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Mirrors > Home > MPE Home > Th. List > asin1 | Structured version Visualization version GIF version |
Description: The arcsine of 1 is π / 2. (Contributed by Mario Carneiro, 2-Apr-2015.) |
Ref | Expression |
---|---|
asin1 | ⊢ (arcsin‘1) = (π / 2) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ax-1cn 10584 | . . 3 ⊢ 1 ∈ ℂ | |
2 | asinval 25387 | . . 3 ⊢ (1 ∈ ℂ → (arcsin‘1) = (-i · (log‘((i · 1) + (√‘(1 − (1↑2))))))) | |
3 | 1, 2 | ax-mp 5 | . 2 ⊢ (arcsin‘1) = (-i · (log‘((i · 1) + (√‘(1 − (1↑2)))))) |
4 | ax-icn 10585 | . . . . . . 7 ⊢ i ∈ ℂ | |
5 | 4 | addid1i 10816 | . . . . . 6 ⊢ (i + 0) = i |
6 | 4 | mulid1i 10634 | . . . . . . 7 ⊢ (i · 1) = i |
7 | sq1 13548 | . . . . . . . . . . 11 ⊢ (1↑2) = 1 | |
8 | 7 | oveq2i 7156 | . . . . . . . . . 10 ⊢ (1 − (1↑2)) = (1 − 1) |
9 | 1m1e0 11698 | . . . . . . . . . 10 ⊢ (1 − 1) = 0 | |
10 | 8, 9 | eqtri 2844 | . . . . . . . . 9 ⊢ (1 − (1↑2)) = 0 |
11 | 10 | fveq2i 6667 | . . . . . . . 8 ⊢ (√‘(1 − (1↑2))) = (√‘0) |
12 | sqrt0 14591 | . . . . . . . 8 ⊢ (√‘0) = 0 | |
13 | 11, 12 | eqtri 2844 | . . . . . . 7 ⊢ (√‘(1 − (1↑2))) = 0 |
14 | 6, 13 | oveq12i 7157 | . . . . . 6 ⊢ ((i · 1) + (√‘(1 − (1↑2)))) = (i + 0) |
15 | efhalfpi 24986 | . . . . . 6 ⊢ (exp‘(i · (π / 2))) = i | |
16 | 5, 14, 15 | 3eqtr4i 2854 | . . . . 5 ⊢ ((i · 1) + (√‘(1 − (1↑2)))) = (exp‘(i · (π / 2))) |
17 | 16 | fveq2i 6667 | . . . 4 ⊢ (log‘((i · 1) + (√‘(1 − (1↑2))))) = (log‘(exp‘(i · (π / 2)))) |
18 | halfpire 24979 | . . . . . . . 8 ⊢ (π / 2) ∈ ℝ | |
19 | 18 | recni 10644 | . . . . . . 7 ⊢ (π / 2) ∈ ℂ |
20 | 4, 19 | mulcli 10637 | . . . . . 6 ⊢ (i · (π / 2)) ∈ ℂ |
21 | pipos 24975 | . . . . . . . . 9 ⊢ 0 < π | |
22 | pire 24973 | . . . . . . . . . 10 ⊢ π ∈ ℝ | |
23 | lt0neg2 11136 | . . . . . . . . . 10 ⊢ (π ∈ ℝ → (0 < π ↔ -π < 0)) | |
24 | 22, 23 | ax-mp 5 | . . . . . . . . 9 ⊢ (0 < π ↔ -π < 0) |
25 | 21, 24 | mpbi 231 | . . . . . . . 8 ⊢ -π < 0 |
26 | pirp 24976 | . . . . . . . . . 10 ⊢ π ∈ ℝ+ | |
27 | rphalfcl 12406 | . . . . . . . . . 10 ⊢ (π ∈ ℝ+ → (π / 2) ∈ ℝ+) | |
28 | 26, 27 | ax-mp 5 | . . . . . . . . 9 ⊢ (π / 2) ∈ ℝ+ |
29 | rpgt0 12391 | . . . . . . . . 9 ⊢ ((π / 2) ∈ ℝ+ → 0 < (π / 2)) | |
30 | 28, 29 | ax-mp 5 | . . . . . . . 8 ⊢ 0 < (π / 2) |
31 | 22 | renegcli 10936 | . . . . . . . . 9 ⊢ -π ∈ ℝ |
32 | 0re 10632 | . . . . . . . . 9 ⊢ 0 ∈ ℝ | |
33 | 31, 32, 18 | lttri 10755 | . . . . . . . 8 ⊢ ((-π < 0 ∧ 0 < (π / 2)) → -π < (π / 2)) |
34 | 25, 30, 33 | mp2an 688 | . . . . . . 7 ⊢ -π < (π / 2) |
35 | 20 | addid2i 10817 | . . . . . . . . 9 ⊢ (0 + (i · (π / 2))) = (i · (π / 2)) |
36 | 35 | fveq2i 6667 | . . . . . . . 8 ⊢ (ℑ‘(0 + (i · (π / 2)))) = (ℑ‘(i · (π / 2))) |
37 | 32, 18 | crimi 14542 | . . . . . . . 8 ⊢ (ℑ‘(0 + (i · (π / 2)))) = (π / 2) |
