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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 15834 | . . . . . 6 ⊢ cos:ℂ⟶ℂ | |
2 | ffn 6600 | . . . . . 6 ⊢ (cos:ℂ⟶ℂ → cos Fn ℂ) | |
3 | 1, 2 | ax-mp 5 | . . . . 5 ⊢ cos Fn ℂ |
4 | 0re 10977 | . . . . . . 7 ⊢ 0 ∈ ℝ | |
5 | pire 25615 | . . . . . . 7 ⊢ π ∈ ℝ | |
6 | iccssre 13161 | . . . . . . 7 ⊢ ((0 ∈ ℝ ∧ π ∈ ℝ) → (0[,]π) ⊆ ℝ) | |
7 | 4, 5, 6 | mp2an 689 | . . . . . 6 ⊢ (0[,]π) ⊆ ℝ |
8 | ax-resscn 10928 | . . . . . 6 ⊢ ℝ ⊆ ℂ | |
9 | 7, 8 | sstri 3930 | . . . . 5 ⊢ (0[,]π) ⊆ ℂ |
10 | fnssres 6555 | . . . . 5 ⊢ ((cos Fn ℂ ∧ (0[,]π) ⊆ ℂ) → (cos ↾ (0[,]π)) Fn (0[,]π)) | |
11 | 3, 9, 10 | mp2an 689 | . . . 4 ⊢ (cos ↾ (0[,]π)) Fn (0[,]π) |
12 | fvres 6793 | . . . . . 6 ⊢ (𝑥 ∈ (0[,]π) → ((cos ↾ (0[,]π))‘𝑥) = (cos‘𝑥)) | |
13 | 7 | sseli 3917 | . . . . . . 7 ⊢ (𝑥 ∈ (0[,]π) → 𝑥 ∈ ℝ) |
14 | cosbnd2 15892 | . . . . . . 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 3074 | . . . 4 ⊢ ∀𝑥 ∈ (0[,]π)((cos ↾ (0[,]π))‘𝑥) ∈ (-1[,]1) |
18 | ffnfv 6992 | . . . 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 708 | . . 3 ⊢ (cos ↾ (0[,]π)):(0[,]π)⟶(-1[,]1) |
20 | fvres 6793 | . . . . . 6 ⊢ (𝑦 ∈ (0[,]π) → ((cos ↾ (0[,]π))‘𝑦) = (cos‘𝑦)) | |
21 | 12, 20 | eqeqan12d 2752 | . . . . 5 ⊢ ((𝑥 ∈ (0[,]π) ∧ 𝑦 ∈ (0[,]π)) → (((cos ↾ (0[,]π))‘𝑥) = ((cos ↾ (0[,]π))‘𝑦) ↔ (cos‘𝑥) = (cos‘𝑦))) |
22 | cos11 25689 | . . . . . 6 ⊢ ((𝑥 ∈ (0[,]π) ∧ 𝑦 ∈ (0[,]π)) → (𝑥 = 𝑦 ↔ (cos‘𝑥) = (cos‘𝑦))) | |
23 | 22 | biimprd 247 | . . . . 5 ⊢ ((𝑥 ∈ (0[,]π) ∧ 𝑦 ∈ (0[,]π)) → ((cos‘𝑥) = (cos‘𝑦) → 𝑥 = 𝑦)) |
24 | 21, 23 | sylbid 239 | . . . 4 ⊢ ((𝑥 ∈ (0[,]π) ∧ 𝑦 ∈ (0[,]π)) → (((cos ↾ (0[,]π))‘𝑥) = ((cos ↾ (0[,]π))‘𝑦) → 𝑥 = 𝑦)) |
25 | 24 | rgen2 3120 | . . 3 ⊢ ∀𝑥 ∈ (0[,]π)∀𝑦 ∈ (0[,]π)(((cos ↾ (0[,]π))‘𝑥) = ((cos ↾ (0[,]π))‘𝑦) → 𝑥 = 𝑦) |
26 | dff13 7128 | . . 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 708 | . 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 12088 | . . . . . . . 8 ⊢ -1 ∈ ℝ | |
31 | 1re 10975 | . . . . . . . 8 ⊢ 1 ∈ ℝ | |
32 | 30, 31 | elicc2i 13145 | . . . . . . 7 ⊢ (𝑥 ∈ (-1[,]1) ↔ (𝑥 ∈ ℝ ∧ -1 ≤ 𝑥 ∧ 𝑥 ≤ 1)) |
33 | 32 | simp1bi 1144 | . . . . . 6 ⊢ (𝑥 ∈ (-1[,]1) → 𝑥 ∈ ℝ) |
34 | pipos 25617 | . . . . . . 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 25604 | . . . . . . 7 ⊢ cos ∈ (ℂ–cn→ℂ) | |
38 | 37 | a1i 11 | . . . . . 6 ⊢ (𝑥 ∈ (-1[,]1) → cos ∈ (ℂ–cn→ℂ)) |
39 | 7 | sseli 3917 | . . . . . . . 8 ⊢ (𝑧 ∈ (0[,]π) → 𝑧 ∈ ℝ) |
40 | 39 | recoscld 15853 | . . . . . . 7 ⊢ (𝑧 ∈ (0[,]π) → (cos‘𝑧) ∈ ℝ) |
41 | 40 | adantl 482 | . . . . . 6 ⊢ ((𝑥 ∈ (-1[,]1) ∧ 𝑧 ∈ (0[,]π)) → (cos‘𝑧) ∈ ℝ) |
