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Definition df-hodif 27757
Description: Define the difference of two Hilbert space operators. Definition of [Beran] p. 111. (Contributed by NM, 9-Nov-2000.) (New usage is discouraged.)
Assertion
Ref Expression
df-hodif op = (𝑓 ∈ ( ℋ ↑𝑚 ℋ), 𝑔 ∈ ( ℋ ↑𝑚 ℋ) ↦ (𝑥 ∈ ℋ ↦ ((𝑓𝑥) − (𝑔𝑥))))
Distinct variable group:   𝑓,𝑔,𝑥

Detailed syntax breakdown of Definition df-hodif
StepHypRef Expression
1 chod 26963 . 2 class op
2 vf . . 3 setvar 𝑓
3 vg . . 3 setvar 𝑔
4 chil 26942 . . . 4 class
5 cmap 7624 . . . 4 class 𝑚
64, 4, 5co 6431 . . 3 class ( ℋ ↑𝑚 ℋ)
7 vx . . . 4 setvar 𝑥
87cv 1473 . . . . . 6 class 𝑥
92cv 1473 . . . . . 6 class 𝑓
108, 9cfv 5694 . . . . 5 class (𝑓𝑥)
113cv 1473 . . . . . 6 class 𝑔
128, 11cfv 5694 . . . . 5 class (𝑔𝑥)
13 cmv 26948 . . . . 5 class
1410, 12, 13co 6431 . . . 4 class ((𝑓𝑥) − (𝑔𝑥))
157, 4, 14cmpt 4541 . . 3 class (𝑥 ∈ ℋ ↦ ((𝑓𝑥) − (𝑔𝑥)))
162, 3, 6, 6, 15cmpt2 6433 . 2 class (𝑓 ∈ ( ℋ ↑𝑚 ℋ), 𝑔 ∈ ( ℋ ↑𝑚 ℋ) ↦ (𝑥 ∈ ℋ ↦ ((𝑓𝑥) − (𝑔𝑥))))
171, 16wceq 1474 1 wff op = (𝑓 ∈ ( ℋ ↑𝑚 ℋ), 𝑔 ∈ ( ℋ ↑𝑚 ℋ) ↦ (𝑥 ∈ ℋ ↦ ((𝑓𝑥) − (𝑔𝑥))))
Colors of variables: wff setvar class
This definition is referenced by:  hodmval  27762
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