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Mirrors > Home > MPE Home > Th. List > cos11 | Structured version Visualization version GIF version |
Description: Cosine is one-to-one over the closed interval from 0 to π. (Contributed by Paul Chapman, 16-Mar-2008.) (Proof shortened by Mario Carneiro, 10-May-2014.) |
Ref | Expression |
---|---|
cos11 | ⊢ ((𝐴 ∈ (0[,]π) ∧ 𝐵 ∈ (0[,]π)) → (𝐴 = 𝐵 ↔ (cos‘𝐴) = (cos‘𝐵))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ancom 462 | . . 3 ⊢ ((¬ 𝐴 < 𝐵 ∧ ¬ 𝐵 < 𝐴) ↔ (¬ 𝐵 < 𝐴 ∧ ¬ 𝐴 < 𝐵)) | |
2 | cosord 26021 | . . . . . 6 ⊢ ((𝐵 ∈ (0[,]π) ∧ 𝐴 ∈ (0[,]π)) → (𝐵 < 𝐴 ↔ (cos‘𝐴) < (cos‘𝐵))) | |
3 | 2 | ancoms 460 | . . . . 5 ⊢ ((𝐴 ∈ (0[,]π) ∧ 𝐵 ∈ (0[,]π)) → (𝐵 < 𝐴 ↔ (cos‘𝐴) < (cos‘𝐵))) |
4 | 3 | notbid 318 | . . . 4 ⊢ ((𝐴 ∈ (0[,]π) ∧ 𝐵 ∈ (0[,]π)) → (¬ 𝐵 < 𝐴 ↔ ¬ (cos‘𝐴) < (cos‘𝐵))) |
5 | cosord 26021 | . . . . 5 ⊢ ((𝐴 ∈ (0[,]π) ∧ 𝐵 ∈ (0[,]π)) → (𝐴 < 𝐵 ↔ (cos‘𝐵) < (cos‘𝐴))) | |
6 | 5 | notbid 318 | . . . 4 ⊢ ((𝐴 ∈ (0[,]π) ∧ 𝐵 ∈ (0[,]π)) → (¬ 𝐴 < 𝐵 ↔ ¬ (cos‘𝐵) < (cos‘𝐴))) |
7 | 4, 6 | anbi12d 632 | . . 3 ⊢ ((𝐴 ∈ (0[,]π) ∧ 𝐵 ∈ (0[,]π)) → ((¬ 𝐵 < 𝐴 ∧ ¬ 𝐴 < 𝐵) ↔ (¬ (cos‘𝐴) < (cos‘𝐵) ∧ ¬ (cos‘𝐵) < (cos‘𝐴)))) |
8 | 1, 7 | bitrid 283 | . 2 ⊢ ((𝐴 ∈ (0[,]π) ∧ 𝐵 ∈ (0[,]π)) → ((¬ 𝐴 < 𝐵 ∧ ¬ 𝐵 < 𝐴) ↔ (¬ (cos‘𝐴) < (cos‘𝐵) ∧ ¬ (cos‘𝐵) < (cos‘𝐴)))) |
9 | 0re 11211 | . . . . 5 ⊢ 0 ∈ ℝ | |
10 | pire 25949 | . . . . 5 ⊢ π ∈ ℝ | |
11 | 9, 10 | elicc2i 13385 | . . . 4 ⊢ (𝐴 ∈ (0[,]π) ↔ (𝐴 ∈ ℝ ∧ 0 ≤ 𝐴 ∧ 𝐴 ≤ π)) |
12 | 11 | simp1bi 1146 | . . 3 ⊢ (𝐴 ∈ (0[,]π) → 𝐴 ∈ ℝ) |
13 | 9, 10 | elicc2i 13385 | . . . 4 ⊢ (𝐵 ∈ (0[,]π) ↔ (𝐵 ∈ ℝ ∧ 0 ≤ 𝐵 ∧ 𝐵 ≤ π)) |
14 | 13 | simp1bi 1146 | . . 3 ⊢ (𝐵 ∈ (0[,]π) → 𝐵 ∈ ℝ) |
15 | lttri3 11292 | . . 3 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 = 𝐵 ↔ (¬ 𝐴 < 𝐵 ∧ ¬ 𝐵 < 𝐴))) | |
16 | 12, 14, 15 | syl2an 597 | . 2 ⊢ ((𝐴 ∈ (0[,]π) ∧ 𝐵 ∈ (0[,]π)) → (𝐴 = 𝐵 ↔ (¬ 𝐴 < 𝐵 ∧ ¬ 𝐵 < 𝐴))) |
17 | recoscl 16079 | . . . 4 ⊢ (𝐴 ∈ ℝ → (cos‘𝐴) ∈ ℝ) | |
18 | recoscl 16079 | . . . 4 ⊢ (𝐵 ∈ ℝ → (cos‘𝐵) ∈ ℝ) | |
