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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 11190 | . . 3 ⊢ 1 ∈ ℂ | |
2 | asinval 26807 | . . 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 11191 | . . . . . . 7 ⊢ i ∈ ℂ | |
5 | 4 | addridi 11425 | . . . . . 6 ⊢ (i + 0) = i |
6 | 4 | mulridi 11242 | . . . . . . 7 ⊢ (i · 1) = i |
7 | sq1 14184 | . . . . . . . . . . 11 ⊢ (1↑2) = 1 | |
8 | 7 | oveq2i 7425 | . . . . . . . . . 10 ⊢ (1 − (1↑2)) = (1 − 1) |
9 | 1m1e0 12308 | . . . . . . . . . 10 ⊢ (1 − 1) = 0 | |
10 | 8, 9 | eqtri 2756 | . . . . . . . . 9 ⊢ (1 − (1↑2)) = 0 |
11 | 10 | fveq2i 6894 | . . . . . . . 8 ⊢ (√‘(1 − (1↑2))) = (√‘0) |
12 | sqrt0 15214 | . . . . . . . 8 ⊢ (√‘0) = 0 | |
13 | 11, 12 | eqtri 2756 | . . . . . . 7 ⊢ (√‘(1 − (1↑2))) = 0 |
14 | 6, 13 | oveq12i 7426 | . . . . . 6 ⊢ ((i · 1) + (√‘(1 − (1↑2)))) = (i + 0) |
15 | efhalfpi 26399 | . . . . . 6 ⊢ (exp‘(i · (π / 2))) = i | |
16 | 5, 14, 15 | 3eqtr4i 2766 | . . . . 5 ⊢ ((i · 1) + (√‘(1 − (1↑2)))) = (exp‘(i · (π / 2))) |
17 | 16 | fveq2i 6894 | . . . 4 ⊢ (log‘((i · 1) + (√‘(1 − (1↑2))))) = (log‘(exp‘(i · (π / 2)))) |
18 | halfpire 26392 | . . . . . . . 8 ⊢ (π / 2) ∈ ℝ | |
19 | 18 | recni 11252 | . . . . . . 7 ⊢ (π / 2) ∈ ℂ |
20 | 4, 19 | mulcli 11245 | . . . . . 6 ⊢ (i · (π / 2)) ∈ ℂ |
21 | pipos 26388 | . . . . . . . . 9 ⊢ 0 < π | |
22 | pire 26386 | . . . . . . . . . 10 ⊢ π ∈ ℝ | |
23 | lt0neg2 11745 | . . . . . . . . . 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 26389 | . . . . . . . . . 10 ⊢ π ∈ ℝ+ | |
27 | rphalfcl 13027 | . . . . . . . . . 10 ⊢ (π ∈ ℝ+ → (π / 2) ∈ ℝ+) | |
28 | 26, 27 | ax-mp 5 | . . . . . . . . 9 ⊢ (π / 2) ∈ ℝ+ |
29 | rpgt0 13012 | . . . . . . . . 9 ⊢ ((π / 2) ∈ ℝ+ → 0 < (π / 2)) | |
30 | 28, 29 | ax-mp 5 | . . . . . . . 8 ⊢ 0 < (π / 2) |
31 | 22 | renegcli 11545 | . . . . . . . . 9 ⊢ -π ∈ ℝ |
32 | 0re 11240 | . . . . . . . . 9 ⊢ 0 ∈ ℝ | |
33 | 31, 32, 18 | lttri 11364 | . . . . . . . 8 ⊢ ((-π < 0 ∧ 0 < (π / 2)) → -π < (π / 2)) |
34 | 25, 30, 33 | mp2an 691 | . . . . . . 7 ⊢ -π < (π / 2) |
35 | 20 | addlidi 11426 | . . . . . . . . 9 ⊢ (0 + (i · (π / 2))) = (i · (π / 2)) |
36 | 35 | fveq2i 6894 | . . . . . . . 8 ⊢ (ℑ‘(0 + (i · (π / 2)))) = (ℑ‘(i · (π / 2))) |
37 | 32, 18 | crimi 15166 | . . . . . . . 8 ⊢ (ℑ‘(0 + (i · (π / 2)))) = (π / 2) |
38 | 36, 37 | eqtr3i 2758 | . . . . . . 7 ⊢ (ℑ‘(i · (π / 2))) = (π / 2) |
39 | 34, 38 | breqtrri 5169 | . . . . . 6 ⊢ -π < (ℑ‘(i · (π / 2))) |
40 | rphalflt 13029 | . . . . . . . . 9 ⊢ (π ∈ ℝ+ → (π / 2) < π) | |
41 | 26, 40 | ax-mp 5 | . . . . . . . 8 ⊢ (π / 2) < π |
