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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 15474 | . 2 ⊢ ((𝐴 ∈ ℂ ∧ (cos‘𝐴) ≠ 0) → (tan‘𝐴) ∈ ℂ) | |
2 | tanval 15473 | . . . . 5 ⊢ ((𝐴 ∈ ℂ ∧ (cos‘𝐴) ≠ 0) → (tan‘𝐴) = ((sin‘𝐴) / (cos‘𝐴))) | |
3 | 2 | oveq1d 7150 | . . . 4 ⊢ ((𝐴 ∈ ℂ ∧ (cos‘𝐴) ≠ 0) → ((tan‘𝐴)↑2) = (((sin‘𝐴) / (cos‘𝐴))↑2)) |
4 | sincl 15471 | . . . . . 6 ⊢ (𝐴 ∈ ℂ → (sin‘𝐴) ∈ ℂ) | |
5 | 4 | adantr 484 | . . . . 5 ⊢ ((𝐴 ∈ ℂ ∧ (cos‘𝐴) ≠ 0) → (sin‘𝐴) ∈ ℂ) |
6 | coscl 15472 | . . . . . 6 ⊢ (𝐴 ∈ ℂ → (cos‘𝐴) ∈ ℂ) | |
7 | 6 | adantr 484 | . . . . 5 ⊢ ((𝐴 ∈ ℂ ∧ (cos‘𝐴) ≠ 0) → (cos‘𝐴) ∈ ℂ) |
8 | simpr 488 | . . . . 5 ⊢ ((𝐴 ∈ ℂ ∧ (cos‘𝐴) ≠ 0) → (cos‘𝐴) ≠ 0) | |
9 | 5, 7, 8 | sqdivd 13519 | . . . 4 ⊢ ((𝐴 ∈ ℂ ∧ (cos‘𝐴) ≠ 0) → (((sin‘𝐴) / (cos‘𝐴))↑2) = (((sin‘𝐴)↑2) / ((cos‘𝐴)↑2))) |
10 | 3, 9 | eqtrd 2833 | . . 3 ⊢ ((𝐴 ∈ ℂ ∧ (cos‘𝐴) ≠ 0) → ((tan‘𝐴)↑2) = (((sin‘𝐴)↑2) / ((cos‘𝐴)↑2))) |
11 | 5 | sqcld 13504 | . . . . . 6 ⊢ ((𝐴 ∈ ℂ ∧ (cos‘𝐴) ≠ 0) → ((sin‘𝐴)↑2) ∈ ℂ) |
12 | 7 | sqcld 13504 | . . . . . . 7 ⊢ ((𝐴 ∈ ℂ ∧ (cos‘𝐴) ≠ 0) → ((cos‘𝐴)↑2) ∈ ℂ) |
13 | 12 | negcld 10973 | . . . . . 6 ⊢ ((𝐴 ∈ ℂ ∧ (cos‘𝐴) ≠ 0) → -((cos‘𝐴)↑2) ∈ ℂ) |
14 | 11, 12 | subnegd 10993 | . . . . . . . 8 ⊢ ((𝐴 ∈ ℂ ∧ (cos‘𝐴) ≠ 0) → (((sin‘𝐴)↑2) − -((cos‘𝐴)↑2)) = (((sin‘𝐴)↑2) + ((cos‘𝐴)↑2))) |
15 | sincossq 15521 | . . . . . . . . 9 ⊢ (𝐴 ∈ ℂ → (((sin‘𝐴)↑2) + ((cos‘𝐴)↑2)) = 1) | |
16 | 15 | adantr 484 | . . . . . . . 8 ⊢ ((𝐴 ∈ ℂ ∧ (cos‘𝐴) ≠ 0) → (((sin‘𝐴)↑2) + ((cos‘𝐴)↑2)) = 1) |
17 | 14, 16 | eqtrd 2833 | . . . . . . 7 ⊢ ((𝐴 ∈ ℂ ∧ (cos‘𝐴) ≠ 0) → (((sin‘𝐴)↑2) − -((cos‘𝐴)↑2)) = 1) |
18 | ax-1ne0 10595 | . . . . . . . 8 ⊢ 1 ≠ 0 | |
19 | 18 | a1i 11 | . . . . . . 7 ⊢ ((𝐴 ∈ ℂ ∧ (cos‘𝐴) ≠ 0) → 1 ≠ 0) |
20 | 17, 19 | eqnetrd 3054 | . . . . . 6 ⊢ ((𝐴 ∈ ℂ ∧ (cos‘𝐴) ≠ 0) → (((sin‘𝐴)↑2) − -((cos‘𝐴)↑2)) ≠ 0) |
21 | 11, 13, 20 | subne0ad 10997 | . . . . 5 ⊢ ((𝐴 ∈ ℂ ∧ (cos‘𝐴) ≠ 0) → ((sin‘𝐴)↑2) ≠ -((cos‘𝐴)↑2)) |
22 | 12 | mulm1d 11081 | . . . . 5 ⊢ ((𝐴 ∈ ℂ ∧ (cos‘𝐴) ≠ 0) → (-1 · ((cos‘𝐴)↑2)) = -((cos‘𝐴)↑2)) |
23 | 21, 22 | neeqtrrd 3061 | . . . 4 ⊢ ((𝐴 ∈ ℂ ∧ (cos‘𝐴) ≠ 0) → ((sin‘𝐴)↑2) ≠ (-1 · ((cos‘𝐴)↑2))) |
24 | neg1cn 11739 | . . . . . . 7 ⊢ -1 ∈ ℂ | |
25 | 24 | a1i 11 | . . . . . 6 ⊢ ((𝐴 ∈ ℂ ∧ (cos‘𝐴) ≠ 0) → -1 ∈ ℂ) |
26 | sqne0 13485 | . . . . . . . 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 481 | . . . . . 6 ⊢ ((𝐴 ∈ ℂ ∧ (cos‘𝐴) ≠ 0) → ((cos‘𝐴)↑2) ≠ 0) |
29 | 11, 25, 12, 28 | divmul3d 11439 | . . . . 5 ⊢ ((𝐴 ∈ ℂ ∧ (cos‘𝐴) ≠ 0) → ((((sin‘𝐴)↑2) / ((cos‘𝐴)↑2)) = -1 ↔ ((sin‘𝐴)↑2) = (-1 · ((cos‘𝐴)↑2)))) |
30 | 29 | necon3bid 3031 | . . . 4 ⊢ ((𝐴 ∈ ℂ ∧ (cos‘𝐴) ≠ 0) → ((((sin‘𝐴)↑2) / ((cos‘𝐴)↑2)) ≠ -1 ↔ ((sin‘𝐴)↑2) ≠ (-1 · ((cos‘𝐴)↑2)))) |
