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Mathbox for Alexander van der Vekens |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > iccpartimp | Structured version Visualization version GIF version |
Description: Implications for a class being a partition. (Contributed by AV, 11-Jul-2020.) |
Ref | Expression |
---|---|
iccpartimp | β’ ((π β β β§ π β (RePartβπ) β§ πΌ β (0..^π)) β (π β (β* βm (0...π)) β§ (πβπΌ) < (πβ(πΌ + 1)))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | iccpart 45698 | . . 3 β’ (π β β β (π β (RePartβπ) β (π β (β* βm (0...π)) β§ βπ β (0..^π)(πβπ) < (πβ(π + 1))))) | |
2 | fveq2 6846 | . . . . . . 7 β’ (π = πΌ β (πβπ) = (πβπΌ)) | |
3 | fvoveq1 7384 | . . . . . . 7 β’ (π = πΌ β (πβ(π + 1)) = (πβ(πΌ + 1))) | |
4 | 2, 3 | breq12d 5122 | . . . . . 6 β’ (π = πΌ β ((πβπ) < (πβ(π + 1)) β (πβπΌ) < (πβ(πΌ + 1)))) |
5 | 4 | rspccv 3580 | . . . . 5 β’ (βπ β (0..^π)(πβπ) < (πβ(π + 1)) β (πΌ β (0..^π) β (πβπΌ) < (πβ(πΌ + 1)))) |
6 | 5 | adantl 483 | . . . 4 β’ ((π β (β* βm (0...π)) β§ βπ β (0..^π)(πβπ) < (πβ(π + 1))) β (πΌ β (0..^π) β (πβπΌ) < (πβ(πΌ + 1)))) |
7 | simpl 484 | . . . 4 β’ ((π β (β* βm (0...π)) β§ βπ β (0..^π)(πβπ) < (πβ(π + 1))) β π β (β* βm (0...π))) | |
8 | 6, 7 | jctild 527 | . . 3 β’ ((π β (β* βm (0...π)) β§ βπ β (0..^π)(πβπ) < (πβ(π + 1))) β (πΌ β (0..^π) β (π β (β* βm (0...π)) β§ (πβπΌ) < (πβ(πΌ + 1))))) |
9 | 1, 8 | syl6bi 253 | . 2 β’ (π β β β (π β (RePartβπ) β (πΌ β (0..^π) β (π β (β* βm (0...π)) β§ (πβπΌ) < (πβ(πΌ + 1)))))) |
10 | 9 | 3imp 1112 | 1 β’ ((π β β β§ π β (RePartβπ) β§ πΌ β (0..^π)) β (π β (β* βm (0...π)) β§ (πβπΌ) < (πβ(πΌ + 1)))) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ wa 397 β§ w3a 1088 = wceq 1542 β wcel 2107 βwral 3061 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: iccpartgtprec 45702 iccpartipre 45703 iccpartiltu 45704 iccpartigtl 45705 iccpartlt 45706 iccpartgt 45709 |
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