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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 6728 | . 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 5673 ⟶wf 6528 ℋchba 30035 |
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 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2702 ax-sep 5292 ax-nul 5299 ax-pr 5420 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ral 3061 df-rex 3070 df-rab 3432 df-v 3475 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-nul 4319 df-if 4523 df-sn 4623 df-pr 4625 df-op 4629 df-br 5142 df-opab 5204 df-id 5567 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-fun 6534 df-fn 6535 df-f 6536 |
This theorem is referenced by: hocofni 30883 hocadddiri 30895 hocsubdiri 30896 ho2coi 30897 ho0coi 30904 hoid1i 30905 hoid1ri 30906 hoddii 31105 lnopcoi 31119 bdopcoi 31214 adjcoi 31216 nmopcoadji 31217 unierri 31220 pjsdii 31271 pjddii 31272 pjsdi2i 31273 pjss1coi 31279 pjss2coi 31280 pjorthcoi 31285 pjinvari 31307 pjclem1 31311 pjclem4 31315 pjadj2coi 31320 pj3lem1 31322 pj3si 31323 pj3cor1i 31325 |
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