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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 11052 | . . . . 5 ⊢ (𝐴 ∈ ℂ → (cos‘𝐴) ∈ ℂ) | |
3 | 2 | adantr 271 | . . . 4 ⊢ ((𝐴 ∈ ℂ ∧ (cos‘𝐴) # 0) → (cos‘𝐴) ∈ ℂ) |
4 | simpr 109 | . . . . 5 ⊢ ((𝐴 ∈ ℂ ∧ (cos‘𝐴) # 0) → (cos‘𝐴) # 0) | |
5 | 0cnd 7535 | . . . . . 6 ⊢ ((𝐴 ∈ ℂ ∧ (cos‘𝐴) # 0) → 0 ∈ ℂ) | |
6 | apne 8154 | . . . . . 6 ⊢ (((cos‘𝐴) ∈ ℂ ∧ 0 ∈ ℂ) → ((cos‘𝐴) # 0 → (cos‘𝐴) ≠ 0)) | |
7 | 3, 5, 6 | syl2anc 404 | . . . . 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 3573 | . . . 4 ⊢ ((cos‘𝐴) ∈ (ℂ ∖ {0}) ↔ ((cos‘𝐴) ∈ ℂ ∧ (cos‘𝐴) ≠ 0)) | |
10 | 3, 8, 9 | sylanbrc 409 | . . 3 ⊢ ((𝐴 ∈ ℂ ∧ (cos‘𝐴) # 0) → (cos‘𝐴) ∈ (ℂ ∖ {0})) |
11 | cosf 11050 | . . . 4 ⊢ cos:ℂ⟶ℂ | |
12 | ffn 5174 | . . . 4 ⊢ (cos:ℂ⟶ℂ → cos Fn ℂ) | |
13 | elpreima 5432 | . . . 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 409 | . 2 ⊢ ((𝐴 ∈ ℂ ∧ (cos‘𝐴) # 0) → 𝐴 ∈ (◡cos “ (ℂ ∖ {0}))) |
16 | sincl 11051 | . . . 4 ⊢ (𝐴 ∈ ℂ → (sin‘𝐴) ∈ ℂ) | |
17 | 16 | adantr 271 | . . 3 ⊢ ((𝐴 ∈ ℂ ∧ (cos‘𝐴) # 0) → (sin‘𝐴) ∈ ℂ) |
18 | 17, 3, 4 | divclapd 8311 | . 2 ⊢ ((𝐴 ∈ ℂ ∧ (cos‘𝐴) # 0) → ((sin‘𝐴) / (cos‘𝐴)) ∈ ℂ) |
19 | fveq2 5318 | . . . 4 ⊢ (𝑥 = 𝐴 → (sin‘𝑥) = (sin‘𝐴)) | |
20 | fveq2 5318 | . . . 4 ⊢ (𝑥 = 𝐴 → (cos‘𝑥) = (cos‘𝐴)) | |
21 | 19, 20 | oveq12d 5684 | . . 3 ⊢ (𝑥 = 𝐴 → ((sin‘𝑥) / (cos‘𝑥)) = ((sin‘𝐴) / (cos‘𝐴))) |
22 | df-tan 10996 | . . 3 ⊢ tan = (𝑥 ∈ (◡cos “ (ℂ ∖ {0})) ↦ ((sin‘𝑥) / (cos‘𝑥))) | |
23 | 21, 22 | fvmptg 5393 | . 2 ⊢ ((𝐴 ∈ (◡cos “ (ℂ ∖ {0})) ∧ ((sin‘𝐴) / (cos‘𝐴)) ∈ ℂ) → (tan‘𝐴) = ((sin‘𝐴) / (cos‘𝐴))) |
24 | 15, 18, 23 | syl2anc 404 | 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 1290 ∈ wcel 1439 ≠ wne 2256 ∖ cdif 2997 {csn 3450 class class class wbr 3851 ◡ccnv 4450 “ cima 4454 Fn wfn 5023 ⟶wf 5024 ‘cfv 5028 (class class class)co 5666 ℂcc 7402 0cc0 7404 # cap 8112 / cdiv 8193 sincsin 10988 cosccos 10989 tanctan 10990 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 580 ax-in2 581 ax-io 666 ax-5 1382 ax-7 1383 ax-gen 1384 ax-ie1 1428 ax-ie2 1429 ax-8 1441 ax-10 1442 ax-11 1443 ax-i12 1444 ax-bndl 1445 ax-4 1446 ax-13 1450 ax-14 1451 ax-17 1465 ax-i9 1469 ax-ial 1473 ax-i5r 1474 ax-ext 2071 ax-coll 