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Mirrors > Home > ILE Home > Th. List > isumss | Unicode version |
Description: Change the index set to a subset in an upper integer sum. (Contributed by Mario Carneiro, 21-Apr-2014.) (Revised by Jim Kingdon, 21-Sep-2022.) |
Ref | Expression |
---|---|
sumss.1 | |
sumss.2 | |
sumss.3 | |
isumss.adc | DECID |
isumss.m | |
sumss.4 | |
isumss.bdc | DECID |
Ref | Expression |
---|---|
isumss |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2137 | . . . 4 | |
2 | isumss.m | . . . 4 | |
3 | sumss.1 | . . . . 5 | |
4 | sumss.4 | . . . . 5 | |
5 | 3, 4 | sstrd 3102 | . . . 4 |
6 | simpr 109 | . . . . . 6 | |
7 | simpr 109 | . . . . . . . 8 | |
8 | sumss.2 | . . . . . . . . . 10 | |
9 | 8 | ralrimiva 2503 | . . . . . . . . 9 |
10 | 9 | ad2antrr 479 | . . . . . . . 8 |
11 | nfcsb1v 3030 | . . . . . . . . . 10 | |
12 | 11 | nfel1 2290 | . . . . . . . . 9 |
13 | csbeq1a 3007 | . . . . . . . . . 10 | |
14 | 13 | eleq1d 2206 | . . . . . . . . 9 |
15 | 12, 14 | rspc 2778 | . . . . . . . 8 |
16 | 7, 10, 15 | sylc 62 | . . . . . . 7 |
17 | 0cnd 7752 | . . . . . . 7 | |
18 | eleq1w 2198 | . . . . . . . . 9 | |
19 | 18 | dcbid 823 | . . . . . . . 8 DECID DECID |
20 | isumss.adc | . . . . . . . . 9 DECID | |
21 | 20 | adantr 274 | . . . . . . . 8 DECID |
22 | 19, 21, 6 | rspcdva 2789 | . . . . . . 7 DECID |
23 | 16, 17, 22 | ifcldadc 3496 | . . . . . 6 |
24 | nfcv 2279 | . . . . . . 7 | |
25 | nfv 1508 | . . . . . . . 8 | |
26 | nfcv 2279 | . . . . . . . 8 | |
27 | 25, 11, 26 | nfif 3495 | . . . . . . 7 |
28 | eleq1w 2198 | . . . . . . . 8 | |
29 | 28, 13 | ifbieq1d 3489 | . . . . . . 7 |
30 | eqid 2137 | . . . . . . 7 | |
31 | 24, 27, 29, 30 | fvmptf 5506 | . . . . . 6 |
32 | 6, 23, 31 | syl2anc 408 | . . . . 5 |
33 | eqid 2137 | . . . . . . . 8 | |
34 | 33 | fvmpts 5492 | . . . . . . 7 |
35 | 7, 16, 34 | syl2anc 408 | . . . . . 6 |
36 | 35, 22 | ifeq1dadc 3497 | . . . . 5 |
37 | 32, 36 | eqtr4d 2173 | . . . 4 |
38 | 8 | fmpttd 5568 | . . . . 5 |
39 | 38 | ffvelrnda 5548 | . . . 4 |
40 | 1, 2, 5, 37, 20, 39 | zsumdc 11146 | . . 3 |
41 | elin 3254 | . . . . . . . 8 | |
42 | dfss1 3275 | . . . . . . . . . 10 | |
43 | 3, 42 | sylib 121 | . . . . . . . . 9 |
44 | 43 | eleq2d 2207 | . . . . . . . 8 |
45 | 41, 44 | syl5rbbr 194 | . . . . . . 7 |
46 | 45 | adantr 274 | . . . . . 6 |
47 | 46 | ifbid 3488 | . . . . 5 |
48 | simplr 519 | . . . . . . . . . 10 | |
49 | 16 | adantlr 468 | . . . . . . . . . 10 |
50 | eqid 2137 | . . . . . . . . . . 11 | |