38 | 36, 37 | eqtr3i 2846 | . . . . . . 7 ⊢ (ℑ‘(i · (π / 2))) = (π / 2) |
39 | 34, 38 | breqtrri 5085 | . . . . . 6 ⊢ -π < (ℑ‘(i · (π / 2))) |
40 | rphalflt 12408 | . . . . . . . . 9 ⊢ (π ∈ ℝ+ → (π / 2) < π) | |
41 | 26, 40 | ax-mp 5 | . . . . . . . 8 ⊢ (π / 2) < π |
42 | 18, 22, 41 | ltleii 10752 | . . . . . . 7 ⊢ (π / 2) ≤ π |
43 | 38, 42 | eqbrtri 5079 | . . . . . 6 ⊢ (ℑ‘(i · (π / 2))) ≤ π |
44 | ellogrn 25070 | . . . . . 6 ⊢ ((i · (π / 2)) ∈ ran log ↔ ((i · (π / 2)) ∈ ℂ ∧ -π < (ℑ‘(i · (π / 2))) ∧ (ℑ‘(i · (π / 2))) ≤ π)) | |
45 | 20, 39, 43, 44 | mpbir3an 1333 | . . . . 5 ⊢ (i · (π / 2)) ∈ ran log |
46 | logef 25092 | . . . . 5 ⊢ ((i · (π / 2)) ∈ ran log → (log‘(exp‘(i · (π / 2)))) = (i · (π / 2))) | |
47 | 45, 46 | ax-mp 5 | . . . 4 ⊢ (log‘(exp‘(i · (π / 2)))) = (i · (π / 2)) |
48 | 17, 47 | eqtri 2844 | . . 3 ⊢ (log‘((i · 1) + (√‘(1 − (1↑2))))) = (i · (π / 2)) |
49 | 48 | oveq2i 7156 | . 2 ⊢ (-i · (log‘((i · 1) + (√‘(1 − (1↑2)))))) = (-i · (i · (π / 2))) |
50 | 4, 4 | mulneg1i 11075 | . . . . . 6 ⊢ (-i · i) = -(i · i) |
51 | ixi 11258 | . . . . . . 7 ⊢ (i · i) = -1 | |
52 | 51 | negeqi 10868 | . . . . . 6 ⊢ -(i · i) = --1 |
53 | negneg1e1 11744 | . . . . . 6 ⊢ --1 = 1 | |
54 | 50, 52, 53 | 3eqtri 2848 | . . . . 5 ⊢ (-i · i) = 1 |
55 | 54 | oveq1i 7155 | . . . 4 ⊢ ((-i · i) · (π / 2)) = (1 · (π / 2)) |
56 | negicn 10876 | . . . . 5 ⊢ -i ∈ ℂ | |
57 | 56, 4, 19 | mulassi 10641 | . . . 4 ⊢ ((-i · i) · (π / 2)) = (-i · (i · (π / 2))) |
58 | 55, 57 | eqtr3i 2846 | . . 3 ⊢ (1 · (π / 2)) = (-i · (i · (π / 2))) |
59 | 19 | mulid2i 10635 | . . 3 ⊢ (1 · (π / 2)) = (π / 2) |
60 | 58, 59 | eqtr3i 2846 | . 2 ⊢ (-i · (i · (π / 2))) = (π / 2) |
61 | 3, 49, 60 | 3eqtri 2848 | 1 ⊢ (arcsin‘1) = (π / 2) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 207 = wceq 1528 ∈ wcel 2105 class class class wbr 5058 ran crn 5550 ‘cfv 6349 (class class class)co 7145 ℂcc 10524 ℝcr 10525 0cc0 10526 1c1 10527 ici 10528 + caddc 10529 · cmul 10531 < clt 10664 ≤ cle 10665 − cmin 10859 -cneg 10860 / cdiv 11286 2c2 11681 ℝ+crp 12379 ↑cexp 13419 ℑcim 14447 √csqrt 14582 expce 15405 πcpi 15410 logclog 25065 arcsincasin 25367 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2793 ax-rep 5182 ax-sep 5195 ax-nul 5202 ax-pow 5258 ax-pr 5321 ax-un 7450 ax-inf2 9093 ax-cnex 10582 ax-resscn 10583 ax-1cn 10584 ax-icn 10585 ax-addcl 10586 ax-addrcl 10587 ax-mulcl 10588 ax-mulrcl 10589 ax-mulcom 10590 ax-addass 10591 ax-mulass 10592 ax-distr 10593 ax-i2m1 10594 ax-1ne0 10595 ax-1rid 10596 ax-rnegex 10597 ax-rrecex 10598 ax-cnre 10599 ax-pre-lttri 10600 ax-pre-lttrn 10601 ax-pre-ltadd 10602 ax-pre-mulgt0 10603 ax-pre-sup 10604 ax-addf 10605 ax-mulf 10606 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3or 1080 df-3an 1081 df-tru 1531 df-fal 1541 df-ex 1772 df-nf 1776 df-sb 