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Theorem sqrtcn 26712

Description: Continuity of the square root function. (Contributed by Mario Carneiro, 2-May-2016.)

**Hypothesis**
Ref	Expression
sqrcn.d	⊢ 𝐷 = (ℂ ∖ (-∞(,]0))

**Assertion**
Ref	Expression
sqrtcn	⊢ (√ ↾ 𝐷) ∈ (𝐷–cn→ℂ)

Proof of Theorem sqrtcn Dummy variables 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		sqrtf 15382	. . . . . . 7 ⊢ √:ℂ⟶ℂ
2	1	a1i 11	. . . . . 6 ⊢ (⊤ → √:ℂ⟶ℂ)
3	2	feqmptd 6947	. . . . 5 ⊢ (⊤ → √ = (𝑥 ∈ ℂ ↦ (√‘𝑥)))
4	3	reseq1d 5965	. . . 4 ⊢ (⊤ → (√ ↾ 𝐷) = ((𝑥 ∈ ℂ ↦ (√‘𝑥)) ↾ 𝐷))
5		sqrcn.d	. . . . . 6 ⊢ 𝐷 = (ℂ ∖ (-∞(,]0))
6		difss 4111	. . . . . 6 ⊢ (ℂ ∖ (-∞(,]0)) ⊆ ℂ
7	5, 6	eqsstri 4005	. . . . 5 ⊢ 𝐷 ⊆ ℂ
8		resmpt 6024	. . . . 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 3954	. . . . . . . 8 ⊢ (𝑥 ∈ 𝐷 → 𝑥 ∈ ℂ)
11	10	adantl 481	. . . . . . 7 ⊢ ((⊤ ∧ 𝑥 ∈ 𝐷) → 𝑥 ∈ ℂ)
12		cxpsqrt 26664	. . . . . . 7 ⊢ (𝑥 ∈ ℂ → (𝑥↑_𝑐(1 / 2)) = (√‘𝑥))
13	11, 12	syl 17	. . . . . 6 ⊢ ((⊤ ∧ 𝑥 ∈ 𝐷) → (𝑥↑_𝑐(1 / 2)) = (√‘𝑥))
14	13	eqcomd 2741	. . . . 5 ⊢ ((⊤ ∧ 𝑥 ∈ 𝐷) → (√‘𝑥) = (𝑥↑_𝑐(1 / 2)))
15	14	mpteq2dva 5214	. . . 4 ⊢ (⊤ → (𝑥 ∈ 𝐷 ↦ (√‘𝑥)) = (𝑥 ∈ 𝐷 ↦ (𝑥↑_𝑐(1 / 2))))
16	4, 9, 15	3eqtrd 2774	. . 3 ⊢ (⊤ → (√ ↾ 𝐷) = (𝑥 ∈ 𝐷 ↦ (𝑥↑_𝑐(1 / 2))))
17		eqid 2735	. . . . . . . 8 ⊢ (TopOpen‘ℂ_fld) = (TopOpen‘ℂ_fld)
18	17	cnfldtopon 24721	. . . . . . 7 ⊢ (TopOpen‘ℂ_fld) ∈ (TopOn‘ℂ)
19	18	a1i 11	. . . . . 6 ⊢ (⊤ → (TopOpen‘ℂ_fld) ∈ (TopOn‘ℂ))
20		resttopon 23099	. . . . . 6 ⊢ (((TopOpen‘ℂ_fld) ∈ (TopOn‘ℂ) ∧ 𝐷 ⊆ ℂ) → ((TopOpen‘ℂ_fld) ↾_t 𝐷) ∈ (TopOn‘𝐷))
21	19, 7, 20	sylancl 586	. . . . 5 ⊢ (⊤ → ((TopOpen‘ℂ_fld) ↾_t 𝐷) ∈ (TopOn‘𝐷))
22	21	cnmptid 23599	. . . . 5 ⊢ (⊤ → (𝑥 ∈ 𝐷 ↦ 𝑥) ∈ (((TopOpen‘ℂ_fld) ↾_t 𝐷) Cn ((TopOpen‘ℂ_fld) ↾_t 𝐷)))
23		ax-1cn 11187	. . . . . . 7 ⊢ 1 ∈ ℂ
24		halfcl 12467	. . . . . . 7 ⊢ (1 ∈ ℂ → (1 / 2) ∈ ℂ)
25	23, 24	mp1i 13	. . . . . 6 ⊢ (⊤ → (1 / 2) ∈ ℂ)
26	21, 19, 25	cnmptc 23600	. . . . 5 ⊢ (⊤ → (𝑥 ∈ 𝐷 ↦ (1 / 2)) ∈ (((TopOpen‘ℂ_fld) ↾_t 𝐷) Cn (TopOpen‘ℂ_fld)))
27		eqid 2735	. . . . . . 7 ⊢ ((TopOpen‘ℂ_fld) ↾_t 𝐷) = ((TopOpen‘ℂ_fld) ↾_t 𝐷)
28	5, 17, 27	cxpcn 26706	. . . . . 6 ⊢ (𝑦 ∈ 𝐷, 𝑧 ∈ ℂ ↦ (𝑦↑_𝑐𝑧)) ∈ ((((TopOpen‘ℂ_fld) ↾_t 𝐷) ×_t (TopOpen‘ℂ_fld)) Cn (TopOpen‘ℂ_fld))
29	28	a1i 11	. . . . 5 ⊢ (⊤ → (𝑦 ∈ 𝐷, 𝑧 ∈ ℂ ↦ (𝑦↑_𝑐𝑧)) ∈ ((((TopOpen‘ℂ_fld) ↾_t 𝐷) ×_t (TopOpen‘ℂ_fld)) Cn (TopOpen‘ℂ_fld)))
30		oveq12 7414	. . . . 5 ⊢ ((𝑦 = 𝑥 ∧ 𝑧 = (1 / 2)) → (𝑦↑_𝑐𝑧) = (𝑥↑_𝑐(1 / 2)))
31	21, 22, 26, 21, 19, 29, 30	cnmpt12 23605	. . . 4 ⊢ (⊤ → (𝑥 ∈ 𝐷 ↦ (𝑥↑_𝑐(1 / 2))) ∈ (((TopOpen‘ℂ_fld) ↾_t 𝐷) Cn (TopOpen‘ℂ_fld)))
32		ssid 3981	. . . . 5 ⊢ ℂ ⊆ ℂ
33	18	toponrestid 22859	. . . . . 6 ⊢ (TopOpen‘ℂ_fld) = ((TopOpen‘ℂ_fld) ↾_t ℂ)
34	17, 27, 33	cncfcn 24854	. . . . 5 ⊢ ((𝐷 ⊆ ℂ ∧ ℂ ⊆ ℂ) → (𝐷–cn→ℂ) = (((TopOpen‘ℂ_fld) ↾_t 𝐷) Cn (TopOpen‘ℂ_fld)))
35	7, 32, 34	mp2an 692	. . . 4 ⊢ (𝐷–cn→ℂ) = (((TopOpen‘ℂ_fld) ↾_t 𝐷) Cn (TopOpen‘ℂ_fld))
36	31, 35	eleqtrrdi 2845	. . 3 ⊢ (⊤ → (𝑥 ∈ 𝐷 ↦ (𝑥↑_𝑐(1 / 2))) ∈ (𝐷–cn→ℂ))
37	16, 36	eqeltrd 2834	. 2 ⊢ (⊤ → (√ ↾ 𝐷) ∈ (𝐷–cn→ℂ))
38	37	mptru 1547	1 ⊢ (√ ↾ 𝐷) ∈ (𝐷–cn→ℂ)

