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Mathbox for Glauco Siliprandi |
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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 6670 | . . . 4 β’ (π β πΉ Fn π΄) |
3 | fsumsupp0.a | . . . 4 β’ (π β π΄ β Fin) | |
4 | 0red 11159 | . . . 4 β’ (π β 0 β β) | |
5 | suppvalfn 8101 | . . . 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 1372 | . . 3 β’ (π β (πΉ supp 0) = {π β π΄ β£ (πΉβπ) β 0}) |
7 | ssrab2 4038 | . . 3 β’ {π β π΄ β£ (πΉβπ) β 0} β π΄ | |
8 | 6, 7 | eqsstrdi 3999 | . 2 β’ (π β (πΉ supp 0) β π΄) |
9 | 1 | adantr 482 | . . 3 β’ ((π β§ π β (πΉ supp 0)) β πΉ:π΄βΆβ) |
10 | 8 | sselda 3945 | . . 3 β’ ((π β§ π β (πΉ supp 0)) β π β π΄) |
11 | 9, 10 | ffvelcdmd 7037 | . 2 β’ ((π β§ π β (πΉ supp 0)) β (πΉβπ) β β) |
12 | eldifi 4087 | . . . . . . . 8 β’ (π β (π΄ β (πΉ supp 0)) β π β π΄) | |
13 | 12 | adantr 482 | . . . . . . 7 β’ ((π β (π΄ β (πΉ supp 0)) β§ Β¬ (πΉβπ) = 0) β π β π΄) |
14 | neqne 2952 | . . . . . . . 8 β’ (Β¬ (πΉβπ) = 0 β (πΉβπ) β 0) | |
15 | 14 | adantl 483 | . . . . . . 7 β’ ((π β (π΄ β (πΉ supp 0)) β§ Β¬ (πΉβπ) = 0) β (πΉβπ) β 0) |
16 | 13, 15 | jca 513 | . . . . . 6 β’ ((π β (π΄ β (πΉ supp 0)) β§ Β¬ (πΉβπ) = 0) β (π β π΄ β§ (πΉβπ) β 0)) |
17 | rabid 3428 | . . . . . 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 713 | . . . 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 2824 | . . . . 5 β’ (π β (π β (πΉ supp 0) β π β {π β π΄ β£ (πΉβπ) β 0})) |
21 | 20 | ad2antrr 725 | . . . 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 4088 | . . . 4 β’ (π β (π΄ β (πΉ supp 0)) β Β¬ π β (πΉ supp 0)) | |
24 | 23 | ad2antlr 726 | . . 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 817 | . 2 β’ ((π β§ π β (π΄ β (πΉ supp 0))) β (πΉβπ) = 0) |
26 | 8, 11, 25, 3 | fsumss 15611 | 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 397 = wceq 1542 β wcel 2107 β wne 2944 {crab 3408 β cdif 3908 Fn wfn 6492 βΆwf 6493 βcfv 6497 (class class class)co 7358 supp csupp 8093 Fincfn 8884 βcc 11050 βcr 11051 0cc0 11052 Ξ£csu 15571 |
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 5243 ax-sep 5257 ax-nul 5264 ax-pow 5321 ax-pr 5385 ax-un 7673 ax-inf2 9578 ax-cnex 11108 ax-resscn 11109 ax-1cn 11110 ax-icn 11111 ax-addcl 11112 ax-addrcl 11113 ax-mulcl 11114 ax-mulrcl 11115 ax-mulcom 11116 ax-addass 11117 ax-mulass 11118 ax-distr 11119 ax-i2m1 11120 ax-1ne0 11121 ax-1rid 11122 ax-rnegex 11123 ax-rrecex 11124 ax-cnre 11125 ax-pre-lttri 11126 ax-pre-lttrn 11127 ax-pre-ltadd 11128 ax-pre-mulgt0 11129 ax-pre-sup 11130 |
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 3354 df-reu 3355 df-rab 3409 df-v 3448 df-sbc 3741 df-csb 3857 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-pss 3930 df-nul 4284 df-if 4488 df-pw 4563 df-sn 4588 df-pr 4590 df-op 4594 df-uni 4867 df-int 4909 df-iun 4957 df-br 5107 df-opab 5169 df-mpt 5190 df-tr 5224 df-id 5532 df-eprel 5538 df-po 5546 df-so 5547 df-fr 5589 df-se 5590 df-we 5591 df-xp 5640 df-rel 5641 df-cnv 5642 df-co 5643 df-dm 5644 df-rn 5645 df-res 5646 df-ima 5647 df-pred 6254 df-ord 6321 df-on 6322 df-lim 6323 df-suc 6324 df-iota 6449 df-fun 6499 df-fn 6500 df-f 6501 df-f1 6502 df-fo 6503 df-f1o 6504 df-fv 6505 df-isom 6506 df-riota 7314 df-ov 7361 df-oprab 7362 df-mpo 7363 df-om 7804 df-1st 7922 df-2nd 7923 df-supp 8094 df-frecs 8213 df-wrecs 8244 df-recs 8318 df-rdg 8357 df-1o 8413 df-er 8649 df-en 8885 df-dom 8886 df-sdom 8887 df-fin 8888 df-sup 9379 df-oi 9447 df-card 9876 df-pnf 11192 df-mnf 11193 df-xr 11194 df-ltxr 11195 df-le 11196 df-sub 11388 df-neg 11389 df-div 11814 df-nn 12155 df-2 12217 df-3 12218 df-n0 12415 df-z 12501 df-uz 12765 df-rp 12917 df-fz 13426 df-fzo 13569 df-seq 13908 df-exp 13969 df-hash 14232 df-cj 14985 df-re 14986 df-im 14987 df-sqrt 15121 df-abs 15122 df-clim 15371 df-sum 15572 |
This theorem is referenced by: rrxtopnfi 44535 |
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