Mathbox for Alexander van der Vekens |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > Mathboxes > iccpval | Structured version Visualization version GIF version |
Description: Partition consisting of a fixed number 𝑀 of parts. (Contributed by AV, 9-Jul-2020.) |
Ref | Expression |
---|---|
iccpval | ⊢ (𝑀 ∈ ℕ → (RePart‘𝑀) = {𝑝 ∈ (ℝ* ↑𝑚 (0...𝑀)) ∣ ∀𝑖 ∈ (0..^𝑀)(𝑝‘𝑖) < (𝑝‘(𝑖 + 1))}) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-iccp 39952 | . . 3 ⊢ RePart = (𝑚 ∈ ℕ ↦ {𝑝 ∈ (ℝ* ↑𝑚 (0...𝑚)) ∣ ∀𝑖 ∈ (0..^𝑚)(𝑝‘𝑖) < (𝑝‘(𝑖 + 1))}) | |
2 | 1 | a1i 11 | . 2 ⊢ (𝑀 ∈ ℕ → RePart = (𝑚 ∈ ℕ ↦ {𝑝 ∈ (ℝ* ↑𝑚 (0...𝑚)) ∣ ∀𝑖 ∈ (0..^𝑚)(𝑝‘𝑖) < (𝑝‘(𝑖 + 1))})) |
3 | oveq2 6557 | . . . . 5 ⊢ (𝑚 = 𝑀 → (0...𝑚) = (0...𝑀)) | |
4 | 3 | oveq2d 6565 | . . . 4 ⊢ (𝑚 = 𝑀 → (ℝ* ↑𝑚 (0...𝑚)) = (ℝ* ↑𝑚 (0...𝑀))) |
5 | oveq2 6557 | . . . . 5 ⊢ (𝑚 = 𝑀 → (0..^𝑚) = (0..^𝑀)) | |
6 | 5 | raleqdv 3121 | . . . 4 ⊢ (𝑚 = 𝑀 → (∀𝑖 ∈ (0..^𝑚)(𝑝‘𝑖) < (𝑝‘(𝑖 + 1)) ↔ ∀𝑖 ∈ (0..^𝑀)(𝑝‘𝑖) < (𝑝‘(𝑖 + 1)))) |
7 | 4, 6 | rabeqbidv 3168 | . . 3 ⊢ (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))}) |
8 | 7 | adantl 481 | . 2 ⊢ ((𝑀 ∈ ℕ ∧ 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))}) |
9 | id 22 | . 2 ⊢ (𝑀 ∈ ℕ → 𝑀 ∈ ℕ) | |
10 | ovex 6577 | . . . 4 ⊢ (ℝ* ↑𝑚 (0...𝑀)) ∈ V | |
11 | 10 | rabex 4740 | . . 3 ⊢ {𝑝 ∈ (ℝ* ↑𝑚 (0...𝑀)) ∣ ∀𝑖 ∈ (0..^𝑀)(𝑝‘𝑖) < (𝑝‘(𝑖 + 1))} ∈ V |
12 | 11 | a1i 11 | . 2 ⊢ (𝑀 ∈ ℕ → {𝑝 ∈ (ℝ* ↑𝑚 (0...𝑀)) ∣ ∀𝑖 ∈ (0..^𝑀)(𝑝‘𝑖) < (𝑝‘(𝑖 + 1))} ∈ V) |
13 | 2, 8, 9, 12 | fvmptd 6197 | 1 ⊢ (𝑀 ∈ ℕ → (RePart‘𝑀) = {𝑝 ∈ (ℝ* ↑𝑚 (0...𝑀)) ∣ ∀𝑖 ∈ (0..^𝑀)(𝑝‘𝑖) < (𝑝‘(𝑖 + 1))}) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1475 ∈ wcel 1977 ∀wral 2896 {crab 2900 Vcvv 3173 class class class wbr 4583 ↦ cmpt 4643 ‘cfv 5804 (class class class)co 6549 ↑𝑚 cmap 7744 0cc0 9815 1c1 9816 + caddc 9818 ℝ*cxr 9952 < clt 9953 ℕcn 10897 ...cfz 12197 ..^cfzo 12334 RePartciccp 39951 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pr 4833 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ral 2901 df-rex 2902 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-br 4584 df-opab 4644 df-mpt 4645 df-id 4953 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-iota 5768 df-fun 5806 df-fv 5812 df-ov 6552 df-iccp 39952 |
This theorem is referenced by: iccpart 39954 |
Copyright terms: Public domain | W3C validator |