31 | 23, 30 | mpbird 260 | . . 3 ⊢ ((𝐴 ∈ ℂ ∧ (cos‘𝐴) ≠ 0) → (((sin‘𝐴)↑2) / ((cos‘𝐴)↑2)) ≠ -1) |
32 | 10, 31 | eqnetrd 3054 | . 2 ⊢ ((𝐴 ∈ ℂ ∧ (cos‘𝐴) ≠ 0) → ((tan‘𝐴)↑2) ≠ -1) |
33 | atandm3 25464 | . 2 ⊢ ((tan‘𝐴) ∈ dom arctan ↔ ((tan‘𝐴) ∈ ℂ ∧ ((tan‘𝐴)↑2) ≠ -1)) | |
34 | 1, 32, 33 | sylanbrc 586 | 1 ⊢ ((𝐴 ∈ ℂ ∧ (cos‘𝐴) ≠ 0) → (tan‘𝐴) ∈ dom arctan) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 ∧ wa 399 = wceq 1538 ∈ wcel 2111 ≠ wne 2987 dom cdm 5519 ‘cfv 6324 (class class class)co 7135 ℂcc 10524 0cc0 10526 1c1 10527 + caddc 10529 · cmul 10531 − cmin 10859 -cneg 10860 / cdiv 11286 2c2 11680 ↑cexp 13425 sincsin 15409 cosccos 15410 tanctan 15411 arctancatan 25450 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-rep 5154 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 ax-inf2 9088 ax-cnex 10582 ax-resscn 10583 ax-1cn 10584 ax-icn 10585 ax-addcl 10586 ax-addrcl 10587 ax-mulcl 10588 ax-mulrcl 10589 ax-mulcom 10590 ax-addass 10591 ax-mulass 10592 ax-distr 10593 ax-i2m1 10594 ax-1ne0 10595 ax-1rid 10596 ax-rnegex 10597 ax-rrecex 10598 ax-cnre 10599 ax-pre-lttri 10600 ax-pre-lttrn 10601 ax-pre-ltadd 10602 ax-pre-mulgt0 10603 ax-pre-sup 10604 ax-addf 10605 ax-mulf 10606 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-fal 1551 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-nel 3092 df-ral 3111 df-rex 3112 df-reu 3113 df-rmo 3114 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-pss 3900 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-tp 4530 df-op 4532 df-uni 4801 df-int 4839 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-tr 5137 df-id 5425 df-eprel 5430 df-po 5438 df-so 5439 df-fr 5478 df-se 5479 df-we 5480 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-pred 6116 df-ord 6162 df-on 6163 df-lim 6164 df-suc 6165 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-isom 6333 df-riota 7093 df-ov 7138 df-oprab 7139 df-mpo 7140 df-om 7561 df-1st 7671 df-2nd 7672 df-wrecs 7930 df-recs 7991 df-rdg 8029 df-1o 8085 df-oadd 8089 df-er 8272 df-pm 8392 df-en 8493 df-dom 8494 df-sdom 8495 df-fin 8496 df-sup 8890 df-inf 8891 df-oi 8958 df-card 9352 df-pnf 10666 df-mnf 10667 df-xr 10668 df-ltxr 10669 df-le 10670 df-sub 10861 df-neg 10862 df-div 11287 df-nn 11626 df-2 11688 df-3 11689 df-n0 11886 df-z 11970 df-uz 12232 df-rp 12378 df-ico 12732 df-fz 12886 df-fzo 13029 df-fl 13157 df-seq 13365 df-exp 13426 df-fac 13630 df-bc 13659 df-hash 13687 df-shft 14418 df-cj 14450 df-re 14451 df-im 14452 df-sqrt 14586 df-abs 14587 df-limsup 14820 df-clim 14837 df-rlim 14838 df-sum 15035 df-ef 15413 df-sin 15415 df-cos 15416 df-tan 15417 df-atan 25453 |
This theorem is referenced by: atantan 25509 |
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