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Mirrors > Home > ILE Home > Th. List > sumeq1 | Unicode version |
Description: Equality theorem for a sum. (Contributed by NM, 11-Dec-2005.) (Revised by Mario Carneiro, 13-Jun-2019.) |
Ref | Expression |
---|---|
sumeq1 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | sseq1 3120 | . . . . . 6 | |
2 | eleq2 2203 | . . . . . . . 8 | |
3 | 2 | dcbid 823 | . . . . . . 7 DECID DECID |
4 | 3 | ralbidv 2437 | . . . . . 6 DECID DECID |
5 | simpl 108 | . . . . . . . . . . 11 | |
6 | 5 | eleq2d 2209 | . . . . . . . . . 10 |
7 | 6 | ifbid 3493 | . . . . . . . . 9 |
8 | 7 | mpteq2dva 4018 | . . . . . . . 8 |
9 | 8 | seqeq3d 10226 | . . . . . . 7 |
10 | 9 | breq1d 3939 | . . . . . 6 |
11 | 1, 4, 10 | 3anbi123d 1290 | . . . . 5 DECID DECID |
12 | 11 | rexbidv 2438 | . . . 4 DECID DECID |
13 | f1oeq3 5358 | . . . . . . 7 | |
14 | 13 | anbi1d 460 | . . . . . 6 |
15 | 14 | exbidv 1797 | . . . . 5 |
16 | 15 | rexbidv 2438 | . . . 4 |
17 | 12, 16 | orbi12d 782 | . . 3 DECID DECID |
18 | 17 | iotabidv 5109 | . 2 DECID DECID |
19 | df-sumdc 11123 | . 2 DECID | |
20 | df-sumdc 11123 | . 2 DECID | |
21 | 18, 19, 20 | 3eqtr4g 2197 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wo 697 DECID wdc 819 w3a 962 wceq 1331 wex 1468 wcel 1480 wral 2416 wrex 2417 csb 3003 wss 3071 cif 3474 class class class wbr 3929 cmpt 3989 cio 5086 wf1o 5122 cfv 5123 (class class class)co 5774 cc0 7620 c1 7621 caddc 7623 cle 7801 cn 8720 cz 9054 cuz 9326 cfz 9790 cseq 10218 cli 11047 csu 11122 |
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-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 |
This theorem depends on definitions: df-bi 116 df-dc 820 df-3an 964 df-tru 1334 df-nf 1437 df-sb 1736 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-ral 2421 df-rex 2422 df-v 2688 df-un 3075 df-in 3077 df-ss 3084 df-if 3475 df-sn 3533 df-pr 3534 df-op 3536 df-uni 3737 df-br 3930 df-opab 3990 df-mpt 3991 df-cnv 4547 df-dm 4549 df-rn 4550 df-res 4551 df-iota 5088 df-f 5127 df-f1 5128 df-fo 5129 df-f1o 5130 df-fv 5131 df-ov 5777 df-oprab 5778 df-mpo 5779 df-recs 6202 df-frec 6288 df-seqfrec 10219 df-sumdc 11123 |
This theorem is referenced by: sumeq1i 11132 sumeq1d 11135 isumz 11158 fsumadd 11175 fsum2d 11204 fisumrev2 11215 fsummulc2 11217 fsumconst 11223 modfsummod 11227 fsumabs 11234 fsumrelem 11240 fsumiun 11246 fsumcncntop 12725 |
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