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Definition df-lnop 29009
Description: Define the set of linear operators on Hilbert space. (See df-hosum 28898 for definition of operator.) (Contributed by NM, 18-Jan-2006.) (New usage is discouraged.)
Assertion
Ref Expression
df-lnop LinOp = {𝑡 ∈ ( ℋ ↑𝑚 ℋ) ∣ ∀𝑥 ∈ ℂ ∀𝑦 ∈ ℋ ∀𝑧 ∈ ℋ (𝑡‘((𝑥 · 𝑦) + 𝑧)) = ((𝑥 · (𝑡𝑦)) + (𝑡𝑧))}
Distinct variable group:   𝑥,𝑡,𝑦,𝑧

Detailed syntax breakdown of Definition df-lnop
StepHypRef Expression
1 clo 28113 . 2 class LinOp
2 vx . . . . . . . . . . 11 setvar 𝑥
32cv 1631 . . . . . . . . . 10 class 𝑥
4 vy . . . . . . . . . . 11 setvar 𝑦
54cv 1631 . . . . . . . . . 10 class 𝑦
6 csm 28087 . . . . . . . . . 10 class ·
73, 5, 6co 6813 . . . . . . . . 9 class (𝑥 · 𝑦)
8 vz . . . . . . . . . 10 setvar 𝑧
98cv 1631 . . . . . . . . 9 class 𝑧
10 cva 28086 . . . . . . . . 9 class +
117, 9, 10co 6813 . . . . . . . 8 class ((𝑥 · 𝑦) + 𝑧)
12 vt . . . . . . . . 9 setvar 𝑡
1312cv 1631 . . . . . . . 8 class 𝑡
1411, 13cfv 6049 . . . . . . 7 class (𝑡‘((𝑥 · 𝑦) + 𝑧))
155, 13cfv 6049 . . . . . . . . 9 class (𝑡𝑦)
163, 15, 6co 6813 . . . . . . . 8 class (𝑥 · (𝑡𝑦))
179, 13cfv 6049 . . . . . . . 8 class (𝑡𝑧)
1816, 17, 10co 6813 . . . . . . 7 class ((𝑥 · (𝑡𝑦)) + (𝑡𝑧))
1914, 18wceq 1632 . . . . . 6 wff (𝑡‘((𝑥 · 𝑦) + 𝑧)) = ((𝑥 · (𝑡𝑦)) + (𝑡𝑧))
20 chil 28085 . . . . . 6 class
2119, 8, 20wral 3050 . . . . 5 wff 𝑧 ∈ ℋ (𝑡‘((𝑥 · 𝑦) + 𝑧)) = ((𝑥 · (𝑡𝑦)) + (𝑡𝑧))
2221, 4, 20wral 3050 . . . 4 wff 𝑦 ∈ ℋ ∀𝑧 ∈ ℋ (𝑡‘((𝑥 · 𝑦) + 𝑧)) = ((𝑥 · (𝑡𝑦)) + (𝑡𝑧))
23 cc 10126 . . . 4 class
2422, 2, 23wral 3050 . . 3 wff 𝑥 ∈ ℂ ∀𝑦 ∈ ℋ ∀𝑧 ∈ ℋ (𝑡‘((𝑥 · 𝑦) + 𝑧)) = ((𝑥 · (𝑡𝑦)) + (𝑡𝑧))
25 cmap 8023 . . . 4 class 𝑚
2620, 20, 25co 6813 . . 3 class ( ℋ ↑𝑚 ℋ)
2724, 12, 26crab 3054 . 2 class {𝑡 ∈ ( ℋ ↑𝑚 ℋ) ∣ ∀𝑥 ∈ ℂ ∀𝑦 ∈ ℋ ∀𝑧 ∈ ℋ (𝑡‘((𝑥 · 𝑦) + 𝑧)) = ((𝑥 · (𝑡𝑦)) + (𝑡𝑧))}
281, 27wceq 1632 1 wff LinOp = {𝑡 ∈ ( ℋ ↑𝑚 ℋ) ∣ ∀𝑥 ∈ ℂ ∀𝑦 ∈ ℋ ∀𝑧 ∈ ℋ (𝑡‘((𝑥 · 𝑦) + 𝑧)) = ((𝑥 · (𝑡𝑦)) + (𝑡𝑧))}
Colors of variables: wff setvar class
This definition is referenced by:  ellnop  29026  hhlnoi  29068
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