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Mirrors > Home > HSE Home > Th. List > hfsmval | Structured version Visualization version GIF version |
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.) |
Ref | Expression |
---|---|
hfsmval | ⊢ ((𝑆: ℋ⟶ℂ ∧ 𝑇: ℋ⟶ℂ) → (𝑆 +fn 𝑇) = (𝑥 ∈ ℋ ↦ ((𝑆‘𝑥) + (𝑇‘𝑥)))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cnex 10209 | . . 3 ⊢ ℂ ∈ V | |
2 | ax-hilex 28165 | . . 3 ⊢ ℋ ∈ V | |
3 | 1, 2 | elmap 8052 | . 2 ⊢ (𝑆 ∈ (ℂ ↑𝑚 ℋ) ↔ 𝑆: ℋ⟶ℂ) |
4 | 1, 2 | elmap 8052 | . 2 ⊢ (𝑇 ∈ (ℂ ↑𝑚 ℋ) ↔ 𝑇: ℋ⟶ℂ) |
5 | fveq1 6351 | . . . . 5 ⊢ (𝑓 = 𝑆 → (𝑓‘𝑥) = (𝑆‘𝑥)) | |
6 | 5 | oveq1d 6828 | . . . 4 ⊢ (𝑓 = 𝑆 → ((𝑓‘𝑥) + (𝑔‘𝑥)) = ((𝑆‘𝑥) + (𝑔‘𝑥))) |
7 | 6 | mpteq2dv 4897 | . . 3 ⊢ (𝑓 = 𝑆 → (𝑥 ∈ ℋ ↦ ((𝑓‘𝑥) + (𝑔‘𝑥))) = (𝑥 ∈ ℋ ↦ ((𝑆‘𝑥) + (𝑔‘𝑥)))) |
8 | fveq1 6351 | . . . . 5 ⊢ (𝑔 = 𝑇 → (𝑔‘𝑥) = (𝑇‘𝑥)) | |
9 | 8 | oveq2d 6829 | . . . 4 ⊢ (𝑔 = 𝑇 → ((𝑆‘𝑥) + (𝑔‘𝑥)) = ((𝑆‘𝑥) + (𝑇‘𝑥))) |
10 | 9 | mpteq2dv 4897 | . . 3 ⊢ (𝑔 = 𝑇 → (𝑥 ∈ ℋ ↦ ((𝑆‘𝑥) + (𝑔‘𝑥))) = (𝑥 ∈ ℋ ↦ ((𝑆‘𝑥) + (𝑇‘𝑥)))) |
11 | df-hfsum 28901 | . . 3 ⊢ +fn = (𝑓 ∈ (ℂ ↑𝑚 ℋ), 𝑔 ∈ (ℂ ↑𝑚 ℋ) ↦ (𝑥 ∈ ℋ ↦ ((𝑓‘𝑥) + (𝑔‘𝑥)))) | |
12 | 2 | mptex 6650 | . . 3 ⊢ (𝑥 ∈ ℋ ↦ ((𝑆‘𝑥) + (𝑇‘𝑥))) ∈ V |
13 | 7, 10, 11, 12 | ovmpt2 6961 | . 2 ⊢ ((𝑆 ∈ (ℂ ↑𝑚 ℋ) ∧ 𝑇 ∈ (ℂ ↑𝑚 ℋ)) → (𝑆 +fn 𝑇) = (𝑥 ∈ ℋ ↦ ((𝑆‘𝑥) + (𝑇‘𝑥)))) |
14 | 3, 4, 13 | syl2anbr 498 | 1 ⊢ ((𝑆: ℋ⟶ℂ ∧ 𝑇: ℋ⟶ℂ) → (𝑆 +fn 𝑇) = (𝑥 ∈ ℋ ↦ ((𝑆‘𝑥) + (𝑇‘𝑥)))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 383 = wceq 1632 ∈ wcel 2139 ↦ cmpt 4881 ⟶wf 6045 ‘cfv 6049 (class class class)co 6813 ↑𝑚 cmap 8023 ℂcc 10126 + caddc 10131 ℋchil 28085 +fn chfs 28107 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1871 ax-4 1886 ax-5 1988 ax-6 2054 ax-7 2090 ax-8 2141 ax-9 2148 ax-10 2168 ax-11 2183 ax-12 2196 ax-13 2391 ax-ext 2740 ax-rep 4923 ax-sep 4933 ax-nul 4941 ax-pow 4992 ax-pr 5055 ax-un 7114 ax-cnex 10184 ax-hilex 28165 |
This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-3an 1074 df-tru 1635 df-ex 1854 df-nf 1859 df-sb 2047 df-eu 2611 df-mo 2612 df-clab 2747 df-cleq 2753 df-clel 2756 df-nfc 2891 df-ne 2933 df-ral 3055 df-rex 3056 df-reu 3057 df-rab 3059 df-v 3342 df-sbc 3577 df-csb 3675 df-dif 3718 df-un 3720 df-in 3722 df-ss 3729 df-nul 4059 df-if 4231 df-pw 4304 df-sn 4322 df-pr 4324 df-op 4328 df-uni 4589 df-iun 4674 df-br 4805 df-opab 4865 df-mpt 4882 df-id 5174 df-xp 5272 df-rel 5273 df-cnv 5274 df-co 5275 df-dm 5276 df-rn 5277 df-res 5278 df-ima 5279 df-iota 6012 df-fun 6051 df-fn 6052 df-f 6053 df-f1 6054 df-fo 6055 df-f1o 6056 df-fv 6057 df-ov 6816 df-oprab 6817 df-mpt2 6818 df-map 8025 df-hfsum 28901 |
This theorem is referenced by: hfsval 28911 |
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