![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > cncfmptc | Structured version Visualization version GIF version |
Description: A constant function is a continuous function on β. (Contributed by Jeff Madsen, 2-Sep-2009.) (Revised by Mario Carneiro, 7-Sep-2015.) |
Ref | Expression |
---|---|
cncfmptc | β’ ((π΄ β π β§ π β β β§ π β β) β (π₯ β π β¦ π΄) β (πβcnβπ)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2726 | . . . . 5 β’ (TopOpenββfld) = (TopOpenββfld) | |
2 | 1 | cnfldtopon 24654 | . . . 4 β’ (TopOpenββfld) β (TopOnββ) |
3 | simp2 1134 | . . . 4 β’ ((π΄ β π β§ π β β β§ π β β) β π β β) | |
4 | resttopon 23020 | . . . 4 β’ (((TopOpenββfld) β (TopOnββ) β§ π β β) β ((TopOpenββfld) βΎt π) β (TopOnβπ)) | |
5 | 2, 3, 4 | sylancr 586 | . . 3 β’ ((π΄ β π β§ π β β β§ π β β) β ((TopOpenββfld) βΎt π) β (TopOnβπ)) |
6 | simp3 1135 | . . . 4 β’ ((π΄ β π β§ π β β β§ π β β) β π β β) | |
7 | resttopon 23020 | . . . 4 β’ (((TopOpenββfld) β (TopOnββ) β§ π β β) β ((TopOpenββfld) βΎt π) β (TopOnβπ)) | |
8 | 2, 6, 7 | sylancr 586 | . . 3 β’ ((π΄ β π β§ π β β β§ π β β) β ((TopOpenββfld) βΎt π) β (TopOnβπ)) |
9 | simp1 1133 | . . 3 β’ ((π΄ β π β§ π β β β§ π β β) β π΄ β π) | |
10 | 5, 8, 9 | cnmptc 23521 | . 2 β’ ((π΄ β π β§ π β β β§ π β β) β (π₯ β π β¦ π΄) β (((TopOpenββfld) βΎt π) Cn ((TopOpenββfld) βΎt π))) |
11 | eqid 2726 | . . . 4 β’ ((TopOpenββfld) βΎt π) = ((TopOpenββfld) βΎt π) | |
12 | eqid 2726 | . . . 4 β’ ((TopOpenββfld) βΎt π) = ((TopOpenββfld) βΎt π) | |
13 | 1, 11, 12 | cncfcn 24785 | . . 3 β’ ((π β β β§ π β β) β (πβcnβπ) = (((TopOpenββfld) βΎt π) Cn ((TopOpenββfld) βΎt π))) |
14 | 13 | 3adant1 1127 | . 2 β’ ((π΄ β π β§ π β β β§ π β β) β (πβcnβπ) = (((TopOpenββfld) βΎt π) Cn ((TopOpenββfld) βΎt π))) |
15 | 10, 14 | eleqtrrd 2830 | 1 β’ ((π΄ β π β§ π β β β§ π β β) β (π₯ β π β¦ π΄) β (πβcnβπ)) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ w3a 1084 = wceq 1533 β wcel 2098 β wss 3943 β¦ cmpt 5224 βcfv 6537 (class class class)co 7405 βcc 11110 βΎt crest 17375 TopOpenctopn 17376 βfldccnfld 21240 TopOnctopon 22767 Cn ccn 23083 βcnβccncf 24751 |
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 2697 ax-rep 5278 ax-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 ax-un 7722 ax-cnex 11168 ax-resscn 11169 ax-1cn 11170 ax-icn 11171 ax-addcl 11172 ax-addrcl 11173 ax-mulcl 11174 ax-mulrcl 11175 ax-mulcom 11176 ax-addass 11177 ax-mulass 11178 ax-distr 11179 ax-i2m1 11180 ax-1ne0 11181 ax-1rid 11182 ax-rnegex 11183 ax-rrecex 11184 ax-cnre 11185 ax-pre-lttri 11186 ax-pre-lttrn 11187 ax-pre-ltadd 11188 ax-pre-mulgt0 11189 ax-pre-sup 11190 |
