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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 25036 | . . . . 5 ⊢ π ∈ ℝ | |
2 | 4nn 11712 | . . . . 5 ⊢ 4 ∈ ℕ | |
3 | nndivre 11670 | . . . . 5 ⊢ ((π ∈ ℝ ∧ 4 ∈ ℕ) → (π / 4) ∈ ℝ) | |
4 | 1, 2, 3 | mp2an 690 | . . . 4 ⊢ (π / 4) ∈ ℝ |
5 | 4 | recni 10647 | . . 3 ⊢ (π / 4) ∈ ℂ |
6 | sincos4thpi 25091 | . . . . 5 ⊢ ((sin‘(π / 4)) = (1 / (√‘2)) ∧ (cos‘(π / 4)) = (1 / (√‘2))) | |
7 | 6 | simpri 488 | . . . 4 ⊢ (cos‘(π / 4)) = (1 / (√‘2)) |
8 | sqrt2re 15595 | . . . . . 6 ⊢ (√‘2) ∈ ℝ | |
9 | 8 | recni 10647 | . . . . 5 ⊢ (√‘2) ∈ ℂ |
10 | 2re 11703 | . . . . . . . 8 ⊢ 2 ∈ ℝ | |
11 | 0le2 11731 | . . . . . . . 8 ⊢ 0 ≤ 2 | |
12 | resqrtth 14607 | . . . . . . . 8 ⊢ ((2 ∈ ℝ ∧ 0 ≤ 2) → ((√‘2)↑2) = 2) | |
13 | 10, 11, 12 | mp2an 690 | . . . . . . 7 ⊢ ((√‘2)↑2) = 2 |
14 | 2ne0 11733 | . . . . . . 7 ⊢ 2 ≠ 0 | |
15 | 13, 14 | eqnetri 3084 | . . . . . 6 ⊢ ((√‘2)↑2) ≠ 0 |
16 | sqne0 13481 | . . . . . . 7 ⊢ ((√‘2) ∈ ℂ → (((√‘2)↑2) ≠ 0 ↔ (√‘2) ≠ 0)) | |
17 | 9, 16 | ax-mp 5 | . . . . . 6 ⊢ (((√‘2)↑2) ≠ 0 ↔ (√‘2) ≠ 0) |
18 | 15, 17 | mpbi 232 | . . . . 5 ⊢ (√‘2) ≠ 0 |
19 | recne0 11303 | . . . . 5 ⊢ (((√‘2) ∈ ℂ ∧ (√‘2) ≠ 0) → (1 / (√‘2)) ≠ 0) | |
20 | 9, 18, 19 | mp2an 690 | . . . 4 ⊢ (1 / (√‘2)) ≠ 0 |
21 | 7, 20 | eqnetri 3084 | . . 3 ⊢ (cos‘(π / 4)) ≠ 0 |
22 | tanval 15473 | . . 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 486 | . . 3 ⊢ (sin‘(π / 4)) = (1 / (√‘2)) |
25 | 24, 7 | oveq12i 7160 | . 2 ⊢ ((sin‘(π / 4)) / (cos‘(π / 4))) = ((1 / (√‘2)) / (1 / (√‘2))) |
26 | 9, 18 | reccli 11362 | . . 3 ⊢ (1 / (√‘2)) ∈ ℂ |
27 | 26, 20 | dividi 11365 | . 2 ⊢ ((1 / (√‘2)) / (1 / (√‘2))) = 1 |
28 | 23, 25, 27 | 3eqtri 2846 | 1 ⊢ (tan‘(π / 4)) = 1 |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 208 = wceq 1531 ∈ wcel 2108 ≠ wne 3014 class class class wbr 5057 ‘cfv 6348 (class class class)co 7148 ℂcc 10527 ℝcr 10528 0cc0 10529 1c1 10530 ≤ cle 10668 / cdiv 11289 ℕcn 11630 2c2 11684 4c4 11686 ↑cexp 13421 √csqrt 14584 sincsin 15409 cosccos 15410 tanctan 15411 πcpi 15412 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1905 ax-6 1964 ax-7 2009 ax-8 2110 ax-9 2118 ax-10 2139 ax-11 2154 ax-12 2170 ax-ext 2791 ax-rep 5181 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7453 ax-inf2 9096 ax-cnex 10585 ax-resscn 10586 ax-1cn 10587 ax-icn 10588 ax-addcl 10589 ax-addrcl 10590 ax-mulcl 10591 ax-mulrcl 10592 ax-mulcom 10593 ax-addass 10594 ax-mulass 10595 ax-distr 10596 ax-i2m1 10597 ax-1ne0 10598 ax-1rid 10599 ax-rnegex 10600 ax-rrecex 10601 ax-cnre 10602 ax-pre-lttri 10603 ax-pre-lttrn 10604 ax-pre-ltadd 10605 ax-pre-mulgt0 10606 ax-pre-sup 10607 ax-addf 10608 ax-mulf 10609 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1083 df-3an 1084 df-tru 1534 df-fal 1544 df-ex 1775 df-nf 1779 