19 | lttri3 11292 | . . . 4 ⊢ (((cos‘𝐴) ∈ ℝ ∧ (cos‘𝐵) ∈ ℝ) → ((cos‘𝐴) = (cos‘𝐵) ↔ (¬ (cos‘𝐴) < (cos‘𝐵) ∧ ¬ (cos‘𝐵) < (cos‘𝐴)))) | |
20 | 17, 18, 19 | syl2an 597 | . . 3 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → ((cos‘𝐴) = (cos‘𝐵) ↔ (¬ (cos‘𝐴) < (cos‘𝐵) ∧ ¬ (cos‘𝐵) < (cos‘𝐴)))) |
21 | 12, 14, 20 | syl2an 597 | . 2 ⊢ ((𝐴 ∈ (0[,]π) ∧ 𝐵 ∈ (0[,]π)) → ((cos‘𝐴) = (cos‘𝐵) ↔ (¬ (cos‘𝐴) < (cos‘𝐵) ∧ ¬ (cos‘𝐵) < (cos‘𝐴)))) |
22 | 8, 16, 21 | 3bitr4d 311 | 1 ⊢ ((𝐴 ∈ (0[,]π) ∧ 𝐵 ∈ (0[,]π)) → (𝐴 = 𝐵 ↔ (cos‘𝐴) = (cos‘𝐵))) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 205 ∧ wa 397 = wceq 1542 ∈ wcel 2107 class class class wbr 5146 ‘cfv 6539 (class class class)co 7403 ℝcr 11104 0cc0 11105 < clt 11243 ≤ cle 11244 [,]cicc 13322 cosccos 16003 πcpi 16005 |
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 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-rep 5283 ax-sep 5297 ax-nul 5304 ax-pow 5361 ax-pr 5425 ax-un 7719 ax-inf2 9631 ax-cnex 11161 ax-resscn 11162 ax-1cn 11163 ax-icn 11164 ax-addcl 11165 ax-addrcl 11166 ax-mulcl 11167 ax-mulrcl 11168 ax-mulcom 11169 ax-addass 11170 ax-mulass 11171 ax-distr 11172 ax-i2m1 11173 ax-1ne0 11174 ax-1rid 11175 ax-rnegex 11176 ax-rrecex 11177 ax-cnre 11178 ax-pre-lttri 11179 ax-pre-lttrn 11180 ax-pre-ltadd 11181 ax-pre-mulgt0 11182 ax-pre-sup 11183 ax-addf 11184 ax-mulf 11185 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-rmo 3377 df-reu 3378 df-rab 3434 df-v 3477 df-sbc 3776 df-csb 3892 df-dif 3949 df-un 3951 df-in 3953 df-ss 3963 df-pss 3965 df-nul 4321 df-if 4527 df-pw 4602 df-sn 4627 df-pr 4629 df-tp 4631 df-op 4633 df-uni 4907 df-int 4949 df-iun 4997 df-iin 4998 df-br 5147 df-opab 5209 df-mpt 5230 df-tr 5264 df-id 5572 df-eprel 5578 df-po 5586 df-so 5587 df-fr 5629 df-se 5630 df-we 5631 df-xp 5680 df-rel 5681 df-cnv 5682 df-co 5683 df-dm 5684 df-rn 5685 df-res 5686 df-ima 5687 df-pred 6296 df-ord 6363 df-on 6364 df-lim 6365 df-suc 6366 df-iota 6491 df-fun 6541 df-fn 6542 df-f 6543 df-f1 6544 df-fo 6545 df-f1o 6546 df-fv 6547 df-isom 6548 df-riota 7359 df-ov 7406 df-oprab 7407 df-mpo 7408 df-of 