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Definition df-dvap 14062
Description: Define the derivative operator. This acts on functions to produce a function that is defined where the original function is differentiable, with value the derivative of the function at these points. The set 𝑠 here is the ambient topological space under which we are evaluating the continuity of the difference quotient. Although the definition is valid for any subset of β„‚ and is well-behaved when 𝑠 contains no isolated points, we will restrict our attention to the cases 𝑠 = ℝ or 𝑠 = β„‚ for the majority of the development, these corresponding respectively to real and complex differentiation. (Contributed by Mario Carneiro, 7-Aug-2014.) (Revised by Jim Kingdon, 25-Jun-2023.)
Assertion
Ref Expression
df-dvap D = (𝑠 ∈ 𝒫 β„‚, 𝑓 ∈ (β„‚ ↑pm 𝑠) ↦ βˆͺ π‘₯ ∈ ((intβ€˜((MetOpenβ€˜(abs ∘ βˆ’ )) β†Ύt 𝑠))β€˜dom 𝑓)({π‘₯} Γ— ((𝑧 ∈ {𝑀 ∈ dom 𝑓 ∣ 𝑀 # π‘₯} ↦ (((π‘“β€˜π‘§) βˆ’ (π‘“β€˜π‘₯)) / (𝑧 βˆ’ π‘₯))) limβ„‚ π‘₯)))
Distinct variable group:   𝑓,𝑠,π‘₯,𝑧,𝑀

Detailed syntax breakdown of Definition df-dvap
StepHypRef Expression
1 cdv 14060 . 2 class D
2 vs . . 3 setvar 𝑠
3 vf . . 3 setvar 𝑓
4 cc 7808 . . . 4 class β„‚
54cpw 3575 . . 3 class 𝒫 β„‚
62cv 1352 . . . 4 class 𝑠
7 cpm 6648 . . . 4 class ↑pm
84, 6, 7co 5874 . . 3 class (β„‚ ↑pm 𝑠)
9 vx . . . 4 setvar π‘₯
103cv 1352 . . . . . 6 class 𝑓
1110cdm 4626 . . . . 5 class dom 𝑓
12 cabs 11005 . . . . . . . . 9 class abs
13 cmin 8127 . . . . . . . . 9 class βˆ’
1412, 13ccom 4630 . . . . . . . 8 class (abs ∘ βˆ’ )
15 cmopn 13381 . . . . . . . 8 class MetOpen
1614, 15cfv 5216 . . . . . . 7 class (MetOpenβ€˜(abs ∘ βˆ’ ))
17 crest 12687 . . . . . . 7 class β†Ύt
1816, 6, 17co 5874 . . . . . 6 class ((MetOpenβ€˜(abs ∘ βˆ’ )) β†Ύt 𝑠)
19 cnt 13529 . . . . . 6 class int
2018, 19cfv 5216 . . . . 5 class (intβ€˜((MetOpenβ€˜(abs ∘ βˆ’ )) β†Ύt 𝑠))
2111, 20cfv 5216 . . . 4 class ((intβ€˜((MetOpenβ€˜(abs ∘ βˆ’ )) β†Ύt 𝑠))β€˜dom 𝑓)
