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Mathbox for Glauco Siliprandi |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > supcnvlimsupmpt | Structured version Visualization version GIF version |
Description: If a function on a set of upper integers has a real superior limit, the supremum of the rightmost parts of the function, converges to that superior limit. (Contributed by Glauco Siliprandi, 23-Oct-2021.) |
Ref | Expression |
---|---|
supcnvlimsupmpt.j | β’ β²ππ |
supcnvlimsupmpt.m | β’ (π β π β β€) |
supcnvlimsupmpt.z | β’ π = (β€β₯βπ) |
supcnvlimsupmpt.b | β’ ((π β§ π β π) β π΅ β β) |
supcnvlimsupmpt.r | β’ (π β (lim supβ(π β π β¦ π΅)) β β) |
Ref | Expression |
---|---|
supcnvlimsupmpt | β’ (π β (π β π β¦ sup(ran (π β (β€β₯βπ) β¦ π΅), β*, < )) β (lim supβ(π β π β¦ π΅))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fveq2 6878 | . . . . . . 7 β’ (π = π β (β€β₯βπ) = (β€β₯βπ)) | |
2 | 1 | mpteq1d 5236 | . . . . . 6 β’ (π = π β (π β (β€β₯βπ) β¦ π΅) = (π β (β€β₯βπ) β¦ π΅)) |
3 | 2 | rneqd 5929 | . . . . 5 β’ (π = π β ran (π β (β€β₯βπ) β¦ π΅) = ran (π β (β€β₯βπ) β¦ π΅)) |
4 | 3 | supeq1d 9423 | . . . 4 β’ (π = π β sup(ran (π β (β€β₯βπ) β¦ π΅), β*, < ) = sup(ran (π β (β€β₯βπ) β¦ π΅), β*, < )) |
5 | 4 | cbvmptv 5254 | . . 3 β’ (π β π β¦ sup(ran (π β (β€β₯βπ) β¦ π΅), β*, < )) = (π β π β¦ sup(ran (π β (β€β₯βπ) β¦ π΅), β*, < )) |
6 | supcnvlimsupmpt.z | . . . . . . . . . 10 β’ π = (β€β₯βπ) | |
7 | 6 | uzssd3 43909 | . . . . . . . . 9 β’ (π β π β (β€β₯βπ) β π) |
8 | 7 | adantl 482 | . . . . . . . 8 β’ ((π β§ π β π) β (β€β₯βπ) β π) |
9 | 8 | resmptd 6030 | . . . . . . 7 β’ ((π β§ π β π) β ((π β π β¦ π΅) βΎ (β€β₯βπ)) = (π β (β€β₯βπ) β¦ π΅)) |
10 | 9 | eqcomd 2737 | . . . . . 6 β’ ((π β§ π β π) β (π β (β€β₯βπ) β¦ π΅) = ((π β π β¦ π΅) βΎ (β€β₯βπ))) |
11 | 10 | rneqd 5929 | . . . . 5 β’ ((π β§ π β π) β ran (π β (β€β₯βπ) β¦ π΅) = ran ((π β π β¦ π΅) βΎ (β€β₯βπ))) |
12 | 11 | supeq1d 9423 | . . . 4 β’ ((π β§ π β π) β sup(ran (π β (β€β₯βπ) β¦ π΅), β*, < ) = sup(ran ((π β π β¦ π΅) βΎ (β€β₯βπ)), β*, < )) |
13 | 12 | mpteq2dva 5241 | . . 3 β’ (π β (π β π β¦ sup(ran (π β (β€β₯βπ) β¦ π΅), β*, < )) = (π β π β¦ sup(ran ((π β π β¦ π΅) βΎ (β€β₯βπ)), β*, < ))) |
14 | 5, 13 | eqtrid 2783 | . 2 β’ (π β (π β π β¦ sup(ran (π β (β€β₯βπ) β¦ π΅), β*, < )) = (π β π β¦ sup(ran ((π β π β¦ π΅) βΎ (β€β₯βπ)), β*, < ))) |
15 | supcnvlimsupmpt.m | . . 3 β’ (π β π β β€) | |
16 | supcnvlimsupmpt.j | . . . 4 β’ β²ππ | |
17 | supcnvlimsupmpt.b | . . . 4 β’ ((π β§ π β π) β π΅ β β) | |
