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Mirrors > Home > MPE Home > Th. List > cncfmptid | Structured version Visualization version GIF version |
Description: The identity function is a continuous function on β. (Contributed by Jeff Madsen, 11-Jun-2010.) (Revised by Mario Carneiro, 17-May-2016.) |
Ref | Expression |
---|---|
cncfmptid | β’ ((π β π β§ π β β) β (π₯ β π β¦ π₯) β (πβcnβπ)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cncfss 24740 | . 2 β’ ((π β π β§ π β β) β (πβcnβπ) β (πβcnβπ)) | |
2 | eqid 2724 | . . . . . 6 β’ (TopOpenββfld) = (TopOpenββfld) | |
3 | 2 | cnfldtopon 24620 | . . . . 5 β’ (TopOpenββfld) β (TopOnββ) |
4 | sstr 3982 | . . . . 5 β’ ((π β π β§ π β β) β π β β) | |
5 | resttopon 22986 | . . . . 5 β’ (((TopOpenββfld) β (TopOnββ) β§ π β β) β ((TopOpenββfld) βΎt π) β (TopOnβπ)) | |
6 | 3, 4, 5 | sylancr 586 | . . . 4 β’ ((π β π β§ π β β) β ((TopOpenββfld) βΎt π) β (TopOnβπ)) |
7 | 6 | cnmptid 23486 | . . 3 β’ ((π β π β§ π β β) β (π₯ β π β¦ π₯) β (((TopOpenββfld) βΎt π) Cn ((TopOpenββfld) βΎt π))) |
8 | eqid 2724 | . . . . 5 β’ ((TopOpenββfld) βΎt π) = ((TopOpenββfld) βΎt π) | |
9 | 2, 8, 8 | cncfcn 24751 | . . . 4 β’ ((π β β β§ π β β) β (πβcnβπ) = (((TopOpenββfld) βΎt π) Cn ((TopOpenββfld) βΎt π))) |
10 | 4, 4, 9 | syl2anc 583 | . . 3 β’ ((π β π β§ π β β) β (πβcnβπ) = (((TopOpenββfld) βΎt π) Cn ((TopOpenββfld) βΎt π))) |
11 | 7, 10 | eleqtrrd 2828 | . 2 β’ ((π β π β§ π β β) β (π₯ β π β¦ π₯) β (πβcnβπ)) |
12 | 1, 11 | sseldd 3975 | 1 β’ ((π β π β§ π β β) β (π₯ β π β¦ π₯) β (πβcnβπ)) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ wa 395 = wceq 1533 β wcel 2098 β wss 3940 β¦ cmpt 5221 βcfv 6533 (class class class)co 7401 βcc 11103 βΎt crest 17364 TopOpenctopn 17365 βfldccnfld 21227 TopOnctopon 22733 Cn ccn 23049 βcnβccncf 24717 |
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 2163 ax-ext 2695 ax-rep 5275 ax-sep 5289 ax-nul 5296 ax-pow 5353 ax-pr 5417 ax-un 7718 ax-cnex 11161 ax-resscn 11162 ax-1cn 11163 ax-icn 11164 ax-addcl 11165 ax-addrcl 11166 ax-mulcl 11167 ax-mulrcl 11168 ax-mulcom 11169 ax-addass 11170 ax-mulass 11171 ax-distr 11172 ax-i2m1 11173 ax-1ne0 11174 ax-1rid 11175 ax-rnegex 11176 ax-rrecex 11177 ax-cnre 11178 ax-pre-lttri 11179 ax-pre-lttrn 11180 ax-pre-ltadd 11181 ax-pre-mulgt0 11182 ax-pre-sup 11183 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2526 df-eu 2555 df-clab 2702 df-cleq 2716 df-clel 2802 df-nfc 2877 df-ne 2933 df-nel 3039 df-ral 3054 df-rex 3063 df-rmo 3368 df-reu 3369 df-rab 3425 df-v 3468 df-sbc 3770 df-csb 3886 df-dif 3943 df-un 3945 df-in 3947 df-ss 3957 df-pss 3959 df-nul 4315 df-if 4521 df-pw 4596 df-sn 4621 df-pr 4623 df-tp 4625 df-op 4627 df-uni 4900 df-int 4941 df-iun 4989 df-br 5139 df-opab 5201 df-mpt 5222 df-tr 5256 df-id 5564 df-eprel 5570 df-po 5578 df-so 5579 df-fr 5621 df-we 5623 df-xp 5672 df-rel 5673 df-cnv 5674 df-co 5675 df-dm 5676 df-rn 5677 df-res 5678 df-ima 5679 df-pred 6290 df-ord 6357 df-on 6358 df-lim 6359 df-suc 6360 df-iota 6485 df-fun 6535 df-fn 6536 df-f 6537 df-f1 6538 df-fo 6539 df-f1o 6540 df-fv 6541 df-riota 7357 df-ov 7404 df-oprab 7405 df-mpo 7406 df-om 7849 df-1st 7968 df-2nd 7969 df-frecs 8261 df-wrecs 8292 df-recs 8366 df-rdg 8405 df-1o 8461 df-er 8698 df-map 8817 df-en 8935 df-dom 8936 df-sdom 8937 df-fin 8938 df-fi 9401 df-sup 9432 df-inf 9433 df-pnf 11246 df-mnf 11247 df-xr 11248 df-ltxr 11249 df-le 11250 df-sub 11442 df-neg 11443 df-div 11868 df-nn 12209 df-2 12271 df-3 12272 df-4 12273 df-5 12274 df-6 12275 df-7 12276 df-8 12277 df-9 12278 df-n0 12469 df-z 12555 df-dec 12674 df-uz 12819 df-q 12929 df-rp 12971 df-xneg 13088 df-xadd 13089 df-xmul 13090 df-fz 13481 df-seq 13963 df-exp 14024 df-cj 15042 df-re 15043 df-im 15044 df-sqrt 15178 df-abs 15179 df-struct 17078 df-slot 17113 df-ndx 17125 df-base 17143 df-plusg 17208 df-mulr 17209 df-starv 17210 df-tset 17214 df-ple 17215 df-ds 17217 df-unif 17218 df-rest 17366 df-topn 17367 df-topgen 17387 df-psmet 21219 df-xmet 21220 df-met 21221 df-bl 21222 df-mopn 21223 df-cnfld 21228 df-top 22717 df-topon 22734 df-topsp 22756 df-bases 22770 df-cn 23052 df-cnp 23053 df-xms 24147 df-ms 24148 df-cncf 24719 |
This theorem is referenced by: addccncf 24758 idcncf 24759 negcncf 24763 negcncfOLD 24764 dvcnp2 25770 dvcnp2OLD 25771 mvth 25846 dvlipcn 25848 dvfsumle 25875 dvfsumleOLD 25876 dvfsumabs 25878 dvfsumlem2 25882 dvfsumlem2OLD 25883 taylthlem2 26226 loglesqrt 26608 lgamgulmlem2 26877 pntlem3 27457 iblidicc 34059 circlemeth 34107 logdivsqrle 34117 gg-taylthlem2 35623 areacirclem4 37035 lcmineqlem12 41364 areaquad 42420 idcncfg 45040 addccncf2 45043 add1cncf 45068 add2cncf 45069 sub1cncfd 45070 sub2cncfd 45071 itgsbtaddcnst 45149 dirkercncflem2 45271 fourierdlem16 45290 fourierdlem22 45296 fourierdlem93 45366 fourierdlem111 45384 |
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