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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 7060 | . 2 β’ (π β (πΉ β (π β π΄ β¦ π΅)) = (π β π΄ β¦ (πΉβπ΅))) |
4 | 2 | fmpttd 7045 | . . 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 15396 | . 2 β’ (π β (πΉ β (π β π΄ β¦ π΅)) βπ (πΉβπΆ)) |
9 | 3, 8 | eqbrtrrd 5116 | 1 β’ (π β (π β π΄ β¦ (πΉβπ΅)) βπ (πΉβπΆ)) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ wa 396 β wcel 2105 βwral 3061 βwrex 3070 class class class wbr 5092 β¦ cmpt 5175 β ccom 5624 βΆwf 6475 βcfv 6479 (class class class)co 7337 βcc 10970 < clt 11110 β cmin 11306 β+crp 12831 abscabs 15044 βπ crli 15293 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2153 ax-12 2170 ax-ext 2707 ax-sep 5243 ax-nul 5250 ax-pow 5308 ax-pr 5372 ax-un 7650 ax-cnex 11028 ax-resscn 11029 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-nf 1785 df-sb 2067 df-mo 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2886 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3404 df-v 3443 df-sbc 3728 df-csb 3844 df-dif 3901 df-un 3903 df-in 3905 df-ss 3915 df-nul 4270 df-if 4474 df-pw 4549 df-sn 4574 df-pr 4576 df-op 4580 df-uni 4853 df-br 5093 df-opab 5155 df-mpt 5176 df-id 5518 df-xp 5626 df-rel 5627 df-cnv 5628 df-co 5629 df-dm 5630 df-rn 5631 df-res 5632 df-ima 5633 df-iota 6431 df-fun 6481 df-fn 6482 df-f 6483 df-fv 6487 df-ov 7340 df-oprab 7341 df-mpo 7342 df-pm 8689 df-rlim 15297 |
This theorem is referenced by: rlimabs 15417 rlimcj 15418 rlimre 15419 rlimim 15420 |
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