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Mirrors > Home > MPE Home > Th. List > cncfmpt2f | Structured version Visualization version GIF version |
Description: Composition of continuous functions. βcnβ analogue of cnmpt12f 23588. (Contributed by Mario Carneiro, 3-Sep-2014.) |
Ref | Expression |
---|---|
cncfmpt2f.1 | β’ π½ = (TopOpenββfld) |
cncfmpt2f.2 | β’ (π β πΉ β ((π½ Γt π½) Cn π½)) |
cncfmpt2f.3 | β’ (π β (π₯ β π β¦ π΄) β (πβcnββ)) |
cncfmpt2f.4 | β’ (π β (π₯ β π β¦ π΅) β (πβcnββ)) |
Ref | Expression |
---|---|
cncfmpt2f | β’ (π β (π₯ β π β¦ (π΄πΉπ΅)) β (πβcnββ)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cncfmpt2f.1 | . . . . 5 β’ π½ = (TopOpenββfld) | |
2 | 1 | cnfldtopon 24717 | . . . 4 β’ π½ β (TopOnββ) |
3 | cncfmpt2f.3 | . . . . 5 β’ (π β (π₯ β π β¦ π΄) β (πβcnββ)) | |
4 | cncfrss 24829 | . . . . 5 β’ ((π₯ β π β¦ π΄) β (πβcnββ) β π β β) | |
5 | 3, 4 | syl 17 | . . . 4 β’ (π β π β β) |
6 | resttopon 23083 | . . . 4 β’ ((π½ β (TopOnββ) β§ π β β) β (π½ βΎt π) β (TopOnβπ)) | |
7 | 2, 5, 6 | sylancr 585 | . . 3 β’ (π β (π½ βΎt π) β (TopOnβπ)) |
8 | ssid 3995 | . . . . 5 β’ β β β | |
9 | eqid 2725 | . . . . . 6 β’ (π½ βΎt π) = (π½ βΎt π) | |
10 | 2 | toponrestid 22841 | . . . . . 6 β’ π½ = (π½ βΎt β) |
11 | 1, 9, 10 | cncfcn 24848 | . . . . 5 β’ ((π β β β§ β β β) β (πβcnββ) = ((π½ βΎt π) Cn π½)) |
12 | 5, 8, 11 | sylancl 584 | . . . 4 β’ (π β (πβcnββ) = ((π½ βΎt π) Cn π½)) |
13 | 3, 12 | eleqtrd 2827 | . . 3 β’ (π β (π₯ β π β¦ π΄) β ((π½ βΎt π) Cn π½)) |
14 | cncfmpt2f.4 | . . . 4 β’ (π β (π₯ β π β¦ π΅) β (πβcnββ)) | |
15 | 14, 12 | eleqtrd 2827 | . . 3 β’ (π β (π₯ β π β¦ π΅) β ((π½ βΎt π) Cn π½)) |
16 | cncfmpt2f.2 | . . 3 β’ (π β πΉ β ((π½ Γt π½) Cn π½)) | |
17 | 7, 13, 15, 16 | cnmpt12f 23588 | . 2 β’ (π β (π₯ β π β¦ (π΄πΉπ΅)) β ((π½ βΎt π) Cn π½)) |
18 | 17, 12 | eleqtrrd 2828 | 1 β’ (π β (π₯ β π β¦ (π΄πΉπ΅)) β (πβcnββ)) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 = wceq 1533 β wcel 2098 β wss 3939 β¦ cmpt 5226 βcfv 6543 (class class class)co 7416 βcc 11136 βΎt crest 17401 TopOpenctopn 17402 βfldccnfld 21283 TopOnctopon 22830 Cn ccn 23146 Γt ctx 23482 βcnβccncf 24814 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2696 ax-rep 5280 ax-sep 5294 ax-nul 5301 ax-pow 5359 ax-pr 5423 ax-un 7738 ax-cnex 11194 ax-resscn 11195 ax-1cn 11196 ax-icn 11197 ax-addcl 11198 ax-addrcl 11199 ax-mulcl 11200 ax-mulrcl 11201 ax-mulcom 11202 ax-addass 11203 ax-mulass 11204 ax-distr 11205 ax-i2m1 11206 ax-1ne0 11207 ax-1rid 11208 ax-rnegex 11209 ax-rrecex 11210 ax-cnre 11211 ax-pre-lttri 11212 ax-pre-lttrn 11213 ax-pre-ltadd 11214 ax-pre-mulgt0 11215 ax-pre-sup 11216 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2931 df-nel 3037 df-ral 3052 df-rex 3061 df-rmo 3364 df-reu 3365 df-rab 3420 df-v 3465 df-sbc 3769 df-csb 3885 df-dif 3942 df-un 3944 df-in 3946 df-ss 3956 df-pss 3959 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-tp 4629 df-op 4631 df-uni 4904 df-int 4945 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5227 df-tr 5261 df-id 5570 df-eprel 5576 df-po 5584 df-so 5585 df-fr 5627 df-we 5629 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-pred 6300 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-riota 7372 df-ov 7419 df-oprab 7420 df-mpo 7421 df-om 7869 df-1st 7991 df-2nd 7992 df-frecs 8285 df-wrecs 8316 df-recs 8390 df-rdg 8429 df-1o 8485 df-er 8723 df-map 8845 df-en 8963 df-dom 8964 df-sdom 8965 df-fin 8966 df-fi 9434 df-sup 9465 df-inf 9466 df-pnf 11280 df-mnf 11281 df-xr 11282 df-ltxr 11283 df-le 11284 df-sub 11476 df-neg 11477 df-div 11902 df-nn 12243 df-2 12305 df-3 12306 df-4 12307 df-5 12308 df-6 12309 df-7 12310 df-8 12311 df-9 12312 df-n0 12503 df-z 12589 df-dec 12708 df-uz 12853 df-q 12963 df-rp 13007 df-xneg 13124 df-xadd 13125 df-xmul 13126 df-fz 13517 df-seq 13999 df-exp 14059 df-cj 15078 df-re 15079 df-im 15080 df-sqrt 15214 df-abs 15215 df-struct 17115 df-slot 17150 df-ndx 17162 df-base 17180 df-plusg 17245 df-mulr 17246 df-starv 17247 df-tset 17251 df-ple 17252 df-ds 17254 df-unif 17255 df-rest 17403 df-topn 17404 df-topgen 17424 df-psmet 21275 df-xmet 21276 df-met 21277 df-bl 21278 df-mopn 21279 df-cnfld 21284 df-top 22814 df-topon 22831 df-topsp 22853 df-bases 22867 df-cn 23149 df-cnp 23150 df-tx 23484 df-xms 24244 df-ms 24245 df-cncf 24816 |
This theorem is referenced by: cncfmpt2ss 24854 addccncf 24855 sub1cncf 24857 sub2cncf 24858 negcncf 24860 negcncfOLD 24861 addcncf 25390 subcncf 25391 mulcncfOLD 25393 dvcnp2OLD 25868 dvlipcn 25945 dvfsumabs 25975 ftc2 25997 itgparts 26000 taylthlem2 26327 taylthlem2OLD 26328 sincn 26399 coscn 26400 logcn 26599 loglesqrt 26711 lgamgulmlem2 26980 pntlem3 27560 logdivsqrle 34339 ftc1cnnclem 37221 ftc2nc 37232 areacirclem4 37241 areaquad 42709 |
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