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Theorem itgoval 42463
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 11190 . . 3 β„‚ ∈ V
21elpw2 5338 . 2 (𝑆 ∈ 𝒫 β„‚ ↔ 𝑆 βŠ† β„‚)
3 fveq2 6884 . . . . 5 (𝑠 = 𝑆 β†’ (Polyβ€˜π‘ ) = (Polyβ€˜π‘†))
43rexeqdv 3320 . . . 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 3434 . . 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 42461 . . 3 IntgOver = (𝑠 ∈ 𝒫 β„‚ ↦ {π‘₯ ∈ β„‚ ∣ βˆƒπ‘ ∈ (Polyβ€˜π‘ )((π‘β€˜π‘₯) = 0 ∧ ((coeffβ€˜π‘)β€˜(degβ€˜π‘)) = 1)})
71rabex 5325 . . 3 {π‘₯ ∈ β„‚ ∣ βˆƒπ‘ ∈ (Polyβ€˜π‘†)((π‘β€˜π‘₯) = 0 ∧ ((coeffβ€˜π‘)β€˜(degβ€˜π‘)) = 1)} ∈ V
85, 6, 7fvmpt 6991 . 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 395   = wceq 1533   ∈ wcel 2098  βˆƒwrex 3064  {crab 3426   βŠ† wss 3943  π’« cpw 4597  β€˜cfv 6536  β„‚cc 11107  0cc0 11109  1c1 11110  Polycply 26068  coeffccoe 26070  degcdgr 26071  IntgOvercitgo 42459
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 2163  ax-ext 2697  ax-sep 5292  ax-nul 5299  ax-pr 5420  ax-cnex 11165
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ral 3056  df-rex 3065  df-rab 3427  df-v 3470  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-pw 4599  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-br 5142  df-opab 5204  df-mpt 5225  df-id 5567  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-iota 6488  df-fun 6538  df-fv 6544  df-itgo 42461
This theorem is referenced by:  aaitgo  42464  itgoss  42465  itgocn  42466
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