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Mirrors > Home > ILE Home > Th. List > cnlimcim | GIF version |
Description: If πΉ is a continuous function, the limit of the function at each point equals the value of the function. (Contributed by Mario Carneiro, 28-Dec-2016.) (Revised by Jim Kingdon, 16-Jun-2023.) |
Ref | Expression |
---|---|
cnlimcim | β’ (π΄ β β β (πΉ β (π΄βcnββ) β (πΉ:π΄βΆβ β§ βπ₯ β π΄ (πΉβπ₯) β (πΉ limβ π₯)))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ssid 3190 | . . . . 5 β’ β β β | |
2 | eqid 2189 | . . . . . 6 β’ (MetOpenβ(abs β β )) = (MetOpenβ(abs β β )) | |
3 | eqid 2189 | . . . . . 6 β’ ((MetOpenβ(abs β β )) βΎt π΄) = ((MetOpenβ(abs β β )) βΎt π΄) | |
4 | 2 | cntoptopon 14429 | . . . . . . 7 β’ (MetOpenβ(abs β β )) β (TopOnββ) |
5 | 4 | toponrestid 13918 | . . . . . 6 β’ (MetOpenβ(abs β β )) = ((MetOpenβ(abs β β )) βΎt β) |
6 | 2, 3, 5 | cncfcncntop 14477 | . . . . 5 β’ ((π΄ β β β§ β β β) β (π΄βcnββ) = (((MetOpenβ(abs β β )) βΎt π΄) Cn (MetOpenβ(abs β β )))) |
7 | 1, 6 | mpan2 425 | . . . 4 β’ (π΄ β β β (π΄βcnββ) = (((MetOpenβ(abs β β )) βΎt π΄) Cn (MetOpenβ(abs β β )))) |
8 | 7 | eleq2d 2259 | . . 3 β’ (π΄ β β β (πΉ β (π΄βcnββ) β πΉ β (((MetOpenβ(abs β β )) βΎt π΄) Cn (MetOpenβ(abs β β ))))) |
9 | resttopon 14068 | . . . . 5 β’ (((MetOpenβ(abs β β )) β (TopOnββ) β§ π΄ β β) β ((MetOpenβ(abs β β )) βΎt π΄) β (TopOnβπ΄)) | |
10 | 4, 9 | mpan 424 | . . . 4 β’ (π΄ β β β ((MetOpenβ(abs β β )) βΎt π΄) β (TopOnβπ΄)) |
11 | cncnp 14127 | . . . 4 β’ ((((MetOpenβ(abs β β )) βΎt π΄) β (TopOnβπ΄) β§ (MetOpenβ(abs β β )) β (TopOnββ)) β (πΉ β (((MetOpenβ(abs β β )) βΎt π΄) Cn (MetOpenβ(abs β β ))) β (πΉ:π΄βΆβ β§ βπ₯ β π΄ πΉ β ((((MetOpenβ(abs β β )) βΎt π΄) CnP (MetOpenβ(abs β β )))βπ₯)))) | |
12 | 10, 4, 11 | sylancl 413 | . . 3 β’ (π΄ β β β (πΉ β (((MetOpenβ(abs β β )) βΎt π΄) Cn (MetOpenβ(abs β β ))) β (πΉ:π΄βΆβ β§ βπ₯ β π΄ πΉ β ((((MetOpenβ(abs β β )) βΎt π΄) CnP (MetOpenβ(abs β β )))βπ₯)))) |
13 | 8, 12 | bitrd 188 | . 2 β’ (π΄ β β β (πΉ β (π΄βcnββ) β (πΉ:π΄βΆβ β§ βπ₯ β π΄ πΉ β ((((MetOpenβ(abs β β )) βΎt π΄) CnP (MetOpenβ(abs β β )))βπ₯)))) |
14 | 2, 3 | cnplimcim 14533 | . . . . 5 β’ ((π΄ β β β§ π₯ β π΄) β (πΉ β ((((MetOpenβ(abs β β )) βΎt π΄) CnP (MetOpenβ(abs β β )))βπ₯) β (πΉ:π΄βΆβ β§ (πΉβπ₯) β (πΉ limβ π₯)))) |
15 | simpr 110 | . . . . 5 β’ ((πΉ:π΄βΆβ β§ (πΉβπ₯) β (πΉ limβ π₯)) β (πΉβπ₯) β (πΉ limβ π₯)) | |
16 | 14, 15 | syl6 33 | . . . 4 β’ ((π΄ β β β§ π₯ β π΄) β (πΉ β ((((MetOpenβ(abs β β )) βΎt π΄) CnP (MetOpenβ(abs β β )))βπ₯) β (πΉβπ₯) β (πΉ limβ π₯))) |
17 | 16 | ralimdva 2557 | . . 3 β’ (π΄ β β β (βπ₯ β π΄ πΉ β ((((MetOpenβ(abs β β )) βΎt π΄) CnP (MetOpenβ(abs β β )))βπ₯) β βπ₯ β π΄ (πΉβπ₯) β (πΉ limβ π₯))) |
18 | 17 | anim2d 337 | . 2 β’ (π΄ β β β ((πΉ:π΄βΆβ β§ βπ₯ β π΄ πΉ β ((((MetOpenβ(abs β β )) βΎt π΄) CnP (MetOpenβ(abs β β )))βπ₯)) β (πΉ:π΄βΆβ β§ βπ₯ β π΄ (πΉβπ₯) β (πΉ limβ π₯)))) |
