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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 2164 | . . . 4 | |
2 | isumss.m | . . . 4 | |
3 | sumss.1 | . . . . 5 | |
4 | sumss.4 | . . . . 5 | |
5 | 3, 4 | sstrd 3147 | . . . 4 |
6 | simpr 109 | . . . . . 6 | |
7 | simpr 109 | . . . . . . . 8 | |
8 | sumss.2 | . . . . . . . . . 10 | |
9 | 8 | ralrimiva 2537 | . . . . . . . . 9 |
10 | 9 | ad2antrr 480 | . . . . . . . 8 |
11 | nfcsb1v 3073 | . . . . . . . . . 10 | |
12 | 11 | nfel1 2317 | . . . . . . . . 9 |
13 | csbeq1a 3049 | . . . . . . . . . 10 | |
14 | 13 | eleq1d 2233 | . . . . . . . . 9 |
15 | 12, 14 | rspc 2819 | . . . . . . . 8 |
16 | 7, 10, 15 | sylc 62 | . . . . . . 7 |
17 | 0cnd 7883 | . . . . . . 7 | |
18 | eleq1w 2225 | . . . . . . . . 9 | |
19 | 18 | dcbid 828 | . . . . . . . 8 DECID DECID |
20 | isumss.adc | . . . . . . . . 9 DECID | |
21 | 20 | adantr 274 | . . . . . . . 8 DECID |
22 | 19, 21, 6 | rspcdva 2830 | . . . . . . 7 DECID |
23 | 16, 17, 22 | ifcldadc 3544 | . . . . . 6 |
24 | nfcv 2306 | . . . . . . 7 | |
25 | nfv 1515 | . . . . . . . 8 | |
26 | nfcv 2306 | . . . . . . . 8 | |
27 | 25, 11, 26 | nfif 3543 | . . . . . . 7 |
28 | eleq1w 2225 | . . . . . . . 8 | |
29 | 28, 13 | ifbieq1d 3537 | . . . . . . 7 |
30 | eqid 2164 | . . . . . . 7 | |
31 | 24, 27, 29, 30 | fvmptf 5572 | . . . . . 6 |
32 | 6, 23, 31 | syl2anc 409 | . . . . 5 |
33 | eqid 2164 | . . . . . . . 8 | |
34 | 33 | fvmpts 5558 | . . . . . . 7 |
35 | 7, 16, 34 | syl2anc 409 | . . . . . 6 |
36 | 35, 22 | ifeq1dadc 3545 | . . . . 5 |
37 | 32, 36 | eqtr4d 2200 | . . . 4 |
38 | 8 | fmpttd 5634 | . . . . 5 |
39 | 38 | ffvelrnda 5614 | . . . 4 |
40 | 1, 2, 5, 37, 20, 39 | zsumdc 11311 | . . 3 |
41 | dfss1 3321 | . . . . . . . . . 10 | |
42 | 3, 41 | sylib 121 | . . . . . . . . 9 |
43 | 42 | eleq2d 2234 | . . . . . . . 8 |
44 | elin 3300 | . . . . . . . 8 | |
45 | 43, 44 | bitr3di 194 | . . . . . . 7 |
46 | 45 | adantr 274 | . . . . . 6 |
47 | 46 | ifbid 3536 | . . . . 5 |
48 | simplr 520 | . . . . . . . . . 10 | |
49 | 16 | adantlr 469 | . . . . . . . . . 10 |
50 | eqid 2164 | . . . . . . . . . . 11 | |
51 | 50 | fvmpts 5558 | . . . . . . . . . 10 |
52 | 48, 49, 51 | syl2anc 409 | . . . . . . . . 9 |
53 | simpr 109 | . . . . . . . . . 10 | |
54 | 53 | iftrued 3522 | . . . . . . . . 9 |
55 | 52, 54 | eqtr4d 2200 | . . . . . . . 8 |
56 | simplr 520 | . . . . . . . . . . 11 | |
57 | simpr 109 | . . . . . . . . . . 11 | |
