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Mirrors > Home > MPE Home > Th. List > cnmpt21f | 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 |
---|---|
cnmpt21.j | β’ (π β π½ β (TopOnβπ)) |
cnmpt21.k | β’ (π β πΎ β (TopOnβπ)) |
cnmpt21.a | β’ (π β (π₯ β π, π¦ β π β¦ π΄) β ((π½ Γt πΎ) Cn πΏ)) |
cnmpt21f.f | β’ (π β πΉ β (πΏ Cn π)) |
Ref | Expression |
---|---|
cnmpt21f | β’ (π β (π₯ β π, π¦ β π β¦ (πΉβπ΄)) β ((π½ Γt πΎ) Cn π)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cnmpt21.j | . 2 β’ (π β π½ β (TopOnβπ)) | |
2 | cnmpt21.k | . 2 β’ (π β πΎ β (TopOnβπ)) | |
3 | cnmpt21.a | . 2 β’ (π β (π₯ β π, π¦ β π β¦ π΄) β ((π½ Γt πΎ) Cn πΏ)) | |
4 | cnmpt21f.f | . . . 4 β’ (π β πΉ β (πΏ Cn π)) | |
5 | cntop1 22614 | . . . 4 β’ (πΉ β (πΏ Cn π) β πΏ β Top) | |
6 | 4, 5 | syl 17 | . . 3 β’ (π β πΏ β Top) |
7 | toptopon2 22290 | . . 3 β’ (πΏ β Top β πΏ β (TopOnββͺ πΏ)) | |
8 | 6, 7 | sylib 217 | . 2 β’ (π β πΏ β (TopOnββͺ πΏ)) |
9 | eqid 2733 | . . . . . 6 β’ βͺ πΏ = βͺ πΏ | |
10 | eqid 2733 | . . . . . 6 β’ βͺ π = βͺ π | |
11 | 9, 10 | cnf 22620 | . . . . 5 β’ (πΉ β (πΏ Cn π) β πΉ:βͺ πΏβΆβͺ π) |
12 | 4, 11 | syl 17 | . . . 4 β’ (π β πΉ:βͺ πΏβΆβͺ π) |
13 | 12 | feqmptd 6914 | . . 3 β’ (π β πΉ = (π§ β βͺ πΏ β¦ (πΉβπ§))) |
14 | 13, 4 | eqeltrrd 2835 | . 2 β’ (π β (π§ β βͺ πΏ β¦ (πΉβπ§)) β (πΏ Cn π)) |
15 | fveq2 6846 | . 2 β’ (π§ = π΄ β (πΉβπ§) = (πΉβπ΄)) | |
16 | 1, 2, 3, 8, 14, 15 | cnmpt21 23045 | 1 β’ (π β (π₯ β π, π¦ β π β¦ (πΉβπ΄)) β ((π½ Γt πΎ) Cn π)) |
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 β cmpo 7363 Topctop 22265 TopOnctopon 22282 Cn ccn 22598 Γt ctx 22934 |
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-iun 4960 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-1st 7925 df-2nd 7926 df-map 8773 df-topgen 17333 df-top 22266 df-topon 22283 df-bases 22319 df-cn 22601 df-tx 22936 |
This theorem is referenced by: cnmpt22 23048 cnmptk2 23060 txhmeo 23177 tgpsubcn 23464 istgp2 23465 dvrcn 23558 htpyid 24363 htpyco1 24364 reparphti 24383 pcocn 24403 pcorevlem 24412 cxpcn 26121 dipcn 29711 mndpluscn 32571 cvxsconn 33901 cvmlift2lem6 33966 cvmlift2lem12 33972 |
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