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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 109 | . . 3 ⊢ ((𝐴 ∈ ℂ ∧ (cos‘𝐴) # 0) → 𝐴 ∈ ℂ) | |
2 | coscl 11746 | . . . . 5 ⊢ (𝐴 ∈ ℂ → (cos‘𝐴) ∈ ℂ) | |
3 | 2 | adantr 276 | . . . 4 ⊢ ((𝐴 ∈ ℂ ∧ (cos‘𝐴) # 0) → (cos‘𝐴) ∈ ℂ) |
4 | simpr 110 | . . . . 5 ⊢ ((𝐴 ∈ ℂ ∧ (cos‘𝐴) # 0) → (cos‘𝐴) # 0) | |
5 | 0cnd 7979 | . . . . . 6 ⊢ ((𝐴 ∈ ℂ ∧ (cos‘𝐴) # 0) → 0 ∈ ℂ) | |
6 | apne 8609 | . . . . . 6 ⊢ (((cos‘𝐴) ∈ ℂ ∧ 0 ∈ ℂ) → ((cos‘𝐴) # 0 → (cos‘𝐴) ≠ 0)) | |
7 | 3, 5, 6 | syl2anc 411 | . . . . 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 3734 | . . . 4 ⊢ ((cos‘𝐴) ∈ (ℂ ∖ {0}) ↔ ((cos‘𝐴) ∈ ℂ ∧ (cos‘𝐴) ≠ 0)) | |
10 | 3, 8, 9 | sylanbrc 417 | . . 3 ⊢ ((𝐴 ∈ ℂ ∧ (cos‘𝐴) # 0) → (cos‘𝐴) ∈ (ℂ ∖ {0})) |
11 | cosf 11744 | . . . 4 ⊢ cos:ℂ⟶ℂ | |
12 | ffn 5384 | . . . 4 ⊢ (cos:ℂ⟶ℂ → cos Fn ℂ) | |
13 | elpreima 5655 | . . . 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 417 | . 2 ⊢ ((𝐴 ∈ ℂ ∧ (cos‘𝐴) # 0) → 𝐴 ∈ (◡cos “ (ℂ ∖ {0}))) |
16 | sincl 11745 | . . . 4 ⊢ (𝐴 ∈ ℂ → (sin‘𝐴) ∈ ℂ) | |
17 | 16 | adantr 276 | . . 3 ⊢ ((𝐴 ∈ ℂ ∧ (cos‘𝐴) # 0) → (sin‘𝐴) ∈ ℂ) |
18 | 17, 3, 4 | divclapd 8776 | . 2 ⊢ ((𝐴 ∈ ℂ ∧ (cos‘𝐴) # 0) → ((sin‘𝐴) / (cos‘𝐴)) ∈ ℂ) |
19 | fveq2 5534 | . . . 4 ⊢ (𝑥 = 𝐴 → (sin‘𝑥) = (sin‘𝐴)) | |
20 | fveq2 5534 | . . . 4 ⊢ (𝑥 = 𝐴 → (cos‘𝑥) = (cos‘𝐴)) | |
21 | 19, 20 | oveq12d 5913 | . . 3 ⊢ (𝑥 = 𝐴 → ((sin‘𝑥) / (cos‘𝑥)) = ((sin‘𝐴) / (cos‘𝐴))) |
22 | df-tan 11691 | . . 3 ⊢ tan = (𝑥 ∈ (◡cos “ (ℂ ∖ {0})) ↦ ((sin‘𝑥) / (cos‘𝑥))) | |
23 | 21, 22 | fvmptg 5612 | . 2 ⊢ ((𝐴 ∈ (◡cos “ (ℂ ∖ {0})) ∧ ((sin‘𝐴) / (cos‘𝐴)) ∈ ℂ) → (tan‘𝐴) = ((sin‘𝐴) / (cos‘𝐴))) |
24 | 15, 18, 23 | syl2anc 411 | 1 ⊢ ((𝐴 ∈ ℂ ∧ (cos‘𝐴) # 0) → (tan‘𝐴) = ((sin‘𝐴) / (cos‘𝐴))) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 104 ↔ wb 105 = wceq 1364 ∈ wcel 2160 ≠ wne 2360 ∖ cdif 3141 {csn 3607 class class class wbr 4018 ◡ccnv 4643 “ cima 4647 Fn wfn 5230 ⟶wf 5231 ‘cfv 5235 (class class class)co 5895 ℂcc 7838 0cc0 7840 # cap 8567 / cdiv 8658 sincsin 11683 cosccos 11684 tanctan 11685 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 615 ax-in2 616 ax-io 710 ax-5 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-13 2162 ax-14 2163 ax-ext 2171 ax-coll 