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Theorem iccpart 44541
Description: A special partition. Corresponds to fourierdlem2 43325 in GS's mathbox. (Contributed by AV, 9-Jul-2020.)
Assertion
Ref Expression
iccpart (𝑀 ∈ ℕ → (𝑃 ∈ (RePart‘𝑀) ↔ (𝑃 ∈ (ℝ*m (0...𝑀)) ∧ ∀𝑖 ∈ (0..^𝑀)(𝑃𝑖) < (𝑃‘(𝑖 + 1)))))
Distinct variable groups:   𝑖,𝑀   𝑃,𝑖

Proof of Theorem iccpart
Dummy variable 𝑝 is distinct from all other variables.
StepHypRef Expression
1 iccpval 44540 . . 3 (𝑀 ∈ ℕ → (RePart‘𝑀) = {𝑝 ∈ (ℝ*m (0...𝑀)) ∣ ∀𝑖 ∈ (0..^𝑀)(𝑝𝑖) < (𝑝‘(𝑖 + 1))})
21eleq2d 2823 . 2 (𝑀 ∈ ℕ → (𝑃 ∈ (RePart‘𝑀) ↔ 𝑃 ∈ {𝑝 ∈ (ℝ*m (0...𝑀)) ∣ ∀𝑖 ∈ (0..^𝑀)(𝑝𝑖) < (𝑝‘(𝑖 + 1))}))
3 fveq1 6716 . . . . 5 (𝑝 = 𝑃 → (𝑝𝑖) = (𝑃𝑖))
4 fveq1 6716 . . . . 5 (𝑝 = 𝑃 → (𝑝‘(𝑖 + 1)) = (𝑃‘(𝑖 + 1)))
53, 4breq12d 5066 . . . 4 (𝑝 = 𝑃 → ((𝑝𝑖) < (𝑝‘(𝑖 + 1)) ↔ (𝑃𝑖) < (𝑃‘(𝑖 + 1))))
65ralbidv 3118 . . 3 (𝑝 = 𝑃 → (∀𝑖 ∈ (0..^𝑀)(𝑝𝑖) < (𝑝‘(𝑖 + 1)) ↔ ∀𝑖 ∈ (0..^𝑀)(𝑃𝑖) < (𝑃‘(𝑖 + 1))))
76elrab 3602 . 2 (𝑃 ∈ {𝑝 ∈ (ℝ*m (0...𝑀)) ∣ ∀𝑖 ∈ (0..^𝑀)(𝑝𝑖) < (𝑝‘(𝑖 + 1))} ↔ (𝑃 ∈ (ℝ*m (0...𝑀)) ∧ ∀𝑖 ∈ (0..^𝑀)(𝑃𝑖) < (𝑃‘(𝑖 + 1))))
82, 7bitrdi 290 1 (𝑀 ∈ ℕ → (𝑃 ∈ (RePart‘𝑀) ↔ (𝑃 ∈ (ℝ*m (0...𝑀)) ∧ ∀𝑖 ∈ (0..^𝑀)(𝑃𝑖) < (𝑃‘(𝑖 + 1)))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wa 399   = wceq 1543  wcel 2110  wral 3061  {crab 3065   class class class wbr 5053  cfv 6380  (class class class)co 7213  m cmap 8508  0cc0 10729  1c1 10730   + caddc 10732  *cxr 10866   < clt 10867  cn 11830  ...cfz 13095  ..^cfzo 13238  RePartciccp 44538
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2158  ax-12 2175  ax-ext 2708  ax-sep 5192  ax-nul 5199  ax-pr 5322
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-nf 1792  df-sb 2071  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2886  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3410  df-dif 3869  df-un 3871  df-in 3873  df-ss 3883  df-nul 4238  df-if 4440  df-sn 4542  df-pr 4544  df-op 4548  df-uni 4820  df-br 5054  df-opab 5116  df-mpt 5136  df-id 5455  df-xp 5557  df-rel 5558  df-cnv 5559  df-co 5560  df-dm 5561  df-iota 6338  df-fun 6382  df-fv 6388  df-ov 7216  df-iccp 44539
This theorem is referenced by:  iccpartimp  44542  iccpartres  44543  iccpartxr  44544  iccpartrn  44555  iccpartf  44556  iccpartnel  44563
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