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Mathbox for Glauco Siliprandi |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > xlimclim2lem | Structured version Visualization version GIF version |
Description: Lemma for xlimclim2 45141. Here it is additionally assumed that the sequence will eventually become (and stay) real. (Contributed by Glauco Siliprandi, 5-Feb-2022.) |
Ref | Expression |
---|---|
xlimclim2lem.z | β’ π = (β€β₯βπ) |
xlimclim2lem.f | β’ (π β πΉ:πβΆβ*) |
xlimclim2lem.a | β’ (π β π΄ β β) |
xlimclim2lem.r | β’ (π β βπ β π (πΉ βΎ (β€β₯βπ)):(β€β₯βπ)βΆβ) |
Ref | Expression |
---|---|
xlimclim2lem | β’ (π β (πΉ~~>*π΄ β πΉ β π΄)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | xlimclim2lem.z | . . . . . 6 β’ π = (β€β₯βπ) | |
2 | xlimclim2lem.f | . . . . . 6 β’ (π β πΉ:πβΆβ*) | |
3 | 1, 2 | fuzxrpmcn 45129 | . . . . 5 β’ (π β πΉ β (β* βpm β)) |
4 | 3 | ad2antrr 725 | . . . 4 β’ (((π β§ π β π) β§ (πΉ βΎ (β€β₯βπ)):(β€β₯βπ)βΆβ) β πΉ β (β* βpm β)) |
5 | 1 | eluzelz2 44698 | . . . . 5 β’ (π β π β π β β€) |
6 | 5 | ad2antlr 726 | . . . 4 β’ (((π β§ π β π) β§ (πΉ βΎ (β€β₯βπ)):(β€β₯βπ)βΆβ) β π β β€) |
7 | 4, 6 | xlimres 45122 | . . 3 β’ (((π β§ π β π) β§ (πΉ βΎ (β€β₯βπ)):(β€β₯βπ)βΆβ) β (πΉ~~>*π΄ β (πΉ βΎ (β€β₯βπ))~~>*π΄)) |
8 | eqid 2727 | . . . 4 β’ (β€β₯βπ) = (β€β₯βπ) | |
9 | simpr 484 | . . . 4 β’ (((π β§ π β π) β§ (πΉ βΎ (β€β₯βπ)):(β€β₯βπ)βΆβ) β (πΉ βΎ (β€β₯βπ)):(β€β₯βπ)βΆβ) | |
10 | xlimclim2lem.a | . . . . 5 β’ (π β π΄ β β) | |
11 | 10 | ad2antrr 725 | . . . 4 β’ (((π β§ π β π) β§ (πΉ βΎ (β€β₯βπ)):(β€β₯βπ)βΆβ) β π΄ β β) |
12 | 6, 8, 9, 11 | xlimclim 45125 | . . 3 β’ (((π β§ π β π) β§ (πΉ βΎ (β€β₯βπ)):(β€β₯βπ)βΆβ) β ((πΉ βΎ (β€β₯βπ))~~>*π΄ β (πΉ βΎ (β€β₯βπ)) β π΄)) |
13 | 1 | fvexi 6905 | . . . . . . 7 β’ π β V |
14 | 13 | a1i 11 | . . . . . 6 β’ (π β π β V) |
15 | 2, 14 | fexd 7233 | . . . . 5 β’ (π β πΉ β V) |
16 | climres 15537 | . . . . 5 β’ ((π β β€ β§ πΉ β V) β ((πΉ βΎ (β€β₯βπ)) β π΄ β πΉ β π΄)) | |
17 | 5, 15, 16 | syl2anr 596 | . . . 4 β’ ((π β§ π β π) β ((πΉ βΎ (β€β₯βπ)) β π΄ β πΉ β π΄)) |
18 | 17 | adantr 480 | . . 3 β’ (((π β§ π β π) β§ (πΉ βΎ (β€β₯βπ)):(β€β₯βπ)βΆβ) β ((πΉ βΎ (β€β₯βπ)) β π΄ β πΉ β π΄)) |
19 | 7, 12, 18 | 3bitrd 305 | . 2 β’ (((π β§ π β π) β§ (πΉ βΎ (β€β₯βπ)):(β€β₯βπ)βΆβ) β (πΉ~~>*π΄ β πΉ β π΄)) |
20 | xlimclim2lem.r | . 2 β’ (π β βπ β π (πΉ βΎ (β€β₯βπ)):(β€β₯βπ)βΆβ) | |
