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Mirrors > Home > HSE Home > Th. List > hfsval | 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, 16-Nov-2013.) (New usage is discouraged.) |
Ref | Expression |
---|---|
hfsval | ⊢ ((𝑆: ℋ⟶ℂ ∧ 𝑇: ℋ⟶ℂ ∧ 𝐴 ∈ ℋ) → ((𝑆 +fn 𝑇)‘𝐴) = ((𝑆‘𝐴) + (𝑇‘𝐴))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | hfsmval 29517 | . . . 4 ⊢ ((𝑆: ℋ⟶ℂ ∧ 𝑇: ℋ⟶ℂ) → (𝑆 +fn 𝑇) = (𝑥 ∈ ℋ ↦ ((𝑆‘𝑥) + (𝑇‘𝑥)))) | |
2 | 1 | fveq1d 6674 | . . 3 ⊢ ((𝑆: ℋ⟶ℂ ∧ 𝑇: ℋ⟶ℂ) → ((𝑆 +fn 𝑇)‘𝐴) = ((𝑥 ∈ ℋ ↦ ((𝑆‘𝑥) + (𝑇‘𝑥)))‘𝐴)) |
3 | fveq2 6672 | . . . . 5 ⊢ (𝑥 = 𝐴 → (𝑆‘𝑥) = (𝑆‘𝐴)) | |
4 | fveq2 6672 | . . . . 5 ⊢ (𝑥 = 𝐴 → (𝑇‘𝑥) = (𝑇‘𝐴)) | |
5 | 3, 4 | oveq12d 7176 | . . . 4 ⊢ (𝑥 = 𝐴 → ((𝑆‘𝑥) + (𝑇‘𝑥)) = ((𝑆‘𝐴) + (𝑇‘𝐴))) |
6 | eqid 2823 | . . . 4 ⊢ (𝑥 ∈ ℋ ↦ ((𝑆‘𝑥) + (𝑇‘𝑥))) = (𝑥 ∈ ℋ ↦ ((𝑆‘𝑥) + (𝑇‘𝑥))) | |
7 | ovex 7191 | . . . 4 ⊢ ((𝑆‘𝐴) + (𝑇‘𝐴)) ∈ V | |
8 | 5, 6, 7 | fvmpt 6770 | . . 3 ⊢ (𝐴 ∈ ℋ → ((𝑥 ∈ ℋ ↦ ((𝑆‘𝑥) + (𝑇‘𝑥)))‘𝐴) = ((𝑆‘𝐴) + (𝑇‘𝐴))) |
9 | 2, 8 | sylan9eq 2878 | . 2 ⊢ (((𝑆: ℋ⟶ℂ ∧ 𝑇: ℋ⟶ℂ) ∧ 𝐴 ∈ ℋ) → ((𝑆 +fn 𝑇)‘𝐴) = ((𝑆‘𝐴) + (𝑇‘𝐴))) |
10 | 9 | 3impa 1106 | 1 ⊢ ((𝑆: ℋ⟶ℂ ∧ 𝑇: ℋ⟶ℂ ∧ 𝐴 ∈ ℋ) → ((𝑆 +fn 𝑇)‘𝐴) = ((𝑆‘𝐴) + (𝑇‘𝐴))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 ∧ w3a 1083 = wceq 1537 ∈ wcel 2114 ↦ cmpt 5148 ⟶wf 6353 ‘cfv 6357 (class class class)co 7158 ℂcc 10537 + caddc 10542 ℋchba 28698 +fn chfs 28720 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-rep 5192 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 ax-cnex 10595 ax-hilex 28778 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-ral 3145 df-rex 3146 df-reu 3147 df-rab 3149 df-v 3498 df-sbc 3775 df-csb 3886 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-op 4576 df-uni 4841 df-iun 4923 df-br 5069 df-opab 5131 df-mpt 5149 df-id 5462 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-f1 6362 df-fo 6363 df-f1o 6364 df-fv 6365 df-ov 7161 df-oprab 7162 df-mpo 7163 df-map 8410 df-hfsum 29512 |
This theorem is referenced by: (None) |
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