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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 11855 | . . 3 ⊢ (1 / 2) ∈ ℂ | |
2 | dvcncxp1.d | . . . 4 ⊢ 𝐷 = (ℂ ∖ (-∞(,]0)) | |
3 | 2 | dvcncxp1 25327 | . . 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 4111 | . . . . . . 7 ⊢ (ℂ ∖ (-∞(,]0)) ⊆ ℂ | |
6 | 2, 5 | eqsstri 4004 | . . . . . 6 ⊢ 𝐷 ⊆ ℂ |
7 | 6 | sseli 3966 | . . . . 5 ⊢ (𝑥 ∈ 𝐷 → 𝑥 ∈ ℂ) |
8 | cxpsqrt 25289 | . . . . 5 ⊢ (𝑥 ∈ ℂ → (𝑥↑𝑐(1 / 2)) = (√‘𝑥)) | |
9 | 7, 8 | syl 17 | . . . 4 ⊢ (𝑥 ∈ 𝐷 → (𝑥↑𝑐(1 / 2)) = (√‘𝑥)) |
10 | 9 | mpteq2ia 5160 | . . 3 ⊢ (𝑥 ∈ 𝐷 ↦ (𝑥↑𝑐(1 / 2))) = (𝑥 ∈ 𝐷 ↦ (√‘𝑥)) |
11 | 10 | oveq2i 7170 | . 2 ⊢ (ℂ D (𝑥 ∈ 𝐷 ↦ (𝑥↑𝑐(1 / 2)))) = (ℂ D (𝑥 ∈ 𝐷 ↦ (√‘𝑥))) |
12 | 1p0e1 11764 | . . . . . . . . . . 11 ⊢ (1 + 0) = 1 | |
13 | ax-1cn 10598 | . . . . . . . . . . . 12 ⊢ 1 ∈ ℂ | |
14 | 2halves 11868 | . . . . . . . . . . . 12 ⊢ (1 ∈ ℂ → ((1 / 2) + (1 / 2)) = 1) | |
15 | 13, 14 | ax-mp 5 | . . . . . . . . . . 11 ⊢ ((1 / 2) + (1 / 2)) = 1 |
16 | 12, 15 | eqtr4i 2850 | . . . . . . . . . 10 ⊢ (1 + 0) = ((1 / 2) + (1 / 2)) |
17 | 0cn 10636 | . . . . . . . . . . 11 ⊢ 0 ∈ ℂ | |
18 | addsubeq4 10904 | . . . . . . . . . . 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 691 | . . . . . . . . . 10 ⊢ ((1 + 0) = ((1 / 2) + (1 / 2)) ↔ ((1 / 2) − 1) = (0 − (1 / 2))) |
20 | 16, 19 | mpbi 232 | . . . . . . . . 9 ⊢ ((1 / 2) − 1) = (0 − (1 / 2)) |
21 | df-neg 10876 | . . . . . . . . 9 ⊢ -(1 / 2) = (0 − (1 / 2)) | |
22 | 20, 21 | eqtr4i 2850 | . . . . . . . 8 ⊢ ((1 / 2) − 1) = -(1 / 2) |
23 | 22 | oveq2i 7170 | . . . . . . 7 ⊢ (𝑥↑𝑐((1 / 2) − 1)) = (𝑥↑𝑐-(1 / 2)) |
24 | 2 | logdmn0 25226 | . . . . . . . 8 ⊢ (𝑥 ∈ 𝐷 → 𝑥 ≠ 0) |
25 | 1 | a1i 11 | . . . . . . . 8 ⊢ (𝑥 ∈ 𝐷 → (1 / 2) ∈ ℂ) |
26 | 7, 24, 25 | cxpnegd 25301 | . . . . . . 7 ⊢ (𝑥 ∈ 𝐷 → (𝑥↑𝑐-(1 / 2)) = (1 / (𝑥↑𝑐(1 / 2)))) |
27 | 23, 26 | syl5eq 2871 | . . . . . 6 ⊢ (𝑥 ∈ 𝐷 → (𝑥↑𝑐((1 / 2) − 1)) = (1 / (𝑥↑𝑐(1 / 2)))) |
28 | 9 | oveq2d 7175 | . . . . . 6 ⊢ (𝑥 ∈ 𝐷 → (1 / (𝑥↑𝑐(1 / 2))) = (1 / (√‘𝑥))) |
29 | 27, 28 | eqtrd 2859 | . . . . 5 ⊢ (𝑥 ∈ 𝐷 → (𝑥↑𝑐((1 / 2) − 1)) = (1 / (√‘𝑥))) |
30 | 29 | oveq2d 7175 | . . . 4 ⊢ (𝑥 ∈ 𝐷 → ((1 / 2) · (𝑥↑𝑐((1 / 2) − 1))) = ((1 / 2) · (1 / (√‘𝑥)))) |
31 | 1cnd 10639 | . . . . . 6 ⊢ (𝑥 ∈ 𝐷 → 1 ∈ ℂ) | |
32 | 2cnd 11718 | . . . . . 6 ⊢ (𝑥 ∈ 𝐷 → 2 ∈ ℂ) | |
33 | 7 | sqrtcld 14800 | . . . . . 6 ⊢ (𝑥 ∈ 𝐷 → (√‘𝑥) ∈ ℂ) |
34 | 2ne0 11744 | . . . . . . 7 ⊢ 2 ≠ 0 | |
35 | 34 | a1i 11 | . . . . . 6 ⊢ (𝑥 ∈ 𝐷 → 2 ≠ 0) |