18 | 16, 17 | fmptd2f 43709 | . . 3 β’ (π β (π β π β¦ π΅):πβΆβ) |
19 | supcnvlimsupmpt.r | . . 3 β’ (π β (lim supβ(π β π β¦ π΅)) β β) | |
20 | 15, 6, 18, 19 | supcnvlimsup 44229 | . 2 β’ (π β (π β π β¦ sup(ran ((π β π β¦ π΅) βΎ (β€β₯βπ)), β*, < )) β (lim supβ(π β π β¦ π΅))) |
21 | 14, 20 | eqbrtrd 5163 | 1 β’ (π β (π β π β¦ sup(ran (π β (β€β₯βπ) β¦ π΅), β*, < )) β (lim supβ(π β π β¦ π΅))) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ wa 396 = wceq 1541 β²wnf 1785 β wcel 2106 β wss 3944 class class class wbr 5141 β¦ cmpt 5224 ran crn 5670 βΎ cres 5671 βcfv 6532 supcsup 9417 βcr 11091 β*cxr 11229 < clt 11230 β€cz 12540 β€β₯cuz 12804 lim supclsp 15396 β cli 15410 |
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 2702 ax-rep 5278 ax-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 ax-un 7708 ax-cnex 11148 ax-resscn 11149 ax-1cn 11150 ax-icn 11151 ax-addcl 11152 ax-addrcl 11153 ax-mulcl 11154 ax-mulrcl 11155 ax-mulcom 11156 ax-addass 11157 ax-mulass 11158 ax-distr 11159 ax-i2m1 11160 ax-1ne0 11161 ax-1rid 11162 ax-rnegex 11163 ax-rrecex 11164 ax-cnre 11165 ax-pre-lttri 11166 ax-pre-lttrn 11167 ax-pre-ltadd 11168 ax-pre-mulgt0 11169 ax-pre-sup 11170 |
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 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ne 2940 df-nel 3046 df-ral 3061 df-rex 3070 df-rmo 3375 df-reu 3376 df-rab 3432 df-v 3475 df-sbc 3774 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-pss 3963 df-nul 4319 df-if 4523 df-pw 4598 df-sn 4623 df-pr 4625 df-op 4629 df-uni 4902 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-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 6289 df-ord 6356 df-on 6357 df-lim 6358 df-suc 6359 df-iota 6484 df-fun 6534 df-fn 6535 df-f 6536 df-f1 6537 df-fo 6538 df-f1o 6539 df-fv 6540 df-riota 7349 df-ov 7396 df-oprab 7397 df-mpo 7398 df-om 7839 df-1st 7957 df-2nd 7958 df-frecs 8248 df-wrecs 8279 df-recs 8353 df-rdg 8392 df-1o 8448 df-er 8686 df-en 8923 df-dom 8924 df-sdom 8925 df-fin 8926 df-sup 9419 df-inf 9420 df-pnf 11232 df-mnf 11233 df-xr 11234 df-ltxr 11235 df-le 11236 df-sub 11428 df-neg 11429 df-div 11854 df-nn 12195 df-2 12257 df-3 12258 df-n0 12455 df-z 12541 df-uz 12805 df-rp 12957 df-ico 13312 df-fz 13467 df-fl 13739 df-ceil 13740 df-seq 13949 df-exp 14010 df-cj 15028 df-re 15029 df-im 15030 df-sqrt 15164 df-abs 15165 df-limsup 15397 df-clim 15414 |
This theorem is referenced by: smflimsuplem5 45313 |
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