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Mirrors > Home > ILE Home > Th. List > isumss2 | Unicode version |
Description: Change the index set of a sum by adding zeroes. The nonzero elements are in the contained set and the added zeroes compose the rest of the containing set which needs to be summable. (Contributed by Mario Carneiro, 15-Jul-2013.) (Revised by Jim Kingdon, 24-Sep-2022.) |
Ref | Expression |
---|---|
isumss2.ss | |
isumss2.adc | DECID |
isumss2.c | |
isumss2.b | DECID |
Ref | Expression |
---|---|
isumss2 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | isumss2.ss | . . . . 5 | |
2 | 1 | adantr 274 | . . . 4 DECID |
3 | isumss2.c | . . . . . 6 | |
4 | iftrue 3530 | . . . . . . . 8 | |
5 | 4 | adantl 275 | . . . . . . 7 |
6 | nfcsb1v 3082 | . . . . . . . . . 10 | |
7 | 6 | nfel1 2323 | . . . . . . . . 9 |
8 | csbeq1a 3058 | . . . . . . . . . 10 | |
9 | 8 | eleq1d 2239 | . . . . . . . . 9 |
10 | 7, 9 | rspc 2828 | . . . . . . . 8 |
11 | 10 | impcom 124 | . . . . . . 7 |
12 | 5, 11 | eqeltrd 2247 | . . . . . 6 |
13 | 3, 12 | sylan 281 | . . . . 5 |
14 | 13 | adantlr 474 | . . . 4 DECID |
15 | eldifn 3250 | . . . . . 6 | |
16 | 15 | iffalsed 3535 | . . . . 5 |
17 | 16 | adantl 275 | . . . 4 DECID |
18 | isumss2.adc | . . . . . . . . . 10 DECID | |
19 | 18 | adantr 274 | . . . . . . . . 9 DECID DECID |
20 | eleq1w 2231 | . . . . . . . . . . 11 | |
21 | 20 | dcbid 833 | . . . . . . . . . 10 DECID DECID |
22 | 21 | cbvralv 2696 | . . . . . . . . 9 DECID DECID |
23 | 19, 22 | sylib 121 | . . . . . . . 8 DECID DECID |
24 | 23 | r19.21bi 2558 | . . . . . . 7 DECID DECID |
25 | 24 | adantlr 474 | . . . . . 6 DECID DECID |
26 | 2 | adantr 274 | . . . . . . . . . 10 DECID |
27 | 26 | ssneld 3149 | . . . . . . . . 9 DECID |
28 | 27 | imp 123 | . . . . . . . 8 DECID |
29 | 28 | olcd 729 | . . . . . . 7 DECID |
30 | df-dc 830 | . . . . . . 7 DECID | |
31 | 29, 30 | sylibr 133 | . . . . . 6 DECID DECID |
32 | eleq1w 2231 | . . . . . . . . 9 | |
33 | 32 | dcbid 833 | . . . . . . . 8 DECID DECID |
34 | simplr3 1036 | . . . . . . . 8 DECID DECID | |
35 | simpr 109 | . . . . . . . 8 DECID | |
36 | 33, 34, 35 | rspcdva 2839 | . . . . . . 7 DECID DECID |
37 | exmiddc 831 | . . . . . . 7 DECID | |
38 | 36, 37 | syl 14 | . . . . . 6 DECID |
39 | 25, 31, 38 | mpjaodan 793 | . . . . 5 DECID DECID |
40 | 39 | ralrimiva 2543 | . . . 4 DECID DECID |
41 | simpr1 998 | . . . 4 DECID | |
42 | simpr2 999 | . . . 4 DECID | |
43 | simpr3 1000 | . . . . 5 DECID DECID | |
44 | 33 | cbvralv 2696 | . . . . 5 DECID DECID |
45 | 43, 44 | sylib 121 | . . . 4 DECID DECID |
46 | 2, 14, 17, 40, 41, 42, 45 | isumss 11341 | . . 3 DECID |
47 | 1 | adantr 274 | . . . 4 |
48 | 13 | adantlr 474 | . . . 4 |
49 | 16 | adantl 275 | . . . 4 |
50 | 18 | adantr 274 | . . . 4 DECID |
51 | simpr 109 | . . . 4 | |
52 | 47, 48, 49, 50, 51 | fisumss 11342 | . . 3 |
53 | isumss2.b | . . 3 DECID | |
54 | 46, 52, 53 | mpjaodan 793 | . 2 |
