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Mirrors > Home > MPE Home > Th. List > sumeq1 | Structured version Visualization version 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 3626 | . . . . . 6 | |
2 | simpl 473 | . . . . . . . . . . 11 | |
3 | 2 | eleq2d 2687 | . . . . . . . . . 10 |
4 | 3 | ifbid 4108 | . . . . . . . . 9 |
5 | 4 | mpteq2dva 4744 | . . . . . . . 8 |
6 | 5 | seqeq3d 12809 | . . . . . . 7 |
7 | 6 | breq1d 4663 | . . . . . 6 |
8 | 1, 7 | anbi12d 747 | . . . . 5 |
9 | 8 | rexbidv 3052 | . . . 4 |
10 | f1oeq3 6129 | . . . . . . 7 | |
11 | 10 | anbi1d 741 | . . . . . 6 |
12 | 11 | exbidv 1850 | . . . . 5 |
13 | 12 | rexbidv 3052 | . . . 4 |
14 | 9, 13 | orbi12d 746 | . . 3 |
15 | 14 | iotabidv 5872 | . 2 |
16 | df-sum 14417 | . 2 | |
17 | df-sum 14417 | . 2 | |
18 | 15, 16, 17 | 3eqtr4g 2681 | 1 |
Colors of variables: wff setvar class |
Syntax hints: wi 4 wo 383 wa 384 wceq 1483 wex 1704 wcel 1990 wrex 2913 csb 3533 wss 3574 cif 4086 class class class wbr 4653 cmpt 4729 cio 5849 wf1o 5887 cfv 5888 (class class class)co 6650 cc0 9936 c1 9937 caddc 9939 cn 11020 cz 11377 cuz 11687 cfz 12326 cseq 12801 cli 14215 csu 14416 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ral 2917 df-rex 2918 df-rab 2921 df-v 3202 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-sn 4178 df-pr 4180 df-op 4184 df-uni 4437 df-br 4654 df-opab 4713 df-mpt 4730 df-xp 5120 df-cnv 5122 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-pred 5680 df-iota 5851 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-seq 12802 df-sum 14417 |
This theorem is referenced by: sumeq1i 14428 sumeq1d 14431 sumz 14453 fsumadd 14470 fsum2d 14502 fsumrev2 14514 fsummulc2 14516 fsumconst 14522 modfsummods 14525 modfsummod 14526 fsumabs 14533 fsumrelem 14539 fsumrlim 14543 fsumo1 14544 fsumiun 14553 sumeven 15110 sumodd 15111 bitsinv2 15165 bitsf1ocnv 15166 bitsinv 15170 prmreclem5 15624 gsumfsum 19813 fsumcn 22673 ovolfiniun 23269 volfiniun 23315 itgfsum 23593 dvmptfsum 23738 pntrsumbnd2 25256 finsumvtxdg2size 26446 esumpcvgval 30140 esumcvg 30148 rrnval 33626 mccl 39830 dvmptfprod 40160 dvnprodlem1 40161 dvnprodlem2 40162 dvnprodlem3 40163 dvnprod 40164 sge0rnn0 40585 sge00 40593 fsumlesge0 40594 sge0sn 40596 sge0cl 40598 sge0f1o 40599 sge0resplit 40623 sge0xaddlem1 40650 sge0xaddlem2 40651 sge0reuz 40664 |
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