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Mirrors > Home > ILE Home > Th. List > iserabs | Unicode version |
Description: Generalized triangle inequality: the absolute value of an infinite sum is less than or equal to the sum of absolute values. (Contributed by Paul Chapman, 10-Sep-2007.) (Revised by Jim Kingdon, 14-Dec-2022.) |
Ref | Expression |
---|---|
iserabs.1 |
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iserabs.2 |
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iserabs.3 |
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iserabs.5 |
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iserabs.6 |
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iserabs.7 |
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Ref | Expression |
---|---|
iserabs |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | iserabs.1 |
. 2
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2 | iserabs.5 |
. 2
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3 | iserabs.2 |
. . 3
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4 | zex 9292 |
. . . . . . 7
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5 | uzssz 9577 |
. . . . . . 7
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6 | 4, 5 | ssexi 4156 |
. . . . . 6
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7 | 1, 6 | eqeltri 2262 |
. . . . 5
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8 | 7 | mptex 5763 |
. . . 4
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9 | 8 | a1i 9 |
. . 3
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10 | iserabs.6 |
. . . . 5
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11 | 1, 2, 10 | serf 10505 |
. . . 4
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12 | 11 | ffvelcdmda 5672 |
. . 3
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13 | simpr 110 |
. . . 4
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14 | 12 | abscld 11222 |
. . . 4
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15 | 2fveq3 5539 |
. . . . 5
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16 | eqid 2189 |
. . . . 5
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
17 | 15, 16 | fvmptg 5613 |
. . . 4
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18 | 13, 14, 17 | syl2anc 411 |
. . 3
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19 | 1, 3, 9, 2, 12, 18 | climabs 11360 |
. 2
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20 | iserabs.3 |
. 2
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21 | 18, 14 | eqeltrd 2266 |
. 2
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
22 | iserabs.7 |
. . . . 5
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
23 | 10 | abscld 11222 |
. . . . 5
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24 | 22, 23 | eqeltrd 2266 |
. . . 4
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25 | 1, 2, 24 | serfre 10506 |
. . 3
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26 | 25 | ffvelcdmda 5672 |
. 2
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27 | 2 | adantr 276 |
. . . . . 6
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28 | eluzelz 9567 |
. . . . . . . 8
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29 | 28, 1 | eleq2s 2284 |
. . . . . . 7
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30 | 29 | adantl 277 |
. . . . . 6
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31 | 27, 30 | fzfigd 10462 |
. . . . 5
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32 | elfzuz 10051 |
. . . . . . . 8
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33 | 32, 1 | eleqtrrdi 2283 |
. . . . . . 7
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34 | 33, 10 | sylan2 286 |
. . . . . 6
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35 | 34 | adantlr 477 |
. . . . 5
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36 | 31, 35 | fsumabs 11505 |
. . . 4
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
37 | eqidd 2190 |
. . . . . 6
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38 | 1 | eleq2i 2256 |
. . . . . . . 8
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39 | 38 | biimpi 120 |
. . . . . . 7
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40 | 39 | adantl 277 |
. . . . . 6
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41 | 1 | eleq2i 2256 |
. . . . . . . 8
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42 | 41, 10 | sylan2br 288 |
. . . . . . 7
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43 | 42 | adantlr 477 |
. . . . . 6
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44 | 37, 40, 43 | fsum3ser 11437 |
. . . . 5
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45 | 44 | fveq2d 5538 |
. . . 4
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46 | 22 | adantlr 477 |
. . . . . 6
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
47 | 41, 46 | sylan2br 288 |
. . . . 5
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48 | 23 | adantlr 477 |
. . . . . . 7
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49 | 41, 48 | sylan2br 288 |
. . . . . 6
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
50 | 49 | recnd 8016 |
. . . . 5
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51 | 47, 40, 50 | fsum3ser 11437 |
. . . 4
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
52 | 36, 45, 51 | 3brtr3d 4049 |
. . 3
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
53 | 18, 52 | eqbrtrd 4040 |
. 2
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
54 | 1, 2, 19, 20, 21, 26, 53 | climle 11374 |
1
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Colors of variables: wff set class |
Syntax hints: ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 615 ax-in2 616 ax-io 710 ax-5 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-13 2162 ax-14 2163 ax-ext 2171 ax-coll 4133 ax-sep 4136 ax-nul 4144 ax-pow 4192 ax-pr 4227 ax-un 4451 ax-setind 4554 ax-iinf 4605 ax-cnex 7932 ax-resscn 7933 ax-1cn 7934 ax-1re 7935 ax-icn 7936 ax-addcl 7937 ax-addrcl 7938 ax-mulcl 7939 ax-mulrcl 7940 ax-addcom 7941 ax-mulcom 7942 ax-addass 7943 ax-mulass 7944 ax-distr 7945 ax-i2m1 7946 ax-0lt1 7947 ax-1rid 7948 ax-0id 7949 ax-rnegex 7950 ax-precex 7951 ax-cnre 7952 ax-pre-ltirr 7953 ax-pre-ltwlin 7954 ax-pre-lttrn 7955 ax-pre-apti 7956 ax-pre-ltadd 7957 ax-pre-mulgt0 7958 ax-pre-mulext 7959 ax-arch 7960 ax-caucvg 7961 |
This theorem depends on definitions: df-bi 117 df-dc 836 df-3or 981 df-3an 982 df-tru 1367 df-fal 1370 df-nf 1472 df-sb 1774 df-eu 2041 df-mo 2042 df-clab 2176 df-cleq 2182 df-clel 2185 df-nfc 2321 df-ne 2361 df-nel 2456 df-ral 2473 df-rex 2474 df-reu 2475 df-rmo 2476 df-rab 2477 df-v 2754 df-sbc 2978 df-csb 3073 df-dif 3146 df-un 3148 df-in 3150 df-ss 3157 df-nul 3438 df-if 3550 df-pw 3592 df-sn 3613 df-pr 3614 df-op 3616 df-uni 3825 df-int 3860 df-iun 3903 df-br 4019 df-opab 4080 df-mpt 4081 df-tr 4117 df-id 4311 df-po 4314 df-iso 4315 df-iord 4384 df-on 4386 df-ilim 4387 df-suc 4389 df-iom 4608 df-xp 4650 df-rel 4651 df-cnv 4652 df-co 4653 df-dm 4654 df-rn 4655 df-res 4656 df-ima 4657 df-iota 5196 df-fun 5237 df-fn 5238 df-f 5239 df-f1 5240 df-fo 5241 df-f1o 5242 df-fv 5243 df-isom 5244 df-riota 5852 df-ov 5899 df-oprab 5900 df-mpo 5901 df-1st 6165 df-2nd 6166 df-recs 6330 df-irdg 6395 df-frec 6416 df-1o 6441 df-oadd 6445 df-er 6559 df-en 6767 df-dom 6768 df-fin 6769 df-pnf 8024 df-mnf 8025 df-xr 8026 df-ltxr 8027 df-le 8028 df-sub 8160 df-neg 8161 df-reap 8562 df-ap 8569 df-div 8660 df-inn 8950 df-2 9008 df-3 9009 df-4 9010 df-n0 9207 df-z 9284 df-uz 9559 df-q 9650 df-rp 9684 df-fz 10039 df-fzo 10173 df-seqfrec 10477 df-exp 10551 df-ihash 10788 df-cj 10883 df-re 10884 df-im 10885 df-rsqrt 11039 df-abs 11040 df-clim 11319 df-sumdc 11394 |
This theorem is referenced by: eftlub 11730 |
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