Theorem hfsmval 31762

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 11259	. . 3 ⊢ ℂ ∈ V
2		ax-hilex 31023	. . 3 ⊢ ℋ ∈ V
3	1, 2	elmap 8923	. 2 ⊢ (𝑆 ∈ (ℂ ↑m ℋ) ↔ 𝑆: ℋ⟶ℂ)
4	1, 2	elmap 8923	. 2 ⊢ (𝑇 ∈ (ℂ ↑m ℋ) ↔ 𝑇: ℋ⟶ℂ)
5		fveq1 6914	. . . . 5 ⊢ (𝑓 = 𝑆 → (𝑓‘𝑥) = (𝑆‘𝑥))
6	5	oveq1d 7458	. . . 4 ⊢ (𝑓 = 𝑆 → ((𝑓‘𝑥) + (𝑔‘𝑥)) = ((𝑆‘𝑥) + (𝑔‘𝑥)))
7	6	mpteq2dv 5268	. . 3 ⊢ (𝑓 = 𝑆 → (𝑥 ∈ ℋ ↦ ((𝑓‘𝑥) + (𝑔‘𝑥))) = (𝑥 ∈ ℋ ↦ ((𝑆‘𝑥) + (𝑔‘𝑥))))
8		fveq1 6914	. . . . 5 ⊢ (𝑔 = 𝑇 → (𝑔‘𝑥) = (𝑇‘𝑥))
9	8	oveq2d 7459	. . . 4 ⊢ (𝑔 = 𝑇 → ((𝑆‘𝑥) + (𝑔‘𝑥)) = ((𝑆‘𝑥) + (𝑇‘𝑥)))
10	9	mpteq2dv 5268	. . 3 ⊢ (𝑔 = 𝑇 → (𝑥 ∈ ℋ ↦ ((𝑆‘𝑥) + (𝑔‘𝑥))) = (𝑥 ∈ ℋ ↦ ((𝑆‘𝑥) + (𝑇‘𝑥))))
11		df-hfsum 31757	. . 3 ⊢ +fn = (𝑓 ∈ (ℂ ↑m ℋ), 𝑔 ∈ (ℂ ↑m ℋ) ↦ (𝑥 ∈ ℋ ↦ ((𝑓‘𝑥) + (𝑔‘𝑥))))
12	2	mptex 7255	. . 3 ⊢ (𝑥 ∈ ℋ ↦ ((𝑆‘𝑥) + (𝑇‘𝑥))) ∈ V
13	7, 10, 11, 12	ovmpo 7604	. 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 395   = wceq 1537   ∈ wcel 2108   ↦ cmpt 5249  ⟶wf 6564  ‘cfv 6568  (class class class)co 7443   ↑_m cmap 8878  ℂcc 11176   + caddc 11181   ℋchba 30943   +_fn chfs 30965

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-rep 5303  ax-sep 5317  ax-nul 5324  ax-pow 5383  ax-pr 5447  ax-un 7764  ax-cnex 11234  ax-hilex 31023

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-ral 3068  df-rex 3077  df-reu 3389  df-rab 3444  df-v 3490  df-sbc 3805  df-csb 3922  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-nul 4353  df-if 4549  df-pw 4624  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-iun 5017  df-br 5167  df-opab 5229  df-mpt 5250  df-id 5593  df-xp 5701  df-rel 5702  df-cnv 5703  df-co 5704  df-dm 5705  df-rn 5706  df-res 5707  df-ima 5708  df-iota 6520  df-fun 6570  df-fn 6571  df-f 6572  df-f1 6573  df-fo 6574  df-f1o 6575  df-fv 6576  df-ov 7446  df-oprab 7447  df-mpo 7448  df-map 8880  df-hfsum 31757

This theorem is referenced by:  hfsval  31767