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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 11210 | . . 3 ⊢ 1 ∈ ℂ | |
2 | asinval 26939 | . . 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 11211 | . . . . . . 7 ⊢ i ∈ ℂ | |
5 | 4 | addridi 11445 | . . . . . 6 ⊢ (i + 0) = i |
6 | 4 | mulridi 11262 | . . . . . . 7 ⊢ (i · 1) = i |
7 | sq1 14230 | . . . . . . . . . . 11 ⊢ (1↑2) = 1 | |
8 | 7 | oveq2i 7441 | . . . . . . . . . 10 ⊢ (1 − (1↑2)) = (1 − 1) |
9 | 1m1e0 12335 | . . . . . . . . . 10 ⊢ (1 − 1) = 0 | |
10 | 8, 9 | eqtri 2762 | . . . . . . . . 9 ⊢ (1 − (1↑2)) = 0 |
11 | 10 | fveq2i 6909 | . . . . . . . 8 ⊢ (√‘(1 − (1↑2))) = (√‘0) |
12 | sqrt0 15276 | . . . . . . . 8 ⊢ (√‘0) = 0 | |
13 | 11, 12 | eqtri 2762 | . . . . . . 7 ⊢ (√‘(1 − (1↑2))) = 0 |
14 | 6, 13 | oveq12i 7442 | . . . . . 6 ⊢ ((i · 1) + (√‘(1 − (1↑2)))) = (i + 0) |
15 | efhalfpi 26527 | . . . . . 6 ⊢ (exp‘(i · (π / 2))) = i | |
16 | 5, 14, 15 | 3eqtr4i 2772 | . . . . 5 ⊢ ((i · 1) + (√‘(1 − (1↑2)))) = (exp‘(i · (π / 2))) |
17 | 16 | fveq2i 6909 | . . . 4 ⊢ (log‘((i · 1) + (√‘(1 − (1↑2))))) = (log‘(exp‘(i · (π / 2)))) |
18 | halfpire 26520 | . . . . . . . 8 ⊢ (π / 2) ∈ ℝ | |
19 | 18 | recni 11272 | . . . . . . 7 ⊢ (π / 2) ∈ ℂ |
20 | 4, 19 | mulcli 11265 | . . . . . 6 ⊢ (i · (π / 2)) ∈ ℂ |
21 | pipos 26516 | . . . . . . . . 9 ⊢ 0 < π | |
22 | pire 26514 | . . . . . . . . . 10 ⊢ π ∈ ℝ | |
23 | lt0neg2 11767 | . . . . . . . . . 10 ⊢ (π ∈ ℝ → (0 < π ↔ -π < 0)) | |
24 | 22, 23 | ax-mp 5 | . . . . . . . . 9 ⊢ (0 < π ↔ -π < 0) |
25 | 21, 24 | mpbi 230 | . . . . . . . 8 ⊢ -π < 0 |
26 | pirp 26517 | . . . . . . . . . 10 ⊢ π ∈ ℝ+ | |
27 | rphalfcl 13059 | . . . . . . . . . 10 ⊢ (π ∈ ℝ+ → (π / 2) ∈ ℝ+) | |
28 | 26, 27 | ax-mp 5 | . . . . . . . . 9 ⊢ (π / 2) ∈ ℝ+ |
29 | rpgt0 13044 | . . . . . . . . 9 ⊢ ((π / 2) ∈ ℝ+ → 0 < (π / 2)) | |
30 | 28, 29 | ax-mp 5 | . . . . . . . 8 ⊢ 0 < (π / 2) |
31 | 22 | renegcli 11567 | . . . . . . . . 9 ⊢ -π ∈ ℝ |
32 | 0re 11260 | . . . . . . . . 9 ⊢ 0 ∈ ℝ | |
33 | 31, 32, 18 | lttri 11384 | . . . . . . . 8 ⊢ ((-π < 0 ∧ 0 < (π / 2)) → -π < (π / 2)) |
34 | 25, 30, 33 | mp2an 692 | . . . . . . 7 ⊢ -π < (π / 2) |
35 | 20 | addlidi 11446 | . . . . . . . . 9 ⊢ (0 + (i · (π / 2))) = (i · (π / 2)) |
36 | 35 | fveq2i 6909 | . . . . . . . 8 ⊢ (ℑ‘(0 + (i · (π / 2)))) = (ℑ‘(i · (π / 2))) |
37 | 32, 18 | crimi 15228 | . . . . . . . 8 ⊢ (ℑ‘(0 + (i · (π / 2)))) = (π / 2) |
38 | 36, 37 | eqtr3i 2764 | . . . . . . 7 ⊢ (ℑ‘(i · (π / 2))) = (π / 2) |
39 | 34, 38 | breqtrri 5174 | . . . . . 6 ⊢ -π < (ℑ‘(i · (π / 2))) |
40 | rphalflt 13061 | . . . . . . . . 9 ⊢ (π ∈ ℝ+ → (π / 2) < π) | |
