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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 10929 | . . 3 ⊢ 1 ∈ ℂ | |
2 | asinval 26032 | . . 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 10930 | . . . . . . 7 ⊢ i ∈ ℂ | |
5 | 4 | addid1i 11162 | . . . . . 6 ⊢ (i + 0) = i |
6 | 4 | mulid1i 10979 | . . . . . . 7 ⊢ (i · 1) = i |
7 | sq1 13912 | . . . . . . . . . . 11 ⊢ (1↑2) = 1 | |
8 | 7 | oveq2i 7286 | . . . . . . . . . 10 ⊢ (1 − (1↑2)) = (1 − 1) |
9 | 1m1e0 12045 | . . . . . . . . . 10 ⊢ (1 − 1) = 0 | |
10 | 8, 9 | eqtri 2766 | . . . . . . . . 9 ⊢ (1 − (1↑2)) = 0 |
11 | 10 | fveq2i 6777 | . . . . . . . 8 ⊢ (√‘(1 − (1↑2))) = (√‘0) |
12 | sqrt0 14953 | . . . . . . . 8 ⊢ (√‘0) = 0 | |
13 | 11, 12 | eqtri 2766 | . . . . . . 7 ⊢ (√‘(1 − (1↑2))) = 0 |
14 | 6, 13 | oveq12i 7287 | . . . . . 6 ⊢ ((i · 1) + (√‘(1 − (1↑2)))) = (i + 0) |
15 | efhalfpi 25628 | . . . . . 6 ⊢ (exp‘(i · (π / 2))) = i | |
16 | 5, 14, 15 | 3eqtr4i 2776 | . . . . 5 ⊢ ((i · 1) + (√‘(1 − (1↑2)))) = (exp‘(i · (π / 2))) |
17 | 16 | fveq2i 6777 | . . . 4 ⊢ (log‘((i · 1) + (√‘(1 − (1↑2))))) = (log‘(exp‘(i · (π / 2)))) |
18 | halfpire 25621 | . . . . . . . 8 ⊢ (π / 2) ∈ ℝ | |
19 | 18 | recni 10989 | . . . . . . 7 ⊢ (π / 2) ∈ ℂ |
20 | 4, 19 | mulcli 10982 | . . . . . 6 ⊢ (i · (π / 2)) ∈ ℂ |
21 | pipos 25617 | . . . . . . . . 9 ⊢ 0 < π | |
22 | pire 25615 | . . . . . . . . . 10 ⊢ π ∈ ℝ | |
23 | lt0neg2 11482 | . . . . . . . . . 10 ⊢ (π ∈ ℝ → (0 < π ↔ -π < 0)) | |
24 | 22, 23 | ax-mp 5 | . . . . . . . . 9 ⊢ (0 < π ↔ -π < 0) |
25 | 21, 24 | mpbi 229 | . . . . . . . 8 ⊢ -π < 0 |
26 | pirp 25618 | . . . . . . . . . 10 ⊢ π ∈ ℝ+ | |
27 | rphalfcl 12757 | . . . . . . . . . 10 ⊢ (π ∈ ℝ+ → (π / 2) ∈ ℝ+) | |
28 | 26, 27 | ax-mp 5 | . . . . . . . . 9 ⊢ (π / 2) ∈ ℝ+ |
29 | rpgt0 12742 | . . . . . . . . 9 ⊢ ((π / 2) ∈ ℝ+ → 0 < (π / 2)) | |
30 | 28, 29 | ax-mp 5 | . . . . . . . 8 ⊢ 0 < (π / 2) |
31 | 22 | renegcli 11282 | . . . . . . . . 9 ⊢ -π ∈ ℝ |
32 | 0re 10977 | . . . . . . . . 9 ⊢ 0 ∈ ℝ | |
33 | 31, 32, 18 | lttri 11101 | . . . . . . . 8 ⊢ ((-π < 0 ∧ 0 < (π / 2)) → -π < (π / 2)) |
34 | 25, 30, 33 | mp2an 689 | . . . . . . 7 ⊢ -π < (π / 2) |
35 | 20 | addid2i 11163 | . . . . . . . . 9 ⊢ (0 + (i · (π / 2))) = (i · (π / 2)) |
36 | 35 | fveq2i 6777 | . . . . . . . 8 ⊢ (ℑ‘(0 + (i · (π / 2)))) = (ℑ‘(i · (π / 2))) |
37 | 32, 18 | crimi 14904 | . . . . . . . 8 ⊢ (ℑ‘(0 + (i · (π / 2)))) = (π / 2) |
38 | 36, 37 | eqtr3i 2768 | . . . . . . 7 ⊢ (ℑ‘(i · (π / 2))) = (π / 2) |
39 | 34, 38 | breqtrri 5101 | . . . . . 6 ⊢ -π < (ℑ‘(i · (π / 2))) |
40 | rphalflt 12759 | . . . . . . . . 9 ⊢ (π ∈ ℝ+ → (π / 2) < π) | |
