|   | Hilbert Space Explorer | < Previous  
      Next > Nearby theorems | |
| Mirrors > Home > HSE Home > Th. List > hocofni | Structured version Visualization version GIF version | ||
| Description: Functionality of composition of Hilbert space operators. (Contributed by NM, 12-Nov-2000.) (New usage is discouraged.) | 
| Ref | Expression | 
|---|---|
| hoeq.1 | ⊢ 𝑆: ℋ⟶ ℋ | 
| hoeq.2 | ⊢ 𝑇: ℋ⟶ ℋ | 
| Ref | Expression | 
|---|---|
| hocofni | ⊢ (𝑆 ∘ 𝑇) Fn ℋ | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | hoeq.1 | . . 3 ⊢ 𝑆: ℋ⟶ ℋ | |
| 2 | hoeq.2 | . . 3 ⊢ 𝑇: ℋ⟶ ℋ | |
| 3 | 1, 2 | hocofi 31785 | . 2 ⊢ (𝑆 ∘ 𝑇): ℋ⟶ ℋ | 
| 4 | ffn 6736 | . 2 ⊢ ((𝑆 ∘ 𝑇): ℋ⟶ ℋ → (𝑆 ∘ 𝑇) Fn ℋ) | |
| 5 | 3, 4 | ax-mp 5 | 1 ⊢ (𝑆 ∘ 𝑇) Fn ℋ | 
| Colors of variables: wff setvar class | 
| Syntax hints: ∘ ccom 5689 Fn wfn 6556 ⟶wf 6557 ℋchba 30938 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pr 5432 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-br 5144 df-opab 5206 df-id 5578 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-fun 6563 df-fn 6564 df-f 6565 | 
| This theorem is referenced by: pjcofni 32181 pjinvari 32210 pj3si 32226 | 
| Copyright terms: Public domain | W3C validator |