36 | 7 | adantr 483 | . . . . . . . . . 10 ⊢ ((𝑥 ∈ 𝐷 ∧ (√‘𝑥) = 0) → 𝑥 ∈ ℂ) |
37 | simpr 487 | . . . . . . . . . 10 ⊢ ((𝑥 ∈ 𝐷 ∧ (√‘𝑥) = 0) → (√‘𝑥) = 0) | |
38 | 36, 37 | sqr00d 14804 | . . . . . . . . 9 ⊢ ((𝑥 ∈ 𝐷 ∧ (√‘𝑥) = 0) → 𝑥 = 0) |
39 | 38 | ex 415 | . . . . . . . 8 ⊢ (𝑥 ∈ 𝐷 → ((√‘𝑥) = 0 → 𝑥 = 0)) |
40 | 39 | necon3d 3040 | . . . . . . 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 11460 | . . . . 5 ⊢ (𝑥 ∈ 𝐷 → ((1 / 2) · (1 / (√‘𝑥))) = ((1 · 1) / (2 · (√‘𝑥)))) |
43 | 1t1e1 11802 | . . . . . 6 ⊢ (1 · 1) = 1 | |
44 | 43 | oveq1i 7169 | . . . . 5 ⊢ ((1 · 1) / (2 · (√‘𝑥))) = (1 / (2 · (√‘𝑥))) |
45 | 42, 44 | syl6eq 2875 | . . . 4 ⊢ (𝑥 ∈ 𝐷 → ((1 / 2) · (1 / (√‘𝑥))) = (1 / (2 · (√‘𝑥)))) |
46 | 30, 45 | eqtrd 2859 | . . 3 ⊢ (𝑥 ∈ 𝐷 → ((1 / 2) · (𝑥↑𝑐((1 / 2) − 1))) = (1 / (2 · (√‘𝑥)))) |
47 | 46 | mpteq2ia 5160 | . 2 ⊢ (𝑥 ∈ 𝐷 ↦ ((1 / 2) · (𝑥↑𝑐((1 / 2) − 1)))) = (𝑥 ∈ 𝐷 ↦ (1 / (2 · (√‘𝑥)))) |
48 | 4, 11, 47 | 3eqtr3i 2855 | 1 ⊢ (ℂ D (𝑥 ∈ 𝐷 ↦ (√‘𝑥))) = (𝑥 ∈ 𝐷 ↦ (1 / (2 · (√‘𝑥)))) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 208 ∧ wa 398 = wceq 1536 ∈ wcel 2113 ≠ wne 3019 ∖ cdif 3936 ↦ cmpt 5149 ‘cfv 6358 (class class class)co 7159 ℂcc 10538 0cc0 10540 1c1 10541 + caddc 10543 · cmul 10545 -∞cmnf 10676 − cmin 10873 -cneg 10874 / cdiv 11300 2c2 11695 (,]cioc 12742 √csqrt 14595 D cdv 24464 ↑𝑐ccxp 25142 |
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 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2796 ax-rep 5193 ax-sep 5206 ax-nul 5213 ax-pow 5269 ax-pr 5333 ax-un 7464 ax-inf2 9107 ax-cnex 10596 ax-resscn 10597 ax-1cn 10598 ax-icn 10599 ax-addcl 10600 ax-addrcl 10601 ax-mulcl 10602 ax-mulrcl 10603 ax-mulcom 10604 ax-addass 10605 ax-mulass 10606 ax-distr 10607 ax-i2m1 10608 ax-1ne0 10609 ax-1rid 10610 ax-rnegex 10611 ax-rrecex 10612 ax-cnre 10613 ax-pre-lttri 10614 ax-pre-lttrn 10615 ax-pre-ltadd 10616 ax-pre-mulgt0 10617 ax-pre-sup 10618 ax-addf 10619 ax-mulf 10620 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1539 df-fal 1549 df-ex 1780 df-nf 1784 df-sb 2069 df-mo 2621 df-eu 2653 df-clab 2803 df-cleq 2817 df-clel 2896 df-nfc 2966 df-ne 3020 df-nel 3127 df-ral 3146 df-rex 3147 df-reu 3148 df-rmo 3149 df-rab 3150 df-v 3499 df-sbc 3776 df-csb 3887 df-dif 3942 df-un 3944 df-in 3946 df-ss 3955 df-pss 3957 df-nul 4295 df-if 4471 df-pw 4544 df-sn 4571 df-pr 4573 df-tp 4575 df-op 4577 df-uni 4842 df-int 4880 df-iun 4924 df-iin 4925 df-br 5070 df-opab 5132 df-mpt 5150 df-tr 5176 df-id 5463 df-eprel 5468 