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Mirrors > Home > MPE Home > Th. List > sqrtcn | Structured version Visualization version GIF version |
Description: Continuity of the square root function. (Contributed by Mario Carneiro, 2-May-2016.) |
Ref | Expression |
---|---|
sqrcn.d | ⊢ 𝐷 = (ℂ ∖ (-∞(,]0)) |
Ref | Expression |
---|---|
sqrtcn | ⊢ (√ ↾ 𝐷) ∈ (𝐷–cn→ℂ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | sqrtf 14587 | . . . . . . 7 ⊢ √:ℂ⟶ℂ | |
2 | 1 | a1i 11 | . . . . . 6 ⊢ (⊤ → √:ℂ⟶ℂ) |
3 | 2 | feqmptd 6564 | . . . . 5 ⊢ (⊤ → √ = (𝑥 ∈ ℂ ↦ (√‘𝑥))) |
4 | 3 | reseq1d 5695 | . . . 4 ⊢ (⊤ → (√ ↾ 𝐷) = ((𝑥 ∈ ℂ ↦ (√‘𝑥)) ↾ 𝐷)) |
5 | sqrcn.d | . . . . . 6 ⊢ 𝐷 = (ℂ ∖ (-∞(,]0)) | |
6 | difss 4000 | . . . . . 6 ⊢ (ℂ ∖ (-∞(,]0)) ⊆ ℂ | |
7 | 5, 6 | eqsstri 3893 | . . . . 5 ⊢ 𝐷 ⊆ ℂ |
8 | resmpt 5752 | . . . . 5 ⊢ (𝐷 ⊆ ℂ → ((𝑥 ∈ ℂ ↦ (√‘𝑥)) ↾ 𝐷) = (𝑥 ∈ 𝐷 ↦ (√‘𝑥))) | |
9 | 7, 8 | mp1i 13 | . . . 4 ⊢ (⊤ → ((𝑥 ∈ ℂ ↦ (√‘𝑥)) ↾ 𝐷) = (𝑥 ∈ 𝐷 ↦ (√‘𝑥))) |
10 | 7 | sseli 3856 | . . . . . . . 8 ⊢ (𝑥 ∈ 𝐷 → 𝑥 ∈ ℂ) |
11 | 10 | adantl 474 | . . . . . . 7 ⊢ ((⊤ ∧ 𝑥 ∈ 𝐷) → 𝑥 ∈ ℂ) |
12 | cxpsqrt 24990 | . . . . . . 7 ⊢ (𝑥 ∈ ℂ → (𝑥↑𝑐(1 / 2)) = (√‘𝑥)) | |
13 | 11, 12 | syl 17 | . . . . . 6 ⊢ ((⊤ ∧ 𝑥 ∈ 𝐷) → (𝑥↑𝑐(1 / 2)) = (√‘𝑥)) |
14 | 13 | eqcomd 2784 | . . . . 5 ⊢ ((⊤ ∧ 𝑥 ∈ 𝐷) → (√‘𝑥) = (𝑥↑𝑐(1 / 2))) |
15 | 14 | mpteq2dva 5023 | . . . 4 ⊢ (⊤ → (𝑥 ∈ 𝐷 ↦ (√‘𝑥)) = (𝑥 ∈ 𝐷 ↦ (𝑥↑𝑐(1 / 2)))) |
16 | 4, 9, 15 | 3eqtrd 2818 | . . 3 ⊢ (⊤ → (√ ↾ 𝐷) = (𝑥 ∈ 𝐷 ↦ (𝑥↑𝑐(1 / 2)))) |
17 | eqid 2778 | . . . . . . . 8 ⊢ (TopOpen‘ℂfld) = (TopOpen‘ℂfld) | |
18 | 17 | cnfldtopon 23097 | . . . . . . 7 ⊢ (TopOpen‘ℂfld) ∈ (TopOn‘ℂ) |
19 | 18 | a1i 11 | . . . . . 6 ⊢ (⊤ → (TopOpen‘ℂfld) ∈ (TopOn‘ℂ)) |
20 | resttopon 21476 | . . . . . 6 ⊢ (((TopOpen‘ℂfld) ∈ (TopOn‘ℂ) ∧ 𝐷 ⊆ ℂ) → ((TopOpen‘ℂfld) ↾t 𝐷) ∈ (TopOn‘𝐷)) | |
21 | 19, 7, 20 | sylancl 577 | . . . . 5 ⊢ (⊤ → ((TopOpen‘ℂfld) ↾t 𝐷) ∈ (TopOn‘𝐷)) |
22 | 21 | cnmptid 21976 | . . . . 5 ⊢ (⊤ → (𝑥 ∈ 𝐷 ↦ 𝑥) ∈ (((TopOpen‘ℂfld) ↾t 𝐷) Cn ((TopOpen‘ℂfld) ↾t 𝐷))) |
23 | ax-1cn 10395 | . . . . . . 7 ⊢ 1 ∈ ℂ | |
24 | halfcl 11675 | . . . . . . 7 ⊢ (1 ∈ ℂ → (1 / 2) ∈ ℂ) | |
25 | 23, 24 | mp1i 13 | . . . . . 6 ⊢ (⊤ → (1 / 2) ∈ ℂ) |
26 | 21, 19, 25 | cnmptc 21977 | . . . . 5 ⊢ (⊤ → (𝑥 ∈ 𝐷 ↦ (1 / 2)) ∈ (((TopOpen‘ℂfld) ↾t 𝐷) Cn (TopOpen‘ℂfld))) |
27 | eqid 2778 | . . . . . . 7 ⊢ ((TopOpen‘ℂfld) ↾t 𝐷) = ((TopOpen‘ℂfld) ↾t 𝐷) | |
28 | 5, 17, 27 | cxpcn 25030 | . . . . . 6 ⊢ (𝑦 ∈ 𝐷, 𝑧 ∈ ℂ ↦ (𝑦↑𝑐𝑧)) ∈ ((((TopOpen‘ℂfld) ↾t 𝐷) ×t (TopOpen‘ℂfld)) Cn (TopOpen‘ℂfld)) |
