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Definition df-lnfn 27873
Description: Define the set of linear functionals on Hilbert space. (Contributed by NM, 11-Feb-2006.) (New usage is discouraged.)
Assertion
Ref Expression
df-lnfn LinFn = {𝑡 ∈ (ℂ ↑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-lnfn
StepHypRef Expression
1 clf 26977 . 2 class LinFn
2 vx . . . . . . . . . . 11 setvar 𝑥
32cv 1473 . . . . . . . . . 10 class 𝑥
4 vy . . . . . . . . . . 11 setvar 𝑦
54cv 1473 . . . . . . . . . 10 class 𝑦
6 csm 26944 . . . . . . . . . 10 class ·
73, 5, 6co 6431 . . . . . . . . 9 class (𝑥 · 𝑦)
8 vz . . . . . . . . . 10 setvar 𝑧
98cv 1473 . . . . . . . . 9 class 𝑧
10 cva 26943 . . . . . . . . 9 class +
117, 9, 10co 6431 . . . . . . . 8 class ((𝑥 · 𝑦) + 𝑧)
12 vt . . . . . . . . 9 setvar 𝑡
1312cv 1473 . . . . . . . 8 class 𝑡
1411, 13cfv 5694 . . . . . . 7 class (𝑡‘((𝑥 · 𝑦) + 𝑧))
155, 13cfv 5694 . . . . . . . . 9 class (𝑡𝑦)
16 cmul 9700 . . . . . . . . 9 class ·
173, 15, 16co 6431 . . . . . . . 8 class (𝑥 · (𝑡𝑦))
189, 13cfv 5694 . . . . . . . 8 class (𝑡𝑧)
19 caddc 9698 . . . . . . . 8 class +
2017, 18, 19co 6431 . . . . . . 7 class ((𝑥 · (𝑡𝑦)) + (𝑡𝑧))
2114, 20wceq 1474 . . . . . 6 wff (𝑡‘((𝑥 · 𝑦) + 𝑧)) = ((𝑥 · (𝑡𝑦)) + (𝑡𝑧))
22 chil 26942 . . . . . 6 class
2321, 8, 22wral 2800 . . . . 5 wff 𝑧 ∈ ℋ (𝑡‘((𝑥 · 𝑦) + 𝑧)) = ((𝑥 · (𝑡𝑦)) + (𝑡𝑧))
2423, 4, 22wral 2800 . . . 4 wff 𝑦 ∈ ℋ ∀𝑧 ∈ ℋ (𝑡‘((𝑥 · 𝑦) + 𝑧)) = ((𝑥 · (𝑡𝑦)) + (𝑡𝑧))
25 cc 9693 . . . 4 class
2624, 2, 25wral 2800 . . 3 wff 𝑥 ∈ ℂ ∀𝑦 ∈ ℋ ∀𝑧 ∈ ℋ (𝑡‘((𝑥 · 𝑦) + 𝑧)) = ((𝑥 · (𝑡𝑦)) + (𝑡𝑧))
27 cmap 7624 . . . 4 class 𝑚
2825, 22, 27co 6431 . . 3 class (ℂ ↑𝑚 ℋ)
2926, 12, 28crab 2804 . 2 class {𝑡 ∈ (ℂ ↑𝑚 ℋ) ∣ ∀𝑥 ∈ ℂ ∀𝑦 ∈ ℋ ∀𝑧 ∈ ℋ (𝑡‘((𝑥 · 𝑦) + 𝑧)) = ((𝑥 · (𝑡𝑦)) + (𝑡𝑧))}
301, 29wceq 1474 1 wff LinFn = {𝑡 ∈ (ℂ ↑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:  ellnfn  27908
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