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Theorem iccpval 43452
Description: Partition consisting of a fixed number 𝑀 of parts. (Contributed by AV, 9-Jul-2020.)
Assertion
Ref Expression
iccpval (𝑀 ∈ ℕ → (RePart‘𝑀) = {𝑝 ∈ (ℝ*m (0...𝑀)) ∣ ∀𝑖 ∈ (0..^𝑀)(𝑝𝑖) < (𝑝‘(𝑖 + 1))})
Distinct variable group:   𝑖,𝑝,𝑀

Proof of Theorem iccpval
Dummy variable 𝑚 is distinct from all other variables.
StepHypRef Expression
1 oveq2 7153 . . . 4 (𝑚 = 𝑀 → (0...𝑚) = (0...𝑀))
21oveq2d 7161 . . 3 (𝑚 = 𝑀 → (ℝ*m (0...𝑚)) = (ℝ*m (0...𝑀)))
3 oveq2 7153 . . . 4 (𝑚 = 𝑀 → (0..^𝑚) = (0..^𝑀))
43raleqdv 3413 . . 3 (𝑚 = 𝑀 → (∀𝑖 ∈ (0..^𝑚)(𝑝𝑖) < (𝑝‘(𝑖 + 1)) ↔ ∀𝑖 ∈ (0..^𝑀)(𝑝𝑖) < (𝑝‘(𝑖 + 1))))
52, 4rabeqbidv 3483 . 2 (𝑚 = 𝑀 → {𝑝 ∈ (ℝ*m (0...𝑚)) ∣ ∀𝑖 ∈ (0..^𝑚)(𝑝𝑖) < (𝑝‘(𝑖 + 1))} = {𝑝 ∈ (ℝ*m (0...𝑀)) ∣ ∀𝑖 ∈ (0..^𝑀)(𝑝𝑖) < (𝑝‘(𝑖 + 1))})
6 df-iccp 43451 . 2 RePart = (𝑚 ∈ ℕ ↦ {𝑝 ∈ (ℝ*m (0...𝑚)) ∣ ∀𝑖 ∈ (0..^𝑚)(𝑝𝑖) < (𝑝‘(𝑖 + 1))})
7 ovex 7178 . . 3 (ℝ*m (0...𝑀)) ∈ V
87rabex 5226 . 2 {𝑝 ∈ (ℝ*m (0...𝑀)) ∣ ∀𝑖 ∈ (0..^𝑀)(𝑝𝑖) < (𝑝‘(𝑖 + 1))} ∈ V
95, 6, 8fvmpt 6761 1 (𝑀 ∈ ℕ → (RePart‘𝑀) = {𝑝 ∈ (ℝ*m (0...𝑀)) ∣ ∀𝑖 ∈ (0..^𝑀)(𝑝𝑖) < (𝑝‘(𝑖 + 1))})
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1528  wcel 2105  wral 3135  {crab 3139   class class class wbr 5057  cfv 6348  (class class class)co 7145  m cmap 8395  0cc0 10525  1c1 10526   + caddc 10528  *cxr 10662   < clt 10663  cn 11626  ...cfz 12880  ..^cfzo 13021  RePartciccp 43450
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-sep 5194  ax-nul 5201  ax-pr 5320
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ral 3140  df-rex 3141  df-rab 3144  df-v 3494  df-sbc 3770  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4831  df-br 5058  df-opab 5120  df-mpt 5138  df-id 5453  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-iota 6307  df-fun 6350  df-fv 6356  df-ov 7148  df-iccp 43451
This theorem is referenced by:  iccpart  43453
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