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Mathbox for Glauco Siliprandi |
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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 24742 | . 2 ⊢ ((𝐵 ∈ 𝐶 ∧ 𝐴 ⊆ ℂ ∧ 𝐶 ⊆ ℂ) → (𝑥 ∈ 𝐴 ↦ 𝐵) ∈ (𝐴–cn→𝐶)) | |
5 | 1, 2, 3, 4 | syl3anc 1368 | 1 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐵) ∈ (𝐴–cn→𝐶)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2098 ⊆ wss 3940 ↦ cmpt 5221 (class class class)co 7401 ℂcc 11103 –cn→ccncf 24706 |
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 17076 df-slot 17111 df-ndx 17123 df-base 17141 df-plusg 17206 df-mulr 17207 df-starv 17208 df-tset 17212 df-ple 17213 df-ds 17215 df-unif 17216 df-rest 17364 df-topn 17365 df-topgen 17385 df-psmet 21215 df-xmet 21216 df-met 21217 df-bl 21218 df-mopn 21219 df-cnfld 21224 df-top 22706 df-topon 22723 df-topsp 22745 df-bases 22759 df-cn 23041 df-cnp 23042 df-xms 24136 df-ms 24137 df-cncf 24708 |
This theorem is referenced by: addccncf2 45043 negcncfg 45048 fprodcncf 45067 itgsinexplem1 45121 itgcoscmulx 45136 itgsincmulx 45141 itgiccshift 45147 itgperiod 45148 itgsbtaddcnst 45149 dirkeritg 45269 dirkercncflem2 45271 dirkercncflem4 45273 fourierdlem16 45290 fourierdlem18 45292 fourierdlem21 45295 fourierdlem22 45296 fourierdlem39 45313 fourierdlem40 45314 fourierdlem58 45331 fourierdlem59 45332 fourierdlem62 45335 fourierdlem68 45341 fourierdlem73 45346 fourierdlem76 45349 fourierdlem78 45351 fourierdlem83 45356 fourierdlem93 45366 fourierdlem111 45384 sqwvfoura 45395 sqwvfourb 45396 fouriersw 45398 etransclem18 45419 etransclem22 45423 etransclem34 45435 etransclem46 45447 |
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