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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 44087, showing that the first hypothesis is needed, if the empty set is a complex number (see 0ncn 11074 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 6724 | . . . 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 43733 | . . . . . . 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 2761 | . . . . 5 β’ π = β |
8 | 2, 7 | feq12i 6662 | . . . 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 15380 | . . . . 5 β’ Rel β | |
12 | 11 | a1i 11 | . . . 4 β’ (β β β β Rel β ) |
13 | 0cnv 44069 | . . . . 5 β’ (β β β β β β β ) | |
14 | 2, 13 | eqbrtrid 5141 | . . . 4 β’ (β β β β πΉ β β ) |
15 | releldm 5900 | . . . 4 β’ ((Rel β β§ πΉ β β ) β πΉ β dom β ) | |
16 | 12, 14, 15 | syl2anc 585 | . . 3 β’ (β β β β πΉ β dom β ) |
17 | 16 | adantr 482 | . 2 β’ ((β β β β§ Β¬ -β β β) β πΉ β dom β ) |
18 | 13 | adantr 482 | . . . 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 486 | . . . . . 6 β’ ((πΉ β (lim supβπΉ) β§ β β β ) β β β β ) | |
21 | 2 | fveq2i 6846 | . . . . . . . . . 10 β’ (lim supβπΉ) = (lim supββ ) |
22 | limsup0 44021 | . . . . . . . . . 10 β’ (lim supββ ) = -β | |
23 | 21, 22 | eqtri 2761 | . . . . . . . . 9 β’ (lim supβπΉ) = -β |
24 | 2, 23 | breq12i 5115 | . . . . . . . 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 482 | . . . . . 6 β’ ((πΉ β (lim supβπΉ) β§ β β β ) β β β -β) |
27 | climuni 15440 | . . . . . 6 β’ ((β β β β§ β β -β) β β = -β) | |
28 | 20, 26, 27 | syl2anc 585 | . . . . 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 2858 | . . . . 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 1129 | 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 397 β§ w3a 1088 = wceq 1542 β wcel 2107 β c0 4283 class class class wbr 5106 dom cdm 5634 Rel wrel 5639 βΆwf 6493 βcfv 6497 βcc 11054 βcr 11055 -βcmnf 11192 β€cz 12504 β€β₯cuz 12768 lim supclsp 15358 β cli 15372 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-sep 5257 ax-nul 5264 ax-pow 5321 ax-pr 5385 ax-un 7673 ax-cnex 11112 ax-resscn 11113 ax-1cn 11114 ax-icn 11115 ax-addcl 11116 ax-addrcl 11117 ax-mulcl 11118 ax-mulrcl 11119 ax-mulcom 11120 ax-addass 11121 ax-mulass 11122 ax-distr 11123 ax-i2m1 11124 ax-1ne0 11125 ax-1rid 11126 ax-rnegex 11127 ax-rrecex 11128 ax-cnre 11129 ax-pre-lttri 11130 ax-pre-lttrn 11131 ax-pre-ltadd 11132 ax-pre-mulgt0 11133 ax-pre-sup 11134 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3352 df-reu 3353 df-rab 3407 df-v 3446 df-sbc 3741 df-csb 3857 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-pss 3930 df-nul 4284 df-if 4488 df-pw 4563 df-sn 4588 df-pr 4590 df-op 4594 df-uni 4867 df-iun 4957 df-br 5107 df-opab 5169 df-mpt 5190 df-tr 5224 df-id 5532 df-eprel 5538 df-po 5546 df-so 5547 df-fr 5589 df-we 5591 df-xp 5640 df-rel 5641 df-cnv 5642 df-co 5643 df-dm 5644 df-rn 5645 df-res 5646 df-ima 5647 df-pred 6254 df-ord 6321 df-on 6322 df-lim 6323 df-suc 6324 df-iota 6449 df-fun 6499 df-fn 6500 df-f 6501 df-f1 6502 df-fo 6503 df-f1o 6504 df-fv 6505 df-riota 7314 df-ov 7361 df-oprab 7362 df-mpo 7363 df-om 7804 df-2nd 7923 df-frecs 8213 df-wrecs 8244 df-recs 8318 df-rdg 8357 df-er 8651 df-en 8887 df-dom 8888 df-sdom 8889 df-sup 9383 df-inf 9384 df-pnf 11196 df-mnf 11197 df-xr 11198 df-ltxr 11199 df-le 11200 df-sub 11392 df-neg 11393 df-div 11818 df-nn 12159 df-2 12221 df-3 12222 df-n0 12419 df-z 12505 df-uz 12769 df-rp 12921 df-seq 13913 df-exp 13974 df-cj 14990 df-re 14991 df-im 14992 df-sqrt 15126 df-abs 15127 df-limsup 15359 df-clim 15376 |
This theorem is referenced by: (None) |
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