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Mirrors > Home > ILE Home > Th. List > fsumf1o | Unicode version |
Description: Re-index a finite sum using a bijection. (Contributed by Mario Carneiro, 20-Apr-2014.) |
Ref | Expression |
---|---|
fsumf1o.1 | |
fsumf1o.2 | |
fsumf1o.3 | |
fsumf1o.4 | |
fsumf1o.5 |
Ref | Expression |
---|---|
fsumf1o |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | sum0 11278 | . . . 4 | |
2 | fsumf1o.3 | . . . . . . . 8 | |
3 | f1oeq2 5403 | . . . . . . . 8 | |
4 | 2, 3 | syl5ibcom 154 | . . . . . . 7 |
5 | 4 | imp 123 | . . . . . 6 |
6 | f1ofo 5420 | . . . . . 6 | |
7 | fo00 5449 | . . . . . . 7 | |
8 | 7 | simprbi 273 | . . . . . 6 |
9 | 5, 6, 8 | 3syl 17 | . . . . 5 |
10 | 9 | sumeq1d 11256 | . . . 4 |
11 | simpr 109 | . . . . . 6 | |
12 | 11 | sumeq1d 11256 | . . . . 5 |
13 | sum0 11278 | . . . . 5 | |
14 | 12, 13 | eqtrdi 2206 | . . . 4 |
15 | 1, 10, 14 | 3eqtr4a 2216 | . . 3 |
16 | 15 | ex 114 | . 2 |
17 | 2fveq3 5472 | . . . . . . . 8 | |
18 | simprl 521 | . . . . . . . 8 ♯ ♯ ♯ | |
19 | simprr 522 | . . . . . . . 8 ♯ ♯ ♯ | |
20 | f1of 5413 | . . . . . . . . . . . 12 | |
21 | 2, 20 | syl 14 | . . . . . . . . . . 11 |
22 | 21 | ffvelrnda 5601 | . . . . . . . . . 10 |
23 | fsumf1o.5 | . . . . . . . . . . . 12 | |
24 | 23 | fmpttd 5621 | . . . . . . . . . . 11 |
25 | 24 | ffvelrnda 5601 | . . . . . . . . . 10 |
26 | 22, 25 | syldan 280 | . . . . . . . . 9 |
27 | 26 | adantlr 469 | . . . . . . . 8 ♯ ♯ |
28 | 2 | adantr 274 | . . . . . . . . . . . 12 ♯ ♯ |
29 | f1oco 5436 | . . . . . . . . . . . 12 ♯ ♯ | |
30 | 28, 19, 29 | syl2anc 409 | . . . . . . . . . . 11 ♯ ♯ ♯ |
31 | f1of 5413 | . . . . . . . . . . 11 ♯ ♯ | |
32 | 30, 31 | syl 14 | . . . . . . . . . 10 ♯ ♯ ♯ |
33 | fvco3 5538 | . . . . . . . . . 10 ♯ ♯ | |
34 | 32, 33 | sylan 281 | . . . . . . . . 9 ♯ ♯ ♯ |
35 | f1of 5413 | . . . . . . . . . . . 12 ♯ ♯ | |
36 | 35 | ad2antll 483 | . . . . . . . . . . 11 ♯ ♯ ♯ |
37 | fvco3 5538 | . . . . . . . . . . 11 ♯ ♯ | |
38 | 36, 37 | sylan 281 | . . . . . . . . . 10 ♯ ♯ ♯ |
39 | 38 | fveq2d 5471 | . . . . . . . . 9 ♯ ♯ ♯ |
40 | 34, 39 | eqtrd 2190 | . . . . . . . 8 ♯ ♯ ♯ |
41 | 17, 18, 19, 27, 40 | fsum3 11277 | . . . . . . 7 ♯ ♯ ♯ ♯ |
42 | eqid 2157 | . . . . . . . . . . . . 13 | |
43 | fsumf1o.1 | . . . . . . . . . . . . 13 | |
44 | fsumf1o.4 | . . . . . . . . . . . . . 14 | |