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 2528 df-eu 2557 df-clab 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ne 2935 df-nel 3041 df-ral 3056 df-rex 3065 df-rmo 3370 df-reu 3371 df-rab 3427 df-v 3470 df-sbc 3773 df-csb 3889 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-pss 3962 df-nul 4318 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-tp 4628 df-op 4630 df-uni 4903 df-int 4944 df-iun 4992 df-br 5142 df-opab 5204 df-mpt 5225 df-tr 5259 df-id 5567 df-eprel 5573 df-po 5581 df-so 5582 df-fr 5624 df-we 5626 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-pred 6294 df-ord 6361 df-on 6362 df-lim 6363 df-suc 6364 df-iota 6489 df-fun 6539 df-fn 6540 df-f 6541 df-f1 6542 df-fo 6543 df-f1o 6544 df-fv 6545 df-riota 7361 df-ov 7408 df-oprab 7409 df-mpo 7410 df-om 7853 df-1st 7974 df-2nd 7975 df-frecs 8267 df-wrecs 8298 df-recs 8372 df-rdg 8411 df-1o 8467 df-er 8705 df-map 8824 df-en 8942 df-dom 8943 df-sdom 8944 df-fin 8945 df-fi 9408 df-sup 9439 df-inf 9440 df-pnf 11254 df-mnf 11255 df-xr 11256 df-ltxr 11257 df-le 11258 df-sub 11450 df-neg 11451 df-div 11876 df-nn 12217 df-2 12279 df-3 12280 df-4 12281 df-5 12282 df-6 12283 df-7 12284 df-8 12285 df-9 12286 df-n0 12477 df-z 12563 df-dec 12682 df-uz 12827 df-q 12937 df-rp 12981 df-xneg 13098 df-xadd 13099 df-xmul 13100 df-fz 13491 df-seq 13973 df-exp 14033 df-cj 15052 df-re 15053 df-im 15054 df-sqrt 15188 df-abs 15189 df-struct 17089 df-slot 17124 df-ndx 17136 df-base 17154 df-plusg 17219 df-mulr 17220 df-starv 17221 df-tset 17225 df-ple 17226 df-ds 17228 df-unif 17229 df-rest 17377 df-topn 17378 df-topgen 17398 df-psmet 21232 df-xmet 21233 df-met 21234 df-bl 21235 df-mopn 21236 df-cnfld 21241 df-top 22751 df-topon 22768 df-topsp 22790 df-bases 22804 df-cn 23086 df-cnp 23087 df-xms 24181 df-ms 24182 df-cncf 24753 |
This theorem is referenced by: addccncf 24792 sub1cncf 24794 sub2cncf 24795 negcncf 24797 negcncfOLD 24798 dvidlem 25799 dvcnp2 25804 dvcnp2OLD 25805 dvmulbr 25824 dvmulbrOLD 25825 cmvth 25878 cmvthOLD 25879 dvlipcn 25882 lhop1lem 25901 dvfsumle 25909 dvfsumleOLD 25910 dvfsumge 25911 dvfsumabs 25912 dvfsumlem2 25916 dvfsumlem2OLD 25917 itgpowd 25940 taylthlem2 26264 taylthlem2OLD 26265 loglesqrt 26648 lgamgulmlem2 26917 pntlem3 27497 efmul2picn 34137 circlemeth 34181 logdivsqrle 34191 ftc1cnnclem 37072 ftc2nc 37083 areacirclem3 37091 areacirclem4 37092 areacirc 37094 constcncf 37143 lcmineqlem10 41419 lcmineqlem12 41421 arearect 42537 areaquad 42538 constcncfg 45157 add1cncf 45186 add2cncf 45187 sub1cncfd 45188 sub2cncfd 45189 itgsbtaddcnst 45267 dirkeritg 45387 |
Copyright terms: Public domain | W3C validator |