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Mirrors > Home > MPE Home > Th. List > Mathboxes > eulerpartlemsf | Structured version Visualization version GIF version |
Description: Lemma for eulerpart 33022. (Contributed by Thierry Arnoux, 8-Aug-2018.) |
Ref | Expression |
---|---|
eulerpartlems.r | β’ π = {π β£ (β‘π β β) β Fin} |
eulerpartlems.s | β’ π = (π β ((β0 βm β) β© π ) β¦ Ξ£π β β ((πβπ) Β· π)) |
Ref | Expression |
---|---|
eulerpartlemsf | β’ π:((β0 βm β) β© π )βΆβ0 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eulerpartlems.s | . 2 β’ π = (π β ((β0 βm β) β© π ) β¦ Ξ£π β β ((πβπ) Β· π)) | |
2 | simpl 484 | . . . . . . 7 β’ ((π = π β§ π β β) β π = π) | |
3 | 2 | fveq1d 6849 | . . . . . 6 β’ ((π = π β§ π β β) β (πβπ) = (πβπ)) |
4 | 3 | oveq1d 7377 | . . . . 5 β’ ((π = π β§ π β β) β ((πβπ) Β· π) = ((πβπ) Β· π)) |
5 | 4 | sumeq2dv 15595 | . . . 4 β’ (π = π β Ξ£π β β ((πβπ) Β· π) = Ξ£π β β ((πβπ) Β· π)) |
6 | 5 | eleq1d 2823 | . . 3 β’ (π = π β (Ξ£π β β ((πβπ) Β· π) β β0 β Ξ£π β β ((πβπ) Β· π) β β0)) |
7 | eulerpartlems.r | . . . . . 6 β’ π = {π β£ (β‘π β β) β Fin} | |
8 | 7, 1 | eulerpartlemsv2 32998 | . . . . 5 β’ (π β ((β0 βm β) β© π ) β (πβπ) = Ξ£π β (β‘π β β)((πβπ) Β· π)) |
9 | 7, 1 | eulerpartlemsv1 32996 | . . . . 5 β’ (π β ((β0 βm β) β© π ) β (πβπ) = Ξ£π β β ((πβπ) Β· π)) |
10 | 8, 9 | eqtr3d 2779 | . . . 4 β’ (π β ((β0 βm β) β© π ) β Ξ£π β (β‘π β β)((πβπ) Β· π) = Ξ£π β β ((πβπ) Β· π)) |
11 | 7, 1 | eulerpartlemelr 32997 | . . . . . 6 β’ (π β ((β0 βm β) β© π ) β (π:ββΆβ0 β§ (β‘π β β) β Fin)) |
12 | 11 | simprd 497 | . . . . 5 β’ (π β ((β0 βm β) β© π ) β (β‘π β β) β Fin) |
13 | 11 | simpld 496 | . . . . . . . 8 β’ (π β ((β0 βm β) β© π ) β π:ββΆβ0) |
14 | 13 | adantr 482 | . . . . . . 7 β’ ((π β ((β0 βm β) β© π ) β§ π β (β‘π β β)) β π:ββΆβ0) |
15 | cnvimass 6038 | . . . . . . . . 9 β’ (β‘π β β) β dom π | |
16 | 15, 13 | fssdm 6693 | . . . . . . . 8 β’ (π β ((β0 βm β) β© π ) β (β‘π β β) β β) |
17 | 16 | sselda 3949 | . . . . . . 7 β’ ((π β ((β0 βm β) β© π ) β§ π β (β‘π β β)) β π β β) |
18 | 14, 17 | ffvelcdmd 7041 | . . . . . 6 β’ ((π β ((β0 βm β) β© π ) β§ π β (β‘π β β)) β (πβπ) β β0) |
19 | 17 | nnnn0d 12480 | . . . . . 6 β’ ((π β ((β0 βm β) β© π ) β§ π β (β‘π β β)) β π β β0) |
20 | 18, 19 | nn0mulcld 12485 | . . . . 5 β’ ((π β ((β0 βm β) β© π ) β§ π β (β‘π β β)) β ((πβπ) Β· π) β β0) |
21 | 12, 20 | fsumnn0cl 15628 | . . . 4 β’ (π β ((β0 βm β) β© π ) β Ξ£π β (β‘π β β)((πβπ) Β· π) β β0) |
