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Mirrors > Home > MPE Home > Th. List > iserabs | Structured version Visualization version GIF 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 | β’ π = (β€β₯βπ) |
iserabs.2 | β’ (π β seqπ( + , πΉ) β π΄) |
iserabs.3 | β’ (π β seqπ( + , πΊ) β π΅) |
iserabs.5 | β’ (π β π β β€) |
iserabs.6 | β’ ((π β§ π β π) β (πΉβπ) β β) |
iserabs.7 | β’ ((π β§ π β π) β (πΊβπ) = (absβ(πΉβπ))) |
Ref | Expression |
---|---|
iserabs | β’ (π β (absβπ΄) β€ π΅) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | iserabs.1 | . 2 β’ π = (β€β₯βπ) | |
2 | iserabs.5 | . 2 β’ (π β π β β€) | |
3 | iserabs.2 | . . 3 β’ (π β seqπ( + , πΉ) β π΄) | |
4 | 1 | fvexi 6916 | . . . . 5 β’ π β V |
5 | 4 | mptex 7241 | . . . 4 β’ (π β π β¦ (absβ(seqπ( + , πΉ)βπ))) β V |
6 | 5 | a1i 11 | . . 3 β’ (π β (π β π β¦ (absβ(seqπ( + , πΉ)βπ))) β V) |
7 | iserabs.6 | . . . . 5 β’ ((π β§ π β π) β (πΉβπ) β β) | |
8 | 1, 2, 7 | serf 14035 | . . . 4 β’ (π β seqπ( + , πΉ):πβΆβ) |
9 | 8 | ffvelcdmda 7099 | . . 3 β’ ((π β§ π β π) β (seqπ( + , πΉ)βπ) β β) |
10 | 2fveq3 6907 | . . . . 5 β’ (π = π β (absβ(seqπ( + , πΉ)βπ)) = (absβ(seqπ( + , πΉ)βπ))) | |
11 | eqid 2728 | . . . . 5 β’ (π β π β¦ (absβ(seqπ( + , πΉ)βπ))) = (π β π β¦ (absβ(seqπ( + , πΉ)βπ))) | |
12 | fvex 6915 | . . . . 5 β’ (absβ(seqπ( + , πΉ)βπ)) β V | |
13 | 10, 11, 12 | fvmpt 7010 | . . . 4 β’ (π β π β ((π β π β¦ (absβ(seqπ( + , πΉ)βπ)))βπ) = (absβ(seqπ( + , πΉ)βπ))) |
14 | 13 | adantl 480 | . . 3 β’ ((π β§ π β π) β ((π β π β¦ (absβ(seqπ( + , πΉ)βπ)))βπ) = (absβ(seqπ( + , πΉ)βπ))) |
15 | 1, 3, 6, 2, 9, 14 | climabs 15588 | . 2 β’ (π β (π β π β¦ (absβ(seqπ( + , πΉ)βπ))) β (absβπ΄)) |
16 | iserabs.3 | . 2 β’ (π β seqπ( + , πΊ) β π΅) | |
17 | 9 | abscld 15423 | . . 3 β’ ((π β§ π β π) β (absβ(seqπ( + , πΉ)βπ)) β β) |
18 | 14, 17 | eqeltrd 2829 | . 2 β’ ((π β§ π β π) β ((π β π β¦ (absβ(seqπ( + , πΉ)βπ)))βπ) β β) |
19 | iserabs.7 | . . . . 5 β’ ((π β§ π β π) β (πΊβπ) = (absβ(πΉβπ))) | |
20 | 7 | abscld 15423 | . . . . 5 β’ ((π β§ π β π) β (absβ(πΉβπ)) β β) |
21 | 19, 20 | eqeltrd 2829 | . . . 4 β’ ((π β§ π β π) β (πΊβπ) β β) |
22 | 1, 2, 21 | serfre 14036 | . . 3 β’ (π β seqπ( + , πΊ):πβΆβ) |
23 | 22 | ffvelcdmda 7099 | . 2 β’ ((π β§ π β π) β (seqπ( + , πΊ)βπ) β β) |
24 | simpr 483 | . . . . 5 β’ ((π β§ π β π) β π β π) | |
25 | 24, 1 | eleqtrdi 2839 | . . . 4 β’ ((π β§ π β π) β π β (β€β₯βπ)) |
26 | elfzuz 13537 | . . . . . . 7 β’ (π β (π...π) β π β (β€β₯βπ)) | |
27 | 26, 1 | eleqtrrdi 2840 | . . . . . 6 β’ (π β (π...π) β π β π) |
28 | 27, 7 | sylan2 591 | . . . . 5 β’ ((π β§ π β (π...π)) β (πΉβπ) β β) |
29 | 28 | adantlr 713 | . . . 4 β’ (((π β§ π β π) β§ π β (π...π)) β (πΉβπ) β β) |