51 | 50 | fvmpts 5492 | . . . . . . . . . 10 |
52 | 48, 49, 51 | syl2anc 408 | . . . . . . . . 9 |
53 | simpr 109 | . . . . . . . . . 10 | |
54 | 53 | iftrued 3476 | . . . . . . . . 9 |
55 | 52, 54 | eqtr4d 2173 | . . . . . . . 8 |
56 | simplr 519 | . . . . . . . . . . 11 | |
57 | simpr 109 | . . . . . . . . . . 11 | |
58 | 56, 57 | eldifd 3076 | . . . . . . . . . 10 |
59 | sumss.3 | . . . . . . . . . . . 12 | |
60 | 59 | ralrimiva 2503 | . . . . . . . . . . 11 |
61 | 60 | ad3antrrr 483 | . . . . . . . . . 10 |
62 | 11 | nfeq1 2289 | . . . . . . . . . . 11 |
63 | 13 | eqeq1d 2146 | . . . . . . . . . . 11 |
64 | 62, 63 | rspc 2778 | . . . . . . . . . 10 |
65 | 58, 61, 64 | sylc 62 | . . . . . . . . 9 |
66 | 0cnd 7752 | . . . . . . . . . . 11 | |
67 | 65, 66 | eqeltrd 2214 | . . . . . . . . . 10 |
68 | 56, 67, 51 | syl2anc 408 | . . . . . . . . 9 |
69 | 57 | iffalsed 3479 | . . . . . . . . 9 |
70 | 65, 68, 69 | 3eqtr4d 2180 | . . . . . . . 8 |
71 | 22 | adantr 274 | . . . . . . . . 9 DECID |
72 | exmiddc 821 | . . . . . . . . 9 DECID | |
73 | 71, 72 | syl 14 | . . . . . . . 8 |
74 | 55, 70, 73 | mpjaodan 787 | . . . . . . 7 |
75 | eleq1w 2198 | . . . . . . . . 9 | |
76 | 75 | dcbid 823 | . . . . . . . 8 DECID DECID |
77 | isumss.bdc | . . . . . . . . 9 DECID | |
78 | 77 | adantr 274 | . . . . . . . 8 DECID |
79 | 76, 78, 6 | rspcdva 2789 | . . . . . . 7 DECID |
80 | 74, 79 | ifeq1dadc 3497 | . . . . . 6 |
81 | ifandc 3503 | . . . . . . 7 DECID | |
82 | 79, 81 | syl 14 | . . . . . 6 |
83 | 80, 82 | eqtr4d 2173 | . . . . 5 |
84 | 47, 32, 83 | 3eqtr4d 2180 | . . . 4 |
85 | 8 | adantlr 468 | . . . . . . 7 |
86 | simpll 518 | . . . . . . . . 9 | |
87 | simplr 519 | . . . . . . . . . 10 | |
88 | simpr 109 | . . . . . . . . . 10 | |
89 | 87, 88 | eldifd 3076 | . . . . . . . . 9 |
90 | 86, 89, 59 | syl2anc 408 | . . . . . . . 8 |
91 | 0cnd 7752 | . . . . . . . 8 | |
92 | 90, 91 | eqeltrd 2214 | . . . . . . 7 |
93 | eleq1w 2198 | . . . . . . . . . 10 | |
94 | 93 | dcbid 823 | . . . . . . . . 9 DECID DECID |
95 | 20 | adantr 274 | . . . . . . . . 9 DECID |
96 | 4 | sselda 3092 | . . . . . . . . 9 |
97 | 94, 95, 96 | rspcdva 2789 | . . . . . . . 8 DECID |
98 | exmiddc 821 | . . . . . . . 8 DECID | |
99 | 97, 98 | syl 14 | . . . . . . 7 |
100 | 85, 92, 99 | mpjaodan 787 | . . . . . 6 |
101 | 100 | fmpttd 5568 | . . . . 5 |
102 | 101 | ffvelrnda 5548 | . . . 4 |
103 | 1, 2, 4, 84, 77, 102 | zsumdc 11146 | . . 3 |
104 | 40, 103 | eqtr4d 2173 | . 2 |
105 | sumfct 11136 | . . 3 | |
106 | 9, 105 | syl 14 | . 2 |
107 | 100 | ralrimiva 2503 | . . 3 |