22 | 10, 21 | eqeltrrd 2839 | . . 3 β’ (π β ((β0 βm β) β© π ) β Ξ£π β β ((πβπ) Β· π) β β0) |
23 | 6, 22 | vtoclga 3537 | . 2 β’ (π β ((β0 βm β) β© π ) β Ξ£π β β ((πβπ) Β· π) β β0) |
24 | 1, 23 | fmpti 7065 | 1 β’ π:((β0 βm β) β© π )βΆβ0 |
Colors of variables: wff setvar class |
Syntax hints: β§ wa 397 = wceq 1542 β wcel 2107 {cab 2714 β© cin 3914 β¦ cmpt 5193 β‘ccnv 5637 β cima 5641 βΆwf 6497 βcfv 6501 (class class class)co 7362 βm cmap 8772 Fincfn 8890 Β· cmul 11063 βcn 12160 β0cn0 12420 Ξ£csu 15577 |
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 2708 ax-rep 5247 ax-sep 5261 ax-nul 5268 ax-pow 5325 ax-pr 5389 ax-un 7677 ax-inf2 9584 ax-cnex 11114 ax-resscn 11115 ax-1cn 11116 ax-icn 11117 ax-addcl 11118 ax-addrcl 11119 ax-mulcl 11120 ax-mulrcl 11121 ax-mulcom 11122 ax-addass 11123 ax-mulass 11124 ax-distr 11125 ax-i2m1 11126 ax-1ne0 11127 ax-1rid 11128 ax-rnegex 11129 ax-rrecex 11130 ax-cnre 11131 ax-pre-lttri 11132 ax-pre-lttrn 11133 ax-pre-ltadd 11134 ax-pre-mulgt0 11135 ax-pre-sup 11136 |
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 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2815 df-nfc 2890 df-ne 2945 df-nel 3051 df-ral 3066 df-rex 3075 df-rmo 3356 df-reu 3357 df-rab 3411 df-v 3450 df-sbc 3745 df-csb 3861 df-dif 3918 df-un 3920 df-in 3922 df-ss 3932 df-pss 3934 df-nul 4288 df-if 4492 df-pw 4567 df-sn 4592 df-pr 4594 df-op 4598 df-uni 4871 df-int 4913 df-iun 4961 df-br 5111 df-opab 5173 df-mpt 5194 df-tr 5228 df-id 5536 df-eprel 5542 df-po 5550 df-so 5551 df-fr 5593 df-se 5594 df-we 5595 df-xp 5644 df-rel 5645 df-cnv 5646 df-co 5647 df-dm 5648 df-rn 5649 df-res 5650 df-ima 5651 df-pred 6258 df-ord 6325 df-on 6326 df-lim 6327 df-suc 6328 df-iota 6453 df-fun 6503 df-fn 6504 df-f 6505 df-f1 6506 df-fo 6507 df-f1o 6508 df-fv 6509 df-isom 6510 df-riota 7318 df-ov 7365 df-oprab 7366 df-mpo 7367 df-om 7808 df-1st 7926 df-2nd 7927 df-frecs 8217 df-wrecs 8248 df-recs 8322 df-rdg 8361 df-1o 8417 df-er 8655 df-map 8774 df-en 8891 df-dom 8892 df-sdom 8893 df-fin 8894 df-sup 9385 df-oi 9453 df-card 9882 df-pnf 11198 df-mnf 11199 df-xr 11200 df-ltxr 11201 df-le 11202 df-sub 11394 df-neg 11395 df-div 11820 df-nn 12161 df-2 12223 df-3 12224 df-n0 12421 df-z 12507 df-uz 12771 df-rp 12923 df-fz 13432 df-fzo 13575 df-seq 13914 df-exp 13975 df-hash 14238 df-cj 14991 df-re 14992 df-im 14993 df-sqrt 15127 df-abs 15128 df-clim 15377 df-sum 15578 |
This theorem is referenced by: eulerpartlems 33000 eulerpartlemsv3 33001 eulerpartlemgc 33002 |
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