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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 6990 | . 2 ⊢ ((𝑇: ℋ⟶ ℋ ∧ 𝐴 ∈ ℋ) → ((𝑆 ∘ 𝑇)‘𝐴) = (𝑆‘(𝑇‘𝐴))) | |
3 | 1, 2 | mpan 687 | 1 ⊢ (𝐴 ∈ ℋ → ((𝑆 ∘ 𝑇)‘𝐴) = (𝑆‘(𝑇‘𝐴))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1540 ∈ wcel 2105 ∘ ccom 5680 ⟶wf 6539 ‘cfv 6543 ℋchba 30604 |
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 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2153 ax-12 2170 ax-ext 2702 ax-sep 5299 ax-nul 5306 ax-pr 5427 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-nf 1785 df-sb 2067 df-mo 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-ne 2940 df-ral 3061 df-rex 3070 df-rab 3432 df-v 3475 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-nul 4323 df-if 4529 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-br 5149 df-opab 5211 df-id 5574 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-fv 6551 |
This theorem is referenced by: hococli 31450 hocadddiri 31464 hocsubdiri 31465 ho2coi 31466 ho0coi 31473 hoid1i 31474 hoid1ri 31475 hoddii 31674 lnopcoi 31688 lnopco0i 31689 nmopcoi 31780 adjcoi 31785 nmopcoadji 31786 hmopidmchi 31836 hmopidmpji 31837 pjsdii 31840 pjddii 31841 pjcoi 31843 pjcohocli 31888 pjadj2coi 31889 pj3lem1 31891 |
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