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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 2670 | . . . . . 6 | |
5 | 4 | anbi1d 465 | . . . . 5 |
6 | sumeq1 11329 | . . . . . . 7 | |
7 | 6 | oveq1d 5880 | . . . . . 6 |
8 | sumeq1 11329 | . . . . . . 7 | |
9 | 8 | oveq1d 5880 | . . . . . 6 |
10 | 7, 9 | eqeq12d 2190 | . . . . 5 |
11 | 5, 10 | imbi12d 234 | . . . 4 |
12 | raleq 2670 | . . . . . 6 | |
13 | 12 | anbi1d 465 | . . . . 5 |
14 | sumeq1 11329 | . . . . . . 7 | |
15 | 14 | oveq1d 5880 | . . . . . 6 |
16 | sumeq1 11329 | . . . . . . 7 | |
17 | 16 | oveq1d 5880 | . . . . . 6 |
18 | 15, 17 | eqeq12d 2190 | . . . . 5 |
19 | 13, 18 | imbi12d 234 | . . . 4 |
20 | raleq 2670 | . . . . . 6 | |
21 | 20 | anbi1d 465 | . . . . 5 |
22 | sumeq1 11329 | . . . . . . 7 | |
23 | 22 | oveq1d 5880 | . . . . . 6 |
24 | sumeq1 11329 | . . . . . . 7 | |
25 | 24 | oveq1d 5880 | . . . . . 6 |
26 | 23, 25 | eqeq12d 2190 | . . . . 5 |
27 | 21, 26 | imbi12d 234 | . . . 4 |
28 | raleq 2670 | . . . . . 6 | |
29 | 28 | anbi1d 465 | . . . . 5 |
30 | sumeq1 11329 | . . . . . . 7 | |
31 | 30 | oveq1d 5880 | . . . . . 6 |
32 | sumeq1 11329 | . . . . . . 7 | |
33 | 32 | oveq1d 5880 | . . . . . 6 |
34 | 31, 33 | eqeq12d 2190 | . . . . 5 |
35 | 29, 34 | imbi12d 234 | . . . 4 |
36 | sum0 11362 | . . . . . . 7 | |
37 | 36 | oveq1i 5875 | . . . . . 6 |
38 | sum0 11362 | . . . . . . . 8 | |
39 | 38 | a1i 9 | . . . . . . 7 |
40 | 39 | oveq1d 5880 | . . . . . 6 |
41 | 37, 40 | eqtr4id 2227 | . . . . 5 |
42 | 41 | adantl 277 | . . . 4 |
43 | simp-4l 541 | . . . . . . . . . 10 | |
44 | simprr 531 | . . . . . . . . . . 11 | |
45 | 44 | ad2antrr 488 | . . . . . . . . . 10 |
46 | simprl 529 | . . . . . . . . . . . 12 | |
47 | 46 | ad2antrr 488 | . . . . . . . . . . 11 |
48 | simplr 528 | . . . . . . . . . . 11 | |
49 | ralun 3315 | . . . . . . . . . . 11 | |
50 | 47, 48, 49 | syl2anc 411 | . . . . . . . . . 10 |
51 | simplr 528 | . . . . . . . . . . 11 | |
52 | 51 | ad2antrr 488 | . . . . . . . . . 10 |
53 | simpr 110 | . . . . . . . . . 10 | |
54 | 43, 45, 50, 52, 53 | modfsummodlemstep 11431 | . . . . . . . . 9 |
55 | 54 | exp31 364 | . . . . . . . 8 |
56 | 55 | com23 78 | . . . . . . 7 |
57 | 56 | ex 115 | . . . . . 6 |
58 | 57 | a2d 26 | . . . . 5 |
59 | ralunb 3314 | . . . . . . . 8 | |
60 | 59 | anbi1i 458 | . . . . . . 7 |
61 | 60 | imbi1i 238 | . . . . . 6 |
62 | an32 562 | . . . . . . 7 | |
63 | 62 | imbi1i 238 | . . . . . 6 |
64 | impexp 263 | . . . . . 6 | |
65 | 61, 63, 64 | 3bitri 206 | . . . . 5 |
66 | 58, 65 | syl6ibr 162 | . . . 4 |
67 | 11, 19, 27, 35, 42, 66 | findcard2s 6880 | . . 3 |
