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Mirrors > Home > MPE Home > Th. List > acosbnd | Structured version Visualization version GIF version |
Description: The arccosine function has range within a vertical strip of the complex plane with real part between 0 and π. (Contributed by Mario Carneiro, 2-Apr-2015.) |
Ref | Expression |
---|---|
acosbnd | ⊢ (𝐴 ∈ ℂ → (ℜ‘(arccos‘𝐴)) ∈ (0[,]π)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | acosval 26941 | . . . 4 ⊢ (𝐴 ∈ ℂ → (arccos‘𝐴) = ((π / 2) − (arcsin‘𝐴))) | |
2 | 1 | fveq2d 6911 | . . 3 ⊢ (𝐴 ∈ ℂ → (ℜ‘(arccos‘𝐴)) = (ℜ‘((π / 2) − (arcsin‘𝐴)))) |
3 | halfpire 26521 | . . . . . 6 ⊢ (π / 2) ∈ ℝ | |
4 | 3 | recni 11273 | . . . . 5 ⊢ (π / 2) ∈ ℂ |
5 | asincl 26931 | . . . . 5 ⊢ (𝐴 ∈ ℂ → (arcsin‘𝐴) ∈ ℂ) | |
6 | resub 15163 | . . . . 5 ⊢ (((π / 2) ∈ ℂ ∧ (arcsin‘𝐴) ∈ ℂ) → (ℜ‘((π / 2) − (arcsin‘𝐴))) = ((ℜ‘(π / 2)) − (ℜ‘(arcsin‘𝐴)))) | |
7 | 4, 5, 6 | sylancr 587 | . . . 4 ⊢ (𝐴 ∈ ℂ → (ℜ‘((π / 2) − (arcsin‘𝐴))) = ((ℜ‘(π / 2)) − (ℜ‘(arcsin‘𝐴)))) |
8 | rere 15158 | . . . . . 6 ⊢ ((π / 2) ∈ ℝ → (ℜ‘(π / 2)) = (π / 2)) | |
9 | 3, 8 | ax-mp 5 | . . . . 5 ⊢ (ℜ‘(π / 2)) = (π / 2) |
10 | 9 | oveq1i 7441 | . . . 4 ⊢ ((ℜ‘(π / 2)) − (ℜ‘(arcsin‘𝐴))) = ((π / 2) − (ℜ‘(arcsin‘𝐴))) |
11 | 7, 10 | eqtrdi 2791 | . . 3 ⊢ (𝐴 ∈ ℂ → (ℜ‘((π / 2) − (arcsin‘𝐴))) = ((π / 2) − (ℜ‘(arcsin‘𝐴)))) |
12 | 2, 11 | eqtrd 2775 | . 2 ⊢ (𝐴 ∈ ℂ → (ℜ‘(arccos‘𝐴)) = ((π / 2) − (ℜ‘(arcsin‘𝐴)))) |
13 | 5 | recld 15230 | . . . 4 ⊢ (𝐴 ∈ ℂ → (ℜ‘(arcsin‘𝐴)) ∈ ℝ) |
14 | resubcl 11571 | . . . 4 ⊢ (((π / 2) ∈ ℝ ∧ (ℜ‘(arcsin‘𝐴)) ∈ ℝ) → ((π / 2) − (ℜ‘(arcsin‘𝐴))) ∈ ℝ) | |
15 | 3, 13, 14 | sylancr 587 | . . 3 ⊢ (𝐴 ∈ ℂ → ((π / 2) − (ℜ‘(arcsin‘𝐴))) ∈ ℝ) |
16 | asinbnd 26957 | . . . . . 6 ⊢ (𝐴 ∈ ℂ → (ℜ‘(arcsin‘𝐴)) ∈ (-(π / 2)[,](π / 2))) | |
17 | neghalfpire 26522 | . . . . . . 7 ⊢ -(π / 2) ∈ ℝ | |
18 | 17, 3 | elicc2i 13450 | . . . . . 6 ⊢ ((ℜ‘(arcsin‘𝐴)) ∈ (-(π / 2)[,](π / 2)) ↔ ((ℜ‘(arcsin‘𝐴)) ∈ ℝ ∧ -(π / 2) ≤ (ℜ‘(arcsin‘𝐴)) ∧ (ℜ‘(arcsin‘𝐴)) ≤ (π / 2))) |
19 | 16, 18 | sylib 218 | . . . . 5 ⊢ (𝐴 ∈ ℂ → ((ℜ‘(arcsin‘𝐴)) ∈ ℝ ∧ -(π / 2) ≤ (ℜ‘(arcsin‘𝐴)) ∧ (ℜ‘(arcsin‘𝐴)) ≤ (π / 2))) |
20 | 19 | simp3d 1143 | . . . 4 ⊢ (𝐴 ∈ ℂ → (ℜ‘(arcsin‘𝐴)) ≤ (π / 2)) |
21 | subge0 11774 | . . . . 5 ⊢ (((π / 2) ∈ ℝ ∧ (ℜ‘(arcsin‘𝐴)) ∈ ℝ) → (0 ≤ ((π / 2) − (ℜ‘(arcsin‘𝐴))) ↔ (ℜ‘(arcsin‘𝐴)) ≤ (π / 2))) | |
22 | 3, 13, 21 | sylancr 587 | . . . 4 ⊢ (𝐴 ∈ ℂ → (0 ≤ ((π / 2) − (ℜ‘(arcsin‘𝐴))) ↔ (ℜ‘(arcsin‘𝐴)) ≤ (π / 2))) |
