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Mirrors > Home > ILE Home > Th. List > tanvalap | GIF version |
Description: Value of the tangent function. (Contributed by Mario Carneiro, 14-Mar-2014.) (Revised by Jim Kingdon, 21-Dec-2022.) |
Ref | Expression |
---|---|
tanvalap | ⊢ ((𝐴 ∈ ℂ ∧ (cos‘𝐴) # 0) → (tan‘𝐴) = ((sin‘𝐴) / (cos‘𝐴))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simpl 108 | . . 3 ⊢ ((𝐴 ∈ ℂ ∧ (cos‘𝐴) # 0) → 𝐴 ∈ ℂ) | |
2 | coscl 11341 | . . . . 5 ⊢ (𝐴 ∈ ℂ → (cos‘𝐴) ∈ ℂ) | |
3 | 2 | adantr 274 | . . . 4 ⊢ ((𝐴 ∈ ℂ ∧ (cos‘𝐴) # 0) → (cos‘𝐴) ∈ ℂ) |
4 | simpr 109 | . . . . 5 ⊢ ((𝐴 ∈ ℂ ∧ (cos‘𝐴) # 0) → (cos‘𝐴) # 0) | |
5 | 0cnd 7727 | . . . . . 6 ⊢ ((𝐴 ∈ ℂ ∧ (cos‘𝐴) # 0) → 0 ∈ ℂ) | |
6 | apne 8353 | . . . . . 6 ⊢ (((cos‘𝐴) ∈ ℂ ∧ 0 ∈ ℂ) → ((cos‘𝐴) # 0 → (cos‘𝐴) ≠ 0)) | |
7 | 3, 5, 6 | syl2anc 408 | . . . . 5 ⊢ ((𝐴 ∈ ℂ ∧ (cos‘𝐴) # 0) → ((cos‘𝐴) # 0 → (cos‘𝐴) ≠ 0)) |
8 | 4, 7 | mpd 13 | . . . 4 ⊢ ((𝐴 ∈ ℂ ∧ (cos‘𝐴) # 0) → (cos‘𝐴) ≠ 0) |
9 | eldifsn 3620 | . . . 4 ⊢ ((cos‘𝐴) ∈ (ℂ ∖ {0}) ↔ ((cos‘𝐴) ∈ ℂ ∧ (cos‘𝐴) ≠ 0)) | |
10 | 3, 8, 9 | sylanbrc 413 | . . 3 ⊢ ((𝐴 ∈ ℂ ∧ (cos‘𝐴) # 0) → (cos‘𝐴) ∈ (ℂ ∖ {0})) |
11 | cosf 11339 | . . . 4 ⊢ cos:ℂ⟶ℂ | |
12 | ffn 5242 | . . . 4 ⊢ (cos:ℂ⟶ℂ → cos Fn ℂ) | |
13 | elpreima 5507 | . . . 4 ⊢ (cos Fn ℂ → (𝐴 ∈ (◡cos “ (ℂ ∖ {0})) ↔ (𝐴 ∈ ℂ ∧ (cos‘𝐴) ∈ (ℂ ∖ {0})))) | |
14 | 11, 12, 13 | mp2b 8 | . . 3 ⊢ (𝐴 ∈ (◡cos “ (ℂ ∖ {0})) ↔ (𝐴 ∈ ℂ ∧ (cos‘𝐴) ∈ (ℂ ∖ {0}))) |
15 | 1, 10, 14 | sylanbrc 413 | . 2 ⊢ ((𝐴 ∈ ℂ ∧ (cos‘𝐴) # 0) → 𝐴 ∈ (◡cos “ (ℂ ∖ {0}))) |
16 | sincl 11340 | . . . 4 ⊢ (𝐴 ∈ ℂ → (sin‘𝐴) ∈ ℂ) | |
17 | 16 | adantr 274 | . . 3 ⊢ ((𝐴 ∈ ℂ ∧ (cos‘𝐴) # 0) → (sin‘𝐴) ∈ ℂ) |
18 | 17, 3, 4 | divclapd 8518 | . 2 ⊢ ((𝐴 ∈ ℂ ∧ (cos‘𝐴) # 0) → ((sin‘𝐴) / (cos‘𝐴)) ∈ ℂ) |
19 | fveq2 5389 | . . . 4 ⊢ (𝑥 = 𝐴 → (sin‘𝑥) = (sin‘𝐴)) | |
20 | fveq2 5389 | . . . 4 ⊢ (𝑥 = 𝐴 → (cos‘𝑥) = (cos‘𝐴)) | |
21 | 19, 20 | oveq12d 5760 | . . 3 ⊢ (𝑥 = 𝐴 → ((sin‘𝑥) / (cos‘𝑥)) = ((sin‘𝐴) / (cos‘𝐴))) |
22 | df-tan 11285 | . . 3 ⊢ tan = (𝑥 ∈ (◡cos “ (ℂ ∖ {0})) ↦ ((sin‘𝑥) / (cos‘𝑥))) | |
23 | 21, 22 | fvmptg 5465 | . 2 ⊢ ((𝐴 ∈ (◡cos “ (ℂ ∖ {0})) ∧ ((sin‘𝐴) / (cos‘𝐴)) ∈ ℂ) → (tan‘𝐴) = ((sin‘𝐴) / (cos‘𝐴))) |