41 | 26, 40 | ax-mp 5 | . . . . . . . 8 ⊢ (π / 2) < π |
42 | 18, 22, 41 | ltleii 11381 | . . . . . . 7 ⊢ (π / 2) ≤ π |
43 | 38, 42 | eqbrtri 5168 | . . . . . 6 ⊢ (ℑ‘(i · (π / 2))) ≤ π |
44 | ellogrn 26615 | . . . . . 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 26637 | . . . . 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 2762 | . . 3 ⊢ (log‘((i · 1) + (√‘(1 − (1↑2))))) = (i · (π / 2)) |
49 | 48 | oveq2i 7441 | . 2 ⊢ (-i · (log‘((i · 1) + (√‘(1 − (1↑2)))))) = (-i · (i · (π / 2))) |
50 | 4, 4 | mulneg1i 11706 | . . . . . 6 ⊢ (-i · i) = -(i · i) |
51 | ixi 11889 | . . . . . . 7 ⊢ (i · i) = -1 | |
52 | 51 | negeqi 11498 | . . . . . 6 ⊢ -(i · i) = --1 |
53 | negneg1e1 12381 | . . . . . 6 ⊢ --1 = 1 | |
54 | 50, 52, 53 | 3eqtri 2766 | . . . . 5 ⊢ (-i · i) = 1 |
55 | 54 | oveq1i 7440 | . . . 4 ⊢ ((-i · i) · (π / 2)) = (1 · (π / 2)) |
56 | negicn 11506 | . . . . 5 ⊢ -i ∈ ℂ | |
57 | 56, 4, 19 | mulassi 11269 | . . . 4 ⊢ ((-i · i) · (π / 2)) = (-i · (i · (π / 2))) |
58 | 55, 57 | eqtr3i 2764 | . . 3 ⊢ (1 · (π / 2)) = (-i · (i · (π / 2))) |
59 | 19 | mullidi 11263 | . . 3 ⊢ (1 · (π / 2)) = (π / 2) |
60 | 58, 59 | eqtr3i 2764 | . 2 ⊢ (-i · (i · (π / 2))) = (π / 2) |
61 | 3, 49, 60 | 3eqtri 2766 | 1 ⊢ (arcsin‘1) = (π / 2) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 206 = wceq 1536 ∈ wcel 2105 class class class wbr 5147 ran crn 5689 ‘cfv 6562 (class class class)co 7430 ℂcc 11150 ℝcr 11151 0cc0 11152 1c1 11153 ici 11154 + caddc 11155 · cmul 11157 < clt 11292 ≤ cle 11293 − cmin 11489 -cneg 11490 / cdiv 11917 2c2 12318 ℝ+crp 13031 ↑cexp 14098 ℑcim 15133 √csqrt 15268 expce 16093 πcpi 16098 logclog 26610 arcsincasin 26919 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1791 ax-4 1805 ax-5 1907 ax-6 1964 ax-7 2004 ax-8 2107 ax-9 2115 ax-10 2138 ax-11 2154 ax-12 2174 ax-ext 2705 ax-rep 5284 ax-sep 5301 ax-nul 5311 ax-pow 5370 ax-pr 5437 ax-un 7753 ax-inf2 9678 ax-cnex 11208 ax-resscn 11209 ax-1cn 11210 ax-icn 11211 ax-addcl 11212 ax-addrcl 11213 ax-mulcl 11214 ax-mulrcl 11215 ax-mulcom 11216 ax-addass 11217 ax-mulass 11218 ax-distr 11219 ax-i2m1 11220 ax-1ne0 11221 ax-1rid 11222 ax-rnegex 11223 ax-rrecex 11224 ax-cnre 11225 ax-pre-lttri 11226 ax-pre-lttrn 11227 ax-pre-ltadd 11228 ax-pre-mulgt0 11229 ax-pre-sup 11230 ax-addf 11231 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1539 df-fal 1549 df-ex 1776 df-nf 1780 df-sb 2062 df-mo 2537 df-eu 2566 df-clab 2712 df-cleq 2726 df-clel 2813 df-nfc 2889 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-rmo 3377 df-reu 3378 df-rab 3433 df-v 3479 df-sbc 3791 df-csb 3908 df-dif 3965 df-un 3967 df-in 3969 df-ss 3979 df-pss 3982 df-nul 4339 