Mathbox for Brendan Leahy |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > dvreasin | Structured version Visualization version GIF version |
Description: Real derivative of arcsine. (Contributed by Brendan Leahy, 3-Aug-2017.) (Proof shortened by Brendan Leahy, 18-Dec-2018.) |
Ref | Expression |
---|---|
dvreasin | ⊢ (ℝ D (arcsin ↾ (-1(,)1))) = (𝑥 ∈ (-1(,)1) ↦ (1 / (√‘(1 − (𝑥↑2))))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | asinf 24399 | . . . . . 6 ⊢ arcsin:ℂ⟶ℂ | |
2 | 1 | a1i 11 | . . . . 5 ⊢ (⊤ → arcsin:ℂ⟶ℂ) |
3 | ioossre 12106 | . . . . . . 7 ⊢ (-1(,)1) ⊆ ℝ | |
4 | ax-resscn 9872 | . . . . . . 7 ⊢ ℝ ⊆ ℂ | |
5 | 3, 4 | sstri 3577 | . . . . . 6 ⊢ (-1(,)1) ⊆ ℂ |
6 | 5 | a1i 11 | . . . . 5 ⊢ (⊤ → (-1(,)1) ⊆ ℂ) |
7 | 2, 6 | feqresmpt 6160 | . . . 4 ⊢ (⊤ → (arcsin ↾ (-1(,)1)) = (𝑥 ∈ (-1(,)1) ↦ (arcsin‘𝑥))) |
8 | 7 | oveq2d 6565 | . . 3 ⊢ (⊤ → (ℝ D (arcsin ↾ (-1(,)1))) = (ℝ D (𝑥 ∈ (-1(,)1) ↦ (arcsin‘𝑥)))) |
9 | eqid 2610 | . . . 4 ⊢ (TopOpen‘ℂfld) = (TopOpen‘ℂfld) | |
10 | reelprrecn 9907 | . . . . 5 ⊢ ℝ ∈ {ℝ, ℂ} | |
11 | 10 | a1i 11 | . . . 4 ⊢ (⊤ → ℝ ∈ {ℝ, ℂ}) |
12 | 9 | recld2 22425 | . . . . . 6 ⊢ ℝ ∈ (Clsd‘(TopOpen‘ℂfld)) |
13 | neg1rr 11002 | . . . . . . . . 9 ⊢ -1 ∈ ℝ | |
14 | iocmnfcld 22382 | . . . . . . . . 9 ⊢ (-1 ∈ ℝ → (-∞(,]-1) ∈ (Clsd‘(topGen‘ran (,)))) | |
15 | 13, 14 | ax-mp 5 | . . . . . . . 8 ⊢ (-∞(,]-1) ∈ (Clsd‘(topGen‘ran (,))) |
16 | 1re 9918 | . . . . . . . . 9 ⊢ 1 ∈ ℝ | |
17 | icopnfcld 22381 | . . . . . . . . 9 ⊢ (1 ∈ ℝ → (1[,)+∞) ∈ (Clsd‘(topGen‘ran (,)))) | |
18 | 16, 17 | ax-mp 5 | . . . . . . . 8 ⊢ (1[,)+∞) ∈ (Clsd‘(topGen‘ran (,))) |
19 | uncld 20655 | . . . . . . . 8 ⊢ (((-∞(,]-1) ∈ (Clsd‘(topGen‘ran (,))) ∧ (1[,)+∞) ∈ (Clsd‘(topGen‘ran (,)))) → ((-∞(,]-1) ∪ (1[,)+∞)) ∈ (Clsd‘(topGen‘ran (,)))) | |
20 | 15, 18, 19 | mp2an 704 | . . . . . . 7 ⊢ ((-∞(,]-1) ∪ (1[,)+∞)) ∈ (Clsd‘(topGen‘ran (,))) |
21 | 9 | tgioo2 22414 | . . . . . . . 8 ⊢ (topGen‘ran (,)) = ((TopOpen‘ℂfld) ↾t ℝ) |
22 | 21 | fveq2i 6106 | . . . . . . 7 ⊢ (Clsd‘(topGen‘ran (,))) = (Clsd‘((TopOpen‘ℂfld) ↾t ℝ)) |