68 | 3, 67 | syl 14 | . 2 |
69 | 1, 2, 68 | mp2and 433 | 1 |
Colors of variables: wff set class |
Syntax hints: wn 3 wi 4 wa 104 wceq 1353 wcel 2146 wral 2453 cun 3125 c0 3420 csn 3589 (class class class)co 5865 cfn 6730 cc0 7786 cn 8890 cz 9224 cmo 10290 csu 11327 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 614 ax-in2 615 ax-io 709 ax-5 1445 ax-7 1446 ax-gen 1447 ax-ie1 1491 ax-ie2 1492 ax-8 1502 ax-10 1503 ax-11 1504 ax-i12 1505 ax-bndl 1507 ax-4 1508 ax-17 1524 ax-i9 1528 ax-ial 1532 ax-i5r 1533 ax-13 2148 ax-14 2149 ax-ext 2157 ax-coll 4113 ax-sep 4116 ax-nul 4124 ax-pow 4169 ax-pr 4203 ax-un 4427 ax-setind 4530 ax-iinf 4581 ax-cnex 7877 ax-resscn 7878 ax-1cn 7879 ax-1re 7880 ax-icn 7881 ax-addcl 7882 ax-addrcl 7883 ax-mulcl 7884 ax-mulrcl 7885 ax-addcom 7886 ax-mulcom 7887 ax-addass 7888 ax-mulass 7889 ax-distr 7890 ax-i2m1 7891 ax-0lt1 7892 ax-1rid 7893 ax-0id 7894 ax-rnegex 7895 ax-precex 7896 ax-cnre 7897 ax-pre-ltirr 7898 ax-pre-ltwlin 7899 ax-pre-lttrn 7900 ax-pre-apti 7901 ax-pre-ltadd 7902 ax-pre-mulgt0 7903 ax-pre-mulext 7904 ax-arch 7905 ax-caucvg 7906 |
This theorem depends on definitions: df-bi 117 df-dc 835 df-3or 979 df-3an 980 df-tru 1356 df-fal 1359 df-nf 1459 df-sb 1761 df-eu 2027 df-mo 2028 df-clab 2162 df-cleq 2168 df-clel 2171 df-nfc 2306 df-ne 2346 df-nel 2441 df-ral 2458 df-rex 2459 df-reu 2460 df-rmo 2461 df-rab 2462 df-v 2737 df-sbc 2961 df-csb 3056 df-dif 3129 df-un 3131 df-in 3133 df-ss 3140 df-nul 3421 df-if 3533 df-pw 3574 df-sn 3595 df-pr 3596 df-op 3598 df-uni 3806 df-int 3841 df-iun 3884 df-br 3999 df-opab 4060 df-mpt 4061 df-tr 4097 df-id 4287 df-po 4290 df-iso 4291 df-iord 4360 df-on 4362 df-ilim 4363 df-suc 4365 df-iom 4584 df-xp 4626 df-rel 4627 df-cnv 4628 df-co 4629 df-dm 4630 df-rn 4631 df-res 4632 df-ima 4633 df-iota 5170 df-fun 5210 df-fn 5211 df-f 5212 df-f1 5213 df-fo 5214 df-f1o 5215 df-fv 5216 df-isom 5217 df-riota 5821 df-ov 5868 df-oprab 5869 df-mpo 5870 df-1st 6131 df-2nd 6132 df-recs 6296 df-irdg 6361 df-frec 6382 df-1o 6407 df-oadd 6411 df-er 6525 df-en 6731 df-dom 6732 df-fin 6733 df-pnf 7968 df-mnf 7969 df-xr 7970 df-ltxr 7971 df-le 7972 df-sub 8104 df-neg 8105 df-reap 8506 df-ap 8513 df-div 8602 df-inn 8891 df-2 8949 df-3 8950 df-4 8951 df-n0 9148 df-z 9225 df-uz 9500 df-q 9591 df-rp 9623 df-fz 9978 df-fzo 10111 df-fl 10238 df-mod 10291 df-seqfrec 10414 df-exp 10488 df-ihash 10722 df-cj 10817 df-re 10818 df-im 10819 df-rsqrt 10973 df-abs 10974 df-clim 11253 df-sumdc 11328 |
This theorem is referenced by: (None) |
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