2061 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 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-pss 3953 df-nul 4291 df-if 4466 df-pw 4539 df-sn 4560 df-pr 4562 df-tp 4564 df-op 4566 df-uni 4833 df-int 4870 df-iun 4914 df-iin 4915 df-br 5059 df-opab 5121 df-mpt 5139 df-tr 5165 df-id 5454 df-eprel 5459 df-po 5468 df-so 5469 df-fr 5508 df-se 5509 df-we 5510 df-xp 5555 df-rel 5556 df-cnv 5557 df-co 5558 df-dm 5559 df-rn 5560 df-res 5561 df-ima 5562 df-pred 6142 df-ord 6188 df-on 6189 df-lim 6190 df-suc 6191 df-iota 6308 df-fun 6351 df-fn 6352 df-f 6353 df-f1 6354 df-fo 6355 df-f1o 6356 df-fv 6357 df-isom 6358 df-riota 7103 df-ov 7148 df-oprab 7149 df-mpo 7150 df-of 7398 df-om 7569 df-1st 7680 df-2nd 7681 df-supp 7822 df-wrecs 7938 df-recs 7999 df-rdg 8037 df-1o 8093 df-2o 8094 df-oadd 8097 df-er 8279 df-map 8398 df-pm 8399 df-ixp 8451 df-en 8499 df-dom 8500 df-sdom 8501 df-fin 8502 df-fsupp 8823 df-fi 8864 df-sup 8895 df-inf 8896 df-oi 8963 df-card 9357 df-pnf 10666 df-mnf 10667 df-xr 10668 df-ltxr 10669 df-le 10670 df-sub 10861 df-neg 10862 df-div 11287 df-nn 11628 df-2 11689 df-3 11690 df-4 11691 df-5 11692 df-6 11693 df-7 11694 df-8 11695 df-9 11696 df-n0 11887 df-z 11971 df-dec 12088 df-uz 12233 df-q 12338 df-rp 12380 df-xneg 12497 df-xadd 12498 df-xmul 12499 df-ioo 12732 df-ioc 12733 df-ico 12734 df-icc 12735 df-fz 12883 df-fzo 13024 df-fl 13152 df-mod 13228 df-seq 13360 df-exp 13420 df-fac 13624 df-bc 13653 df-hash 13681 df-shft 14416 df-cj 14448 df-re 14449 df-im 14450 df-sqrt 14584 df-abs 14585 df-limsup 14818 df-clim 14835 df-rlim 14836 df-sum 15033 df-ef 15411 df-sin 15413 df-cos 15414 df-pi 15416 df-struct 16475 df-ndx 16476 df-slot 16477 df-base 16479 df-sets 16480 df-ress 16481 df-plusg 16568 df-mulr 16569 df-starv 16570 df-sca 16571 df-vsca 16572 df-ip 16573 df-tset 16574 df-ple 16575 df-ds 16577 df-unif 16578 df-hom 16579 df-cco 16580 df-rest 16686 df-topn 16687 df-0g 16705 df-gsum 16706 df-topgen 16707 df-pt 16708 df-prds 16711 df-xrs 16765 df-qtop 16770 df-imas 16771 df-xps 16773 df-mre 16847 df-mrc 16848 df-acs 16850 df-mgm 17842 df-sgrp 17891 df-mnd 17902 df-submnd 17947 df-mulg 18165 df-cntz 18387 df-cmn 18839 df-psmet 20467 df-xmet 20468 df-met 20469 df-bl 20470 df-mopn 20471 df-fbas 20472 df-fg 20473 df-cnfld 20476 df-top 21432 df-topon 21449 df-topsp 21471 df-bases 21484 df-cld 21557 df-ntr 21558 df-cls 21559 df-nei 21636 df-lp 21674 df-perf 21675 df-cn 21765 df-cnp 21766 df-haus 21853 df-tx 22100 df-hmeo 22293 df-fil 22384 df-fm 22476 df-flim 22477 df-flf 22478 df-xms 22859 df-ms 22860 df-tms 22861 df-cncf 23415 df-limc 24393 df-dv 24394 df-log 25067 df-asin 25370 |
This theorem is referenced by: acos1 25400 reasinsin 25401 areacirc 34869 |
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