Colors of variables: wff setvar class

Syntax hints: ∧ wa 395 = wceq 1540 ⊤wtru 1541 ∈ wcel 2108 ∖ cdif 3923 ⊆ wss 3926 ↦ cmpt 5201 ↾ cres 5656 ⟶wf 6527 ‘cfv 6531 (class class class)co 7405 ∈ cmpo 7407 ℂcc 11127 0cc0 11129 1c1 11130 -∞cmnf 11267 / cdiv 11894 2c2 12295 (,]cioc 13363 √csqrt 15252 ↾_t crest 17434 TopOpenctopn 17435 ℂ_fldccnfld 21315 TopOnctopon 22848 Cn ccn 23162 ×_t ctx 23498 –cn→ccncf 24820 ↑_𝑐ccxp 26516

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2707 ax-rep 5249 ax-sep 5266 ax-nul 5276 ax-pow 5335 ax-pr 5402 ax-un 7729 ax-inf2 9655 ax-cnex 11185 ax-resscn 11186 ax-1cn 11187 ax-icn 11188 ax-addcl 11189 ax-addrcl 11190 ax-mulcl 11191 ax-mulrcl 11192 ax-mulcom 11193 ax-addass 11194 ax-mulass 11195 ax-distr 11196 ax-i2m1 11197 ax-1ne0 11198 ax-1rid 11199 ax-rnegex 11200 ax-rrecex 11201 ax-cnre 11202 ax-pre-lttri 11203 ax-pre-lttrn 11204 ax-pre-ltadd 11205 ax-pre-mulgt0 11206 ax-pre-sup 11207 ax-addf 11208

