![]() |
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 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 2738 | . . . . 5 ⊢ (TopOpen‘ℂfld) = (TopOpen‘ℂfld) | |
2 | 1 | cnfldtopon 24098 | . . . 4 ⊢ (TopOpen‘ℂfld) ∈ (TopOn‘ℂ) |
3 | simp2 1138 | . . . 4 ⊢ ((𝐴 ∈ 𝑇 ∧ 𝑆 ⊆ ℂ ∧ 𝑇 ⊆ ℂ) → 𝑆 ⊆ ℂ) | |
4 | resttopon 22464 | . . . 4 ⊢ (((TopOpen‘ℂfld) ∈ (TopOn‘ℂ) ∧ 𝑆 ⊆ ℂ) → ((TopOpen‘ℂfld) ↾t 𝑆) ∈ (TopOn‘𝑆)) | |
5 | 2, 3, 4 | sylancr 588 | . . 3 ⊢ ((𝐴 ∈ 𝑇 ∧ 𝑆 ⊆ ℂ ∧ 𝑇 ⊆ ℂ) → ((TopOpen‘ℂfld) ↾t 𝑆) ∈ (TopOn‘𝑆)) |
6 | simp3 1139 | . . . 4 ⊢ ((𝐴 ∈ 𝑇 ∧ 𝑆 ⊆ ℂ ∧ 𝑇 ⊆ ℂ) → 𝑇 ⊆ ℂ) | |
7 | resttopon 22464 | . . . 4 ⊢ (((TopOpen‘ℂfld) ∈ (TopOn‘ℂ) ∧ 𝑇 ⊆ ℂ) → ((TopOpen‘ℂfld) ↾t 𝑇) ∈ (TopOn‘𝑇)) | |
8 | 2, 6, 7 | sylancr 588 | . . 3 ⊢ ((𝐴 ∈ 𝑇 ∧ 𝑆 ⊆ ℂ ∧ 𝑇 ⊆ ℂ) → ((TopOpen‘ℂfld) ↾t 𝑇) ∈ (TopOn‘𝑇)) |
9 | simp1 1137 | . . 3 ⊢ ((𝐴 ∈ 𝑇 ∧ 𝑆 ⊆ ℂ ∧ 𝑇 ⊆ ℂ) → 𝐴 ∈ 𝑇) | |
10 | 5, 8, 9 | cnmptc 22965 | . 2 ⊢ ((𝐴 ∈ 𝑇 ∧ 𝑆 ⊆ ℂ ∧ 𝑇 ⊆ ℂ) → (𝑥 ∈ 𝑆 ↦ 𝐴) ∈ (((TopOpen‘ℂfld) ↾t 𝑆) Cn ((TopOpen‘ℂfld) ↾t 𝑇))) |
11 | eqid 2738 | . . . 4 ⊢ ((TopOpen‘ℂfld) ↾t 𝑆) = ((TopOpen‘ℂfld) ↾t 𝑆) | |
12 | eqid 2738 | . . . 4 ⊢ ((TopOpen‘ℂfld) ↾t 𝑇) = ((TopOpen‘ℂfld) ↾t 𝑇) | |
13 | 1, 11, 12 | cncfcn 24225 | . . 3 ⊢ ((𝑆 ⊆ ℂ ∧ 𝑇 ⊆ ℂ) → (𝑆–cn→𝑇) = (((TopOpen‘ℂfld) ↾t 𝑆) Cn ((TopOpen‘ℂfld) ↾t 𝑇))) |
14 | 13 | 3adant1 1131 | . 2 ⊢ ((𝐴 ∈ 𝑇 ∧ 𝑆 ⊆ ℂ ∧ 𝑇 ⊆ ℂ) → (𝑆–cn→𝑇) = (((TopOpen‘ℂfld) ↾t 𝑆) Cn ((TopOpen‘ℂfld) ↾t 𝑇))) |
15 | 10, 14 | eleqtrrd 2842 | 1 ⊢ ((𝐴 ∈ 𝑇 ∧ 𝑆 ⊆ ℂ ∧ 𝑇 ⊆ ℂ) → (𝑥 ∈ 𝑆 ↦ 𝐴) ∈ (𝑆–cn→𝑇)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ w3a 1088 = wceq 1542 ∈ wcel 2107 ⊆ wss 3909 ↦ cmpt 5187 ‘cfv 6494 (class class class)co 7352 ℂcc 11008 ↾t crest 17262 TopOpenctopn 17263 ℂfldccnfld 20749 TopOnctopon 22211 Cn ccn 22527 –cn→ccncf 24191 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2709 ax-rep 5241 ax-sep 5255 ax-nul 5262 ax-pow 5319 ax-pr 5383 ax-un 7665 ax-cnex 11066 ax-resscn 11067 ax-1cn 11068 ax-icn 11069 ax-addcl 11070 ax-addrcl 11071 ax-mulcl 11072 ax-mulrcl 11073 ax-mulcom 11074 ax-addass 11075 ax-mulass 11076 ax-distr 11077 ax-i2m1 11078 ax-1ne0 11079 ax-1rid 11080 ax-rnegex 11081 ax-rrecex 11082 ax-cnre 11083 ax-pre-lttri 11084 ax-pre-lttrn 11085 ax-pre-ltadd 11086 ax-pre-mulgt0 11087 ax-pre-sup 11088 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2888 df-ne 2943 df-nel 3049 df-ral 3064 df-rex 3073 df-rmo 3352 df-reu 3353 df-rab 