![]() |
Mathbox for Glauco Siliprandi |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > Mathboxes > fsumsupp0 | Structured version Visualization version GIF version |
Description: Finite sum of function values, for a function of finite support. (Contributed by Glauco Siliprandi, 24-Dec-2020.) |
Ref | Expression |
---|---|
fsumsupp0.a | β’ (π β π΄ β Fin) |
fsumsupp0.f | β’ (π β πΉ:π΄βΆβ) |
Ref | Expression |
---|---|
fsumsupp0 | β’ (π β Ξ£π β (πΉ supp 0)(πΉβπ) = Ξ£π β π΄ (πΉβπ)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fsumsupp0.f | . . . . 5 β’ (π β πΉ:π΄βΆβ) | |
2 | 1 | ffnd 6712 | . . . 4 β’ (π β πΉ Fn π΄) |
3 | fsumsupp0.a | . . . 4 β’ (π β π΄ β Fin) | |
4 | 0red 11221 | . . . 4 β’ (π β 0 β β) | |
5 | suppvalfn 8154 | . . . 4 β’ ((πΉ Fn π΄ β§ π΄ β Fin β§ 0 β β) β (πΉ supp 0) = {π β π΄ β£ (πΉβπ) β 0}) | |
6 | 2, 3, 4, 5 | syl3anc 1368 | . . 3 β’ (π β (πΉ supp 0) = {π β π΄ β£ (πΉβπ) β 0}) |
7 | ssrab2 4072 | . . 3 β’ {π β π΄ β£ (πΉβπ) β 0} β π΄ | |
8 | 6, 7 | eqsstrdi 4031 | . 2 β’ (π β (πΉ supp 0) β π΄) |
9 | 1 | adantr 480 | . . 3 β’ ((π β§ π β (πΉ supp 0)) β πΉ:π΄βΆβ) |
10 | 8 | sselda 3977 | . . 3 β’ ((π β§ π β (πΉ supp 0)) β π β π΄) |
11 | 9, 10 | ffvelcdmd 7081 | . 2 β’ ((π β§ π β (πΉ supp 0)) β (πΉβπ) β β) |
12 | eldifi 4121 | . . . . . . . 8 β’ (π β (π΄ β (πΉ supp 0)) β π β π΄) | |
13 | 12 | adantr 480 | . . . . . . 7 β’ ((π β (π΄ β (πΉ supp 0)) β§ Β¬ (πΉβπ) = 0) β π β π΄) |
14 | neqne 2942 | . . . . . . . 8 β’ (Β¬ (πΉβπ) = 0 β (πΉβπ) β 0) | |
15 | 14 | adantl 481 | . . . . . . 7 β’ ((π β (π΄ β (πΉ supp 0)) β§ Β¬ (πΉβπ) = 0) β (πΉβπ) β 0) |
16 | 13, 15 | jca 511 | . . . . . 6 β’ ((π β (π΄ β (πΉ supp 0)) β§ Β¬ (πΉβπ) = 0) β (π β π΄ β§ (πΉβπ) β 0)) |
17 | rabid 3446 | . . . . . 6 β’ (π β {π β π΄ β£ (πΉβπ) β 0} β (π β π΄ β§ (πΉβπ) β 0)) | |
18 | 16, 17 | sylibr 233 | . . . . 5 β’ ((π β (π΄ β (πΉ supp 0)) β§ Β¬ (πΉβπ) = 0) β π β {π β π΄ β£ (πΉβπ) β 0}) |
19 | 18 | adantll 711 | . . . 4 β’ (((π β§ π β (π΄ β (πΉ supp 0))) β§ Β¬ (πΉβπ) = 0) β π β {π β π΄ β£ (πΉβπ) β 0}) |
20 | 6 | eleq2d 2813 | . . . . 5 β’ (π β (π β (πΉ supp 0) β π β {π β π΄ β£ (πΉβπ) β 0})) |
21 | 20 | ad2antrr 723 | . . . 4 β’ (((π β§ π β (π΄ β (πΉ supp 0))) β§ Β¬ (πΉβπ) = 0) β (π β (πΉ supp 0) β π β {π β π΄ β£ (πΉβπ) β 0})) |
22 | 19, 21 | mpbird 257 | . . 3 β’ (((π β§ π β (π΄ β (πΉ supp 0))) β§ Β¬ (πΉβπ) = 0) β π β (πΉ supp 0)) |
23 | eldifn 4122 | . . . 4 β’ (π β (π΄ β (πΉ supp 0)) β Β¬ π β (πΉ supp 0)) | |