42 | cospi 25629 | . . . . . . . 8 ⊢ (cos‘π) = -1 | |
43 | 32 | simp2bi 1145 | . . . . . . . 8 ⊢ (𝑥 ∈ (-1[,]1) → -1 ≤ 𝑥) |
44 | 42, 43 | eqbrtrid 5109 | . . . . . . 7 ⊢ (𝑥 ∈ (-1[,]1) → (cos‘π) ≤ 𝑥) |
45 | 32 | simp3bi 1146 | . . . . . . . 8 ⊢ (𝑥 ∈ (-1[,]1) → 𝑥 ≤ 1) |
46 | cos0 15859 | . . . . . . . 8 ⊢ (cos‘0) = 1 | |
47 | 45, 46 | breqtrrdi 5116 | . . . . . . 7 ⊢ (𝑥 ∈ (-1[,]1) → 𝑥 ≤ (cos‘0)) |
48 | 44, 47 | jca 512 | . . . . . 6 ⊢ (𝑥 ∈ (-1[,]1) → ((cos‘π) ≤ 𝑥 ∧ 𝑥 ≤ (cos‘0))) |
49 | 28, 29, 33, 35, 36, 38, 41, 48 | ivthle2 24621 | . . . . 5 ⊢ (𝑥 ∈ (-1[,]1) → ∃𝑦 ∈ (0[,]π)(cos‘𝑦) = 𝑥) |
50 | eqcom 2745 | . . . . . . 7 ⊢ (𝑥 = ((cos ↾ (0[,]π))‘𝑦) ↔ ((cos ↾ (0[,]π))‘𝑦) = 𝑥) | |
51 | 20 | eqeq1d 2740 | . . . . . . 7 ⊢ (𝑦 ∈ (0[,]π) → (((cos ↾ (0[,]π))‘𝑦) = 𝑥 ↔ (cos‘𝑦) = 𝑥)) |
52 | 50, 51 | bitrid 282 | . . . . . 6 ⊢ (𝑦 ∈ (0[,]π) → (𝑥 = ((cos ↾ (0[,]π))‘𝑦) ↔ (cos‘𝑦) = 𝑥)) |
53 | 52 | rexbiia 3180 | . . . . 5 ⊢ (∃𝑦 ∈ (0[,]π)𝑥 = ((cos ↾ (0[,]π))‘𝑦) ↔ ∃𝑦 ∈ (0[,]π)(cos‘𝑦) = 𝑥) |
54 | 49, 53 | sylibr 233 | . . . 4 ⊢ (𝑥 ∈ (-1[,]1) → ∃𝑦 ∈ (0[,]π)𝑥 = ((cos ↾ (0[,]π))‘𝑦)) |
55 | 54 | rgen 3074 | . . 3 ⊢ ∀𝑥 ∈ (-1[,]1)∃𝑦 ∈ (0[,]π)𝑥 = ((cos ↾ (0[,]π))‘𝑦) |
56 | dffo3 6978 | . . 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 708 | . 2 ⊢ (cos ↾ (0[,]π)):(0[,]π)–onto→(-1[,]1) |
58 | df-f1o 6440 | . 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 708 | 1 ⊢ (cos ↾ (0[,]π)):(0[,]π)–1-1-onto→(-1[,]1) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 = wceq 1539 ∈ wcel 2106 ∀wral 3064 ∃wrex 3065 ⊆ wss 3887 class class class wbr 5074 ↾ cres 5591 Fn wfn 6428 ⟶wf 6429 –1-1→wf1 6430 –onto→wfo 6431 –1-1-onto→wf1o 6432 ‘cfv 6433 (class class class)co 7275 ℂcc 10869 ℝcr 10870 0cc0 10871 1c1 10872 < clt 11009 ≤ cle 11010 -cneg 11206 [,]cicc 13082 cosccos 15774 πcpi 15776 –cn→ccncf 24039 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2709 ax-rep 5209 ax-sep 5223 ax-nul 5230 ax-pow 5288 ax-pr 5352 ax-un 7588 ax-inf2 9399 ax-cnex 10927 ax-resscn 10928 ax-1cn 10929 ax-icn 10930 ax-addcl 10931 ax-addrcl 10932 ax-mulcl 10933 ax-mulrcl 10934 ax-mulcom 10935 ax-addass 10936 ax-mulass 10937 ax-distr 10938 ax-i2m1 10939 ax-1ne0 10940 ax-1rid 10941 ax-rnegex 10942 ax-rrecex 10943 ax-cnre 10944 ax-pre-lttri 10945 ax-pre-lttrn 10946 ax-pre-ltadd 10947 ax-pre-mulgt0 10948 ax-pre-sup 10949 ax-addf 10950 ax-mulf 10951 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2068 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2889 df-ne 2944 df-nel 3050 df-ral 3069 df-rex 3070 df-rmo 3071 df-reu 3072 df-rab 3073 df-v 3434 df-sbc 3717 df-csb 3833 df-dif 3890 df-un 3892 df-in 3894 df-ss 3904 df-pss 3906 