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Mirrors > Home > MPE Home > Th. List > tan4thpi | Structured version Visualization version GIF version |
Description: The tangent of π / 4. (Contributed by Mario Carneiro, 5-Apr-2015.) |
Ref | Expression |
---|---|
tan4thpi | ⊢ (tan‘(π / 4)) = 1 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | pire 26409 | . . . . 5 ⊢ π ∈ ℝ | |
2 | 4nn 12323 | . . . . 5 ⊢ 4 ∈ ℕ | |
3 | nndivre 12281 | . . . . 5 ⊢ ((π ∈ ℝ ∧ 4 ∈ ℕ) → (π / 4) ∈ ℝ) | |
4 | 1, 2, 3 | mp2an 690 | . . . 4 ⊢ (π / 4) ∈ ℝ |
5 | 4 | recni 11256 | . . 3 ⊢ (π / 4) ∈ ℂ |
6 | sincos4thpi 26464 | . . . . 5 ⊢ ((sin‘(π / 4)) = (1 / (√‘2)) ∧ (cos‘(π / 4)) = (1 / (√‘2))) | |
7 | 6 | simpri 484 | . . . 4 ⊢ (cos‘(π / 4)) = (1 / (√‘2)) |
8 | sqrt2re 16224 | . . . . . 6 ⊢ (√‘2) ∈ ℝ | |
9 | 8 | recni 11256 | . . . . 5 ⊢ (√‘2) ∈ ℂ |
10 | 2re 12314 | . . . . . . . 8 ⊢ 2 ∈ ℝ | |
11 | 0le2 12342 | . . . . . . . 8 ⊢ 0 ≤ 2 | |
12 | resqrtth 15232 | . . . . . . . 8 ⊢ ((2 ∈ ℝ ∧ 0 ≤ 2) → ((√‘2)↑2) = 2) | |
13 | 10, 11, 12 | mp2an 690 | . . . . . . 7 ⊢ ((√‘2)↑2) = 2 |
14 | 2ne0 12344 | . . . . . . 7 ⊢ 2 ≠ 0 | |
15 | 13, 14 | eqnetri 3001 | . . . . . 6 ⊢ ((√‘2)↑2) ≠ 0 |
16 | sqne0 14117 | . . . . . . 7 ⊢ ((√‘2) ∈ ℂ → (((√‘2)↑2) ≠ 0 ↔ (√‘2) ≠ 0)) | |
17 | 9, 16 | ax-mp 5 | . . . . . 6 ⊢ (((√‘2)↑2) ≠ 0 ↔ (√‘2) ≠ 0) |
18 | 15, 17 | mpbi 229 | . . . . 5 ⊢ (√‘2) ≠ 0 |
19 | recne0 11913 | . . . . 5 ⊢ (((√‘2) ∈ ℂ ∧ (√‘2) ≠ 0) → (1 / (√‘2)) ≠ 0) | |
20 | 9, 18, 19 | mp2an 690 | . . . 4 ⊢ (1 / (√‘2)) ≠ 0 |
21 | 7, 20 | eqnetri 3001 | . . 3 ⊢ (cos‘(π / 4)) ≠ 0 |
22 | tanval 16102 | . . 3 ⊢ (((π / 4) ∈ ℂ ∧ (cos‘(π / 4)) ≠ 0) → (tan‘(π / 4)) = ((sin‘(π / 4)) / (cos‘(π / 4)))) | |
23 | 5, 21, 22 | mp2an 690 | . 2 ⊢ (tan‘(π / 4)) = ((sin‘(π / 4)) / (cos‘(π / 4))) |
24 | 6 | simpli 482 | . . 3 ⊢ (sin‘(π / 4)) = (1 / (√‘2)) |
25 | 24, 7 | oveq12i 7426 | . 2 ⊢ ((sin‘(π / 4)) / (cos‘(π / 4))) = ((1 / (√‘2)) / (1 / (√‘2))) |
26 | 9, 18 | reccli 11972 | . . 3 ⊢ (1 / (√‘2)) ∈ ℂ |
27 | 26, 20 | dividi 11975 | . 2 ⊢ ((1 / (√‘2)) / (1 / (√‘2))) = 1 |
28 | 23, 25, 27 | 3eqtri 2757 | 1 ⊢ (tan‘(π / 4)) = 1 |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 205 = wceq 1533 ∈ wcel 2098 ≠ wne 2930 class class class wbr 5141 ‘cfv 6541 (class class class)co 7414 ℂcc 11134 ℝcr 11135 0cc0 11136 1c1 11137 ≤ cle 11277 / cdiv 11899 ℕcn 12240 2c2 12295 4c4 12297 ↑cexp 14056 √csqrt 15210 sincsin 16037 cosccos 16038 tanctan 16039 πcpi 16040 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2696 ax-rep 5278 ax-sep 5292 ax-nul 5299 ax-pow 5357 ax-pr 5421 ax-un 7736 ax-inf2 9662 ax-cnex 11192 ax-resscn 11193 ax-1cn 11194 ax-icn 11195 ax-addcl 11196 ax-addrcl 11197 ax-mulcl 11198 ax-mulrcl 11199 ax-mulcom 11200 ax-addass 11201 ax-mulass 11202 ax-distr 11203 ax-i2m1 11204 ax-1ne0 11205 ax-1rid 11206 ax-rnegex 11207 ax-rrecex 11208 ax-cnre 11209 ax-pre-lttri 11210 ax-pre-lttrn 11211 