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Theorem iccpart 43948
 Description: A special partition. Corresponds to fourierdlem2 42766 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 43947 . . 3 (𝑀 ∈ ℕ → (RePart‘𝑀) = {𝑝 ∈ (ℝ*m (0...𝑀)) ∣ ∀𝑖 ∈ (0..^𝑀)(𝑝𝑖) < (𝑝‘(𝑖 + 1))})
21eleq2d 2875 . 2 (𝑀 ∈ ℕ → (𝑃 ∈ (RePart‘𝑀) ↔ 𝑃 ∈ {𝑝 ∈ (ℝ*m (0...𝑀)) ∣ ∀𝑖 ∈ (0..^𝑀)(𝑝𝑖) < (𝑝‘(𝑖 + 1))}))
3 fveq1 6644 . . . . 5 (𝑝 = 𝑃 → (𝑝𝑖) = (𝑃𝑖))
4 fveq1 6644 . . . . 5 (𝑝 = 𝑃 → (𝑝‘(𝑖 + 1)) = (𝑃‘(𝑖 + 1)))
53, 4breq12d 5043 . . . 4 (𝑝 = 𝑃 → ((𝑝𝑖) < (𝑝‘(𝑖 + 1)) ↔ (𝑃𝑖) < (𝑃‘(𝑖 + 1))))
65ralbidv 3162 . . 3 (𝑝 = 𝑃 → (∀𝑖 ∈ (0..^𝑀)(𝑝𝑖) < (𝑝‘(𝑖 + 1)) ↔ ∀𝑖 ∈ (0..^𝑀)(𝑃𝑖) < (𝑃‘(𝑖 + 1))))
76elrab 3628 . 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 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 1538   ∈ wcel 2111  ∀wral 3106  {crab 3110   class class class wbr 5030  ‘cfv 6324  (class class class)co 7135   ↑m cmap 8391  0cc0 10528  1c1 10529   + caddc 10531  ℝ*cxr 10665   < clt 10666  ℕcn 11627  ...cfz 12887  ..^cfzo 13030  RePartciccp 43945 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-sep 5167  ax-nul 5174  ax-pr 5295 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ral 3111  df-rex 3112  df-rab 3115  df-v 3443  df-sbc 3721  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-nul 4244  df-if 4426  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4801  df-br 5031  df-opab 5093  df-mpt 5111  df-id 5425  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-iota 6283  df-fun 6326  df-fv 6332  df-ov 7138  df-iccp 43946 This theorem is referenced by:  iccpartimp  43949  iccpartres  43950  iccpartxr  43951  iccpartrn  43962  iccpartf  43963  iccpartnel  43970
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