn 𝑇) = (𝑥 ∈ ℋ ↦ ((𝑆𝑥) + (𝑇𝑥))))
21fveq1d 6352 . . 3 ((𝑆: ℋ⟶ℂ ∧ 𝑇: ℋ⟶ℂ) → ((𝑆 +fn 𝑇)‘𝐴) = ((𝑥 ∈ ℋ ↦ ((𝑆𝑥) + (𝑇𝑥)))‘𝐴))
3 fveq2 6350 . . . . 5 (𝑥 = 𝐴 → (𝑆𝑥) = (𝑆𝐴))
4 fveq2 6350 . . . . 5 (𝑥 = 𝐴 → (𝑇𝑥) = (𝑇𝐴))
53, 4oveq12d 6829 . . . 4 (𝑥 = 𝐴 → ((𝑆𝑥) + (𝑇𝑥)) = ((𝑆𝐴) + (𝑇𝐴)))
6 eqid 2758 . . . 4 (𝑥 ∈ ℋ ↦ ((𝑆𝑥) + (𝑇𝑥))) = (𝑥 ∈ ℋ ↦ ((𝑆𝑥) + (𝑇𝑥)))
7 ovex 6839 . . . 4 ((𝑆𝐴) + (𝑇𝐴)) ∈ V
85, 6, 7fvmpt 6442 . . 3 (𝐴 ∈ ℋ → ((𝑥 ∈ ℋ ↦ ((𝑆𝑥) + (𝑇𝑥)))‘𝐴) = ((𝑆𝐴) + (𝑇𝐴)))
92, 8sylan9eq 2812 . 2 (((𝑆: ℋ⟶ℂ ∧ 𝑇: ℋ⟶ℂ) ∧ 𝐴 ∈ ℋ) → ((𝑆 +fn 𝑇)‘𝐴) = ((𝑆𝐴) + (𝑇𝐴)))
1093impa 1101 1 ((𝑆: ℋ⟶ℂ ∧ 𝑇: ℋ⟶ℂ ∧ 𝐴 ∈ ℋ) → ((𝑆 +fn 𝑇)‘𝐴) = ((𝑆𝐴) + (𝑇𝐴)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383  w3a 1072   = wceq 1630  wcel 2137  cmpt 4879  wf 6043  cfv 6047  (class class class)co 6811  cc 10124   + caddc 10129  chil 28083   +fn chfs 28105
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1869  ax-4 1884  ax-5 1986  ax-6 2052  ax-7 2088  ax-8 2139  ax-9 2146  ax-10 2166  ax-11 2181  ax-12 2194  ax-13 2389  ax-ext 2738  ax-rep 4921  ax-sep 4931  ax-nul 4939  ax-pow 4990  ax-pr 5053  ax-un 7112  ax-cnex 10182  ax-hilex 28163
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1633  df-ex 1852  df-nf 1857  df-sb 2045  df-eu 2609  df-mo 2610  df-clab 2745  df-cleq 2751  df-clel 2754  df-nfc 2889  df-ne 2931  df-ral 3053  df-rex 3054  df-reu 3055  df-rab 3057  df-v 3340  df-sbc 3575  df-csb 3673  df-dif 3716  df-un 3718  df-in 3720  df-ss 3727  df-nul 4057  df-if 4229  df-pw 4302  df-sn 4320  df-pr 4322  df-op 4326  df-uni 4587  df-iun 4672  df-br 4803  df-opab 4863  df-mpt 4880  df-id 5172  df-xp 5270  df-rel 5271  df-cnv 5272  df-co 5273  df-dm 5274  df-rn 5275  df-res 5276  df-ima 5277  df-iota 6010  df-fun 6049  df-fn 6050  df-f 6051  df-f1 6052  df-fo 6053  df-f1o 6054  df-fv 6055  df-ov 6814  df-oprab 6815  df-mpt2 6816  df-map 8023  df-hfsum 28899
This theorem is referenced by: (None)
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