Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > atandmtan | Structured version Visualization version GIF version |
Description: The tangent function has range contained in the domain of the arctangent. (Contributed by Mario Carneiro, 31-Mar-2015.) |
Ref | Expression |
---|---|
atandmtan | ⊢ ((𝐴 ∈ ℂ ∧ (cos‘𝐴) ≠ 0) → (tan‘𝐴) ∈ dom arctan) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | tancl 15482 | . 2 ⊢ ((𝐴 ∈ ℂ ∧ (cos‘𝐴) ≠ 0) → (tan‘𝐴) ∈ ℂ) | |
2 | tanval 15481 | . . . . 5 ⊢ ((𝐴 ∈ ℂ ∧ (cos‘𝐴) ≠ 0) → (tan‘𝐴) = ((sin‘𝐴) / (cos‘𝐴))) | |
3 | 2 | oveq1d 7171 | . . . 4 ⊢ ((𝐴 ∈ ℂ ∧ (cos‘𝐴) ≠ 0) → ((tan‘𝐴)↑2) = (((sin‘𝐴) / (cos‘𝐴))↑2)) |
4 | sincl 15479 | . . . . . 6 ⊢ (𝐴 ∈ ℂ → (sin‘𝐴) ∈ ℂ) | |
5 | 4 | adantr 483 | . . . . 5 ⊢ ((𝐴 ∈ ℂ ∧ (cos‘𝐴) ≠ 0) → (sin‘𝐴) ∈ ℂ) |
6 | coscl 15480 | . . . . . 6 ⊢ (𝐴 ∈ ℂ → (cos‘𝐴) ∈ ℂ) | |
7 | 6 | adantr 483 | . . . . 5 ⊢ ((𝐴 ∈ ℂ ∧ (cos‘𝐴) ≠ 0) → (cos‘𝐴) ∈ ℂ) |
8 | simpr 487 | . . . . 5 ⊢ ((𝐴 ∈ ℂ ∧ (cos‘𝐴) ≠ 0) → (cos‘𝐴) ≠ 0) | |
9 | 5, 7, 8 | sqdivd 13524 | . . . 4 ⊢ ((𝐴 ∈ ℂ ∧ (cos‘𝐴) ≠ 0) → (((sin‘𝐴) / (cos‘𝐴))↑2) = (((sin‘𝐴)↑2) / ((cos‘𝐴)↑2))) |
10 | 3, 9 | eqtrd 2856 | . . 3 ⊢ ((𝐴 ∈ ℂ ∧ (cos‘𝐴) ≠ 0) → ((tan‘𝐴)↑2) = (((sin‘𝐴)↑2) / ((cos‘𝐴)↑2))) |
11 | 5 | sqcld 13509 | . . . . . 6 ⊢ ((𝐴 ∈ ℂ ∧ (cos‘𝐴) ≠ 0) → ((sin‘𝐴)↑2) ∈ ℂ) |
12 | 7 | sqcld 13509 | . . . . . . 7 ⊢ ((𝐴 ∈ ℂ ∧ (cos‘𝐴) ≠ 0) → ((cos‘𝐴)↑2) ∈ ℂ) |
13 | 12 | negcld 10984 | . . . . . 6 ⊢ ((𝐴 ∈ ℂ ∧ (cos‘𝐴) ≠ 0) → -((cos‘𝐴)↑2) ∈ ℂ) |
14 | 11, 12 | subnegd 11004 | . . . . . . . 8 ⊢ ((𝐴 ∈ ℂ ∧ (cos‘𝐴) ≠ 0) → (((sin‘𝐴)↑2) − -((cos‘𝐴)↑2)) = (((sin‘𝐴)↑2) + ((cos‘𝐴)↑2))) |
15 | sincossq 15529 | . . . . . . . . 9 ⊢ (𝐴 ∈ ℂ → (((sin‘𝐴)↑2) + ((cos‘𝐴)↑2)) = 1) | |
16 | 15 | adantr 483 | . . . . . . . 8 ⊢ ((𝐴 ∈ ℂ ∧ (cos‘𝐴) ≠ 0) → (((sin‘𝐴)↑2) + ((cos‘𝐴)↑2)) = 1) |
17 | 14, 16 | eqtrd 2856 | . . . . . . 7 ⊢ ((𝐴 ∈ ℂ ∧ (cos‘𝐴) ≠ 0) → (((sin‘𝐴)↑2) − -((cos‘𝐴)↑2)) = 1) |
18 | ax-1ne0 10606 | . . . . . . . 8 ⊢ 1 ≠ 0 | |