23 | 20, 22 | eleqtri 2686 | . . . . . 6 ⊢ ((-∞(,]-1) ∪ (1[,)+∞)) ∈ (Clsd‘((TopOpen‘ℂfld) ↾t ℝ)) |
24 | restcldr 20788 | . . . . . 6 ⊢ ((ℝ ∈ (Clsd‘(TopOpen‘ℂfld)) ∧ ((-∞(,]-1) ∪ (1[,)+∞)) ∈ (Clsd‘((TopOpen‘ℂfld) ↾t ℝ))) → ((-∞(,]-1) ∪ (1[,)+∞)) ∈ (Clsd‘(TopOpen‘ℂfld))) | |
25 | 12, 23, 24 | mp2an 704 | . . . . 5 ⊢ ((-∞(,]-1) ∪ (1[,)+∞)) ∈ (Clsd‘(TopOpen‘ℂfld)) |
26 | 9 | cnfldtopon 22396 | . . . . . . 7 ⊢ (TopOpen‘ℂfld) ∈ (TopOn‘ℂ) |
27 | 26 | toponunii 20547 | . . . . . 6 ⊢ ℂ = ∪ (TopOpen‘ℂfld) |
28 | 27 | cldopn 20645 | . . . . 5 ⊢ (((-∞(,]-1) ∪ (1[,)+∞)) ∈ (Clsd‘(TopOpen‘ℂfld)) → (ℂ ∖ ((-∞(,]-1) ∪ (1[,)+∞))) ∈ (TopOpen‘ℂfld)) |
29 | 25, 28 | mp1i 13 | . . . 4 ⊢ (⊤ → (ℂ ∖ ((-∞(,]-1) ∪ (1[,)+∞))) ∈ (TopOpen‘ℂfld)) |
30 | incom 3767 | . . . . . 6 ⊢ (ℝ ∩ (ℂ ∖ ((-∞(,]-1) ∪ (1[,)+∞)))) = ((ℂ ∖ ((-∞(,]-1) ∪ (1[,)+∞))) ∩ ℝ) | |
31 | eqid 2610 | . . . . . . 7 ⊢ (ℂ ∖ ((-∞(,]-1) ∪ (1[,)+∞))) = (ℂ ∖ ((-∞(,]-1) ∪ (1[,)+∞))) | |
32 | 31 | asindmre 32665 | . . . . . 6 ⊢ ((ℂ ∖ ((-∞(,]-1) ∪ (1[,)+∞))) ∩ ℝ) = (-1(,)1) |
33 | 30, 32 | eqtri 2632 | . . . . 5 ⊢ (ℝ ∩ (ℂ ∖ ((-∞(,]-1) ∪ (1[,)+∞)))) = (-1(,)1) |
34 | 33 | a1i 11 | . . . 4 ⊢ (⊤ → (ℝ ∩ (ℂ ∖ ((-∞(,]-1) ∪ (1[,)+∞)))) = (-1(,)1)) |
35 | eldifi 3694 | . . . . . 6 ⊢ (𝑥 ∈ (ℂ ∖ ((-∞(,]-1) ∪ (1[,)+∞))) → 𝑥 ∈ ℂ) | |
36 | asincl 24400 | . . . . . 6 ⊢ (𝑥 ∈ ℂ → (arcsin‘𝑥) ∈ ℂ) | |
37 | 35, 36 | syl 17 | . . . . 5 ⊢ (𝑥 ∈ (ℂ ∖ ((-∞(,]-1) ∪ (1[,)+∞))) → (arcsin‘𝑥) ∈ ℂ) |
38 | 37 | adantl 481 | . . . 4 ⊢ ((⊤ ∧ 𝑥 ∈ (ℂ ∖ ((-∞(,]-1) ∪ (1[,)+∞)))) → (arcsin‘𝑥) ∈ ℂ) |
39 | ovex 6577 | . . . . 5 ⊢ (1 / (√‘(1 − (𝑥↑2)))) ∈ V | |
40 | 39 | a1i 11 | . . . 4 ⊢ ((⊤ ∧ 𝑥 ∈ (ℂ ∖ ((-∞(,]-1) ∪ (1[,)+∞)))) → (1 / (√‘(1 − (𝑥↑2)))) ∈ V) |
41 | 31 | dvasin 32666 | . . . . 5 ⊢ (ℂ D (arcsin ↾ (ℂ ∖ ((-∞(,]-1) ∪ (1[,)+∞))))) = (𝑥 ∈ (ℂ ∖ ((-∞(,]-1) ∪ (1[,)+∞))) ↦ (1 / (√‘(1 − (𝑥↑2))))) |
42 | difssd 3700 | . . . . . . 7 ⊢ (⊤ → (ℂ ∖ ((-∞(,]-1) ∪ (1[,)+∞))) ⊆ ℂ) | |
43 | 2, 42 | feqresmpt 6160 | . . . . . 6 ⊢ (⊤ → (arcsin ↾ (ℂ ∖ ((-∞(,]-1) ∪ (1[,)+∞)))) = (𝑥 ∈ (ℂ ∖ ((-∞(,]-1) ∪ (1[,)+∞))) ↦ (arcsin‘𝑥))) |
