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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 6219 | . 2 ⊢ ((𝑆: ℋ⟶ ℋ ∧ 𝑇: ℋ⟶ ℋ) → (𝑆 ∘ 𝑇): ℋ⟶ ℋ) | |
4 | 1, 2, 3 | mp2an 710 | 1 ⊢ (𝑆 ∘ 𝑇): ℋ⟶ ℋ |
Colors of variables: wff setvar class |
Syntax hints: ∘ ccom 5270 ⟶wf 6045 ℋchil 28085 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1871 ax-4 1886 ax-5 1988 ax-6 2054 ax-7 2090 ax-9 2148 ax-10 2168 ax-11 2183 ax-12 2196 ax-13 2391 ax-ext 2740 ax-sep 4933 ax-nul 4941 ax-pr 5055 |
This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-3an 1074 df-tru 1635 df-ex 1854 df-nf 1859 df-sb 2047 df-eu 2611 df-mo 2612 df-clab 2747 df-cleq 2753 df-clel 2756 df-nfc 2891 df-ral 3055 df-rex 3056 df-rab 3059 df-v 3342 df-dif 3718 df-un 3720 df-in 3722 df-ss 3729 df-nul 4059 df-if 4231 df-sn 4322 df-pr 4324 df-op 4328 df-br 4805 df-opab 4865 df-id 5174 df-xp 5272 df-rel 5273 df-cnv 5274 df-co 5275 df-dm 5276 df-rn 5277 df-fun 6051 df-fn 6052 df-f 6053 |
This theorem is referenced by: hocofni 28935 hocadddiri 28947 hocsubdiri 28948 ho2coi 28949 ho0coi 28956 hoid1i 28957 hoid1ri 28958 hoddii 29157 lnopcoi 29171 bdopcoi 29266 adjcoi 29268 nmopcoadji 29269 unierri 29272 pjsdii 29323 pjddii 29324 pjsdi2i 29325 pjss1coi 29331 pjss2coi 29332 pjorthcoi 29337 pjinvari 29359 pjclem1 29363 pjclem4 29367 pjadj2coi 29372 pj3lem1 29374 pj3si 29375 pj3cor1i 29377 |
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