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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 31524 | . . 3 ⊢ (𝑅 ∘ 𝑆): ℋ⟶ ℋ |
4 | hods.3 | . . 3 ⊢ 𝑇: ℋ⟶ ℋ | |
5 | 3, 4 | hocoi 31522 | . 2 ⊢ (𝐴 ∈ ℋ → (((𝑅 ∘ 𝑆) ∘ 𝑇)‘𝐴) = ((𝑅 ∘ 𝑆)‘(𝑇‘𝐴))) |
6 | 4 | ffvelcdmi 7078 | . . 3 ⊢ (𝐴 ∈ ℋ → (𝑇‘𝐴) ∈ ℋ) |
7 | 1, 2 | hocoi 31522 | . . 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 2766 | 1 ⊢ (𝐴 ∈ ℋ → (((𝑅 ∘ 𝑆) ∘ 𝑇)‘𝐴) = (𝑅‘(𝑆‘(𝑇‘𝐴)))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1533 ∈ wcel 2098 ∘ ccom 5673 ⟶wf 6532 ‘cfv 6536 ℋchba 30677 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2697 ax-sep 5292 ax-nul 5299 ax-pr 5420 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ne 2935 df-ral 3056 df-rex 3065 df-rab 3427 df-v 3470 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-nul 4318 df-if 4524 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4903 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-iota 6488 df-fun 6538 df-fn 6539 df-f 6540 df-fv 6544 |
This theorem is referenced by: pj2cocli 31963 |
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