44 | 43 | oveq2d 6565 | . . . . 5 ⊢ (⊤ → (ℂ D (arcsin ↾ (ℂ ∖ ((-∞(,]-1) ∪ (1[,)+∞))))) = (ℂ D (𝑥 ∈ (ℂ ∖ ((-∞(,]-1) ∪ (1[,)+∞))) ↦ (arcsin‘𝑥)))) |
45 | 41, 44 | syl5reqr 2659 | . . . 4 ⊢ (⊤ → (ℂ D (𝑥 ∈ (ℂ ∖ ((-∞(,]-1) ∪ (1[,)+∞))) ↦ (arcsin‘𝑥))) = (𝑥 ∈ (ℂ ∖ ((-∞(,]-1) ∪ (1[,)+∞))) ↦ (1 / (√‘(1 − (𝑥↑2)))))) |
46 | 9, 11, 29, 34, 38, 40, 45 | dvmptres3 23525 | . . 3 ⊢ (⊤ → (ℝ D (𝑥 ∈ (-1(,)1) ↦ (arcsin‘𝑥))) = (𝑥 ∈ (-1(,)1) ↦ (1 / (√‘(1 − (𝑥↑2)))))) |
47 | 8, 46 | eqtrd 2644 | . 2 ⊢ (⊤ → (ℝ D (arcsin ↾ (-1(,)1))) = (𝑥 ∈ (-1(,)1) ↦ (1 / (√‘(1 − (𝑥↑2)))))) |
48 | 47 | trud 1484 | 1 ⊢ (ℝ D (arcsin ↾ (-1(,)1))) = (𝑥 ∈ (-1(,)1) ↦ (1 / (√‘(1 − (𝑥↑2))))) |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 383 = wceq 1475 ⊤wtru 1476 ∈ wcel 1977 Vcvv 3173 ∖ cdif 3537 ∪ cun 3538 ∩ cin 3539 ⊆ wss 3540 {cpr 4127 ↦ cmpt 4643 ran crn 5039 ↾ cres 5040 ⟶wf 5800 ‘cfv 5804 (class class class)co 6549 ℂcc 9813 ℝcr 9814 1c1 9816 +∞cpnf 9950 -∞cmnf 9951 − cmin 10145 -cneg 10146 / cdiv 10563 2c2 10947 (,)cioo 12046 (,]cioc 12047 [,)cico 12048 ↑cexp 12722 √csqrt 13821 ↾t crest 15904 TopOpenctopn 15905 topGenctg 15921 ℂfldccnfld 19567 Clsdccld 20630 D cdv 23433 arcsincasin 24389 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-inf2 8421 ax-cnex 9871 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-mulcom 9879 ax-addass 9880 ax-mulass 9881 ax-distr 9882 ax-i2m1 9883 ax-1ne0 9884 ax-1rid 9885 ax-rnegex 9886 ax-rrecex 9887 ax-cnre 9888 ax-pre-lttri 9889 ax-pre-lttrn 9890 ax-pre-ltadd 9891 ax-pre-mulgt0 9892 ax-pre-sup 9893 ax-addf 9894 ax-mulf 9895 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-fal 1481 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-reu 2903 df-rmo 2904 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-int 4411 df-iun 4457 df-iin 4458 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-se 4998 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-isom 5813 df-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-of 6795 df-om 6958 df-1st 7059 df-2nd 7060 df-supp 7183 