Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE 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 Mario Carneiro, 27-May-2014.) |
Ref | Expression |
---|---|
iserabs.1 | |
iserabs.2 | |
iserabs.3 | |
iserabs.5 | |
iserabs.6 | |
iserabs.7 |
Ref | Expression |
---|---|
iserabs |
M
, ,Step | Hyp | Ref | Expression |
---|---|---|---|
1 | iserabs.1 | . 2 | |
2 | iserabs.5 | . 2 | |
3 | iserabs.2 | . . 3 | |
4 | fvex 5881 | . . . . . 6 | |
5 | 1, 4 | eqeltri 2541 | . . . . 5 |
6 | 5 | mptex 6143 | . . . 4 |
7 | 6 | a1i 11 | . . 3 |
8 | iserabs.6 | . . . . 5 | |
9 | 1, 2, 8 | serf 12135 | . . . 4 |
10 | 9 | ffvelrnda 6031 | . . 3 |
11 | fveq2 5871 | . . . . . 6 | |
12 | 11 | fveq2d 5875 | . . . . 5 |
13 | eqid 2457 | . . . . 5 | |
14 | fvex 5881 | . . . . 5 | |
15 | 12, 13, 14 | fvmpt 5956 | . . . 4 |
16 | 15 | adantl 466 | . . 3 |
17 | 1, 3, 7, 2, 10, 16 | climabs 13426 | . 2 |
18 | iserabs.3 | . 2 | |
19 | 10 | abscld 13267 | . . 3 |
20 | 16, 19 | eqeltrd 2545 | . 2 |
21 | iserabs.7 | . . . . 5 | |
22 | 8 | abscld 13267 | . . . . 5 |
23 | 21, 22 | eqeltrd 2545 | . . . 4 |
24 | 1, 2, 23 | serfre 12136 | . . 3 |
25 | 24 | ffvelrnda 6031 | . 2 |
26 | simpr 461 | . . . . 5 | |
27 | 26, 1 | syl6eleq 2555 | . . . 4 |
28 | elfzuz 11713 | . . . . . . 7 | |
29 | 28, 1 | syl6eleqr 2556 | . . . . . 6 |
30 | 29, 8 | sylan2 474 | . . . . 5 |
31 | 30 | adantlr 714 | . . . 4 |
32 | 29, 21 | sylan2 474 | . . . . 5 |
33 | 32 | adantlr 714 | . . . 4 |
34 | 27, 31, 33 | seqabs 13628 | . . 3 |
35 | 16, 34 | eqbrtrd 4472 | . 2 |
36 | 1, 2, 17, 18, 20, 25, 35 | climle 13462 | 1 |
Colors of variables: wff setvar class |
Syntax hints: -> wi 4 /\ wa 369
= wceq 1395 e. wcel 1818 cvv 3109
class class class wbr 4452 e. cmpt 4510
` cfv 5593 (class class class)co 6296
cc 9511 cr 9512 caddc 9516 cle 9650 cz 10889 cuz 11110
cfz 11701 seq cseq 12107 cabs 13067 cli 13307 |
This theorem is referenced by: eftlub 13844 abelthlem7 22833 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1618 ax-4 1631 ax-5 1704 ax-6 1747 ax-7 1790 ax-8 1820 ax-9 1822 ax-10 1837 ax-11 1842 ax-12 1854 ax-13 1999 ax-ext 2435 ax-rep 4563 ax-sep 4573 ax-nul 4581 ax-pow 4630 ax-pr 4691 ax-un 6592 ax-inf2 8079 ax-cnex 9569 ax-resscn 9570 ax-1cn 9571 ax-icn 9572 ax-addcl 9573 ax-addrcl 9574 ax-mulcl 9575 ax-mulrcl 9576 ax-mulcom 9577 ax-addass 9578 ax-mulass 9579 ax-distr 9580 ax-i2m1 9581 ax-1ne0 9582 ax-1rid 9583 ax-rnegex 9584 ax-rrecex 9585 ax-cnre 9586 ax-pre-lttri 9587 ax-pre-lttrn 9588 ax-pre-ltadd 9589 ax-pre-mulgt0 9590 ax-pre-sup 9591 |
This theorem depends on definitions: df-bi 185 df-or 370 df-an 371 df-3or 974 df-3an 975 df-tru 1398 df-fal 1401 df-ex 1613 df-nf 1617 df-sb 1740 df-eu 2286 df-mo 2287 df-clab 2443 df-cleq 2449 df-clel 2452 df-nfc 2607 df-ne 2654 df-nel 2655 df-ral 2812 df-rex 2813 df-reu 2814 df-rmo 2815 df-rab 2816 df-v 3111 df-sbc 3328 df-csb 3435 df-dif 3478 df-un 3480 df-in 3482 df-ss 3489 df-pss 3491 df-nul 3785 df-if 3942 df-pw 4014 df-sn 4030 df-pr 4032 df-tp 4034 df-op 4036 df-uni 4250 df-int 4287 df-iun 4332 df-br 4453 df-opab 4511 df-mpt 4512 df-tr 4546 df-eprel 4796 df-id 4800 df-po 4805 df-so 4806 df-fr 4843 df-se 4844 df-we 4845 df-ord 4886 df-on 4887 df-lim 4888 df-suc 4889 df-xp 5010 df-rel 5011 df-cnv 5012 df-co 5013 df-dm 5014 df-rn 5015 df-res 5016 df-ima 5017 df-iota 5556 df-fun 5595 df-fn 5596 df-f 5597 df-f1 5598 df-fo 5599 df-f1o 5600 df-fv 5601 df-isom 5602 df-riota 6257 df-ov 6299 df-oprab 6300 df-mpt2 6301 df-om 6701 df-1st 6800 df-2nd 6801 df-recs 7061 df-rdg 7095 df-1o 7149 df-oadd 7153 df-er 7330 df-pm 7442 df-en 7537 df-dom 7538 df-sdom 7539 df-fin 7540 df-sup 7921 df-oi 7956 df-card 8341 df-pnf 9651 df-mnf 9652 df-xr 9653 df-ltxr 9654 df-le 9655 df-sub 9830 df-neg 9831 df-div 10232 df-nn 10562 df-2 10619 df-3 10620 df-n0 10821 df-z 10890 df-uz 11111 df-rp 11250 df-fz 11702 df-fzo 11825 df-fl 11929 df-seq 12108 df-exp 12167 df-hash 12406 df-cj 12932 df-re 12933 df-im 12934 df-sqrt 13068 df-abs 13069 df-clim 13311 df-rlim 13312 df-sum 13509 |
Copyright terms: Public domain | W3C validator |