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Theorem iccpval 47340
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 7439 . . . 4 (𝑚 = 𝑀 → (0...𝑚) = (0...𝑀))
21oveq2d 7447 . . 3 (𝑚 = 𝑀 → (ℝ*m (0...𝑚)) = (ℝ*m (0...𝑀)))
3 oveq2 7439 . . . 4 (𝑚 = 𝑀 → (0..^𝑚) = (0..^𝑀))
43raleqdv 3324 . . 3 (𝑚 = 𝑀 → (∀𝑖 ∈ (0..^𝑚)(𝑝𝑖) < (𝑝‘(𝑖 + 1)) ↔ ∀𝑖 ∈ (0..^𝑀)(𝑝𝑖) < (𝑝‘(𝑖 + 1))))
52, 4rabeqbidv 3452 . 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 47339 . 2 RePart = (𝑚 ∈ ℕ ↦ {𝑝 ∈ (ℝ*m (0...𝑚)) ∣ ∀𝑖 ∈ (0..^𝑚)(𝑝𝑖) < (𝑝‘(𝑖 + 1))})
7 ovex 7464 . . 3 (ℝ*m (0...𝑀)) ∈ V
87rabex 5345 . 2 {𝑝 ∈ (ℝ*m (0...𝑀)) ∣ ∀𝑖 ∈ (0..^𝑀)(𝑝𝑖) < (𝑝‘(𝑖 + 1))} ∈ V
95, 6, 8fvmpt 7016 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 1537  wcel 2106  wral 3059  {crab 3433   class class class wbr 5148  cfv 6563  (class class class)co 7431  m cmap 8865  0cc0 11153  1c1 11154   + caddc 11156  *cxr 11292   < clt 11293  cn 12264  ...cfz 13544  ..^cfzo 13691  RePartciccp 47338
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-iota 6516  df-fun 6565  df-fv 6571  df-ov 7434  df-iccp 47339
This theorem is referenced by:  iccpart  47341
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