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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 3170 | . . . . . 6 | |
2 | eleq2 2234 | . . . . . . . 8 | |
3 | 2 | dcbid 833 | . . . . . . 7 DECID DECID |
4 | 3 | ralbidv 2470 | . . . . . 6 DECID DECID |
5 | simpl 108 | . . . . . . . . . . 11 | |
6 | 5 | eleq2d 2240 | . . . . . . . . . 10 |
7 | 6 | ifbid 3547 | . . . . . . . . 9 |
8 | 7 | mpteq2dva 4079 | . . . . . . . 8 |
9 | 8 | seqeq3d 10409 | . . . . . . 7 |
10 | 9 | breq1d 3999 | . . . . . 6 |
11 | 1, 4, 10 | 3anbi123d 1307 | . . . . 5 DECID DECID |
12 | 11 | rexbidv 2471 | . . . 4 DECID DECID |
13 | f1oeq3 5433 | . . . . . . 7 | |
14 | 13 | anbi1d 462 | . . . . . 6 |
15 | 14 | exbidv 1818 | . . . . 5 |
16 | 15 | rexbidv 2471 | . . . 4 |
17 | 12, 16 | orbi12d 788 | . . 3 DECID DECID |
18 | 17 | iotabidv 5181 | . 2 DECID DECID |
19 | df-sumdc 11317 | . 2 DECID | |
20 | df-sumdc 11317 | . 2 DECID | |
21 | 18, 19, 20 | 3eqtr4g 2228 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wo 703 DECID wdc 829 w3a 973 wceq 1348 wex 1485 wcel 2141 wral 2448 wrex 2449 csb 3049 wss 3121 cif 3526 class class class wbr 3989 cmpt 4050 cio 5158 wf1o 5197 cfv 5198 (class class class)co 5853 cc0 7774 c1 7775 caddc 7777 cle 7955 cn 8878 cz 9212 cuz 9487 cfz 9965 cseq 10401 cli 11241 csu 11316 |
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-ext 2152 |
This theorem depends on definitions: df-bi 116 df-dc 830 df-3an 975 df-tru 1351 df-nf 1454 df-sb 1756 df-clab 2157 df-cleq 2163 df-clel 2166 df-nfc 2301 df-ral 2453 df-rex 2454 df-v 2732 df-un 3125 df-in 3127 df-ss 3134 df-if 3527 df-sn 3589 df-pr 3590 df-op 3592 df-uni 3797 df-br 3990 df-opab 4051 df-mpt 4052 df-cnv 4619 df-dm 4621 df-rn 4622 df-res 4623 df-iota 5160 df-f 5202 df-f1 5203 df-fo 5204 df-f1o 5205 df-fv 5206 df-ov 5856 df-oprab 5857 df-mpo 5858 df-recs 6284 df-frec 6370 df-seqfrec 10402 df-sumdc 11317 |
This theorem is referenced by: sumeq1i 11326 sumeq1d 11329 isumz 11352 fsumadd 11369 fsum2d 11398 fisumrev2 11409 fsummulc2 11411 fsumconst 11417 modfsummod 11421 fsumabs 11428 fsumrelem 11434 fsumiun 11440 fsumcncntop 13350 |
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