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Mirrors > Home > HSE Home > Th. List > hosval | Structured version Visualization version GIF version |
Description: Value of the sum of two Hilbert space operators. (Contributed by NM, 10-Nov-2000.) (Revised by Mario Carneiro, 16-Nov-2013.) (New usage is discouraged.) |
Ref | Expression |
---|---|
hosval | ⊢ ((𝑆: ℋ⟶ ℋ ∧ 𝑇: ℋ⟶ ℋ ∧ 𝐴 ∈ ℋ) → ((𝑆 +op 𝑇)‘𝐴) = ((𝑆‘𝐴) +ℎ (𝑇‘𝐴))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | hosmval 29439 | . . . 4 ⊢ ((𝑆: ℋ⟶ ℋ ∧ 𝑇: ℋ⟶ ℋ) → (𝑆 +op 𝑇) = (𝑥 ∈ ℋ ↦ ((𝑆‘𝑥) +ℎ (𝑇‘𝑥)))) | |
2 | 1 | fveq1d 6665 | . . 3 ⊢ ((𝑆: ℋ⟶ ℋ ∧ 𝑇: ℋ⟶ ℋ) → ((𝑆 +op 𝑇)‘𝐴) = ((𝑥 ∈ ℋ ↦ ((𝑆‘𝑥) +ℎ (𝑇‘𝑥)))‘𝐴)) |
3 | fveq2 6663 | . . . . 5 ⊢ (𝑥 = 𝐴 → (𝑆‘𝑥) = (𝑆‘𝐴)) | |
4 | fveq2 6663 | . . . . 5 ⊢ (𝑥 = 𝐴 → (𝑇‘𝑥) = (𝑇‘𝐴)) | |
5 | 3, 4 | oveq12d 7163 | . . . 4 ⊢ (𝑥 = 𝐴 → ((𝑆‘𝑥) +ℎ (𝑇‘𝑥)) = ((𝑆‘𝐴) +ℎ (𝑇‘𝐴))) |
6 | eqid 2818 | . . . 4 ⊢ (𝑥 ∈ ℋ ↦ ((𝑆‘𝑥) +ℎ (𝑇‘𝑥))) = (𝑥 ∈ ℋ ↦ ((𝑆‘𝑥) +ℎ (𝑇‘𝑥))) | |
7 | ovex 7178 | . . . 4 ⊢ ((𝑆‘𝐴) +ℎ (𝑇‘𝐴)) ∈ V | |
8 | 5, 6, 7 | fvmpt 6761 | . . 3 ⊢ (𝐴 ∈ ℋ → ((𝑥 ∈ ℋ ↦ ((𝑆‘𝑥) +ℎ (𝑇‘𝑥)))‘𝐴) = ((𝑆‘𝐴) +ℎ (𝑇‘𝐴))) |
9 | 2, 8 | sylan9eq 2873 | . 2 ⊢ (((𝑆: ℋ⟶ ℋ ∧ 𝑇: ℋ⟶ ℋ) ∧ 𝐴 ∈ ℋ) → ((𝑆 +op 𝑇)‘𝐴) = ((𝑆‘𝐴) +ℎ (𝑇‘𝐴))) |
10 | 9 | 3impa 1102 | 1 ⊢ ((𝑆: ℋ⟶ ℋ ∧ 𝑇: ℋ⟶ ℋ ∧ 𝐴 ∈ ℋ) → ((𝑆 +op 𝑇)‘𝐴) = ((𝑆‘𝐴) +ℎ (𝑇‘𝐴))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 ∧ w3a 1079 = wceq 1528 ∈ wcel 2105 ↦ cmpt 5137 ⟶wf 6344 ‘cfv 6348 (class class class)co 7145 ℋchba 28623 +ℎ cva 28624 +op chos 28642 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-rep 5181 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7450 ax-hilex 28703 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-ral 3140 df-rex 3141 df-reu 3142 df-rab 3144 df-v 3494 df-sbc 3770 df-csb 3881 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-nul 4289 df-if 4464 df-pw 4537 df-sn 4558 df-pr 4560 df-op 4564 df-uni 4831 df-iun 4912 df-br 5058 df-opab 5120 df-mpt 5138 df-id 5453 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-ov 7148 df-oprab 7149 df-mpo 7150 df-map 8397 df-hosum 29434 |
This theorem is referenced by: hoscl 29449 hoaddcomi 29476 hodsi 29479 hoaddassi 29480 hocadddiri 29483 hoaddid1i 29490 honegsubi 29500 hoadddi 29507 hoadddir 29508 lnophsi 29705 hmops 29724 adjadd 29797 nmoptrii 29798 leopadd 29836 pjsdii 29859 pjscji 29874 pjtoi 29883 |
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