ax-pre-ltadd 11212 ax-pre-mulgt0 11213 ax-pre-sup 11214 ax-addf 11215 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2931 df-nel 3037 df-ral 3052 df-rex 3061 df-rmo 3364 df-reu 3365 df-rab 3420 df-v 3465 df-sbc 3769 df-csb 3885 df-dif 3942 df-un 3944 df-in 3946 df-ss 3956 df-pss 3958 df-nul 4317 df-if 4523 df-pw 4598 df-sn 4623 df-pr 4625 df-tp 4627 df-op 4629 df-uni 4902 df-int 4943 df-iun 4991 df-iin 4992 df-br 5142 df-opab 5204 df-mpt 5225 df-tr 5259 df-id 5568 df-eprel 5574 df-po 5582 df-so 5583 df-fr 5625 df-se 5626 df-we 5627 df-xp 5676 df-rel 5677 df-cnv 5678 df-co 5679 df-dm 5680 df-rn 5681 df-res 5682 df-ima 5683 df-pred 6298 df-ord 6365 df-on 6366 df-lim 6367 df-suc 6368 df-iota 6493 df-fun 6543 df-fn 6544 df-f 6545 df-f1 6546 df-fo 6547 df-f1o 6548 df-fv 6549 df-isom 6550 df-riota 7370 df-ov 7417 df-oprab 7418 df-mpo 7419 df-of 7680 df-om 7867 df-1st 7989 df-2nd 7990 df-supp 8162 df-frecs 8283 df-wrecs 8314 df-recs 8388 df-rdg 8427 df-1o 8483 df-2o 8484 df-er 8721 df-map 8843 df-pm 8844 df-ixp 8913 df-en 8961 df-dom 8962 df-sdom 8963 df-fin 8964 df-fsupp 9384 df-fi 9432 df-sup 9463 df-inf 9464 df-oi 9531 df-card 9960 df-pnf 11278 df-mnf 11279 df-xr 11280 df-ltxr 11281 df-le 11282 df-sub 11474 df-neg 11475 df-div 11900 df-nn 12241 df-2 12303 df-3 12304 df-4 12305 df-5 12306 df-6 12307 df-7 12308 df-8 12309 df-9 12310 df-n0 12501 df-z 12587 df-dec 12706 df-uz 12851 df-q 12961 df-rp 13005 df-xneg 13122 df-xadd 13123 df-xmul 13124 df-ioo 13358 df-ioc 13359 df-ico 13360 df-icc 13361 df-fz 13515 df-fzo 13658 df-fl 13787 df-seq 13997 df-exp 14057 df-fac 14263 df-bc 14292 df-hash 14320 df-shft 15044 df-cj 15076 df-re 15077 df-im 15078 df-sqrt 15212 df-abs 15213 df-limsup 15445 df-clim 15462 df-rlim 15463 df-sum 15663 df-ef 16041 df-sin 16043 df-cos 16044 df-tan 16045 df-pi 16046 df-struct 17113 df-sets 17130 df-slot 17148 df-ndx 17160 df-base 17178 df-ress 17207 df-plusg 17243 df-mulr 17244 df-starv 17245 df-sca 17246 df-vsca 17247 df-ip 17248 df-tset 17249 df-ple 17250 df-ds 17252 df-unif 17253 df-hom 17254 df-cco 17255 df-rest 17401 df-topn 17402 df-0g 17420 df-gsum 17421 df-topgen 17422 df-pt 17423 df-prds 17426 df-xrs 17481 df-qtop 17486 df-imas 17487 df-xps 17489 df-mre 17563 df-mrc 17564 df-acs 17566 df-mgm 18597 df-sgrp 18676 df-mnd 18692 df-submnd 18738 df-mulg 19026 df-cntz 19270 df-cmn 19739 df-psmet 21273 df-xmet 21274 df-met 21275 df-bl 21276 df-mopn 21277 df-fbas 21278 df-fg 21279 df-cnfld 21282 df-top 22812 df-topon 22829 df-topsp 22851 df-bases 22865 df-cld 22939 df-ntr 22940 df-cls 22941 df-nei 23018 df-lp 23056 df-perf 23057 df-cn 23147 df-cnp 23148 df-haus 23235 df-tx 23482 df-hmeo 23675 df-fil 23766 df-fm 23858 df-flim 23859 df-flf 23860 df-xms 24242 df-ms 24243 df-tms 24244 df-cncf 24814 df-limc 25811 df-dv 25812 |
This theorem is referenced by: atan1 26876 |
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