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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 9873 | . . 3 ⊢ 1 ∈ ℂ | |
2 | asinval 24409 | . . 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 9874 | . . . . . . 7 ⊢ i ∈ ℂ | |
5 | 4 | addid1i 10102 | . . . . . 6 ⊢ (i + 0) = i |
6 | 4 | mulid1i 9921 | . . . . . . 7 ⊢ (i · 1) = i |
7 | sq1 12820 | . . . . . . . . . . 11 ⊢ (1↑2) = 1 | |
8 | 7 | oveq2i 6560 | . . . . . . . . . 10 ⊢ (1 − (1↑2)) = (1 − 1) |
9 | 1m1e0 10966 | . . . . . . . . . 10 ⊢ (1 − 1) = 0 | |
10 | 8, 9 | eqtri 2632 | . . . . . . . . 9 ⊢ (1 − (1↑2)) = 0 |
11 | 10 | fveq2i 6106 | . . . . . . . 8 ⊢ (√‘(1 − (1↑2))) = (√‘0) |
12 | sqrt0 13830 | . . . . . . . 8 ⊢ (√‘0) = 0 | |
13 | 11, 12 | eqtri 2632 | . . . . . . 7 ⊢ (√‘(1 − (1↑2))) = 0 |
14 | 6, 13 | oveq12i 6561 | . . . . . 6 ⊢ ((i · 1) + (√‘(1 − (1↑2)))) = (i + 0) |
15 | efhalfpi 24027 | . . . . . 6 ⊢ (exp‘(i · (π / 2))) = i | |
16 | 5, 14, 15 | 3eqtr4i 2642 | . . . . 5 ⊢ ((i · 1) + (√‘(1 − (1↑2)))) = (exp‘(i · (π / 2))) |
17 | 16 | fveq2i 6106 | . . . 4 ⊢ (log‘((i · 1) + (√‘(1 − (1↑2))))) = (log‘(exp‘(i · (π / 2)))) |
18 | halfpire 24020 | . . . . . . . 8 ⊢ (π / 2) ∈ ℝ | |
19 | 18 | recni 9931 | . . . . . . 7 ⊢ (π / 2) ∈ ℂ |
20 | 4, 19 | mulcli 9924 | . . . . . 6 ⊢ (i · (π / 2)) ∈ ℂ |
21 | pipos 24016 | . . . . . . . . 9 ⊢ 0 < π | |
22 | pire 24014 | . . . . . . . . . 10 ⊢ π ∈ ℝ | |
23 | lt0neg2 10414 | . . . . . . . . . 10 ⊢ (π ∈ ℝ → (0 < π ↔ -π < 0)) | |
24 | 22, 23 | ax-mp 5 | . . . . . . . . 9 ⊢ (0 < π ↔ -π < 0) |
25 | 21, 24 | mpbi 219 | . . . . . . . 8 ⊢ -π < 0 |
26 | pirp 24017 | . . . . . . . . . 10 ⊢ π ∈ ℝ+ | |
27 | rphalfcl 11734 | . . . . . . . . . 10 ⊢ (π ∈ ℝ+ → (π / 2) ∈ ℝ+) | |
28 | 26, 27 | ax-mp 5 | . . . . . . . . 9 ⊢ (π / 2) ∈ ℝ+ |
29 | rpgt0 11720 | . . . . . . . . 9 ⊢ ((π / 2) ∈ ℝ+ → 0 < (π / 2)) | |
30 | 28, 29 | ax-mp 5 | . . . . . . . 8 ⊢ 0 < (π / 2) |
31 | 22 | renegcli 10221 | . . . . . . . . 9 ⊢ -π ∈ ℝ |
32 | 0re 9919 | . . . . . . . . 9 ⊢ 0 ∈ ℝ | |
33 | 31, 32, 18 | lttri 10042 | . . . . . . . 8 ⊢ ((-π < 0 ∧ 0 < (π / 2)) → -π < (π / 2)) |
34 | 25, 30, 33 | mp2an 704 | . . . . . . 7 ⊢ -π < (π / 2) |
35 | 20 | addid2i 10103 | . . . . . . . . 9 ⊢ (0 + (i · (π / 2))) = (i · (π / 2)) |
36 | 35 | fveq2i 6106 | . . . . . . . 8 ⊢ (ℑ‘(0 + (i · (π / 2)))) = (ℑ‘(i · (π / 2))) |
