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| Mirrors > Home > HSE Home > Th. List > hocoi | Structured version Visualization version GIF version | ||
| Description: Composition of Hilbert space operators. (Contributed by NM, 12-Nov-2000.) (New usage is discouraged.) | 
| Ref | Expression | 
|---|---|
| hoeq.1 | ⊢ 𝑆: ℋ⟶ ℋ | 
| hoeq.2 | ⊢ 𝑇: ℋ⟶ ℋ | 
| Ref | Expression | 
|---|---|
| hocoi | ⊢ (𝐴 ∈ ℋ → ((𝑆 ∘ 𝑇)‘𝐴) = (𝑆‘(𝑇‘𝐴))) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | hoeq.2 | . 2 ⊢ 𝑇: ℋ⟶ ℋ | |
| 2 | fvco3 7007 | . 2 ⊢ ((𝑇: ℋ⟶ ℋ ∧ 𝐴 ∈ ℋ) → ((𝑆 ∘ 𝑇)‘𝐴) = (𝑆‘(𝑇‘𝐴))) | |
| 3 | 1, 2 | mpan 690 | 1 ⊢ (𝐴 ∈ ℋ → ((𝑆 ∘ 𝑇)‘𝐴) = (𝑆‘(𝑇‘𝐴))) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 = wceq 1539 ∈ wcel 2107 ∘ ccom 5688 ⟶wf 6556 ‘cfv 6560 ℋchba 30939 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-10 2140 ax-11 2156 ax-12 2176 ax-ext 2707 ax-sep 5295 ax-nul 5305 ax-pr 5431 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-nf 1783 df-sb 2064 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2728 df-clel 2815 df-ne 2940 df-ral 3061 df-rex 3070 df-rab 3436 df-v 3481 df-dif 3953 df-un 3955 df-in 3957 df-ss 3967 df-nul 4333 df-if 4525 df-sn 4626 df-pr 4628 df-op 4632 df-uni 4907 df-br 5143 df-opab 5205 df-id 5577 df-xp 5690 df-rel 5691 df-cnv 5692 df-co 5693 df-dm 5694 df-rn 5695 df-res 5696 df-ima 5697 df-iota 6513 df-fun 6562 df-fn 6563 df-f 6564 df-fv 6568 | 
| This theorem is referenced by: hococli 31785 hocadddiri 31799 hocsubdiri 31800 ho2coi 31801 ho0coi 31808 hoid1i 31809 hoid1ri 31810 hoddii 32009 lnopcoi 32023 lnopco0i 32024 nmopcoi 32115 adjcoi 32120 nmopcoadji 32121 hmopidmchi 32171 hmopidmpji 32172 pjsdii 32175 pjddii 32176 pjcoi 32178 pjcohocli 32223 pjadj2coi 32224 pj3lem1 32226 | 
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