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Mathbox for Thierry Arnoux |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > reprf | Structured version Visualization version GIF version |
Description: Members of the representation of π as the sum of π nonnegative integers from set π΄ as functions. (Contributed by Thierry Arnoux, 5-Dec-2021.) |
Ref | Expression |
---|---|
reprval.a | β’ (π β π΄ β β) |
reprval.m | β’ (π β π β β€) |
reprval.s | β’ (π β π β β0) |
reprf.c | β’ (π β πΆ β (π΄(reprβπ)π)) |
Ref | Expression |
---|---|
reprf | β’ (π β πΆ:(0..^π)βΆπ΄) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | reprf.c | . . 3 β’ (π β πΆ β (π΄(reprβπ)π)) | |
2 | reprval.a | . . . 4 β’ (π β π΄ β β) | |
3 | reprval.m | . . . 4 β’ (π β π β β€) | |
4 | reprval.s | . . . 4 β’ (π β π β β0) | |
5 | 2, 3, 4 | reprval 33611 | . . 3 β’ (π β (π΄(reprβπ)π) = {π β (π΄ βm (0..^π)) β£ Ξ£π β (0..^π)(πβπ) = π}) |
6 | 1, 5 | eleqtrd 2836 | . 2 β’ (π β πΆ β {π β (π΄ βm (0..^π)) β£ Ξ£π β (0..^π)(πβπ) = π}) |
7 | elrabi 3677 | . 2 β’ (πΆ β {π β (π΄ βm (0..^π)) β£ Ξ£π β (0..^π)(πβπ) = π} β πΆ β (π΄ βm (0..^π))) | |
8 | elmapi 8840 | . 2 β’ (πΆ β (π΄ βm (0..^π)) β πΆ:(0..^π)βΆπ΄) | |
9 | 6, 7, 8 | 3syl 18 | 1 β’ (π β πΆ:(0..^π)βΆπ΄) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 = wceq 1542 β wcel 2107 {crab 3433 β wss 3948 βΆwf 6537 βcfv 6541 (class class class)co 7406 βm cmap 8817 0cc0 11107 βcn 12209 β0cn0 12469 β€cz 12555 ..^cfzo 13624 Ξ£csu 15629 reprcrepr 33609 |
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-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7722 ax-cnex 11163 ax-resscn 11164 ax-1cn 11165 ax-addcl 11167 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 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 2942 df-ral 3063 df-rex 3072 df-reu 3378 df-rab 3434 df-v 3477 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-pss 3967 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-iun 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5574 df-eprel 5580 df-po 5588 df-so 5589 df-fr 5631 df-we 5633 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-pred 6298 df-ord 6365 df-on 6366 df-lim 6367 df-suc 6368 df-iota 6493 df-fun 6543 df-fn 6544 df-f 6545 df-f1 6546 df-fo 6547 df-f1o 6548 df-fv 6549 df-ov 7409 df-oprab 7410 df-mpo 7411 df-om 7853 df-1st 7972 df-2nd 7973 df-frecs 8263 df-wrecs 8294 df-recs 8368 df-rdg 8407 df-map 8819 df-neg 11444 df-nn 12210 df-z 12556 df-seq 13964 df-sum 15630 df-repr 33610 |
This theorem is referenced by: reprle 33615 reprsuc 33616 hashreprin 33621 reprpmtf1o 33627 reprdifc 33628 breprexplema 33631 breprexplemc 33633 breprexpnat 33635 circlemeth 33641 circlevma 33643 circlemethhgt 33644 hgt750lemb 33657 hgt750lema 33658 hgt750leme 33659 tgoldbachgtde 33661 tgoldbachgt 33664 |
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