19 | 18 | a1i 11 | . . . . . . 7 ⊢ ((𝐴 ∈ ℂ ∧ (cos‘𝐴) ≠ 0) → 1 ≠ 0) |
20 | 17, 19 | eqnetrd 3083 | . . . . . 6 ⊢ ((𝐴 ∈ ℂ ∧ (cos‘𝐴) ≠ 0) → (((sin‘𝐴)↑2) − -((cos‘𝐴)↑2)) ≠ 0) |
21 | 11, 13, 20 | subne0ad 11008 | . . . . 5 ⊢ ((𝐴 ∈ ℂ ∧ (cos‘𝐴) ≠ 0) → ((sin‘𝐴)↑2) ≠ -((cos‘𝐴)↑2)) |
22 | 12 | mulm1d 11092 | . . . . 5 ⊢ ((𝐴 ∈ ℂ ∧ (cos‘𝐴) ≠ 0) → (-1 · ((cos‘𝐴)↑2)) = -((cos‘𝐴)↑2)) |
23 | 21, 22 | neeqtrrd 3090 | . . . 4 ⊢ ((𝐴 ∈ ℂ ∧ (cos‘𝐴) ≠ 0) → ((sin‘𝐴)↑2) ≠ (-1 · ((cos‘𝐴)↑2))) |
24 | neg1cn 11752 | . . . . . . 7 ⊢ -1 ∈ ℂ | |
25 | 24 | a1i 11 | . . . . . 6 ⊢ ((𝐴 ∈ ℂ ∧ (cos‘𝐴) ≠ 0) → -1 ∈ ℂ) |
26 | sqne0 13490 | . . . . . . . 8 ⊢ ((cos‘𝐴) ∈ ℂ → (((cos‘𝐴)↑2) ≠ 0 ↔ (cos‘𝐴) ≠ 0)) | |
27 | 6, 26 | syl 17 | . . . . . . 7 ⊢ (𝐴 ∈ ℂ → (((cos‘𝐴)↑2) ≠ 0 ↔ (cos‘𝐴) ≠ 0)) |
28 | 27 | biimpar 480 | . . . . . 6 ⊢ ((𝐴 ∈ ℂ ∧ (cos‘𝐴) ≠ 0) → ((cos‘𝐴)↑2) ≠ 0) |
29 | 11, 25, 12, 28 | divmul3d 11450 | . . . . 5 ⊢ ((𝐴 ∈ ℂ ∧ (cos‘𝐴) ≠ 0) → ((((sin‘𝐴)↑2) / ((cos‘𝐴)↑2)) = -1 ↔ ((sin‘𝐴)↑2) = (-1 · ((cos‘𝐴)↑2)))) |
30 | 29 | necon3bid 3060 | . . . 4 ⊢ ((𝐴 ∈ ℂ ∧ (cos‘𝐴) ≠ 0) → ((((sin‘𝐴)↑2) / ((cos‘𝐴)↑2)) ≠ -1 ↔ ((sin‘𝐴)↑2) ≠ (-1 · ((cos‘𝐴)↑2)))) |
31 | 23, 30 | mpbird 259 | . . 3 ⊢ ((𝐴 ∈ ℂ ∧ (cos‘𝐴) ≠ 0) → (((sin‘𝐴)↑2) / ((cos‘𝐴)↑2)) ≠ -1) |
32 | 10, 31 | eqnetrd 3083 | . 2 ⊢ ((𝐴 ∈ ℂ ∧ (cos‘𝐴) ≠ 0) → ((tan‘𝐴)↑2) ≠ -1) |
33 | atandm3 25456 | . 2 ⊢ ((tan‘𝐴) ∈ dom arctan ↔ ((tan‘𝐴) ∈ ℂ ∧ ((tan‘𝐴)↑2) ≠ -1)) | |
34 | 1, 32, 33 | sylanbrc 585 | 1 ⊢ ((𝐴 ∈ ℂ ∧ (cos‘𝐴) ≠ 0) → (tan‘𝐴) ∈ dom arctan) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 = wceq 1537 ∈ wcel 2114 ≠ wne 3016 dom cdm 5555 ‘cfv 6355 (class class class)co 7156 ℂcc 10535 0cc0 10537 1c1 10538 + caddc 10540 · cmul 10542 − cmin 10870 -cneg 10871 / cdiv 11297 2c2 11693 ↑cexp 13430 sincsin 15417 cosccos 15418 tanctan 15419 arctancatan 25442 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-rep 5190 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 