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Mirrors > Home > ILE Home > Th. List > modfsummod | 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 2626 | . . . . . 6 | |
5 | 4 | anbi1d 460 | . . . . 5 |
6 | sumeq1 11124 | . . . . . . 7 | |
7 | 6 | oveq1d 5789 | . . . . . 6 |
8 | sumeq1 11124 | . . . . . . 7 | |
9 | 8 | oveq1d 5789 | . . . . . 6 |
10 | 7, 9 | eqeq12d 2154 | . . . . 5 |
11 | 5, 10 | imbi12d 233 | . . . 4 |
12 | raleq 2626 | . . . . . 6 | |
13 | 12 | anbi1d 460 | . . . . 5 |
14 | sumeq1 11124 | . . . . . . 7 | |
15 | 14 | oveq1d 5789 | . . . . . 6 |
16 | sumeq1 11124 | . . . . . . 7 | |
17 | 16 | oveq1d 5789 | . . . . . 6 |
18 | 15, 17 | eqeq12d 2154 | . . . . 5 |
19 | 13, 18 | imbi12d 233 | . . . 4 |
20 | raleq 2626 | . . . . . 6 | |
21 | 20 | anbi1d 460 | . . . . 5 |
22 | sumeq1 11124 | . . . . . . 7 | |
23 | 22 | oveq1d 5789 | . . . . . 6 |
24 | sumeq1 11124 | . . . . . . 7 | |
25 | 24 | oveq1d 5789 | . . . . . 6 |
26 | 23, 25 | eqeq12d 2154 | . . . . 5 |
27 | 21, 26 | imbi12d 233 | . . . 4 |
28 | raleq 2626 | . . . . . 6 | |
29 | 28 | anbi1d 460 | . . . . 5 |
30 | sumeq1 11124 | . . . . . . 7 | |
31 | 30 | oveq1d 5789 | . . . . . 6 |
32 | sumeq1 11124 | . . . . . . 7 | |
33 | 32 | oveq1d 5789 | . . . . . 6 |
34 | 31, 33 | eqeq12d 2154 | . . . . 5 |
35 | 29, 34 | imbi12d 233 | . . . 4 |
36 | sum0 11157 | . . . . . . . 8 | |
37 | 36 | a1i 9 | . . . . . . 7 |
38 | 37 | oveq1d 5789 | . . . . . 6 |
39 | sum0 11157 | . . . . . . 7 | |
40 | 39 | oveq1i 5784 | . . . . . 6 |
41 | 38, 40 | syl6reqr 2191 | . . . . 5 |
42 | 41 | adantl 275 | . . . 4 |
43 | simp-4l 530 | . . . . . . . . . 10 | |
44 | simprr 521 | . . . . . . . . . . 11 | |
45 | 44 | ad2antrr 479 | . . . . . . . . . 10 |
46 | simprl 520 | . . . . . . . . . . . 12 | |
47 | 46 | ad2antrr 479 | . . . . . . . . . . 11 |
48 | simplr 519 | . . . . . . . . . . 11 | |
49 | ralun 3258 | . . . . . . . . . . 11 | |
50 | 47, 48, 49 | syl2anc 408 | . . . . . . . . . 10 |
51 | simplr 519 | . . . . . . . . . . 11 | |
52 | 51 | ad2antrr 479 | . . . . . . . . . 10 |
53 | simpr 109 | . . . . . . . . . 10 | |
54 | 43, 45, 50, 52, 53 | modfsummodlemstep 11226 | . . . . . . . . 9 |
55 | 54 | exp31 361 | . . . . . . . 8 |
56 | 55 | com23 78 | . . . . . . 7 |
57 | 56 | ex 114 | . . . . . 6 |
58 | 57 | a2d 26 | . . . . 5 |
59 | ralunb 3257 | . . . . . . . 8 | |
60 | 59 | anbi1i 453 | . . . . . . 7 |
61 | 60 | imbi1i 237 | . . . . . 6 |
62 | an32 551 | . . . . . . 7 | |
63 | 62 | imbi1i 237 | . . . . . 6 |
64 | impexp 261 | . . . . . 6 | |
