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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 15475 | . . 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 7088 | . . 3 β’ (π§ β β β (π»βπ§) β β) |
8 | 7 | adantl 480 | . 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 578 | . 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 15568 | 1 β’ (π β πΊ β (π»βπ΄)) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ wa 394 = wceq 1533 β wcel 2098 βwral 3051 βwrex 3060 class class class wbr 5143 βΆwf 6539 βcfv 6543 (class class class)co 7416 βcc 11136 < clt 11278 β cmin 11474 β€cz 12588 β€β₯cuz 12852 β+crp 13006 abscabs 15213 β cli 15460 |
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 2166 ax-ext 2696 ax-sep 5294 ax-nul 5301 ax-pow 5359 ax-pr 5423 ax-un 7738 ax-cnex 11194 ax-resscn 11195 ax-pre-lttri 11212 ax-pre-lttrn 11213 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 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 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2931 df-nel 3037 df-ral 3052 df-rex 3061 df-rab 3420 df-v 3465 df-sbc 3769 df-csb 3885 df-dif 3942 df-un 3944 df-in 3946 df-ss 3956 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4904 df-br 5144 df-opab 5206 df-mpt 5227 df-id 5570 df-po 5584 df-so 5585 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-ov 7419 df-er 8723 df-en 8963 df-dom 8964 df-sdom 8965 df-pnf 11280 df-mnf 11281 df-xr 11282 df-ltxr 11283 df-le 11284 df-neg 11477 df-z 12589 df-uz 12853 df-clim 15464 |
This theorem is referenced by: climabs 15580 climcj 15581 climre 15582 climim 15583 sinccvglem 35333 |
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