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Mirrors > Home > ILE Home > Th. List > fsumrev | Unicode version |
Description: Reversal of a finite sum. (Contributed by NM, 26-Nov-2005.) (Revised by Mario Carneiro, 24-Apr-2014.) |
Ref | Expression |
---|---|
fsumrev.1 | |
fsumrev.2 | |
fsumrev.3 | |
fsumrev.4 | |
fsumrev.5 |
Ref | Expression |
---|---|
fsumrev |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fsumrev.5 | . 2 | |
2 | fsumrev.1 | . . . 4 | |
3 | fsumrev.3 | . . . 4 | |
4 | 2, 3 | zsubcld 9171 | . . 3 |
5 | fsumrev.2 | . . . 4 | |
6 | 2, 5 | zsubcld 9171 | . . 3 |
7 | 4, 6 | fzfigd 10197 | . 2 |
8 | eqid 2137 | . . 3 | |
9 | 2 | adantr 274 | . . . 4 |
10 | elfzelz 9799 | . . . . 5 | |
11 | 10 | adantl 275 | . . . 4 |
12 | 9, 11 | zsubcld 9171 | . . 3 |
13 | 2 | adantr 274 | . . . 4 |
14 | elfzelz 9799 | . . . . 5 | |
15 | 14 | adantl 275 | . . . 4 |
16 | 13, 15 | zsubcld 9171 | . . 3 |
17 | simprr 521 | . . . . . 6 | |
18 | simprl 520 | . . . . . . 7 | |
19 | 5 | adantr 274 | . . . . . . . 8 |
20 | 3 | adantr 274 | . . . . . . . 8 |
21 | 2 | adantr 274 | . . . . . . . 8 |
22 | 18, 10 | syl 14 | . . . . . . . 8 |
23 | fzrev 9857 | . . . . . . . 8 | |
24 | 19, 20, 21, 22, 23 | syl22anc 1217 | . . . . . . 7 |
25 | 18, 24 | mpbid 146 | . . . . . 6 |
26 | 17, 25 | eqeltrd 2214 | . . . . 5 |
27 | 17 | oveq2d 5783 | . . . . . 6 |
28 | zcn 9052 | . . . . . . . 8 | |
29 | zcn 9052 | . . . . . . . 8 | |
30 | nncan 7984 | . . . . . . . 8 | |
31 | 28, 29, 30 | syl2an 287 | . . . . . . 7 |
32 | 21, 22, 31 | syl2anc 408 | . . . . . 6 |
33 | 27, 32 | eqtr2d 2171 | . . . . 5 |
34 | 26, 33 | jca 304 | . . . 4 |
35 | simprr 521 | . . . . . 6 | |
36 | simprl 520 | . . . . . . 7 | |
37 | 5 | adantr 274 | . . . . . . . 8 |
38 | 3 | adantr 274 | . . . . . . . 8 |
39 | 2 | adantr 274 | . . . . . . . 8 |
40 | 36, 14 | syl 14 | . . . . . . . 8 |
41 | fzrev2 9858 | . . . . . . . 8 | |
42 | 37, 38, 39, 40, 41 | syl22anc 1217 | . . . . . . 7 |
43 | 36, 42 | mpbid 146 | . . . . . 6 |
44 | 35, 43 | eqeltrd 2214 | . . . . 5 |
45 | 35 | oveq2d 5783 | . . . . . 6 |
46 | zcn 9052 | . . . . . . . 8 | |
47 | nncan 7984 | . . . . . . . 8 | |
48 | 28, 46, 47 | syl2an 287 | . . . . . . 7 |
49 | 39, 40, 48 | syl2anc 408 | . . . . . 6 |
50 | 45, 49 | eqtr2d 2171 | . . . . 5 |
51 | 44, 50 | jca 304 | . . . 4 |
52 | 34, 51 | impbida 585 | . . 3 |
53 | 8, 12, 16, 52 | f1od 5966 | . 2 |
54 | simpr 109 | . . 3 | |
