Users' Mathboxes Mathbox for Alexander van der Vekens < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  iccpval Structured version   Visualization version   GIF version

Theorem iccpval 45919
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 7402 . . . 4 (𝑚 = 𝑀 → (0...𝑚) = (0...𝑀))
21oveq2d 7410 . . 3 (𝑚 = 𝑀 → (ℝ*m (0...𝑚)) = (ℝ*m (0...𝑀)))
3 oveq2 7402 . . . 4 (𝑚 = 𝑀 → (0..^𝑚) = (0..^𝑀))
43raleqdv 3325 . . 3 (𝑚 = 𝑀 → (∀𝑖 ∈ (0..^𝑚)(𝑝𝑖) < (𝑝‘(𝑖 + 1)) ↔ ∀𝑖 ∈ (0..^𝑀)(𝑝𝑖) < (𝑝‘(𝑖 + 1))))
52, 4rabeqbidv 3449 . 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 45918 . 2 RePart = (𝑚 ∈ ℕ ↦ {𝑝 ∈ (ℝ*m (0...𝑚)) ∣ ∀𝑖 ∈ (0..^𝑚)(𝑝𝑖) < (𝑝‘(𝑖 + 1))})
7 ovex 7427 . . 3 (ℝ*m (0...𝑀)) ∈ V
87rabex 5326 . 2 {𝑝 ∈ (ℝ*m (0...𝑀)) ∣ ∀𝑖 ∈ (0..^𝑀)(𝑝𝑖) < (𝑝‘(𝑖 + 1))} ∈ V
95, 6, 8fvmpt 6985 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 1541  wcel 2106  wral 3061  {crab 3432   class class class wbr 5142  cfv 6533  (class class class)co 7394  m cmap 8805  0cc0 11094  1c1 11095   + caddc 11097  *cxr 11231   < clt 11232  cn 12196  ...cfz 13468  ..^cfzo 13611  RePartciccp 45917
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-sep 5293  ax-nul 5300  ax-pr 5421
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3433  df-v 3476  df-dif 3948  df-un 3950  df-in 3952  df-ss 3962  df-nul 4320  df-if 4524  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-br 5143  df-opab 5205  df-mpt 5226  df-id 5568  df-xp 5676  df-rel 5677  df-cnv 5678  df-co 5679  df-dm 5680  df-iota 6485  df-fun 6535  df-fv 6541  df-ov 7397  df-iccp 45918
This theorem is referenced by:  iccpart  45920
  Copyright terms: Public domain W3C validator