![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > recosf1o | Structured version Visualization version GIF version |
Description: The cosine function is a bijection when restricted to its principal domain. (Contributed by Mario Carneiro, 12-May-2014.) |
Ref | Expression |
---|---|
recosf1o | ⊢ (cos ↾ (0[,]π)):(0[,]π)–1-1-onto→(-1[,]1) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cosf 15054 | . . . . . 6 ⊢ cos:ℂ⟶ℂ | |
2 | ffn 6206 | . . . . . 6 ⊢ (cos:ℂ⟶ℂ → cos Fn ℂ) | |
3 | 1, 2 | ax-mp 5 | . . . . 5 ⊢ cos Fn ℂ |
4 | 0re 10232 | . . . . . . 7 ⊢ 0 ∈ ℝ | |
5 | pire 24409 | . . . . . . 7 ⊢ π ∈ ℝ | |
6 | iccssre 12448 | . . . . . . 7 ⊢ ((0 ∈ ℝ ∧ π ∈ ℝ) → (0[,]π) ⊆ ℝ) | |
7 | 4, 5, 6 | mp2an 710 | . . . . . 6 ⊢ (0[,]π) ⊆ ℝ |
8 | ax-resscn 10185 | . . . . . 6 ⊢ ℝ ⊆ ℂ | |
9 | 7, 8 | sstri 3753 | . . . . 5 ⊢ (0[,]π) ⊆ ℂ |
10 | fnssres 6165 | . . . . 5 ⊢ ((cos Fn ℂ ∧ (0[,]π) ⊆ ℂ) → (cos ↾ (0[,]π)) Fn (0[,]π)) | |
11 | 3, 9, 10 | mp2an 710 | . . . 4 ⊢ (cos ↾ (0[,]π)) Fn (0[,]π) |
12 | fvres 6368 | . . . . . 6 ⊢ (𝑥 ∈ (0[,]π) → ((cos ↾ (0[,]π))‘𝑥) = (cos‘𝑥)) | |
13 | 7 | sseli 3740 | . . . . . . 7 ⊢ (𝑥 ∈ (0[,]π) → 𝑥 ∈ ℝ) |
14 | cosbnd2 15112 | . . . . . . 7 ⊢ (𝑥 ∈ ℝ → (cos‘𝑥) ∈ (-1[,]1)) | |
15 | 13, 14 | syl 17 | . . . . . 6 ⊢ (𝑥 ∈ (0[,]π) → (cos‘𝑥) ∈ (-1[,]1)) |
16 | 12, 15 | eqeltrd 2839 | . . . . 5 ⊢ (𝑥 ∈ (0[,]π) → ((cos ↾ (0[,]π))‘𝑥) ∈ (-1[,]1)) |
17 | 16 | rgen 3060 | . . . 4 ⊢ ∀𝑥 ∈ (0[,]π)((cos ↾ (0[,]π))‘𝑥) ∈ (-1[,]1) |
18 | ffnfv 6551 | . . . 4 ⊢ ((cos ↾ (0[,]π)):(0[,]π)⟶(-1[,]1) ↔ ((cos ↾ (0[,]π)) Fn (0[,]π) ∧ ∀𝑥 ∈ (0[,]π)((cos ↾ (0[,]π))‘𝑥) ∈ (-1[,]1))) | |
19 | 11, 17, 18 | mpbir2an 993 | . . 3 ⊢ (cos ↾ (0[,]π)):(0[,]π)⟶(-1[,]1) |
20 | fvres 6368 | . . . . . 6 ⊢ (𝑦 ∈ (0[,]π) → ((cos ↾ (0[,]π))‘𝑦) = (cos‘𝑦)) | |
21 | 12, 20 | eqeqan12d 2776 | . . . . 5 ⊢ ((𝑥 ∈ (0[,]π) ∧ 𝑦 ∈ (0[,]π)) → (((cos ↾ (0[,]π))‘𝑥) = ((cos ↾ (0[,]π))‘𝑦) ↔ (cos‘𝑥) = (cos‘𝑦))) |
22 | cos11 24478 | . . . . . 6 ⊢ ((𝑥 ∈ (0[,]π) ∧ 𝑦 ∈ (0[,]π)) → (𝑥 = 𝑦 ↔ (cos‘𝑥) = (cos‘𝑦))) | |
23 | 22 | biimprd 238 | . . . . 5 ⊢ ((𝑥 ∈ (0[,]π) ∧ 𝑦 ∈ (0[,]π)) → ((cos‘𝑥) = (cos‘𝑦) → 𝑥 = 𝑦)) |
24 | 21, 23 | sylbid 230 | . . . 4 ⊢ ((𝑥 ∈ (0[,]π) ∧ 𝑦 ∈ (0[,]π)) → (((cos ↾ (0[,]π))‘𝑥) = ((cos ↾ (0[,]π))‘𝑦) → 𝑥 = 𝑦)) |
25 | 24 | rgen2a 3115 | . . 3 ⊢ ∀𝑥 ∈ (0[,]π)∀𝑦 ∈ (0[,]π)(((cos ↾ (0[,]π))‘𝑥) = ((cos ↾ (0[,]π))‘𝑦) → 𝑥 = 𝑦) |
