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Mathbox for Glauco Siliprandi |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > limcdm0 | Structured version Visualization version GIF version |
Description: If a function has empty domain, every complex number is a limit. (Contributed by Glauco Siliprandi, 11-Dec-2019.) |
Ref | Expression |
---|---|
limcdm0.f | β’ (π β πΉ:β βΆβ) |
limcdm0.b | β’ (π β π΅ β β) |
Ref | Expression |
---|---|
limcdm0 | β’ (π β (πΉ limβ π΅) = β) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | limccl 25242 | . . . . 5 β’ (πΉ limβ π΅) β β | |
2 | 1 | sseli 3941 | . . . 4 β’ (π₯ β (πΉ limβ π΅) β π₯ β β) |
3 | 2 | adantl 483 | . . 3 β’ ((π β§ π₯ β (πΉ limβ π΅)) β π₯ β β) |
4 | simpr 486 | . . . 4 β’ ((π β§ π₯ β β) β π₯ β β) | |
5 | 1rp 12920 | . . . . . . 7 β’ 1 β β+ | |
6 | ral0 4471 | . . . . . . 7 β’ βπ§ β β ((π§ β π΅ β§ (absβ(π§ β π΅)) < 1) β (absβ((πΉβπ§) β π₯)) < π¦) | |
7 | brimralrspcev 5167 | . . . . . . 7 β’ ((1 β β+ β§ βπ§ β β ((π§ β π΅ β§ (absβ(π§ β π΅)) < 1) β (absβ((πΉβπ§) β π₯)) < π¦)) β βπ€ β β+ βπ§ β β ((π§ β π΅ β§ (absβ(π§ β π΅)) < π€) β (absβ((πΉβπ§) β π₯)) < π¦)) | |
8 | 5, 6, 7 | mp2an 691 | . . . . . 6 β’ βπ€ β β+ βπ§ β β ((π§ β π΅ β§ (absβ(π§ β π΅)) < π€) β (absβ((πΉβπ§) β π₯)) < π¦) |
9 | 8 | rgenw 3069 | . . . . 5 β’ βπ¦ β β+ βπ€ β β+ βπ§ β β ((π§ β π΅ β§ (absβ(π§ β π΅)) < π€) β (absβ((πΉβπ§) β π₯)) < π¦) |
10 | 9 | a1i 11 | . . . 4 β’ ((π β§ π₯ β β) β βπ¦ β β+ βπ€ β β+ βπ§ β β ((π§ β π΅ β§ (absβ(π§ β π΅)) < π€) β (absβ((πΉβπ§) β π₯)) < π¦)) |
11 | limcdm0.f | . . . . . 6 β’ (π β πΉ:β βΆβ) | |
12 | 11 | adantr 482 | . . . . 5 β’ ((π β§ π₯ β β) β πΉ:β βΆβ) |
13 | 0ss 4357 | . . . . . 6 β’ β β β | |
14 | 13 | a1i 11 | . . . . 5 β’ ((π β§ π₯ β β) β β β β) |
15 | limcdm0.b | . . . . . 6 β’ (π β π΅ β β) | |
16 | 15 | adantr 482 | . . . . 5 β’ ((π β§ π₯ β β) β π΅ β β) |
17 | 12, 14, 16 | ellimc3 25246 | . . . 4 β’ ((π β§ π₯ β β) β (π₯ β (πΉ limβ π΅) β (π₯ β β β§ βπ¦ β β+ βπ€ β β+ βπ§ β β ((π§ β π΅ β§ (absβ(π§ β π΅)) < π€) β (absβ((πΉβπ§) β π₯)) < π¦)))) |
18 | 4, 10, 17 | mpbir2and 712 | . . 3 β’ ((π β§ π₯ β β) β π₯ β (πΉ limβ π΅)) |
19 | 3, 18 | impbida 800 | . 2 β’ (π β (π₯ β (πΉ limβ π΅) β π₯ β β)) |
20 | 19 | eqrdv 2735 | 1 β’ (π β (πΉ limβ π΅) = β) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ wa 397 = wceq 1542 β wcel 2107 β wne 2944 βwral 3065 βwrex 3074 β wss 3911 β c0 4283 class class class wbr 5106 βΆwf 6493 βcfv 6497 (class class class)co 7358 βcc 11050 1c1 11053 < clt 11190 β cmin 11386 β+crp 12916 abscabs 15120 limβ climc 25229 |
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 2708 ax-rep 5243 ax-sep 5257 ax-nul 5264 ax-pow 5321 ax-pr 5385 ax-un 7673 ax-cnex 11108 ax-resscn 11109 ax-1cn 11110 ax-icn 11111 ax-addcl 11112 ax-addrcl 11113 ax-mulcl 11114 ax-mulrcl 11115 ax-mulcom 11116 ax-addass 11117 ax-mulass 11118 ax-distr 11119 ax-i2m1 11120 ax-1ne0 11121 ax-1rid 11122 ax-rnegex 11123 ax-rrecex 11124 ax-cnre 11125 ax-pre-lttri 11126 ax-pre-lttrn 11127 ax-pre-ltadd 11128 ax-pre-mulgt0 11129 ax-pre-sup 11130 |
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 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2815 df-nfc 2890 df-ne 2945 df-nel 3051 df-ral 3066 df-rex 3075 df-rmo 3354 df-reu 3355 df-rab 3409 df-v 3448 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-tp 4592 df-op 4594 df-uni 4867 df-int 4909 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-1st 7922 df-2nd 7923 df-frecs 8213 df-wrecs 8244 df-recs 8318 df-rdg 8357 df-1o 8413 df-er 8649 df-map 8768 df-pm 8769 df-en 8885 df-dom 8886 df-sdom 8887 df-fin 8888 df-fi 9348 df-sup 9379 df-inf 9380 df-pnf 11192 df-mnf 11193 df-xr 11194 df-ltxr 11195 df-le 11196 df-sub 11388 df-neg 11389 df-div 11814 df-nn 12155 df-2 12217 df-3 12218 df-4 12219 df-5 12220 df-6 12221 df-7 12222 df-8 12223 df-9 12224 df-n0 12415 df-z 12501 df-dec 12620 df-uz 12765 df-q 12875 df-rp 12917 df-xneg 13034 df-xadd 13035 df-xmul 13036 df-fz 13426 df-seq 13908 df-exp 13969 df-cj 14985 df-re 14986 df-im 14987 df-sqrt 15121 df-abs 15122 df-struct 17020 df-slot 17055 df-ndx 17067 df-base 17085 df-plusg 17147 df-mulr 17148 df-starv 17149 df-tset 17153 df-ple 17154 df-ds 17156 df-unif 17157 df-rest 17305 df-topn 17306 df-topgen 17326 df-psmet 20791 df-xmet 20792 df-met 20793 df-bl 20794 df-mopn 20795 df-cnfld 20800 df-top 22246 df-topon 22263 df-topsp 22285 df-bases 22299 df-cnp 22582 df-xms 23676 df-ms 23677 df-limc 25233 |
This theorem is referenced by: ioodvbdlimc1 44181 ioodvbdlimc2 44183 |
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