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Mirrors > Home > HSE Home > Th. List > ho2coi | Structured version Visualization version GIF version |
Description: Double composition of Hilbert space operators. (Contributed by NM, 1-Dec-2000.) (New usage is discouraged.) |
Ref | Expression |
---|---|
hods.1 | ⊢ 𝑅: ℋ⟶ ℋ |
hods.2 | ⊢ 𝑆: ℋ⟶ ℋ |
hods.3 | ⊢ 𝑇: ℋ⟶ ℋ |
Ref | Expression |
---|---|
ho2coi | ⊢ (𝐴 ∈ ℋ → (((𝑅 ∘ 𝑆) ∘ 𝑇)‘𝐴) = (𝑅‘(𝑆‘(𝑇‘𝐴)))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | hods.1 | . . . 4 ⊢ 𝑅: ℋ⟶ ℋ | |
2 | hods.2 | . . . 4 ⊢ 𝑆: ℋ⟶ ℋ | |
3 | 1, 2 | hocofi 30750 | . . 3 ⊢ (𝑅 ∘ 𝑆): ℋ⟶ ℋ |
4 | hods.3 | . . 3 ⊢ 𝑇: ℋ⟶ ℋ | |
5 | 3, 4 | hocoi 30748 | . 2 ⊢ (𝐴 ∈ ℋ → (((𝑅 ∘ 𝑆) ∘ 𝑇)‘𝐴) = ((𝑅 ∘ 𝑆)‘(𝑇‘𝐴))) |
6 | 4 | ffvelcdmi 7035 | . . 3 ⊢ (𝐴 ∈ ℋ → (𝑇‘𝐴) ∈ ℋ) |
7 | 1, 2 | hocoi 30748 | . . 3 ⊢ ((𝑇‘𝐴) ∈ ℋ → ((𝑅 ∘ 𝑆)‘(𝑇‘𝐴)) = (𝑅‘(𝑆‘(𝑇‘𝐴)))) |
8 | 6, 7 | syl 17 | . 2 ⊢ (𝐴 ∈ ℋ → ((𝑅 ∘ 𝑆)‘(𝑇‘𝐴)) = (𝑅‘(𝑆‘(𝑇‘𝐴)))) |
9 | 5, 8 | eqtrd 2773 | 1 ⊢ (𝐴 ∈ ℋ → (((𝑅 ∘ 𝑆) ∘ 𝑇)‘𝐴) = (𝑅‘(𝑆‘(𝑇‘𝐴)))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1542 ∈ wcel 2107 ∘ ccom 5638 ⟶wf 6493 ‘cfv 6497 ℋchba 29903 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-sep 5257 ax-nul 5264 ax-pr 5385 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3407 df-v 3446 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-nul 4284 df-if 4488 df-sn 4588 df-pr 4590 df-op 4594 df-uni 4867 df-br 5107 df-opab 5169 df-id 5532 df-xp 5640 df-rel 5641 df-cnv 5642 df-co 5643 df-dm 5644 df-rn 5645 df-res 5646 df-ima 5647 df-iota 6449 df-fun 6499 df-fn 6500 df-f 6501 df-fv 6505 |
This theorem is referenced by: pj2cocli 31189 |
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