55 | 2 | adantr 274 | . . . 4 |
56 | elfzelz 9799 | . . . . 5 | |
57 | 56 | adantl 275 | . . . 4 |
58 | 55, 57 | zsubcld 9171 | . . 3 |
59 | oveq2 5775 | . . . 4 | |
60 | 59, 8 | fvmptg 5490 | . . 3 |
61 | 54, 58, 60 | syl2anc 408 | . 2 |
62 | fsumrev.4 | . 2 | |
63 | 1, 7, 53, 61, 62 | fsumf1o 11152 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wb 104 wceq 1331 wcel 1480 cmpt 3984 cfv 5118 (class class class)co 5767 cc 7611 cmin 7926 cz 9047 cfz 9783 csu 11115 |
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 2119 ax-coll 4038 ax-sep 4041 ax-nul 4049 ax-pow 4093 ax-pr 4126 ax-un 4350 ax-setind 4447 ax-iinf 4497 ax-cnex 7704 ax-resscn 7705 ax-1cn 7706 ax-1re 7707 ax-icn 7708 ax-addcl 7709 ax-addrcl 7710 ax-mulcl 7711 ax-mulrcl 7712 ax-addcom 7713 ax-mulcom 7714 ax-addass 7715 ax-mulass 7716 ax-distr 7717 ax-i2m1 7718 ax-0lt1 7719 ax-1rid 7720 ax-0id 7721 ax-rnegex 7722 ax-precex 7723 ax-cnre 7724 ax-pre-ltirr 7725 ax-pre-ltwlin 7726 ax-pre-lttrn 7727 ax-pre-apti 7728 ax-pre-ltadd 7729 ax-pre-mulgt0 7730 ax-pre-mulext 7731 ax-arch 7732 ax-caucvg 7733 |
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 2000 df-mo 2001 df-clab 2124 df-cleq 2130 df-clel 2133 df-nfc 2268 df-ne 2307 df-nel 2402 df-ral 2419 df-rex 2420 df-reu 2421 df-rmo 2422 df-rab 2423 df-v 2683 df-sbc 2905 df-csb 2999 df-dif 3068 df-un 3070 df-in 3072 df-ss 3079 df-nul 3359 df-if 3470 df-pw 3507 df-sn 3528 df-pr 3529 df-op 3531 df-uni 3732 df-int 3767 df-iun 3810 df-br 3925 df-opab 3985 df-mpt 3986 df-tr 4022 df-id 4210 df-po 4213 df-iso 4214 df-iord 4283 df-on 4285 df-ilim 4286 df-suc 4288 df-iom 4500 df-xp 4540 df-rel 4541 df-cnv 4542 df-co 4543 df-dm 4544 df-rn 4545 df-res 4546 df-ima 4547 df-iota 5083 df-fun 5120 df-fn 5121 df-f 5122 df-f1 5123 df-fo 5124 df-f1o 5125 df-fv 5126 df-isom 5127 df-riota 5723 df-ov 5770 df-oprab 5771 df-mpo 5772 df-1st 6031 df-2nd 6032 df-recs 6195 df-irdg 6260 df-frec 6281 df-1o 6306 df-oadd 6310 df-er 6422 df-en 6628 df-dom 6629 df-fin 6630 df-pnf 7795 df-mnf 7796 df-xr 7797 df-ltxr 7798 df-le 7799 df-sub 7928 df-neg 7929 df-reap 8330 df-ap 8337 df-div 8426 df-inn 8714 df-2 8772 df-3 8773 df-4 8774 df-n0 8971 df-z 9048 df-uz 9320 df-q 9405 df-rp 9435 df-fz 9784 df-fzo 9913 df-seqfrec 10212 df-exp 10286 df-ihash 10515 df-cj 10607 df-re 10608 df-im 10609 df-rsqrt 10763 df-abs 10764 df-clim 11041 df-sumdc 11116 |
This theorem is referenced by: fisumrev2 11208 |
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