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Mirrors > Home > MPE Home > Th. List > Mathboxes > eulerpartlemsv2 | Structured version Unicode version |
Description: Lemma for eulerpart 26901. Value of the sum of a finite partition
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Ref | Expression |
---|---|
eulerpartlems.r |
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eulerpartlems.s |
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Ref | Expression |
---|---|
eulerpartlemsv2 |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eulerpartlems.r |
. . 3
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2 | eulerpartlems.s |
. . 3
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3 | 1, 2 | eulerpartlemsv1 26875 |
. 2
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4 | cnvimass 5289 |
. . . 4
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5 | 1, 2 | eulerpartlemelr 26876 |
. . . . . 6
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6 | 5 | simpld 459 |
. . . . 5
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7 | fdm 5663 |
. . . . 5
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8 | 6, 7 | syl 16 |
. . . 4
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9 | 4, 8 | syl5sseq 3504 |
. . 3
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10 | 6 | adantr 465 |
. . . . . 6
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11 | 9 | sselda 3456 |
. . . . . 6
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12 | 10, 11 | ffvelrnd 5945 |
. . . . 5
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13 | 11 | nnnn0d 10739 |
. . . . 5
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14 | 12, 13 | nn0mulcld 10744 |
. . . 4
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15 | 14 | nn0cnd 10741 |
. . 3
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16 | simpr 461 |
. . . . . . . 8
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17 | 16 | eldifad 3440 |
. . . . . . 7
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18 | 16 | eldifbd 3441 |
. . . . . . . . 9
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19 | 6 | adantr 465 |
. . . . . . . . . 10
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20 | ffn 5659 |
. . . . . . . . . 10
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21 | elpreima 5924 |
. . . . . . . . . 10
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22 | 19, 20, 21 | 3syl 20 |
. . . . . . . . 9
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23 | 18, 22 | mtbid 300 |
. . . . . . . 8
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24 | imnan 422 |
. . . . . . . 8
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25 | 23, 24 | sylibr 212 |
. . . . . . 7
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26 | 17, 25 | mpd 15 |
. . . . . 6
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27 | 19, 17 | ffvelrnd 5945 |
. . . . . . 7
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28 | elnn0 10684 |
. . . . . . 7
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29 | 27, 28 | sylib 196 |
. . . . . 6
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30 | orel1 382 |
. . . . . 6
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31 | 26, 29, 30 | sylc 60 |
. . . . 5
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32 | 31 | oveq1d 6207 |
. . . 4
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33 | 17 | nncnd 10441 |
. . . . 5
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34 | 33 | mul02d 9670 |
. . . 4
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35 | 32, 34 | eqtrd 2492 |
. . 3
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36 | nnuz 10999 |
. . . . 5
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37 | 36 | eqimssi 3510 |
. . . 4
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38 | 37 | a1i 11 |
. . 3
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39 | 9, 15, 35, 38 | sumss 13305 |
. 2
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40 | 3, 39 | eqtr4d 2495 |
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 1952 ax-ext 2430 ax-rep 4503 ax-sep 4513 ax-nul 4521 ax-pow 4570 ax-pr 4631 ax-un 6474 ax-inf2 7950 ax-cnex 9441 ax-resscn 9442 ax-1cn 9443 ax-icn 9444 ax-addcl 9445 ax-addrcl 9446 ax-mulcl 9447 ax-mulrcl 9448 ax-mulcom 9449 ax-addass 9450 ax-mulass 9451 ax-distr 9452 ax-i2m1 9453 ax-1ne0 9454 ax-1rid 9455 ax-rnegex 9456 ax-rrecex 9457 ax-cnre 9458 ax-pre-lttri 9459 ax-pre-lttrn 9460 ax-pre-ltadd 9461 ax-pre-mulgt0 9462 |
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 2264 df-mo 2265 df-clab 2437 df-cleq 2443 df-clel 2446 df-nfc 2601 df-ne 2646 df-nel 2647 df-ral 2800 df-rex 2801 df-reu 2802 df-rmo 2803 df-rab 2804 df-v 3072 df-sbc 3287 df-csb 3389 df-dif 3431 df-un 3433 df-in 3435 df-ss 3442 df-pss 3444 df-nul 3738 df-if 3892 df-pw 3962 df-sn 3978 df-pr 3980 df-tp 3982 df-op 3984 df-uni 4192 df-int 4229 df-iun 4273 df-br 4393 df-opab 4451 df-mpt 4452 df-tr 4486 df-eprel 4732 df-id 4736 df-po 4741 df-so 4742 df-fr 4779 df-se 4780 df-we 4781 df-ord 4822 df-on 4823 df-lim 4824 df-suc 4825 df-xp 4946 df-rel 4947 df-cnv 4948 df-co 4949 df-dm 4950 df-rn 4951 df-res 4952 df-ima 4953 df-iota 5481 df-fun 5520 df-fn 5521 df-f 5522 df-f1 5523 df-fo 5524 df-f1o 5525 df-fv 5526 df-isom 5527 df-riota 6153 df-ov 6195 df-oprab 6196 df-mpt2 6197 df-om 6579 df-1st 6679 df-2nd 6680 df-recs 6934 df-rdg 6968 df-1o 7022 df-oadd 7026 df-er 7203 df-map 7318 df-en 7413 df-dom 7414 df-sdom 7415 df-fin 7416 df-oi 7827 df-card 8212 df-pnf 9523 df-mnf 9524 df-xr 9525 df-ltxr 9526 df-le 9527 df-sub 9700 df-neg 9701 df-div 10097 df-nn 10426 df-2 10483 df-n0 10683 df-z 10750 df-uz 10965 df-rp 11095 df-fz 11541 df-fzo 11652 df-seq 11910 df-exp 11969 df-hash 12207 df-cj 12692 df-re 12693 df-im 12694 df-sqr 12828 df-abs 12829 df-clim 13070 df-sum 13268 |
This theorem is referenced by: eulerpartlemsf 26878 eulerpartlemgs2 26899 |
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