41 | 26, 40 | ax-mp 5 | . . . . . . . 8 ⊢ (π / 2) < π |
42 | 18, 22, 41 | ltleii 11098 | . . . . . . 7 ⊢ (π / 2) ≤ π |
43 | 38, 42 | eqbrtri 5095 | . . . . . 6 ⊢ (ℑ‘(i · (π / 2))) ≤ π |
44 | ellogrn 25715 | . . . . . 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 1340 | . . . . 5 ⊢ (i · (π / 2)) ∈ ran log |
46 | logef 25737 | . . . . 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 2766 | . . 3 ⊢ (log‘((i · 1) + (√‘(1 − (1↑2))))) = (i · (π / 2)) |
49 | 48 | oveq2i 7286 | . 2 ⊢ (-i · (log‘((i · 1) + (√‘(1 − (1↑2)))))) = (-i · (i · (π / 2))) |
50 | 4, 4 | mulneg1i 11421 | . . . . . 6 ⊢ (-i · i) = -(i · i) |
51 | ixi 11604 | . . . . . . 7 ⊢ (i · i) = -1 | |
52 | 51 | negeqi 11214 | . . . . . 6 ⊢ -(i · i) = --1 |
53 | negneg1e1 12091 | . . . . . 6 ⊢ --1 = 1 | |
54 | 50, 52, 53 | 3eqtri 2770 | . . . . 5 ⊢ (-i · i) = 1 |
55 | 54 | oveq1i 7285 | . . . 4 ⊢ ((-i · i) · (π / 2)) = (1 · (π / 2)) |
56 | negicn 11222 | . . . . 5 ⊢ -i ∈ ℂ | |
57 | 56, 4, 19 | mulassi 10986 | . . . 4 ⊢ ((-i · i) · (π / 2)) = (-i · (i · (π / 2))) |
58 | 55, 57 | eqtr3i 2768 | . . 3 ⊢ (1 · (π / 2)) = (-i · (i · (π / 2))) |
59 | 19 | mulid2i 10980 | . . 3 ⊢ (1 · (π / 2)) = (π / 2) |
60 | 58, 59 | eqtr3i 2768 | . 2 ⊢ (-i · (i · (π / 2))) = (π / 2) |
61 | 3, 49, 60 | 3eqtri 2770 | 1 ⊢ (arcsin‘1) = (π / 2) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 205 = wceq 1539 ∈ wcel 2106 class class class wbr 5074 ran crn 5590 ‘cfv 6433 (class class class)co 7275 ℂcc 10869 ℝcr 10870 0cc0 10871 1c1 10872 ici 10873 + caddc 10874 · cmul 10876 < clt 11009 ≤ cle 11010 − cmin 11205 -cneg 11206 / cdiv 11632 2c2 12028 ℝ+crp 12730 ↑cexp 13782 ℑcim 14809 √csqrt 14944 expce 15771 πcpi 15776 logclog 25710 arcsincasin 26012 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2709 ax-rep 5209 ax-sep 5223 ax-nul 5230 ax-pow 5288 ax-pr 5352 ax-un 7588 ax-inf2 9399 ax-cnex 10927 ax-resscn 10928 ax-1cn 10929 ax-icn 10930 ax-addcl 10931 ax-addrcl 10932 ax-mulcl 10933 ax-mulrcl 10934 ax-mulcom 10935 ax-addass 10936 ax-mulass 10937 ax-distr 10938 ax-i2m1 10939 ax-1ne0 10940 ax-1rid 10941 ax-rnegex 10942 ax-rrecex 10943 ax-cnre 10944 ax-pre-lttri 10945 ax-pre-lttrn 10946 ax-pre-ltadd 10947 ax-pre-mulgt0 10948 ax-pre-sup 10949 ax-addf 10950 ax-mulf 10951 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2068 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2889 df-ne 2944 df-nel 3050 df-ral 3069 df-rex 3070 df-rmo 3071 df-reu 3072 df-rab 3073 df-v 3434 df-sbc 3717 df-csb 3833 df-dif 3890 df-un 3892 df-in 3894 df-ss 3904 df-pss 3906 df-nul 4257 df-if 4460 df-pw 4535 df-sn 4562 df-pr 4564 