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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 | oveq2 7153 | . . . 4 ⊢ (𝑚 = 𝑀 → (0...𝑚) = (0...𝑀)) | |
2 | 1 | oveq2d 7161 | . . 3 ⊢ (𝑚 = 𝑀 → (ℝ* ↑m (0...𝑚)) = (ℝ* ↑m (0...𝑀))) |
3 | oveq2 7153 | . . . 4 ⊢ (𝑚 = 𝑀 → (0..^𝑚) = (0..^𝑀)) | |
4 | 3 | raleqdv 3413 | . . 3 ⊢ (𝑚 = 𝑀 → (∀𝑖 ∈ (0..^𝑚)(𝑝‘𝑖) < (𝑝‘(𝑖 + 1)) ↔ ∀𝑖 ∈ (0..^𝑀)(𝑝‘𝑖) < (𝑝‘(𝑖 + 1)))) |
5 | 2, 4 | rabeqbidv 3483 | . 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 43451 | . 2 ⊢ RePart = (𝑚 ∈ ℕ ↦ {𝑝 ∈ (ℝ* ↑m (0...𝑚)) ∣ ∀𝑖 ∈ (0..^𝑚)(𝑝‘𝑖) < (𝑝‘(𝑖 + 1))}) | |
7 | ovex 7178 | . . 3 ⊢ (ℝ* ↑m (0...𝑀)) ∈ V | |
8 | 7 | rabex 5226 | . 2 ⊢ {𝑝 ∈ (ℝ* ↑m (0...𝑀)) ∣ ∀𝑖 ∈ (0..^𝑀)(𝑝‘𝑖) < (𝑝‘(𝑖 + 1))} ∈ V |
9 | 5, 6, 8 | fvmpt 6761 | 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 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 ℝ*cxr 10662 < clt 10663 ℕcn 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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