58 | 56, 57 | eldifd 3121 | . . . . . . . . . 10 |
59 | sumss.3 | . . . . . . . . . . . 12 | |
60 | 59 | ralrimiva 2537 | . . . . . . . . . . 11 |
61 | 60 | ad3antrrr 484 | . . . . . . . . . 10 |
62 | 11 | nfeq1 2316 | . . . . . . . . . . 11 |
63 | 13 | eqeq1d 2173 | . . . . . . . . . . 11 |
64 | 62, 63 | rspc 2819 | . . . . . . . . . 10 |
65 | 58, 61, 64 | sylc 62 | . . . . . . . . 9 |
66 | 0cnd 7883 | . . . . . . . . . . 11 | |
67 | 65, 66 | eqeltrd 2241 | . . . . . . . . . 10 |
68 | 56, 67, 51 | syl2anc 409 | . . . . . . . . 9 |
69 | 57 | iffalsed 3525 | . . . . . . . . 9 |
70 | 65, 68, 69 | 3eqtr4d 2207 | . . . . . . . 8 |
71 | 22 | adantr 274 | . . . . . . . . 9 DECID |
72 | exmiddc 826 | . . . . . . . . 9 DECID | |
73 | 71, 72 | syl 14 | . . . . . . . 8 |
74 | 55, 70, 73 | mpjaodan 788 | . . . . . . 7 |
75 | eleq1w 2225 | . . . . . . . . 9 | |
76 | 75 | dcbid 828 | . . . . . . . 8 DECID DECID |
77 | isumss.bdc | . . . . . . . . 9 DECID | |
78 | 77 | adantr 274 | . . . . . . . 8 DECID |
79 | 76, 78, 6 | rspcdva 2830 | . . . . . . 7 DECID |
80 | 74, 79 | ifeq1dadc 3545 | . . . . . 6 |
81 | ifandc 3551 | . . . . . . 7 DECID | |
82 | 79, 81 | syl 14 | . . . . . 6 |
83 | 80, 82 | eqtr4d 2200 | . . . . 5 |
84 | 47, 32, 83 | 3eqtr4d 2207 | . . . 4 |
85 | 8 | adantlr 469 | . . . . . . 7 |
86 | simpll 519 | . . . . . . . . 9 | |
87 | simplr 520 | . . . . . . . . . 10 | |
88 | simpr 109 | . . . . . . . . . 10 | |
89 | 87, 88 | eldifd 3121 | . . . . . . . . 9 |
90 | 86, 89, 59 | syl2anc 409 | . . . . . . . 8 |
91 | 0cnd 7883 | . . . . . . . 8 | |
92 | 90, 91 | eqeltrd 2241 | . . . . . . 7 |
93 | eleq1w 2225 | . . . . . . . . . 10 | |
94 | 93 | dcbid 828 | . . . . . . . . 9 DECID DECID |
95 | 20 | adantr 274 | . . . . . . . . 9 DECID |
96 | 4 | sselda 3137 | . . . . . . . . 9 |
97 | 94, 95, 96 | rspcdva 2830 | . . . . . . . 8 DECID |
98 | exmiddc 826 | . . . . . . . 8 DECID | |
99 | 97, 98 | syl 14 | . . . . . . 7 |
100 | 85, 92, 99 | mpjaodan 788 | . . . . . 6 |
101 | 100 | fmpttd 5634 | . . . . 5 |
102 | 101 | ffvelrnda 5614 | . . . 4 |
103 | 1, 2, 4, 84, 77, 102 | zsumdc 11311 | . . 3 |
104 | 40, 103 | eqtr4d 2200 | . 2 |
105 | sumfct 11301 | . . 3 | |
106 | 9, 105 | syl 14 | . 2 |
107 | 100 | ralrimiva 2537 | . . 3 |
108 | sumfct 11301 | . . 3 | |
109 | 107, 108 | syl 14 | . 2 |
110 | 104, 106, 109 | 3eqtr3d 2205 | 1 |
Colors of variables: wff set class |
