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Theorem iccpart 42240
Description: A special partition. Corresponds to fourierdlem2 41120 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 42239 . . 3 (𝑀 ∈ ℕ → (RePart‘𝑀) = {𝑝 ∈ (ℝ*𝑚 (0...𝑀)) ∣ ∀𝑖 ∈ (0..^𝑀)(𝑝𝑖) < (𝑝‘(𝑖 + 1))})
21eleq2d 2892 . 2 (𝑀 ∈ ℕ → (𝑃 ∈ (RePart‘𝑀) ↔ 𝑃 ∈ {𝑝 ∈ (ℝ*𝑚 (0...𝑀)) ∣ ∀𝑖 ∈ (0..^𝑀)(𝑝𝑖) < (𝑝‘(𝑖 + 1))}))
3 fveq1 6432 . . . . 5 (𝑝 = 𝑃 → (𝑝𝑖) = (𝑃𝑖))
4 fveq1 6432 . . . . 5 (𝑝 = 𝑃 → (𝑝‘(𝑖 + 1)) = (𝑃‘(𝑖 + 1)))
53, 4breq12d 4886 . . . 4 (𝑝 = 𝑃 → ((𝑝𝑖) < (𝑝‘(𝑖 + 1)) ↔ (𝑃𝑖) < (𝑃‘(𝑖 + 1))))
65ralbidv 3195 . . 3 (𝑝 = 𝑃 → (∀𝑖 ∈ (0..^𝑀)(𝑝𝑖) < (𝑝‘(𝑖 + 1)) ↔ ∀𝑖 ∈ (0..^𝑀)(𝑃𝑖) < (𝑃‘(𝑖 + 1))))
76elrab 3585 . 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, 7syl6bb 279 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 198  wa 386   = wceq 1658  wcel 2166  wral 3117  {crab 3121   class class class wbr 4873  cfv 6123  (class class class)co 6905  𝑚 cmap 8122  0cc0 10252  1c1 10253   + caddc 10255  *cxr 10390   < clt 10391  cn 11350  ...cfz 12619  ..^cfzo 12760  RePartciccp 42237
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1896  ax-4 1910  ax-5 2011  ax-6 2077  ax-7 2114  ax-9 2175  ax-10 2194  ax-11 2209  ax-12 2222  ax-13 2391  ax-ext 2803  ax-sep 5005  ax-nul 5013  ax-pr 5127
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 881  df-3an 1115  df-tru 1662  df-ex 1881  df-nf 1885  df-sb 2070  df-mo 2605  df-eu 2640  df-clab 2812  df-cleq 2818  df-clel 2821  df-nfc 2958  df-ral 3122  df-rex 3123  df-rab 3126  df-v 3416  df-sbc 3663  df-csb 3758  df-dif 3801  df-un 3803  df-in 3805  df-ss 3812  df-nul 4145  df-if 4307  df-sn 4398  df-pr 4400  df-op 4404  df-uni 4659  df-br 4874  df-opab 4936  df-mpt 4953  df-id 5250  df-xp 5348  df-rel 5349  df-cnv 5350  df-co 5351  df-dm 5352  df-iota 6086  df-fun 6125  df-fv 6131  df-ov 6908  df-iccp 42238
This theorem is referenced by:  iccpartimp  42241  iccpartres  42242  iccpartxr  42243  iccpartrn  42254  iccpartf  42255  iccpartnel  42262
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