![]() |
Hilbert Space Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > HSE Home > Th. List > hodmval | Structured version Visualization version GIF version |
Description: Value of the difference of two Hilbert space operators. (Contributed by NM, 9-Nov-2000.) (Revised by Mario Carneiro, 23-Aug-2014.) (New usage is discouraged.) |
Ref | Expression |
---|---|
hodmval | ⊢ ((𝑆: ℋ⟶ ℋ ∧ 𝑇: ℋ⟶ ℋ) → (𝑆 −op 𝑇) = (𝑥 ∈ ℋ ↦ ((𝑆‘𝑥) −ℎ (𝑇‘𝑥)))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ax-hilex 30756 | . . 3 ⊢ ℋ ∈ V | |
2 | 1, 1 | elmap 8864 | . 2 ⊢ (𝑆 ∈ ( ℋ ↑m ℋ) ↔ 𝑆: ℋ⟶ ℋ) |
3 | 1, 1 | elmap 8864 | . 2 ⊢ (𝑇 ∈ ( ℋ ↑m ℋ) ↔ 𝑇: ℋ⟶ ℋ) |
4 | fveq1 6883 | . . . . 5 ⊢ (𝑓 = 𝑆 → (𝑓‘𝑥) = (𝑆‘𝑥)) | |
5 | 4 | oveq1d 7419 | . . . 4 ⊢ (𝑓 = 𝑆 → ((𝑓‘𝑥) −ℎ (𝑔‘𝑥)) = ((𝑆‘𝑥) −ℎ (𝑔‘𝑥))) |
6 | 5 | mpteq2dv 5243 | . . 3 ⊢ (𝑓 = 𝑆 → (𝑥 ∈ ℋ ↦ ((𝑓‘𝑥) −ℎ (𝑔‘𝑥))) = (𝑥 ∈ ℋ ↦ ((𝑆‘𝑥) −ℎ (𝑔‘𝑥)))) |
7 | fveq1 6883 | . . . . 5 ⊢ (𝑔 = 𝑇 → (𝑔‘𝑥) = (𝑇‘𝑥)) | |
8 | 7 | oveq2d 7420 | . . . 4 ⊢ (𝑔 = 𝑇 → ((𝑆‘𝑥) −ℎ (𝑔‘𝑥)) = ((𝑆‘𝑥) −ℎ (𝑇‘𝑥))) |
9 | 8 | mpteq2dv 5243 | . . 3 ⊢ (𝑔 = 𝑇 → (𝑥 ∈ ℋ ↦ ((𝑆‘𝑥) −ℎ (𝑔‘𝑥))) = (𝑥 ∈ ℋ ↦ ((𝑆‘𝑥) −ℎ (𝑇‘𝑥)))) |
10 | df-hodif 31489 | . . 3 ⊢ −op = (𝑓 ∈ ( ℋ ↑m ℋ), 𝑔 ∈ ( ℋ ↑m ℋ) ↦ (𝑥 ∈ ℋ ↦ ((𝑓‘𝑥) −ℎ (𝑔‘𝑥)))) | |
11 | 1 | mptex 7219 | . . 3 ⊢ (𝑥 ∈ ℋ ↦ ((𝑆‘𝑥) −ℎ (𝑇‘𝑥))) ∈ V |
12 | 6, 9, 10, 11 | ovmpo 7563 | . 2 ⊢ ((𝑆 ∈ ( ℋ ↑m ℋ) ∧ 𝑇 ∈ ( ℋ ↑m ℋ)) → (𝑆 −op 𝑇) = (𝑥 ∈ ℋ ↦ ((𝑆‘𝑥) −ℎ (𝑇‘𝑥)))) |
13 | 2, 3, 12 | syl2anbr 598 | 1 ⊢ ((𝑆: ℋ⟶ ℋ ∧ 𝑇: ℋ⟶ ℋ) → (𝑆 −op 𝑇) = (𝑥 ∈ ℋ ↦ ((𝑆‘𝑥) −ℎ (𝑇‘𝑥)))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 = wceq 1533 ∈ wcel 2098 ↦ cmpt 5224 ⟶wf 6532 ‘cfv 6536 (class class class)co 7404 ↑m cmap 8819 ℋchba 30676 −ℎ cmv 30682 −op chod 30697 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2697 ax-rep 5278 ax-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 ax-un 7721 ax-hilex 30756 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ne 2935 df-ral 3056 df-rex 3065 df-reu 3371 df-rab 3427 df-v 3470 df-sbc 3773 df-csb 3889 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-nul 4318 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4903 df-iun 4992 df-br 5142 df-opab 5204 df-mpt 5225 df-id 5567 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-iota 6488 df-fun 6538 df-fn 6539 df-f 6540 df-f1 6541 df-fo 6542 df-f1o 6543 df-fv 6544 df-ov 7407 df-oprab 7408 df-mpo 7409 df-map 8821 df-hodif 31489 |
This theorem is referenced by: hodval 31499 hosubcli 31526 |
Copyright terms: Public domain | W3C validator |