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Mirrors > Home > MPE Home > Th. List > cnlimci | Structured version Visualization version GIF version |
Description: If πΉ is a continuous function, then the limit of the function at any point equals its value. (Contributed by Mario Carneiro, 28-Dec-2016.) |
Ref | Expression |
---|---|
cnlimci.f | β’ (π β πΉ β (π΄βcnβπ·)) |
cnlimci.c | β’ (π β π΅ β π΄) |
Ref | Expression |
---|---|
cnlimci | β’ (π β (πΉβπ΅) β (πΉ limβ π΅)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fveq2 6884 | . . 3 β’ (π₯ = π΅ β (πΉβπ₯) = (πΉβπ΅)) | |
2 | oveq2 7412 | . . 3 β’ (π₯ = π΅ β (πΉ limβ π₯) = (πΉ limβ π΅)) | |
3 | 1, 2 | eleq12d 2821 | . 2 β’ (π₯ = π΅ β ((πΉβπ₯) β (πΉ limβ π₯) β (πΉβπ΅) β (πΉ limβ π΅))) |
4 | cnlimci.f | . . . 4 β’ (π β πΉ β (π΄βcnβπ·)) | |
5 | cncfrss 24761 | . . . 4 β’ (πΉ β (π΄βcnβπ·) β π΄ β β) | |
6 | 4, 5 | syl 17 | . . 3 β’ (π β π΄ β β) |
7 | cncfrss2 24762 | . . . . . 6 β’ (πΉ β (π΄βcnβπ·) β π· β β) | |
8 | 4, 7 | syl 17 | . . . . 5 β’ (π β π· β β) |
9 | ssid 3999 | . . . . 5 β’ β β β | |
10 | cncfss 24769 | . . . . 5 β’ ((π· β β β§ β β β) β (π΄βcnβπ·) β (π΄βcnββ)) | |
11 | 8, 9, 10 | sylancl 585 | . . . 4 β’ (π β (π΄βcnβπ·) β (π΄βcnββ)) |
12 | 11, 4 | sseldd 3978 | . . 3 β’ (π β πΉ β (π΄βcnββ)) |
13 | cnlimc 25767 | . . . 4 β’ (π΄ β β β (πΉ β (π΄βcnββ) β (πΉ:π΄βΆβ β§ βπ₯ β π΄ (πΉβπ₯) β (πΉ limβ π₯)))) | |
14 | 13 | simplbda 499 | . . 3 β’ ((π΄ β β β§ πΉ β (π΄βcnββ)) β βπ₯ β π΄ (πΉβπ₯) β (πΉ limβ π₯)) |
15 | 6, 12, 14 | syl2anc 583 | . 2 β’ (π β βπ₯ β π΄ (πΉβπ₯) β (πΉ limβ π₯)) |
16 | cnlimci.c | . 2 β’ (π β π΅ β π΄) | |
17 | 3, 15, 16 | rspcdva 3607 | 1 β’ (π β (πΉβπ΅) β (πΉ limβ π΅)) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 = wceq 1533 β wcel 2098 βwral 3055 β wss 3943 βΆwf 6532 βcfv 6536 (class class class)co 7404 βcc 11107 βcnβccncf 24746 limβ climc 25741 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2697 ax-rep 5278 ax-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 ax-un 7721 ax-cnex 11165 ax-resscn 11166 ax-1cn 11167 ax-icn 11168 ax-addcl 11169 ax-addrcl 11170 ax-mulcl 11171 ax-mulrcl 11172 ax-mulcom 11173 ax-addass 11174 ax-mulass 11175 ax-distr 11176 ax-i2m1 11177 ax-1ne0 11178 ax-1rid 11179 ax-rnegex 11180 ax-rrecex 11181 ax-cnre 11182 ax-pre-lttri 11183 ax-pre-lttrn 11184 ax-pre-ltadd 11185 ax-pre-mulgt0 11186 ax-pre-sup 11187 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ne 2935 df-nel 3041 df-ral 3056 df-rex 3065 df-rmo 3370 df-reu 3371 df-rab 3427 df-v 3470 df-sbc 3773 df-csb 3889 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-pss 3962 df-nul 4318 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-tp 4628 df-op 4630 df-uni 4903 df-int 4944 df-iun 4992 df-br 5142 df-opab 5204 df-mpt 5225 df-tr 5259 df-id 5567 df-eprel 5573 df-po 5581 df-so 5582 df-fr 5624 df-we 5626 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-pred 6293 df-ord 6360 df-on 6361 df-lim 6362 df-suc 6363 df-iota 6488 df-fun 6538 df-fn 6539 df-f 6540 df-f1 6541 df-fo 6542 df-f1o 6543 df-fv 6544 df-riota 7360 df-ov 7407 df-oprab 7408 df-mpo 7409 df-om 7852 df-1st 7971 df-2nd 7972 df-frecs 8264 df-wrecs 8295 df-recs 8369 df-rdg 8408 df-1o 8464 df-er 8702 df-map 8821 df-pm 8822 df-en 8939 df-dom 8940 df-sdom 8941 df-fin 8942 df-fi 9405 df-sup 9436 df-inf 9437 df-pnf 11251 df-mnf 11252 df-xr 11253 df-ltxr 11254 df-le 11255 df-sub 11447 df-neg 11448 df-div 11873 df-nn 12214 df-2 12276 df-3 12277 df-4 12278 df-5 12279 df-6 12280 df-7 12281 df-8 12282 df-9 12283 df-n0 12474 df-z 12560 df-dec 12679 df-uz 12824 df-q 12934 df-rp 12978 df-xneg 13095 df-xadd 13096 df-xmul 13097 df-fz 13488 df-seq 13970 df-exp 14030 df-cj 15049 df-re 15050 df-im 15051 df-sqrt 15185 df-abs 15186 df-struct 17086 df-slot 17121 df-ndx 17133 df-base 17151 df-plusg 17216 df-mulr 17217 df-starv 17218 df-tset 17222 df-ple 17223 df-ds 17225 df-unif 17226 df-rest 17374 df-topn 17375 df-topgen 17395 df-psmet 21227 df-xmet 21228 df-met 21229 df-bl 21230 df-mopn 21231 df-cnfld 21236 df-top 22746 df-topon 22763 df-topsp 22785 df-bases 22799 df-cn 23081 df-cnp 23082 df-xms 24176 df-ms 24177 df-cncf 24748 df-limc 25745 |
This theorem is referenced by: cnmptlimc 25769 dvcnvlem 25858 ioccncflimc 45155 icocncflimc 45159 dirkercncflem2 45374 fourierdlem84 45460 fourierdlem85 45461 fourierdlem88 45464 fourierdlem111 45487 fouriercn 45502 |
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