3960 ax-sep 3963 ax-nul 3971 ax-pow 4015 ax-pr 4045 ax-un 4269 ax-setind 4366 ax-iinf 4416 ax-cnex 7490 ax-resscn 7491 ax-1cn 7492 ax-1re 7493 ax-icn 7494 ax-addcl 7495 ax-addrcl 7496 ax-mulcl 7497 ax-mulrcl 7498 ax-addcom 7499 ax-mulcom 7500 ax-addass 7501 ax-mulass 7502 ax-distr 7503 ax-i2m1 7504 ax-0lt1 7505 ax-1rid 7506 ax-0id 7507 ax-rnegex 7508 ax-precex 7509 ax-cnre 7510 ax-pre-ltirr 7511 ax-pre-ltwlin 7512 ax-pre-lttrn 7513 ax-pre-apti 7514 ax-pre-ltadd 7515 ax-pre-mulgt0 7516 ax-pre-mulext 7517 ax-arch 7518 ax-caucvg 7519 |
This theorem depends on definitions: df-bi 116 df-dc 782 df-3or 926 df-3an 927 df-tru 1293 df-fal 1296 df-nf 1396 df-sb 1694 df-eu 1952 df-mo 1953 df-clab 2076 df-cleq 2082 df-clel 2085 df-nfc 2218 df-ne 2257 df-nel 2352 df-ral 2365 df-rex 2366 df-reu 2367 df-rmo 2368 df-rab 2369 df-v 2622 df-sbc 2842 df-csb 2935 df-dif 3002 df-un 3004 df-in 3006 df-ss 3013 df-nul 3288 df-if 3398 df-pw 3435 df-sn 3456 df-pr 3457 df-op 3459 df-uni 3660 df-int 3695 df-iun 3738 df-br 3852 df-opab 3906 df-mpt 3907 df-tr 3943 df-id 4129 df-po 4132 df-iso 4133 df-iord 4202 df-on 4204 df-ilim 4205 df-suc 4207 df-iom 4419 df-xp 4457 df-rel 4458 df-cnv 4459 df-co 4460 df-dm 4461 df-rn 4462 df-res 4463 df-ima 4464 df-iota 4993 df-fun 5030 df-fn 5031 df-f 5032 df-f1 5033 df-fo 5034 df-f1o 5035 df-fv 5036 df-isom 5037 df-riota 5622 df-ov 5669 df-oprab 5670 df-mpt2 5671 df-1st 5925 df-2nd 5926 df-recs 6084 df-irdg 6149 df-frec 6170 df-1o 6195 df-oadd 6199 df-er 6306 df-en 6512 df-dom 6513 df-fin 6514 df-pnf 7578 df-mnf 7579 df-xr 7580 df-ltxr 7581 df-le 7582 df-sub 7709 df-neg 7710 df-reap 8106 df-ap 8113 df-div 8194 df-inn 8477 df-2 8535 df-3 8536 df-4 8537 df-n0 8728 df-z 8805 df-uz 9074 df-q 9159 df-rp 9189 df-ico 9366 df-fz 9479 df-fzo 9608 df-iseq 9907 df-seq3 9908 df-exp 10009 df-fac 10188 df-ihash 10238 df-cj 10330 df-re 10331 df-im 10332 df-rsqrt 10485 df-abs 10486 df-clim 10721 df-isum 10797 df-ef 10992 df-sin 10994 df-cos 10995 df-tan 10996 |
This theorem is referenced by: tanclap 11054 tanval2ap 11058 retanclap 11067 tannegap 11073 tan0 11076 tanaddaplem 11083 tanaddap 11084 |
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