65 | 61, 63, 64 | 3bitri 205 | . . . . 5 |
66 | 58, 65 | syl6ibr 161 | . . . 4 |
67 | 11, 19, 27, 35, 42, 66 | findcard2s 6784 | . . 3 |
68 | 3, 67 | syl 14 | . 2 |
69 | 1, 2, 68 | mp2and 429 | 1 |
Colors of variables: wff set class |
Syntax hints: wn 3 wi 4 wa 103 wceq 1331 wcel 1480 wral 2416 cun 3069 c0 3363 csn 3527 (class class class)co 5774 cfn 6634 cc0 7620 cn 8720 cz 9054 cmo 10095 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-13 1491 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 ax-coll 4043 ax-sep 4046 ax-nul 4054 ax-pow 4098 ax-pr 4131 ax-un 4355 ax-setind 4452 ax-iinf 4502 ax-cnex 7711 ax-resscn 7712 ax-1cn 7713 ax-1re 7714 ax-icn 7715 ax-addcl 7716 ax-addrcl 7717 ax-mulcl 7718 ax-mulrcl 7719 ax-addcom 7720 ax-mulcom 7721 ax-addass 7722 ax-mulass 7723 ax-distr 7724 ax-i2m1 7725 ax-0lt1 7726 ax-1rid 7727 ax-0id 7728 ax-rnegex 7729 ax-precex 7730 ax-cnre 7731 ax-pre-ltirr 7732 ax-pre-ltwlin 7733 ax-pre-lttrn 7734 ax-pre-apti 7735 ax-pre-ltadd 7736 ax-pre-mulgt0 7737 ax-pre-mulext 7738 ax-arch 7739 ax-caucvg 7740 |
This theorem depends on definitions: df-bi 116 df-dc 820 df-3or 963 df-3an 964 df-tru 1334 df-fal 1337 df-nf 1437 df-sb 1736 df-eu 2002 df-mo 2003 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-ne 2309 df-nel 2404 df-ral 2421 df-rex 2422 df-reu 2423 df-rmo 2424 df-rab 2425 df-v 2688 df-sbc 2910 df-csb 3004 df-dif 3073 df-un 3075 df-in 3077 df-ss 3084 df-nul 3364 df-if 3475 df-pw 3512 df-sn 3533 df-pr 3534 df-op 3536 df-uni 3737 df-int 3772 df-iun 3815 df-br 3930 df-opab 3990 df-mpt 3991 df-tr 4027 df-id 4215 df-po 4218 df-iso 4219 df-iord 4288 df-on 4290 df-ilim 4291 df-suc 4293 df-iom 4505 df-xp 4545 df-rel 4546 df-cnv 4547 df-co 4548 df-dm 4549 df-rn 4550 df-res 4551 df-ima 4552 df-iota 5088 df-fun 5125 df-fn 5126 df-f 5127 df-f1 5128 df-fo 5129 df-f1o 5130 df-fv 5131 df-isom 5132 df-riota 5730 df-ov 5777 df-oprab 5778 df-mpo 5779 df-1st 6038 df-2nd 6039 df-recs 6202 df-irdg 6267 df-frec 6288 df-1o 6313 df-oadd 6317 df-er 6429 df-en 6635 df-dom 6636 df-fin 6637 df-pnf 7802 df-mnf 7803 df-xr 7804 df-ltxr 7805 df-le 7806 df-sub 7935 df-neg 7936 df-reap 8337 df-ap 8344 df-div 8433 df-inn 8721 df-2 8779 df-3 8780 df-4 8781 df-n0 8978 df-z 9055 df-uz 9327 df-q 9412 df-rp 9442 df-fz 9791 df-fzo 9920 df-fl 10043 df-mod 10096 df-seqfrec 10219 df-exp 10293 df-ihash 10522 df-cj 10614 df-re 10615 df-im 10616 df-rsqrt 10770 df-abs 10771 df-clim 11048 df-sumdc 11123 |
This theorem is referenced by: (None) |
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