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Mirrors > Home > HSE Home > Th. List > hocofi | Structured version Visualization version GIF version |
Description: Mapping of composition of Hilbert space operators. (Contributed by NM, 14-Nov-2000.) (New usage is discouraged.) |
Ref | Expression |
---|---|
hoeq.1 | ⊢ 𝑆: ℋ⟶ ℋ |
hoeq.2 | ⊢ 𝑇: ℋ⟶ ℋ |
Ref | Expression |
---|---|
hocofi | ⊢ (𝑆 ∘ 𝑇): ℋ⟶ ℋ |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | hoeq.1 | . 2 ⊢ 𝑆: ℋ⟶ ℋ | |
2 | hoeq.2 | . 2 ⊢ 𝑇: ℋ⟶ ℋ | |
3 | fco 6761 | . 2 ⊢ ((𝑆: ℋ⟶ ℋ ∧ 𝑇: ℋ⟶ ℋ) → (𝑆 ∘ 𝑇): ℋ⟶ ℋ) | |
4 | 1, 2, 3 | mp2an 692 | 1 ⊢ (𝑆 ∘ 𝑇): ℋ⟶ ℋ |
Colors of variables: wff setvar class |
Syntax hints: ∘ ccom 5693 ⟶wf 6559 ℋchba 30948 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pr 5438 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ral 3060 df-rex 3069 df-rab 3434 df-v 3480 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-nul 4340 df-if 4532 df-sn 4632 df-pr 4634 df-op 4638 df-br 5149 df-opab 5211 df-id 5583 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-fun 6565 df-fn 6566 df-f 6567 |
This theorem is referenced by: hocofni 31796 hocadddiri 31808 hocsubdiri 31809 ho2coi 31810 ho0coi 31817 hoid1i 31818 hoid1ri 31819 hoddii 32018 lnopcoi 32032 bdopcoi 32127 adjcoi 32129 nmopcoadji 32130 unierri 32133 pjsdii 32184 pjddii 32185 pjsdi2i 32186 pjss1coi 32192 pjss2coi 32193 pjorthcoi 32198 pjinvari 32220 pjclem1 32224 pjclem4 32228 pjadj2coi 32233 pj3lem1 32235 pj3si 32236 pj3cor1i 32238 |
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