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Mirrors > Home > MPE Home > Th. List > dvcnsqrt | Structured version Visualization version GIF version |
Description: Derivative of square root function. (Contributed by Brendan Leahy, 18-Dec-2018.) |
Ref | Expression |
---|---|
dvcncxp1.d | ⊢ 𝐷 = (ℂ ∖ (-∞(,]0)) |
Ref | Expression |
---|---|
dvcnsqrt | ⊢ (ℂ D (𝑥 ∈ 𝐷 ↦ (√‘𝑥))) = (𝑥 ∈ 𝐷 ↦ (1 / (2 · (√‘𝑥)))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | halfcn 11247 | . . 3 ⊢ (1 / 2) ∈ ℂ | |
2 | dvcncxp1.d | . . . 4 ⊢ 𝐷 = (ℂ ∖ (-∞(,]0)) | |
3 | 2 | dvcncxp1 24484 | . . 3 ⊢ ((1 / 2) ∈ ℂ → (ℂ D (𝑥 ∈ 𝐷 ↦ (𝑥↑𝑐(1 / 2)))) = (𝑥 ∈ 𝐷 ↦ ((1 / 2) · (𝑥↑𝑐((1 / 2) − 1))))) |
4 | 1, 3 | ax-mp 5 | . 2 ⊢ (ℂ D (𝑥 ∈ 𝐷 ↦ (𝑥↑𝑐(1 / 2)))) = (𝑥 ∈ 𝐷 ↦ ((1 / 2) · (𝑥↑𝑐((1 / 2) − 1)))) |
5 | difss 3737 | . . . . . . 7 ⊢ (ℂ ∖ (-∞(,]0)) ⊆ ℂ | |
6 | 2, 5 | eqsstri 3635 | . . . . . 6 ⊢ 𝐷 ⊆ ℂ |
7 | 6 | sseli 3599 | . . . . 5 ⊢ (𝑥 ∈ 𝐷 → 𝑥 ∈ ℂ) |
8 | cxpsqrt 24449 | . . . . 5 ⊢ (𝑥 ∈ ℂ → (𝑥↑𝑐(1 / 2)) = (√‘𝑥)) | |
9 | 7, 8 | syl 17 | . . . 4 ⊢ (𝑥 ∈ 𝐷 → (𝑥↑𝑐(1 / 2)) = (√‘𝑥)) |
10 | 9 | mpteq2ia 4740 | . . 3 ⊢ (𝑥 ∈ 𝐷 ↦ (𝑥↑𝑐(1 / 2))) = (𝑥 ∈ 𝐷 ↦ (√‘𝑥)) |
11 | 10 | oveq2i 6661 | . 2 ⊢ (ℂ D (𝑥 ∈ 𝐷 ↦ (𝑥↑𝑐(1 / 2)))) = (ℂ D (𝑥 ∈ 𝐷 ↦ (√‘𝑥))) |
12 | 1p0e1 11133 | . . . . . . . . . . 11 ⊢ (1 + 0) = 1 | |
13 | ax-1cn 9994 | . . . . . . . . . . . 12 ⊢ 1 ∈ ℂ | |
14 | 2halves 11260 | . . . . . . . . . . . 12 ⊢ (1 ∈ ℂ → ((1 / 2) + (1 / 2)) = 1) | |
15 | 13, 14 | ax-mp 5 | . . . . . . . . . . 11 ⊢ ((1 / 2) + (1 / 2)) = 1 |
16 | 12, 15 | eqtr4i 2647 | . . . . . . . . . 10 ⊢ (1 + 0) = ((1 / 2) + (1 / 2)) |
17 | 0cn 10032 | . . . . . . . . . . 11 ⊢ 0 ∈ ℂ | |
18 | addsubeq4 10296 | . . . . . . . . . . 11 ⊢ (((1 ∈ ℂ ∧ 0 ∈ ℂ) ∧ ((1 / 2) ∈ ℂ ∧ (1 / 2) ∈ ℂ)) → ((1 + 0) = ((1 / 2) + (1 / 2)) ↔ ((1 / 2) − 1) = (0 − (1 / 2)))) | |
19 | 13, 17, 1, 1, 18 | mp4an 709 | . . . . . . . . . 10 ⊢ ((1 + 0) = ((1 / 2) + (1 / 2)) ↔ ((1 / 2) − 1) = (0 − (1 / 2))) |
20 | 16, 19 | mpbi 220 | . . . . . . . . 9 ⊢ ((1 / 2) − 1) = (0 − (1 / 2)) |
21 | df-neg 10269 | . . . . . . . . 9 ⊢ -(1 / 2) = (0 − (1 / 2)) | |
22 | 20, 21 | eqtr4i 2647 | . . . . . . . 8 ⊢ ((1 / 2) − 1) = -(1 / 2) |
23 | 22 | oveq2i 6661 | . . . . . . 7 ⊢ (𝑥↑𝑐((1 / 2) − 1)) = (𝑥↑𝑐-(1 / 2)) |
24 | 2 | logdmn0 24386 | . . . . . . . 8 ⊢ (𝑥 ∈ 𝐷 → 𝑥 ≠ 0) |
25 | 1 | a1i 11 | . . . . . . . 8 ⊢ (𝑥 ∈ 𝐷 → (1 / 2) ∈ ℂ) |
26 | 7, 24, 25 | cxpnegd 24461 | . . . . . . 7 ⊢ (𝑥 ∈ 𝐷 → (𝑥↑𝑐-(1 / 2)) = (1 / (𝑥↑𝑐(1 / 2)))) |
27 | 23, 26 | syl5eq 2668 | . . . . . 6 ⊢ (𝑥 ∈ 𝐷 → (𝑥↑𝑐((1 / 2) − 1)) = (1 / (𝑥↑𝑐(1 / 2)))) |
