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Mathbox for Alexander van der Vekens |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > fsumsplitsndif | Structured version Visualization version Unicode version |
Description: Separate out a term in a finite sum by splitting the sum into two parts. (Contributed by Alexander van der Vekens, 31-Aug-2018.) |
Ref | Expression |
---|---|
fsumsplitsndif |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | neldifsnd 4069 |
. . . . 5
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2 | disjsn 4000 |
. . . . 5
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3 | 1, 2 | sylibr 217 |
. . . 4
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4 | uncom 3546 |
. . . . 5
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
5 | simp2 1010 |
. . . . . . 7
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
6 | 5 | snssd 4086 |
. . . . . 6
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7 | undif 3816 |
. . . . . 6
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8 | 6, 7 | sylib 201 |
. . . . 5
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9 | 4, 8 | syl5req 2499 |
. . . 4
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10 | simp1 1009 |
. . . 4
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11 | rspcsbela 3763 |
. . . . . . . 8
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
12 | 11 | zcnd 11031 |
. . . . . . 7
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13 | 12 | expcom 441 |
. . . . . 6
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14 | 13 | 3ad2ant3 1032 |
. . . . 5
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15 | 14 | imp 435 |
. . . 4
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16 | 3, 9, 10, 15 | fsumsplit 13817 |
. . 3
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17 | nfcv 2593 |
. . . 4
![]() ![]() ![]() ![]() | |
18 | nfcsb1v 3347 |
. . . 4
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
19 | csbeq1a 3340 |
. . . 4
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20 | 17, 18, 19 | cbvsumi 13774 |
. . 3
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21 | 17, 18, 19 | cbvsumi 13774 |
. . . 4
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22 | 17, 18, 19 | cbvsumi 13774 |
. . . 4
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23 | 21, 22 | oveq12i 6288 |
. . 3
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24 | 16, 20, 23 | 3eqtr4g 2511 |
. 2
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25 | rspcsbela 3763 |
. . . . . 6
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
26 | 25 | zcnd 11031 |
. . . . 5
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27 | 26 | 3adant1 1027 |
. . . 4
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28 | sumsns 13822 |
. . . 4
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29 | 5, 27, 28 | syl2anc 671 |
. . 3
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30 | 29 | oveq2d 6292 |
. 2
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31 | 24, 30 | eqtrd 2486 |
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 1673 ax-4 1686 ax-5 1762 ax-6 1809 ax-7 1855 ax-8 1893 ax-9 1900 ax-10 1919 ax-11 1924 ax-12 1937 ax-13 2092 ax-ext 2432 ax-rep 4487 ax-sep 4497 ax-nul 4506 ax-pow 4554 ax-pr 4612 ax-un 6571 ax-inf2 8133 ax-cnex 9582 ax-resscn 9583 ax-1cn 9584 ax-icn 9585 ax-addcl 9586 ax-addrcl 9587 ax-mulcl 9588 ax-mulrcl 9589 ax-mulcom 9590 ax-addass 9591 ax-mulass 9592 ax-distr 9593 ax-i2m1 9594 ax-1ne0 9595 ax-1rid 9596 ax-rnegex 9597 ax-rrecex 9598 ax-cnre 9599 ax-pre-lttri 9600 ax-pre-lttrn 9601 ax-pre-ltadd 9602 ax-pre-mulgt0 9603 ax-pre-sup 9604 |
This theorem depends on definitions: df-bi 190 df-or 376 df-an 377 df-3or 987 df-3an 988 df-tru 1451 df-fal 1454 df-ex 1668 df-nf 1672 df-sb 1802 df-eu 2304 df-mo 2305 df-clab 2439 df-cleq 2445 df-clel 2448 df-nfc 2582 df-ne 2624 df-nel 2625 df-ral 2742 df-rex 2743 df-reu 2744 df-rmo 2745 df-rab 2746 df-v 3015 df-sbc 3236 df-csb 3332 df-dif 3375 df-un 3377 df-in 3379 df-ss 3386 df-pss 3388 df-nul 3700 df-if 3850 df-pw 3921 df-sn 3937 df-pr 3939 df-tp 3941 df-op 3943 df-uni 4169 df-int 4205 df-iun 4250 df-br 4375 df-opab 4434 df-mpt 4435 df-tr 4470 df-eprel 4723 df-id 4727 df-po 4733 df-so 4734 df-fr 4771 df-se 4772 df-we 4773 df-xp 4818 df-rel 4819 df-cnv 4820 df-co 4821 df-dm 4822 df-rn 4823 df-res 4824 df-ima 4825 df-pred 5359 df-ord 5405 df-on 5406 df-lim 5407 df-suc 5408 df-iota 5525 df-fun 5563 df-fn 5564 df-f 5565 df-f1 5566 df-fo 5567 df-f1o 5568 df-fv 5569 df-isom 5570 df-riota 6238 df-ov 6279 df-oprab 6280 df-mpt2 6281 df-om 6681 df-1st 6781 df-2nd 6782 df-wrecs 7015 df-recs 7077 df-rdg 7115 df-1o 7169 df-oadd 7173 df-er 7350 df-en 7557 df-dom 7558 df-sdom 7559 df-fin 7560 df-sup 7943 df-oi 8012 df-card 8360 df-pnf 9664 df-mnf 9665 df-xr 9666 df-ltxr 9667 df-le 9668 df-sub 9849 df-neg 9850 df-div 10259 df-nn 10599 df-2 10657 df-3 10658 df-n0 10860 df-z 10928 df-uz 11150 df-rp 11293 df-fz 11776 df-fzo 11909 df-seq 12208 df-exp 12267 df-hash 12510 df-cj 13173 df-re 13174 df-im 13175 df-sqrt 13309 df-abs 13310 df-clim 13563 df-sum 13764 |
This theorem is referenced by: (None) |
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