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Theorem itgoval 42616
Description: Value of the integral-over function. (Contributed by Stefan O'Rear, 27-Nov-2014.)
Assertion
Ref Expression
itgoval (𝑆 βŠ† β„‚ β†’ (IntgOverβ€˜π‘†) = {π‘₯ ∈ β„‚ ∣ βˆƒπ‘ ∈ (Polyβ€˜π‘†)((π‘β€˜π‘₯) = 0 ∧ ((coeffβ€˜π‘)β€˜(degβ€˜π‘)) = 1)})
Distinct variable group:   π‘₯,𝑆,𝑝

Proof of Theorem itgoval
Dummy variable 𝑠 is distinct from all other variables.
StepHypRef Expression
1 cnex 11227 . . 3 β„‚ ∈ V
21elpw2 5351 . 2 (𝑆 ∈ 𝒫 β„‚ ↔ 𝑆 βŠ† β„‚)
3 fveq2 6902 . . . . 5 (𝑠 = 𝑆 β†’ (Polyβ€˜π‘ ) = (Polyβ€˜π‘†))
43rexeqdv 3324 . . . 4 (𝑠 = 𝑆 β†’ (βˆƒπ‘ ∈ (Polyβ€˜π‘ )((π‘β€˜π‘₯) = 0 ∧ ((coeffβ€˜π‘)β€˜(degβ€˜π‘)) = 1) ↔ βˆƒπ‘ ∈ (Polyβ€˜π‘†)((π‘β€˜π‘₯) = 0 ∧ ((coeffβ€˜π‘)β€˜(degβ€˜π‘)) = 1)))
54rabbidv 3438 . . 3 (𝑠 = 𝑆 β†’ {π‘₯ ∈ β„‚ ∣ βˆƒπ‘ ∈ (Polyβ€˜π‘ )((π‘β€˜π‘₯) = 0 ∧ ((coeffβ€˜π‘)β€˜(degβ€˜π‘)) = 1)} = {π‘₯ ∈ β„‚ ∣ βˆƒπ‘ ∈ (Polyβ€˜π‘†)((π‘β€˜π‘₯) = 0 ∧ ((coeffβ€˜π‘)β€˜(degβ€˜π‘)) = 1)})
6 df-itgo 42614 . . 3 IntgOver = (𝑠 ∈ 𝒫 β„‚ ↦ {π‘₯ ∈ β„‚ ∣ βˆƒπ‘ ∈ (Polyβ€˜π‘ )((π‘β€˜π‘₯) = 0 ∧ ((coeffβ€˜π‘)β€˜(degβ€˜π‘)) = 1)})
71rabex 5338 . . 3 {π‘₯ ∈ β„‚ ∣ βˆƒπ‘ ∈ (Polyβ€˜π‘†)((π‘β€˜π‘₯) = 0 ∧ ((coeffβ€˜π‘)β€˜(degβ€˜π‘)) = 1)} ∈ V
85, 6, 7fvmpt 7010 . 2 (𝑆 ∈ 𝒫 β„‚ β†’ (IntgOverβ€˜π‘†) = {π‘₯ ∈ β„‚ ∣ βˆƒπ‘ ∈ (Polyβ€˜π‘†)((π‘β€˜π‘₯) = 0 ∧ ((coeffβ€˜π‘)β€˜(degβ€˜π‘)) = 1)})
92, 8sylbir 234 1 (𝑆 βŠ† β„‚ β†’ (IntgOverβ€˜π‘†) = {π‘₯ ∈ β„‚ ∣ βˆƒπ‘ ∈ (Polyβ€˜π‘†)((π‘β€˜π‘₯) = 0 ∧ ((coeffβ€˜π‘)β€˜(degβ€˜π‘)) = 1)})
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ wa 394   = wceq 1533   ∈ wcel 2098  βˆƒwrex 3067  {crab 3430   βŠ† wss 3949  π’« cpw 4606  β€˜cfv 6553  β„‚cc 11144  0cc0 11146  1c1 11147  Polycply 26138  coeffccoe 26140  degcdgr 26141  IntgOvercitgo 42612
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2699  ax-sep 5303  ax-nul 5310  ax-pr 5433  ax-cnex 11202
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2529  df-eu 2558  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ral 3059  df-rex 3068  df-rab 3431  df-v 3475  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4327  df-if 4533  df-pw 4608  df-sn 4633  df-pr 4635  df-op 4639  df-uni 4913  df-br 5153  df-opab 5215  df-mpt 5236  df-id 5580  df-xp 5688  df-rel 5689  df-cnv 5690  df-co 5691  df-dm 5692  df-iota 6505  df-fun 6555  df-fv 6561  df-itgo 42614
This theorem is referenced by:  aaitgo  42617  itgoss  42618  itgocn  42619
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