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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 22743 | . . . 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 22419 | . . 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 2732 | . . . . . 6 β’ βͺ πΏ = βͺ πΏ | |
10 | eqid 2732 | . . . . . 6 β’ βͺ π = βͺ π | |
11 | 9, 10 | cnf 22749 | . . . . 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 6960 | . . 3 β’ (π β πΉ = (π§ β βͺ πΏ β¦ (πΉβπ§))) |
14 | 13, 4 | eqeltrrd 2834 | . 2 β’ (π β (π§ β βͺ πΏ β¦ (πΉβπ§)) β (πΏ Cn π)) |
15 | fveq2 6891 | . 2 β’ (π§ = π΄ β (πΉβπ§) = (πΉβπ΄)) | |
16 | 1, 2, 3, 8, 14, 15 | cnmpt21 23174 | 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 2106 βͺ cuni 4908 β¦ cmpt 5231 βΆwf 6539 βcfv 6543 (class class class)co 7408 β cmpo 7410 Topctop 22394 TopOnctopon 22411 Cn ccn 22727 Γt ctx 23063 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2703 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7724 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3433 df-v 3476 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-iun 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5574 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-fv 6551 df-ov 7411 df-oprab 7412 df-mpo 7413 df-1st 7974 df-2nd 7975 df-map 8821 df-topgen 17388 df-top 22395 df-topon 22412 df-bases 22448 df-cn 22730 df-tx 23065 |
This theorem is referenced by: cnmpt22 23177 cnmptk2 23189 txhmeo 23306 tgpsubcn 23593 istgp2 23594 dvrcn 23687 htpyid 24492 htpyco1 24493 reparphti 24512 pcocn 24532 pcorevlem 24541 cxpcn 26250 dipcn 29968 mndpluscn 32901 cvxsconn 34229 cvmlift2lem6 34294 cvmlift2lem12 34300 gg-reparphti 35167 gg-cxpcn 35179 |
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