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Theorem iccpart 43080
Description: A special partition. Corresponds to fourierdlem2 41958 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 43079 . . 3 (𝑀 ∈ ℕ → (RePart‘𝑀) = {𝑝 ∈ (ℝ*𝑚 (0...𝑀)) ∣ ∀𝑖 ∈ (0..^𝑀)(𝑝𝑖) < (𝑝‘(𝑖 + 1))})
21eleq2d 2870 . 2 (𝑀 ∈ ℕ → (𝑃 ∈ (RePart‘𝑀) ↔ 𝑃 ∈ {𝑝 ∈ (ℝ*𝑚 (0...𝑀)) ∣ ∀𝑖 ∈ (0..^𝑀)(𝑝𝑖) < (𝑝‘(𝑖 + 1))}))
3 fveq1 6544 . . . . 5 (𝑝 = 𝑃 → (𝑝𝑖) = (𝑃𝑖))
4 fveq1 6544 . . . . 5 (𝑝 = 𝑃 → (𝑝‘(𝑖 + 1)) = (𝑃‘(𝑖 + 1)))
53, 4breq12d 4981 . . . 4 (𝑝 = 𝑃 → ((𝑝𝑖) < (𝑝‘(𝑖 + 1)) ↔ (𝑃𝑖) < (𝑃‘(𝑖 + 1))))
65ralbidv 3166 . . 3 (𝑝 = 𝑃 → (∀𝑖 ∈ (0..^𝑀)(𝑝𝑖) < (𝑝‘(𝑖 + 1)) ↔ ∀𝑖 ∈ (0..^𝑀)(𝑃𝑖) < (𝑃‘(𝑖 + 1))))
76elrab 3621 . 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 288 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 207  wa 396   = wceq 1525  wcel 2083  wral 3107  {crab 3111   class class class wbr 4968  cfv 6232  (class class class)co 7023  𝑚 cmap 8263  0cc0 10390  1c1 10391   + caddc 10393  *cxr 10527   < clt 10528  cn 11492  ...cfz 12746  ..^cfzo 12887  RePartciccp 43077
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1781  ax-4 1795  ax-5 1892  ax-6 1951  ax-7 1996  ax-8 2085  ax-9 2093  ax-10 2114  ax-11 2128  ax-12 2143  ax-13 2346  ax-ext 2771  ax-sep 5101  ax-nul 5108  ax-pr 5228
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 843  df-3an 1082  df-tru 1528  df-ex 1766  df-nf 1770  df-sb 2045  df-mo 2578  df-eu 2614  df-clab 2778  df-cleq 2790  df-clel 2865  df-nfc 2937  df-ral 3112  df-rex 3113  df-rab 3116  df-v 3442  df-sbc 3712  df-dif 3868  df-un 3870  df-in 3872  df-ss 3880  df-nul 4218  df-if 4388  df-sn 4479  df-pr 4481  df-op 4485  df-uni 4752  df-br 4969  df-opab 5031  df-mpt 5048  df-id 5355  df-xp 5456  df-rel 5457  df-cnv 5458  df-co 5459  df-dm 5460  df-iota 6196  df-fun 6234  df-fv 6240  df-ov 7026  df-iccp 43078
This theorem is referenced by:  iccpartimp  43081  iccpartres  43082  iccpartxr  43083  iccpartrn  43094  iccpartf  43095  iccpartnel  43102
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