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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 14694 | . . . . . 6 ⊢ cos:ℂ⟶ℂ | |
2 | ffn 5958 | . . . . . 6 ⊢ (cos:ℂ⟶ℂ → cos Fn ℂ) | |
3 | 1, 2 | ax-mp 5 | . . . . 5 ⊢ cos Fn ℂ |
4 | 0re 9919 | . . . . . . 7 ⊢ 0 ∈ ℝ | |
5 | pire 24014 | . . . . . . 7 ⊢ π ∈ ℝ | |
6 | iccssre 12126 | . . . . . . 7 ⊢ ((0 ∈ ℝ ∧ π ∈ ℝ) → (0[,]π) ⊆ ℝ) | |
7 | 4, 5, 6 | mp2an 704 | . . . . . 6 ⊢ (0[,]π) ⊆ ℝ |
8 | ax-resscn 9872 | . . . . . 6 ⊢ ℝ ⊆ ℂ | |
9 | 7, 8 | sstri 3577 | . . . . 5 ⊢ (0[,]π) ⊆ ℂ |
10 | fnssres 5918 | . . . . 5 ⊢ ((cos Fn ℂ ∧ (0[,]π) ⊆ ℂ) → (cos ↾ (0[,]π)) Fn (0[,]π)) | |
11 | 3, 9, 10 | mp2an 704 | . . . 4 ⊢ (cos ↾ (0[,]π)) Fn (0[,]π) |
12 | fvres 6117 | . . . . . 6 ⊢ (𝑥 ∈ (0[,]π) → ((cos ↾ (0[,]π))‘𝑥) = (cos‘𝑥)) | |
13 | 7 | sseli 3564 | . . . . . . 7 ⊢ (𝑥 ∈ (0[,]π) → 𝑥 ∈ ℝ) |
14 | cosbnd2 14752 | . . . . . . 7 ⊢ (𝑥 ∈ ℝ → (cos‘𝑥) ∈ (-1[,]1)) | |
15 | 13, 14 | syl 17 | . . . . . 6 ⊢ (𝑥 ∈ (0[,]π) → (cos‘𝑥) ∈ (-1[,]1)) |
16 | 12, 15 | eqeltrd 2688 | . . . . 5 ⊢ (𝑥 ∈ (0[,]π) → ((cos ↾ (0[,]π))‘𝑥) ∈ (-1[,]1)) |
17 | 16 | rgen 2906 | . . . 4 ⊢ ∀𝑥 ∈ (0[,]π)((cos ↾ (0[,]π))‘𝑥) ∈ (-1[,]1) |
18 | ffnfv 6295 | . . . 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 957 | . . 3 ⊢ (cos ↾ (0[,]π)):(0[,]π)⟶(-1[,]1) |
20 | fvres 6117 | . . . . . 6 ⊢ (𝑦 ∈ (0[,]π) → ((cos ↾ (0[,]π))‘𝑦) = (cos‘𝑦)) | |
21 | 12, 20 | eqeqan12d 2626 | . . . . 5 ⊢ ((𝑥 ∈ (0[,]π) ∧ 𝑦 ∈ (0[,]π)) → (((cos ↾ (0[,]π))‘𝑥) = ((cos ↾ (0[,]π))‘𝑦) ↔ (cos‘𝑥) = (cos‘𝑦))) |
22 | cos11 24083 | . . . . . 6 ⊢ ((𝑥 ∈ (0[,]π) ∧ 𝑦 ∈ (0[,]π)) → (𝑥 = 𝑦 ↔ (cos‘𝑥) = (cos‘𝑦))) | |
23 | 22 | biimprd 237 | . . . . 5 ⊢ ((𝑥 ∈ (0[,]π) ∧ 𝑦 ∈ (0[,]π)) → ((cos‘𝑥) = (cos‘𝑦) → 𝑥 = 𝑦)) |
24 | 21, 23 | sylbid 229 | . . . 4 ⊢ ((𝑥 ∈ (0[,]π) ∧ 𝑦 ∈ (0[,]π)) → (((cos ↾ (0[,]π))‘𝑥) = ((cos ↾ (0[,]π))‘𝑦) → 𝑥 = 𝑦)) |
25 | 24 | rgen2a 2960 | . . 3 ⊢ ∀𝑥 ∈ (0[,]π)∀𝑦 ∈ (0[,]π)(((cos ↾ (0[,]π))‘𝑥) = ((cos ↾ (0[,]π))‘𝑦) → 𝑥 = 𝑦) |
26 | dff13 6416 | . . 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 957 | . 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 11002 | . . . . . . . 8 ⊢ -1 ∈ ℝ | |
31 | 1re 9918 | . . . . . . . 8 ⊢ 1 ∈ ℝ | |
32 | 30, 31 | elicc2i 12110 | . . . . . . 7 ⊢ (𝑥 ∈ (-1[,]1) ↔ (𝑥 ∈ ℝ ∧ -1 ≤ 𝑥 ∧ 𝑥 ≤ 1)) |
33 | 32 | simp1bi 1069 | . . . . . 6 ⊢ (𝑥 ∈ (-1[,]1) → 𝑥 ∈ ℝ) |
34 | pipos 24016 | . . . . . . 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 24003 | . . . . . . 7 ⊢ cos ∈ (ℂ–cn→ℂ) | |