3407 df-v 3446 df-sbc 3739 df-csb 3855 df-dif 3912 df-un 3914 df-in 3916 df-ss 3926 df-pss 3928 df-nul 4282 df-if 4486 df-pw 4561 df-sn 4586 df-pr 4588 df-tp 4590 df-op 4592 df-uni 4865 df-int 4907 df-iun 4955 df-br 5105 df-opab 5167 df-mpt 5188 df-tr 5222 df-id 5530 df-eprel 5536 df-po 5544 df-so 5545 df-fr 5587 df-we 5589 df-xp 5638 df-rel 5639 df-cnv 5640 df-co 5641 df-dm 5642 df-rn 5643 df-res 5644 df-ima 5645 df-pred 6252 df-ord 6319 df-on 6320 df-lim 6321 df-suc 6322 df-iota 6446 df-fun 6496 df-fn 6497 df-f 6498 df-f1 6499 df-fo 6500 df-f1o 6501 df-fv 6502 df-riota 7308 df-ov 7355 df-oprab 7356 df-mpo 7357 df-om 7796 df-1st 7914 df-2nd 7915 df-frecs 8205 df-wrecs 8236 df-recs 8310 df-rdg 8349 df-1o 8405 df-er 8607 df-map 8726 df-en 8843 df-dom 8844 df-sdom 8845 df-fin 8846 df-fi 9306 df-sup 9337 df-inf 9338 df-pnf 11150 df-mnf 11151 df-xr 11152 df-ltxr 11153 df-le 11154 df-sub 11346 df-neg 11347 df-div 11772 df-nn 12113 df-2 12175 df-3 12176 df-4 12177 df-5 12178 df-6 12179 df-7 12180 df-8 12181 df-9 12182 df-n0 12373 df-z 12459 df-dec 12578 df-uz 12723 df-q 12829 df-rp 12871 df-xneg 12988 df-xadd 12989 df-xmul 12990 df-fz 13380 df-seq 13862 df-exp 13923 df-cj 14944 df-re 14945 df-im 14946 df-sqrt 15080 df-abs 15081 df-struct 16979 df-slot 17014 df-ndx 17026 df-base 17044 df-plusg 17106 df-mulr 17107 df-starv 17108 df-tset 17112 df-ple 17113 df-ds 17115 df-unif 17116 df-rest 17264 df-topn 17265 df-topgen 17285 df-psmet 20741 df-xmet 20742 df-met 20743 df-bl 20744 df-mopn 20745 df-cnfld 20750 df-top 22195 df-topon 22212 df-topsp 22234 df-bases 22248 df-cn 22530 df-cnp 22531 df-xms 23625 df-ms 23626 df-cncf 24193 |
This theorem is referenced by: addccncf 24232 sub1cncf 24234 sub2cncf 24235 negcncf 24237 dvidlem 25231 dvcnp2 25236 dvmulbr 25255 cmvth 25307 dvlipcn 25310 lhop1lem 25329 dvfsumle 25337 dvfsumge 25338 dvfsumabs 25339 dvfsumlem2 25343 itgpowd 25366 taylthlem2 25685 loglesqrt 26063 lgamgulmlem2 26331 pntlem3 26909 efmul2picn 33013 circlemeth 33057 logdivsqrle 33067 ftc1cnnclem 36081 ftc2nc 36092 areacirclem3 36100 areacirclem4 36101 areacirc 36103 constcncf 36153 lcmineqlem10 40427 lcmineqlem12 40429 arearect 41452 areaquad 41453 constcncfg 44008 add1cncf 44037 add2cncf 44038 sub1cncfd 44039 sub2cncfd 44040 itgsbtaddcnst 44118 dirkeritg 44238 |
Copyright terms: Public domain | W3C validator |