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Theorem iccpart 41116
Description: A special partition. Corresponds to fourierdlem2 40089 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 41115 . . 3 (𝑀 ∈ ℕ → (RePart‘𝑀) = {𝑝 ∈ (ℝ*𝑚 (0...𝑀)) ∣ ∀𝑖 ∈ (0..^𝑀)(𝑝𝑖) < (𝑝‘(𝑖 + 1))})
21eleq2d 2685 . 2 (𝑀 ∈ ℕ → (𝑃 ∈ (RePart‘𝑀) ↔ 𝑃 ∈ {𝑝 ∈ (ℝ*𝑚 (0...𝑀)) ∣ ∀𝑖 ∈ (0..^𝑀)(𝑝𝑖) < (𝑝‘(𝑖 + 1))}))
3 fveq1 6177 . . . . 5 (𝑝 = 𝑃 → (𝑝𝑖) = (𝑃𝑖))
4 fveq1 6177 . . . . 5 (𝑝 = 𝑃 → (𝑝‘(𝑖 + 1)) = (𝑃‘(𝑖 + 1)))
53, 4breq12d 4657 . . . 4 (𝑝 = 𝑃 → ((𝑝𝑖) < (𝑝‘(𝑖 + 1)) ↔ (𝑃𝑖) < (𝑃‘(𝑖 + 1))))
65ralbidv 2983 . . 3 (𝑝 = 𝑃 → (∀𝑖 ∈ (0..^𝑀)(𝑝𝑖) < (𝑝‘(𝑖 + 1)) ↔ ∀𝑖 ∈ (0..^𝑀)(𝑃𝑖) < (𝑃‘(𝑖 + 1))))
76elrab 3357 . 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 276 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 196  wa 384   = wceq 1481  wcel 1988  wral 2909  {crab 2913   class class class wbr 4644  cfv 5876  (class class class)co 6635  𝑚 cmap 7842  0cc0 9921  1c1 9922   + caddc 9924  *cxr 10058   < clt 10059  cn 11005  ...cfz 12311  ..^cfzo 12449  RePartciccp 41113
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1720  ax-4 1735  ax-5 1837  ax-6 1886  ax-7 1933  ax-9 1997  ax-10 2017  ax-11 2032  ax-12 2045  ax-13 2244  ax-ext 2600  ax-sep 4772  ax-nul 4780  ax-pr 4897
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1484  df-ex 1703  df-nf 1708  df-sb 1879  df-eu 2472  df-mo 2473  df-clab 2607  df-cleq 2613  df-clel 2616  df-nfc 2751  df-ral 2914  df-rex 2915  df-rab 2918  df-v 3197  df-sbc 3430  df-csb 3527  df-dif 3570  df-un 3572  df-in 3574  df-ss 3581  df-nul 3908  df-if 4078  df-sn 4169  df-pr 4171  df-op 4175  df-uni 4428  df-br 4645  df-opab 4704  df-mpt 4721  df-id 5014  df-xp 5110  df-rel 5111  df-cnv 5112  df-co 5113  df-dm 5114  df-iota 5839  df-fun 5878  df-fv 5884  df-ov 6638  df-iccp 41114
This theorem is referenced by:  iccpartimp  41117  iccpartres  41118  iccpartxr  41119  iccpartrn  41130  iccpartf  41131  iccpartnel  41138
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