42 | 18, 22, 41 | ltleii 11361 | . . . . . . 7 ⊢ (π / 2) ≤ π |
43 | 38, 42 | eqbrtri 5163 | . . . . . 6 ⊢ (ℑ‘(i · (π / 2))) ≤ π |
44 | ellogrn 26486 | . . . . . 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 1339 | . . . . 5 ⊢ (i · (π / 2)) ∈ ran log |
46 | logef 26508 | . . . . 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 2756 | . . 3 ⊢ (log‘((i · 1) + (√‘(1 − (1↑2))))) = (i · (π / 2)) |
49 | 48 | oveq2i 7425 | . 2 ⊢ (-i · (log‘((i · 1) + (√‘(1 − (1↑2)))))) = (-i · (i · (π / 2))) |
50 | 4, 4 | mulneg1i 11684 | . . . . . 6 ⊢ (-i · i) = -(i · i) |
51 | ixi 11867 | . . . . . . 7 ⊢ (i · i) = -1 | |
52 | 51 | negeqi 11477 | . . . . . 6 ⊢ -(i · i) = --1 |
53 | negneg1e1 12354 | . . . . . 6 ⊢ --1 = 1 | |
54 | 50, 52, 53 | 3eqtri 2760 | . . . . 5 ⊢ (-i · i) = 1 |
55 | 54 | oveq1i 7424 | . . . 4 ⊢ ((-i · i) · (π / 2)) = (1 · (π / 2)) |
56 | negicn 11485 | . . . . 5 ⊢ -i ∈ ℂ | |
57 | 56, 4, 19 | mulassi 11249 | . . . 4 ⊢ ((-i · i) · (π / 2)) = (-i · (i · (π / 2))) |
58 | 55, 57 | eqtr3i 2758 | . . 3 ⊢ (1 · (π / 2)) = (-i · (i · (π / 2))) |
59 | 19 | mullidi 11243 | . . 3 ⊢ (1 · (π / 2)) = (π / 2) |
60 | 58, 59 | eqtr3i 2758 | . 2 ⊢ (-i · (i · (π / 2))) = (π / 2) |
61 | 3, 49, 60 | 3eqtri 2760 | 1 ⊢ (arcsin‘1) = (π / 2) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 205 = wceq 1534 ∈ wcel 2099 class class class wbr 5142 ran crn 5673 ‘cfv 6542 (class class class)co 7414 ℂcc 11130 ℝcr 11131 0cc0 11132 1c1 11133 ici 11134 + caddc 11135 · cmul 11137 < clt 11272 ≤ cle 11273 − cmin 11468 -cneg 11469 / cdiv 11895 2c2 12291 ℝ+crp 13000 ↑cexp 14052 ℑcim 15071 √csqrt 15206 expce 16031 πcpi 16036 logclog 26481 arcsincasin 26787 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2699 ax-rep 5279 ax-sep 5293 ax-nul 5300 ax-pow 5359 ax-pr 5423 ax-un 7734 ax-inf2 9658 ax-cnex 11188 ax-resscn 11189 ax-1cn 11190 ax-icn 11191 ax-addcl 11192 ax-addrcl 11193 ax-mulcl 11194 ax-mulrcl 11195 ax-mulcom 11196 ax-addass 11197 ax-mulass 11198 ax-distr 11199 ax-i2m1 11200 ax-1ne0 11201 ax-1rid 11202 ax-rnegex 11203 ax-rrecex 11204 ax-cnre 11205 ax-pre-lttri 11206 ax-pre-lttrn 11207 ax-pre-ltadd 11208 ax-pre-mulgt0 11209 ax-pre-sup 11210 ax-addf 11211 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2530 df-eu 2559 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2937 df-nel 3043 df-ral 3058 df-rex 3067 df-rmo 3372 df-reu 3373 df-rab 3429 df-v 3472 df-sbc 3776 df-csb 3891 df-dif 3948 df-un 3950 df-in 3952 df-ss 3962 df-pss 3964 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-tp 4629 df-op 