![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > dvsqrt | Structured version Visualization version GIF version |
Description: The derivative of the real square root function. (Contributed by Mario Carneiro, 1-May-2016.) |
Ref | Expression |
---|---|
dvsqrt | ⊢ (ℝ D (𝑥 ∈ ℝ+ ↦ (√‘𝑥))) = (𝑥 ∈ ℝ+ ↦ (1 / (2 · (√‘𝑥)))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | halfcn 11459 | . . 3 ⊢ (1 / 2) ∈ ℂ | |
2 | dvcxp1 24701 | . . 3 ⊢ ((1 / 2) ∈ ℂ → (ℝ D (𝑥 ∈ ℝ+ ↦ (𝑥↑𝑐(1 / 2)))) = (𝑥 ∈ ℝ+ ↦ ((1 / 2) · (𝑥↑𝑐((1 / 2) − 1))))) | |
3 | 1, 2 | ax-mp 5 | . 2 ⊢ (ℝ D (𝑥 ∈ ℝ+ ↦ (𝑥↑𝑐(1 / 2)))) = (𝑥 ∈ ℝ+ ↦ ((1 / 2) · (𝑥↑𝑐((1 / 2) − 1)))) |
4 | rpcn 12054 | . . . . 5 ⊢ (𝑥 ∈ ℝ+ → 𝑥 ∈ ℂ) | |
5 | cxpsqrt 24669 | . . . . 5 ⊢ (𝑥 ∈ ℂ → (𝑥↑𝑐(1 / 2)) = (√‘𝑥)) | |
6 | 4, 5 | syl 17 | . . . 4 ⊢ (𝑥 ∈ ℝ+ → (𝑥↑𝑐(1 / 2)) = (√‘𝑥)) |
7 | 6 | mpteq2ia 4892 | . . 3 ⊢ (𝑥 ∈ ℝ+ ↦ (𝑥↑𝑐(1 / 2))) = (𝑥 ∈ ℝ+ ↦ (√‘𝑥)) |
8 | 7 | oveq2i 6825 | . 2 ⊢ (ℝ D (𝑥 ∈ ℝ+ ↦ (𝑥↑𝑐(1 / 2)))) = (ℝ D (𝑥 ∈ ℝ+ ↦ (√‘𝑥))) |
9 | 1p0e1 11345 | . . . . . . . . . . 11 ⊢ (1 + 0) = 1 | |
10 | ax-1cn 10206 | . . . . . . . . . . . 12 ⊢ 1 ∈ ℂ | |
11 | 2halves 11472 | . . . . . . . . . . . 12 ⊢ (1 ∈ ℂ → ((1 / 2) + (1 / 2)) = 1) | |
12 | 10, 11 | ax-mp 5 | . . . . . . . . . . 11 ⊢ ((1 / 2) + (1 / 2)) = 1 |
13 | 9, 12 | eqtr4i 2785 | . . . . . . . . . 10 ⊢ (1 + 0) = ((1 / 2) + (1 / 2)) |
14 | 0cn 10244 | . . . . . . . . . . 11 ⊢ 0 ∈ ℂ | |
15 | addsubeq4 10508 | . . . . . . . . . . 11 ⊢ (((1 ∈ ℂ ∧ 0 ∈ ℂ) ∧ ((1 / 2) ∈ ℂ ∧ (1 / 2) ∈ ℂ)) → ((1 + 0) = ((1 / 2) + (1 / 2)) ↔ ((1 / 2) − 1) = (0 − (1 / 2)))) | |
16 | 10, 14, 1, 1, 15 | mp4an 711 | . . . . . . . . . 10 ⊢ ((1 + 0) = ((1 / 2) + (1 / 2)) ↔ ((1 / 2) − 1) = (0 − (1 / 2))) |
17 | 13, 16 | mpbi 220 | . . . . . . . . 9 ⊢ ((1 / 2) − 1) = (0 − (1 / 2)) |
18 | df-neg 10481 | . . . . . . . . 9 ⊢ -(1 / 2) = (0 − (1 / 2)) | |
19 | 17, 18 | eqtr4i 2785 | . . . . . . . 8 ⊢ ((1 / 2) − 1) = -(1 / 2) |
20 | 19 | oveq2i 6825 | . . . . . . 7 ⊢ (𝑥↑𝑐((1 / 2) − 1)) = (𝑥↑𝑐-(1 / 2)) |
21 | rpne0 12061 | . . . . . . . 8 ⊢ (𝑥 ∈ ℝ+ → 𝑥 ≠ 0) | |
22 | 1 | a1i 11 | . . . . . . . 8 ⊢ (𝑥 ∈ ℝ+ → (1 / 2) ∈ ℂ) |
23 | 4, 21, 22 | cxpnegd 24681 | . . . . . . 7 ⊢ (𝑥 ∈ ℝ+ → (𝑥↑𝑐-(1 / 2)) = (1 / (𝑥↑𝑐(1 / 2)))) |
24 | 20, 23 | syl5eq 2806 | . . . . . 6 ⊢ (𝑥 ∈ ℝ+ → (𝑥↑𝑐((1 / 2) − 1)) = (1 / (𝑥↑𝑐(1 / 2)))) |
25 | 6 | oveq2d 6830 | . . . . . 6 ⊢ (𝑥 ∈ ℝ+ → (1 / (𝑥↑𝑐(1 / 2))) = (1 / (√‘𝑥))) |
26 | 24, 25 | eqtrd 2794 | . . . . 5 ⊢ (𝑥 ∈ ℝ+ → (𝑥↑𝑐((1 / 2) − 1)) = (1 / (√‘𝑥))) |
27 | 26 | oveq2d 6830 | . . . 4 ⊢ (𝑥 ∈ ℝ+ → ((1 / 2) · (𝑥↑𝑐((1 / 2) − 1))) = ((1 / 2) · (1 / (√‘𝑥)))) |
