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Mathbox for Thierry Arnoux |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > esumpr2 | Structured version Unicode version |
Description: Extended sum over a pair, with a relaxed condition compared to esumpr 26654. (Contributed by Thierry Arnoux, 2-Jan-2017.) |
Ref | Expression |
---|---|
esumpr.1 |
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esumpr.2 |
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esumpr.3 |
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esumpr.4 |
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esumpr.5 |
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esumpr.6 |
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esumpr2.1 |
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Ref | Expression |
---|---|
esumpr2 |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simpr 461 |
. . . . 5
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2 | dfsn2 3991 |
. . . . . 6
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3 | preq2 4056 |
. . . . . 6
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
4 | 2, 3 | syl5req 2505 |
. . . . 5
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5 | esumeq1 26628 |
. . . . 5
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6 | 1, 4, 5 | 3syl 20 |
. . . 4
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7 | esumpr.1 |
. . . . . 6
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8 | esumpr.3 |
. . . . . 6
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9 | esumpr.5 |
. . . . . 6
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10 | 7, 8, 9 | esumsn 26653 |
. . . . 5
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11 | 10 | adantr 465 |
. . . 4
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12 | 6, 11 | eqtrd 2492 |
. . 3
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13 | esumpr2.1 |
. . . . 5
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14 | oveq2 6201 |
. . . . . . 7
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
15 | 0xr 9534 |
. . . . . . . . 9
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16 | eleq1 2523 |
. . . . . . . . 9
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17 | 15, 16 | mpbiri 233 |
. . . . . . . 8
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18 | xaddid1 11313 |
. . . . . . . 8
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19 | 17, 18 | syl 16 |
. . . . . . 7
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20 | 14, 19 | eqtrd 2492 |
. . . . . 6
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
21 | pnfxr 11196 |
. . . . . . . . 9
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22 | eleq1 2523 |
. . . . . . . . 9
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23 | 21, 22 | mpbiri 233 |
. . . . . . . 8
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
24 | pnfnemnf 11201 |
. . . . . . . . 9
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25 | neeq1 2729 |
. . . . . . . . 9
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26 | 24, 25 | mpbiri 233 |
. . . . . . . 8
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27 | xaddpnf1 11300 |
. . . . . . . 8
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
28 | 23, 26, 27 | syl2anc 661 |
. . . . . . 7
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29 | oveq2 6201 |
. . . . . . 7
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30 | id 22 |
. . . . . . 7
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31 | 28, 29, 30 | 3eqtr4d 2502 |
. . . . . 6
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32 | 20, 31 | jaoi 379 |
. . . . 5
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33 | 13, 32 | syl6 33 |
. . . 4
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34 | 33 | imp 429 |
. . 3
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35 | simpll 753 |
. . . . . . 7
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36 | eqeq2 2466 |
. . . . . . . . . 10
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
37 | 36 | biimprd 223 |
. . . . . . . . 9
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38 | 1, 37 | syl 16 |
. . . . . . . 8
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39 | 38 | imp 429 |
. . . . . . 7
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
40 | 35, 39, 7 | syl2anc 661 |
. . . . . 6
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41 | esumpr.4 |
. . . . . . 7
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42 | 41 | adantr 465 |
. . . . . 6
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43 | 9 | adantr 465 |
. . . . . 6
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44 | 40, 42, 43 | esumsn 26653 |
. . . . 5
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45 | esumpr.2 |
. . . . . . 7
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46 | esumpr.6 |