26 | dff13 6675 | . . 3 ⊢ ((cos ↾ (0[,]π)):(0[,]π)–1-1→(-1[,]1) ↔ ((cos ↾ (0[,]π)):(0[,]π)⟶(-1[,]1) ∧ ∀𝑥 ∈ (0[,]π)∀𝑦 ∈ (0[,]π)(((cos ↾ (0[,]π))‘𝑥) = ((cos ↾ (0[,]π))‘𝑦) → 𝑥 = 𝑦))) | |
27 | 19, 25, 26 | mpbir2an 993 | . 2 ⊢ (cos ↾ (0[,]π)):(0[,]π)–1-1→(-1[,]1) |
28 | 4 | a1i 11 | . . . . . 6 ⊢ (𝑥 ∈ (-1[,]1) → 0 ∈ ℝ) |
29 | 5 | a1i 11 | . . . . . 6 ⊢ (𝑥 ∈ (-1[,]1) → π ∈ ℝ) |
30 | neg1rr 11317 | . . . . . . . 8 ⊢ -1 ∈ ℝ | |
31 | 1re 10231 | . . . . . . . 8 ⊢ 1 ∈ ℝ | |
32 | 30, 31 | elicc2i 12432 | . . . . . . 7 ⊢ (𝑥 ∈ (-1[,]1) ↔ (𝑥 ∈ ℝ ∧ -1 ≤ 𝑥 ∧ 𝑥 ≤ 1)) |
33 | 32 | simp1bi 1140 | . . . . . 6 ⊢ (𝑥 ∈ (-1[,]1) → 𝑥 ∈ ℝ) |
34 | pipos 24411 | . . . . . . 7 ⊢ 0 < π | |
35 | 34 | a1i 11 | . . . . . 6 ⊢ (𝑥 ∈ (-1[,]1) → 0 < π) |
36 | 9 | a1i 11 | . . . . . 6 ⊢ (𝑥 ∈ (-1[,]1) → (0[,]π) ⊆ ℂ) |
37 | coscn 24398 | . . . . . . 7 ⊢ cos ∈ (ℂ–cn→ℂ) | |
38 | 37 | a1i 11 | . . . . . 6 ⊢ (𝑥 ∈ (-1[,]1) → cos ∈ (ℂ–cn→ℂ)) |
39 | 7 | sseli 3740 | . . . . . . . 8 ⊢ (𝑧 ∈ (0[,]π) → 𝑧 ∈ ℝ) |
40 | 39 | recoscld 15073 | . . . . . . 7 ⊢ (𝑧 ∈ (0[,]π) → (cos‘𝑧) ∈ ℝ) |
41 | 40 | adantl 473 | . . . . . 6 ⊢ ((𝑥 ∈ (-1[,]1) ∧ 𝑧 ∈ (0[,]π)) → (cos‘𝑧) ∈ ℝ) |
42 | cospi 24423 | . . . . . . . 8 ⊢ (cos‘π) = -1 | |
43 | 32 | simp2bi 1141 | . . . . . . . 8 ⊢ (𝑥 ∈ (-1[,]1) → -1 ≤ 𝑥) |
44 | 42, 43 | syl5eqbr 4839 | . . . . . . 7 ⊢ (𝑥 ∈ (-1[,]1) → (cos‘π) ≤ 𝑥) |
45 | 32 | simp3bi 1142 | . . . . . . . 8 ⊢ (𝑥 ∈ (-1[,]1) → 𝑥 ≤ 1) |
46 | cos0 15079 | . . . . . . . 8 ⊢ (cos‘0) = 1 | |
47 | 45, 46 | syl6breqr 4846 | . . . . . . 7 ⊢ (𝑥 ∈ (-1[,]1) → 𝑥 ≤ (cos‘0)) |
48 | 44, 47 | jca 555 | . . . . . 6 ⊢ (𝑥 ∈ (-1[,]1) → ((cos‘π) ≤ 𝑥 ∧ 𝑥 ≤ (cos‘0))) |
49 | 28, 29, 33, 35, 36, 38, 41, 48 | ivthle2 23426 | . . . . 5 ⊢ (𝑥 ∈ (-1[,]1) → ∃𝑦 ∈ (0[,]π)(cos‘𝑦) = 𝑥) |
50 | eqcom 2767 | . . . . . . 7 ⊢ (𝑥 = ((cos ↾ (0[,]π))‘𝑦) ↔ ((cos ↾ (0[,]π))‘𝑦) = 𝑥) | |
51 | 20 | eqeq1d 2762 | . . . . . . 7 ⊢ (𝑦 ∈ (0[,]π) → (((cos ↾ (0[,]π))‘𝑦) = 𝑥 ↔ (cos‘𝑦) = 𝑥)) |
52 | 50, 51 | syl5bb 272 | . . . . . 6 ⊢ (𝑦 ∈ (0[,]π) → (𝑥 = ((cos ↾ (0[,]π))‘𝑦) ↔ (cos‘𝑦) = 𝑥)) |
53 | 52 | rexbiia 3178 | . . . . 5 ⊢ (∃𝑦 ∈ (0[,]π)𝑥 = ((cos ↾ (0[,]π))‘𝑦) ↔ ∃𝑦 ∈ (0[,]π)(cos‘𝑦) = 𝑥) |
54 | 49, 53 | sylibr 224 | . . . 4 ⊢ (𝑥 ∈ (-1[,]1) → ∃𝑦 ∈ (0[,]π)𝑥 = ((cos ↾ (0[,]π))‘𝑦)) |