4133 ax-sep 4136 ax-nul 4144 ax-pow 4192 ax-pr 4227 ax-un 4451 ax-setind 4554 ax-iinf 4605 ax-cnex 7931 ax-resscn 7932 ax-1cn 7933 ax-1re 7934 ax-icn 7935 ax-addcl 7936 ax-addrcl 7937 ax-mulcl 7938 ax-mulrcl 7939 ax-addcom 7940 ax-mulcom 7941 ax-addass 7942 ax-mulass 7943 ax-distr 7944 ax-i2m1 7945 ax-0lt1 7946 ax-1rid 7947 ax-0id 7948 ax-rnegex 7949 ax-precex 7950 ax-cnre 7951 ax-pre-ltirr 7952 ax-pre-ltwlin 7953 ax-pre-lttrn 7954 ax-pre-apti 7955 ax-pre-ltadd 7956 ax-pre-mulgt0 7957 ax-pre-mulext 7958 ax-arch 7959 ax-caucvg 7960 |
This theorem depends on definitions: df-bi 117 df-dc 836 df-3or 981 df-3an 982 df-tru 1367 df-fal 1370 df-nf 1472 df-sb 1774 df-eu 2041 df-mo 2042 df-clab 2176 df-cleq 2182 df-clel 2185 df-nfc 2321 df-ne 2361 df-nel 2456 df-ral 2473 df-rex 2474 df-reu 2475 df-rmo 2476 df-rab 2477 df-v 2754 df-sbc 2978 df-csb 3073 df-dif 3146 df-un 3148 df-in 3150 df-ss 3157 df-nul 3438 df-if 3550 df-pw 3592 df-sn 3613 df-pr 3614 df-op 3616 df-uni 3825 df-int 3860 df-iun 3903 df-br 4019 df-opab 4080 df-mpt 4081 df-tr 4117 df-id 4311 df-po 4314 df-iso 4315 df-iord 4384 df-on 4386 df-ilim 4387 df-suc 4389 df-iom 4608 df-xp 4650 df-rel 4651 df-cnv 4652 df-co 4653 df-dm 4654 df-rn 4655 df-res 4656 df-ima 4657 df-iota 5196 df-fun 5237 df-fn 5238 df-f 5239 df-f1 5240 df-fo 5241 df-f1o 5242 df-fv 5243 df-isom 5244 df-riota 5851 df-ov 5898 df-oprab 5899 df-mpo 5900 df-1st 6164 df-2nd 6165 df-recs 6329 df-irdg 6394 df-frec 6415 df-1o 6440 df-oadd 6444 df-er 6558 df-en 6766 df-dom 6767 df-fin 6768 df-pnf 8023 df-mnf 8024 df-xr 8025 df-ltxr 8026 df-le 8027 df-sub 8159 df-neg 8160 df-reap 8561 df-ap 8568 df-div 8659 df-inn 8949 df-2 9007 df-3 9008 df-4 9009 df-n0 9206 df-z 9283 df-uz 9558 df-q 9649 df-rp 9683 df-ico 9923 df-fz 10038 df-fzo 10172 df-seqfrec 10476 df-exp 10550 df-fac 10737 df-ihash 10787 df-cj 10882 df-re 10883 df-im 10884 df-rsqrt 11038 df-abs 11039 df-clim 11318 df-sumdc 11393 df-ef 11687 df-sin 11689 df-cos 11690 df-tan 11691 |
This theorem is referenced by: tanclap 11748 tanval2ap 11752 retanclap 11761 tannegap 11767 tan0 11770 tanaddaplem 11777 tanaddap 11778 tanrpcl 14710 tangtx 14711 tan4thpi 14714 |
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