. . . . . . 7
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47 | 45, 41, 46 | esumsn 26653 |
. . . . . 6
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48 | 47 | adantr 465 |
. . . . 5
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
49 | 44, 48 | eqtr3d 2494 |
. . . 4
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50 | 49 | oveq2d 6209 |
. . 3
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51 | 12, 34, 50 | 3eqtr2d 2498 |
. 2
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52 | 7 | adantlr 714 |
. . 3
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
53 | 45 | adantlr 714 |
. . 3
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54 | 8 | adantr 465 |
. . 3
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55 | 41 | adantr 465 |
. . 3
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56 | 9 | adantr 465 |
. . 3
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
57 | 46 | adantr 465 |
. . 3
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
58 | simpr 461 |
. . 3
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
59 | 52, 53, 54, 55, 56, 57, 58 | esumpr 26654 |
. 2
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
60 | 51, 59 | pm2.61dane 2766 |
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 4504 ax-sep 4514 ax-nul 4522 ax-pow 4571 ax-pr 4632 ax-un 6475 ax-inf2 7951 ax-cnex 9442 ax-resscn 9443 ax-1cn 9444 ax-icn 9445 ax-addcl 9446 ax-addrcl 9447 ax-mulcl 9448 ax-mulrcl 9449 ax-mulcom 9450 ax-addass 9451 ax-mulass 9452 ax-distr 9453 ax-i2m1 9454 ax-1ne0 9455 ax-1rid 9456 ax-rnegex 9457 ax-rrecex 9458 ax-cnre 9459 ax-pre-lttri 9460 ax-pre-lttrn 9461 ax-pre-ltadd 9462 ax-pre-mulgt0 9463 ax-pre-sup 9464 ax-addf 9465 ax-mulf 9466 |
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 3073 df-sbc 3288 df-csb 3390 df-dif 3432 df-un 3434 df-in 3436 df-ss 3443 df-pss 3445 df-nul 3739 df-if 3893 df-pw 3963 df-sn 3979 df-pr 3981 df-tp 3983 df-op 3985 df-uni 4193 df-int 4230 df-iun 4274 df-iin 4275 df-br 4394 df-opab 4452 df-mpt 4453 df-tr 4487 df-eprel 4733 df-id 4737 df-po 4742 df-so 4743 df-fr 4780 df-se 4781 df-we 4782 df-ord 4823 df-on 4824 df-lim 4825 df-suc 4826 df-xp 4947 df-rel 4948 df-cnv 4949 df-co 4950 df-dm 4951 df-rn 4952 df-res 4953 df-ima 4954 df-iota 5482 df-fun 5521 df-fn 5522 df-f 5523 df-f1 5524 df-fo 5525 df-f1o 5526 df-fv 5527 df-isom 5528 df-riota 6154 df-ov 6196 df-oprab 6197 df-mpt2 6198 df-of 6423 df-om 6580 df-1st 6680 df-2nd 6681 df-supp 6794 df-recs 6935 df-rdg 6969 df-1o 7023 df-2o 7024 df-oadd 7027 df-er 7204 df-map 7319 df-pm 7320 df-ixp 7367 df-en 7414 df-dom 7415 df-sdom 7416 df-fin 7417 df-fsupp 7725 df-fi 7765 df-sup 7795 df-oi 7828 df-card 8213 df-cda 8441 df-pnf 9524 df-mnf 9525 df-xr 9526 df-ltxr 9527 df-le 9528 df-sub 9701 df-neg 9702 df-div 10098 df-nn 10427 df-2 10484 df-3 10485 df-4 10486 df-5 10487 df-6 10488 df-7 10489 df-8 10490 df-9 10491 df-10 10492 df-n0 10684 df-z 10751 df-dec 10860 df-uz 10966 df-q 11058 df-rp 11096 df-xneg 11193 df-xadd 11194 df-xmul 11195 df-ioo 11408 df-ioc 11409 df-ico 11410 df-icc 11411 df-fz 11548 df-fzo 11659 df-fl 11752 df-mod 11819 df-seq 11917 df-exp 11976 df-fac 12162 df-bc 12189 df-hash 12214 df-shft 12667 df-cj 12699 df-re 12700 df-im 12701 df-sqr 12835 df-abs 12836 df-limsup 13060 df-clim 13077 df-rlim 13078 df-sum 13275 df-ef 13464 df-sin 13466 df-cos 13467 df-pi 13469 df-struct 14287 df-ndx 14288 df-slot 14289 df-base 14290 df-sets 14291 df-ress 14292 df-plusg 14362 df-mulr 14363 df-starv 14364 df-sca 14365 df-vsca 14366 df-ip 14367 df-tset 14368 df-ple 14369 df-ds 14371 df-unif 14372 df-hom 14373 df-cco 14374 df-rest 14472 df-topn 14473 df-0g 14491 df-gsum 14492 df-topgen 14493 df-pt 14494 df-prds 14497 df-ordt 14550 df-xrs 14551 df-qtop 14556 df-imas 14557 df-xps 14559 df-mre 14635 df-mrc 14636 df-acs 14638 df-ps 15481 df-tsr 15482 df-mnd 15526 df-plusf 15527 df-mhm 15575 df-submnd 15576 df-grp 15656 df-minusg 15657 df-sbg 15658 df-mulg 15659 df-subg 15789 df-cntz 15946 df-cmn 16392 df-abl 16393 df-mgp 16706 df-ur 16718 df-rng 16762 df-cring 16763 df-subrg 16978 df-abv 17017 df-lmod 17065 df-scaf 17066 df-sra 17368 df-rgmod 17369 df-psmet 17927 df-xmet 17928 df-met 17929 df-bl 17930 df-mopn 17931 df-fbas 17932 df-fg 17933 df-cnfld 17937 df-top 18628 df-bases 18630 df-topon 18631 df-topsp 18632 df-cld 18748 df-ntr 18749 df-cls 18750 df-nei 18827 df-lp 18865 df-perf 18866 df-cn 18956 df-cnp 18957 df-haus 19044 df-tx 19260 df-hmeo 19453 df-fil 19544 df-fm 19636 df-flim 19637 df-flf 19638 df-tmd 19768 df-tgp 19769 df-tsms 19822 df-trg 19859 df-xms 20020 df-ms 20021 df-tms 20022 df-nm 20300 df-ngp 20301 df-nrg 20303 df-nlm 20304 df-ii 20578 df-cncf 20579 df-limc 21467 df-dv 21468 df-log 22134 df-esum 26622 |
This theorem is referenced by: measxun2 26762 measssd 26767 |
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