38 | 37 | a1i 11 | . . . . . 6 ⊢ (𝑥 ∈ (-1[,]1) → cos ∈ (ℂ–cn→ℂ)) |
39 | 7 | sseli 3564 | . . . . . . . 8 ⊢ (𝑧 ∈ (0[,]π) → 𝑧 ∈ ℝ) |
40 | 39 | recoscld 14713 | . . . . . . 7 ⊢ (𝑧 ∈ (0[,]π) → (cos‘𝑧) ∈ ℝ) |
41 | 40 | adantl 481 | . . . . . 6 ⊢ ((𝑥 ∈ (-1[,]1) ∧ 𝑧 ∈ (0[,]π)) → (cos‘𝑧) ∈ ℝ) |
42 | cospi 24028 | . . . . . . . 8 ⊢ (cos‘π) = -1 | |
43 | 32 | simp2bi 1070 | . . . . . . . 8 ⊢ (𝑥 ∈ (-1[,]1) → -1 ≤ 𝑥) |
44 | 42, 43 | syl5eqbr 4618 | . . . . . . 7 ⊢ (𝑥 ∈ (-1[,]1) → (cos‘π) ≤ 𝑥) |
45 | 32 | simp3bi 1071 | . . . . . . . 8 ⊢ (𝑥 ∈ (-1[,]1) → 𝑥 ≤ 1) |
46 | cos0 14719 | . . . . . . . 8 ⊢ (cos‘0) = 1 | |
47 | 45, 46 | syl6breqr 4625 | . . . . . . 7 ⊢ (𝑥 ∈ (-1[,]1) → 𝑥 ≤ (cos‘0)) |
48 | 44, 47 | jca 553 | . . . . . 6 ⊢ (𝑥 ∈ (-1[,]1) → ((cos‘π) ≤ 𝑥 ∧ 𝑥 ≤ (cos‘0))) |
49 | 28, 29, 33, 35, 36, 38, 41, 48 | ivthle2 23033 | . . . . 5 ⊢ (𝑥 ∈ (-1[,]1) → ∃𝑦 ∈ (0[,]π)(cos‘𝑦) = 𝑥) |
50 | eqcom 2617 | . . . . . . 7 ⊢ (𝑥 = ((cos ↾ (0[,]π))‘𝑦) ↔ ((cos ↾ (0[,]π))‘𝑦) = 𝑥) | |
51 | 20 | eqeq1d 2612 | . . . . . . 7 ⊢ (𝑦 ∈ (0[,]π) → (((cos ↾ (0[,]π))‘𝑦) = 𝑥 ↔ (cos‘𝑦) = 𝑥)) |
52 | 50, 51 | syl5bb 271 | . . . . . 6 ⊢ (𝑦 ∈ (0[,]π) → (𝑥 = ((cos ↾ (0[,]π))‘𝑦) ↔ (cos‘𝑦) = 𝑥)) |
53 | 52 | rexbiia 3022 | . . . . 5 ⊢ (∃𝑦 ∈ (0[,]π)𝑥 = ((cos ↾ (0[,]π))‘𝑦) ↔ ∃𝑦 ∈ (0[,]π)(cos‘𝑦) = 𝑥) |
54 | 49, 53 | sylibr 223 | . . . 4 ⊢ (𝑥 ∈ (-1[,]1) → ∃𝑦 ∈ (0[,]π)𝑥 = ((cos ↾ (0[,]π))‘𝑦)) |
55 | 54 | rgen 2906 | . . 3 ⊢ ∀𝑥 ∈ (-1[,]1)∃𝑦 ∈ (0[,]π)𝑥 = ((cos ↾ (0[,]π))‘𝑦) |
56 | dffo3 6282 | . . 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 957 | . 2 ⊢ (cos ↾ (0[,]π)):(0[,]π)–onto→(-1[,]1) |
58 | df-f1o 5811 | . 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 957 | 1 ⊢ (cos ↾ (0[,]π)):(0[,]π)–1-1-onto→(-1[,]1) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 383 = wceq 1475 ∈ wcel 1977 ∀wral 2896 ∃wrex 2897 ⊆ wss 3540 class class class wbr 4583 ↾ cres 5040 Fn wfn 5799 ⟶wf 5800 –1-1→wf1 5801 –onto→wfo 5802 –1-1-onto→wf1o 5803 ‘cfv 5804 (class class class)co 6549 ℂcc 9813 ℝcr 9814 0cc0 9815 1c1 9816 < clt 9953 ≤ cle 9954 -cneg 10146 [,]cicc 12049 cosccos 14634 πcpi 14636 –cn→ccncf 22487 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-inf2 8421 ax-cnex 9871 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-mulcom 9879 ax-addass 9880 ax-mulass 9881 ax-distr 9882 ax-i2m1 9883 ax-1ne0 9884 ax-1rid 9885 ax-rnegex 9886 ax-rrecex 9887 ax-cnre 9888 ax-pre-lttri 9889 ax-pre-lttrn 9890 ax-pre-ltadd 9891 ax-pre-mulgt0 9892 ax-pre-sup 9893 ax-addf 9894 ax-mulf 9895 