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Mathbox for Thierry Arnoux |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > esumdivc | Structured version Visualization version Unicode version |
Description: An extended sum divided by a constant. (Contributed by Thierry Arnoux, 6-Jul-2017.) |
Ref | Expression |
---|---|
esumdivc.a |
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esumdivc.b |
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esumdivc.c |
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Ref | Expression |
---|---|
esumdivc |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | esumdivc.a |
. . 3
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2 | esumdivc.b |
. . 3
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3 | 1red 9684 |
. . . . 5
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4 | esumdivc.c |
. . . . . 6
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5 | 4 | rpred 11370 |
. . . . 5
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6 | 4 | rpne0d 11375 |
. . . . 5
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7 | rexdiv 28444 |
. . . . 5
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8 | 3, 5, 6, 7 | syl3anc 1276 |
. . . 4
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9 | ioorp 11741 |
. . . . . 6
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10 | ioossico 11752 |
. . . . . 6
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11 | 9, 10 | eqsstr3i 3475 |
. . . . 5
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12 | 4 | rpreccld 11380 |
. . . . 5
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13 | 11, 12 | sseldi 3442 |
. . . 4
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14 | 8, 13 | eqeltrd 2540 |
. . 3
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15 | 1, 2, 14 | esummulc1 28951 |
. 2
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16 | iccssxr 11746 |
. . . 4
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17 | 2 | ralrimiva 2814 |
. . . . 5
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18 | nfcv 2603 |
. . . . . 6
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19 | 18 | esumcl 28900 |
. . . . 5
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20 | 1, 17, 19 | syl2anc 671 |
. . . 4
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21 | 16, 20 | sseldi 3442 |
. . 3
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22 | xdivrec 28445 |
. . 3
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23 | 21, 5, 6, 22 | syl3anc 1276 |
. 2
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24 | 16, 2 | sseldi 3442 |
. . . 4
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25 | 5 | adantr 471 |
. . . 4
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26 | 6 | adantr 471 |
. . . 4
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27 | xdivrec 28445 |
. . . 4
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
28 | 24, 25, 26, 27 | syl3anc 1276 |
. . 3
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
29 | 28 | esumeq2dv 28908 |
. 2
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
30 | 15, 23, 29 | 3eqtr4d 2506 |
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 1680 ax-4 1693 ax-5 1769 ax-6 1816 ax-7 1862 ax-8 1900 ax-9 1907 ax-10 1926 ax-11 1931 ax-12 1944 ax-13 2102 ax-ext 2442 ax-rep 4529 ax-sep 4539 ax-nul 4548 ax-pow 4595 ax-pr 4653 ax-un 6610 ax-cnex 9621 ax-resscn 9622 ax-1cn 9623 ax-icn 9624 ax-addcl 9625 ax-addrcl 9626 ax-mulcl 9627 ax-mulrcl 9628 ax-mulcom 9629 ax-addass 9630 ax-mulass 9631 ax-distr 9632 ax-i2m1 9633 ax-1ne0 9634 ax-1rid 9635 ax-rnegex 9636 ax-rrecex 9637 ax-cnre 9638 ax-pre-lttri 9639 ax-pre-lttrn 9640 ax-pre-ltadd 9641 ax-pre-mulgt0 9642 ax-pre-sup 9643 |
This theorem depends on definitions: df-bi 190 df-or 376 df-an 377 df-3or 992 df-3an 993 df-tru 1458 df-fal 1461 df-ex 1675 df-nf 1679 df-sb 1809 df-eu 2314 df-mo 2315 df-clab 2449 df-cleq 2455 df-clel 2458 df-nfc 2592 df-ne 2635 df-nel 2636 df-ral 2754 df-rex 2755 df-reu 2756 df-rmo 2757 df-rab 2758 df-v 3059 df-sbc 3280 df-csb 3376 df-dif 3419 df-un 3421 df-in 3423 df-ss 3430 df-pss 3432 df-nul 3744 df-if 3894 df-pw 3965 df-sn 3981 df-pr 3983 df-tp 3985 df-op 3987 df-uni 4213 df-int 4249 df-iun 4294 df-iin 4295 df-br 4417 df-opab 4476 df-mpt 4477 df-tr 4512 df-eprel 4764 df-id 4768 df-po 4774 df-so 4775 df-fr 4812 df-se 4813 df-we 4814 df-xp 4859 df-rel 4860 df-cnv 4861 df-co 4862 df-dm 4863 df-rn 4864 df-res 4865 df-ima 4866 df-pred 5399 df-ord 5445 df-on 5446 df-lim 5447 df-suc 5448 df-iota 5565 df-fun 5603 df-fn 5604 df-f 5605 df-f1 5606 df-fo 5607 df-f1o 5608 df-fv 5609 df-isom 5610 df-riota 6277 df-ov 6318 df-oprab 6319 df-mpt2 6320 df-of 6558 df-om 6720 df-1st 6820 df-2nd 6821 df-supp 6942 df-wrecs 7054 df-recs 7116 df-rdg 7154 df-1o 7208 df-oadd 7212 df-er 7389 df-map 7500 df-en 7596 df-dom 7597 df-sdom 7598 df-fin 7599 df-fsupp 7910 df-fi 7951 df-sup 7982 df-inf 7983 df-oi 8051 df-card 8399 df-pnf 9703 df-mnf 9704 df-xr 9705 df-ltxr 9706 df-le 9707 df-sub 9888 df-neg 9889 df-div 10298 df-nn 10638 df-2 10696 df-3 10697 df-4 10698 df-5 10699 df-6 10700 df-7 10701 df-8 10702 df-9 10703 df-10 10704 df-n0 10899 df-z 10967 df-dec 11081 df-uz 11189 df-q 11294 df-rp 11332 df-xneg 11438 df-xadd 11439 df-xmul 11440 df-ioo 11668 df-ioc 11669 df-ico 11670 df-icc 11671 df-fz 11814 df-fzo 11947 df-seq 12246 df-hash 12548 df-struct 15172 df-ndx 15173 df-slot 15174 df-base 15175 df-sets 15176 df-ress 15177 df-plusg 15252 df-mulr 15253 df-tset 15258 df-ple 15259 df-ds 15261 df-rest 15370 df-topn 15371 df-0g 15389 df-gsum 15390 df-topgen 15391 df-ordt 15448 df-xrs 15449 df-mre 15541 df-mrc 15542 df-acs 15544 df-ps 16495 df-tsr 16496 df-mgm 16537 df-sgrp 16576 df-mnd 16586 df-mhm 16631 df-submnd 16632 df-cntz 17020 df-cmn 17481 df-fbas 19016 df-fg 19017 df-top 19970 df-bases 19971 df-topon 19972 df-topsp 19973 df-ntr 20084 df-nei 20163 df-cn 20292 df-cnp 20293 df-haus 20380 df-fil 20910 df-fm 21002 df-flim 21003 df-flf 21004 df-tsms 21190 df-xdiv 28436 df-esum 28898 |
This theorem is referenced by: measdivcstOLD 29095 measdivcst 29096 |
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