229cv 1352 . . . . . 6 class π‘₯
2322csn 3592 . . . . 5 class {π‘₯}
24 vz . . . . . . 7 setvar 𝑧
25 vw . . . . . . . . . 10 setvar 𝑀
2625cv 1352 . . . . . . . . 9 class 𝑀
27 cap 8537 . . . . . . . . 9 class #
2826, 22, 27wbr 4003 . . . . . . . 8 wff 𝑀 # π‘₯
2928, 25, 11crab 2459 . . . . . . 7 class {𝑀 ∈ dom 𝑓 ∣ 𝑀 # π‘₯}
3024cv 1352 . . . . . . . . . 10 class 𝑧
3130, 10cfv 5216 . . . . . . . . 9 class (π‘“β€˜π‘§)
3222, 10cfv 5216 . . . . . . . . 9 class (π‘“β€˜π‘₯)
3331, 32, 13co 5874 . . . . . . . 8 class ((π‘“β€˜π‘§) βˆ’ (π‘“β€˜π‘₯))
3430, 22, 13co 5874 . . . . . . . 8 class (𝑧 βˆ’ π‘₯)
35 cdiv 8628 . . . . . . . 8 class /
3633, 34, 35co 5874 . . . . . . 7 class (((π‘“β€˜π‘§) βˆ’ (π‘“β€˜π‘₯)) / (𝑧 βˆ’ π‘₯))
3724, 29, 36cmpt 4064 . . . . . 6 class (𝑧 ∈ {𝑀 ∈ dom 𝑓 ∣ 𝑀 # π‘₯} ↦ (((π‘“β€˜π‘§) βˆ’ (π‘“β€˜π‘₯)) / (𝑧 βˆ’ π‘₯)))
38 climc 14059 . . . . . 6 class limβ„‚
3937, 22, 38co 5874 . . . . 5 class ((𝑧 ∈ {𝑀 ∈ dom 𝑓 ∣ 𝑀 # π‘₯} ↦ (((π‘“β€˜π‘§) βˆ’ (π‘“β€˜π‘₯)) / (𝑧 βˆ’ π‘₯))) limβ„‚ π‘₯)
4023, 39cxp 4624 . . . 4 class ({π‘₯} Γ— ((𝑧 ∈ {𝑀 ∈ dom 𝑓 ∣ 𝑀 # π‘₯} ↦ (((π‘“β€˜π‘§) βˆ’ (π‘“β€˜π‘₯)) / (𝑧 βˆ’ π‘₯))) limβ„‚ π‘₯))
419, 21, 40ciun 3886 . . 3 class βˆͺ π‘₯ ∈ ((intβ€˜((MetOpenβ€˜(abs ∘ βˆ’ )) β†Ύt 𝑠))β€˜dom 𝑓)({π‘₯} Γ— ((𝑧 ∈ {𝑀 ∈ dom 𝑓 ∣ 𝑀 # π‘₯} ↦ (((π‘“β€˜π‘§) βˆ’ (π‘“β€˜π‘₯)) / (𝑧 βˆ’ π‘₯))) limβ„‚ π‘₯))
422, 3, 5, 8, 41cmpo 5876 . 2 class (𝑠 ∈ 𝒫 β„‚, 𝑓 ∈ (β„‚ ↑pm 𝑠) ↦ βˆͺ π‘₯ ∈ ((intβ€˜((MetOpenβ€˜(abs ∘ βˆ’ )) β†Ύt 𝑠))β€˜dom 𝑓)({π‘₯} Γ— ((𝑧 ∈ {𝑀 ∈ dom 𝑓 ∣ 𝑀 # π‘₯} ↦ (((π‘“β€˜π‘§) βˆ’ (π‘“β€˜π‘₯)) / (𝑧 βˆ’ π‘₯))) limβ„‚ π‘₯)))
431, 42wceq 1353 1 wff D = (𝑠 ∈ 𝒫 β„‚, 𝑓 ∈ (β„‚ ↑pm 𝑠) ↦ βˆͺ π‘₯ ∈ ((intβ€˜((MetOpenβ€˜(abs ∘ βˆ’ )) β†Ύt 𝑠))β€˜dom 𝑓)({π‘₯} Γ— ((𝑧 ∈ {𝑀 ∈ dom 𝑓 ∣ 𝑀 # π‘₯} ↦ (((π‘“β€˜π‘§) βˆ’ (π‘“β€˜π‘₯)) / (𝑧 βˆ’ π‘₯))) limβ„‚ π‘₯)))
Colors of variables: wff set class
This definition is referenced by:  reldvg  14084  dvfvalap  14086
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