![]() |
Mathbox for Glauco Siliprandi |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > Mathboxes > idcncfg | Structured version Visualization version GIF version |
Description: The identity function is a continuous function on ℂ. (Contributed by Glauco Siliprandi, 11-Dec-2019.) |
Ref | Expression |
---|---|
idcncfg.a | ⊢ (𝜑 → 𝐴 ⊆ 𝐵) |
idcncfg.b | ⊢ (𝜑 → 𝐵 ⊆ ℂ) |
Ref | Expression |
---|---|
idcncfg | ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝑥) ∈ (𝐴–cn→𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | idcncfg.a | . 2 ⊢ (𝜑 → 𝐴 ⊆ 𝐵) | |
2 | idcncfg.b | . 2 ⊢ (𝜑 → 𝐵 ⊆ ℂ) | |
3 | cncfmptid 23040 | . 2 ⊢ ((𝐴 ⊆ 𝐵 ∧ 𝐵 ⊆ ℂ) → (𝑥 ∈ 𝐴 ↦ 𝑥) ∈ (𝐴–cn→𝐵)) | |
4 | 1, 2, 3 | syl2anc 580 | 1 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝑥) ∈ (𝐴–cn→𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2157 ⊆ wss 3768 ↦ cmpt 4921 (class class class)co 6877 ℂcc 10221 –cn→ccncf 23004 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1891 ax-4 1905 ax-5 2006 ax-6 2072 ax-7 2107 ax-8 2159 ax-9 2166 ax-10 2185 ax-11 2200 ax-12 2213 ax-13 2377 ax-ext 2776 ax-rep 4963 ax-sep 4974 ax-nul 4982 ax-pow 5034 ax-pr 5096 ax-un 7182 ax-cnex 10279 ax-resscn 10280 ax-1cn 10281 ax-icn 10282 ax-addcl 10283 ax-addrcl 10284 ax-mulcl 10285 ax-mulrcl 10286 ax-mulcom 10287 ax-addass 10288 ax-mulass 10289 ax-distr 10290 ax-i2m1 10291 ax-1ne0 10292 ax-1rid 10293 ax-rnegex 10294 ax-rrecex 10295 ax-cnre 10296 ax-pre-lttri 10297 ax-pre-lttrn 10298 ax-pre-ltadd 10299 ax-pre-mulgt0 10300 ax-pre-sup 10301 |
This theorem depends on definitions: df-bi 199 df-an 386 df-or 875 df-3or 1109 df-3an 1110 df-tru 1657 df-ex 1876 df-nf 1880 df-sb 2065 df-mo 2591 df-eu 2609 df-clab 2785 df-cleq 2791 df-clel 2794 df-nfc 2929 df-ne 2971 df-nel 3074 df-ral 3093 df-rex 3094 df-reu 3095 df-rmo 3096 df-rab 3097 df-v 3386 df-sbc 3633 df-csb 3728 df-dif 3771 df-un 3773 df-in 3775 df-ss 3782 df-pss 3784 df-nul 4115 df-if 4277 df-pw 4350 df-sn 4368 df-pr 4370 df-tp 4372 df-op 4374 df-uni 4628 df-int 4667 df-iun 4711 df-br 4843 df-opab 4905 df-mpt 4922 df-tr 4945 df-id 5219 df-eprel 5224 df-po 5232 df-so 5233 df-fr 5270 df-we 5272 df-xp 5317 df-rel 5318 df-cnv 5319 df-co 5320 df-dm 5321 df-rn 5322 df-res 5323 df-ima 5324 df-pred 5897 df-ord 5943 df-on 5944 df-lim 5945 df-suc 5946 df-iota 6063 df-fun 6102 df-fn 6103 df-f 6104 df-f1 6105 df-fo 6106 df-f1o 6107 df-fv 6108 df-riota 6838 df-ov 6880 df-oprab 6881 df-mpt2 6882 df-om 7299 df-1st 7400 df-2nd 7401 df-wrecs 7644 df-recs 7706 df-rdg 7744 df-1o 7798 df-oadd 7802 df-er 7981 df-map 8096 df-en 8195 df-dom 8196 df-sdom 8197 df-fin 8198 df-fi 8558 df-sup 8589 df-inf 8590 df-pnf 10364 df-mnf 10365 df-xr 10366 df-ltxr 10367 df-le 10368 df-sub 10557 df-neg 10558 df-div 10976 df-nn 11312 df-2 11373 df-3 11374 df-4 11375 df-5 11376 df-6 11377 df-7 11378 df-8 11379 df-9 11380 df-n0 11578 df-z 11664 df-dec 11781 df-uz 11928 df-q 12031 df-rp 12072 df-xneg 12190 df-xadd 12191 df-xmul 12192 df-fz 12578 df-seq 13053 df-exp 13112 df-cj 14177 df-re 14178 df-im 14179 df-sqrt 14313 df-abs 14314 df-struct 16183 df-ndx 16184 df-slot 16185 df-base 16187 df-plusg 16277 df-mulr 16278 df-starv 16279 df-tset 16283 df-ple 16284 df-ds 16286 df-unif 16287 df-rest 16395 df-topn 16396 df-topgen 16416 df-psmet 20057 df-xmet 20058 df-met 20059 df-bl 20060 df-mopn 20061 df-cnfld 20066 df-top 21024 df-topon 21041 df-topsp 21063 df-bases 21076 df-cn 21357 df-cnp 21358 df-xms 22450 df-ms 22451 df-cncf 23006 |
This theorem is referenced by: itgcoscmulx 40917 itgsincmulx 40922 dirkeritg 41051 dirkercncflem2 41053 dirkercncflem4 41055 fourierdlem18 41074 fourierdlem21 41077 fourierdlem39 41095 fourierdlem58 41113 fourierdlem62 41117 fourierdlem68 41123 fourierdlem73 41128 fourierdlem76 41131 fourierdlem78 41133 fourierdlem83 41138 sqwvfoura 41177 sqwvfourb 41178 etransclem18 41201 etransclem22 41205 etransclem46 41229 |
Copyright terms: Public domain | W3C validator |