Theorem iccpart 44868

Description: A special partition. Corresponds to fourierdlem2 43650 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)(𝑃‘𝑖) < (𝑃‘(𝑖 + 1)))))

Distinct variable groups:   𝑖,𝑀   𝑃,𝑖

Proof of Theorem iccpart

Dummy variable 𝑝 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		iccpval 44867	. . 3 ⊢ (𝑀 ∈ ℕ → (RePart‘𝑀) = {𝑝 ∈ (ℝ* ↑m (0...𝑀)) ∣ ∀𝑖 ∈ (0..^𝑀)(𝑝‘𝑖) < (𝑝‘(𝑖 + 1))})
2	1	eleq2d 2824	. 2 ⊢ (𝑀 ∈ ℕ → (𝑃 ∈ (RePart‘𝑀) ↔ 𝑃 ∈ {𝑝 ∈ (ℝ* ↑m (0...𝑀)) ∣ ∀𝑖 ∈ (0..^𝑀)(𝑝‘𝑖) < (𝑝‘(𝑖 + 1))}))
3		fveq1 6773	. . . . 5 ⊢ (𝑝 = 𝑃 → (𝑝‘𝑖) = (𝑃‘𝑖))
4		fveq1 6773	. . . . 5 ⊢ (𝑝 = 𝑃 → (𝑝‘(𝑖 + 1)) = (𝑃‘(𝑖 + 1)))
5	3, 4	breq12d 5087	. . . 4 ⊢ (𝑝 = 𝑃 → ((𝑝‘𝑖) < (𝑝‘(𝑖 + 1)) ↔ (𝑃‘𝑖) < (𝑃‘(𝑖 + 1))))
6	5	ralbidv 3112	. . 3 ⊢ (𝑝 = 𝑃 → (∀𝑖 ∈ (0..^𝑀)(𝑝‘𝑖) < (𝑝‘(𝑖 + 1)) ↔ ∀𝑖 ∈ (0..^𝑀)(𝑃‘𝑖) < (𝑃‘(𝑖 + 1))))
7	6	elrab 3624	. 2 ⊢ (𝑃 ∈ {𝑝 ∈ (ℝ* ↑m (0...𝑀)) ∣ ∀𝑖 ∈ (0..^𝑀)(𝑝‘𝑖) < (𝑝‘(𝑖 + 1))} ↔ (𝑃 ∈ (ℝ* ↑m (0...𝑀)) ∧ ∀𝑖 ∈ (0..^𝑀)(𝑃‘𝑖) < (𝑃‘(𝑖 + 1))))
8	2, 7	bitrdi 287	1 ⊢ (𝑀 ∈ ℕ → (𝑃 ∈ (RePart‘𝑀) ↔ (𝑃 ∈ (ℝ* ↑m (0...𝑀)) ∧ ∀𝑖 ∈ (0..^𝑀)(𝑃‘𝑖) < (𝑃‘(𝑖 + 1)))))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 396   = wceq 1539   ∈ wcel 2106  ∀wral 3064  {crab 3068   class class class wbr 5074  ‘cfv 6433  (class class class)co 7275   ↑_m cmap 8615  0cc0 10871  1c1 10872   + caddc 10874  ℝ^*cxr 11008   < clt 11009  ℕcn 11973  ...cfz 13239  ..^cfzo 13382  RePartciccp 44865

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-sep 5223  ax-nul 5230  ax-pr 5352

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ral 3069  df-rex 3070  df-rab 3073  df-v 3434  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-br 5075  df-opab 5137  df-mpt 5158  df-id 5489  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-iota 6391  df-fun 6435  df-fv 6441  df-ov 7278  df-iccp 44866

This theorem is referenced by:  iccpartimp  44869  iccpartres  44870  iccpartxr  44871  iccpartrn  44882  iccpartf  44883  iccpartnel  44890