4631 df-uni 4904 df-int 4945 df-iun 4993 df-iin 4994 df-br 5143 df-opab 5205 df-mpt 5226 df-tr 5260 df-id 5570 df-eprel 5576 df-po 5584 df-so 5585 df-fr 5627 df-se 5628 df-we 5629 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-pred 6299 df-ord 6366 df-on 6367 df-lim 6368 df-suc 6369 df-iota 6494 df-fun 6544 df-fn 6545 df-f 6546 df-f1 6547 df-fo 6548 df-f1o 6549 df-fv 6550 df-isom 6551 df-riota 7370 df-ov 7417 df-oprab 7418 df-mpo 7419 df-of 7679 df-om 7865 df-1st 7987 df-2nd 7988 df-supp 8160 df-frecs 8280 df-wrecs 8311 df-recs 8385 df-rdg 8424 df-1o 8480 df-2o 8481 df-er 8718 df-map 8840 df-pm 8841 df-ixp 8910 df-en 8958 df-dom 8959 df-sdom 8960 df-fin 8961 df-fsupp 9380 df-fi 9428 df-sup 9459 df-inf 9460 df-oi 9527 df-card 9956 df-pnf 11274 df-mnf 11275 df-xr 11276 df-ltxr 11277 df-le 11278 df-sub 11470 df-neg 11471 df-div 11896 df-nn 12237 df-2 12299 df-3 12300 df-4 12301 df-5 12302 df-6 12303 df-7 12304 df-8 12305 df-9 12306 df-n0 12497 df-z 12583 df-dec 12702 df-uz 12847 df-q 12957 df-rp 13001 df-xneg 13118 df-xadd 13119 df-xmul 13120 df-ioo 13354 df-ioc 13355 df-ico 13356 df-icc 13357 df-fz 13511 df-fzo 13654 df-fl 13783 df-mod 13861 df-seq 13993 df-exp 14053 df-fac 14259 df-bc 14288 df-hash 14316 df-shft 15040 df-cj 15072 df-re 15073 df-im 15074 df-sqrt 15208 df-abs 15209 df-limsup 15441 df-clim 15458 df-rlim 15459 df-sum 15659 df-ef 16037 df-sin 16039 df-cos 16040 df-pi 16042 df-struct 17109 df-sets 17126 df-slot 17144 df-ndx 17156 df-base 17174 df-ress 17203 df-plusg 17239 df-mulr 17240 df-starv 17241 df-sca 17242 df-vsca 17243 df-ip 17244 df-tset 17245 df-ple 17246 df-ds 17248 df-unif 17249 df-hom 17250 df-cco 17251 df-rest 17397 df-topn 17398 df-0g 17416 df-gsum 17417 df-topgen 17418 df-pt 17419 df-prds 17422 df-xrs 17477 df-qtop 17482 df-imas 17483 df-xps 17485 df-mre 17559 df-mrc 17560 df-acs 17562 df-mgm 18593 df-sgrp 18672 df-mnd 18688 df-submnd 18734 df-mulg 19017 df-cntz 19261 df-cmn 19730 df-psmet 21264 df-xmet 21265 df-met 21266 df-bl 21267 df-mopn 21268 df-fbas 21269 df-fg 21270 df-cnfld 21273 df-top 22789 df-topon 22806 df-topsp 22828 df-bases 22842 df-cld 22916 df-ntr 22917 df-cls 22918 df-nei 22995 df-lp 23033 df-perf 23034 df-cn 23124 df-cnp 23125 df-haus 23212 df-tx 23459 df-hmeo 23652 df-fil 23743 df-fm 23835 df-flim 23836 df-flf 23837 df-xms 24219 df-ms 24220 df-tms 24221 df-cncf 24791 df-limc 25788 df-dv 25789 df-log 26483 df-asin 26790 |
This theorem is referenced by: acos1 26820 reasinsin 26821 areacirc 37180 |
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