19 | 13, 18 | sylbid 150 | 1 β’ (π΄ β β β (πΉ β (π΄βcnββ) β (πΉ:π΄βΆβ β§ βπ₯ β π΄ (πΉβπ₯) β (πΉ limβ π₯)))) |
Colors of variables: wff set class |
Syntax hints: β wi 4 β§ wa 104 β wb 105 = wceq 1364 β wcel 2160 βwral 2468 β wss 3144 β ccom 4645 βΆwf 5227 βcfv 5231 (class class class)co 5891 βcc 7827 β cmin 8146 abscabs 11024 βΎt crest 12710 MetOpencmopn 13815 TopOnctopon 13907 Cn ccn 14082 CnP ccnp 14083 βcnβccncf 14454 limβ climc 14520 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 615 ax-in2 616 ax-io 710 ax-5 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-13 2162 ax-14 2163 ax-ext 2171 ax-coll 4133 ax-sep 4136 ax-nul 4144 ax-pow 4189 ax-pr 4224 ax-un 4448 ax-setind 4551 ax-iinf 4602 ax-cnex 7920 ax-resscn 7921 ax-1cn 7922 ax-1re 7923 ax-icn 7924 ax-addcl 7925 ax-addrcl 7926 ax-mulcl 7927 ax-mulrcl 7928 ax-addcom 7929 ax-mulcom 7930 ax-addass 7931 ax-mulass 7932 ax-distr 7933 ax-i2m1 7934 ax-0lt1 7935 ax-1rid 7936 ax-0id 7937 ax-rnegex 7938 ax-precex 7939 ax-cnre 7940 ax-pre-ltirr 7941 ax-pre-ltwlin 7942 ax-pre-lttrn 7943 ax-pre-apti 7944 ax-pre-ltadd 7945 ax-pre-mulgt0 7946 ax-pre-mulext 7947 ax-arch 7948 ax-caucvg 7949 |
This theorem depends on definitions: df-bi 117 df-stab 832 df-dc 836 df-3or 981 df-3an 982 df-tru 1367 df-fal 1370 df-nf 1472 df-sb 1774 df-eu 2041 df-mo 2042 df-clab 2176 df-cleq 2182 df-clel 2185 df-nfc 2321 df-ne 2361 df-nel 2456 df-ral 2473 df-rex 2474 df-reu 2475 df-rmo 2476 df-rab 2477 df-v 2754 df-sbc 2978 df-csb 3073 df-dif 3146 df-un 3148 df-in 3150 df-ss 3157 df-nul 3438 df-if 3550 df-pw 3592 df-sn 3613 df-pr 3614 df-op 3616 df-uni 3825 df-int 3860 df-iun 3903 df-br 4019 df-opab 4080 df-mpt 4081 df-tr 4117 df-id 4308 df-po 4311 df-iso 4312 df-iord 4381 df-on 4383 df-ilim 4384 df-suc 4386 df-iom 4605 df-xp 4647 df-rel 4648 df-cnv 4649 df-co 4650 df-dm 4651 df-rn 4652 df-res 4653 df-ima 4654 df-iota 5193 df-fun 5233 df-fn 5234 df-f 5235 df-f1 5236 df-fo 5237 df-f1o 5238 df-fv 5239 df-isom 5240 df-riota 5847 df-ov 5894 df-oprab 5895 df-mpo 5896 df-1st 6159 df-2nd 6160 df-recs 6324 df-frec 6410 df-map 6668 df-pm 6669 df-sup 7001 df-inf 7002 df-pnf 8012 df-mnf 8013 df-xr 8014 df-ltxr 8015 df-le 8016 df-sub 8148 df-neg 8149 df-reap 8550 df-ap 8557 df-div 8648 df-inn 8938 df-2 8996 df-3 8997 df-4 8998 df-n0 9195 df-z 9272 df-uz 9547 df-q 9638 df-rp 9672 df-xneg 9790 df-xadd 9791 df-seqfrec 10464 df-exp 10538 df-cj 10869 df-re 10870 df-im 10871 df-rsqrt 11025 df-abs 11026 df-rest 12712 df-topgen 12731 df-psmet 13817 df-xmet 13818 df-met 13819 df-bl 13820 df-mopn 13821 df-top 13895 df-topon 13908 df-bases 13940 df-cn 14085 df-cnp 14086 df-cncf 14455 df-limced 14522 |
This theorem is referenced by: cnlimci 14539 |
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