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Mirrors > Home > MPE Home > Th. List > cncfmptid | Structured version Visualization version GIF version |
Description: The identity function is a continuous function on ℂ. (Contributed by Jeff Madsen, 11-Jun-2010.) (Revised by Mario Carneiro, 17-May-2016.) |
Ref | Expression |
---|---|
cncfmptid | ⊢ ((𝑆 ⊆ 𝑇 ∧ 𝑇 ⊆ ℂ) → (𝑥 ∈ 𝑆 ↦ 𝑥) ∈ (𝑆–cn→𝑇)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cncfss 24384 | . 2 ⊢ ((𝑆 ⊆ 𝑇 ∧ 𝑇 ⊆ ℂ) → (𝑆–cn→𝑆) ⊆ (𝑆–cn→𝑇)) | |
2 | eqid 2733 | . . . . . 6 ⊢ (TopOpen‘ℂfld) = (TopOpen‘ℂfld) | |
3 | 2 | cnfldtopon 24268 | . . . . 5 ⊢ (TopOpen‘ℂfld) ∈ (TopOn‘ℂ) |
4 | sstr 3988 | . . . . 5 ⊢ ((𝑆 ⊆ 𝑇 ∧ 𝑇 ⊆ ℂ) → 𝑆 ⊆ ℂ) | |
5 | resttopon 22634 | . . . . 5 ⊢ (((TopOpen‘ℂfld) ∈ (TopOn‘ℂ) ∧ 𝑆 ⊆ ℂ) → ((TopOpen‘ℂfld) ↾t 𝑆) ∈ (TopOn‘𝑆)) | |
6 | 3, 4, 5 | sylancr 588 | . . . 4 ⊢ ((𝑆 ⊆ 𝑇 ∧ 𝑇 ⊆ ℂ) → ((TopOpen‘ℂfld) ↾t 𝑆) ∈ (TopOn‘𝑆)) |
7 | 6 | cnmptid 23134 | . . 3 ⊢ ((𝑆 ⊆ 𝑇 ∧ 𝑇 ⊆ ℂ) → (𝑥 ∈ 𝑆 ↦ 𝑥) ∈ (((TopOpen‘ℂfld) ↾t 𝑆) Cn ((TopOpen‘ℂfld) ↾t 𝑆))) |
8 | eqid 2733 | . . . . 5 ⊢ ((TopOpen‘ℂfld) ↾t 𝑆) = ((TopOpen‘ℂfld) ↾t 𝑆) | |
9 | 2, 8, 8 | cncfcn 24395 | . . . 4 ⊢ ((𝑆 ⊆ ℂ ∧ 𝑆 ⊆ ℂ) → (𝑆–cn→𝑆) = (((TopOpen‘ℂfld) ↾t 𝑆) Cn ((TopOpen‘ℂfld) ↾t 𝑆))) |
10 | 4, 4, 9 | syl2anc 585 | . . 3 ⊢ ((𝑆 ⊆ 𝑇 ∧ 𝑇 ⊆ ℂ) → (𝑆–cn→𝑆) = (((TopOpen‘ℂfld) ↾t 𝑆) Cn ((TopOpen‘ℂfld) ↾t 𝑆))) |
11 | 7, 10 | eleqtrrd 2837 | . 2 ⊢ ((𝑆 ⊆ 𝑇 ∧ 𝑇 ⊆ ℂ) → (𝑥 ∈ 𝑆 ↦ 𝑥) ∈ (𝑆–cn→𝑆)) |
12 | 1, 11 | sseldd 3981 | 1 ⊢ ((𝑆 ⊆ 𝑇 ∧ 𝑇 ⊆ ℂ) → (𝑥 ∈ 𝑆 ↦ 𝑥) ∈ (𝑆–cn→𝑇)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 397 = wceq 1542 ∈ wcel 2107 ⊆ wss 3946 ↦ cmpt 5227 ‘cfv 6535 (class class class)co 7396 ℂcc 11095 ↾t crest 17353 TopOpenctopn 17354 ℂfldccnfld 20918 TopOnctopon 22381 Cn ccn 22697 –cn→ccncf 24361 |
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 2704 ax-rep 5281 ax-sep 5295 ax-nul 5302 ax-pow 5359 ax-pr 5423 ax-un 7712 ax-cnex 11153 ax-resscn 11154 ax-1cn 11155 ax-icn 11156 ax-addcl 11157 ax-addrcl 11158 ax-mulcl 11159 ax-mulrcl 11160 ax-mulcom 11161 ax-addass 11162 ax-mulass 11163 ax-distr 11164 ax-i2m1 11165 ax-1ne0 11166 ax-1rid 11167 ax-rnegex 11168 ax-rrecex 11169 ax-cnre 11170 ax-pre-lttri 11171 ax-pre-lttrn 11172 ax-pre-ltadd 11173 ax-pre-mulgt0 11174 ax-pre-sup 11175 |