45 | 21 | ffvelrnda 5601 | . . . . . . . . . . . . . 14 |
46 | 44, 45 | eqeltrrd 2235 | . . . . . . . . . . . . 13 |
47 | 43 | eleq1d 2226 | . . . . . . . . . . . . . 14 |
48 | 23 | ralrimiva 2530 | . . . . . . . . . . . . . . 15 |
49 | 48 | adantr 274 | . . . . . . . . . . . . . 14 |
50 | 47, 49, 46 | rspcdva 2821 | . . . . . . . . . . . . 13 |
51 | 42, 43, 46, 50 | fvmptd3 5560 | . . . . . . . . . . . 12 |
52 | 44 | fveq2d 5471 | . . . . . . . . . . . 12 |
53 | simpr 109 | . . . . . . . . . . . . 13 | |
54 | eqid 2157 | . . . . . . . . . . . . . 14 | |
55 | 54 | fvmpt2 5550 | . . . . . . . . . . . . 13 |
56 | 53, 50, 55 | syl2anc 409 | . . . . . . . . . . . 12 |
57 | 51, 52, 56 | 3eqtr4rd 2201 | . . . . . . . . . . 11 |
58 | 57 | ralrimiva 2530 | . . . . . . . . . 10 |
59 | nffvmpt1 5478 | . . . . . . . . . . . 12 | |
60 | 59 | nfeq1 2309 | . . . . . . . . . . 11 |
61 | fveq2 5467 | . . . . . . . . . . . 12 | |
62 | 2fveq3 5472 | . . . . . . . . . . . 12 | |
63 | 61, 62 | eqeq12d 2172 | . . . . . . . . . . 11 |
64 | 60, 63 | rspc 2810 | . . . . . . . . . 10 |
65 | 58, 64 | mpan9 279 | . . . . . . . . 9 |
66 | 65 | adantlr 469 | . . . . . . . 8 ♯ ♯ |
67 | 66 | sumeq2dv 11258 | . . . . . . 7 ♯ ♯ |
68 | fveq2 5467 | . . . . . . . 8 | |
69 | 24 | adantr 274 | . . . . . . . . 9 ♯ ♯ |
70 | 69 | ffvelrnda 5601 | . . . . . . . 8 ♯ ♯ |
71 | 68, 18, 30, 70, 34 | fsum3 11277 | . . . . . . 7 ♯ ♯ ♯ ♯ |
72 | 41, 67, 71 | 3eqtr4rd 2201 | . . . . . 6 ♯ ♯ |
73 | sumfct 11264 | . . . . . . . 8 | |
74 | 48, 73 | syl 14 | . . . . . . 7 |
75 | 74 | adantr 274 | . . . . . 6 ♯ ♯ |
76 | 50 | ralrimiva 2530 | . . . . . . . 8 |
77 | sumfct 11264 | . . . . . . . 8 | |
78 | 76, 77 | syl 14 | . . . . . . 7 |
79 | 78 | adantr 274 | . . . . . 6 ♯ ♯ |
80 | 72, 75, 79 | 3eqtr3d 2198 | . . . . 5 ♯ ♯ |
81 | 80 | expr 373 | . . . 4 ♯ ♯ |
82 | 81 | exlimdv 1799 | . . 3 ♯ ♯ |
83 | 82 | expimpd 361 | . 2 ♯ ♯ |
84 | fsumf1o.2 | . . 3 | |
85 | fz1f1o 11265 | . . 3 ♯ ♯ | |
86 | 84, 85 | syl 14 | . 2 ♯ ♯ |
87 | 16, 83, 86 | mpjaod 708 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wo 698 wceq 1335 wex 1472 wcel 2128 wral 2435 c0 3394 cif 3505 class class class wbr 3965 cmpt 4025 ccom 4589 wf 5165 wfo 5167 wf1o 5168 cfv 5169 (class class class)co 5821 cfn 6682 cc 7724 cc0 7726 c1 7727 caddc 7729 cle 7907 cn 8827 cfz 9905 cseq 10337 ♯chash 10642 csu 11243 |
