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Mirrors > Home > MPE Home > Th. List > cncfmptc | Structured version Visualization version GIF version |
Description: A constant function is a continuous function on ℂ. (Contributed by Jeff Madsen, 2-Sep-2009.) (Revised by Mario Carneiro, 7-Sep-2015.) |
Ref | Expression |
---|---|
cncfmptc | ⊢ ((𝐴 ∈ 𝑇 ∧ 𝑆 ⊆ ℂ ∧ 𝑇 ⊆ ℂ) → (𝑥 ∈ 𝑆 ↦ 𝐴) ∈ (𝑆–cn→𝑇)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2726 | . . . . 5 ⊢ (TopOpen‘ℂfld) = (TopOpen‘ℂfld) | |
2 | 1 | cnfldtopon 24787 | . . . 4 ⊢ (TopOpen‘ℂfld) ∈ (TopOn‘ℂ) |
3 | simp2 1134 | . . . 4 ⊢ ((𝐴 ∈ 𝑇 ∧ 𝑆 ⊆ ℂ ∧ 𝑇 ⊆ ℂ) → 𝑆 ⊆ ℂ) | |
4 | resttopon 23153 | . . . 4 ⊢ (((TopOpen‘ℂfld) ∈ (TopOn‘ℂ) ∧ 𝑆 ⊆ ℂ) → ((TopOpen‘ℂfld) ↾t 𝑆) ∈ (TopOn‘𝑆)) | |
5 | 2, 3, 4 | sylancr 585 | . . 3 ⊢ ((𝐴 ∈ 𝑇 ∧ 𝑆 ⊆ ℂ ∧ 𝑇 ⊆ ℂ) → ((TopOpen‘ℂfld) ↾t 𝑆) ∈ (TopOn‘𝑆)) |
6 | simp3 1135 | . . . 4 ⊢ ((𝐴 ∈ 𝑇 ∧ 𝑆 ⊆ ℂ ∧ 𝑇 ⊆ ℂ) → 𝑇 ⊆ ℂ) | |
7 | resttopon 23153 | . . . 4 ⊢ (((TopOpen‘ℂfld) ∈ (TopOn‘ℂ) ∧ 𝑇 ⊆ ℂ) → ((TopOpen‘ℂfld) ↾t 𝑇) ∈ (TopOn‘𝑇)) | |
8 | 2, 6, 7 | sylancr 585 | . . 3 ⊢ ((𝐴 ∈ 𝑇 ∧ 𝑆 ⊆ ℂ ∧ 𝑇 ⊆ ℂ) → ((TopOpen‘ℂfld) ↾t 𝑇) ∈ (TopOn‘𝑇)) |
9 | simp1 1133 | . . 3 ⊢ ((𝐴 ∈ 𝑇 ∧ 𝑆 ⊆ ℂ ∧ 𝑇 ⊆ ℂ) → 𝐴 ∈ 𝑇) | |
10 | 5, 8, 9 | cnmptc 23654 | . 2 ⊢ ((𝐴 ∈ 𝑇 ∧ 𝑆 ⊆ ℂ ∧ 𝑇 ⊆ ℂ) → (𝑥 ∈ 𝑆 ↦ 𝐴) ∈ (((TopOpen‘ℂfld) ↾t 𝑆) Cn ((TopOpen‘ℂfld) ↾t 𝑇))) |
11 | eqid 2726 | . . . 4 ⊢ ((TopOpen‘ℂfld) ↾t 𝑆) = ((TopOpen‘ℂfld) ↾t 𝑆) | |
12 | eqid 2726 | . . . 4 ⊢ ((TopOpen‘ℂfld) ↾t 𝑇) = ((TopOpen‘ℂfld) ↾t 𝑇) | |
13 | 1, 11, 12 | cncfcn 24918 | . . 3 ⊢ ((𝑆 ⊆ ℂ ∧ 𝑇 ⊆ ℂ) → (𝑆–cn→𝑇) = (((TopOpen‘ℂfld) ↾t 𝑆) Cn ((TopOpen‘ℂfld) ↾t 𝑇))) |
14 | 13 | 3adant1 1127 | . 2 ⊢ ((𝐴 ∈ 𝑇 ∧ 𝑆 ⊆ ℂ ∧ 𝑇 ⊆ ℂ) → (𝑆–cn→𝑇) = (((TopOpen‘ℂfld) ↾t 𝑆) Cn ((TopOpen‘ℂfld) ↾t 𝑇))) |
15 | 10, 14 | eleqtrrd 2829 | 1 ⊢ ((𝐴 ∈ 𝑇 ∧ 𝑆 ⊆ ℂ ∧ 𝑇 ⊆ ℂ) → (𝑥 ∈ 𝑆 ↦ 𝐴) ∈ (𝑆–cn→𝑇)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ w3a 1084 = wceq 1534 ∈ wcel 2099 ⊆ wss 3946 ↦ cmpt 5228 ‘cfv 6546 (class class class)co 7416 ℂcc 11147 ↾t crest 17430 TopOpenctopn 17431 ℂfldccnfld 21339 TopOnctopon 22900 Cn ccn 23216 –cn→ccncf 24884 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2697 ax-rep 5282 ax-sep 5296 ax-nul 5303 ax-pow 5361 ax-pr 5425 ax-un 7738 ax-cnex 11205 ax-resscn 11206 ax-1cn 11207 ax-icn 11208 ax-addcl 11209 ax-addrcl 11210 ax-mulcl 11211 ax-mulrcl 11212 ax-mulcom 11213 ax-addass 11214 ax-mulass 11215 ax-distr 11216 ax-i2m1 11217 ax-1ne0 11218 ax-1rid 11219 ax-rnegex 11220 ax-rrecex 11221 ax-cnre 11222 ax-pre-lttri 11223 ax-pre-lttrn 11224 ax-pre-ltadd 11225 ax-pre-mulgt0 11226 ax-pre-sup 11227 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2704 