Syntax hints: wn 3 wi 4 wa 103 wb 104 wo 698 DECID wdc 824 wceq 1342 wcel 2135 wral 2442 csb 3040 cdif 3108 cin 3110 wss 3111 cif 3515 cmpt 4037 cfv 5182 cc 7742 cc0 7744 caddc 7747 cz 9182 cuz 9457 cseq 10370 cli 11205 csu 11280 |
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 604 ax-in2 605 ax-io 699 ax-5 1434 ax-7 1435 ax-gen 1436 ax-ie1 1480 ax-ie2 1481 ax-8 1491 ax-10 1492 ax-11 1493 ax-i12 1494 ax-bndl 1496 ax-4 1497 ax-17 1513 ax-i9 1517 ax-ial 1521 ax-i5r 1522 ax-13 2137 ax-14 2138 ax-ext 2146 ax-coll 4091 ax-sep 4094 ax-nul 4102 ax-pow 4147 ax-pr 4181 ax-un 4405 ax-setind 4508 ax-iinf 4559 ax-cnex 7835 ax-resscn 7836 ax-1cn 7837 ax-1re 7838 ax-icn 7839 ax-addcl 7840 ax-addrcl 7841 ax-mulcl 7842 ax-mulrcl 7843 ax-addcom 7844 ax-mulcom 7845 ax-addass 7846 ax-mulass 7847 ax-distr 7848 ax-i2m1 7849 ax-0lt1 7850 ax-1rid 7851 ax-0id 7852 ax-rnegex 7853 ax-precex 7854 ax-cnre 7855 ax-pre-ltirr 7856 ax-pre-ltwlin 7857 ax-pre-lttrn 7858 ax-pre-apti 7859 ax-pre-ltadd 7860 ax-pre-mulgt0 7861 ax-pre-mulext 7862 |
This theorem depends on definitions: df-bi 116 df-dc 825 df-3or 968 df-3an 969 df-tru 1345 df-fal 1348 df-nf 1448 df-sb 1750 df-eu 2016 df-mo 2017 df-clab 2151 df-cleq 2157 df-clel 2160 df-nfc 2295 df-ne 2335 df-nel 2430 df-ral 2447 df-rex 2448 df-reu 2449 df-rmo 2450 df-rab 2451 df-v 2723 df-sbc 2947 df-csb 3041 df-dif 3113 df-un 3115 df-in 3117 df-ss 3124 df-nul 3405 df-if 3516 df-pw 3555 df-sn 3576 df-pr 3577 df-op 3579 df-uni 3784 df-int 3819 df-iun 3862 df-br 3977 df-opab 4038 df-mpt 4039 df-tr 4075 df-id 4265 df-po 4268 df-iso 4269 df-iord 4338 df-on 4340 df-ilim 4341 df-suc 4343 df-iom 4562 df-xp 4604 df-rel 4605 df-cnv 4606 df-co 4607 df-dm 4608 df-rn 4609 df-res 4610 df-ima 4611 df-iota 5147 df-fun 5184 df-fn 5185 df-f 5186 df-f1 5187 df-fo 5188 df-f1o 5189 df-fv 5190 df-isom 5191 df-riota 5792 df-ov 5839 df-oprab 5840 df-mpo 5841 df-1st 6100 df-2nd 6101 df-recs 6264 df-irdg 6329 df-frec 6350 df-1o 6375 df-oadd 6379 df-er 6492 df-en 6698 df-dom 6699 df-fin 6700 df-pnf 7926 df-mnf 7927 df-xr 7928 df-ltxr 7929 df-le 7930 df-sub 8062 df-neg 8063 df-reap 8464 df-ap 8471 df-div 8560 df-inn 8849 df-2 8907 df-n0 9106 df-z 9183 df-uz 9458 df-q 9549 df-rp 9581 df-fz 9936 df-fzo 10068 df-seqfrec 10371 df-exp 10445 df-ihash 10678 df-cj 10770 df-rsqrt 10926 df-abs 10927 df-clim 11206 df-sumdc 11281 |
This theorem is referenced by: fisumss 11319 isumss2 11320 binomlem 11410 |
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