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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 14717 | . . . . . . 7 ⊢ √:ℂ⟶ℂ | |
2 | 1 | a1i 11 | . . . . . 6 ⊢ (⊤ → √:ℂ⟶ℂ) |
3 | 2 | feqmptd 6728 | . . . . 5 ⊢ (⊤ → √ = (𝑥 ∈ ℂ ↦ (√‘𝑥))) |
4 | 3 | reseq1d 5847 | . . . 4 ⊢ (⊤ → (√ ↾ 𝐷) = ((𝑥 ∈ ℂ ↦ (√‘𝑥)) ↾ 𝐷)) |
5 | sqrcn.d | . . . . . 6 ⊢ 𝐷 = (ℂ ∖ (-∞(,]0)) | |
6 | difss 4108 | . . . . . 6 ⊢ (ℂ ∖ (-∞(,]0)) ⊆ ℂ | |
7 | 5, 6 | eqsstri 4001 | . . . . 5 ⊢ 𝐷 ⊆ ℂ |
8 | resmpt 5900 | . . . . 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 3963 | . . . . . . . 8 ⊢ (𝑥 ∈ 𝐷 → 𝑥 ∈ ℂ) |
11 | 10 | adantl 484 | . . . . . . 7 ⊢ ((⊤ ∧ 𝑥 ∈ 𝐷) → 𝑥 ∈ ℂ) |
12 | cxpsqrt 25280 | . . . . . . 7 ⊢ (𝑥 ∈ ℂ → (𝑥↑𝑐(1 / 2)) = (√‘𝑥)) | |
13 | 11, 12 | syl 17 | . . . . . 6 ⊢ ((⊤ ∧ 𝑥 ∈ 𝐷) → (𝑥↑𝑐(1 / 2)) = (√‘𝑥)) |
14 | 13 | eqcomd 2827 | . . . . 5 ⊢ ((⊤ ∧ 𝑥 ∈ 𝐷) → (√‘𝑥) = (𝑥↑𝑐(1 / 2))) |
15 | 14 | mpteq2dva 5154 | . . . 4 ⊢ (⊤ → (𝑥 ∈ 𝐷 ↦ (√‘𝑥)) = (𝑥 ∈ 𝐷 ↦ (𝑥↑𝑐(1 / 2)))) |
16 | 4, 9, 15 | 3eqtrd 2860 | . . 3 ⊢ (⊤ → (√ ↾ 𝐷) = (𝑥 ∈ 𝐷 ↦ (𝑥↑𝑐(1 / 2)))) |
17 | eqid 2821 | . . . . . . . 8 ⊢ (TopOpen‘ℂfld) = (TopOpen‘ℂfld) | |
18 | 17 | cnfldtopon 23385 | . . . . . . 7 ⊢ (TopOpen‘ℂfld) ∈ (TopOn‘ℂ) |
19 | 18 | a1i 11 | . . . . . 6 ⊢ (⊤ → (TopOpen‘ℂfld) ∈ (TopOn‘ℂ)) |
20 | resttopon 21763 | . . . . . 6 ⊢ (((TopOpen‘ℂfld) ∈ (TopOn‘ℂ) ∧ 𝐷 ⊆ ℂ) → ((TopOpen‘ℂfld) ↾t 𝐷) ∈ (TopOn‘𝐷)) | |
21 | 19, 7, 20 | sylancl 588 | . . . . 5 ⊢ (⊤ → ((TopOpen‘ℂfld) ↾t 𝐷) ∈ (TopOn‘𝐷)) |
22 | 21 | cnmptid 22263 | . . . . 5 ⊢ (⊤ → (𝑥 ∈ 𝐷 ↦ 𝑥) ∈ (((TopOpen‘ℂfld) ↾t 𝐷) Cn ((TopOpen‘ℂfld) ↾t 𝐷))) |
23 | ax-1cn 10589 | . . . . . . 7 ⊢ 1 ∈ ℂ | |
24 | halfcl 11856 | . . . . . . 7 ⊢ (1 ∈ ℂ → (1 / 2) ∈ ℂ) | |
25 | 23, 24 | mp1i 13 | . . . . . 6 ⊢ (⊤ → (1 / 2) ∈ ℂ) |
26 | 21, 19, 25 | cnmptc 22264 | . . . . 5 ⊢ (⊤ → (𝑥 ∈ 𝐷 ↦ (1 / 2)) ∈ (((TopOpen‘ℂfld) ↾t 𝐷) Cn (TopOpen‘ℂfld))) |
27 | eqid 2821 | . . . . . . 7 ⊢ ((TopOpen‘ℂfld) ↾t 𝐷) = ((TopOpen‘ℂfld) ↾t 𝐷) | |
28 | 5, 17, 27 | cxpcn 25320 | . . . . . 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 7159 | . . . . 5 ⊢ ((𝑦 = 𝑥 ∧ 𝑧 = (1 / 2)) → (𝑦↑𝑐𝑧) = (𝑥↑𝑐(1 / 2))) | |
31 | 21, 22, 26, 21, 19, 29, 30 | cnmpt12 22269 | . . . 4 ⊢ (⊤ → (𝑥 ∈ 𝐷 ↦ (𝑥↑𝑐(1 / 2))) ∈ (((TopOpen‘ℂfld) ↾t 𝐷) Cn (TopOpen‘ℂfld))) |
