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Theorem sqrtcn 25903

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 15075	. . . . . . 7 ⊢ √:ℂ⟶ℂ
2	1	a1i 11	. . . . . 6 ⊢ (⊤ → √:ℂ⟶ℂ)
3	2	feqmptd 6837	. . . . 5 ⊢ (⊤ → √ = (𝑥 ∈ ℂ ↦ (√‘𝑥)))
4	3	reseq1d 5890	. . . 4 ⊢ (⊤ → (√ ↾ 𝐷) = ((𝑥 ∈ ℂ ↦ (√‘𝑥)) ↾ 𝐷))
5		sqrcn.d	. . . . . 6 ⊢ 𝐷 = (ℂ ∖ (-∞(,]0))
6		difss 4066	. . . . . 6 ⊢ (ℂ ∖ (-∞(,]0)) ⊆ ℂ
7	5, 6	eqsstri 3955	. . . . 5 ⊢ 𝐷 ⊆ ℂ
8		resmpt 5945	. . . . 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 3917	. . . . . . . 8 ⊢ (𝑥 ∈ 𝐷 → 𝑥 ∈ ℂ)
11	10	adantl 482	. . . . . . 7 ⊢ ((⊤ ∧ 𝑥 ∈ 𝐷) → 𝑥 ∈ ℂ)
12		cxpsqrt 25858	. . . . . . 7 ⊢ (𝑥 ∈ ℂ → (𝑥↑_𝑐(1 / 2)) = (√‘𝑥))
13	11, 12	syl 17	. . . . . 6 ⊢ ((⊤ ∧ 𝑥 ∈ 𝐷) → (𝑥↑_𝑐(1 / 2)) = (√‘𝑥))
14	13	eqcomd 2744	. . . . 5 ⊢ ((⊤ ∧ 𝑥 ∈ 𝐷) → (√‘𝑥) = (𝑥↑_𝑐(1 / 2)))
15	14	mpteq2dva 5174	. . . 4 ⊢ (⊤ → (𝑥 ∈ 𝐷 ↦ (√‘𝑥)) = (𝑥 ∈ 𝐷 ↦ (𝑥↑_𝑐(1 / 2))))
16	4, 9, 15	3eqtrd 2782	. . 3 ⊢ (⊤ → (√ ↾ 𝐷) = (𝑥 ∈ 𝐷 ↦ (𝑥↑_𝑐(1 / 2))))
17		eqid 2738	. . . . . . . 8 ⊢ (TopOpen‘ℂ_fld) = (TopOpen‘ℂ_fld)
18	17	cnfldtopon 23946	. . . . . . 7 ⊢ (TopOpen‘ℂ_fld) ∈ (TopOn‘ℂ)
19	18	a1i 11	. . . . . 6 ⊢ (⊤ → (TopOpen‘ℂ_fld) ∈ (TopOn‘ℂ))
20		resttopon 22312	. . . . . 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 22812	. . . . 5 ⊢ (⊤ → (𝑥 ∈ 𝐷 ↦ 𝑥) ∈ (((TopOpen‘ℂ_fld) ↾_t 𝐷) Cn ((TopOpen‘ℂ_fld) ↾_t 𝐷)))
23		ax-1cn 10929	. . . . . . 7 ⊢ 1 ∈ ℂ
24		halfcl 12198	. . . . . . 7 ⊢ (1 ∈ ℂ → (1 / 2) ∈ ℂ)
25	23, 24	mp1i 13	. . . . . 6 ⊢ (⊤ → (1 / 2) ∈ ℂ)
26	21, 19, 25	cnmptc 22813	. . . . 5 ⊢ (⊤ → (𝑥 ∈ 𝐷 ↦ (1 / 2)) ∈ (((TopOpen‘ℂ_fld) ↾_t 𝐷) Cn (TopOpen‘ℂ_fld)))
27		eqid 2738	. . . . . . 7 ⊢ ((TopOpen‘ℂ_fld) ↾_t 𝐷) = ((TopOpen‘ℂ_fld) ↾_t 𝐷)
28	5, 17, 27	cxpcn 25898	. . . . . 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 7284	. . . . 5 ⊢ ((𝑦 = 𝑥 ∧ 𝑧 = (1 / 2)) → (𝑦↑_𝑐𝑧) = (𝑥↑_𝑐(1 / 2)))
31	21, 22, 26, 21, 19, 29, 30	cnmpt12 22818	. . . 4 ⊢ (⊤ → (𝑥 ∈ 𝐷 ↦ (𝑥↑_𝑐(1 / 2))) ∈ (((TopOpen‘ℂ_fld) ↾_t 𝐷) Cn (TopOpen‘ℂ_fld)))
32		ssid 3943	. . . . 5 ⊢ ℂ ⊆ ℂ
33	18	toponrestid 22070	. . . . . 6 ⊢ (TopOpen‘ℂ_fld) = ((TopOpen‘ℂ_fld) ↾_t ℂ)
34	17, 27, 33	cncfcn 24073	. . . . 5 ⊢ ((𝐷 ⊆ ℂ ∧ ℂ ⊆ ℂ) → (𝐷–cn→ℂ) = (((TopOpen‘ℂ_fld) ↾_t 𝐷) Cn (TopOpen‘ℂ_fld)))
35	7, 32, 34	mp2an 689	. . . 4 ⊢ (𝐷–cn→ℂ) = (((TopOpen‘ℂ_fld) ↾_t 𝐷) Cn (TopOpen‘ℂ_fld))
36	31, 35	eleqtrrdi 2850	. . 3 ⊢ (⊤ → (𝑥 ∈ 𝐷 ↦ (𝑥↑_𝑐(1 / 2))) ∈ (𝐷–cn→ℂ))
37	16, 36	eqeltrd 2839	. 2 ⊢ (⊤ → (√ ↾ 𝐷) ∈ (𝐷–cn→ℂ))
38	37	mptru 1546	1 ⊢ (√ ↾ 𝐷) ∈ (𝐷–cn→ℂ)