ax-inf2 9104 ax-cnex 10593 ax-resscn 10594 ax-1cn 10595 ax-icn 10596 ax-addcl 10597 ax-addrcl 10598 ax-mulcl 10599 ax-mulrcl 10600 ax-mulcom 10601 ax-addass 10602 ax-mulass 10603 ax-distr 10604 ax-i2m1 10605 ax-1ne0 10606 ax-1rid 10607 ax-rnegex 10608 ax-rrecex 10609 ax-cnre 10610 ax-pre-lttri 10611 ax-pre-lttrn 10612 ax-pre-ltadd 10613 ax-pre-mulgt0 10614 ax-pre-sup 10615 ax-addf 10616 ax-mulf 10617 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-fal 1550 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3496 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-pss 3954 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-tp 4572 df-op 4574 df-uni 4839 df-int 4877 df-iun 4921 df-br 5067 df-opab 5129 df-mpt 5147 df-tr 5173 df-id 5460 df-eprel 5465 df-po 5474 df-so 5475 df-fr 5514 df-se 5515 df-we 5516 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-pred 6148 df-ord 6194 df-on 6195 df-lim 6196 df-suc 6197 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-f1 6360 df-fo 6361 df-f1o 6362 df-fv 6363 df-isom 6364 df-riota 7114 df-ov 7159 df-oprab 7160 df-mpo 7161 df-om 7581 df-1st 7689 df-2nd 7690 df-wrecs 7947 df-recs 8008 df-rdg 8046 df-1o 8102 df-oadd 8106 df-er 8289 df-pm 8409 df-en 8510 df-dom 8511 df-sdom 8512 df-fin 8513 df-sup 8906 df-inf 8907 df-oi 8974 df-card 9368 df-pnf 10677 df-mnf 10678 df-xr 10679 df-ltxr 10680 df-le 10681 df-sub 10872 df-neg 10873 df-div 11298 df-nn 11639 df-2 11701 df-3 11702 df-n0 11899 df-z 11983 df-uz 12245 df-rp 12391 df-ico 12745 df-fz 12894 df-fzo 13035 df-fl 13163 df-seq 13371 df-exp 13431 df-fac 13635 df-bc 13664 df-hash 13692 df-shft 14426 df-cj 14458 df-re 14459 df-im 14460 df-sqrt 14594 df-abs 14595 df-limsup 14828 df-clim 14845 df-rlim 14846 df-sum 15043 df-ef 15421 df-sin 15423 df-cos 15424 df-tan 15425 df-atan 25445 |
This theorem is referenced by: atantan 25501 |
Copyright terms: Public domain | W3C validator |