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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 11213 | . . 3 ⊢ 1 ∈ ℂ | |
| 2 | asinval 26925 | . . 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 11214 | . . . . . . 7 ⊢ i ∈ ℂ | |
| 5 | 4 | addridi 11448 | . . . . . 6 ⊢ (i + 0) = i |
| 6 | 4 | mulridi 11265 | . . . . . . 7 ⊢ (i · 1) = i |
| 7 | sq1 14234 | . . . . . . . . . . 11 ⊢ (1↑2) = 1 | |
| 8 | 7 | oveq2i 7442 | . . . . . . . . . 10 ⊢ (1 − (1↑2)) = (1 − 1) |
| 9 | 1m1e0 12338 | . . . . . . . . . 10 ⊢ (1 − 1) = 0 | |
| 10 | 8, 9 | eqtri 2765 | . . . . . . . . 9 ⊢ (1 − (1↑2)) = 0 |
| 11 | 10 | fveq2i 6909 | . . . . . . . 8 ⊢ (√‘(1 − (1↑2))) = (√‘0) |
| 12 | sqrt0 15280 | . . . . . . . 8 ⊢ (√‘0) = 0 | |
| 13 | 11, 12 | eqtri 2765 | . . . . . . 7 ⊢ (√‘(1 − (1↑2))) = 0 |
| 14 | 6, 13 | oveq12i 7443 | . . . . . 6 ⊢ ((i · 1) + (√‘(1 − (1↑2)))) = (i + 0) |
| 15 | efhalfpi 26513 | . . . . . 6 ⊢ (exp‘(i · (π / 2))) = i | |
| 16 | 5, 14, 15 | 3eqtr4i 2775 | . . . . 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 26506 | . . . . . . . 8 ⊢ (π / 2) ∈ ℝ | |
| 19 | 18 | recni 11275 | . . . . . . 7 ⊢ (π / 2) ∈ ℂ |
| 20 | 4, 19 | mulcli 11268 | . . . . . 6 ⊢ (i · (π / 2)) ∈ ℂ |
| 21 | pipos 26502 | . . . . . . . . 9 ⊢ 0 < π | |
| 22 | pire 26500 | . . . . . . . . . 10 ⊢ π ∈ ℝ | |
| 23 | lt0neg2 11770 | . . . . . . . . . 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 26503 | . . . . . . . . . 10 ⊢ π ∈ ℝ+ | |
| 27 | rphalfcl 13062 | . . . . . . . . . 10 ⊢ (π ∈ ℝ+ → (π / 2) ∈ ℝ+) | |
| 28 | 26, 27 | ax-mp 5 | . . . . . . . . 9 ⊢ (π / 2) ∈ ℝ+ |
| 29 | rpgt0 13047 | . . . . . . . . 9 ⊢ ((π / 2) ∈ ℝ+ → 0 < (π / 2)) | |
| 30 | 28, 29 | ax-mp 5 | . . . . . . . 8 ⊢ 0 < (π / 2) |
| 31 | 22 | renegcli 11570 | . . . . . . . . 9 ⊢ -π ∈ ℝ |
| 32 | 0re 11263 | . . . . . . . . 9 ⊢ 0 ∈ ℝ | |
| 33 | 31, 32, 18 | lttri 11387 | . . . . . . . 8 ⊢ ((-π < 0 ∧ 0 < (π / 2)) → -π < (π / 2)) |
| 34 | 25, 30, 33 | mp2an 692 | . . . . . . 7 ⊢ -π < (π / 2) |
| 35 | 20 | addlidi 11449 | . . . . . . . . 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 15232 | . . . . . . . 8 ⊢ (ℑ‘(0 + (i · (π / 2)))) = (π / 2) |
| 38 | 36, 37 | eqtr3i 2767 | . . . . . . 7 ⊢ (ℑ‘(i · (π / 2))) = (π / 2) |
| 39 | 34, 38 | breqtrri 5170 | . . . . . 6 ⊢ -π < (ℑ‘(i · (π / 2))) |
| 40 | rphalflt 13064 | . . . . . . . . 9 ⊢ (π ∈ ℝ+ → (π / 2) < π) | |
| 41 | 26, 40 | ax-mp 5 | . . . . . . . 8 ⊢ (π / 2) < π |
| 42 | 18, 22, 41 | ltleii 11384 | . . . . . . 7 ⊢ (π / 2) ≤ π |
| 43 | 38, 42 | eqbrtri 5164 | . . . . . 6 ⊢ (ℑ‘(i · (π / 2))) ≤ π |
| 44 | ellogrn 26601 | . . . . . 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 1342 | . . . . 5 ⊢ (i · (π / 2)) ∈ ran log |
| 46 | logef 26623 | . . . . 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 2765 | . . 3 ⊢ (log‘((i · 1) + (√‘(1 − (1↑2))))) = (i · (π / 2)) |
| 49 | 48 | oveq2i 7442 | . 2 ⊢ (-i · (log‘((i · 1) + (√‘(1 − (1↑2)))))) = (-i · (i · (π / 2))) |
| 50 | 4, 4 | mulneg1i 11709 | . . . . . 6 ⊢ (-i · i) = -(i · i) |
| 51 | ixi 11892 | . . . . . . 7 ⊢ (i · i) = -1 | |
| 52 | 51 | negeqi 11501 | . . . . . 6 ⊢ -(i · i) = --1 |
| 53 | negneg1e1 12384 | . . . . . 6 ⊢ --1 = 1 | |
| 54 | 50, 52, 53 | 3eqtri 2769 | . . . . 5 ⊢ (-i · i) = 1 |
| 55 | 54 | oveq1i 7441 | . . . 4 ⊢ ((-i · i) · (π / 2)) = (1 · (π / 2)) |
| 56 | negicn 11509 | . . . . 5 ⊢ -i ∈ ℂ | |
| 57 | 56, 4, 19 | mulassi 11272 | . . . 4 ⊢ ((-i · i) · (π / 2)) = (-i · (i · (π / 2))) |
| 58 | 55, 57 | eqtr3i 2767 | . . 3 ⊢ (1 · (π / 2)) = (-i · (i · (π / 2))) |
| 59 | 19 | mullidi 11266 | . . 3 ⊢ (1 · (π / 2)) = (π / 2) |