28 | 9 | oveq2d 6666 | . . . . . 6 ⊢ (𝑥 ∈ 𝐷 → (1 / (𝑥↑𝑐(1 / 2))) = (1 / (√‘𝑥))) |
29 | 27, 28 | eqtrd 2656 | . . . . 5 ⊢ (𝑥 ∈ 𝐷 → (𝑥↑𝑐((1 / 2) − 1)) = (1 / (√‘𝑥))) |
30 | 29 | oveq2d 6666 | . . . 4 ⊢ (𝑥 ∈ 𝐷 → ((1 / 2) · (𝑥↑𝑐((1 / 2) − 1))) = ((1 / 2) · (1 / (√‘𝑥)))) |
31 | 1cnd 10056 | . . . . . 6 ⊢ (𝑥 ∈ 𝐷 → 1 ∈ ℂ) | |
32 | 2cnd 11093 | . . . . . 6 ⊢ (𝑥 ∈ 𝐷 → 2 ∈ ℂ) | |
33 | 7 | sqrtcld 14176 | . . . . . 6 ⊢ (𝑥 ∈ 𝐷 → (√‘𝑥) ∈ ℂ) |
34 | 2ne0 11113 | . . . . . . 7 ⊢ 2 ≠ 0 | |
35 | 34 | a1i 11 | . . . . . 6 ⊢ (𝑥 ∈ 𝐷 → 2 ≠ 0) |
36 | 7 | adantr 481 | . . . . . . . . . 10 ⊢ ((𝑥 ∈ 𝐷 ∧ (√‘𝑥) = 0) → 𝑥 ∈ ℂ) |
37 | simpr 477 | . . . . . . . . . 10 ⊢ ((𝑥 ∈ 𝐷 ∧ (√‘𝑥) = 0) → (√‘𝑥) = 0) | |
38 | 36, 37 | sqr00d 14180 | . . . . . . . . 9 ⊢ ((𝑥 ∈ 𝐷 ∧ (√‘𝑥) = 0) → 𝑥 = 0) |
39 | 38 | ex 450 | . . . . . . . 8 ⊢ (𝑥 ∈ 𝐷 → ((√‘𝑥) = 0 → 𝑥 = 0)) |
40 | 39 | necon3d 2815 | . . . . . . 7 ⊢ (𝑥 ∈ 𝐷 → (𝑥 ≠ 0 → (√‘𝑥) ≠ 0)) |
41 | 24, 40 | mpd 15 | . . . . . 6 ⊢ (𝑥 ∈ 𝐷 → (√‘𝑥) ≠ 0) |
42 | 31, 32, 31, 33, 35, 41 | divmuldivd 10842 | . . . . 5 ⊢ (𝑥 ∈ 𝐷 → ((1 / 2) · (1 / (√‘𝑥))) = ((1 · 1) / (2 · (√‘𝑥)))) |
43 | 1t1e1 11175 | . . . . . 6 ⊢ (1 · 1) = 1 | |
44 | 43 | oveq1i 6660 | . . . . 5 ⊢ ((1 · 1) / (2 · (√‘𝑥))) = (1 / (2 · (√‘𝑥))) |
45 | 42, 44 | syl6eq 2672 | . . . 4 ⊢ (𝑥 ∈ 𝐷 → ((1 / 2) · (1 / (√‘𝑥))) = (1 / (2 · (√‘𝑥)))) |
46 | 30, 45 | eqtrd 2656 | . . 3 ⊢ (𝑥 ∈ 𝐷 → ((1 / 2) · (𝑥↑𝑐((1 / 2) − 1))) = (1 / (2 · (√‘𝑥)))) |
47 | 46 | mpteq2ia 4740 | . 2 ⊢ (𝑥 ∈ 𝐷 ↦ ((1 / 2) · (𝑥↑𝑐((1 / 2) − 1)))) = (𝑥 ∈ 𝐷 ↦ (1 / (2 · (√‘𝑥)))) |
48 | 4, 11, 47 | 3eqtr3i 2652 | 1 ⊢ (ℂ D (𝑥 ∈ 𝐷 ↦ (√‘𝑥))) = (𝑥 ∈ 𝐷 ↦ (1 / (2 · (√‘𝑥)))) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 196 ∧ wa 384 = wceq 1483 ∈ wcel 1990 ≠ wne 2794 ∖ cdif 3571 ↦ cmpt 4729 ‘cfv 5888 (class class class)co 6650 ℂcc 9934 0cc0 9936 1c1 9937 + caddc 9939 · cmul 9941 -∞cmnf 10072 − cmin 10266 -cneg 10267 / cdiv 10684 2c2 11070 (,]cioc 12176 √csqrt 13973 D cdv 23627 ↑𝑐ccxp 24302 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-rep 4771 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-inf2 8538 ax-cnex 9992 ax-resscn 9993 ax-1cn 9994 ax-icn 9995 ax-addcl 9996 ax-addrcl 9997 ax-mulcl 9998 ax-mulrcl 9999 ax-mulcom 10000 ax-addass 10001 ax-mulass 10002 ax-distr 10003 ax-i2m1 10004 ax-1ne0 10005 ax-1rid 10006 ax-rnegex 10007 ax-rrecex 10008 ax-cnre 10009 ax-pre-lttri 10010 ax-pre-lttrn 10011 ax-pre-ltadd 10012 ax-pre-mulgt0 10013 ax-pre-sup 10014 ax-addf 10015 ax-mulf 10016 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-fal 1489 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-nel 