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Mirrors > Home > MPE Home > Th. List > climcn1lem | Structured version Visualization version GIF version |
Description: The limit of a continuous function, theorem form. (Contributed by Mario Carneiro, 9-Feb-2014.) |
Ref | Expression |
---|---|
climcn1lem.1 | β’ π = (β€β₯βπ) |
climcn1lem.2 | β’ (π β πΉ β π΄) |
climcn1lem.4 | β’ (π β πΊ β π) |
climcn1lem.5 | β’ (π β π β β€) |
climcn1lem.6 | β’ ((π β§ π β π) β (πΉβπ) β β) |
climcn1lem.7 | β’ π»:ββΆβ |
climcn1lem.8 | β’ ((π΄ β β β§ π₯ β β+) β βπ¦ β β+ βπ§ β β ((absβ(π§ β π΄)) < π¦ β (absβ((π»βπ§) β (π»βπ΄))) < π₯)) |
climcn1lem.9 | β’ ((π β§ π β π) β (πΊβπ) = (π»β(πΉβπ))) |
Ref | Expression |
---|---|
climcn1lem | β’ (π β πΊ β (π»βπ΄)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | climcn1lem.1 | . 2 β’ π = (β€β₯βπ) | |
2 | climcn1lem.5 | . 2 β’ (π β π β β€) | |
3 | climcn1lem.2 | . . 3 β’ (π β πΉ β π΄) | |
4 | climcl 15449 | . . 3 β’ (πΉ β π΄ β π΄ β β) | |
5 | 3, 4 | syl 17 | . 2 β’ (π β π΄ β β) |
6 | climcn1lem.7 | . . . 4 β’ π»:ββΆβ | |
7 | 6 | ffvelcdmi 7079 | . . 3 β’ (π§ β β β (π»βπ§) β β) |
8 | 7 | adantl 481 | . 2 β’ ((π β§ π§ β β) β (π»βπ§) β β) |
9 | climcn1lem.4 | . 2 β’ (π β πΊ β π) | |
10 | climcn1lem.8 | . . 3 β’ ((π΄ β β β§ π₯ β β+) β βπ¦ β β+ βπ§ β β ((absβ(π§ β π΄)) < π¦ β (absβ((π»βπ§) β (π»βπ΄))) < π₯)) | |
11 | 5, 10 | sylan 579 | . 2 β’ ((π β§ π₯ β β+) β βπ¦ β β+ βπ§ β β ((absβ(π§ β π΄)) < π¦ β (absβ((π»βπ§) β (π»βπ΄))) < π₯)) |
12 | climcn1lem.6 | . 2 β’ ((π β§ π β π) β (πΉβπ) β β) | |
13 | climcn1lem.9 | . 2 β’ ((π β§ π β π) β (πΊβπ) = (π»β(πΉβπ))) | |
14 | 1, 2, 5, 8, 3, 9, 11, 12, 13 | climcn1 15542 | 1 β’ (π β πΊ β (π»βπ΄)) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ wa 395 = wceq 1533 β wcel 2098 βwral 3055 βwrex 3064 class class class wbr 5141 βΆwf 6533 βcfv 6537 (class class class)co 7405 βcc 11110 < clt 11252 β cmin 11448 β€cz 12562 β€β₯cuz 12826 β+crp 12980 abscabs 15187 β cli 15434 |
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-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 ax-un 7722 ax-cnex 11168 ax-resscn 11169 ax-pre-lttri 11186 ax-pre-lttrn 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-rab 3427 df-v 3470 df-sbc 3773 df-csb 3889 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-nul 4318 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4903 df-br 5142 df-opab 5204 df-mpt 5225 df-id 5567 df-po 5581 df-so 5582 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-iota 6489 df-fun 6539 df-fn 6540 df-f 6541 df-f1 6542 df-fo 6543 df-f1o 6544 df-fv 6545 df-ov 7408 df-er 8705 df-en 8942 df-dom 8943 df-sdom 8944 df-pnf 11254 df-mnf 11255 df-xr 11256 df-ltxr 11257 df-le 11258 df-neg 11451 df-z 12563 df-uz 12827 df-clim 15438 |
This theorem is referenced by: climabs 15554 climcj 15555 climre 15556 climim 15557 sinccvglem 35185 |
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