55 | 54 | rgen 3060 | . . 3 ⊢ ∀𝑥 ∈ (-1[,]1)∃𝑦 ∈ (0[,]π)𝑥 = ((cos ↾ (0[,]π))‘𝑦) |
56 | dffo3 6537 | . . 3 ⊢ ((cos ↾ (0[,]π)):(0[,]π)–onto→(-1[,]1) ↔ ((cos ↾ (0[,]π)):(0[,]π)⟶(-1[,]1) ∧ ∀𝑥 ∈ (-1[,]1)∃𝑦 ∈ (0[,]π)𝑥 = ((cos ↾ (0[,]π))‘𝑦))) | |
57 | 19, 55, 56 | mpbir2an 993 | . 2 ⊢ (cos ↾ (0[,]π)):(0[,]π)–onto→(-1[,]1) |
58 | df-f1o 6056 | . 2 ⊢ ((cos ↾ (0[,]π)):(0[,]π)–1-1-onto→(-1[,]1) ↔ ((cos ↾ (0[,]π)):(0[,]π)–1-1→(-1[,]1) ∧ (cos ↾ (0[,]π)):(0[,]π)–onto→(-1[,]1))) | |
59 | 27, 57, 58 | mpbir2an 993 | 1 ⊢ (cos ↾ (0[,]π)):(0[,]π)–1-1-onto→(-1[,]1) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 383 = wceq 1632 ∈ wcel 2139 ∀wral 3050 ∃wrex 3051 ⊆ wss 3715 class class class wbr 4804 ↾ cres 5268 Fn wfn 6044 ⟶wf 6045 –1-1→wf1 6046 –onto→wfo 6047 –1-1-onto→wf1o 6048 ‘cfv 6049 (class class class)co 6813 ℂcc 10126 ℝcr 10127 0cc0 10128 1c1 10129 < clt 10266 ≤ cle 10267 -cneg 10459 [,]cicc 12371 cosccos 14994 πcpi 14996 –cn→ccncf 22880 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1871 ax-4 1886 ax-5 1988 ax-6 2054 ax-7 2090 ax-8 2141 ax-9 2148 ax-10 2168 ax-11 2183 ax-12 2196 ax-13 2391 ax-ext 2740 ax-rep 4923 ax-sep 4933 ax-nul 4941 ax-pow 4992 ax-pr 5055 ax-un 7114 ax-inf2 8711 ax-cnex 10184 ax-resscn 10185 ax-1cn 10186 ax-icn 10187 ax-addcl 10188 ax-addrcl 10189 ax-mulcl 10190 ax-mulrcl 10191 ax-mulcom 10192 ax-addass 10193 ax-mulass 10194 ax-distr 10195 ax-i2m1 10196 ax-1ne0 10197 ax-1rid 10198 ax-rnegex 10199 ax-rrecex 10200 ax-cnre 10201 ax-pre-lttri 10202 ax-pre-lttrn 10203 ax-pre-ltadd 10204 ax-pre-mulgt0 10205 ax-pre-sup 10206 ax-addf 10207 ax-mulf 10208 |
This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-3or 1073 df-3an 1074 df-tru 1635 df-fal 1638 df-ex 1854 df-nf 1859 df-sb 2047 df-eu 2611 df-mo 2612 df-clab 2747 df-cleq 2753 df-clel 2756 df-nfc 2891 df-ne 2933 df-nel 3036 df-ral 3055 df-rex 3056 df-reu 3057 df-rmo 3058 df-rab 3059 df-v 3342 df-sbc 3577 df-csb 3675 df-dif 3718 df-un 3720 df-in 3722 df-ss 3729 df-pss 3731 df-nul 4059 df-if 4231 df-pw 4304 df-sn 4322 df-pr 4324 df-tp 4326 df-op 4328 df-uni 4589 df-int 4628 df-iun 4674 df-iin 4675 df-br 4805 df-opab 4865 df-mpt 4882 df-tr 4905 df-id 5174 df-eprel 5179 df-po 5187 df-so 5188 df-fr 5225 df-se 5226 df-we 5227 df-xp 5272 df-rel 5273 df-cnv 5274 df-co 5275 df-dm 5276 df-rn 5277 