Theorem hfsmval 31766

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 11233	. . 3 ⊢ ℂ ∈ V
2		ax-hilex 31027	. . 3 ⊢ ℋ ∈ V
3	1, 2	elmap 8909	. 2 ⊢ (𝑆 ∈ (ℂ ↑m ℋ) ↔ 𝑆: ℋ⟶ℂ)
4	1, 2	elmap 8909	. 2 ⊢ (𝑇 ∈ (ℂ ↑m ℋ) ↔ 𝑇: ℋ⟶ℂ)
5		fveq1 6905	. . . . 5 ⊢ (𝑓 = 𝑆 → (𝑓‘𝑥) = (𝑆‘𝑥))
6	5	oveq1d 7445	. . . 4 ⊢ (𝑓 = 𝑆 → ((𝑓‘𝑥) + (𝑔‘𝑥)) = ((𝑆‘𝑥) + (𝑔‘𝑥)))
7	6	mpteq2dv 5249	. . 3 ⊢ (𝑓 = 𝑆 → (𝑥 ∈ ℋ ↦ ((𝑓‘𝑥) + (𝑔‘𝑥))) = (𝑥 ∈ ℋ ↦ ((𝑆‘𝑥) + (𝑔‘𝑥))))
8		fveq1 6905	. . . . 5 ⊢ (𝑔 = 𝑇 → (𝑔‘𝑥) = (𝑇‘𝑥))
9	8	oveq2d 7446	. . . 4 ⊢ (𝑔 = 𝑇 → ((𝑆‘𝑥) + (𝑔‘𝑥)) = ((𝑆‘𝑥) + (𝑇‘𝑥)))
10	9	mpteq2dv 5249	. . 3 ⊢ (𝑔 = 𝑇 → (𝑥 ∈ ℋ ↦ ((𝑆‘𝑥) + (𝑔‘𝑥))) = (𝑥 ∈ ℋ ↦ ((𝑆‘𝑥) + (𝑇‘𝑥))))
11		df-hfsum 31761	. . 3 ⊢ +fn = (𝑓 ∈ (ℂ ↑m ℋ), 𝑔 ∈ (ℂ ↑m ℋ) ↦ (𝑥 ∈ ℋ ↦ ((𝑓‘𝑥) + (𝑔‘𝑥))))
12	2	mptex 7242	. . 3 ⊢ (𝑥 ∈ ℋ ↦ ((𝑆‘𝑥) + (𝑇‘𝑥))) ∈ V
13	7, 10, 11, 12	ovmpo 7592	. 2 ⊢ ((𝑆 ∈ (ℂ ↑m ℋ) ∧ 𝑇 ∈ (ℂ ↑m ℋ)) → (𝑆 +fn 𝑇) = (𝑥 ∈ ℋ ↦ ((𝑆‘𝑥) + (𝑇‘𝑥))))
14	3, 4, 13	syl2anbr 599	1 ⊢ ((𝑆: ℋ⟶ℂ ∧ 𝑇: ℋ⟶ℂ) → (𝑆 +fn 𝑇) = (𝑥 ∈ ℋ ↦ ((𝑆‘𝑥) + (𝑇‘𝑥))))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1536   ∈ wcel 2105   ↦ cmpt 5230  ⟶wf 6558  ‘cfv 6562  (class class class)co 7430   ↑_m cmap 8864  ℂcc 11150   + caddc 11155   ℋchba 30947   +_fn chfs 30969

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-10 2138  ax-11 2154  ax-12 2174  ax-ext 2705  ax-rep 5284  ax-sep 5301  ax-nul 5311  ax-pow 5370  ax-pr 5437  ax-un 7753  ax-cnex 11208  ax-hilex 31027

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1539  df-fal 1549  df-ex 1776  df-nf 1780  df-sb 2062  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2726  df-clel 2813  df-nfc 2889  df-ne 2938  df-ral 3059  df-rex 3068  df-reu 3378  df-rab 3433  df-v 3479  df-sbc 3791  df-csb 3908  df-dif 3965  df-un 3967  df-in 3969  df-ss 3979  df-nul 4339  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4912  df-iun 4997  df-br 5148  df-opab 5210  df-mpt 5231  df-id 5582  df-xp 5694  df-rel 5695  df-cnv 5696  df-co 5697  df-dm 5698  df-rn 5699  df-res 5700  df-ima 5701  df-iota 6515  df-fun 6564  df-fn 6565  df-f 6566  df-f1 6567  df-fo 6568  df-f1o 6569  df-fv 6570  df-ov 7433  df-oprab 7434  df-mpo 7435  df-map 8866  df-hfsum 31761

This theorem is referenced by:  hfsval  31771