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-fal 1481 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-reu 2903 df-rmo 2904 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-int 4411 df-iun 4457 df-iin 4458 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-se 4998 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-isom 5813 df-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-of 6795 df-om 6958 df-1st 7059 df-2nd 7060 df-supp 7183 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-1o 7447 df-2o 7448 df-oadd 7451 df-er 7629 df-map 7746 df-pm 7747 df-ixp 7795 df-en 7842 df-dom 7843 df-sdom 7844 df-fin 7845 df-fsupp 8159 df-fi 8200 df-sup 8231 df-inf 8232 df-oi 8298 df-card 8648 df-cda 8873 df-pnf 9955 df-mnf 9956 df-xr 9957 df-ltxr 9958 df-le 9959 df-sub 10147 df-neg 10148 df-div 10564 df-nn 10898 df-2 10956 df-3 10957 df-4 10958 df-5 10959 df-6 10960 df-7 10961 df-8 10962 df-9 10963 df-n0 11170 df-z 11255 df-dec 11370 df-uz 11564 df-q 11665 df-rp 11709 df-xneg 11822 df-xadd 11823 df-xmul 11824 df-ioo 12050 df-ioc 12051 df-ico 12052 df-icc 12053 df-fz 12198 df-fzo 12335 df-fl 12455 df-seq 12664 df-exp 12723 df-fac 12923 df-bc 12952 df-hash 12980 df-shft 13655 df-cj 13687 df-re 13688 df-im 13689 df-sqrt 13823 df-abs 13824 df-limsup 14050 df-clim 14067 df-rlim 14068 df-sum 14265 df-ef 14637 df-sin 14639 df-cos 14640 df-pi 14642 df-struct 15697 df-ndx 15698 df-slot 15699 df-base 15700 df-sets 15701 df-ress 15702 df-plusg 15781 df-mulr 15782 df-starv 15783 df-sca 15784 df-vsca 15785 df-ip 15786 df-tset 15787 df-ple 15788 df-ds 15791 df-unif 15792 df-hom 15793 df-cco 15794 df-rest 15906 df-topn 15907 df-0g 15925 df-gsum 15926 df-topgen 15927 df-pt 15928 df-prds 15931 df-xrs 15985 df-qtop 15990 df-imas 15991 df-xps 15993 df-mre 16069 df-mrc 16070 df-acs 16072 df-mgm 17065 df-sgrp 17107 df-mnd 17118 df-submnd 17159 df-mulg 17364 df-cntz 17573 df-cmn 18018 df-psmet 19559 df-xmet 19560 df-met 19561 df-bl 19562 df-mopn 19563 df-fbas 19564 df-fg 19565 df-cnfld 19568 df-top 20521 df-bases 20522 df-topon 20523 df-topsp 20524 df-cld 20633 df-ntr 20634 df-cls 20635 df-nei 20712 df-lp 20750 df-perf 20751 df-cn 20841 df-cnp 20842 df-haus 20929 df-tx 21175 df-hmeo 21368 df-fil 21460 df-fm 21552 df-flim 21553 df-flf 21554 df-xms 21935 df-ms 21936 df-tms 21937 df-cncf 22489 df-limc 23436 df-dv 23437 |
This theorem is referenced by: resinf1o 24086 |
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