108 | sumfct 11136 | . . 3 | |
109 | 107, 108 | syl 14 | . 2 |
110 | 104, 106, 109 | 3eqtr3d 2178 | 1 |
Colors of variables: wff set class |
Syntax hints: wn 3 wi 4 wa 103 wb 104 wo 697 DECID wdc 819 wceq 1331 wcel 1480 wral 2414 csb 2998 cdif 3063 cin 3065 wss 3066 cif 3469 cmpt 3984 cfv 5118 cc 7611 cc0 7613 caddc 7616 cz 9047 cuz 9319 cseq 10211 cli 11040 csu 11115 |
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 603 ax-in2 604 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-13 1491 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2119 ax-coll 4038 ax-sep 4041 ax-nul 4049 ax-pow 4093 ax-pr 4126 ax-un 4350 ax-setind 4447 ax-iinf 4497 ax-cnex 7704 ax-resscn 7705 ax-1cn 7706 ax-1re 7707 ax-icn 7708 ax-addcl 7709 ax-addrcl 7710 ax-mulcl 7711 ax-mulrcl 7712 ax-addcom 7713 ax-mulcom 7714 ax-addass 7715 ax-mulass 7716 ax-distr 7717 ax-i2m1 7718 ax-0lt1 7719 ax-1rid 7720 ax-0id 7721 ax-rnegex 7722 ax-precex 7723 ax-cnre 7724 ax-pre-ltirr 7725 ax-pre-ltwlin 7726 ax-pre-lttrn 7727 ax-pre-apti 7728 ax-pre-ltadd 7729 ax-pre-mulgt0 7730 ax-pre-mulext 7731 |
This theorem depends on definitions: df-bi 116 df-dc 820 df-3or 963 df-3an 964 df-tru 1334 df-fal 1337 df-nf 1437 df-sb 1736 df-eu 2000 df-mo 2001 df-clab 2124 df-cleq 2130 df-clel 2133 df-nfc 2268 df-ne 2307 df-nel 2402 df-ral 2419 df-rex 2420 df-reu 2421 df-rmo 2422 df-rab 2423 df-v 2683 df-sbc 2905 df-csb 2999 df-dif 3068 df-un 3070 df-in 3072 df-ss 3079 df-nul 3359 df-if 3470 df-pw 3507 df-sn 3528 df-pr 3529 df-op 3531 df-uni 3732 df-int 3767 df-iun 3810 df-br 3925 df-opab 3985 df-mpt 3986 df-tr 4022 df-id 4210 df-po 4213 df-iso 4214 df-iord 4283 df-on 4285 df-ilim 4286 df-suc 4288 df-iom 4500 df-xp 4540 df-rel 4541 df-cnv 4542 df-co 4543 df-dm 4544 df-rn 4545 df-res 4546 df-ima 4547 df-iota 5083 df-fun 5120 df-fn 5121 df-f 5122 df-f1 5123 df-fo 5124 df-f1o 5125 df-fv 5126 df-isom 5127 df-riota 5723 df-ov 5770 df-oprab 5771 df-mpo 5772 df-1st 6031 df-2nd 6032 df-recs 6195 df-irdg 6260 df-frec 6281 df-1o 6306 df-oadd 6310 df-er 6422 df-en 6628 df-dom 6629 df-fin 6630 df-pnf 7795 df-mnf 7796 df-xr 7797 df-ltxr 7798 df-le 7799 df-sub 7928 df-neg 7929 df-reap 8330 df-ap 8337 df-div 8426 df-inn 8714 df-2 8772 df-n0 8971 df-z 9048 df-uz 9320 df-q 9405 df-rp 9435 df-fz 9784 df-fzo 9913 df-seqfrec 10212 df-exp 10286 df-ihash 10515 df-cj 10607 df-rsqrt 10763 df-abs 10764 df-clim 11041 df-sumdc 11116 |
This theorem is referenced by: fisumss 11154 isumss2 11155 binomlem 11245 |
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