df-nul 4257 df-if 4460 df-pw 4535 df-sn 4562 df-pr 4564 df-tp 4566 df-op 4568 df-uni 4840 df-int 4880 df-iun 4926 df-iin 4927 df-br 5075 df-opab 5137 df-mpt 5158 df-tr 5192 df-id 5489 df-eprel 5495 df-po 5503 df-so 5504 df-fr 5544 df-se 5545 df-we 5546 df-xp 5595 df-rel 5596 df-cnv 5597 df-co 5598 df-dm 5599 df-rn 5600 df-res 5601 df-ima 5602 df-pred 6202 df-ord 6269 df-on 6270 df-lim 6271 df-suc 6272 df-iota 6391 df-fun 6435 df-fn 6436 df-f 6437 df-f1 6438 df-fo 6439 df-f1o 6440 df-fv 6441 df-isom 6442 df-riota 7232 df-ov 7278 df-oprab 7279 df-mpo 7280 df-of 7533 df-om 7713 df-1st 7831 df-2nd 7832 df-supp 7978 df-frecs 8097 df-wrecs 8128 df-recs 8202 df-rdg 8241 df-1o 8297 df-2o 8298 df-er 8498 df-map 8617 df-pm 8618 df-ixp 8686 df-en 8734 df-dom 8735 df-sdom 8736 df-fin 8737 df-fsupp 9129 df-fi 9170 df-sup 9201 df-inf 9202 df-oi 9269 df-card 9697 df-pnf 11011 df-mnf 11012 df-xr 11013 df-ltxr 11014 df-le 11015 df-sub 11207 df-neg 11208 df-div 11633 df-nn 11974 df-2 12036 df-3 12037 df-4 12038 df-5 12039 df-6 12040 df-7 12041 df-8 12042 df-9 12043 df-n0 12234 df-z 12320 df-dec 12438 df-uz 12583 df-q 12689 df-rp 12731 df-xneg 12848 df-xadd 12849 df-xmul 12850 df-ioo 13083 df-ioc 13084 df-ico 13085 df-icc 13086 df-fz 13240 df-fzo 13383 df-fl 13512 df-seq 13722 df-exp 13783 df-fac 13988 df-bc 14017 df-hash 14045 df-shft 14778 df-cj 14810 df-re 14811 df-im 14812 df-sqrt 14946 df-abs 14947 df-limsup 15180 df-clim 15197 df-rlim 15198 df-sum 15398 df-ef 15777 df-sin 15779 df-cos 15780 df-pi 15782 df-struct 16848 df-sets 16865 df-slot 16883 df-ndx 16895 df-base 16913 df-ress 16942 df-plusg 16975 df-mulr 16976 df-starv 16977 df-sca 16978 df-vsca 16979 df-ip 16980 df-tset 16981 df-ple 16982 df-ds 16984 df-unif 16985 df-hom 16986 df-cco 16987 df-rest 17133 df-topn 17134 df-0g 17152 df-gsum 17153 df-topgen 17154 df-pt 17155 df-prds 17158 df-xrs 17213 df-qtop 17218 df-imas 17219 df-xps 17221 df-mre 17295 df-mrc 17296 df-acs 17298 df-mgm 18326 df-sgrp 18375 df-mnd 18386 df-submnd 18431 df-mulg 18701 df-cntz 18923 df-cmn 19388 df-psmet 20589 df-xmet 20590 df-met 20591 df-bl 20592 df-mopn 20593 df-fbas 20594 df-fg 20595 df-cnfld 20598 df-top 22043 df-topon 22060 df-topsp 22082 df-bases 22096 df-cld 22170 df-ntr 22171 df-cls 22172 df-nei 22249 df-lp 22287 df-perf 22288 df-cn 22378 df-cnp 22379 df-haus 22466 df-tx 22713 df-hmeo 22906 df-fil 22997 df-fm 23089 df-flim 23090 df-flf 23091 df-xms 23473 df-ms 23474 df-tms 23475 df-cncf 24041 df-limc 25030 df-dv 25031 |
This theorem is referenced by: resinf1o 25692 |
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