Theorem iccpval 48315

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)(𝑝‘𝑖) < (𝑝‘(𝑖 + 1))})

Distinct variable group:   𝑖,𝑝,𝑀

Proof of Theorem iccpval

Dummy variable 𝑚 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		oveq2 7421	. . . 4 ⊢ (𝑚 = 𝑀 → (0...𝑚) = (0...𝑀))
2	1	oveq2d 7429	. . 3 ⊢ (𝑚 = 𝑀 → (ℝ* ↑m (0...𝑚)) = (ℝ* ↑m (0...𝑀)))
3		oveq2 7421	. . . 4 ⊢ (𝑚 = 𝑀 → (0..^𝑚) = (0..^𝑀))
4	3	raleqdv 3319	. . 3 ⊢ (𝑚 = 𝑀 → (∀𝑖 ∈ (0..^𝑚)(𝑝‘𝑖) < (𝑝‘(𝑖 + 1)) ↔ ∀𝑖 ∈ (0..^𝑀)(𝑝‘𝑖) < (𝑝‘(𝑖 + 1))))
5	2, 4	rabeqbidv 3429	. 2 ⊢ (𝑚 = 𝑀 → {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))})
6		df-iccp 48314	. 2 ⊢ RePart = (𝑚 ∈ ℕ ↦ {𝑝 ∈ (ℝ* ↑m (0...𝑚)) ∣ ∀𝑖 ∈ (0..^𝑚)(𝑝‘𝑖) < (𝑝‘(𝑖 + 1))})
7		ovex 7446	. . 3 ⊢ (ℝ* ↑m (0...𝑀)) ∈ V
8	7	rabex 5303	. 2 ⊢ {𝑝 ∈ (ℝ* ↑m (0...𝑀)) ∣ ∀𝑖 ∈ (0..^𝑀)(𝑝‘𝑖) < (𝑝‘(𝑖 + 1))} ∈ V
9	5, 6, 8	fvmpt 6986	1 ⊢ (𝑀 ∈ ℕ → (RePart‘𝑀) = {𝑝 ∈ (ℝ* ↑m (0...𝑀)) ∣ ∀𝑖 ∈ (0..^𝑀)(𝑝‘𝑖) < (𝑝‘(𝑖 + 1))})

This proof depends on syntax axioms:   → wi 4   = wceq 1570   ∈ wcel 2145  ∀wral 3076  {crab 3412   class class class wbr 5103  ‘cfv 6533  (class class class)co 7413   ↑_m cmap 8826  0cc0 11124  1c1 11125   + caddc 11127  ℝ^*cxr 11266   < clt 11267  ℕcn 12257  ...cfz 13561  ..^cfzo 13709  RePartciccp 48313

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2732  ax-sep 5251  ax-nul 5263  ax-pr 5398

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-rab 3413  df-v 3452  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-br 5104  df-opab 5168  df-mpt 5187  df-id 5550  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-iota 6489  df-fun 6535  df-fv 6541  df-ov 7416  df-iccp 48314

This theorem is used by:  iccpart  48316