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Mirrors > Home > HSE Home > Th. List > hodval | Structured version Visualization version GIF version |
Description: Value of the difference of two Hilbert space operators. (Contributed by NM, 10-Nov-2000.) (Revised by Mario Carneiro, 16-Nov-2013.) (New usage is discouraged.) |
Ref | Expression |
---|---|
hodval | ⊢ ((𝑆: ℋ⟶ ℋ ∧ 𝑇: ℋ⟶ ℋ ∧ 𝐴 ∈ ℋ) → ((𝑆 −op 𝑇)‘𝐴) = ((𝑆‘𝐴) −ℎ (𝑇‘𝐴))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | hodmval 27980 | . . . 4 ⊢ ((𝑆: ℋ⟶ ℋ ∧ 𝑇: ℋ⟶ ℋ) → (𝑆 −op 𝑇) = (𝑥 ∈ ℋ ↦ ((𝑆‘𝑥) −ℎ (𝑇‘𝑥)))) | |
2 | 1 | fveq1d 6105 | . . 3 ⊢ ((𝑆: ℋ⟶ ℋ ∧ 𝑇: ℋ⟶ ℋ) → ((𝑆 −op 𝑇)‘𝐴) = ((𝑥 ∈ ℋ ↦ ((𝑆‘𝑥) −ℎ (𝑇‘𝑥)))‘𝐴)) |
3 | fveq2 6103 | . . . . 5 ⊢ (𝑥 = 𝐴 → (𝑆‘𝑥) = (𝑆‘𝐴)) | |
4 | fveq2 6103 | . . . . 5 ⊢ (𝑥 = 𝐴 → (𝑇‘𝑥) = (𝑇‘𝐴)) | |
5 | 3, 4 | oveq12d 6567 | . . . 4 ⊢ (𝑥 = 𝐴 → ((𝑆‘𝑥) −ℎ (𝑇‘𝑥)) = ((𝑆‘𝐴) −ℎ (𝑇‘𝐴))) |
6 | eqid 2610 | . . . 4 ⊢ (𝑥 ∈ ℋ ↦ ((𝑆‘𝑥) −ℎ (𝑇‘𝑥))) = (𝑥 ∈ ℋ ↦ ((𝑆‘𝑥) −ℎ (𝑇‘𝑥))) | |
7 | ovex 6577 | . . . 4 ⊢ ((𝑆‘𝐴) −ℎ (𝑇‘𝐴)) ∈ V | |
8 | 5, 6, 7 | fvmpt 6191 | . . 3 ⊢ (𝐴 ∈ ℋ → ((𝑥 ∈ ℋ ↦ ((𝑆‘𝑥) −ℎ (𝑇‘𝑥)))‘𝐴) = ((𝑆‘𝐴) −ℎ (𝑇‘𝐴))) |
9 | 2, 8 | sylan9eq 2664 | . 2 ⊢ (((𝑆: ℋ⟶ ℋ ∧ 𝑇: ℋ⟶ ℋ) ∧ 𝐴 ∈ ℋ) → ((𝑆 −op 𝑇)‘𝐴) = ((𝑆‘𝐴) −ℎ (𝑇‘𝐴))) |
10 | 9 | 3impa 1251 | 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 383 ∧ w3a 1031 = wceq 1475 ∈ wcel 1977 ↦ cmpt 4643 ⟶wf 5800 ‘cfv 5804 (class class class)co 6549 ℋchil 27160 −ℎ cmv 27166 −op chod 27181 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-hilex 27240 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-ral 2901 df-rex 2902 df-reu 2903 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-id 4953 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-map 7746 df-hodif 27975 |
This theorem is referenced by: hodcl 27990 hodsi 28018 hocsubdiri 28023 honegsubi 28039 hoddii 28232 lnopeqi 28251 leop2 28367 pjddii 28399 pjssposi 28415 pjssdif2i 28417 |
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