30 | 27, 19 | sylan2 591 | . . . . 5 β’ ((π β§ π β (π...π)) β (πΊβπ) = (absβ(πΉβπ))) |
31 | 30 | adantlr 713 | . . . 4 β’ (((π β§ π β π) β§ π β (π...π)) β (πΊβπ) = (absβ(πΉβπ))) |
32 | 25, 29, 31 | seqabs 15800 | . . 3 β’ ((π β§ π β π) β (absβ(seqπ( + , πΉ)βπ)) β€ (seqπ( + , πΊ)βπ)) |
33 | 14, 32 | eqbrtrd 5174 | . 2 β’ ((π β§ π β π) β ((π β π β¦ (absβ(seqπ( + , πΉ)βπ)))βπ) β€ (seqπ( + , πΊ)βπ)) |
34 | 1, 2, 15, 16, 18, 23, 33 | climle 15624 | 1 β’ (π β (absβπ΄) β€ π΅) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ wa 394 = wceq 1533 β wcel 2098 Vcvv 3473 class class class wbr 5152 β¦ cmpt 5235 βcfv 6553 (class class class)co 7426 βcc 11144 βcr 11145 + caddc 11149 β€ cle 11287 β€cz 12596 β€β₯cuz 12860 ...cfz 13524 seqcseq 14006 abscabs 15221 β cli 15468 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2699 ax-rep 5289 ax-sep 5303 ax-nul 5310 ax-pow 5369 ax-pr 5433 ax-un 7746 ax-inf2 9672 ax-cnex 11202 ax-resscn 11203 ax-1cn 11204 ax-icn 11205 ax-addcl 11206 ax-addrcl 11207 ax-mulcl 11208 ax-mulrcl 11209 ax-mulcom 11210 ax-addass 11211 ax-mulass 11212 ax-distr 11213 ax-i2m1 11214 ax-1ne0 11215 ax-1rid 11216 ax-rnegex 11217 ax-rrecex 11218 ax-cnre 11219 ax-pre-lttri 11220 ax-pre-lttrn 11221 ax-pre-ltadd 11222 ax-pre-mulgt0 11223 ax-pre-sup 11224 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2529 df-eu 2558 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-rmo 3374 df-reu 3375 df-rab 3431 df-v 3475 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-pss 3968 df-nul 4327 df-if 4533 df-pw 4608 df-sn 4633 df-pr 4635 df-op 4639 df-uni 4913 df-int 4954 df-iun 5002 df-br 5153 df-opab 5215 df-mpt 5236 df-tr 5270 df-id 5580 df-eprel 5586 df-po 5594 df-so 5595 df-fr 5637 df-se 5638 df-we 5639 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-rn 5693 df-res 5694 df-ima 5695 df-pred 6310 df-ord 6377 df-on 6378 df-lim 6379 df-suc 6380 df-iota 6505 df-fun 6555 df-fn 6556 df-f 6557 df-f1 6558 df-fo 6559 df-f1o 6560 df-fv 6561 df-isom 6562 df-riota 7382 df-ov 7429 df-oprab 7430 df-mpo 7431 df-om 7877 df-1st 7999 df-2nd 8000 df-frecs 8293 df-wrecs 8324 df-recs 8398 df-rdg 8437 df-1o 8493 df-er 8731 df-pm 8854 df-en 8971 df-dom 8972 df-sdom 8973 df-fin 8974 df-sup 9473 df-inf 9474 df-oi 9541 df-card 9970 df-pnf 11288 df-mnf 11289 df-xr 11290 df-ltxr 11291 df-le 11292 df-sub 11484 df-neg 11485 df-div 11910 df-nn 12251 df-2 12313 df-3 12314 df-n0 12511 df-z 12597 df-uz 12861 df-rp 13015 df-fz 13525 df-fzo 13668 df-fl 13797 df-seq 14007 df-exp 14067 df-hash 14330 df-cj 15086 df-re 15087 df-im 15088 df-sqrt 15222 df-abs 15223 df-clim 15472 df-rlim 15473 df-sum 15673 |
This theorem is referenced by: eftlub 16093 abelthlem7 26395 |
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