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Mathbox for Alexander van der Vekens |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > iccpartxr | Structured version Visualization version GIF version |
Description: If there is a partition, then all intermediate points and bounds are extended real numbers. (Contributed by AV, 11-Jul-2020.) |
Ref | Expression |
---|---|
iccpartgtprec.m | β’ (π β π β β) |
iccpartgtprec.p | β’ (π β π β (RePartβπ)) |
iccpartxr.i | β’ (π β πΌ β (0...π)) |
Ref | Expression |
---|---|
iccpartxr | β’ (π β (πβπΌ) β β*) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | iccpartgtprec.p | . . . . 5 β’ (π β π β (RePartβπ)) | |
2 | iccpartgtprec.m | . . . . . 6 β’ (π β π β β) | |
3 | iccpart 46637 | . . . . . 6 β’ (π β β β (π β (RePartβπ) β (π β (β* βm (0...π)) β§ βπ β (0..^π)(πβπ) < (πβ(π + 1))))) | |
4 | 2, 3 | syl 17 | . . . . 5 β’ (π β (π β (RePartβπ) β (π β (β* βm (0...π)) β§ βπ β (0..^π)(πβπ) < (πβ(π + 1))))) |
5 | 1, 4 | mpbid 231 | . . . 4 β’ (π β (π β (β* βm (0...π)) β§ βπ β (0..^π)(πβπ) < (πβ(π + 1)))) |
6 | 5 | simpld 494 | . . 3 β’ (π β π β (β* βm (0...π))) |
7 | elmapi 8842 | . . 3 β’ (π β (β* βm (0...π)) β π:(0...π)βΆβ*) | |
8 | 6, 7 | syl 17 | . 2 β’ (π β π:(0...π)βΆβ*) |
9 | iccpartxr.i | . 2 β’ (π β πΌ β (0...π)) | |
10 | 8, 9 | ffvelcdmd 7080 | 1 β’ (π β (πβπΌ) β β*) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β wb 205 β§ wa 395 β wcel 2098 βwral 3055 class class class wbr 5141 βΆwf 6532 βcfv 6536 (class class class)co 7404 βm cmap 8819 0cc0 11109 1c1 11110 + caddc 11112 β*cxr 11248 < clt 11249 βcn 12213 ...cfz 13487 ..^cfzo 13630 RePartciccp 46634 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2697 ax-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 ax-un 7721 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ne 2935 df-ral 3056 df-rex 3065 df-rab 3427 df-v 3470 df-sbc 3773 df-csb 3889 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-nul 4318 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4903 df-iun 4992 df-br 5142 df-opab 5204 df-mpt 5225 df-id 5567 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-iota 6488 df-fun 6538 df-fn 6539 df-f 6540 df-fv 6544 df-ov 7407 df-oprab 7408 df-mpo 7409 df-1st 7971 df-2nd 7972 df-map 8821 df-iccp 46635 |
This theorem is referenced by: iccpartipre 46642 iccpartiltu 46643 iccpartigtl 46644 iccpartlt 46645 iccpartleu 46649 iccpartgel 46650 iccpartrn 46651 iccelpart 46654 iccpartiun 46655 icceuelpartlem 46656 icceuelpart 46657 iccpartdisj 46658 iccpartnel 46659 bgoldbtbndlem2 47027 |
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