7664 df-om 7850 df-1st 7969 df-2nd 7970 df-supp 8141 df-frecs 8260 df-wrecs 8291 df-recs 8365 df-rdg 8404 df-1o 8460 df-2o 8461 df-er 8698 df-map 8817 df-pm 8818 df-ixp 8887 df-en 8935 df-dom 8936 df-sdom 8937 df-fin 8938 df-fsupp 9357 df-fi 9401 df-sup 9432 df-inf 9433 df-oi 9500 df-card 9929 df-pnf 11245 df-mnf 11246 df-xr 11247 df-ltxr 11248 df-le 11249 df-sub 11441 df-neg 11442 df-div 11867 df-nn 12208 df-2 12270 df-3 12271 df-4 12272 df-5 12273 df-6 12274 df-7 12275 df-8 12276 df-9 12277 df-n0 12468 df-z 12554 df-dec 12673 df-uz 12818 df-q 12928 df-rp 12970 df-xneg 13087 df-xadd 13088 df-xmul 13089 df-ioo 13323 df-ioc 13324 df-ico 13325 df-icc 13326 df-fz 13480 df-fzo 13623 df-fl 13752 df-seq 13962 df-exp 14023 df-fac 14229 df-bc 14258 df-hash 14286 df-shft 15009 df-cj 15041 df-re 15042 df-im 15043 df-sqrt 15177 df-abs 15178 df-limsup 15410 df-clim 15427 df-rlim 15428 df-sum 15628 df-ef 16006 df-sin 16008 df-cos 16009 df-pi 16011 df-struct 17075 df-sets 17092 df-slot 17110 df-ndx 17122 df-base 17140 df-ress 17169 df-plusg 17205 df-mulr 17206 df-starv 17207 df-sca 17208 df-vsca 17209 df-ip 17210 df-tset 17211 df-ple 17212 df-ds 17214 df-unif 17215 df-hom 17216 df-cco 17217 df-rest 17363 df-topn 17364 df-0g 17382 df-gsum 17383 df-topgen 17384 df-pt 17385 df-prds 17388 df-xrs 17443 df-qtop 17448 df-imas 17449 df-xps 17451 df-mre 17525 df-mrc 17526 df-acs 17528 df-mgm 18556 df-sgrp 18605 df-mnd 18621 df-submnd 18667 df-mulg 18944 df-cntz 19174 df-cmn 19642 df-psmet 20920 df-xmet 20921 df-met 20922 df-bl 20923 df-mopn 20924 df-fbas 20925 df-fg 20926 df-cnfld 20929 df-top 22377 df-topon 22394 df-topsp 22416 df-bases 22430 df-cld 22504 df-ntr 22505 df-cls 22506 df-nei 22583 df-lp 22621 df-perf 22622 df-cn 22712 df-cnp 22713 df-haus 22800 df-tx 23047 df-hmeo 23240 df-fil 23331 df-fm 23423 df-flim 23424 df-flf 23425 df-xms 23807 df-ms 23808 df-tms 23809 df-cncf 24375 df-limc 25364 df-dv 25365 |
This theorem is referenced by: recosf1o 26025 |
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