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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 22744 | . . . 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 22420 | . . 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 22750 | . . . . 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 6961 | . . 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 6892 | . 2 β’ (π§ = π΄ β (πΉβπ§) = (πΉβπ΄)) | |
16 | 1, 2, 3, 8, 14, 15 | cnmpt21 23175 | 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 4909 β¦ cmpt 5232 βΆwf 6540 βcfv 6544 (class class class)co 7409 β cmpo 7411 Topctop 22395 TopOnctopon 22412 Cn ccn 22728 Γt ctx 23064 |
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 5300 ax-nul 5307 ax-pow 5364 ax-pr 5428 ax-un 7725 |
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 2942 df-ral 3063 df-rex 3072 df-rab 3434 df-v 3477 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4910 df-iun 5000 df-br 5150 df-opab 5212 df-mpt 5233 df-id 5575 df-xp 5683 df-rel 5684 df-cnv 5685 df-co 5686 df-dm 5687 df-rn 5688 df-res 5689 df-ima 5690 df-iota 6496 df-fun 6546 df-fn 6547 df-f 6548 df-fv 6552 df-ov 7412 df-oprab 7413 df-mpo 7414 df-1st 7975 df-2nd 7976 df-map 8822 df-topgen 17389 df-top 22396 df-topon 22413 df-bases 22449 df-cn 22731 df-tx 23066 |
This theorem is referenced by: cnmpt22 23178 cnmptk2 23190 txhmeo 23307 tgpsubcn 23594 istgp2 23595 dvrcn 23688 htpyid 24493 htpyco1 24494 reparphti 24513 pcocn 24533 pcorevlem 24542 cxpcn 26253 dipcn 29973 mndpluscn 32906 cvxsconn 34234 cvmlift2lem6 34299 cvmlift2lem12 34305 gg-reparphti 35172 gg-cxpcn 35184 |
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