24 | 23 | ad2antlr 724 | . . 3 β’ (((π β§ π β (π΄ β (πΉ supp 0))) β§ Β¬ (πΉβπ) = 0) β Β¬ π β (πΉ supp 0)) |
25 | 22, 24 | condan 815 | . 2 β’ ((π β§ π β (π΄ β (πΉ supp 0))) β (πΉβπ) = 0) |
26 | 8, 11, 25, 3 | fsumss 15677 | 1 β’ (π β Ξ£π β (πΉ supp 0)(πΉβπ) = Ξ£π β π΄ (πΉβπ)) |
Colors of variables: wff setvar class |
Syntax hints: Β¬ wn 3 β wi 4 β wb 205 β§ wa 395 = wceq 1533 β wcel 2098 β wne 2934 {crab 3426 β cdif 3940 Fn wfn 6532 βΆwf 6533 βcfv 6537 (class class class)co 7405 supp csupp 8146 Fincfn 8941 βcc 11110 βcr 11111 0cc0 11112 Ξ£csu 15638 |
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-rep 5278 ax-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 ax-un 7722 ax-inf2 9638 ax-cnex 11168 ax-resscn 11169 ax-1cn 11170 ax-icn 11171 ax-addcl 11172 ax-addrcl 11173 ax-mulcl 11174 ax-mulrcl 11175 ax-mulcom 11176 ax-addass 11177 ax-mulass 11178 ax-distr 11179 ax-i2m1 11180 ax-1ne0 11181 ax-1rid 11182 ax-rnegex 11183 ax-rrecex 11184 ax-cnre 11185 ax-pre-lttri 11186 ax-pre-lttrn 11187 ax-pre-ltadd 11188 ax-pre-mulgt0 11189 ax-pre-sup 11190 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1085 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-nel 3041 df-ral 3056 df-rex 3065 df-rmo 3370 df-reu 3371 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-pss 3962 df-nul 4318 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4903 df-int 4944 df-iun 4992 df-br 5142 df-opab 5204 df-mpt 5225 df-tr 5259 df-id 5567 df-eprel 5573 df-po 5581 df-so 5582 df-fr 5624 df-se 5625 df-we 5626 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-pred 6294 df-ord 6361 df-on 6362 df-lim 6363 df-suc 6364 df-iota 6489 df-fun 6539 df-fn 6540 df-f 6541 df-f1 6542 df-fo 6543 df-f1o 6544 df-fv 6545 df-isom 6546 df-riota 7361 df-ov 7408 df-oprab 7409 df-mpo 7410 df-om 7853 df-1st 7974 df-2nd 7975 df-supp 8147 df-frecs 8267 df-wrecs 8298 df-recs 8372 df-rdg 8411 df-1o 8467 df-er 8705 df-en 8942 df-dom 8943 df-sdom 8944 df-fin 8945 df-sup 9439 df-oi 9507 df-card 9936 df-pnf 11254 df-mnf 11255 df-xr 11256 df-ltxr 11257 df-le 11258 df-sub 11450 df-neg 11451 df-div 11876 df-nn 12217 df-2 12279 df-3 12280 df-n0 12477 df-z 12563 df-uz 12827 df-rp 12981 df-fz 13491 df-fzo 13634 df-seq 13973 df-exp 14033 df-hash 14296 df-cj 15052 df-re 15053 df-im 15054 df-sqrt 15188 df-abs 15189 df-clim 15438 df-sum 15639 |
This theorem is referenced by: rrxtopnfi 45575 |
Copyright terms: Public domain | W3C validator |