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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 15387 | . . 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 7035 | . . 3 β’ (π§ β β β (π»βπ§) β β) |
8 | 7 | adantl 483 | . 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 581 | . 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 15480 | 1 β’ (π β πΊ β (π»βπ΄)) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ wa 397 = wceq 1542 β wcel 2107 βwral 3061 βwrex 3070 class class class wbr 5106 βΆwf 6493 βcfv 6497 (class class class)co 7358 βcc 11054 < clt 11194 β cmin 11390 β€cz 12504 β€β₯cuz 12768 β+crp 12920 abscabs 15125 β cli 15372 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-sep 5257 ax-nul 5264 ax-pow 5321 ax-pr 5385 ax-un 7673 ax-cnex 11112 ax-resscn 11113 ax-pre-lttri 11130 ax-pre-lttrn 11131 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rab 3407 df-v 3446 df-sbc 3741 df-csb 3857 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-nul 4284 df-if 4488 df-pw 4563 df-sn 4588 df-pr 4590 df-op 4594 df-uni 4867 df-br 5107 df-opab 5169 df-mpt 5190 df-id 5532 df-po 5546 df-so 5547 df-xp 5640 df-rel 5641 df-cnv 5642 df-co 5643 df-dm 5644 df-rn 5645 df-res 5646 df-ima 5647 df-iota 6449 df-fun 6499 df-fn 6500 df-f 6501 df-f1 6502 df-fo 6503 df-f1o 6504 df-fv 6505 df-ov 7361 df-er 8651 df-en 8887 df-dom 8888 df-sdom 8889 df-pnf 11196 df-mnf 11197 df-xr 11198 df-ltxr 11199 df-le 11200 df-neg 11393 df-z 12505 df-uz 12769 df-clim 15376 |
This theorem is referenced by: climabs 15492 climcj 15493 climre 15494 climim 15495 sinccvglem 34317 |
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