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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 7361 | . . 3 β’ (π΄πΉπ΅) = (πΉββ¨π΄, π΅β©) | |
2 | 1 | mpteq2i 5211 | . 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 23032 | . . 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 23031 | . 2 β’ (π β (π₯ β π β¦ (πΉββ¨π΄, π΅β©)) β (π½ Cn π)) |
9 | 2, 8 | eqeltrid 2838 | 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 2107 β¨cop 4593 β¦ cmpt 5189 βcfv 6497 (class class class)co 7358 TopOnctopon 22275 Cn ccn 22591 Γt ctx 22927 |
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 5257 ax-nul 5264 ax-pow 5321 ax-pr 5385 ax-un 7673 |
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 3446 df-sbc 3741 df-csb 3857 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-nul 4284 df-if 4488 df-pw 4563 df-sn 4588 df-pr 4590 df-op 4594 df-uni 4867 df-iun 4957 df-br 5107 df-opab 5169 df-mpt 5190 df-id 5532 df-xp 5640 df-rel 5641 df-cnv 5642 df-co 5643 df-dm 5644 df-rn 5645 df-res 5646 df-ima 5647 df-iota 6449 df-fun 6499 df-fn 6500 df-f 6501 df-fv 6505 df-ov 7361 df-oprab 7362 df-mpo 7363 df-1st 7922 df-2nd 7923 df-map 8770 df-topgen 17330 df-top 22259 df-topon 22276 df-bases 22312 df-cn 22594 df-tx 22929 |
This theorem is referenced by: cnmpt12 23034 cnmpt1plusg 23454 istgp2 23458 clsnsg 23477 tgpt0 23486 cnmpt1vsca 23561 cnmpt1ds 24221 fsumcn 24249 expcn 24251 divccn 24252 cncfmpt2f 24294 cdivcncf 24300 iirevcn 24309 iihalf1cn 24311 iihalf2cn 24313 icchmeo 24320 evth 24338 evth2 24339 pcoass 24403 cnmpt1ip 24627 dvcnvlem 25356 plycn 25638 psercn2 25798 atansopn 26298 efrlim 26335 ipasslem7 29820 occllem 30287 hmopidmchi 31135 cvxpconn 33893 cvmlift2lem2 33955 cvmlift2lem3 33956 cvmliftphtlem 33968 sinccvglem 34317 knoppcnlem10 35011 broucube 36158 areacirclem2 36213 fprodcnlem 43926 |
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