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Mirrors > Home > MPE Home > Th. List > fzosump1 | Structured version Unicode version |
Description: Separate out the last term in a finite sum. (Contributed by Mario Carneiro, 13-Apr-2016.) |
Ref | Expression |
---|---|
fsumm1.1 |
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fsumm1.2 |
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fsumm1.3 |
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Ref | Expression |
---|---|
fzosump1 |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fsumm1.1 |
. . . . . 6
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2 | eluzelz 10980 |
. . . . . 6
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3 | 1, 2 | syl 16 |
. . . . 5
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4 | fzoval 11670 |
. . . . 5
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5 | 3, 4 | syl 16 |
. . . 4
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6 | 5 | sumeq1d 13295 |
. . 3
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7 | 6 | oveq1d 6214 |
. 2
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8 | fsumm1.2 |
. . 3
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9 | fsumm1.3 |
. . 3
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10 | 1, 8, 9 | fsumm1 13337 |
. 2
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11 | fzval3 11721 |
. . . 4
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12 | 3, 11 | syl 16 |
. . 3
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13 | 12 | sumeq1d 13295 |
. 2
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14 | 7, 10, 13 | 3eqtr2rd 2502 |
1
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Colors of variables: wff setvar class |
Syntax hints: ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1592 ax-4 1603 ax-5 1671 ax-6 1710 ax-7 1730 ax-8 1760 ax-9 1762 ax-10 1777 ax-11 1782 ax-12 1794 ax-13 1955 ax-ext 2432 ax-rep 4510 ax-sep 4520 ax-nul 4528 ax-pow 4577 ax-pr 4638 ax-un 6481 ax-inf2 7957 ax-cnex 9448 ax-resscn 9449 ax-1cn 9450 ax-icn 9451 ax-addcl 9452 ax-addrcl 9453 ax-mulcl 9454 ax-mulrcl 9455 ax-mulcom 9456 ax-addass 9457 ax-mulass 9458 ax-distr 9459 ax-i2m1 9460 ax-1ne0 9461 ax-1rid 9462 ax-rnegex 9463 ax-rrecex 9464 ax-cnre 9465 ax-pre-lttri 9466 ax-pre-lttrn 9467 ax-pre-ltadd 9468 ax-pre-mulgt0 9469 ax-pre-sup 9470 |
This theorem depends on definitions: df-bi 185 df-or 370 df-an 371 df-3or 966 df-3an 967 df-tru 1373 df-fal 1376 df-ex 1588 df-nf 1591 df-sb 1703 df-eu 2266 df-mo 2267 df-clab 2440 df-cleq 2446 df-clel 2449 df-nfc 2604 df-ne 2649 df-nel 2650 df-ral 2803 df-rex 2804 df-reu 2805 df-rmo 2806 df-rab 2807 df-v 3078 df-sbc 3293 df-csb 3395 df-dif 3438 df-un 3440 df-in 3442 df-ss 3449 df-pss 3451 df-nul 3745 df-if 3899 df-pw 3969 df-sn 3985 df-pr 3987 df-tp 3989 df-op 3991 df-uni 4199 df-int 4236 df-iun 4280 df-br 4400 df-opab 4458 df-mpt 4459 df-tr 4493 df-eprel 4739 df-id 4743 df-po 4748 df-so 4749 df-fr 4786 df-se 4787 df-we 4788 df-ord 4829 df-on 4830 df-lim 4831 df-suc 4832 df-xp 4953 df-rel 4954 df-cnv 4955 df-co 4956 df-dm 4957 df-rn 4958 df-res 4959 df-ima 4960 df-iota 5488 df-fun 5527 df-fn 5528 df-f 5529 df-f1 5530 df-fo 5531 df-f1o 5532 df-fv 5533 df-isom 5534 df-riota 6160 df-ov 6202 df-oprab 6203 df-mpt2 6204 df-om 6586 df-1st 6686 df-2nd 6687 df-recs 6941 df-rdg 6975 df-1o 7029 df-oadd 7033 df-er 7210 df-en 7420 df-dom 7421 df-sdom 7422 df-fin 7423 df-sup 7801 df-oi 7834 df-card 8219 df-pnf 9530 df-mnf 9531 df-xr 9532 df-ltxr 9533 df-le 9534 df-sub 9707 df-neg 9708 df-div 10104 df-nn 10433 df-2 10490 df-3 10491 df-n0 10690 df-z 10757 df-uz 10972 df-rp 11102 df-fz 11554 df-fzo 11665 df-seq 11923 df-exp 11982 df-hash 12220 df-cj 12705 df-re 12706 df-im 12707 df-sqr 12841 df-abs 12842 df-clim 13083 df-sum 13281 |
This theorem is referenced by: dchrisumlem2 22871 signsvfn 27126 |
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