Theorem hfsmval 27981

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 ↦ ((𝑆‘𝑥) + (𝑇‘𝑥))))

Distinct variable groups:   𝑥,𝑆   𝑥,𝑇

Proof of Theorem hfsmval

Dummy variables 𝑓 𝑔 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		cnex 9896	. . 3 ⊢ ℂ ∈ V
2		ax-hilex 27240	. . 3 ⊢ ℋ ∈ V
3	1, 2	elmap 7772	. 2 ⊢ (𝑆 ∈ (ℂ ↑𝑚 ℋ) ↔ 𝑆: ℋ⟶ℂ)
4	1, 2	elmap 7772	. 2 ⊢ (𝑇 ∈ (ℂ ↑𝑚 ℋ) ↔ 𝑇: ℋ⟶ℂ)
5		fveq1 6102	. . . . 5 ⊢ (𝑓 = 𝑆 → (𝑓‘𝑥) = (𝑆‘𝑥))
6	5	oveq1d 6564	. . . 4 ⊢ (𝑓 = 𝑆 → ((𝑓‘𝑥) + (𝑔‘𝑥)) = ((𝑆‘𝑥) + (𝑔‘𝑥)))
7	6	mpteq2dv 4673	. . 3 ⊢ (𝑓 = 𝑆 → (𝑥 ∈ ℋ ↦ ((𝑓‘𝑥) + (𝑔‘𝑥))) = (𝑥 ∈ ℋ ↦ ((𝑆‘𝑥) + (𝑔‘𝑥))))
8		fveq1 6102	. . . . 5 ⊢ (𝑔 = 𝑇 → (𝑔‘𝑥) = (𝑇‘𝑥))
9	8	oveq2d 6565	. . . 4 ⊢ (𝑔 = 𝑇 → ((𝑆‘𝑥) + (𝑔‘𝑥)) = ((𝑆‘𝑥) + (𝑇‘𝑥)))
10	9	mpteq2dv 4673	. . 3 ⊢ (𝑔 = 𝑇 → (𝑥 ∈ ℋ ↦ ((𝑆‘𝑥) + (𝑔‘𝑥))) = (𝑥 ∈ ℋ ↦ ((𝑆‘𝑥) + (𝑇‘𝑥))))
11		df-hfsum 27976	. . 3 ⊢ +fn = (𝑓 ∈ (ℂ ↑𝑚 ℋ), 𝑔 ∈ (ℂ ↑𝑚 ℋ) ↦ (𝑥 ∈ ℋ ↦ ((𝑓‘𝑥) + (𝑔‘𝑥))))
12	2	mptex 6390	. . 3 ⊢ (𝑥 ∈ ℋ ↦ ((𝑆‘𝑥) + (𝑇‘𝑥))) ∈ V
13	7, 10, 11, 12	ovmpt2 6694	. 2 ⊢ ((𝑆 ∈ (ℂ ↑𝑚 ℋ) ∧ 𝑇 ∈ (ℂ ↑𝑚 ℋ)) → (𝑆 +fn 𝑇) = (𝑥 ∈ ℋ ↦ ((𝑆‘𝑥) + (𝑇‘𝑥))))
14	3, 4, 13	syl2anbr 496	1 ⊢ ((𝑆: ℋ⟶ℂ ∧ 𝑇: ℋ⟶ℂ) → (𝑆 +fn 𝑇) = (𝑥 ∈ ℋ ↦ ((𝑆‘𝑥) + (𝑇‘𝑥))))

Syntax hints:   → wi 4   ∧ wa 383   = wceq 1475   ∈ wcel 1977   ↦ cmpt 4643  ⟶wf 5800  ‘cfv 5804  (class class class)co 6549   ↑_𝑚 cmap 7744  ℂcc 9813   + caddc 9818   ℋchil 27160   +_fn chfs 27182

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-cnex 9871  ax-hilex 27240

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-map 7746  df-hfsum 27976

This theorem is referenced by:  hfsval  27986