23 | 20, 22 | mpbird 257 | . . 3 ⊢ (𝐴 ∈ ℂ → 0 ≤ ((π / 2) − (ℜ‘(arcsin‘𝐴)))) |
24 | 3 | a1i 11 | . . . 4 ⊢ (𝐴 ∈ ℂ → (π / 2) ∈ ℝ) |
25 | pire 26515 | . . . . 5 ⊢ π ∈ ℝ | |
26 | 25 | a1i 11 | . . . 4 ⊢ (𝐴 ∈ ℂ → π ∈ ℝ) |
27 | 25 | recni 11273 | . . . . . 6 ⊢ π ∈ ℂ |
28 | 17 | recni 11273 | . . . . . 6 ⊢ -(π / 2) ∈ ℂ |
29 | 27, 4 | negsubi 11585 | . . . . . . 7 ⊢ (π + -(π / 2)) = (π − (π / 2)) |
30 | pidiv2halves 26524 | . . . . . . . 8 ⊢ ((π / 2) + (π / 2)) = π | |
31 | 27, 4, 4, 30 | subaddrii 11596 | . . . . . . 7 ⊢ (π − (π / 2)) = (π / 2) |
32 | 29, 31 | eqtri 2763 | . . . . . 6 ⊢ (π + -(π / 2)) = (π / 2) |
33 | 4, 27, 28, 32 | subaddrii 11596 | . . . . 5 ⊢ ((π / 2) − π) = -(π / 2) |
34 | 19 | simp2d 1142 | . . . . 5 ⊢ (𝐴 ∈ ℂ → -(π / 2) ≤ (ℜ‘(arcsin‘𝐴))) |
35 | 33, 34 | eqbrtrid 5183 | . . . 4 ⊢ (𝐴 ∈ ℂ → ((π / 2) − π) ≤ (ℜ‘(arcsin‘𝐴))) |
36 | 24, 26, 13, 35 | subled 11864 | . . 3 ⊢ (𝐴 ∈ ℂ → ((π / 2) − (ℜ‘(arcsin‘𝐴))) ≤ π) |
37 | 0re 11261 | . . . 4 ⊢ 0 ∈ ℝ | |
38 | 37, 25 | elicc2i 13450 | . . 3 ⊢ (((π / 2) − (ℜ‘(arcsin‘𝐴))) ∈ (0[,]π) ↔ (((π / 2) − (ℜ‘(arcsin‘𝐴))) ∈ ℝ ∧ 0 ≤ ((π / 2) − (ℜ‘(arcsin‘𝐴))) ∧ ((π / 2) − (ℜ‘(arcsin‘𝐴))) ≤ π)) |
39 | 15, 23, 36, 38 | syl3anbrc 1342 | . 2 ⊢ (𝐴 ∈ ℂ → ((π / 2) − (ℜ‘(arcsin‘𝐴))) ∈ (0[,]π)) |
40 | 12, 39 | eqeltrd 2839 | 1 ⊢ (𝐴 ∈ ℂ → (ℜ‘(arccos‘𝐴)) ∈ (0[,]π)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 206 ∧ w3a 1086 = wceq 1537 ∈ wcel 2106 class class class wbr 5148 ‘cfv 6563 (class class class)co 7431 ℂcc 11151 ℝcr 11152 0cc0 11153 + caddc 11156 ≤ cle 11294 − cmin 11490 -cneg 11491 / cdiv 11918 2c2 12319 [,]cicc 13387 ℜcre 15133 πcpi 16099 arcsincasin 26920 arccoscacos 26921 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-rep 5285 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 ax-inf2 9679 ax-cnex 11209 ax-resscn 11210 ax-1cn 11211 ax-icn 11212 ax-addcl 11213 ax-addrcl 11214 ax-mulcl 11215 ax-mulrcl 11216 ax-mulcom 11217 ax-addass 11218 ax-mulass 11219 ax-distr 11220 ax-i2m1 11221 ax-1ne0 11222 ax-1rid 11223 ax-rnegex 11224 ax-rrecex 11225 ax-cnre 11226 ax-pre-lttri 11227 ax-pre-lttrn 11228 ax-pre-ltadd 11229 ax-pre-mulgt0 11230 ax-pre-sup 11231 ax-addf 11232 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-rmo 3378 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-pss 3983 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-tp 4636 df-op 4638 df-uni 4913 df-int 4952 df-iun 4998 df-iin 