df-sb 2064 df-mo 2616 df-eu 2648 df-clab 2798 df-cleq 2812 df-clel 2891 df-nfc 2961 df-ne 3015 df-nel 3122 df-ral 3141 df-rex 3142 df-reu 3143 df-rmo 3144 df-rab 3145 df-v 3495 df-sbc 3771 df-csb 3882 df-dif 3937 df-un 3939 df-in 3941 df-ss 3950 df-pss 3952 df-nul 4290 df-if 4466 df-pw 4539 df-sn 4560 df-pr 4562 df-tp 4564 df-op 4566 df-uni 4831 df-int 4868 df-iun 4912 df-iin 4913 df-br 5058 df-opab 5120 df-mpt 5138 df-tr 5164 df-id 5453 df-eprel 5458 df-po 5467 df-so 5468 df-fr 5507 df-se 5508 df-we 5509 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-pred 6141 df-ord 6187 df-on 6188 df-lim 6189 df-suc 6190 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-isom 6357 df-riota 7106 df-ov 7151 df-oprab 7152 df-mpo 7153 df-of 7401 df-om 7573 df-1st 7681 df-2nd 7682 df-supp 7823 df-wrecs 7939 df-recs 8000 df-rdg 8038 df-1o 8094 df-2o 8095 df-oadd 8098 df-er 8281 df-map 8400 df-pm 8401 df-ixp 8454 df-en 8502 df-dom 8503 df-sdom 8504 df-fin 8505 df-fsupp 8826 df-fi 8867 df-sup 8898 df-inf 8899 df-oi 8966 df-card 9360 df-pnf 10669 df-mnf 10670 df-xr 10671 df-ltxr 10672 df-le 10673 df-sub 10864 df-neg 10865 df-div 11290 df-nn 11631 df-2 11692 df-3 11693 df-4 11694 df-5 11695 df-6 11696 df-7 11697 df-8 11698 df-9 11699 df-n0 11890 df-z 11974 df-dec 12091 df-uz 12236 df-q 12341 df-rp 12382 df-xneg 12499 df-xadd 12500 df-xmul 12501 df-ioo 12734 df-ioc 12735 df-ico 12736 df-icc 12737 df-fz 12885 df-fzo 13026 df-fl 13154 df-seq 13362 df-exp 13422 df-fac 13626 df-bc 13655 df-hash 13683 df-shft 14418 df-cj 14450 df-re 14451 df-im 14452 df-sqrt 14586 df-abs 14587 df-limsup 14820 df-clim 14837 df-rlim 14838 df-sum 15035 df-ef 15413 df-sin 15415 df-cos 15416 df-tan 15417 df-pi 15418 df-struct 16477 df-ndx 16478 df-slot 16479 df-base 16481 df-sets 16482 df-ress 16483 df-plusg 16570 df-mulr 16571 df-starv 16572 df-sca 16573 df-vsca 16574 df-ip 16575 df-tset 16576 df-ple 16577 df-ds 16579 df-unif 16580 df-hom 16581 df-cco 16582 df-rest 16688 df-topn 16689 df-0g 16707 df-gsum 16708 df-topgen 16709 df-pt 16710 df-prds 16713 df-xrs 16767 df-qtop 16772 df-imas 16773 df-xps 16775 df-mre 16849 df-mrc 16850 df-acs 16852 df-mgm 17844 df-sgrp 17893 df-mnd 17904 df-submnd 17949 df-mulg 18217 df-cntz 18439 df-cmn 18900 df-psmet 20529 df-xmet 20530 df-met 20531 df-bl 20532 df-mopn 20533 df-fbas 20534 df-fg 20535 df-cnfld 20538 df-top 21494 df-topon 21511 df-topsp 21533 df-bases 21546 df-cld 21619 df-ntr 21620 df-cls 21621 df-nei 21698 df-lp 21736 df-perf 21737 df-cn 21827 df-cnp 21828 df-haus 21915 df-tx 22162 df-hmeo 22355 df-fil 22446 df-fm 22538 df-flim 22539 df-flf 22540 df-xms 22922 df-ms 22923 df-tms 22924 df-cncf 23478 df-limc 24456 df-dv 24457 |
This theorem is referenced by: atan1 25498 |
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