Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > tanval | Structured version Visualization version GIF version |
Description: Value of the tangent function. (Contributed by Mario Carneiro, 14-Mar-2014.) |
Ref | Expression |
---|---|
tanval | ⊢ ((𝐴 ∈ ℂ ∧ (cos‘𝐴) ≠ 0) → (tan‘𝐴) = ((sin‘𝐴) / (cos‘𝐴))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simpl 486 | . . 3 ⊢ ((𝐴 ∈ ℂ ∧ (cos‘𝐴) ≠ 0) → 𝐴 ∈ ℂ) | |
2 | coscl 15651 | . . . . 5 ⊢ (𝐴 ∈ ℂ → (cos‘𝐴) ∈ ℂ) | |
3 | 2 | anim1i 618 | . . . 4 ⊢ ((𝐴 ∈ ℂ ∧ (cos‘𝐴) ≠ 0) → ((cos‘𝐴) ∈ ℂ ∧ (cos‘𝐴) ≠ 0)) |
4 | eldifsn 4686 | . . . 4 ⊢ ((cos‘𝐴) ∈ (ℂ ∖ {0}) ↔ ((cos‘𝐴) ∈ ℂ ∧ (cos‘𝐴) ≠ 0)) | |
5 | 3, 4 | sylibr 237 | . . 3 ⊢ ((𝐴 ∈ ℂ ∧ (cos‘𝐴) ≠ 0) → (cos‘𝐴) ∈ (ℂ ∖ {0})) |
6 | cosf 15649 | . . . 4 ⊢ cos:ℂ⟶ℂ | |
7 | ffn 6523 | . . . 4 ⊢ (cos:ℂ⟶ℂ → cos Fn ℂ) | |
8 | elpreima 6856 | . . . 4 ⊢ (cos Fn ℂ → (𝐴 ∈ (◡cos “ (ℂ ∖ {0})) ↔ (𝐴 ∈ ℂ ∧ (cos‘𝐴) ∈ (ℂ ∖ {0})))) | |
9 | 6, 7, 8 | mp2b 10 | . . 3 ⊢ (𝐴 ∈ (◡cos “ (ℂ ∖ {0})) ↔ (𝐴 ∈ ℂ ∧ (cos‘𝐴) ∈ (ℂ ∖ {0}))) |
10 | 1, 5, 9 | sylanbrc 586 | . 2 ⊢ ((𝐴 ∈ ℂ ∧ (cos‘𝐴) ≠ 0) → 𝐴 ∈ (◡cos “ (ℂ ∖ {0}))) |
11 | fveq2 6695 | . . . 4 ⊢ (𝑥 = 𝐴 → (sin‘𝑥) = (sin‘𝐴)) | |
12 | fveq2 6695 | . . . 4 ⊢ (𝑥 = 𝐴 → (cos‘𝑥) = (cos‘𝐴)) | |
13 | 11, 12 | oveq12d 7209 | . . 3 ⊢ (𝑥 = 𝐴 → ((sin‘𝑥) / (cos‘𝑥)) = ((sin‘𝐴) / (cos‘𝐴))) |
14 | df-tan 15596 | . . 3 ⊢ tan = (𝑥 ∈ (◡cos “ (ℂ ∖ {0})) ↦ ((sin‘𝑥) / (cos‘𝑥))) | |
15 | ovex 7224 | . . 3 ⊢ ((sin‘𝐴) / (cos‘𝐴)) ∈ V | |
16 | 13, 14, 15 | fvmpt 6796 | . 2 ⊢ (𝐴 ∈ (◡cos “ (ℂ ∖ {0})) → (tan‘𝐴) = ((sin‘𝐴) / (cos‘𝐴))) |
17 | 10, 16 | syl 17 | 1 ⊢ ((𝐴 ∈ ℂ ∧ (cos‘𝐴) ≠ 0) → (tan‘𝐴) = ((sin‘𝐴) / (cos‘𝐴))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 ∧ wa 399 = wceq 1543 ∈ wcel 2112 ≠ wne 2932 ∖ cdif 3850 {csn 4527 ◡ccnv 5535 “ cima 5539 Fn wfn 6353 ⟶wf 6354 ‘cfv 6358 (class class class)co 7191 ℂcc 10692 0cc0 10694 / cdiv 11454 sincsin 15588 cosccos 15589 tanctan 15590 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2018 ax-8 2114 ax-9 2122 ax-10 2143 ax-11 2160 ax-12 2177 ax-ext 2708 ax-rep 5164 ax-sep 5177 ax-nul 5184 ax-pow 5243 ax-pr 5307 ax-un 7501 ax-inf2 9234 ax-cnex 10750 ax-resscn 10751 ax-1cn 10752 ax-icn 10753 ax-addcl 10754 ax-addrcl 10755 ax-mulcl 10756 ax-mulrcl 