29 | 28 | a1i 11 | . . . . 5 ⊢ (⊤ → (𝑦 ∈ 𝐷, 𝑧 ∈ ℂ ↦ (𝑦↑𝑐𝑧)) ∈ ((((TopOpen‘ℂfld) ↾t 𝐷) ×t (TopOpen‘ℂfld)) Cn (TopOpen‘ℂfld))) |
30 | oveq12 6987 | . . . . 5 ⊢ ((𝑦 = 𝑥 ∧ 𝑧 = (1 / 2)) → (𝑦↑𝑐𝑧) = (𝑥↑𝑐(1 / 2))) | |
31 | 21, 22, 26, 21, 19, 29, 30 | cnmpt12 21982 | . . . 4 ⊢ (⊤ → (𝑥 ∈ 𝐷 ↦ (𝑥↑𝑐(1 / 2))) ∈ (((TopOpen‘ℂfld) ↾t 𝐷) Cn (TopOpen‘ℂfld))) |
32 | ssid 3881 | . . . . 5 ⊢ ℂ ⊆ ℂ | |
33 | 18 | toponrestid 21236 | . . . . . 6 ⊢ (TopOpen‘ℂfld) = ((TopOpen‘ℂfld) ↾t ℂ) |
34 | 17, 27, 33 | cncfcn 23223 | . . . . 5 ⊢ ((𝐷 ⊆ ℂ ∧ ℂ ⊆ ℂ) → (𝐷–cn→ℂ) = (((TopOpen‘ℂfld) ↾t 𝐷) Cn (TopOpen‘ℂfld))) |
35 | 7, 32, 34 | mp2an 679 | . . . 4 ⊢ (𝐷–cn→ℂ) = (((TopOpen‘ℂfld) ↾t 𝐷) Cn (TopOpen‘ℂfld)) |
36 | 31, 35 | syl6eleqr 2877 | . . 3 ⊢ (⊤ → (𝑥 ∈ 𝐷 ↦ (𝑥↑𝑐(1 / 2))) ∈ (𝐷–cn→ℂ)) |
37 | 16, 36 | eqeltrd 2866 | . 2 ⊢ (⊤ → (√ ↾ 𝐷) ∈ (𝐷–cn→ℂ)) |
38 | 37 | mptru 1514 | 1 ⊢ (√ ↾ 𝐷) ∈ (𝐷–cn→ℂ) |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 387 = wceq 1507 ⊤wtru 1508 ∈ wcel 2050 ∖ cdif 3828 ⊆ wss 3831 ↦ cmpt 5009 ↾ cres 5410 ⟶wf 6186 ‘cfv 6190 (class class class)co 6978 ∈ cmpo 6980 ℂcc 10335 0cc0 10337 1c1 10338 -∞cmnf 10474 / cdiv 11100 2c2 11498 (,]cioc 12558 √csqrt 14456 ↾t crest 16553 TopOpenctopn 16554 ℂfldccnfld 20250 TopOnctopon 21225 Cn ccn 21539 ×t ctx 21875 –cn→ccncf 23190 ↑𝑐ccxp 24843 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1758 ax-4 1772 ax-5 1869 ax-6 1928 ax-7 1965 ax-8 2052 ax-9 2059 ax-10 2079 ax-11 2093 ax-12 2106 ax-13 2301 ax-ext 2750 ax-rep 5050 ax-sep 5061 ax-nul 5068 ax-pow 5120 ax-pr 5187 ax-un 7281 ax-inf2 8900 ax-cnex 10393 ax-resscn 10394 ax-1cn 10395 ax-icn 10396 ax-addcl 10397 ax-addrcl 10398 ax-mulcl 10399 ax-mulrcl 10400 ax-mulcom 10401 ax-addass 10402 ax-mulass 10403 ax-distr 10404 ax-i2m1 10405 ax-1ne0 10406 ax-1rid 10407 ax-rnegex 10408 ax-rrecex 10409 ax-cnre 10410 ax-pre-lttri 10411 ax-pre-lttrn 10412 ax-pre-ltadd 10413 ax-pre-mulgt0 10414 ax-pre-sup 10415 ax-addf 10416 ax-mulf 10417 |
This theorem depends on definitions: df-bi 199 df-an 388 df-or 834 df-3or 1069 df-3an 1070 df-tru 1510 df-fal 1520 df-ex 1743 df-nf 1747 df-sb 2016 df-mo 2547 df-eu 2583 df-clab 2759 df-cleq 2771 df-clel 2846 df-nfc 2918 df-ne 2968 df-nel 3074 df-ral 3093 df-rex 3094 df-reu 3095 df-rmo 3096 df-rab 3097 df-v 3417 df-sbc 3684 df-csb 3789 df-dif 3834 df-un 3836 df-in 3838 df-ss 3845 df-pss 3847 df-nul 4181 df-if 4352 df-pw 4425 df-sn 4443 df-pr 4445 df-tp 4447 df-op 4449 df-uni 4714 df-int 4751 df-iun 4795 df-iin 4796 df-br 4931 df-opab 4993 df-mpt 5010 