28 | 10 | a1i 11 | . . . . . 6 ⊢ (𝑥 ∈ ℝ+ → 1 ∈ ℂ) |
29 | 2cnne0 11454 | . . . . . . 7 ⊢ (2 ∈ ℂ ∧ 2 ≠ 0) | |
30 | 29 | a1i 11 | . . . . . 6 ⊢ (𝑥 ∈ ℝ+ → (2 ∈ ℂ ∧ 2 ≠ 0)) |
31 | rpsqrtcl 14224 | . . . . . . 7 ⊢ (𝑥 ∈ ℝ+ → (√‘𝑥) ∈ ℝ+) | |
32 | 31 | rpcnne0d 12094 | . . . . . 6 ⊢ (𝑥 ∈ ℝ+ → ((√‘𝑥) ∈ ℂ ∧ (√‘𝑥) ≠ 0)) |
33 | divmuldiv 10937 | . . . . . 6 ⊢ (((1 ∈ ℂ ∧ 1 ∈ ℂ) ∧ ((2 ∈ ℂ ∧ 2 ≠ 0) ∧ ((√‘𝑥) ∈ ℂ ∧ (√‘𝑥) ≠ 0))) → ((1 / 2) · (1 / (√‘𝑥))) = ((1 · 1) / (2 · (√‘𝑥)))) | |
34 | 28, 28, 30, 32, 33 | syl22anc 1478 | . . . . 5 ⊢ (𝑥 ∈ ℝ+ → ((1 / 2) · (1 / (√‘𝑥))) = ((1 · 1) / (2 · (√‘𝑥)))) |
35 | 1t1e1 11387 | . . . . . 6 ⊢ (1 · 1) = 1 | |
36 | 35 | oveq1i 6824 | . . . . 5 ⊢ ((1 · 1) / (2 · (√‘𝑥))) = (1 / (2 · (√‘𝑥))) |
37 | 34, 36 | syl6eq 2810 | . . . 4 ⊢ (𝑥 ∈ ℝ+ → ((1 / 2) · (1 / (√‘𝑥))) = (1 / (2 · (√‘𝑥)))) |
38 | 27, 37 | eqtrd 2794 | . . 3 ⊢ (𝑥 ∈ ℝ+ → ((1 / 2) · (𝑥↑𝑐((1 / 2) − 1))) = (1 / (2 · (√‘𝑥)))) |
39 | 38 | mpteq2ia 4892 | . 2 ⊢ (𝑥 ∈ ℝ+ ↦ ((1 / 2) · (𝑥↑𝑐((1 / 2) − 1)))) = (𝑥 ∈ ℝ+ ↦ (1 / (2 · (√‘𝑥)))) |
40 | 3, 8, 39 | 3eqtr3i 2790 | 1 ⊢ (ℝ D (𝑥 ∈ ℝ+ ↦ (√‘𝑥))) = (𝑥 ∈ ℝ+ ↦ (1 / (2 · (√‘𝑥)))) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 196 ∧ wa 383 = wceq 1632 ∈ wcel 2139 ≠ wne 2932 ↦ cmpt 4881 ‘cfv 6049 (class class class)co 6814 ℂcc 10146 ℝcr 10147 0cc0 10148 1c1 10149 + caddc 10151 · cmul 10153 − cmin 10478 -cneg 10479 / cdiv 10896 2c2 11282 ℝ+crp 12045 √csqrt 14192 D cdv 23846 ↑𝑐ccxp 24522 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1871 ax-4 1886 ax-5 1988 ax-6 2054 ax-7 2090 ax-8 2141 ax-9 2148 ax-10 2168 ax-11 2183 ax-12 2196 ax-13 2391 ax-ext 2740 ax-rep 4923 ax-sep 4933 ax-nul 4941 ax-pow 4992 ax-pr 5055 ax-un 7115 ax-inf2 8713 ax-cnex 10204 ax-resscn 10205 ax-1cn 10206 ax-icn 10207 ax-addcl 10208 ax-addrcl 10209 ax-mulcl 10210 ax-mulrcl 10211 ax-mulcom 10212 ax-addass 10213 ax-mulass 10214 ax-distr 10215 ax-i2m1 10216 ax-1ne0 10217 ax-1rid 10218 ax-rnegex 10219 ax-rrecex 10220 ax-cnre 10221 ax-pre-lttri 10222 ax-pre-lttrn 10223 ax-pre-ltadd 10224 ax-pre-mulgt0 10225 ax-pre-sup 10226 ax-addf 10227 ax-mulf 10228 |
This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-3or 1073 df-3an 1074 df-tru 1635 df-fal 1638 df-ex 1854 df-nf 1859 df-sb 2047 df-eu 2611 df-mo 2612 df-clab 2747 df-cleq 2753 df-clel 2756 df-nfc 2891 df-ne 2933 df-nel 3036 df-ral 3055 df-rex 3056 df-reu 3057 df-rmo 3058 df-rab 3059 df-v 3342 df-sbc 3577 df-csb 3675 df-dif 3718 df-un 3720 df-in 3722 df-ss 3729 df-pss 3731 df-nul 4059 df-if 4231 df-pw 4304 df-sn 4322 df-pr 4324 df-tp 4326 df-op 4328 df-uni 4589 df-int 4628 df-iun 4674 df-iin 