df-res 5278 df-ima 5279 df-pred 5841 df-ord 5887 df-on 5888 df-lim 5889 df-suc 5890 df-iota 6012 df-fun 6051 df-fn 6052 df-f 6053 df-f1 6054 df-fo 6055 df-f1o 6056 df-fv 6057 df-isom 6058 df-riota 6774 df-ov 6816 df-oprab 6817 df-mpt2 6818 df-of 7062 df-om 7231 df-1st 7333 df-2nd 7334 df-supp 7464 df-wrecs 7576 df-recs 7637 df-rdg 7675 df-1o 7729 df-2o 7730 df-oadd 7733 df-er 7911 df-map 8025 df-pm 8026 df-ixp 8075 df-en 8122 df-dom 8123 df-sdom 8124 df-fin 8125 df-fsupp 8441 df-fi 8482 df-sup 8513 df-inf 8514 df-oi 8580 df-card 8955 df-cda 9182 df-pnf 10268 df-mnf 10269 df-xr 10270 df-ltxr 10271 df-le 10272 df-sub 10460 df-neg 10461 df-div 10877 df-nn 11213 df-2 11271 df-3 11272 df-4 11273 df-5 11274 df-6 11275 df-7 11276 df-8 11277 df-9 11278 df-n0 11485 df-z 11570 df-dec 11686 df-uz 11880 df-q 11982 df-rp 12026 df-xneg 12139 df-xadd 12140 df-xmul 12141 df-ioo 12372 df-ioc 12373 df-ico 12374 df-icc 12375 df-fz 12520 df-fzo 12660 df-fl 12787 df-seq 12996 df-exp 13055 df-fac 13255 df-bc 13284 df-hash 13312 df-shft 14006 df-cj 14038 df-re 14039 df-im 14040 df-sqrt 14174 df-abs 14175 df-limsup 14401 df-clim 14418 df-rlim 14419 df-sum 14616 df-ef 14997 df-sin 14999 df-cos 15000 df-pi 15002 df-struct 16061 df-ndx 16062 df-slot 16063 df-base 16065 df-sets 16066 df-ress 16067 df-plusg 16156 df-mulr 16157 df-starv 16158 df-sca 16159 df-vsca 16160 df-ip 16161 df-tset 16162 df-ple 16163 df-ds 16166 df-unif 16167 df-hom 16168 df-cco 16169 df-rest 16285 df-topn 16286 df-0g 16304 df-gsum 16305 df-topgen 16306 df-pt 16307 df-prds 16310 df-xrs 16364 df-qtop 16369 df-imas 16370 df-xps 16372 df-mre 16448 df-mrc 16449 df-acs 16451 df-mgm 17443 df-sgrp 17485 df-mnd 17496 df-submnd 17537 df-mulg 17742 df-cntz 17950 df-cmn 18395 df-psmet 19940 df-xmet 19941 df-met 19942 df-bl 19943 df-mopn 19944 df-fbas 19945 df-fg 19946 df-cnfld 19949 df-top 20901 df-topon 20918 df-topsp 20939 df-bases 20952 df-cld 21025 df-ntr 21026 df-cls 21027 df-nei 21104 df-lp 21142 df-perf 21143 df-cn 21233 df-cnp 21234 df-haus 21321 df-tx 21567 df-hmeo 21760 df-fil 21851 df-fm 21943 df-flim 21944 df-flf 21945 df-xms 22326 df-ms 22327 df-tms 22328 df-cncf 22882 df-limc 23829 df-dv 23830 |
This theorem is referenced by: resinf1o 24481 |
Copyright terms: Public domain | W3C validator |