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-tp 4635 df-op 4637 df-uni 4912 df-int 4951 df-iun 4997 df-iin 4998 df-br 5148 df-opab 5210 df-mpt 5231 df-tr 5265 df-id 5582 df-eprel 5588 df-po 5596 df-so 5597 df-fr 5640 df-se 5641 df-we 5642 df-xp 5694 df-rel 5695 df-cnv 5696 df-co 5697 df-dm 5698 df-rn 5699 df-res 5700 df-ima 5701 df-pred 6322 df-ord 6388 df-on 6389 df-lim 6390 df-suc 6391 df-iota 6515 df-fun 6564 df-fn 6565 df-f 6566 df-f1 6567 df-fo 6568 df-f1o 6569 df-fv 6570 df-isom 6571 df-riota 7387 df-ov 7433 df-oprab 7434 df-mpo 7435 df-of 7696 df-om 7887 df-1st 8012 df-2nd 8013 df-supp 8184 df-frecs 8304 df-wrecs 8335 df-recs 8409 df-rdg 8448 df-1o 8504 df-2o 8505 df-er 8743 df-map 8866 df-pm 8867 df-ixp 8936 df-en 8984 df-dom 8985 df-sdom 8986 df-fin 8987 df-fsupp 9399 df-fi 9448 df-sup 9479 df-inf 9480 df-oi 9547 df-card 9976 df-pnf 11294 df-mnf 11295 df-xr 11296 df-ltxr 11297 df-le 11298 df-sub 11491 df-neg 11492 df-div 11918 df-nn 12264 df-2 12326 df-3 12327 df-4 12328 df-5 12329 df-6 12330 df-7 12331 df-8 12332 df-9 12333 df-n0 12524 df-z 12611 df-dec 12731 df-uz 12876 df-q 12988 df-rp 13032 df-xneg 13151 df-xadd 13152 df-xmul 13153 df-ioo 13387 df-ioc 13388 df-ico 13389 df-icc 13390 df-fz 13544 df-fzo 13691 df-fl 13828 df-mod 13906 df-seq 14039 df-exp 14099 df-fac 14309 df-bc 14338 df-hash 14366 df-shft 15102 df-cj 15134 df-re 15135 df-im 15136 df-sqrt 15270 df-abs 15271 df-limsup 15503 df-clim 15520 df-rlim 15521 df-sum 15719 df-ef 16099 df-sin 16101 df-cos 16102 df-pi 16104 df-struct 17180 df-sets 17197 df-slot 17215 df-ndx 17227 df-base 17245 df-ress 17274 df-plusg 17310 df-mulr 17311 df-starv 17312 df-sca 17313 df-vsca 17314 df-ip 17315 df-tset 17316 df-ple 17317 df-ds 17319 df-unif 17320 df-hom 17321 df-cco 17322 df-rest 17468 df-topn 17469 df-0g 17487 df-gsum 17488 df-topgen 17489 df-pt 17490 df-prds 17493 df-xrs 17548 df-qtop 17553 df-imas 17554 df-xps 17556 df-mre 17630 df-mrc 17631 df-acs 17633 df-mgm 18665 df-sgrp 18744 df-mnd 18760 df-submnd 18809 df-mulg 19098 df-cntz 19347 df-cmn 19814 df-psmet 21373 df-xmet 21374 df-met 21375 df-bl 21376 df-mopn 21377 df-fbas 21378 df-fg 21379 df-cnfld 21382 df-top 22915 df-topon 22932 df-topsp 22954 df-bases 22968 df-cld 23042 df-ntr 23043 df-cls 23044 df-nei 23121 df-lp 23159 df-perf 23160 df-cn 23250 df-cnp 23251 df-haus 23338 df-tx 23585 df-hmeo 23778 df-fil 23869 df-fm 23961 df-flim 23962 df-flf 23963 df-xms 24345 df-ms 24346 df-tms 24347 df-cncf 24917 df-limc 25915 df-dv 25916 df-log 26612 df-asin 26922 |
This theorem is referenced by: acos1 26952 reasinsin 26953 areacirc 37699 |
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