21 | 19, 20 | r19.29a 3157 | 1 β’ (π β (πΉ~~>*π΄ β πΉ β π΄)) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β wb 205 β§ wa 395 = wceq 1534 β wcel 2099 βwrex 3065 Vcvv 3469 class class class wbr 5142 βΎ cres 5674 βΆwf 6538 βcfv 6542 (class class class)co 7414 βpm cpm 8835 βcc 11122 βcr 11123 β*cxr 11263 β€cz 12574 β€β₯cuz 12838 β cli 15446 ~~>*clsxlim 45119 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2164 ax-ext 2698 ax-rep 5279 ax-sep 5293 ax-nul 5300 ax-pow 5359 ax-pr 5423 ax-un 7732 ax-cnex 11180 ax-resscn 11181 ax-1cn 11182 ax-icn 11183 ax-addcl 11184 ax-addrcl 11185 ax-mulcl 11186 ax-mulrcl 11187 ax-mulcom 11188 ax-addass 11189 ax-mulass 11190 ax-distr 11191 ax-i2m1 11192 ax-1ne0 11193 ax-1rid 11194 ax-rnegex 11195 ax-rrecex 11196 ax-cnre 11197 ax-pre-lttri 11198 ax-pre-lttrn 11199 ax-pre-ltadd 11200 ax-pre-mulgt0 11201 ax-pre-sup 11202 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2705 df-cleq 2719 df-clel 2805 df-nfc 2880 df-ne 2936 df-nel 3042 df-ral 3057 df-rex 3066 df-rmo 3371 df-reu 3372 df-rab 3428 df-v 3471 df-sbc 3775 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-pss 3963 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-tp 4629 df-op 4631 df-uni 4904 df-int 4945 df-iun 4993 df-br 5143 df-opab 5205 df-mpt 5226 df-tr 5260 df-id 5570 df-eprel 5576 df-po 5584 df-so 5585 df-fr 5627 df-we 5629 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-pred 6299 df-ord 6366 df-on 6367 df-lim 6368 df-suc 6369 df-iota 6494 df-fun 6544 df-fn 6545 df-f 6546 df-f1 6547 df-fo 6548 df-f1o 6549 df-fv 6550 df-riota 7370 df-ov 7417 df-oprab 7418 df-mpo 7419 df-om 7863 df-1st 7985 df-2nd 7986 df-frecs 8278 df-wrecs 8309 df-recs 8383 df-rdg 8422 df-1o 8478 df-er 8716 df-map 8836 df-pm 8837 df-en 8954 df-dom 8955 df-sdom 8956 df-fin 8957 df-fi 9420 df-sup 9451 df-inf 9452 df-pnf 11266 df-mnf 11267 df-xr 11268 df-ltxr 11269 df-le 11270 df-sub 11462 df-neg 11463 df-div 11888 df-nn 12229 df-2 12291 df-3 12292 df-4 12293 df-5 12294 df-6 12295 df-7 12296 df-8 12297 df-9 12298 df-n0 12489 df-z 12575 df-dec 12694 df-uz 12839 df-q 12949 df-rp 12993 df-xneg 13110 df-xadd 13111 df-xmul 13112 df-ioo 13346 df-ioc 13347 df-ico 13348 df-icc 13349 df-fz 13503 df-fl 13775 df-seq 13985 df-exp 14045 df-cj 15064 df-re 15065 df-im 15066 df-sqrt 15200 df-abs 15201 df-clim 15450 df-rlim 15451 df-struct 17101 df-slot 17136 df-ndx 17148 df-base 17166 df-plusg 17231 df-mulr 17232 df-starv 17233 df-tset 17237 df-ple 17238 df-ds 17240 df-unif 17241 df-rest 17389 df-topn 17390 df-topgen 17410 df-ordt 17468 df-ps 18543 df-tsr 18544 df-psmet 21251 df-xmet 21252 df-met 21253 df-bl 21254 df-mopn 21255 df-cnfld 21260 df-top 22770 df-topon 22787 df-topsp 22809 df-bases 22823 df-lm 23107 df-xms 24200 df-ms 24201 df-xlim 45120 |
This theorem is referenced by: xlimclim2 45141 |
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