df-cleq 2718 df-clel 2803 df-nfc 2878 df-ne 2931 df-nel 3037 df-ral 3052 df-rex 3061 df-rmo 3364 df-reu 3365 df-rab 3420 df-v 3464 df-sbc 3776 df-csb 3892 df-dif 3949 df-un 3951 df-in 3953 df-ss 3963 df-pss 3966 df-nul 4323 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-tp 4628 df-op 4630 df-uni 4906 df-int 4947 df-iun 4995 df-br 5146 df-opab 5208 df-mpt 5229 df-tr 5263 df-id 5572 df-eprel 5578 df-po 5586 df-so 5587 df-fr 5629 df-we 5631 df-xp 5680 df-rel 5681 df-cnv 5682 df-co 5683 df-dm 5684 df-rn 5685 df-res 5686 df-ima 5687 df-pred 6304 df-ord 6371 df-on 6372 df-lim 6373 df-suc 6374 df-iota 6498 df-fun 6548 df-fn 6549 df-f 6550 df-f1 6551 df-fo 6552 df-f1o 6553 df-fv 6554 df-riota 7372 df-ov 7419 df-oprab 7420 df-mpo 7421 df-om 7869 df-1st 7995 df-2nd 7996 df-frecs 8288 df-wrecs 8319 df-recs 8393 df-rdg 8432 df-1o 8488 df-er 8726 df-map 8849 df-en 8967 df-dom 8968 df-sdom 8969 df-fin 8970 df-fi 9447 df-sup 9478 df-inf 9479 df-pnf 11291 df-mnf 11292 df-xr 11293 df-ltxr 11294 df-le 11295 df-sub 11487 df-neg 11488 df-div 11913 df-nn 12259 df-2 12321 df-3 12322 df-4 12323 df-5 12324 df-6 12325 df-7 12326 df-8 12327 df-9 12328 df-n0 12519 df-z 12605 df-dec 12724 df-uz 12869 df-q 12979 df-rp 13023 df-xneg 13140 df-xadd 13141 df-xmul 13142 df-fz 13533 df-seq 14016 df-exp 14076 df-cj 15099 df-re 15100 df-im 15101 df-sqrt 15235 df-abs 15236 df-struct 17144 df-slot 17179 df-ndx 17191 df-base 17209 df-plusg 17274 df-mulr 17275 df-starv 17276 df-tset 17280 df-ple 17281 df-ds 17283 df-unif 17284 df-rest 17432 df-topn 17433 df-topgen 17453 df-psmet 21331 df-xmet 21332 df-met 21333 df-bl 21334 df-mopn 21335 df-cnfld 21340 df-top 22884 df-topon 22901 df-topsp 22923 df-bases 22937 df-cn 23219 df-cnp 23220 df-xms 24314 df-ms 24315 df-cncf 24886 |
This theorem is referenced by: addccncf 24925 sub1cncf 24927 sub2cncf 24928 negcncf 24930 negcncfOLD 24931 dvidlem 25932 dvcnp2 25937 dvcnp2OLD 25938 dvmulbr 25957 dvmulbrOLD 25958 cmvth 26011 cmvthOLD 26012 dvlipcn 26015 lhop1lem 26034 dvfsumle 26042 dvfsumleOLD 26043 dvfsumge 26044 dvfsumabs 26045 dvfsumlem2 26049 dvfsumlem2OLD 26050 itgpowd 26073 taylthlem2 26399 taylthlem2OLD 26400 loglesqrt 26786 lgamgulmlem2 27055 pntlem3 27635 efmul2picn 34455 circlemeth 34499 logdivsqrle 34509 ftc1cnnclem 37405 ftc2nc 37416 areacirclem3 37424 areacirclem4 37425 areacirc 37427 constcncf 37476 lcmineqlem10 41750 lcmineqlem12 41752 arearect 42917 areaquad 42918 constcncfg 45529 add1cncf 45558 add2cncf 45559 sub1cncfd 45560 sub2cncfd 45561 itgsbtaddcnst 45639 dirkeritg 45759 |
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