Colors of variables: wff setvar class

Syntax hints: ∧ wa 396 = wceq 1539 ⊤wtru 1540 ∈ wcel 2106 ∖ cdif 3884 ⊆ wss 3887 ↦ cmpt 5157 ↾ cres 5591 ⟶wf 6429 ‘cfv 6433 (class class class)co 7275 ∈ cmpo 7277 ℂcc 10869 0cc0 10871 1c1 10872 -∞cmnf 11007 / cdiv 11632 2c2 12028 (,]cioc 13080 √csqrt 14944 ↾_t crest 17131 TopOpenctopn 17132 ℂ_fldccnfld 20597 TopOnctopon 22059 Cn ccn 22375 ×_t ctx 22711 –cn→ccncf 24039 ↑_𝑐ccxp 25711

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 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2709 ax-rep 5209 ax-sep 5223 ax-nul 5230 ax-pow 5288 ax-pr 5352 ax-un 7588 ax-inf2 9399 ax-cnex 10927 ax-resscn 10928 ax-1cn 10929 ax-icn 10930 ax-addcl 10931 ax-addrcl 10932 ax-mulcl 10933 ax-mulrcl 10934 ax-mulcom 10935 ax-addass 10936 ax-mulass 10937 ax-distr 10938 ax-i2m1 10939 ax-1ne0 10940 ax-1rid 10941 ax-rnegex 10942 ax-rrecex 10943 ax-cnre 10944 ax-pre-lttri 10945 ax-pre-lttrn 10946 ax-pre-ltadd 10947 ax-pre-mulgt0 10948 ax-pre-sup 10949 ax-addf 10950 ax-mulf 10951