32 | ssid 3989 | . . . . 5 ⊢ ℂ ⊆ ℂ | |
33 | 18 | toponrestid 21523 | . . . . . 6 ⊢ (TopOpen‘ℂfld) = ((TopOpen‘ℂfld) ↾t ℂ) |
34 | 17, 27, 33 | cncfcn 23511 | . . . . 5 ⊢ ((𝐷 ⊆ ℂ ∧ ℂ ⊆ ℂ) → (𝐷–cn→ℂ) = (((TopOpen‘ℂfld) ↾t 𝐷) Cn (TopOpen‘ℂfld))) |
35 | 7, 32, 34 | mp2an 690 | . . . 4 ⊢ (𝐷–cn→ℂ) = (((TopOpen‘ℂfld) ↾t 𝐷) Cn (TopOpen‘ℂfld)) |
36 | 31, 35 | eleqtrrdi 2924 | . . 3 ⊢ (⊤ → (𝑥 ∈ 𝐷 ↦ (𝑥↑𝑐(1 / 2))) ∈ (𝐷–cn→ℂ)) |
37 | 16, 36 | eqeltrd 2913 | . 2 ⊢ (⊤ → (√ ↾ 𝐷) ∈ (𝐷–cn→ℂ)) |
38 | 37 | mptru 1540 | 1 ⊢ (√ ↾ 𝐷) ∈ (𝐷–cn→ℂ) |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 398 = wceq 1533 ⊤wtru 1534 ∈ wcel 2110 ∖ cdif 3933 ⊆ wss 3936 ↦ cmpt 5139 ↾ cres 5552 ⟶wf 6346 ‘cfv 6350 (class class class)co 7150 ∈ cmpo 7152 ℂcc 10529 0cc0 10531 1c1 10532 -∞cmnf 10667 / cdiv 11291 2c2 11686 (,]cioc 12733 √csqrt 14586 ↾t crest 16688 TopOpenctopn 16689 ℂfldccnfld 20539 TopOnctopon 21512 Cn ccn 21826 ×t ctx 22162 –cn→ccncf 23478 ↑𝑐ccxp 25133 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2156 ax-12 2172 ax-ext 2793 ax-rep 5183 ax-sep 5196 ax-nul 5203 ax-pow 5259 ax-pr 5322 ax-un 7455 ax-inf2 9098 ax-cnex 10587 ax-resscn 10588 ax-1cn 10589 ax-icn 10590 ax-addcl 10591 ax-addrcl 10592 ax-mulcl 10593 ax-mulrcl 10594 ax-mulcom 10595 ax-addass 10596 ax-mulass 10597 ax-distr 10598 ax-i2m1 10599 ax-1ne0 10600 ax-1rid 10601 ax-rnegex 10602 ax-rrecex 10603 ax-cnre 10604 ax-pre-lttri 10605 ax-pre-lttrn 10606 ax-pre-ltadd 10607 ax-pre-mulgt0 10608 ax-pre-sup 10609 ax-addf 10610 ax-mulf 10611 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1536 df-fal 1546 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3497 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-pss 3954 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4562 df-pr 4564 df-tp 4566 df-op 4568 df-uni 4833 df-int 4870 df-iun 4914 df-iin 4915 df-br 5060 df-opab 5122 df-mpt 5140 df-tr 5166 df-id 5455 df-eprel 5460 df-po 5469 df-so 5470 df-fr 5509 df-se 5510 df-we 5511 df-xp 5556 df-rel 5557 df-cnv 5558 df-co 5559 df-dm 5560 df-rn 5561 df-res 5562 df-ima 5563 df-pred 6143 df-ord 6189 df-on 6190 df-lim 6191 df-suc 6192 df-iota 6309 df-fun 6352 df-fn 6353 df-f 6354 