Theorem hfsmval 29194

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 10453	. . 3 ⊢ ℂ ∈ V
2		ax-hilex 28455	. . 3 ⊢ ℋ ∈ V
3	1, 2	elmap 8276	. 2 ⊢ (𝑆 ∈ (ℂ ↑𝑚 ℋ) ↔ 𝑆: ℋ⟶ℂ)
4	1, 2	elmap 8276	. 2 ⊢ (𝑇 ∈ (ℂ ↑𝑚 ℋ) ↔ 𝑇: ℋ⟶ℂ)
5		fveq1 6529	. . . . 5 ⊢ (𝑓 = 𝑆 → (𝑓‘𝑥) = (𝑆‘𝑥))
6	5	oveq1d 7022	. . . 4 ⊢ (𝑓 = 𝑆 → ((𝑓‘𝑥) + (𝑔‘𝑥)) = ((𝑆‘𝑥) + (𝑔‘𝑥)))
7	6	mpteq2dv 5050	. . 3 ⊢ (𝑓 = 𝑆 → (𝑥 ∈ ℋ ↦ ((𝑓‘𝑥) + (𝑔‘𝑥))) = (𝑥 ∈ ℋ ↦ ((𝑆‘𝑥) + (𝑔‘𝑥))))
8		fveq1 6529	. . . . 5 ⊢ (𝑔 = 𝑇 → (𝑔‘𝑥) = (𝑇‘𝑥))
9	8	oveq2d 7023	. . . 4 ⊢ (𝑔 = 𝑇 → ((𝑆‘𝑥) + (𝑔‘𝑥)) = ((𝑆‘𝑥) + (𝑇‘𝑥)))
10	9	mpteq2dv 5050	. . 3 ⊢ (𝑔 = 𝑇 → (𝑥 ∈ ℋ ↦ ((𝑆‘𝑥) + (𝑔‘𝑥))) = (𝑥 ∈ ℋ ↦ ((𝑆‘𝑥) + (𝑇‘𝑥))))
11		df-hfsum 29189	. . 3 ⊢ +fn = (𝑓 ∈ (ℂ ↑𝑚 ℋ), 𝑔 ∈ (ℂ ↑𝑚 ℋ) ↦ (𝑥 ∈ ℋ ↦ ((𝑓‘𝑥) + (𝑔‘𝑥))))
12	2	mptex 6843	. . 3 ⊢ (𝑥 ∈ ℋ ↦ ((𝑆‘𝑥) + (𝑇‘𝑥))) ∈ V
13	7, 10, 11, 12	ovmpo 7157	. 2 ⊢ ((𝑆 ∈ (ℂ ↑𝑚 ℋ) ∧ 𝑇 ∈ (ℂ ↑𝑚 ℋ)) → (𝑆 +fn 𝑇) = (𝑥 ∈ ℋ ↦ ((𝑆‘𝑥) + (𝑇‘𝑥))))
14	3, 4, 13	syl2anbr 598	1 ⊢ ((𝑆: ℋ⟶ℂ ∧ 𝑇: ℋ⟶ℂ) → (𝑆 +fn 𝑇) = (𝑥 ∈ ℋ ↦ ((𝑆‘𝑥) + (𝑇‘𝑥))))

Syntax hints:   → wi 4   ∧ wa 396   = wceq 1520   ∈ wcel 2079   ↦ cmpt 5035  ⟶wf 6213  ‘cfv 6217  (class class class)co 7007   ↑_𝑚 cmap 8247  ℂcc 10370   + caddc 10375   ℋchba 28375   +_fn chfs 28397

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1775  ax-4 1789  ax-5 1886  ax-6 1945  ax-7 1990  ax-8 2081  ax-9 2089  ax-10 2110  ax-11 2124  ax-12 2139  ax-13 2342  ax-ext 2767  ax-rep 5075  ax-sep 5088  ax-nul 5095  ax-pow 5150  ax-pr 5214  ax-un 7310  ax-cnex 10428  ax-hilex 28455

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 843  df-3an 1080  df-tru 1523  df-ex 1760  df-nf 1764  df-sb 2041  df-mo 2574  df-eu 2610  df-clab 2774  df-cleq 2786  df-clel 2861  df-nfc 2933  df-ne 2983  df-ral 3108  df-rex 3109  df-reu 3110  df-rab 3112  df-v 3434  df-sbc 3702  df-csb 3807  df-dif 3857  df-un 3859  df-in 3861  df-ss 3869  df-nul 4207  df-if 4376  df-pw 4449  df-sn 4467  df-pr 4469  df-op 4473  df-uni 4740  df-iun 4821  df-br 4957  df-opab 5019  df-mpt 5036  df-id 5340  df-xp 5441  df-rel 5442  df-cnv 5443  df-co 5444  df-dm 5445  df-rn 5446  df-res 5447  df-ima 5448  df-iota 6181  df-fun 6219  df-fn 6220  df-f 6221  df-f1 6222  df-fo 6223  df-f1o 6224  df-fv 6225  df-ov 7010  df-oprab 7011  df-mpo 7012  df-map 8249  df-hfsum 29189

This theorem is referenced by:  hfsval  29199