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Mirrors > Home > HSE Home > Th. List > hococli | Structured version Visualization version GIF version |
Description: Closure 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 |
---|---|
hococli | ⊢ (𝐴 ∈ ℋ → ((𝑆 ∘ 𝑇)‘𝐴) ∈ ℋ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | hoeq.1 | . . 3 ⊢ 𝑆: ℋ⟶ ℋ | |
2 | hoeq.2 | . . 3 ⊢ 𝑇: ℋ⟶ ℋ | |
3 | 1, 2 | hocoi 30994 | . 2 ⊢ (𝐴 ∈ ℋ → ((𝑆 ∘ 𝑇)‘𝐴) = (𝑆‘(𝑇‘𝐴))) |
4 | 2 | ffvelcdmi 7080 | . . 3 ⊢ (𝐴 ∈ ℋ → (𝑇‘𝐴) ∈ ℋ) |
5 | 1 | ffvelcdmi 7080 | . . 3 ⊢ ((𝑇‘𝐴) ∈ ℋ → (𝑆‘(𝑇‘𝐴)) ∈ ℋ) |
6 | 4, 5 | syl 17 | . 2 ⊢ (𝐴 ∈ ℋ → (𝑆‘(𝑇‘𝐴)) ∈ ℋ) |
7 | 3, 6 | eqeltrd 2834 | 1 ⊢ (𝐴 ∈ ℋ → ((𝑆 ∘ 𝑇)‘𝐴) ∈ ℋ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2107 ∘ ccom 5678 ⟶wf 6535 ‘cfv 6539 ℋchba 30149 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-sep 5297 ax-nul 5304 ax-pr 5425 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-ne 2942 df-ral 3063 df-rex 3072 df-rab 3434 df-v 3477 df-dif 3949 df-un 3951 df-in 3953 df-ss 3963 df-nul 4321 df-if 4527 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4907 df-br 5147 df-opab 5209 df-id 5572 df-xp 5680 df-rel 5681 df-cnv 5682 df-co 5683 df-dm 5684 df-rn 5685 df-res 5686 df-ima 5687 df-iota 6491 df-fun 6541 df-fn 6542 df-f 6543 df-fv 6547 |
This theorem is referenced by: nmopcoadji 31331 pjcohcli 31390 pj3si 31437 pj3cor1i 31439 |
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