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-1o 7447 df-2o 7448 df-oadd 7451 df-er 7629 df-map 7746 df-pm 7747 df-ixp 7795 df-en 7842 df-dom 7843 df-sdom 7844 df-fin 7845 df-fsupp 8159 df-fi 8200 df-sup 8231 df-inf 8232 df-oi 8298 df-card 8648 df-cda 8873 df-pnf 9955 df-mnf 9956 df-xr 9957 df-ltxr 9958 df-le 9959 df-sub 10147 df-neg 10148 df-div 10564 df-nn 10898 df-2 10956 df-3 10957 df-4 10958 df-5 10959 df-6 10960 df-7 10961 df-8 10962 df-9 10963 df-n0 11170 df-z 11255 df-dec 11370 df-uz 11564 df-q 11665 df-rp 11709 df-xneg 11822 df-xadd 11823 df-xmul 11824 df-ioo 12050 df-ioc 12051 df-ico 12052 df-icc 12053 df-fz 12198 df-fzo 12335 df-fl 12455 df-mod 12531 df-seq 12664 df-exp 12723 df-fac 12923 df-bc 12952 df-hash 12980 df-shft 13655 df-cj 13687 df-re 13688 df-im 13689 df-sqrt 13823 df-abs 13824 df-limsup 14050 df-clim 14067 df-rlim 14068 df-sum 14265 df-ef 14637 df-sin 14639 df-cos 14640 df-tan 14641 df-pi 14642 df-struct 15697 df-ndx 15698 df-slot 15699 df-base 15700 df-sets 15701 df-ress 15702 df-plusg 15781 df-mulr 15782 df-starv 15783 df-sca 15784 df-vsca 15785 df-ip 15786 df-tset 15787 df-ple 15788 df-ds 15791 df-unif 15792 df-hom 15793 df-cco 15794 df-rest 15906 df-topn 15907 df-0g 15925 df-gsum 15926 df-topgen 15927 df-pt 15928 df-prds 15931 df-xrs 15985 df-qtop 15990 df-imas 15991 df-xps 15993 df-mre 16069 df-mrc 16070 df-acs 16072 df-mgm 17065 df-sgrp 17107 df-mnd 17118 df-submnd 17159 df-mulg 17364 df-cntz 17573 df-cmn 18018 df-psmet 19559 df-xmet 19560 df-met 19561 df-bl 19562 df-mopn 19563 df-fbas 19564 df-fg 19565 df-cnfld 19568 df-top 20521 df-bases 20522 df-topon 20523 df-topsp 20524 df-cld 20633 df-ntr 20634 df-cls 20635 df-nei 20712 df-lp 20750 df-perf 20751 df-cn 20841 df-cnp 20842 df-haus 20929 df-cmp 21000 df-tx 21175 df-hmeo 21368 df-fil 21460 df-fm 21552 df-flim 21553 df-flf 21554 df-xms 21935 df-ms 21936 df-tms 21937 df-cncf 22489 df-limc 23436 df-dv 23437 df-log 24107 df-cxp 24108 df-asin 24392 |
This theorem is referenced by: areacirclem1 32670 |
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