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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 22965 | . . . 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 22641 | . . 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 2731 | . . . . . 6 β’ βͺ πΏ = βͺ πΏ | |
10 | eqid 2731 | . . . . . 6 β’ βͺ π = βͺ π | |
11 | 9, 10 | cnf 22971 | . . . . 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 2833 | . 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 23396 | 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 2105 βͺ cuni 4909 β¦ cmpt 5232 βΆwf 6540 βcfv 6544 (class class class)co 7412 β cmpo 7414 Topctop 22616 TopOnctopon 22633 Cn ccn 22949 Γt ctx 23285 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2153 ax-12 2170 ax-ext 2702 ax-sep 5300 ax-nul 5307 ax-pow 5364 ax-pr 5428 ax-un 7728 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-nf 1785 df-sb 2067 df-mo 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ne 2940 df-ral 3061 df-rex 3070 df-rab 3432 df-v 3475 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 7415 df-oprab 7416 df-mpo 7417 df-1st 7978 df-2nd 7979 df-map 8825 df-topgen 17394 df-top 22617 df-topon 22634 df-bases 22670 df-cn 22952 df-tx 23287 |
This theorem is referenced by: cnmpt22 23399 cnmptk2 23411 txhmeo 23528 tgpsubcn 23815 istgp2 23816 dvrcn 23909 htpyid 24724 htpyco1 24725 reparphti 24744 reparphtiOLD 24745 pcocn 24765 pcorevlem 24774 cxpcn 26486 dipcn 30237 mndpluscn 33201 cvxsconn 34529 cvmlift2lem6 34594 cvmlift2lem12 34600 gg-cxpcn 35471 |
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