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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 7416 | . . . 4 β’ (π = π β (0...π) = (0...π)) | |
2 | 1 | oveq2d 7424 | . . 3 β’ (π = π β (β* βm (0...π)) = (β* βm (0...π))) |
3 | oveq2 7416 | . . . 4 β’ (π = π β (0..^π) = (0..^π)) | |
4 | 3 | raleqdv 3325 | . . 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 3449 | . 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 46072 | . 2 β’ RePart = (π β β β¦ {π β (β* βm (0...π)) β£ βπ β (0..^π)(πβπ) < (πβ(π + 1))}) | |
7 | ovex 7441 | . . 3 β’ (β* βm (0...π)) β V | |
8 | 7 | rabex 5332 | . 2 β’ {π β (β* βm (0...π)) β£ βπ β (0..^π)(πβπ) < (πβ(π + 1))} β V |
9 | 5, 6, 8 | fvmpt 6998 | 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 1541 β wcel 2106 βwral 3061 {crab 3432 class class class wbr 5148 βcfv 6543 (class class class)co 7408 βm cmap 8819 0cc0 11109 1c1 11110 + caddc 11112 β*cxr 11246 < clt 11247 βcn 12211 ...cfz 13483 ..^cfzo 13626 RePartciccp 46071 |
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 5299 ax-nul 5306 ax-pr 5427 |
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 3951 df-un 3953 df-in 3955 df-ss 3965 df-nul 4323 df-if 4529 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5574 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-iota 6495 df-fun 6545 df-fv 6551 df-ov 7411 df-iccp 46072 |
This theorem is referenced by: iccpart 46074 |
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