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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 6531 | . 2 ⊢ ((𝑆: ℋ⟶ ℋ ∧ 𝑇: ℋ⟶ ℋ) → (𝑆 ∘ 𝑇): ℋ⟶ ℋ) | |
4 | 1, 2, 3 | mp2an 690 | 1 ⊢ (𝑆 ∘ 𝑇): ℋ⟶ ℋ |
Colors of variables: wff setvar class |
Syntax hints: ∘ ccom 5559 ⟶wf 6351 ℋchba 28696 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-sep 5203 ax-nul 5210 ax-pr 5330 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3496 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-sn 4568 df-pr 4570 df-op 4574 df-br 5067 df-opab 5129 df-id 5460 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-fun 6357 df-fn 6358 df-f 6359 |
This theorem is referenced by: hocofni 29544 hocadddiri 29556 hocsubdiri 29557 ho2coi 29558 ho0coi 29565 hoid1i 29566 hoid1ri 29567 hoddii 29766 lnopcoi 29780 bdopcoi 29875 adjcoi 29877 nmopcoadji 29878 unierri 29881 pjsdii 29932 pjddii 29933 pjsdi2i 29934 pjss1coi 29940 pjss2coi 29941 pjorthcoi 29946 pjinvari 29968 pjclem1 29972 pjclem4 29976 pjadj2coi 29981 pj3lem1 29983 pj3si 29984 pj3cor1i 29986 |
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