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Theorem itgoval 39259
Description: Value of the integral-over function. (Contributed by Stefan O'Rear, 27-Nov-2014.)
Assertion
Ref Expression
itgoval (𝑆 ⊆ ℂ → (IntgOver‘𝑆) = {𝑥 ∈ ℂ ∣ ∃𝑝 ∈ (Poly‘𝑆)((𝑝𝑥) = 0 ∧ ((coeff‘𝑝)‘(deg‘𝑝)) = 1)})
Distinct variable group:   𝑥,𝑆,𝑝

Proof of Theorem itgoval
Dummy variable 𝑠 is distinct from all other variables.
StepHypRef Expression
1 cnex 10467 . . 3 ℂ ∈ V
21elpw2 5142 . 2 (𝑆 ∈ 𝒫 ℂ ↔ 𝑆 ⊆ ℂ)
3 fveq2 6541 . . . . 5 (𝑠 = 𝑆 → (Poly‘𝑠) = (Poly‘𝑆))
43rexeqdv 3375 . . . 4 (𝑠 = 𝑆 → (∃𝑝 ∈ (Poly‘𝑠)((𝑝𝑥) = 0 ∧ ((coeff‘𝑝)‘(deg‘𝑝)) = 1) ↔ ∃𝑝 ∈ (Poly‘𝑆)((𝑝𝑥) = 0 ∧ ((coeff‘𝑝)‘(deg‘𝑝)) = 1)))
54rabbidv 3424 . . 3 (𝑠 = 𝑆 → {𝑥 ∈ ℂ ∣ ∃𝑝 ∈ (Poly‘𝑠)((𝑝𝑥) = 0 ∧ ((coeff‘𝑝)‘(deg‘𝑝)) = 1)} = {𝑥 ∈ ℂ ∣ ∃𝑝 ∈ (Poly‘𝑆)((𝑝𝑥) = 0 ∧ ((coeff‘𝑝)‘(deg‘𝑝)) = 1)})
6 df-itgo 39257 . . 3 IntgOver = (𝑠 ∈ 𝒫 ℂ ↦ {𝑥 ∈ ℂ ∣ ∃𝑝 ∈ (Poly‘𝑠)((𝑝𝑥) = 0 ∧ ((coeff‘𝑝)‘(deg‘𝑝)) = 1)})
71rabex 5129 . . 3 {𝑥 ∈ ℂ ∣ ∃𝑝 ∈ (Poly‘𝑆)((𝑝𝑥) = 0 ∧ ((coeff‘𝑝)‘(deg‘𝑝)) = 1)} ∈ V
85, 6, 7fvmpt 6638 . 2 (𝑆 ∈ 𝒫 ℂ → (IntgOver‘𝑆) = {𝑥 ∈ ℂ ∣ ∃𝑝 ∈ (Poly‘𝑆)((𝑝𝑥) = 0 ∧ ((coeff‘𝑝)‘(deg‘𝑝)) = 1)})
92, 8sylbir 236 1 (𝑆 ⊆ ℂ → (IntgOver‘𝑆) = {𝑥 ∈ ℂ ∣ ∃𝑝 ∈ (Poly‘𝑆)((𝑝𝑥) = 0 ∧ ((coeff‘𝑝)‘(deg‘𝑝)) = 1)})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396   = wceq 1522  wcel 2080  wrex 3105  {crab 3108  wss 3861  𝒫 cpw 4455  cfv 6228  cc 10384  0cc0 10386  1c1 10387  Polycply 24457  coeffccoe 24459  degcdgr 24460  IntgOvercitgo 39255
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1778  ax-4 1792  ax-5 1889  ax-6 1948  ax-7 1993  ax-8 2082  ax-9 2090  ax-10 2111  ax-11 2125  ax-12 2140  ax-13 2343  ax-ext 2768  ax-sep 5097  ax-nul 5104  ax-pr 5224  ax-cnex 10442
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 843  df-3an 1082  df-tru 1525  df-ex 1763  df-nf 1767  df-sb 2042  df-mo 2575  df-eu 2611  df-clab 2775  df-cleq 2787  df-clel 2862  df-nfc 2934  df-ral 3109  df-rex 3110  df-rab 3113  df-v 3438  df-sbc 3708  df-dif 3864  df-un 3866  df-in 3868  df-ss 3876  df-nul 4214  df-if 4384  df-pw 4457  df-sn 4475  df-pr 4477  df-op 4481  df-uni 4748  df-br 4965  df-opab 5027  df-mpt 5044  df-id 5351  df-xp 5452  df-rel 5453  df-cnv 5454  df-co 5455  df-dm 5456  df-iota 6192  df-fun 6230  df-fv 6236  df-itgo 39257
This theorem is referenced by:  aaitgo  39260  itgoss  39261  itgocn  39262
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