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Mathbox for Glauco Siliprandi |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > limsupvaluzmpt | Structured version Visualization version GIF version |
Description: The superior limit, when the domain of the function is a set of upper integers (the first condition is needed, otherwise the l.h.s. would be -β and the r.h.s. would be +β). (Contributed by Glauco Siliprandi, 23-Oct-2021.) |
Ref | Expression |
---|---|
limsupvaluzmpt.j | β’ β²ππ |
limsupvaluzmpt.m | β’ (π β π β β€) |
limsupvaluzmpt.z | β’ π = (β€β₯βπ) |
limsupvaluzmpt.b | β’ ((π β§ π β π) β π΅ β β*) |
Ref | Expression |
---|---|
limsupvaluzmpt | β’ (π β (lim supβ(π β π β¦ π΅)) = inf(ran (π β π β¦ sup(ran (π β (β€β₯βπ) β¦ π΅), β*, < )), β*, < )) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | limsupvaluzmpt.m | . . 3 β’ (π β π β β€) | |
2 | limsupvaluzmpt.z | . . 3 β’ π = (β€β₯βπ) | |
3 | limsupvaluzmpt.j | . . . 4 β’ β²ππ | |
4 | limsupvaluzmpt.b | . . . 4 β’ ((π β§ π β π) β π΅ β β*) | |
5 | 3, 4 | fmptd2f 43614 | . . 3 β’ (π β (π β π β¦ π΅):πβΆβ*) |
6 | 1, 2, 5 | limsupvaluz 44102 | . 2 β’ (π β (lim supβ(π β π β¦ π΅)) = inf(ran (π β π β¦ sup(ran ((π β π β¦ π΅) βΎ (β€β₯βπ)), β*, < )), β*, < )) |
7 | 2 | uzssd3 43814 | . . . . . . . . 9 β’ (π β π β (β€β₯βπ) β π) |
8 | 7 | resmptd 6014 | . . . . . . . 8 β’ (π β π β ((π β π β¦ π΅) βΎ (β€β₯βπ)) = (π β (β€β₯βπ) β¦ π΅)) |
9 | 8 | rneqd 5913 | . . . . . . 7 β’ (π β π β ran ((π β π β¦ π΅) βΎ (β€β₯βπ)) = ran (π β (β€β₯βπ) β¦ π΅)) |
10 | 9 | supeq1d 9406 | . . . . . 6 β’ (π β π β sup(ran ((π β π β¦ π΅) βΎ (β€β₯βπ)), β*, < ) = sup(ran (π β (β€β₯βπ) β¦ π΅), β*, < )) |
11 | 10 | mpteq2ia 5228 | . . . . 5 β’ (π β π β¦ sup(ran ((π β π β¦ π΅) βΎ (β€β₯βπ)), β*, < )) = (π β π β¦ sup(ran (π β (β€β₯βπ) β¦ π΅), β*, < )) |
12 | 11 | a1i 11 | . . . 4 β’ (π β (π β π β¦ sup(ran ((π β π β¦ π΅) βΎ (β€β₯βπ)), β*, < )) = (π β π β¦ sup(ran (π β (β€β₯βπ) β¦ π΅), β*, < ))) |
13 | 12 | rneqd 5913 | . . 3 β’ (π β ran (π β π β¦ sup(ran ((π β π β¦ π΅) βΎ (β€β₯βπ)), β*, < )) = ran (π β π β¦ sup(ran (π β (β€β₯βπ) β¦ π΅), β*, < ))) |
14 | 13 | infeq1d 9437 | . 2 β’ (π β inf(ran (π β π β¦ sup(ran ((π β π β¦ π΅) βΎ (β€β₯βπ)), β*, < )), β*, < ) = inf(ran (π β π β¦ sup(ran (π β (β€β₯βπ) β¦ π΅), β*, < )), β*, < )) |
15 | 6, 14 | eqtrd 2771 | 1 β’ (π β (lim supβ(π β π β¦ π΅)) = inf(ran (π β π β¦ sup(ran (π β (β€β₯βπ) β¦ π΅), β*, < )), β*, < )) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ wa 396 = wceq 1541 β²wnf 1785 β wcel 2106 β¦ cmpt 5208 ran crn 5654 βΎ cres 5655 βcfv 6516 supcsup 9400 infcinf 9401 β*cxr 11212 < clt 11213 β€cz 12523 β€β₯cuz 12787 lim supclsp 15379 |
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 5262 ax-sep 5276 ax-nul 5283 ax-pow 5340 ax-pr 5404 ax-un 7692 ax-cnex 11131 ax-resscn 11132 ax-1cn 11133 ax-icn 11134 ax-addcl 11135 ax-addrcl 11136 ax-mulcl 11137 ax-mulrcl 11138 ax-mulcom 11139 ax-addass 11140 ax-mulass 11141 ax-distr 11142 ax-i2m1 11143 ax-1ne0 11144 ax-1rid 11145 ax-rnegex 11146 ax-rrecex 11147 ax-cnre 11148 ax-pre-lttri 11149 ax-pre-lttrn 11150 ax-pre-ltadd 11151 ax-pre-mulgt0 11152 ax-pre-sup 11153 |
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 3364 df-reu 3365 df-rab 3419 df-v 3461 df-sbc 3758 df-csb 3874 df-dif 3931 df-un 3933 df-in 3935 df-ss 3945 df-pss 3947 df-nul 4303 df-if 4507 df-pw 4582 df-sn 4607 df-pr 4609 df-op 4613 df-uni 4886 df-iun 4976 df-br 5126 df-opab 5188 df-mpt 5209 df-tr 5243 df-id 5551 df-eprel 5557 df-po 5565 df-so 5566 df-fr 5608 df-we 5610 df-xp 5659 df-rel 5660 df-cnv 5661 df-co 5662 df-dm 5663 df-rn 5664 df-res 5665 df-ima 5666 df-pred 6273 df-ord 6340 df-on 6341 df-lim 6342 df-suc 6343 df-iota 6468 df-fun 6518 df-fn 6519 df-f 6520 df-f1 6521 df-fo 6522 df-f1o 6523 df-fv 6524 df-riota 7333 df-ov 7380 df-oprab 7381 df-mpo 7382 df-om 7823 df-1st 7941 df-2nd 7942 df-frecs 8232 df-wrecs 8263 df-recs 8337 df-rdg 8376 df-er 8670 df-en 8906 df-dom 8907 df-sdom 8908 df-sup 9402 df-inf 9403 df-pnf 11215 df-mnf 11216 df-xr 11217 df-ltxr 11218 df-le 11219 df-sub 11411 df-neg 11412 df-nn 12178 df-n0 12438 df-z 12524 df-uz 12788 df-ico 13295 df-fl 13722 df-limsup 15380 |
This theorem is referenced by: smflimsuplem4 45217 |
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