| 60 | 58, 59 | eqtr3i 2767 | . 2 ⊢ (-i · (i · (π / 2))) = (π / 2) |
| 61 | 3, 49, 60 | 3eqtri 2769 | 1 ⊢ (arcsin‘1) = (π / 2) |
| Colors of variables: wff setvar class |
| Syntax hints: ↔ wb 206 = wceq 1540 ∈ wcel 2108 class class class wbr 5143 ran crn 5686 ‘cfv 6561 (class class class)co 7431 ℂcc 11153 ℝcr 11154 0cc0 11155 1c1 11156 ici 11157 + caddc 11158 · cmul 11160 < clt 11295 ≤ cle 11296 − cmin 11492 -cneg 11493 / cdiv 11920 2c2 12321 ℝ+crp 13034 ↑cexp 14102 ℑcim 15137 √csqrt 15272 expce 16097 πcpi 16102 logclog 26596 arcsincasin 26905 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-rep 5279 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-inf2 9681 ax-cnex 11211 ax-resscn 11212 ax-1cn 11213 ax-icn 11214 ax-addcl 11215 ax-addrcl 11216 ax-mulcl 11217 ax-mulrcl 11218 ax-mulcom 11219 ax-addass 11220 ax-mulass 11221 ax-distr 11222 ax-i2m1 11223 ax-1ne0 11224 ax-1rid 11225 ax-rnegex 11226 ax-rrecex 11227 ax-cnre 11228 ax-pre-lttri 11229 ax-pre-lttrn 11230 ax-pre-ltadd 11231 ax-pre-mulgt0 11232 ax-pre-sup 11233 ax-addf 11234 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3380 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-tp 4631 df-op 4633 df-uni 4908 df-int 4947 df-iun 4993 df-iin 4994 df-br 5144 df-opab 5206 df-mpt 5226 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-se 5638 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-pred 6321 df-ord 6387 df-on 6388 df-lim 6389 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-isom 6570 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-of 7697 df-om 7888 df-1st 8014 df-2nd 8015 df-supp 8186 df-frecs 8306 df-wrecs 8337 df-recs 8411 df-rdg 8450 df-1o 8506 df-2o 8507 df-er 8745 df-map 8868 df-pm 8869 df-ixp 8938 df-en 8986 df-dom 8987 df-sdom 8988 df-fin 8989 df-fsupp 9402 df-fi 9451 df-sup 9482 df-inf 9483 df-oi 9550 df-card 9979 df-pnf 11297 df-mnf 11298 df-xr 11299 df-ltxr 11300 df-le 11301 df-sub 11494 df-neg 11495 df-div 11921 df-nn 12267 df-2 12329 df-3 12330 df-4 12331 df-5 12332 df-6 12333 df-7 12334 df-8 12335 df-9 12336 df-n0 12527 df-z 12614 df-dec 12734 df-uz 12879 df-q 12991 df-rp 13035 df-xneg 13154 df-xadd 13155 df-xmul 13156 df-ioo 13391 df-ioc 13392 df-ico 13393 df-icc 13394 df-fz 13548 df-fzo 13695 df-fl 13832 df-mod 13910 df-seq 14043 df-exp 14103 df-fac 14313 df-bc 14342 df-hash 14370 df-shft 15106 df-cj 15138 df-re 15139 df-im 15140 df-sqrt 15274 df-abs 15275 df-limsup 15507 df-clim 15524 df-rlim 15525 df-sum 15723 df-ef 16103 df-sin 16105 df-cos 16106 df-pi 16108 df-struct 17184 df-sets 17201 df-slot 17219 df-ndx 17231 df-base 17248 df-ress 17275 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 17467 df-topn 17468 df-0g 17486 df-gsum 17487 df-topgen 17488 df-pt 17489 df-prds 17492 df-xrs 17547 df-qtop 17552 df-imas 17553 df-xps 17555 df-mre 17629 df-mrc 17630 df-acs 17632 df-mgm 18653 df-sgrp 18732 df-mnd 18748 df-submnd 18797 df-mulg 19086 df-cntz 19335 df-cmn 19800 df-psmet 21356 df-xmet 21357 df-met 21358 df-bl 21359 df-mopn 21360 df-fbas 21361 df-fg 21362 df-cnfld 21365 df-top 22900 df-topon 22917 df-topsp 22939 df-bases 22953 df-cld 23027 df-ntr 23028 df-cls 23029 df-nei 23106 df-lp 23144 df-perf 23145 df-cn 23235 df-cnp 23236 df-haus 23323 df-tx 23570 df-hmeo 23763 df-fil 23854 df-fm 23946 df-flim 23947 df-flf 23948 df-xms 24330 df-ms 24331 df-tms 24332 df-cncf 24904 df-limc 25901 df-dv 25902 df-log 26598 df-asin 26908 |
| This theorem is referenced by: acos1 26938 reasinsin 26939 areacirc 37720 |
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