10757 ax-mulcom 10758 ax-addass 10759 ax-mulass 10760 ax-distr 10761 ax-i2m1 10762 ax-1ne0 10763 ax-1rid 10764 ax-rnegex 10765 ax-rrecex 10766 ax-cnre 10767 ax-pre-lttri 10768 ax-pre-lttrn 10769 ax-pre-ltadd 10770 ax-pre-mulgt0 10771 ax-pre-sup 10772 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3or 1090 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2073 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2728 df-clel 2809 df-nfc 2879 df-ne 2933 df-nel 3037 df-ral 3056 df-rex 3057 df-reu 3058 df-rmo 3059 df-rab 3060 df-v 3400 df-sbc 3684 df-csb 3799 df-dif 3856 df-un 3858 df-in 3860 df-ss 3870 df-pss 3872 df-nul 4224 df-if 4426 df-pw 4501 df-sn 4528 df-pr 4530 df-tp 4532 df-op 4534 df-uni 4806 df-int 4846 df-iun 4892 df-br 5040 df-opab 5102 df-mpt 5121 df-tr 5147 df-id 5440 df-eprel 5445 df-po 5453 df-so 5454 df-fr 5494 df-se 5495 df-we 5496 df-xp 5542 df-rel 5543 df-cnv 5544 df-co 5545 df-dm 5546 df-rn 5547 df-res 5548 df-ima 5549 df-pred 6140 df-ord 6194 df-on 6195 df-lim 6196 df-suc 6197 df-iota 6316 df-fun 6360 df-fn 6361 df-f 6362 df-f1 6363 df-fo 6364 df-f1o 6365 df-fv 6366 df-isom 6367 df-riota 7148 df-ov 7194 df-oprab 7195 df-mpo 7196 df-om 7623 df-1st 7739 df-2nd 7740 df-wrecs 8025 df-recs 8086 df-rdg 8124 df-1o 8180 df-er 8369 df-pm 8489 df-en 8605 df-dom 8606 df-sdom 8607 df-fin 8608 df-sup 9036 df-inf 9037 df-oi 9104 df-card 9520 df-pnf 10834 df-mnf 10835 df-xr 10836 df-ltxr 10837 df-le 10838 df-sub 11029 df-neg 11030 df-div 11455 df-nn 11796 df-2 11858 df-3 11859 df-n0 12056 df-z 12142 df-uz 12404 df-rp 12552 df-ico 12906 df-fz 13061 df-fzo 13204 df-fl 13332 df-seq 13540 df-exp 13601 df-fac 13805 df-hash 13862 df-shft 14595 df-cj 14627 df-re 14628 df-im 14629 df-sqrt 14763 df-abs 14764 df-limsup 14997 df-clim 15014 df-rlim 15015 df-sum 15215 df-ef 15592 df-cos 15595 df-tan 15596 |
This theorem is referenced by: tancl 15653 tanval2 15657 retancl 15666 tanneg 15672 tan0 15675 retanhcl 15683 tanhlt1 15684 tanaddlem 15690 tanadd 15691 tanrpcl 25348 tangtx 25349 tan4thpi 25358 tanord1 25380 atandmtan 25757 atantan 25760 basellem3 25919 basellem8 25924 tan2h 35455 reccot 46074 rectan 46075 onetansqsecsq 46077 |
Copyright terms: Public domain | W3C validator |