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Mathbox for Glauco Siliprandi |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > climlimsupcex | Structured version Visualization version GIF version |
Description: Counterexample for climlimsup 45071, showing that the first hypothesis is needed, if the empty set is a complex number (see 0ncn 11148 and its comment). (Contributed by Glauco Siliprandi, 2-Jan-2022.) |
Ref | Expression |
---|---|
climlimsupcex.1 | β’ Β¬ π β β€ |
climlimsupcex.2 | β’ π = (β€β₯βπ) |
climlimsupcex.3 | β’ πΉ = β |
Ref | Expression |
---|---|
climlimsupcex | β’ ((β β β β§ Β¬ -β β β) β (πΉ:πβΆβ β§ πΉ β dom β β§ Β¬ πΉ β (lim supβπΉ))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | f0 6772 | . . . 4 β’ β :β βΆβ | |
2 | climlimsupcex.3 | . . . . 5 β’ πΉ = β | |
3 | climlimsupcex.2 | . . . . . 6 β’ π = (β€β₯βπ) | |
4 | climlimsupcex.1 | . . . . . . 7 β’ Β¬ π β β€ | |
5 | uz0 44717 | . . . . . . 7 β’ (Β¬ π β β€ β (β€β₯βπ) = β ) | |
6 | 4, 5 | ax-mp 5 | . . . . . 6 β’ (β€β₯βπ) = β |
7 | 3, 6 | eqtri 2755 | . . . . 5 β’ π = β |
8 | 2, 7 | feq12i 6709 | . . . 4 β’ (πΉ:πβΆβ β β :β βΆβ) |
9 | 1, 8 | mpbir 230 | . . 3 β’ πΉ:πβΆβ |
10 | 9 | a1i 11 | . 2 β’ ((β β β β§ Β¬ -β β β) β πΉ:πβΆβ) |
11 | climrel 15460 | . . . . 5 β’ Rel β | |
12 | 11 | a1i 11 | . . . 4 β’ (β β β β Rel β ) |
13 | 0cnv 45053 | . . . . 5 β’ (β β β β β β β ) | |
14 | 2, 13 | eqbrtrid 5177 | . . . 4 β’ (β β β β πΉ β β ) |
15 | releldm 5940 | . . . 4 β’ ((Rel β β§ πΉ β β ) β πΉ β dom β ) | |
16 | 12, 14, 15 | syl2anc 583 | . . 3 β’ (β β β β πΉ β dom β ) |
17 | 16 | adantr 480 | . 2 β’ ((β β β β§ Β¬ -β β β) β πΉ β dom β ) |
18 | 13 | adantr 480 | . . . 4 β’ ((β β β β§ πΉ β (lim supβπΉ)) β β β β ) |
19 | 18 | adantlr 714 | . . 3 β’ (((β β β β§ Β¬ -β β β) β§ πΉ β (lim supβπΉ)) β β β β ) |
20 | simpr 484 | . . . . . 6 β’ ((πΉ β (lim supβπΉ) β§ β β β ) β β β β ) | |
21 | 2 | fveq2i 6894 | . . . . . . . . . 10 β’ (lim supβπΉ) = (lim supββ ) |
22 | limsup0 45005 | . . . . . . . . . 10 β’ (lim supββ ) = -β | |
23 | 21, 22 | eqtri 2755 | . . . . . . . . 9 β’ (lim supβπΉ) = -β |
24 | 2, 23 | breq12i 5151 | . . . . . . . 8 β’ (πΉ β (lim supβπΉ) β β β -β) |
25 | 24 | biimpi 215 | . . . . . . 7 β’ (πΉ β (lim supβπΉ) β β β -β) |
26 | 25 | adantr 480 | . . . . . 6 β’ ((πΉ β (lim supβπΉ) β§ β β β ) β β β -β) |
27 | climuni 15520 | . . . . . 6 β’ ((β β β β§ β β -β) β β = -β) | |
28 | 20, 26, 27 | syl2anc 583 | . . . . 5 β’ ((πΉ β (lim supβπΉ) β§ β β β ) β β = -β) |
29 | 28 | adantll 713 | . . . 4 β’ ((((β β β β§ Β¬ -β β β) β§ πΉ β (lim supβπΉ)) β§ β β β ) β β = -β) |
30 | nelneq 2852 | . . . . 5 β’ ((β β β β§ Β¬ -β β β) β Β¬ β = -β) | |
31 | 30 | ad2antrr 725 | . . . 4 β’ ((((β β β β§ Β¬ -β β β) β§ πΉ β (lim supβπΉ)) β§ β β β ) β Β¬ β = -β) |
32 | 29, 31 | pm2.65da 816 | . . 3 β’ (((β β β β§ Β¬ -β β β) β§ πΉ β (lim supβπΉ)) β Β¬ β β β ) |
33 | 19, 32 | pm2.65da 816 | . 2 β’ ((β β β β§ Β¬ -β β β) β Β¬ πΉ β (lim supβπΉ)) |
34 | 10, 17, 33 | 3jca 1126 | 1 β’ ((β β β β§ Β¬ -β β β) β (πΉ:πβΆβ β§ πΉ β dom β β§ Β¬ πΉ β (lim supβπΉ))) |
Colors of variables: wff setvar class |
Syntax hints: Β¬ wn 3 β wi 4 β§ wa 395 β§ w3a 1085 = wceq 1534 β wcel 2099 β c0 4318 class class class wbr 5142 dom cdm 5672 Rel wrel 5677 βΆwf 6538 βcfv 6542 βcc 11128 βcr 11129 -βcmnf 11268 β€cz 12580 β€β₯cuz 12844 lim supclsp 15438 β cli 15452 |
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-sep 5293 ax-nul 5300 ax-pow 5359 ax-pr 5423 ax-un 7734 ax-cnex 11186 ax-resscn 11187 ax-1cn 11188 ax-icn 11189 ax-addcl 11190 ax-addrcl 11191 ax-mulcl 11192 ax-mulrcl 11193 ax-mulcom 11194 ax-addass 11195 ax-mulass 11196 ax-distr 11197 ax-i2m1 11198 ax-1ne0 11199 ax-1rid 11200 ax-rnegex 11201 ax-rrecex 11202 ax-cnre 11203 ax-pre-lttri 11204 ax-pre-lttrn 11205 ax-pre-ltadd 11206 ax-pre-mulgt0 11207 ax-pre-sup 11208 |
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-op 4631 df-uni 4904 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 7865 df-2nd 7988 df-frecs 8280 df-wrecs 8311 df-recs 8385 df-rdg 8424 df-er 8718 df-en 8956 df-dom 8957 df-sdom 8958 df-sup 9457 df-inf 9458 df-pnf 11272 df-mnf 11273 df-xr 11274 df-ltxr 11275 df-le 11276 df-sub 11468 df-neg 11469 df-div 11894 df-nn 12235 df-2 12297 df-3 12298 df-n0 12495 df-z 12581 df-uz 12845 df-rp 12999 df-seq 13991 df-exp 14051 df-cj 15070 df-re 15071 df-im 15072 df-sqrt 15206 df-abs 15207 df-limsup 15439 df-clim 15456 |
This theorem is referenced by: (None) |
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