df-f1 6355 df-fo 6356 df-f1o 6357 df-fv 6358 df-isom 6359 df-riota 7108 df-ov 7153 df-oprab 7154 df-mpo 7155 df-of 7403 df-om 7575 df-1st 7683 df-2nd 7684 df-supp 7825 df-wrecs 7941 df-recs 8002 df-rdg 8040 df-1o 8096 df-2o 8097 df-oadd 8100 df-er 8283 df-map 8402 df-pm 8403 df-ixp 8456 df-en 8504 df-dom 8505 df-sdom 8506 df-fin 8507 df-fsupp 8828 df-fi 8869 df-sup 8900 df-inf 8901 df-oi 8968 df-card 9362 df-pnf 10671 df-mnf 10672 df-xr 10673 df-ltxr 10674 df-le 10675 df-sub 10866 df-neg 10867 df-div 11292 df-nn 11633 df-2 11694 df-3 11695 df-4 11696 df-5 11697 df-6 11698 df-7 11699 df-8 11700 df-9 11701 df-n0 11892 df-z 11976 df-dec 12093 df-uz 12238 df-q 12343 df-rp 12384 df-xneg 12501 df-xadd 12502 df-xmul 12503 df-ioo 12736 df-ioc 12737 df-ico 12738 df-icc 12739 df-fz 12887 df-fzo 13028 df-fl 13156 df-mod 13232 df-seq 13364 df-exp 13424 df-fac 13628 df-bc 13657 df-hash 13685 df-shft 14420 df-cj 14452 df-re 14453 df-im 14454 df-sqrt 14588 df-abs 14589 df-limsup 14822 df-clim 14839 df-rlim 14840 df-sum 15037 df-ef 15415 df-sin 15417 df-cos 15418 df-tan 15419 df-pi 15420 df-struct 16479 df-ndx 16480 df-slot 16481 df-base 16483 df-sets 16484 df-ress 16485 df-plusg 16572 df-mulr 16573 df-starv 16574 df-sca 16575 df-vsca 16576 df-ip 16577 df-tset 16578 df-ple 16579 df-ds 16581 df-unif 16582 df-hom 16583 df-cco 16584 df-rest 16690 df-topn 16691 df-0g 16709 df-gsum 16710 df-topgen 16711 df-pt 16712 df-prds 16715 df-xrs 16769 df-qtop 16774 df-imas 16775 df-xps 16777 df-mre 16851 df-mrc 16852 df-acs 16854 df-mgm 17846 df-sgrp 17895 df-mnd 17906 df-submnd 17951 df-mulg 18219 df-cntz 18441 df-cmn 18902 df-psmet 20531 df-xmet 20532 df-met 20533 df-bl 20534 df-mopn 20535 df-fbas 20536 df-fg 20537 df-cnfld 20540 df-top 21496 df-topon 21513 df-topsp 21535 df-bases 21548 df-cld 21621 df-ntr 21622 df-cls 21623 df-nei 21700 df-lp 21738 df-perf 21739 df-cn 21829 df-cnp 21830 df-haus 21917 df-cmp 21989 df-tx 22164 df-hmeo 22357 df-fil 22448 df-fm 22540 df-flim 22541 df-flf 22542 df-xms 22924 df-ms 22925 df-tms 22926 df-cncf 23480 df-limc 24458 df-dv 24459 df-log 25134 df-cxp 25135 |
This theorem is referenced by: (None) |
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