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 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-rmo 3377 df-reu 3378 df-rab 3434 df-v 3477 df-sbc 3776 df-csb 3892 df-dif 3949 df-un 3951 df-in 3953 df-ss 3963 df-pss 3965 df-nul 4321 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-tp 4629 df-op 4631 df-uni 4905 df-int 4947 df-iun 4995 df-br 5145 df-opab 5207 df-mpt 5228 df-tr 5262 df-id 5570 df-eprel 5576 df-po 5584 df-so 5585 df-fr 5627 df-we 5629 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-pred 6292 df-ord 6359 df-on 6360 df-lim 6361 df-suc 6362 df-iota 6487 df-fun 6537 df-fn 6538 df-f 6539 df-f1 6540 df-fo 6541 df-f1o 6542 df-fv 6543 df-riota 7352 df-ov 7399 df-oprab 7400 df-mpo 7401 df-om 7843 df-1st 7962 df-2nd 7963 df-frecs 8253 df-wrecs 8284 df-recs 8358 df-rdg 8397 df-1o 8453 df-er 8691 df-map 8810 df-en 8928 df-dom 8929 df-sdom 8930 df-fin 8931 df-fi 9393 df-sup 9424 df-inf 9425 df-pnf 11237 df-mnf 11238 df-xr 11239 df-ltxr 11240 df-le 11241 df-sub 11433 df-neg 11434 df-div 11859 df-nn 12200 df-2 12262 df-3 12263 df-4 12264 df-5 12265 df-6 12266 df-7 12267 df-8 12268 df-9 12269 df-n0 12460 df-z 12546 df-dec 12665 df-uz 12810 df-q 12920 df-rp 12962 df-xneg 13079 df-xadd 13080 df-xmul 13081 df-fz 13472 df-seq 13954 df-exp 14015 df-cj 15033 df-re 15034 df-im 15035 df-sqrt 15169 df-abs 15170 df-struct 17067 df-slot 17102 df-ndx 17114 df-base 17132 df-plusg 17197 df-mulr 17198 df-starv 17199 df-tset 17203 df-ple 17204 df-ds 17206 df-unif 17207 df-rest 17355 df-topn 17356 df-topgen 17376 df-psmet 20910 df-xmet 20911 df-met 20912 df-bl 20913 df-mopn 20914 df-cnfld 20919 df-top 22365 df-topon 22382 df-topsp 22404 df-bases 22418 df-cn 22700 df-cnp 22701 df-xms 23795 df-ms 23796 df-cncf 24363 |
This theorem is referenced by: addccncf 24402 idcncf 24403 negcncf 24407 dvcnp2 25406 mvth 25478 dvlipcn 25480 dvfsumle 25507 dvfsumabs 25509 dvfsumlem2 25513 taylthlem2 25855 loglesqrt 26233 lgamgulmlem2 26501 pntlem3 27079 iblidicc 33535 circlemeth 33583 logdivsqrle 33593 areacirclem4 36484 lcmineqlem12 40811 areaquad 41836 idcncfg 44462 addccncf2 44465 add1cncf 44490 add2cncf 44491 sub1cncfd 44492 sub2cncfd 44493 itgsbtaddcnst 44571 dirkercncflem2 44693 fourierdlem16 44712 fourierdlem22 44718 fourierdlem93 44788 fourierdlem111 44806 |
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