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Mirrors > Home > MPE Home > Th. List > rlimcn1b | Structured version Visualization version GIF version |
Description: Image of a limit under a continuous map. (Contributed by Mario Carneiro, 10-May-2016.) |
Ref | Expression |
---|---|
rlimcn1b.1 | β’ ((π β§ π β π΄) β π΅ β π) |
rlimcn1b.2 | β’ (π β πΆ β π) |
rlimcn1b.3 | β’ (π β (π β π΄ β¦ π΅) βπ πΆ) |
rlimcn1b.4 | β’ (π β πΉ:πβΆβ) |
rlimcn1b.5 | β’ ((π β§ π₯ β β+) β βπ¦ β β+ βπ§ β π ((absβ(π§ β πΆ)) < π¦ β (absβ((πΉβπ§) β (πΉβπΆ))) < π₯)) |
Ref | Expression |
---|---|
rlimcn1b | β’ (π β (π β π΄ β¦ (πΉβπ΅)) βπ (πΉβπΆ)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | rlimcn1b.4 | . . 3 β’ (π β πΉ:πβΆβ) | |
2 | rlimcn1b.1 | . . 3 β’ ((π β§ π β π΄) β π΅ β π) | |
3 | 1, 2 | cofmpt 7125 | . 2 β’ (π β (πΉ β (π β π΄ β¦ π΅)) = (π β π΄ β¦ (πΉβπ΅))) |
4 | 2 | fmpttd 7109 | . . 3 β’ (π β (π β π΄ β¦ π΅):π΄βΆπ) |
5 | rlimcn1b.2 | . . 3 β’ (π β πΆ β π) | |
6 | rlimcn1b.3 | . . 3 β’ (π β (π β π΄ β¦ π΅) βπ πΆ) | |
7 | rlimcn1b.5 | . . 3 β’ ((π β§ π₯ β β+) β βπ¦ β β+ βπ§ β π ((absβ(π§ β πΆ)) < π¦ β (absβ((πΉβπ§) β (πΉβπΆ))) < π₯)) | |
8 | 4, 5, 6, 1, 7 | rlimcn1 15535 | . 2 β’ (π β (πΉ β (π β π΄ β¦ π΅)) βπ (πΉβπΆ)) |
9 | 3, 8 | eqbrtrrd 5165 | 1 β’ (π β (π β π΄ β¦ (πΉβπ΅)) βπ (πΉβπΆ)) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ wa 395 β wcel 2098 βwral 3055 βwrex 3064 class class class wbr 5141 β¦ cmpt 5224 β ccom 5673 βΆwf 6532 βcfv 6536 (class class class)co 7404 βcc 11107 < clt 11249 β cmin 11445 β+crp 12977 abscabs 15184 βπ crli 15432 |
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 7721 ax-cnex 11165 ax-resscn 11166 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 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-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-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 6488 df-fun 6538 df-fn 6539 df-f 6540 df-fv 6544 df-ov 7407 df-oprab 7408 df-mpo 7409 df-pm 8822 df-rlim 15436 |
This theorem is referenced by: rlimabs 15556 rlimcj 15557 rlimre 15558 rlimim 15559 |
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