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5583 df-eprel 5589 df-po 5597 df-so 5598 df-fr 5641 df-se 5642 df-we 5643 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-pred 6323 df-ord 6389 df-on 6390 df-lim 6391 df-suc 6392 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-isom 6572 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-of 7697 df-om 7888 df-1st 8013 df-2nd 8014 df-supp 8185 df-frecs 8305 df-wrecs 8336 df-recs 8410 df-rdg 8449 df-1o 8505 df-2o 8506 df-er 8744 df-map 8867 df-pm 8868 df-ixp 8937 df-en 8985 df-dom 8986 df-sdom 8987 df-fin 8988 df-fsupp 9400 df-fi 9449 df-sup 9480 df-inf 9481 df-oi 9548 df-card 9977 df-pnf 11295 df-mnf 11296 df-xr 11297 df-ltxr 11298 df-le 11299 df-sub 11492 df-neg 11493 df-div 11919 df-nn 12265 df-2 12327 df-3 12328 df-4 12329 df-5 12330 df-6 12331 df-7 12332 df-8 12333 df-9 12334 df-n0 12525 df-z 12612 df-dec 12732 df-uz 12877 df-q 12989 df-rp 13033 df-xneg 13152 df-xadd 13153 df-xmul 13154 df-ioo 13388 df-ioc 13389 df-ico 13390 df-icc 13391 df-fz 13545 df-fzo 13692 df-fl 13829 df-mod 13907 df-seq 14040 df-exp 14100 df-fac 14310 df-bc 14339 df-hash 14367 df-shft 15103 df-cj 15135 df-re 15136 df-im 15137 df-sqrt 15271 df-abs 15272 df-limsup 15504 df-clim 15521 df-rlim 15522 df-sum 15720 df-ef 16100 df-sin 16102 df-cos 16103 df-pi 16105 df-struct 17181 df-sets 17198 df-slot 17216 df-ndx 17228 df-base 17246 df-ress 17275 df-plusg 17311 df-mulr 17312 df-starv 17313 df-sca 17314 df-vsca 17315 df-ip 17316 df-tset 17317 df-ple 17318 df-ds 17320 df-unif 17321 df-hom 17322 df-cco 17323 df-rest 17469 df-topn 17470 df-0g 17488 df-gsum 17489 df-topgen 17490 df-pt 17491 df-prds 17494 df-xrs 17549 df-qtop 17554 df-imas 17555 df-xps 17557 df-mre 17631 df-mrc 17632 df-acs 17634 df-mgm 18666 df-sgrp 18745 df-mnd 18761 df-submnd 18810 df-mulg 19099 df-cntz 19348 df-cmn 19815 df-psmet 21374 df-xmet 21375 df-met 21376 df-bl 21377 df-mopn 21378 df-fbas 21379 df-fg 21380 df-cnfld 21383 df-top 22916 df-topon 22933 df-topsp 22955 df-bases 22969 df-cld 23043 df-ntr 23044 df-cls 23045 df-nei 23122 df-lp 23160 df-perf 23161 df-cn 23251 df-cnp 23252 df-haus 23339 df-tx 23586 df-hmeo 23779 df-fil 23870 df-fm 23962 df-flim 23963 df-flf 23964 df-xms 24346 df-ms 24347 df-tms 24348 df-cncf 24918 df-limc 25916 df-dv 25917 df-log 26613 df-asin 26923 df-acos 26924 |
This theorem is referenced by: (None) |
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