df-tr 5032 df-id 5313 df-eprel 5318 df-po 5327 df-so 5328 df-fr 5367 df-se 5368 df-we 5369 df-xp 5414 df-rel 5415 df-cnv 5416 df-co 5417 df-dm 5418 df-rn 5419 df-res 5420 df-ima 5421 df-pred 5988 df-ord 6034 df-on 6035 df-lim 6036 df-suc 6037 df-iota 6154 df-fun 6192 df-fn 6193 df-f 6194 df-f1 6195 df-fo 6196 df-f1o 6197 df-fv 6198 df-isom 6199 df-riota 6939 df-ov 6981 df-oprab 6982 df-mpo 6983 df-of 7229 df-om 7399 df-1st 7503 df-2nd 7504 df-supp 7636 df-wrecs 7752 df-recs 7814 df-rdg 7852 df-1o 7907 df-2o 7908 df-oadd 7911 df-er 8091 df-map 8210 df-pm 8211 df-ixp 8262 df-en 8309 df-dom 8310 df-sdom 8311 df-fin 8312 df-fsupp 8631 df-fi 8672 df-sup 8703 df-inf 8704 df-oi 8771 df-card 9164 df-cda 9390 df-pnf 10478 df-mnf 10479 df-xr 10480 df-ltxr 10481 df-le 10482 df-sub 10674 df-neg 10675 df-div 11101 df-nn 11442 df-2 11506 df-3 11507 df-4 11508 df-5 11509 df-6 11510 df-7 11511 df-8 11512 df-9 11513 df-n0 11711 df-z 11797 df-dec 11915 df-uz 12062 df-q 12166 df-rp 12208 df-xneg 12327 df-xadd 12328 df-xmul 12329 df-ioo 12561 df-ioc 12562 df-ico 12563 df-icc 12564 df-fz 12712 df-fzo 12853 df-fl 12980 df-mod 13056 df-seq 13188 df-exp 13248 df-fac 13452 df-bc 13481 df-hash 13509 df-shft 14290 df-cj 14322 df-re 14323 df-im 14324 df-sqrt 14458 df-abs 14459 df-limsup 14692 df-clim 14709 df-rlim 14710 df-sum 14907 df-ef 15284 df-sin 15286 df-cos 15287 df-tan 15288 df-pi 15289 df-struct 16344 df-ndx 16345 df-slot 16346 df-base 16348 df-sets 16349 df-ress 16350 df-plusg 16437 df-mulr 16438 df-starv 16439 df-sca 16440 df-vsca 16441 df-ip 16442 df-tset 16443 df-ple 16444 df-ds 16446 df-unif 16447 df-hom 16448 df-cco 16449 df-rest 16555 df-topn 16556 df-0g 16574 df-gsum 16575 df-topgen 16576 df-pt 16577 df-prds 16580 df-xrs 16634 df-qtop 16639 df-imas 16640 df-xps 16642 df-mre 16718 df-mrc 16719 df-acs 16721 df-mgm 17713 df-sgrp 17755 df-mnd 17766 df-submnd 17807 df-mulg 18015 df-cntz 18221 df-cmn 18671 df-psmet 20242 df-xmet 20243 df-met 20244 df-bl 20245 df-mopn 20246 df-fbas 20247 df-fg 20248 df-cnfld 20251 df-top 21209 df-topon 21226 df-topsp 21248 df-bases 21261 df-cld 21334 df-ntr 21335 df-cls 21336 df-nei 21413 df-lp 21451 df-perf 21452 df-cn 21542 df-cnp 21543 df-haus 21630 df-cmp 21702 df-tx 21877 df-hmeo 22070 df-fil 22161 df-fm 22253 df-flim 22254 df-flf 22255 df-xms 22636 df-ms 22637 df-tms 22638 df-cncf 23192 df-limc 24170 df-dv 24171 df-log 24844 df-cxp 24845 |
This theorem is referenced by: (None) |
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