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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 3187 | . . . . 5 β’ β β β | |
2 | eqid 2187 | . . . . . 6 β’ (MetOpenβ(abs β β )) = (MetOpenβ(abs β β )) | |
3 | eqid 2187 | . . . . . 6 β’ ((MetOpenβ(abs β β )) βΎt π΄) = ((MetOpenβ(abs β β )) βΎt π΄) | |
4 | 2 | cntoptopon 14272 | . . . . . . 7 β’ (MetOpenβ(abs β β )) β (TopOnββ) |
5 | 4 | toponrestid 13761 | . . . . . 6 β’ (MetOpenβ(abs β β )) = ((MetOpenβ(abs β β )) βΎt β) |
6 | 2, 3, 5 | cncfcncntop 14320 | . . . . 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 2257 | . . 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 13911 | . . . . 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 13970 | . . . 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 14376 | . . . . 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 2554 | . . 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 1363 β wcel 2158 βwral 2465 β wss 3141 β ccom 4642 βΆwf 5224 βcfv 5228 (class class class)co 5888 βcc 7822 β cmin 8141 abscabs 11019 βΎt crest 12705 MetOpencmopn 13671 TopOnctopon 13750 Cn ccn 13925 CnP ccnp 13926 βcnβccncf 14297 limβ climc 14363 |
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 1457 ax-7 1458 ax-gen 1459 ax-ie1 1503 ax-ie2 1504 ax-8 1514 ax-10 1515 ax-11 1516 ax-i12 1517 ax-bndl 1519 ax-4 1520 ax-17 1536 ax-i9 1540 ax-ial 1544 ax-i5r 1545 ax-13 2160 ax-14 2161 ax-ext 2169 ax-coll 4130 ax-sep 4133 ax-nul 4141 ax-pow 4186 ax-pr 4221 ax-un 4445 ax-setind 4548 ax-iinf 4599 ax-cnex 7915 ax-resscn 7916 ax-1cn 7917 ax-1re 7918 ax-icn 7919 ax-addcl 7920 ax-addrcl 7921 ax-mulcl 7922 ax-mulrcl 7923 ax-addcom 7924 ax-mulcom 7925 ax-addass 7926 ax-mulass 7927 ax-distr 7928 ax-i2m1 7929 ax-0lt1 7930 ax-1rid 7931 ax-0id 7932 ax-rnegex 7933 ax-precex 7934 ax-cnre 7935 ax-pre-ltirr 7936 ax-pre-ltwlin 7937 ax-pre-lttrn 7938 ax-pre-apti 7939 ax-pre-ltadd 7940 ax-pre-mulgt0 7941 ax-pre-mulext 7942 ax-arch 7943 ax-caucvg 7944 |
This theorem depends on definitions: df-bi 117 df-stab 832 df-dc 836 df-3or 980 df-3an 981 df-tru 1366 df-fal 1369 df-nf 1471 df-sb 1773 df-eu 2039 df-mo 2040 df-clab 2174 df-cleq 2180 df-clel 2183 df-nfc 2318 df-ne 2358 df-nel 2453 df-ral 2470 df-rex 2471 df-reu 2472 df-rmo 2473 df-rab 2474 df-v 2751 df-sbc 2975 df-csb 3070 df-dif 3143 df-un 3145 df-in 3147 df-ss 3154 df-nul 3435 df-if 3547 df-pw 3589 df-sn 3610 df-pr 3611 df-op 3613 df-uni 3822 df-int 3857 df-iun 3900 df-br 4016 df-opab 4077 df-mpt 4078 df-tr 4114 df-id 4305 df-po 4308 df-iso 4309 df-iord 4378 df-on 4380 df-ilim 4381 df-suc 4383 df-iom 4602 df-xp 4644 df-rel 4645 df-cnv 4646 df-co 4647 df-dm 4648 df-rn 4649 df-res 4650 df-ima 4651 df-iota 5190 df-fun 5230 df-fn 5231 df-f 5232 df-f1 5233 df-fo 5234 df-f1o 5235 df-fv 5236 df-isom 5237 df-riota 5844 df-ov 5891 df-oprab 5892 df-mpo 5893 df-1st 6154 df-2nd 6155 df-recs 6319 df-frec 6405 df-map 6663 df-pm 6664 df-sup 6996 df-inf 6997 df-pnf 8007 df-mnf 8008 df-xr 8009 df-ltxr 8010 df-le 8011 df-sub 8143 df-neg 8144 df-reap 8545 df-ap 8552 df-div 8643 df-inn 8933 df-2 8991 df-3 8992 df-4 8993 df-n0 9190 df-z 9267 df-uz 9542 df-q 9633 df-rp 9667 df-xneg 9785 df-xadd 9786 df-seqfrec 10459 df-exp 10533 df-cj 10864 df-re 10865 df-im 10866 df-rsqrt 11020 df-abs 11021 df-rest 12707 df-topgen 12726 df-psmet 13673 df-xmet 13674 df-met 13675 df-bl 13676 df-mopn 13677 df-top 13738 df-topon 13751 df-bases 13783 df-cn 13928 df-cnp 13929 df-cncf 14298 df-limced 14365 |
This theorem is referenced by: cnlimci 14382 |
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