37 | 32, 18 | crimi 13781 | . . . . . . . 8 ⊢ (ℑ‘(0 + (i · (π / 2)))) = (π / 2) |
38 | 36, 37 | eqtr3i 2634 | . . . . . . 7 ⊢ (ℑ‘(i · (π / 2))) = (π / 2) |
39 | 34, 38 | breqtrri 4610 | . . . . . 6 ⊢ -π < (ℑ‘(i · (π / 2))) |
40 | rphalflt 11736 | . . . . . . . . 9 ⊢ (π ∈ ℝ+ → (π / 2) < π) | |
41 | 26, 40 | ax-mp 5 | . . . . . . . 8 ⊢ (π / 2) < π |
42 | 18, 22, 41 | ltleii 10039 | . . . . . . 7 ⊢ (π / 2) ≤ π |
43 | 38, 42 | eqbrtri 4604 | . . . . . 6 ⊢ (ℑ‘(i · (π / 2))) ≤ π |
44 | ellogrn 24110 | . . . . . 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 1237 | . . . . 5 ⊢ (i · (π / 2)) ∈ ran log |
46 | logef 24132 | . . . . 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 2632 | . . 3 ⊢ (log‘((i · 1) + (√‘(1 − (1↑2))))) = (i · (π / 2)) |
49 | 48 | oveq2i 6560 | . 2 ⊢ (-i · (log‘((i · 1) + (√‘(1 − (1↑2)))))) = (-i · (i · (π / 2))) |
50 | 4, 4 | mulneg1i 10355 | . . . . . 6 ⊢ (-i · i) = -(i · i) |
51 | ixi 10535 | . . . . . . 7 ⊢ (i · i) = -1 | |
52 | 51 | negeqi 10153 | . . . . . 6 ⊢ -(i · i) = --1 |
53 | negneg1e1 11005 | . . . . . 6 ⊢ --1 = 1 | |
54 | 50, 52, 53 | 3eqtri 2636 | . . . . 5 ⊢ (-i · i) = 1 |
55 | 54 | oveq1i 6559 | . . . 4 ⊢ ((-i · i) · (π / 2)) = (1 · (π / 2)) |
56 | negicn 10161 | . . . . 5 ⊢ -i ∈ ℂ | |
57 | 56, 4, 19 | mulassi 9928 | . . . 4 ⊢ ((-i · i) · (π / 2)) = (-i · (i · (π / 2))) |
58 | 55, 57 | eqtr3i 2634 | . . 3 ⊢ (1 · (π / 2)) = (-i · (i · (π / 2))) |
59 | 19 | mulid2i 9922 | . . 3 ⊢ (1 · (π / 2)) = (π / 2) |
60 | 58, 59 | eqtr3i 2634 | . 2 ⊢ (-i · (i · (π / 2))) = (π / 2) |
61 | 3, 49, 60 | 3eqtri 2636 | 1 ⊢ (arcsin‘1) = (π / 2) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 195 = wceq 1475 ∈ wcel 1977 class class class wbr 4583 ran crn 5039 ‘cfv 5804 (class class class)co 6549 ℂcc 9813 ℝcr 9814 0cc0 9815 1c1 9816 ici 9817 + caddc 9818 · cmul 9820 < clt 9953 ≤ cle 9954 − cmin 10145 -cneg 10146 / cdiv 10563 2c2 10947 ℝ+crp 11708 ↑cexp 12722 ℑcim 13686 √csqrt 13821 expce 14631 πcpi 14636 logclog 24105 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-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-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-asin 24392 |
This theorem is referenced by: acos1 24422 reasinsin 24423 areacirc 32675 |
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