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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 7147 | . 2 β’ (π β (πΉ β (π β π΄ β¦ π΅)) = (π β π΄ β¦ (πΉβπ΅))) |
4 | 2 | fmpttd 7130 | . . 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 15572 | . 2 β’ (π β (πΉ β (π β π΄ β¦ π΅)) βπ (πΉβπΆ)) |
9 | 3, 8 | eqbrtrrd 5176 | 1 β’ (π β (π β π΄ β¦ (πΉβπ΅)) βπ (πΉβπΆ)) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ wa 394 β wcel 2098 βwral 3058 βwrex 3067 class class class wbr 5152 β¦ cmpt 5235 β ccom 5686 βΆwf 6549 βcfv 6553 (class class class)co 7426 βcc 11144 < clt 11286 β cmin 11482 β+crp 13014 abscabs 15221 βπ crli 15469 |
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 2699 ax-sep 5303 ax-nul 5310 ax-pow 5369 ax-pr 5433 ax-un 7746 ax-cnex 11202 ax-resscn 11203 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2529 df-eu 2558 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-ral 3059 df-rex 3068 df-rab 3431 df-v 3475 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-nul 4327 df-if 4533 df-pw 4608 df-sn 4633 df-pr 4635 df-op 4639 df-uni 4913 df-br 5153 df-opab 5215 df-mpt 5236 df-id 5580 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-rn 5693 df-res 5694 df-ima 5695 df-iota 6505 df-fun 6555 df-fn 6556 df-f 6557 df-fv 6561 df-ov 7429 df-oprab 7430 df-mpo 7431 df-pm 8854 df-rlim 15473 |
This theorem is referenced by: rlimabs 15593 rlimcj 15594 rlimre 15595 rlimim 15596 |
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