4675 df-br 4805 df-opab 4865 df-mpt 4882 df-tr 4905 df-id 5174 df-eprel 5179 df-po 5187 df-so 5188 df-fr 5225 df-se 5226 df-we 5227 df-xp 5272 df-rel 5273 df-cnv 5274 df-co 5275 df-dm 5276 df-rn 5277 df-res 5278 df-ima 5279 df-pred 5841 df-ord 5887 df-on 5888 df-lim 5889 df-suc 5890 df-iota 6012 df-fun 6051 df-fn 6052 df-f 6053 df-f1 6054 df-fo 6055 df-f1o 6056 df-fv 6057 df-isom 6058 df-riota 6775 df-ov 6817 df-oprab 6818 df-mpt2 6819 df-of 7063 df-om 7232 df-1st 7334 df-2nd 7335 df-supp 7465 df-wrecs 7577 df-recs 7638 df-rdg 7676 df-1o 7730 df-2o 7731 df-oadd 7734 df-er 7913 df-map 8027 df-pm 8028 df-ixp 8077 df-en 8124 df-dom 8125 df-sdom 8126 df-fin 8127 df-fsupp 8443 df-fi 8484 df-sup 8515 df-inf 8516 df-oi 8582 df-card 8975 df-cda 9202 df-pnf 10288 df-mnf 10289 df-xr 10290 df-ltxr 10291 df-le 10292 df-sub 10480 df-neg 10481 df-div 10897 df-nn 11233 df-2 11291 df-3 11292 df-4 11293 df-5 11294 df-6 11295 df-7 11296 df-8 11297 df-9 11298 df-n0 11505 df-z 11590 df-dec 11706 df-uz 11900 df-q 12002 df-rp 12046 df-xneg 12159 df-xadd 12160 df-xmul 12161 df-ioo 12392 df-ioc 12393 df-ico 12394 df-icc 12395 df-fz 12540 df-fzo 12680 df-fl 12807 df-mod 12883 df-seq 13016 df-exp 13075 df-fac 13275 df-bc 13304 df-hash 13332 df-shft 14026 df-cj 14058 df-re 14059 df-im 14060 df-sqrt 14194 df-abs 14195 df-limsup 14421 df-clim 14438 df-rlim 14439 df-sum 14636 df-ef 15017 df-sin 15019 df-cos 15020 df-pi 15022 df-struct 16081 df-ndx 16082 df-slot 16083 df-base 16085 df-sets 16086 df-ress 16087 df-plusg 16176 df-mulr 16177 df-starv 16178 df-sca 16179 df-vsca 16180 df-ip 16181 df-tset 16182 df-ple 16183 df-ds 16186 df-unif 16187 df-hom 16188 df-cco 16189 df-rest 16305 df-topn 16306 df-0g 16324 df-gsum 16325 df-topgen 16326 df-pt 16327 df-prds 16330 df-xrs 16384 df-qtop 16389 df-imas 16390 df-xps 16392 df-mre 16468 df-mrc 16469 df-acs 16471 df-mgm 17463 df-sgrp 17505 df-mnd 17516 df-submnd 17557 df-mulg 17762 df-cntz 17970 df-cmn 18415 df-psmet 19960 df-xmet 19961 df-met 19962 df-bl 19963 df-mopn 19964 df-fbas 19965 df-fg 19966 df-cnfld 19969 df-top 20921 df-topon 20938 df-topsp 20959 df-bases 20972 df-cld 21045 df-ntr 21046 df-cls 21047 df-nei 21124 df-lp 21162 df-perf 21163 df-cn 21253 df-cnp 21254 df-haus 21341 df-cmp 21412 df-tx 21587 df-hmeo 21780 df-fil 21871 df-fm 21963 df-flim 21964 df-flf 21965 df-xms 22346 df-ms 22347 df-tms 22348 df-cncf 22902 df-limc 23849 df-dv 23850 df-log 24523 df-cxp 24524 |
This theorem is referenced by: loglesqrt 24719 divsqrtsumlem 24926 areacirclem1 33831 |
Copyright terms: Public domain | W3C validator |