2898 df-ral 2917 df-rex 2918 df-reu 2919 df-rmo 2920 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-int 4476 df-iun 4522 df-iin 4523 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-se 5074 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-pred 5680 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-isom 5897 df-riota 6611 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-of 6897 df-om 7066 df-1st 7168 df-2nd 7169 df-supp 7296 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-1o 7560 df-2o 7561 df-oadd 7564 df-er 7742 df-map 7859 df-pm 7860 df-ixp 7909 df-en 7956 df-dom 7957 df-sdom 7958 df-fin 7959 df-fsupp 8276 df-fi 8317 df-sup 8348 df-inf 8349 df-oi 8415 df-card 8765 df-cda 8990 df-pnf 10076 df-mnf 10077 df-xr 10078 df-ltxr 10079 df-le 10080 df-sub 10268 df-neg 10269 df-div 10685 df-nn 11021 df-2 11079 df-3 11080 df-4 11081 df-5 11082 df-6 11083 df-7 11084 df-8 11085 df-9 11086 df-n0 11293 df-z 11378 df-dec 11494 df-uz 11688 df-q 11789 df-rp 11833 df-xneg 11946 df-xadd 11947 df-xmul 11948 df-ioo 12179 df-ioc 12180 df-ico 12181 df-icc 12182 df-fz 12327 df-fzo 12466 df-fl 12593 df-mod 12669 df-seq 12802 df-exp 12861 df-fac 13061 df-bc 13090 df-hash 13118 df-shft 13807 df-cj 13839 df-re 13840 df-im 13841 df-sqrt 13975 df-abs 13976 df-limsup 14202 df-clim 14219 df-rlim 14220 df-sum 14417 df-ef 14798 df-sin 14800 df-cos 14801 df-tan 14802 df-pi 14803 df-struct 15859 df-ndx 15860 df-slot 15861 df-base 15863 df-sets 15864 df-ress 15865 df-plusg 15954 df-mulr 15955 df-starv 15956 df-sca 15957 df-vsca 15958 df-ip 15959 df-tset 15960 df-ple 15961 df-ds 15964 df-unif 15965 df-hom 15966 df-cco 15967 df-rest 16083 df-topn 16084 df-0g 16102 df-gsum 16103 df-topgen 16104 df-pt 16105 df-prds 16108 df-xrs 16162 df-qtop 16167 df-imas 16168 df-xps 16170 df-mre 16246 df-mrc 16247 df-acs 16249 df-mgm 17242 df-sgrp 17284 df-mnd 17295 df-submnd 17336 df-mulg 17541 df-cntz 17750 df-cmn 18195 df-psmet 19738 df-xmet 19739 df-met 19740 df-bl 19741 df-mopn 19742 df-fbas 19743 df-fg 19744 df-cnfld 19747 df-top 20699 df-topon 20716 df-topsp 20737 df-bases 20750 df-cld 20823 df-ntr 20824 df-cls 20825 df-nei 20902 df-lp 20940 df-perf 20941 df-cn 21031 df-cnp 21032 df-haus 21119 df-cmp 21190 df-tx 21365 df-hmeo 21558 df-fil 21650 df-fm 21742 df-flim 21743 df-flf 21744 df-xms 22125 df-ms 22126 df-tms 22127 df-cncf 22681 df-limc 23630 df-dv 23631 df-log 24303 df-cxp 24304 |
This theorem is referenced by: dvasin 33496 |
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