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Mirrors > Home > MPE Home > Th. List > cnmpt12f | 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)) |
cnmpt1t.b | β’ (π β (π₯ β π β¦ π΅) β (π½ Cn πΏ)) |
cnmpt12f.f | β’ (π β πΉ β ((πΎ Γt πΏ) Cn π)) |
Ref | Expression |
---|---|
cnmpt12f | β’ (π β (π₯ β π β¦ (π΄πΉπ΅)) β (π½ Cn π)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-ov 7415 | . . 3 β’ (π΄πΉπ΅) = (πΉββ¨π΄, π΅β©) | |
2 | 1 | mpteq2i 5253 | . 2 β’ (π₯ β π β¦ (π΄πΉπ΅)) = (π₯ β π β¦ (πΉββ¨π΄, π΅β©)) |
3 | cnmptid.j | . . 3 β’ (π β π½ β (TopOnβπ)) | |
4 | cnmpt11.a | . . . 4 β’ (π β (π₯ β π β¦ π΄) β (π½ Cn πΎ)) | |
5 | cnmpt1t.b | . . . 4 β’ (π β (π₯ β π β¦ π΅) β (π½ Cn πΏ)) | |
6 | 3, 4, 5 | cnmpt1t 23390 | . . 3 β’ (π β (π₯ β π β¦ β¨π΄, π΅β©) β (π½ Cn (πΎ Γt πΏ))) |
7 | cnmpt12f.f | . . 3 β’ (π β πΉ β ((πΎ Γt πΏ) Cn π)) | |
8 | 3, 6, 7 | cnmpt11f 23389 | . 2 β’ (π β (π₯ β π β¦ (πΉββ¨π΄, π΅β©)) β (π½ Cn π)) |
9 | 2, 8 | eqeltrid 2836 | 1 β’ (π β (π₯ β π β¦ (π΄πΉπ΅)) β (π½ Cn π)) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β wcel 2105 β¨cop 4634 β¦ cmpt 5231 βcfv 6543 (class class class)co 7412 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 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 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 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 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: cnmpt12 23392 cnmpt1plusg 23812 istgp2 23816 clsnsg 23835 tgpt0 23844 cnmpt1vsca 23919 cnmpt1ds 24579 fsumcn 24609 expcn 24611 expcnOLD 24613 divccnOLD 24614 cncfmpt2f 24656 cdivcncf 24662 iirevcn 24672 iihalf1cnOLD 24675 iihalf2cn 24677 iihalf2cnOLD 24678 icchmeo 24686 icchmeoOLD 24687 evth 24706 evth2 24707 pcoass 24772 cnmpt1ip 24996 dvcnvlem 25729 plycnOLD 26012 psercn2 26172 atansopn 26674 efrlim 26711 ipasslem7 30357 occllem 30824 hmopidmchi 31672 cvxpconn 34532 cvmlift2lem2 34594 cvmlift2lem3 34595 cvmliftphtlem 34607 sinccvglem 34956 knoppcnlem10 35682 broucube 36826 areacirclem2 36881 fprodcnlem 44614 |
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