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Mirrors > Home > MPE Home > Th. List > modfsummod | Structured version Visualization version Unicode version |
Description: A finite sum modulo a positive integer equals the finite sum of their summands modulo the positive integer, modulo the positive integer. (Contributed by Alexander van der Vekens, 1-Sep-2018.) |
Ref | Expression |
---|---|
modfsummod.n | |
modfsummod.1 | |
modfsummod.2 |
Ref | Expression |
---|---|
modfsummod |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | modfsummod.2 | . 2 | |
2 | modfsummod.n | . 2 | |
3 | modfsummod.1 | . . 3 | |
4 | raleq 3138 | . . . . . 6 | |
5 | 4 | anbi1d 741 | . . . . 5 |
6 | sumeq1 14419 | . . . . . . 7 | |
7 | 6 | oveq1d 6665 | . . . . . 6 |
8 | sumeq1 14419 | . . . . . . 7 | |
9 | 8 | oveq1d 6665 | . . . . . 6 |
10 | 7, 9 | eqeq12d 2637 | . . . . 5 |
11 | 5, 10 | imbi12d 334 | . . . 4 |
12 | raleq 3138 | . . . . . 6 | |
13 | 12 | anbi1d 741 | . . . . 5 |
14 | sumeq1 14419 | . . . . . . 7 | |
15 | 14 | oveq1d 6665 | . . . . . 6 |
16 | sumeq1 14419 | . . . . . . 7 | |
17 | 16 | oveq1d 6665 | . . . . . 6 |
18 | 15, 17 | eqeq12d 2637 | . . . . 5 |
19 | 13, 18 | imbi12d 334 | . . . 4 |
20 | raleq 3138 | . . . . . 6 | |
21 | 20 | anbi1d 741 | . . . . 5 |
22 | sumeq1 14419 | . . . . . . 7 | |
23 | 22 | oveq1d 6665 | . . . . . 6 |
24 | sumeq1 14419 | . . . . . . 7 | |
25 | 24 | oveq1d 6665 | . . . . . 6 |
26 | 23, 25 | eqeq12d 2637 | . . . . 5 |
27 | 21, 26 | imbi12d 334 | . . . 4 |
28 | raleq 3138 | . . . . . 6 | |
29 | 28 | anbi1d 741 | . . . . 5 |
30 | sumeq1 14419 | . . . . . . 7 | |
31 | 30 | oveq1d 6665 | . . . . . 6 |
32 | sumeq1 14419 | . . . . . . 7 | |
33 | 32 | oveq1d 6665 | . . . . . 6 |
34 | 31, 33 | eqeq12d 2637 | . . . . 5 |
35 | 29, 34 | imbi12d 334 | . . . 4 |
36 | sum0 14452 | . . . . . . . 8 | |
37 | 36 | a1i 11 | . . . . . . 7 |
38 | 37 | oveq1d 6665 | . . . . . 6 |
39 | sum0 14452 | . . . . . . 7 | |
40 | 39 | oveq1i 6660 | . . . . . 6 |
41 | 38, 40 | syl6reqr 2675 | . . . . 5 |
42 | 41 | adantl 482 | . . . 4 |
43 | simpll 790 | . . . . . . . . . 10 | |
44 | simplrr 801 | . . . . . . . . . 10 | |
45 | ralun 3795 | . . . . . . . . . . . . 13 | |
46 | 45 | ex 450 | . . . . . . . . . . . 12 |
47 | 46 | ad2antrl 764 | . . . . . . . . . . 11 |
48 | 47 | imp 445 | . . . . . . . . . 10 |
49 | modfsummods 14525 | . . . . . . . . . 10 | |
50 | 43, 44, 48, 49 | syl3anc 1326 | . . . . . . . . 9 |
51 | 50 | ex 450 | . . . . . . . 8 |
52 | 51 | com23 86 | . . . . . . 7 |
53 | 52 | ex 450 | . . . . . 6 |
54 | 53 | a2d 29 | . . . . 5 |
55 | ralunb 3794 | . . . . . . . 8 | |
56 | 55 | anbi1i 731 | . . . . . . 7 |
57 | 56 | imbi1i 339 | . . . . . 6 |
58 | an32 839 | . . . . . . 7 | |
59 | 58 | imbi1i 339 | . . . . . 6 |
60 | impexp 462 | . . . . . 6 | |
61 | 57, 59, 60 | 3bitri 286 | . . . . 5 |
62 | 54, 61 | syl6ibr 242 | . . . 4 |
63 | 11, 19, 27, 35, 42, 62 | findcard2 8200 | . . 3 |
64 | 3, 63 | syl 17 | . 2 |
65 | 1, 2, 64 | mp2and 715 | 1 |
Colors of variables: wff setvar class |
Syntax hints: wi 4 wa 384 wceq 1483 wcel 1990 wral 2912 cun 3572 c0 3915 csn 4177 (class class class)co 6650 cfn 7955 cc0 9936 cn 11020 cz 11377 cmo 12668 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-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-rep 4771 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-inf2 8538 ax-cnex 9992 ax-resscn 9993 ax-1cn 9994 ax-icn 9995 ax-addcl 9996 ax-addrcl 9997 ax-mulcl 9998 ax-mulrcl 9999 ax-mulcom 10000 ax-addass 10001 ax-mulass 10002 ax-distr 10003 ax-i2m1 10004 ax-1ne0 10005 ax-1rid 10006 ax-rnegex 10007 ax-rrecex 10008 ax-cnre 10009 ax-pre-lttri 10010 ax-pre-lttrn 10011 ax-pre-ltadd 10012 ax-pre-mulgt0 10013 ax-pre-sup 10014 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-fal 1489 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-nel 2898 df-ral 2917 df-rex 2918 df-reu 2919 df-rmo 2920 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-int 4476 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-se 5074 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-pred 5680 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-isom 5897 df-riota 6611 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-om 7066 df-1st 7168 df-2nd 7169 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-1o 7560 df-oadd 7564 df-er 7742 df-en 7956 df-dom 7957 df-sdom 7958 df-fin 7959 df-sup 8348 df-inf 8349 df-oi 8415 df-card 8765 df-pnf 10076 df-mnf 10077 df-xr 10078 df-ltxr 10079 df-le 10080 df-sub 10268 df-neg 10269 df-div 10685 df-nn 11021 df-2 11079 df-3 11080 df-n0 11293 df-z 11378 df-uz 11688 df-rp 11833 df-fz 12327 df-fzo 12466 df-fl 12593 df-mod 12669 df-seq 12802 df-exp 12861 df-hash 13118 df-cj 13839 df-re 13840 df-im 13841 df-sqrt 13975 df-abs 13976 df-clim 14219 df-sum 14417 |
This theorem is referenced by: numclwwlk6 27248 |
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