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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 6505 | . 2 ⊢ ((𝑆: ℋ⟶ ℋ ∧ 𝑇: ℋ⟶ ℋ) → (𝑆 ∘ 𝑇): ℋ⟶ ℋ) | |
4 | 1, 2, 3 | mp2an 691 | 1 ⊢ (𝑆 ∘ 𝑇): ℋ⟶ ℋ |
Colors of variables: wff setvar class |
Syntax hints: ∘ ccom 5523 ⟶wf 6320 ℋchba 28702 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pr 5295 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ral 3111 df-rex 3112 df-v 3443 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4244 df-if 4426 df-sn 4526 df-pr 4528 df-op 4532 df-br 5031 df-opab 5093 df-id 5425 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-fun 6326 df-fn 6327 df-f 6328 |
This theorem is referenced by: hocofni 29550 hocadddiri 29562 hocsubdiri 29563 ho2coi 29564 ho0coi 29571 hoid1i 29572 hoid1ri 29573 hoddii 29772 lnopcoi 29786 bdopcoi 29881 adjcoi 29883 nmopcoadji 29884 unierri 29887 pjsdii 29938 pjddii 29939 pjsdi2i 29940 pjss1coi 29946 pjss2coi 29947 pjorthcoi 29952 pjinvari 29974 pjclem1 29978 pjclem4 29982 pjadj2coi 29987 pj3lem1 29989 pj3si 29990 pj3cor1i 29992 |
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