![]() |
Mathbox for Glauco Siliprandi |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > Mathboxes > constcncfg | Structured version Visualization version GIF version |
Description: A constant function is a continuous function on ℂ. (Contributed by Glauco Siliprandi, 11-Dec-2019.) |
Ref | Expression |
---|---|
constcncfg.a | ⊢ (𝜑 → 𝐴 ⊆ ℂ) |
constcncfg.b | ⊢ (𝜑 → 𝐵 ∈ 𝐶) |
constcncfg.c | ⊢ (𝜑 → 𝐶 ⊆ ℂ) |
Ref | Expression |
---|---|
constcncfg | ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐵) ∈ (𝐴–cn→𝐶)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | constcncfg.b | . 2 ⊢ (𝜑 → 𝐵 ∈ 𝐶) | |
2 | constcncfg.a | . 2 ⊢ (𝜑 → 𝐴 ⊆ ℂ) | |
3 | constcncfg.c | . 2 ⊢ (𝜑 → 𝐶 ⊆ ℂ) | |
4 | cncfmptc 23126 | . 2 ⊢ ((𝐵 ∈ 𝐶 ∧ 𝐴 ⊆ ℂ ∧ 𝐶 ⊆ ℂ) → (𝑥 ∈ 𝐴 ↦ 𝐵) ∈ (𝐴–cn→𝐶)) | |
5 | 1, 2, 3, 4 | syl3anc 1439 | 1 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐵) ∈ (𝐴–cn→𝐶)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2107 ⊆ wss 3792 ↦ cmpt 4967 (class class class)co 6924 ℂcc 10272 –cn→ccncf 23091 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1839 ax-4 1853 ax-5 1953 ax-6 2021 ax-7 2055 ax-8 2109 ax-9 2116 ax-10 2135 ax-11 2150 ax-12 2163 ax-13 2334 ax-ext 2754 ax-rep 5008 ax-sep 5019 ax-nul 5027 ax-pow 5079 ax-pr 5140 ax-un 7228 ax-cnex 10330 ax-resscn 10331 ax-1cn 10332 ax-icn 10333 ax-addcl 10334 ax-addrcl 10335 ax-mulcl 10336 ax-mulrcl 10337 ax-mulcom 10338 ax-addass 10339 ax-mulass 10340 ax-distr 10341 ax-i2m1 10342 ax-1ne0 10343 ax-1rid 10344 ax-rnegex 10345 ax-rrecex 10346 ax-cnre 10347 ax-pre-lttri 10348 ax-pre-lttrn 10349 ax-pre-ltadd 10350 ax-pre-mulgt0 10351 ax-pre-sup 10352 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 837 df-3or 1072 df-3an 1073 df-tru 1605 df-ex 1824 df-nf 1828 df-sb 2012 df-mo 2551 df-eu 2587 df-clab 2764 df-cleq 2770 df-clel 2774 df-nfc 2921 df-ne 2970 df-nel 3076 df-ral 3095 df-rex 3096 df-reu 3097 df-rmo 3098 df-rab 3099 df-v 3400 df-sbc 3653 df-csb 3752 df-dif 3795 df-un 3797 df-in 3799 df-ss 3806 df-pss 3808 df-nul 4142 df-if 4308 df-pw 4381 df-sn 4399 df-pr 4401 df-tp 4403 df-op 4405 df-uni 4674 df-int 4713 df-iun 4757 df-br 4889 df-opab 4951 df-mpt 4968 df-tr 4990 df-id 5263 df-eprel 5268 df-po 5276 df-so 5277 df-fr 5316 df-we 5318 df-xp 5363 df-rel 5364 df-cnv 5365 df-co 5366 df-dm 5367 df-rn 5368 df-res 5369 df-ima 5370 df-pred 5935 df-ord 5981 df-on 5982 df-lim 5983 df-suc 5984 df-iota 6101 df-fun 6139 df-fn 6140 df-f 6141 df-f1 6142 df-fo 6143 df-f1o 6144 df-fv 6145 df-riota 6885 df-ov 6927 df-oprab 6928 df-mpt2 6929 df-om 7346 df-1st 7447 df-2nd 7448 df-wrecs 7691 df-recs 7753 df-rdg 7791 df-1o 7845 df-oadd 7849 df-er 8028 df-map 8144 df-en 8244 df-dom 8245 df-sdom 8246 df-fin 8247 df-fi 8607 df-sup 8638 df-inf 8639 df-pnf 10415 df-mnf 10416 df-xr 10417 df-ltxr 10418 df-le 10419 df-sub 10610 df-neg 10611 df-div 11035 df-nn 11379 df-2 11442 df-3 11443 df-4 11444 df-5 11445 df-6 11446 df-7 11447 df-8 11448 df-9 11449 df-n0 11647 df-z 11733 df-dec 11850 df-uz 11997 df-q 12100 df-rp 12142 df-xneg 12261 df-xadd 12262 df-xmul 12263 df-fz 12648 df-seq 13124 df-exp 13183 df-cj 14250 df-re 14251 df-im 14252 df-sqrt 14386 df-abs 14387 df-struct 16261 df-ndx 16262 df-slot 16263 df-base 16265 df-plusg 16355 df-mulr 16356 df-starv 16357 df-tset 16361 df-ple 16362 df-ds 16364 df-unif 16365 df-rest 16473 df-topn 16474 df-topgen 16494 df-psmet 20138 df-xmet 20139 df-met 20140 df-bl 20141 df-mopn 20142 df-cnfld 20147 df-top 21110 df-topon 21127 df-topsp 21149 df-bases 21162 df-cn 21443 df-cnp 21444 df-xms 22537 df-ms 22538 df-cncf 23093 |
This theorem is referenced by: addccncf2 41027 negcncfg 41032 fprodcncf 41052 itgsinexplem1 41107 itgcoscmulx 41122 itgsincmulx 41127 itgiccshift 41133 itgperiod 41134 itgsbtaddcnst 41135 dirkeritg 41256 dirkercncflem2 41258 dirkercncflem4 41260 fourierdlem16 41277 fourierdlem18 41279 fourierdlem21 41282 fourierdlem22 41283 fourierdlem39 41300 fourierdlem40 41301 fourierdlem58 41318 fourierdlem59 41319 fourierdlem62 41322 fourierdlem68 41328 fourierdlem73 41333 fourierdlem76 41336 fourierdlem78 41338 fourierdlem83 41343 fourierdlem93 41353 fourierdlem111 41371 sqwvfoura 41382 sqwvfourb 41383 fouriersw 41385 etransclem18 41406 etransclem22 41410 etransclem34 41422 etransclem46 41434 |
Copyright terms: Public domain | W3C validator |