55 | iftrue 3530 | . . . 4 | |
56 | 55 | sumeq2i 11314 | . . 3 |
57 | nfcv 2312 | . . . 4 | |
58 | nfv 1521 | . . . . 5 | |
59 | nfcv 2312 | . . . . 5 | |
60 | 58, 6, 59 | nfif 3553 | . . . 4 |
61 | eleq1w 2231 | . . . . 5 | |
62 | 61, 8 | ifbieq1d 3547 | . . . 4 |
63 | 57, 60, 62 | cbvsumi 11312 | . . 3 |
64 | 56, 63 | eqtr3i 2193 | . 2 |
65 | 57, 60, 62 | cbvsumi 11312 | . 2 |
66 | 54, 64, 65 | 3eqtr4g 2228 | 1 |
Colors of variables: wff set class |
Syntax hints: wn 3 wi 4 wa 103 wo 703 DECID wdc 829 w3a 973 wceq 1348 wcel 2141 wral 2448 csb 3049 cdif 3118 wss 3121 cif 3525 cfv 5196 cfn 6714 cc 7759 cc0 7761 cz 9199 cuz 9474 csu 11303 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 609 ax-in2 610 ax-io 704 ax-5 1440 ax-7 1441 ax-gen 1442 ax-ie1 1486 ax-ie2 1487 ax-8 1497 ax-10 1498 ax-11 1499 ax-i12 1500 ax-bndl 1502 ax-4 1503 ax-17 1519 ax-i9 1523 ax-ial 1527 ax-i5r 1528 ax-13 2143 ax-14 2144 ax-ext 2152 ax-coll 4102 ax-sep 4105 ax-nul 4113 ax-pow 4158 ax-pr 4192 ax-un 4416 ax-setind 4519 ax-iinf 4570 ax-cnex 7852 ax-resscn 7853 ax-1cn 7854 ax-1re 7855 ax-icn 7856 ax-addcl 7857 ax-addrcl 7858 ax-mulcl 7859 ax-mulrcl 7860 ax-addcom 7861 ax-mulcom 7862 ax-addass 7863 ax-mulass 7864 ax-distr 7865 ax-i2m1 7866 ax-0lt1 7867 ax-1rid 7868 ax-0id 7869 ax-rnegex 7870 ax-precex 7871 ax-cnre 7872 ax-pre-ltirr 7873 ax-pre-ltwlin 7874 ax-pre-lttrn 7875 ax-pre-apti 7876 ax-pre-ltadd 7877 ax-pre-mulgt0 7878 ax-pre-mulext 7879 ax-arch 7880 ax-caucvg 7881 |
This theorem depends on definitions: df-bi 116 df-dc 830 df-3or 974 df-3an 975 df-tru 1351 df-fal 1354 df-nf 1454 df-sb 1756 df-eu 2022 df-mo 2023 df-clab 2157 df-cleq 2163 df-clel 2166 df-nfc 2301 df-ne 2341 df-nel 2436 df-ral 2453 df-rex 2454 df-reu 2455 df-rmo 2456 df-rab 2457 df-v 2732 df-sbc 2956 df-csb 3050 df-dif 3123 df-un 3125 df-in 3127 df-ss 3134 df-nul 3415 df-if 3526 df-pw 3566 df-sn 3587 df-pr 3588 df-op 3590 df-uni 3795 df-int 3830 df-iun 3873 df-br 3988 df-opab 4049 df-mpt 4050 df-tr 4086 df-id 4276 df-po 4279 df-iso 4280 df-iord 4349 df-on 4351 df-ilim 4352 df-suc 4354 df-iom 4573 df-xp 4615 df-rel 4616 df-cnv 4617 df-co 4618 df-dm 4619 df-rn 4620 df-res 4621 df-ima 4622 df-iota 5158 df-fun 5198 df-fn 5199 df-f 5200 df-f1 5201 df-fo 5202 df-f1o 5203 df-fv 5204 df-isom 5205 df-riota 5806 df-ov 5853 df-oprab 5854 df-mpo 5855 df-1st 6116 df-2nd 6117 df-recs 6281 df-irdg 6346 df-frec 6367 df-1o 6392 df-oadd 6396 df-er 6509 df-en 6715 df-dom 6716 df-fin 6717 df-pnf 7943 df-mnf 7944 df-xr 7945 df-ltxr 7946 df-le 7947 df-sub 8079 df-neg 8080 df-reap 8481 df-ap 8488 df-div 8577 df-inn 8866 df-2 8924 df-3 8925 df-4 8926 df-n0 9123 df-z 9200 df-uz 9475 df-q 9566 df-rp 9598 df-fz 9953 df-fzo 10086 df-seqfrec 10389 df-exp 10463 df-ihash 10697 df-cj 10793 df-re 10794 df-im 10795 df-rsqrt 10949 df-abs 10950 df-clim 11229 df-sumdc 11304 |
This theorem is referenced by: fsumsplit 11357 sumsplitdc 11382 isumlessdc 11446 sumhashdc 12286 |
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