![]() |
Hilbert Space Explorer |
< Previous
Next >
Nearby theorems |
|
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 31800 | . . 3 ⊢ (𝑅 ∘ 𝑆): ℋ⟶ ℋ |
4 | hods.3 | . . 3 ⊢ 𝑇: ℋ⟶ ℋ | |
5 | 3, 4 | hocoi 31798 | . 2 ⊢ (𝐴 ∈ ℋ → (((𝑅 ∘ 𝑆) ∘ 𝑇)‘𝐴) = ((𝑅 ∘ 𝑆)‘(𝑇‘𝐴))) |
6 | 4 | ffvelcdmi 7119 | . . 3 ⊢ (𝐴 ∈ ℋ → (𝑇‘𝐴) ∈ ℋ) |
7 | 1, 2 | hocoi 31798 | . . 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 2780 | 1 ⊢ (𝐴 ∈ ℋ → (((𝑅 ∘ 𝑆) ∘ 𝑇)‘𝐴) = (𝑅‘(𝑆‘(𝑇‘𝐴)))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1537 ∈ wcel 2108 ∘ ccom 5704 ⟶wf 6571 ‘cfv 6575 ℋchba 30953 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1793 ax-4 1807 ax-5 1909 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2158 ax-12 2178 ax-ext 2711 ax-sep 5317 ax-nul 5324 ax-pr 5447 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 847 df-3an 1089 df-tru 1540 df-fal 1550 df-ex 1778 df-nf 1782 df-sb 2065 df-mo 2543 df-eu 2572 df-clab 2718 df-cleq 2732 df-clel 2819 df-nfc 2895 df-ne 2947 df-ral 3068 df-rex 3077 df-rab 3444 df-v 3490 df-dif 3979 df-un 3981 df-in 3983 df-ss 3993 df-nul 4353 df-if 4549 df-sn 4649 df-pr 4651 df-op 4655 df-uni 4932 df-br 5167 df-opab 5229 df-id 5593 df-xp 5706 df-rel 5707 df-cnv 5708 df-co 5709 df-dm 5710 df-rn 5711 df-res 5712 df-ima 5713 df-iota 6527 df-fun 6577 df-fn 6578 df-f 6579 df-fv 6583 |
This theorem is referenced by: pj2cocli 32239 |
Copyright terms: Public domain | W3C validator |