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Theorem itgoval 37550
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 10002 . . 3 ℂ ∈ V
21elpw2 4819 . 2 (𝑆 ∈ 𝒫 ℂ ↔ 𝑆 ⊆ ℂ)
3 fveq2 6178 . . . . 5 (𝑠 = 𝑆 → (Poly‘𝑠) = (Poly‘𝑆))
43rexeqdv 3140 . . . 4 (𝑠 = 𝑆 → (∃𝑝 ∈ (Poly‘𝑠)((𝑝𝑥) = 0 ∧ ((coeff‘𝑝)‘(deg‘𝑝)) = 1) ↔ ∃𝑝 ∈ (Poly‘𝑆)((𝑝𝑥) = 0 ∧ ((coeff‘𝑝)‘(deg‘𝑝)) = 1)))
54rabbidv 3184 . . 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 37548 . . 3 IntgOver = (𝑠 ∈ 𝒫 ℂ ↦ {𝑥 ∈ ℂ ∣ ∃𝑝 ∈ (Poly‘𝑠)((𝑝𝑥) = 0 ∧ ((coeff‘𝑝)‘(deg‘𝑝)) = 1)})
71rabex 4804 . . 3 {𝑥 ∈ ℂ ∣ ∃𝑝 ∈ (Poly‘𝑆)((𝑝𝑥) = 0 ∧ ((coeff‘𝑝)‘(deg‘𝑝)) = 1)} ∈ V
85, 6, 7fvmpt 6269 . 2 (𝑆 ∈ 𝒫 ℂ → (IntgOver‘𝑆) = {𝑥 ∈ ℂ ∣ ∃𝑝 ∈ (Poly‘𝑆)((𝑝𝑥) = 0 ∧ ((coeff‘𝑝)‘(deg‘𝑝)) = 1)})
92, 8sylbir 225 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 384   = wceq 1481  wcel 1988  wrex 2910  {crab 2913  wss 3567  𝒫 cpw 4149  cfv 5876  cc 9919  0cc0 9921  1c1 9922  Polycply 23921  coeffccoe 23923  degcdgr 23924  IntgOvercitgo 37546
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1720  ax-4 1735  ax-5 1837  ax-6 1886  ax-7 1933  ax-9 1997  ax-10 2017  ax-11 2032  ax-12 2045  ax-13 2244  ax-ext 2600  ax-sep 4772  ax-nul 4780  ax-pr 4897  ax-cnex 9977
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1484  df-ex 1703  df-nf 1708  df-sb 1879  df-eu 2472  df-mo 2473  df-clab 2607  df-cleq 2613  df-clel 2616  df-nfc 2751  df-ral 2914  df-rex 2915  df-rab 2918  df-v 3197  df-sbc 3430  df-dif 3570  df-un 3572  df-in 3574  df-ss 3581  df-nul 3908  df-if 4078  df-pw 4151  df-sn 4169  df-pr 4171  df-op 4175  df-uni 4428  df-br 4645  df-opab 4704  df-mpt 4721  df-id 5014  df-xp 5110  df-rel 5111  df-cnv 5112  df-co 5113  df-dm 5114  df-iota 5839  df-fun 5878  df-fv 5884  df-itgo 37548
This theorem is referenced by:  aaitgo  37551  itgoss  37552  itgocn  37553
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