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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 π of parts. (Contributed by AV, 9-Jul-2020.) |
Ref | Expression |
---|---|
iccpval | β’ (π β β β (RePartβπ) = {π β (β* βm (0...π)) β£ βπ β (0..^π)(πβπ) < (πβ(π + 1))}) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | oveq2 7369 | . . . 4 β’ (π = π β (0...π) = (0...π)) | |
2 | 1 | oveq2d 7377 | . . 3 β’ (π = π β (β* βm (0...π)) = (β* βm (0...π))) |
3 | oveq2 7369 | . . . 4 β’ (π = π β (0..^π) = (0..^π)) | |
4 | 3 | raleqdv 3312 | . . 3 β’ (π = π β (βπ β (0..^π)(πβπ) < (πβ(π + 1)) β βπ β (0..^π)(πβπ) < (πβ(π + 1)))) |
5 | 2, 4 | rabeqbidv 3423 | . 2 β’ (π = π β {π β (β* βm (0...π)) β£ βπ β (0..^π)(πβπ) < (πβ(π + 1))} = {π β (β* βm (0...π)) β£ βπ β (0..^π)(πβπ) < (πβ(π + 1))}) |
6 | df-iccp 45696 | . 2 β’ RePart = (π β β β¦ {π β (β* βm (0...π)) β£ βπ β (0..^π)(πβπ) < (πβ(π + 1))}) | |
7 | ovex 7394 | . . 3 β’ (β* βm (0...π)) β V | |
8 | 7 | rabex 5293 | . 2 β’ {π β (β* βm (0...π)) β£ βπ β (0..^π)(πβπ) < (πβ(π + 1))} β V |
9 | 5, 6, 8 | fvmpt 6952 | 1 β’ (π β β β (RePartβπ) = {π β (β* βm (0...π)) β£ βπ β (0..^π)(πβπ) < (πβ(π + 1))}) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 = wceq 1542 β wcel 2107 βwral 3061 {crab 3406 class class class wbr 5109 βcfv 6500 (class class class)co 7361 βm cmap 8771 0cc0 11059 1c1 11060 + caddc 11062 β*cxr 11196 < clt 11197 βcn 12161 ...cfz 13433 ..^cfzo 13576 RePartciccp 45695 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-sep 5260 ax-nul 5267 ax-pr 5388 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3407 df-v 3449 df-dif 3917 df-un 3919 df-in 3921 df-ss 3931 df-nul 4287 df-if 4491 df-sn 4591 df-pr 4593 df-op 4597 df-uni 4870 df-br 5110 df-opab 5172 df-mpt 5193 df-id 5535 df-xp 5643 df-rel 5644 df-cnv 5645 df-co 5646 df-dm 5647 df-iota 6452 df-fun 6502 df-fv 6508 df-ov 7364 df-iccp 45696 |
This theorem is referenced by: iccpart 45698 |
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