df-po 5477 df-so 5478 df-fr 5517 df-se 5518 df-we 5519 df-xp 5564 df-rel 5565 df-cnv 5566 df-co 5567 df-dm 5568 df-rn 5569 df-res 5570 df-ima 5571 df-pred 6151 df-ord 6197 df-on 6198 df-lim 6199 df-suc 6200 df-iota 6317 df-fun 6360 df-fn 6361 df-f 6362 df-f1 6363 df-fo 6364 df-f1o 6365 df-fv 6366 df-isom 6367 df-riota 7117 df-ov 7162 df-oprab 7163 df-mpo 7164 df-of 7412 df-om 7584 df-1st 7692 df-2nd 7693 df-supp 7834 df-wrecs 7950 df-recs 8011 df-rdg 8049 df-1o 8105 df-2o 8106 df-oadd 8109 df-er 8292 df-map 8411 df-pm 8412 df-ixp 8465 df-en 8513 df-dom 8514 df-sdom 8515 df-fin 8516 df-fsupp 8837 df-fi 8878 df-sup 8909 df-inf 8910 df-oi 8977 df-card 9371 df-pnf 10680 df-mnf 10681 df-xr 10682 df-ltxr 10683 df-le 10684 df-sub 10875 df-neg 10876 df-div 11301 df-nn 11642 df-2 11703 df-3 11704 df-4 11705 df-5 11706 df-6 11707 df-7 11708 df-8 11709 df-9 11710 df-n0 11901 df-z 11985 df-dec 12102 df-uz 12247 df-q 12352 df-rp 12393 df-xneg 12510 df-xadd 12511 df-xmul 12512 df-ioo 12745 df-ioc 12746 df-ico 12747 df-icc 12748 df-fz 12896 df-fzo 13037 df-fl 13165 df-mod 13241 df-seq 13373 df-exp 13433 df-fac 13637 df-bc 13666 df-hash 13694 df-shft 14429 df-cj 14461 df-re 14462 df-im 14463 df-sqrt 14597 df-abs 14598 df-limsup 14831 df-clim 14848 df-rlim 14849 df-sum 15046 df-ef 15424 df-sin 15426 df-cos 15427 df-tan 15428 df-pi 15429 df-struct 16488 df-ndx 16489 df-slot 16490 df-base 16492 df-sets 16493 df-ress 16494 df-plusg 16581 df-mulr 16582 df-starv 16583 df-sca 16584 df-vsca 16585 df-ip 16586 df-tset 16587 df-ple 16588 df-ds 16590 df-unif 16591 df-hom 16592 df-cco 16593 df-rest 16699 df-topn 16700 df-0g 16718 df-gsum 16719 df-topgen 16720 df-pt 16721 df-prds 16724 df-xrs 16778 df-qtop 16783 df-imas 16784 df-xps 16786 df-mre 16860 df-mrc 16861 df-acs 16863 df-mgm 17855 df-sgrp 17904 df-mnd 17915 df-submnd 17960 df-mulg 18228 df-cntz 18450 df-cmn 18911 df-psmet 20540 df-xmet 20541 df-met 20542 df-bl 20543 df-mopn 20544 df-fbas 20545 df-fg 20546 df-cnfld 20549 df-top 21505 df-topon 21522 df-topsp 21544 df-bases 21557 df-cld 21630 df-ntr 21631 df-cls 21632 df-nei 21709 df-lp 21747 df-perf 21748 df-cn 21838 df-cnp 21839 df-haus 21926 df-cmp 21998 df-tx 22173 df-hmeo 22366 df-fil 22457 df-fm 22549 df-flim 22550 df-flf 22551 df-xms 22933 df-ms 22934 df-tms 22935 df-cncf 23489 df-limc 24467 df-dv 24468 df-log 25143 df-cxp 25144 |
This theorem is referenced by: dvasin 34982 |
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