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 604 ax-in2 605 ax-io 699 ax-5 1427 ax-7 1428 ax-gen 1429 ax-ie1 1473 ax-ie2 1474 ax-8 1484 ax-10 1485 ax-11 1486 ax-i12 1487 ax-bndl 1489 ax-4 1490 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-13 2130 ax-14 2131 ax-ext 2139 ax-coll 4079 ax-sep 4082 ax-nul 4090 ax-pow 4135 ax-pr 4169 ax-un 4393 ax-setind 4495 ax-iinf 4546 ax-cnex 7817 ax-resscn 7818 ax-1cn 7819 ax-1re 7820 ax-icn 7821 ax-addcl 7822 ax-addrcl 7823 ax-mulcl 7824 ax-mulrcl 7825 ax-addcom 7826 ax-mulcom 7827 ax-addass 7828 ax-mulass 7829 ax-distr 7830 ax-i2m1 7831 ax-0lt1 7832 ax-1rid 7833 ax-0id 7834 ax-rnegex 7835 ax-precex 7836 ax-cnre 7837 ax-pre-ltirr 7838 ax-pre-ltwlin 7839 ax-pre-lttrn 7840 ax-pre-apti 7841 ax-pre-ltadd 7842 ax-pre-mulgt0 7843 ax-pre-mulext 7844 ax-arch 7845 ax-caucvg 7846 |
This theorem depends on definitions: df-bi 116 df-dc 821 df-3or 964 df-3an 965 df-tru 1338 df-fal 1341 df-nf 1441 df-sb 1743 df-eu 2009 df-mo 2010 df-clab 2144 df-cleq 2150 df-clel 2153 df-nfc 2288 df-ne 2328 df-nel 2423 df-ral 2440 df-rex 2441 df-reu 2442 df-rmo 2443 df-rab 2444 df-v 2714 df-sbc 2938 df-csb 3032 df-dif 3104 df-un 3106 df-in 3108 df-ss 3115 df-nul 3395 df-if 3506 df-pw 3545 df-sn 3566 df-pr 3567 df-op 3569 df-uni 3773 df-int 3808 df-iun 3851 df-br 3966 df-opab 4026 df-mpt 4027 df-tr 4063 df-id 4253 df-po 4256 df-iso 4257 df-iord 4326 df-on 4328 df-ilim 4329 df-suc 4331 df-iom 4549 df-xp 4591 df-rel 4592 df-cnv 4593 df-co 4594 df-dm 4595 df-rn 4596 df-res 4597 df-ima 4598 df-iota 5134 df-fun 5171 df-fn 5172 df-f 5173 df-f1 5174 df-fo 5175 df-f1o 5176 df-fv 5177 df-isom 5178 df-riota 5777 df-ov 5824 df-oprab 5825 df-mpo 5826 df-1st 6085 df-2nd 6086 df-recs 6249 df-irdg 6314 df-frec 6335 df-1o 6360 df-oadd 6364 df-er 6477 df-en 6683 df-dom 6684 df-fin 6685 df-pnf 7908 df-mnf 7909 df-xr 7910 df-ltxr 7911 df-le 7912 df-sub 8042 df-neg 8043 df-reap 8444 df-ap 8451 df-div 8540 df-inn 8828 df-2 8886 df-3 8887 df-4 8888 df-n0 9085 df-z 9162 df-uz 9434 df-q 9522 df-rp 9554 df-fz 9906 df-fzo 10035 df-seqfrec 10338 df-exp 10412 df-ihash 10643 df-cj 10735 df-re 10736 df-im 10737 df-rsqrt 10891 df-abs 10892 df-clim 11169 df-sumdc 11244 |
This theorem is referenced by: fisumss 11282 fsum2dlemstep 11324 fsumcnv 11327 fsumrev 11333 fsumshft 11334 phisum 12103 |
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