24 | 15, 18, 23 | syl2anc 408 | 1 ⊢ ((𝐴 ∈ ℂ ∧ (cos‘𝐴) # 0) → (tan‘𝐴) = ((sin‘𝐴) / (cos‘𝐴))) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 ↔ wb 104 = wceq 1316 ∈ wcel 1465 ≠ wne 2285 ∖ cdif 3038 {csn 3497 class class class wbr 3899 ◡ccnv 4508 “ cima 4512 Fn wfn 5088 ⟶wf 5089 ‘cfv 5093 (class class class)co 5742 ℂcc 7586 0cc0 7588 # cap 8311 / cdiv 8400 sincsin 11277 cosccos 11278 tanctan 11279 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 588 ax-in2 589 ax-io 683 ax-5 1408 ax-7 1409 ax-gen 1410 ax-ie1 1454 ax-ie2 1455 ax-8 1467 ax-10 1468 ax-11 1469 ax-i12 1470 ax-bndl 1471 ax-4 1472 ax-13 1476 ax-14 1477 ax-17 1491 ax-i9 1495 ax-ial 1499 ax-i5r 1500 ax-ext 2099 ax-coll 4013 ax-sep 4016 ax-nul 4024 ax-pow 4068 ax-pr 4101 ax-un 4325 ax-setind 4422 ax-iinf 4472 ax-cnex 7679 ax-resscn 7680 ax-1cn 7681 ax-1re 7682 ax-icn 7683 ax-addcl 7684 ax-addrcl 7685 ax-mulcl 7686 ax-mulrcl 7687 ax-addcom 7688 ax-mulcom 7689 ax-addass 7690 ax-mulass 7691 ax-distr 7692 ax-i2m1 7693 ax-0lt1 7694 ax-1rid 7695 ax-0id 7696 ax-rnegex 7697 ax-precex 7698 ax-cnre 7699 ax-pre-ltirr 7700 ax-pre-ltwlin 7701 ax-pre-lttrn 7702 ax-pre-apti 7703 ax-pre-ltadd 7704 ax-pre-mulgt0 7705 ax-pre-mulext 7706 ax-arch 7707 ax-caucvg 7708 |
This theorem depends on definitions: df-bi 116 df-dc 805 df-3or 948 df-3an 949 df-tru 1319 df-fal 1322 df-nf 1422 df-sb 1721 df-eu 1980 df-mo 1981 df-clab 2104 df-cleq 2110 df-clel 2113 df-nfc 2247 df-ne 2286 df-nel 2381 df-ral 2398 df-rex 2399 df-reu 2400 df-rmo 2401 df-rab 2402 df-v 2662 df-sbc 2883 df-csb 2976 df-dif 3043 df-un 3045 df-in 3047 df-ss 3054 df-nul 3334 df-if 3445 df-pw 3482 df-sn 3503 df-pr 3504 df-op 3506 df-uni 3707 df-int 3742 df-iun 3785 df-br 3900 df-opab 3960 df-mpt 3961 df-tr 3997 df-id 4185 df-po 4188 df-iso 4189 df-iord 4258 df-on 4260 df-ilim 4261 df-suc 4263 df-iom 4475 df-xp 4515 df-rel 4516 df-cnv 4517 df-co 4518 df-dm 4519 df-rn 4520 df-res 4521 df-ima 4522 df-iota 5058 df-fun 5095 df-fn 5096 df-f 5097 df-f1 5098 df-fo 5099 df-f1o 5100 df-fv 5101 df-isom 5102 df-riota 5698 df-ov 5745 df-oprab 5746 df-mpo 5747 df-1st 6006 df-2nd 6007 df-recs 6170 df-irdg 6235 df-frec 6256 df-1o 6281 df-oadd 6285 df-er 6397 df-en 6603 df-dom 6604 df-fin 6605 df-pnf 7770 df-mnf 7771 df-xr 7772 df-ltxr 7773 df-le 7774 df-sub 7903 df-neg 7904 df-reap 8305 df-ap 8312 df-div 8401 df-inn 8689 df-2 8747 df-3 8748 df-4 8749 df-n0 8946 df-z 9023 df-uz 9295 df-q 9380 df-rp 9410 df-ico 9645 df-fz 9759 df-fzo 9888 df-seqfrec 10187 df-exp 10261 df-fac 10440 df-ihash 10490 df-cj 10582 df-re 10583 df-im 10584 df-rsqrt 10738 df-abs 10739 df-clim 11016 df-sumdc 11091 df-ef 11281 df-sin 11283 df-cos 11284 df-tan 11285 |
This theorem is referenced by: tanclap 11343 tanval2ap 11347 retanclap 11356 tannegap 11362 tan0 11365 tanaddaplem 11372 tanaddap 11373 tanrpcl 12845 tangtx 12846 tan4thpi 12849 |
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