![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
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 22615 | . . . 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 22290 | . . 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 2733 | . . . . . 6 β’ βͺ πΎ = βͺ πΎ | |
9 | eqid 2733 | . . . . . 6 β’ βͺ πΏ = βͺ πΏ | |
10 | 8, 9 | cnf 22620 | . . . . 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 6914 | . . 3 β’ (π β πΉ = (π¦ β βͺ πΎ β¦ (πΉβπ¦))) |
13 | 12, 7 | eqeltrrd 2835 | . 2 β’ (π β (π¦ β βͺ πΎ β¦ (πΉβπ¦)) β (πΎ Cn πΏ)) |
14 | fveq2 6846 | . 2 β’ (π¦ = π΄ β (πΉβπ¦) = (πΉβπ΄)) | |
15 | 1, 2, 6, 13, 14 | cnmpt11 23037 | 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 2107 βͺ cuni 4869 β¦ cmpt 5192 βΆwf 6496 βcfv 6500 (class class class)co 7361 Topctop 22265 TopOnctopon 22282 Cn ccn 22598 |
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 5260 ax-nul 5267 ax-pow 5324 ax-pr 5388 ax-un 7676 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 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-ral 3062 df-rex 3071 df-rab 3407 df-v 3449 df-sbc 3744 df-csb 3860 df-dif 3917 df-un 3919 df-in 3921 df-ss 3931 df-nul 4287 df-if 4491 df-pw 4566 df-sn 4591 df-pr 4593 df-op 4597 df-uni 4870 df-br 5110 df-opab 5172 df-mpt 5193 df-id 5535 df-xp 5643 df-rel 5644 df-cnv 5645 df-co 5646 df-dm 5647 df-rn 5648 df-res 5649 df-ima 5650 df-iota 6452 df-fun 6502 df-fn 6503 df-f 6504 df-fv 6508 df-ov 7364 df-oprab 7365 df-mpo 7366 df-map 8773 df-top 22266 df-topon 22283 df-cn 22601 |
This theorem is referenced by: cnmpt12f 23040 tgpmulg 23467 prdstgpd 23499 pcorevcl 24411 pcorevlem 24412 logcn 26025 loglesqrt 26134 efrlim 26342 cvmliftlem8 33950 knoppcnlem10 35018 areacirclem2 36217 areacirclem4 36219 |
Copyright terms: Public domain | W3C validator |