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Mirrors > Home > ILE Home > Th. List > fsum2d | Unicode version |
Description: Write a double sum as a sum over a two-dimensional region. Note that is a function of . (Contributed by Mario Carneiro, 27-Apr-2014.) |
Ref | Expression |
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fsum2d.1 | |
fsum2d.2 | |
fsum2d.3 | |
fsum2d.4 |
Ref | Expression |
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fsum2d |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ssid 3117 | . 2 | |
2 | fsum2d.2 | . . 3 | |
3 | sseq1 3120 | . . . . . 6 | |
4 | sumeq1 11124 | . . . . . . 7 | |
5 | iuneq1 3826 | . . . . . . . 8 | |
6 | 5 | sumeq1d 11135 | . . . . . . 7 |
7 | 4, 6 | eqeq12d 2154 | . . . . . 6 |
8 | 3, 7 | imbi12d 233 | . . . . 5 |
9 | 8 | imbi2d 229 | . . . 4 |
10 | sseq1 3120 | . . . . . 6 | |
11 | sumeq1 11124 | . . . . . . 7 | |
12 | iuneq1 3826 | . . . . . . . 8 | |
13 | 12 | sumeq1d 11135 | . . . . . . 7 |
14 | 11, 13 | eqeq12d 2154 | . . . . . 6 |
15 | 10, 14 | imbi12d 233 | . . . . 5 |
16 | 15 | imbi2d 229 | . . . 4 |
17 | sseq1 3120 | . . . . . 6 | |
18 | sumeq1 11124 | . . . . . . 7 | |
19 | iuneq1 3826 | . . . . . . . 8 | |
20 | 19 | sumeq1d 11135 | . . . . . . 7 |
21 | 18, 20 | eqeq12d 2154 | . . . . . 6 |
22 | 17, 21 | imbi12d 233 | . . . . 5 |
23 | 22 | imbi2d 229 | . . . 4 |
24 | sseq1 3120 | . . . . . 6 | |
25 | sumeq1 11124 | . . . . . . 7 | |
26 | iuneq1 3826 | . . . . . . . 8 | |
27 | 26 | sumeq1d 11135 | . . . . . . 7 |
28 | 25, 27 | eqeq12d 2154 | . . . . . 6 |
29 | 24, 28 | imbi12d 233 | . . . . 5 |
30 | 29 | imbi2d 229 | . . . 4 |
31 | sum0 11157 | . . . . . 6 | |
32 | 0iun 3870 | . . . . . . 7 | |
33 | 32 | sumeq1i 11132 | . . . . . 6 |
34 | sum0 11157 | . . . . . 6 | |
35 | 31, 33, 34 | 3eqtr4ri 2171 | . . . . 5 |
36 | 35 | 2a1i 27 | . . . 4 |
37 | ssun1 3239 | . . . . . . . . 9 | |
38 | sstr 3105 | . . . . . . . . 9 | |
39 | 37, 38 | mpan 420 | . . . . . . . 8 |
40 | 39 | imim1i 60 | . . . . . . 7 |
41 | fsum2d.1 | . . . . . . . . . 10 | |
42 | 2 | ad2antrr 479 | . . . . . . . . . 10 |
43 | simpll 518 | . . . . . . . . . . 11 | |
44 | fsum2d.3 | . . . . . . . . . . 11 | |
45 | 43, 44 | sylan 281 | . . . . . . . . . 10 |
46 | fsum2d.4 | . . . . . . . . . . 11 | |
47 | 43, 46 | sylan 281 | . . . . . . . . . 10 |
48 | simplrr 525 | . . . . . . . . . 10 | |
49 | simpr 109 | . . . . . . . . . 10 | |
50 | simplrl 524 | . . . . . . . . . 10 | |
51 | biid 170 | . . . . . . . . . 10 | |
52 | 41, 42, 45, 47, 48, 49, 50, 51 | fsum2dlemstep 11203 | . . . . . . . . 9 |
53 | 52 | exp31 361 | . . . . . . . 8 |
54 | 53 | a2d 26 | . . . . . . 7 |
55 | 40, 54 | syl5 32 | . . . . . 6 |
56 | 55 | expcom 115 | . . . . 5 |
57 | 56 | a2d 26 | . . . 4 |
58 | 9, 16, 23, 30, 36, 57 | findcard2s 6784 | . . 3 |
59 | 2, 58 | mpcom 36 | . 2 |
60 | 1, 59 | mpi 15 | 1 |
Colors of variables: wff set class |
Syntax hints: wn 3 wi 4 wa 103 wceq 1331 wcel 1480 cun 3069 wss 3071 c0 3363 csn 3527 cop 3530 ciun 3813 cxp 4537 cfn 6634 cc 7618 cc0 7620 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-disj 3907 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-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: fsumxp 11205 fisumcom2 11207 |
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