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Mirrors > Home > MPE Home > Th. List > cnmpt11f | Structured version Visualization version GIF version |
Description: The composition of continuous functions is continuous. (Contributed by Mario Carneiro, 5-May-2014.) (Revised by Mario Carneiro, 22-Aug-2015.) |
Ref | Expression |
---|---|
cnmptid.j | β’ (π β π½ β (TopOnβπ)) |
cnmpt11.a | β’ (π β (π₯ β π β¦ π΄) β (π½ Cn πΎ)) |
cnmpt11f.f | β’ (π β πΉ β (πΎ Cn πΏ)) |
Ref | Expression |
---|---|
cnmpt11f | β’ (π β (π₯ β π β¦ (πΉβπ΄)) β (π½ Cn πΏ)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cnmptid.j | . 2 β’ (π β π½ β (TopOnβπ)) | |
2 | cnmpt11.a | . 2 β’ (π β (π₯ β π β¦ π΄) β (π½ Cn πΎ)) | |
3 | cntop2 23165 | . . . 4 β’ ((π₯ β π β¦ π΄) β (π½ Cn πΎ) β πΎ β Top) | |
4 | 2, 3 | syl 17 | . . 3 β’ (π β πΎ β Top) |
5 | toptopon2 22840 | . . 3 β’ (πΎ β Top β πΎ β (TopOnββͺ πΎ)) | |
6 | 4, 5 | sylib 217 | . 2 β’ (π β πΎ β (TopOnββͺ πΎ)) |
7 | cnmpt11f.f | . . . . 5 β’ (π β πΉ β (πΎ Cn πΏ)) | |
8 | eqid 2728 | . . . . . 6 β’ βͺ πΎ = βͺ πΎ | |
9 | eqid 2728 | . . . . . 6 β’ βͺ πΏ = βͺ πΏ | |
10 | 8, 9 | cnf 23170 | . . . . 5 β’ (πΉ β (πΎ Cn πΏ) β πΉ:βͺ πΎβΆβͺ πΏ) |
11 | 7, 10 | syl 17 | . . . 4 β’ (π β πΉ:βͺ πΎβΆβͺ πΏ) |
12 | 11 | feqmptd 6972 | . . 3 β’ (π β πΉ = (π¦ β βͺ πΎ β¦ (πΉβπ¦))) |
13 | 12, 7 | eqeltrrd 2830 | . 2 β’ (π β (π¦ β βͺ πΎ β¦ (πΉβπ¦)) β (πΎ Cn πΏ)) |
14 | fveq2 6902 | . 2 β’ (π¦ = π΄ β (πΉβπ¦) = (πΉβπ΄)) | |
15 | 1, 2, 6, 13, 14 | cnmpt11 23587 | 1 β’ (π β (π₯ β π β¦ (πΉβπ΄)) β (π½ Cn πΏ)) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β wcel 2098 βͺ cuni 4912 β¦ cmpt 5235 βΆwf 6549 βcfv 6553 (class class class)co 7426 Topctop 22815 TopOnctopon 22832 Cn ccn 23148 |
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 |
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-map 8853 df-top 22816 df-topon 22833 df-cn 23151 |
This theorem is referenced by: cnmpt12f 23590 tgpmulg 24017 prdstgpd 24049 pcorevcl 24972 pcorevlem 24973 logcn 26601 loglesqrt 26713 efrlim 26921 efrlimOLD 26922 cvmliftlem8 34935 knoppcnlem10 36010 areacirclem2 37215 areacirclem4 37217 |
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