This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2068 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2889 df-ne 2944 df-nel 3050 df-ral 3069 df-rex 3070 df-rmo 3071 df-reu 3072 df-rab 3073 df-v 3434 df-sbc 3717 df-csb 3833 df-dif 3890 df-un 3892 df-in 3894 df-ss 3904 df-pss 3906 df-nul 4257 df-if 4460 df-pw 4535 df-sn 4562 df-pr 4564 df-tp 4566 df-op 4568 df-uni 4840 df-int 4880 df-iun 4926 df-iin 4927 df-br 5075 df-opab 5137 df-mpt 5158 df-tr 5192 df-id 5489 df-eprel 5495 df-po 5503 df-so 5504 df-fr 5544 df-se 5545 df-we 5546 df-xp 5595 df-rel 5596 df-cnv 5597 df-co 5598 df-dm 5599 df-rn 5600 df-res 5601 df-ima 5602 df-pred 6202 df-ord 6269 df-on 6270 df-lim 6271 df-suc 6272 df-iota 6391 df-fun 6435 df-fn 6436 df-f 6437 df-f1 6438 df-fo 6439 df-f1o 6440 df-fv 6441 df-isom 6442 df-riota 7232 df-ov 7278 df-oprab 7279 df-mpo 7280 df-of 7533 df-om 7713 df-1st 7831 df-2nd 7832 df-supp 7978 df-frecs 8097 df-wrecs 8128 df-recs 8202 df-rdg 8241 df-1o 8297 df-2o 8298 df-er 8498 df-map 8617 df-pm 8618 df-ixp 8686 df-en 8734 df-dom 8735 df-sdom 8736 df-fin 8737 df-fsupp 9129 df-fi 9170 df-sup 9201 df-inf 9202 df-oi 9269 df-card 9697 df-pnf 11011 df-mnf 11012 df-xr 11013 df-ltxr 11014 df-le 11015 df-sub 11207 df-neg 11208 df-div 11633 df-nn 11974 df-2 12036 df-3 12037 df-4 12038 df-5 12039 df-6 12040 df-7 12041 df-8 12042 df-9 12043 df-n0 12234 df-z 12320 df-dec 12438 df-uz 12583 df-q 12689 df-rp 12731 df-xneg 12848 df-xadd 12849 df-xmul 12850 df-ioo 13083 df-ioc 13084 df-ico 13085 df-icc 13086 df-fz 13240 df-fzo 13383 df-fl 13512 df-mod 13590 df-seq 13722 df-exp 13783 df-fac 13988 df-bc 14017 df-hash 14045 df-shft 14778 df-cj 14810 df-re 14811 df-im 14812 df-sqrt 14946 df-abs 14947 df-limsup 15180 df-clim 15197 df-rlim 15198 df-sum 15398 df-ef 15777 df-sin 15779 df-cos 15780 df-tan 15781 df-pi 15782 df-struct 16848 df-sets 16865 df-slot 16883 df-ndx 16895 df-base 16913 df-ress 16942 df-plusg 16975 df-mulr 16976 df-starv 16977 df-sca 16978 df-vsca 16979 df-ip 16980 df-tset 16981 df-ple 16982 df-ds 16984 df-unif 16985 df-hom 16986 df-cco 16987 df-rest 17133 df-topn 17134 df-0g 17152 df-gsum 17153 df-topgen 17154 df-pt 17155 df-prds 17158 df-xrs 17213 df-qtop 17218 df-imas 17219 df-xps 17221 df-mre 17295 df-mrc 17296 df-acs 17298 df-mgm 18326 df-sgrp 18375 df-mnd 18386 df-submnd 18431 df-mulg 18701 df-cntz 18923 df-cmn 19388 df-psmet 20589 df-xmet 20590 df-met 20591 df-bl 20592 df-mopn 20593 df-fbas 20594 df-fg 20595 df-cnfld 20598 df-top 22043 df-topon 22060 df-topsp 22082 df-bases 22096 df-cld 22170 df-ntr 22171 df-cls 22172 df-nei 22249 df-lp 22287 df-perf 22288 df-cn 22378 df-cnp 22379 df-haus 22466 df-cmp 22538 df-tx 22713 df-hmeo 22906 df-fil 22997 df-fm 23089 df-flim 23090 df-flf 23091 df-xms 23473 df-ms 23474 df-tms 23475 df-cncf 24041 df-limc 25030 df-dv 25031 df-log 25712 df-cxp 25713

This theorem is referenced by: (None)

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