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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 6898 | . . . . 5 β’ π β V |
5 | 4 | mptex 7219 | . . . 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 13998 | . . . 4 β’ (π β seqπ( + , πΉ):πβΆβ) |
9 | 8 | ffvelcdmda 7079 | . . 3 β’ ((π β§ π β π) β (seqπ( + , πΉ)βπ) β β) |
10 | 2fveq3 6889 | . . . . 5 β’ (π = π β (absβ(seqπ( + , πΉ)βπ)) = (absβ(seqπ( + , πΉ)βπ))) | |
11 | eqid 2726 | . . . . 5 β’ (π β π β¦ (absβ(seqπ( + , πΉ)βπ))) = (π β π β¦ (absβ(seqπ( + , πΉ)βπ))) | |
12 | fvex 6897 | . . . . 5 β’ (absβ(seqπ( + , πΉ)βπ)) β V | |
13 | 10, 11, 12 | fvmpt 6991 | . . . 4 β’ (π β π β ((π β π β¦ (absβ(seqπ( + , πΉ)βπ)))βπ) = (absβ(seqπ( + , πΉ)βπ))) |
14 | 13 | adantl 481 | . . 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 15551 | . 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 15386 | . . 3 β’ ((π β§ π β π) β (absβ(seqπ( + , πΉ)βπ)) β β) |
18 | 14, 17 | eqeltrd 2827 | . 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 15386 | . . . . 5 β’ ((π β§ π β π) β (absβ(πΉβπ)) β β) |
21 | 19, 20 | eqeltrd 2827 | . . . 4 β’ ((π β§ π β π) β (πΊβπ) β β) |
22 | 1, 2, 21 | serfre 13999 | . . 3 β’ (π β seqπ( + , πΊ):πβΆβ) |
23 | 22 | ffvelcdmda 7079 | . 2 β’ ((π β§ π β π) β (seqπ( + , πΊ)βπ) β β) |
24 | simpr 484 | . . . . 5 β’ ((π β§ π β π) β π β π) | |
25 | 24, 1 | eleqtrdi 2837 | . . . 4 β’ ((π β§ π β π) β π β (β€β₯βπ)) |
26 | elfzuz 13500 | . . . . . . 7 β’ (π β (π...π) β π β (β€β₯βπ)) | |
27 | 26, 1 | eleqtrrdi 2838 | . . . . . 6 β’ (π β (π...π) β π β π) |
28 | 27, 7 | sylan2 592 | . . . . 5 β’ ((π β§ π β (π...π)) β (πΉβπ) β β) |
29 | 28 | adantlr 712 | . . . 4 β’ (((π β§ π β π) β§ π β (π...π)) β (πΉβπ) β β) |
30 | 27, 19 | sylan2 592 | . . . . 5 β’ ((π β§ π β (π...π)) β (πΊβπ) = (absβ(πΉβπ))) |
31 | 30 | adantlr 712 | . . . 4 β’ (((π β§ π β π) β§ π β (π...π)) β (πΊβπ) = (absβ(πΉβπ))) |
32 | 25, 29, 31 | seqabs 15763 | . . 3 β’ ((π β§ π β π) β (absβ(seqπ( + , πΉ)βπ)) β€ (seqπ( + , πΊ)βπ)) |
33 | 14, 32 | eqbrtrd 5163 | . 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 15587 | 1 β’ (π β (absβπ΄) β€ π΅) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ wa 395 = wceq 1533 β wcel 2098 Vcvv 3468 class class class wbr 5141 β¦ cmpt 5224 βcfv 6536 (class class class)co 7404 βcc 11107 βcr 11108 + caddc 11112 β€ cle 11250 β€cz 12559 β€β₯cuz 12823 ...cfz 13487 seqcseq 13969 abscabs 15184 β cli 15431 |
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 2163 ax-ext 2697 ax-rep 5278 ax-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 ax-un 7721 ax-inf2 9635 ax-cnex 11165 ax-resscn 11166 ax-1cn 11167 ax-icn 11168 ax-addcl 11169 ax-addrcl 11170 ax-mulcl 11171 ax-mulrcl 11172 ax-mulcom 11173 ax-addass 11174 ax-mulass 11175 ax-distr 11176 ax-i2m1 11177 ax-1ne0 11178 ax-1rid 11179 ax-rnegex 11180 ax-rrecex 11181 ax-cnre 11182 ax-pre-lttri 11183 ax-pre-lttrn 11184 ax-pre-ltadd 11185 ax-pre-mulgt0 11186 ax-pre-sup 11187 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ne 2935 df-nel 3041 df-ral 3056 df-rex 3065 df-rmo 3370 df-reu 3371 df-rab 3427 df-v 3470 df-sbc 3773 df-csb 3889 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-pss 3962 df-nul 4318 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4903 df-int 4944 df-iun 4992 df-br 5142 df-opab 5204 df-mpt 5225 df-tr 5259 df-id 5567 df-eprel 5573 df-po 5581 df-so 5582 df-fr 5624 df-se 5625 df-we 5626 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-pred 6293 df-ord 6360 df-on 6361 df-lim 6362 df-suc 6363 df-iota 6488 df-fun 6538 df-fn 6539 df-f 6540 df-f1 6541 df-fo 6542 df-f1o 6543 df-fv 6544 df-isom 6545 df-riota 7360 df-ov 7407 df-oprab 7408 df-mpo 7409 df-om 7852 df-1st 7971 df-2nd 7972 df-frecs 8264 df-wrecs 8295 df-recs 8369 df-rdg 8408 df-1o 8464 df-er 8702 df-pm 8822 df-en 8939 df-dom 8940 df-sdom 8941 df-fin 8942 df-sup 9436 df-inf 9437 df-oi 9504 df-card 9933 df-pnf 11251 df-mnf 11252 df-xr 11253 df-ltxr 11254 df-le 11255 df-sub 11447 df-neg 11448 df-div 11873 df-nn 12214 df-2 12276 df-3 12277 df-n0 12474 df-z 12560 df-uz 12824 df-rp 12978 df-fz 13488 df-fzo 13631 df-fl 13760 df-seq 13970 df-exp 14030 df-hash 14293 df-cj 15049 df-re 15050 df-im 15051 df-sqrt 15185 df-abs 15186 df-clim 15435 df-rlim 15436 df-sum 15636 |
This theorem is referenced by: eftlub 16056 abelthlem7 26325 |
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