df-tp 4566 df-op 4568 df-uni 4840 df-int 4880 df-iun 4926 df-iin 4927 df-br 5075 df-opab 5137 df-mpt 5158 df-tr 5192 df-id 5489 df-eprel 5495 df-po 5503 df-so 5504 df-fr 5544 df-se 5545 df-we 5546 df-xp 5595 df-rel 5596 df-cnv 5597 df-co 5598 df-dm 5599 df-rn 5600 df-res 5601 df-ima 5602 df-pred 6202 df-ord 6269 df-on 6270 df-lim 6271 df-suc 6272 df-iota 6391 df-fun 6435 df-fn 6436 df-f 6437 df-f1 6438 df-fo 6439 df-f1o 6440 df-fv 6441 df-isom 6442 df-riota 7232 df-ov 7278 df-oprab 7279 df-mpo 7280 df-of 7533 df-om 7713 df-1st 7831 df-2nd 7832 df-supp 7978 df-frecs 8097 df-wrecs 8128 df-recs 8202 df-rdg 8241 df-1o 8297 df-2o 8298 df-er 8498 df-map 8617 df-pm 8618 df-ixp 8686 df-en 8734 df-dom 8735 df-sdom 8736 df-fin 8737 df-fsupp 9129 df-fi 9170 df-sup 9201 df-inf 9202 df-oi 9269 df-card 9697 df-pnf 11011 df-mnf 11012 df-xr 11013 df-ltxr 11014 df-le 11015 df-sub 11207 df-neg 11208 df-div 11633 df-nn 11974 df-2 12036 df-3 12037 df-4 12038 df-5 12039 df-6 12040 df-7 12041 df-8 12042 df-9 12043 df-n0 12234 df-z 12320 df-dec 12438 df-uz 12583 df-q 12689 df-rp 12731 df-xneg 12848 df-xadd 12849 df-xmul 12850 df-ioo 13083 df-ioc 13084 df-ico 13085 df-icc 13086 df-fz 13240 df-fzo 13383 df-fl 13512 df-mod 13590 df-seq 13722 df-exp 13783 df-fac 13988 df-bc 14017 df-hash 14045 df-shft 14778 df-cj 14810 df-re 14811 df-im 14812 df-sqrt 14946 df-abs 14947 df-limsup 15180 df-clim 15197 df-rlim 15198 df-sum 15398 df-ef 15777 df-sin 15779 df-cos 15780 df-pi 15782 df-struct 16848 df-sets 16865 df-slot 16883 df-ndx 16895 df-base 16913 df-ress 16942 df-plusg 16975 df-mulr 16976 df-starv 16977 df-sca 16978 df-vsca 16979 df-ip 16980 df-tset 16981 df-ple 16982 df-ds 16984 df-unif 16985 df-hom 16986 df-cco 16987 df-rest 17133 df-topn 17134 df-0g 17152 df-gsum 17153 df-topgen 17154 df-pt 17155 df-prds 17158 df-xrs 17213 df-qtop 17218 df-imas 17219 df-xps 17221 df-mre 17295 df-mrc 17296 df-acs 17298 df-mgm 18326 df-sgrp 18375 df-mnd 18386 df-submnd 18431 df-mulg 18701 df-cntz 18923 df-cmn 19388 df-psmet 20589 df-xmet 20590 df-met 20591 df-bl 20592 df-mopn 20593 df-fbas 20594 df-fg 20595 df-cnfld 20598 df-top 22043 df-topon 22060 df-topsp 22082 df-bases 22096 df-cld 22170 df-ntr 22171 df-cls 22172 df-nei 22249 df-lp 22287 df-perf 22288 df-cn 22378 df-cnp 22379 df-haus 22466 df-tx 22713 df-hmeo 22906 df-fil 22997 df-fm 23089 df-flim 23090 df-flf 23091 df-xms 23473 df-ms 23474 df-tms 23475 df-cncf 24041 df-limc 25030 df-dv 25031 df-log 25712 df-asin 26015 |
This theorem is referenced by: acos1 26045 reasinsin 26046 areacirc 35870 |
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