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2727 df-clel 2809 df-nfc 2885 df-ne 2933 df-nel 3037 df-ral 3052 df-rex 3061 df-rmo 3359 df-reu 3360 df-rab 3416 df-v 3461 df-sbc 3766 df-csb 3875 df-dif 3929 df-un 3931 df-in 3933 df-ss 3943 df-pss 3946 df-nul 4309 df-if 4501 df-pw 4577 df-sn 4602 df-pr 4604 df-tp 4606 df-op 4608 df-uni 4884 df-int 4923 df-iun 4969 df-iin 4970 df-br 5120 df-opab 5182 df-mpt 5202 df-tr 5230 df-id 5548 df-eprel 5553 df-po 5561 df-so 5562 df-fr 5606 df-se 5607 df-we 5608 df-xp 5660 df-rel 5661 df-cnv 5662 df-co 5663 df-dm 5664 df-rn 5665 df-res 5666 df-ima 5667 df-pred 6290 df-ord 6355 df-on 6356 df-lim 6357 df-suc 6358 df-iota 6484 df-fun 6533 df-fn 6534 df-f 6535 df-f1 6536 df-fo 6537 df-f1o 6538 df-fv 6539 df-isom 6540 df-riota 7362 df-ov 7408 df-oprab 7409 df-mpo 7410 df-of 7671 df-om 7862 df-1st 7988 df-2nd 7989 df-supp 8160 df-frecs 8280 df-wrecs 8311 df-recs 8385 df-rdg 8424 df-1o 8480 df-2o 8481 df-er 8719 df-map 8842 df-pm 8843 df-ixp 8912 df-en 8960 df-dom 8961 df-sdom 8962 df-fin 8963 df-fsupp 9374 df-fi 9423 df-sup 9454 df-inf 9455 df-oi 9524 df-card 9953 df-pnf 11271 df-mnf 11272 df-xr 11273 df-ltxr 11274 df-le 11275 df-sub 11468 df-neg 11469 df-div 11895 df-nn 12241 df-2 12303 df-3 12304 df-4 12305 df-5 12306 df-6 12307 df-7 12308 df-8 12309 df-9 12310 df-n0 12502 df-z 12589 df-dec 12709 df-uz 12853 df-q 12965 df-rp 13009 df-xneg 13128 df-xadd 13129 df-xmul 13130 df-ioo 13366 df-ioc 13367 df-ico 13368 df-icc 13369 df-fz 13525 df-fzo 13672 df-fl 13809 df-mod 13887 df-seq 14020 df-exp 14080 df-fac 14292 df-bc 14321 df-hash 14349 df-shft 15086 df-cj 15118 df-re 15119 df-im 15120 df-sqrt 15254 df-abs 15255 df-limsup 15487 df-clim 15504 df-rlim 15505 df-sum 15703 df-ef 16083 df-sin 16085 df-cos 16086 df-tan 16087 df-pi 16088 df-struct 17166 df-sets 17183 df-slot 17201 df-ndx 17213 df-base 17229 df-ress 17252 df-plusg 17284 df-mulr 17285 df-starv 17286 df-sca 17287 df-vsca 17288 df-ip 17289 df-tset 17290 df-ple 17291 df-ds 17293 df-unif 17294 df-hom 17295 df-cco 17296 df-rest 17436 df-topn 17437 df-0g 17455 df-gsum 17456 df-topgen 17457 df-pt 17458 df-prds 17461 df-xrs 17516 df-qtop 17521 df-imas 17522 df-xps 17524 df-mre 17598 df-mrc 17599 df-acs 17601 df-mgm 18618 df-sgrp 18697 df-mnd 18713 df-submnd 18762 df-mulg 19051 df-cntz 19300 df-cmn 19763 df-psmet 21307 df-xmet 21308 df-met 21309 df-bl 21310 df-mopn 21311 df-fbas 21312 df-fg 21313 df-cnfld 21316 df-top 22832 df-topon 22849 df-topsp 22871 df-bases 22884 df-cld 22957 df-ntr 22958 df-cls 22959 df-nei 23036 df-lp 23074 df-perf 23075 df-cn 23165 df-cnp 23166 df-haus 23253 df-cmp 23325 df-tx 23500 df-hmeo 23693 df-fil 23784 df-fm 23876 df-flim 23877 df-flf 23878 df-xms 24259 df-ms 24260 df-tms 24261 df-cncf 24822 df-limc 25819 df-dv 25820 df-log 26517 df-cxp 26518

This theorem is referenced by: (None)

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