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Mathbox for Glauco Siliprandi |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > climliminf | Structured version Visualization version GIF version |
Description: A sequence of real numbers converges if and only if it converges to its inferior limit. (Contributed by Glauco Siliprandi, 2-Jan-2022.) |
Ref | Expression |
---|---|
climliminf.1 | β’ (π β π β β€) |
climliminf.2 | β’ π = (β€β₯βπ) |
climliminf.3 | β’ (π β πΉ:πβΆβ) |
Ref | Expression |
---|---|
climliminf | β’ (π β (πΉ β dom β β πΉ β (lim infβπΉ))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | climliminf.1 | . . . . . 6 β’ (π β π β β€) | |
2 | climliminf.2 | . . . . . 6 β’ π = (β€β₯βπ) | |
3 | climliminf.3 | . . . . . 6 β’ (π β πΉ:πβΆβ) | |
4 | 1, 2, 3 | climlimsup 44555 | . . . . 5 β’ (π β (πΉ β dom β β πΉ β (lim supβπΉ))) |
5 | 4 | biimpd 228 | . . . 4 β’ (π β (πΉ β dom β β πΉ β (lim supβπΉ))) |
6 | 5 | imp 407 | . . 3 β’ ((π β§ πΉ β dom β ) β πΉ β (lim supβπΉ)) |
7 | 1 | adantr 481 | . . . 4 β’ ((π β§ πΉ β dom β ) β π β β€) |
8 | 3 | adantr 481 | . . . 4 β’ ((π β§ πΉ β dom β ) β πΉ:πβΆβ) |
9 | simpr 485 | . . . 4 β’ ((π β§ πΉ β dom β ) β πΉ β dom β ) | |
10 | 7, 2, 8, 9 | climliminflimsupd 44596 | . . 3 β’ ((π β§ πΉ β dom β ) β (lim infβπΉ) = (lim supβπΉ)) |
11 | 6, 10 | breqtrrd 5176 | . 2 β’ ((π β§ πΉ β dom β ) β πΉ β (lim infβπΉ)) |
12 | climrel 15438 | . . . 4 β’ Rel β | |
13 | 12 | releldmi 5947 | . . 3 β’ (πΉ β (lim infβπΉ) β πΉ β dom β ) |
14 | 13 | adantl 482 | . 2 β’ ((π β§ πΉ β (lim infβπΉ)) β πΉ β dom β ) |
15 | 11, 14 | impbida 799 | 1 β’ (π β (πΉ β dom β β πΉ β (lim infβπΉ))) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β wb 205 β§ wa 396 = wceq 1541 β wcel 2106 class class class wbr 5148 dom cdm 5676 βΆwf 6539 βcfv 6543 βcr 11111 β€cz 12560 β€β₯cuz 12824 lim supclsp 15416 β cli 15430 lim infclsi 44546 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2703 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7727 ax-cnex 11168 ax-resscn 11169 ax-1cn 11170 ax-icn 11171 ax-addcl 11172 ax-addrcl 11173 ax-mulcl 11174 ax-mulrcl 11175 ax-mulcom 11176 ax-addass 11177 ax-mulass 11178 ax-distr 11179 ax-i2m1 11180 ax-1ne0 11181 ax-1rid 11182 ax-rnegex 11183 ax-rrecex 11184 ax-cnre 11185 ax-pre-lttri 11186 ax-pre-lttrn 11187 ax-pre-ltadd 11188 ax-pre-mulgt0 11189 ax-pre-sup 11190 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3376 df-reu 3377 df-rab 3433 df-v 3476 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-pss 3967 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-iun 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5574 df-eprel 5580 df-po 5588 df-so 5589 df-fr 5631 df-we 5633 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-pred 6300 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-isom 6552 df-riota 7367 df-ov 7414 df-oprab 7415 df-mpo 7416 df-om 7858 df-1st 7977 df-2nd 7978 df-frecs 8268 df-wrecs 8299 df-recs 8373 df-rdg 8412 df-er 8705 df-pm 8825 df-en 8942 df-dom 8943 df-sdom 8944 df-sup 9439 df-inf 9440 df-pnf 11252 df-mnf 11253 df-xr 11254 df-ltxr 11255 df-le 11256 df-sub 11448 df-neg 11449 df-div 11874 df-nn 12215 df-2 12277 df-3 12278 df-n0 12475 df-z 12561 df-uz 12825 df-q 12935 df-rp 12977 df-xneg 13094 df-ico 13332 df-fl 13759 df-seq 13969 df-exp 14030 df-cj 15048 df-re 15049 df-im 15050 df-sqrt 15184 df-abs 15185 df-limsup 15417 df-clim 15434 df-rlim 15435 df-liminf 44547 |
This theorem is referenced by: climliminflimsup 44603 dmclimxlim 44646 |
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