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Mirrors > Home > MPE Home > Th. List > atandm4 | Structured version Visualization version GIF version |
Description: A compact form of atandm 26933. (Contributed by Mario Carneiro, 3-Apr-2015.) |
Ref | Expression |
---|---|
atandm4 | ⊢ (𝐴 ∈ dom arctan ↔ (𝐴 ∈ ℂ ∧ (1 + (𝐴↑2)) ≠ 0)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | atandm3 26935 | . 2 ⊢ (𝐴 ∈ dom arctan ↔ (𝐴 ∈ ℂ ∧ (𝐴↑2) ≠ -1)) | |
2 | sqcl 14154 | . . . . . 6 ⊢ (𝐴 ∈ ℂ → (𝐴↑2) ∈ ℂ) | |
3 | neg1cn 12377 | . . . . . 6 ⊢ -1 ∈ ℂ | |
4 | subeq0 11532 | . . . . . 6 ⊢ (((𝐴↑2) ∈ ℂ ∧ -1 ∈ ℂ) → (((𝐴↑2) − -1) = 0 ↔ (𝐴↑2) = -1)) | |
5 | 2, 3, 4 | sylancl 586 | . . . . 5 ⊢ (𝐴 ∈ ℂ → (((𝐴↑2) − -1) = 0 ↔ (𝐴↑2) = -1)) |
6 | ax-1cn 11210 | . . . . . . . 8 ⊢ 1 ∈ ℂ | |
7 | subneg 11555 | . . . . . . . 8 ⊢ (((𝐴↑2) ∈ ℂ ∧ 1 ∈ ℂ) → ((𝐴↑2) − -1) = ((𝐴↑2) + 1)) | |
8 | 2, 6, 7 | sylancl 586 | . . . . . . 7 ⊢ (𝐴 ∈ ℂ → ((𝐴↑2) − -1) = ((𝐴↑2) + 1)) |
9 | addcom 11444 | . . . . . . . 8 ⊢ (((𝐴↑2) ∈ ℂ ∧ 1 ∈ ℂ) → ((𝐴↑2) + 1) = (1 + (𝐴↑2))) | |
10 | 2, 6, 9 | sylancl 586 | . . . . . . 7 ⊢ (𝐴 ∈ ℂ → ((𝐴↑2) + 1) = (1 + (𝐴↑2))) |
11 | 8, 10 | eqtrd 2774 | . . . . . 6 ⊢ (𝐴 ∈ ℂ → ((𝐴↑2) − -1) = (1 + (𝐴↑2))) |
12 | 11 | eqeq1d 2736 | . . . . 5 ⊢ (𝐴 ∈ ℂ → (((𝐴↑2) − -1) = 0 ↔ (1 + (𝐴↑2)) = 0)) |
13 | 5, 12 | bitr3d 281 | . . . 4 ⊢ (𝐴 ∈ ℂ → ((𝐴↑2) = -1 ↔ (1 + (𝐴↑2)) = 0)) |
14 | 13 | necon3bid 2982 | . . 3 ⊢ (𝐴 ∈ ℂ → ((𝐴↑2) ≠ -1 ↔ (1 + (𝐴↑2)) ≠ 0)) |
15 | 14 | pm5.32i 574 | . 2 ⊢ ((𝐴 ∈ ℂ ∧ (𝐴↑2) ≠ -1) ↔ (𝐴 ∈ ℂ ∧ (1 + (𝐴↑2)) ≠ 0)) |
16 | 1, 15 | bitri 275 | 1 ⊢ (𝐴 ∈ dom arctan ↔ (𝐴 ∈ ℂ ∧ (1 + (𝐴↑2)) ≠ 0)) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 206 ∧ wa 395 = wceq 1536 ∈ wcel 2105 ≠ wne 2937 dom cdm 5688 (class class class)co 7430 ℂcc 11150 0cc0 11152 1c1 11153 + caddc 11155 − cmin 11489 -cneg 11490 2c2 12318 ↑cexp 14098 arctancatan 26921 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1791 ax-4 1805 ax-5 1907 ax-6 1964 ax-7 2004 ax-8 2107 ax-9 2115 ax-10 2138 ax-11 2154 ax-12 2174 ax-ext 2705 ax-sep 5301 ax-nul 5311 ax-pow 5370 ax-pr 5437 ax-un 7753 ax-cnex 11208 ax-resscn 11209 ax-1cn 11210 ax-icn 11211 ax-addcl 11212 ax-addrcl 11213 ax-mulcl 11214 ax-mulrcl 11215 ax-mulcom 11216 ax-addass 11217 ax-mulass 11218 ax-distr 11219 ax-i2m1 11220 ax-1ne0 11221 ax-1rid 11222 ax-rnegex 11223 ax-rrecex 11224 ax-cnre 11225 ax-pre-lttri 11226 ax-pre-lttrn 11227 ax-pre-ltadd 11228 ax-pre-mulgt0 11229 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1539 df-fal 1549 df-ex 1776 df-nf 1780 df-sb 2062 df-mo 2537 df-eu 2566 df-clab 2712 df-cleq 2726 df-clel 2813 df-nfc 2889 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-reu 3378 df-rab 3433 df-v 3479 df-sbc 3791 df-csb 3908 df-dif 3965 df-un 3967 df-in 3969 df-ss 3979 df-pss 3982 df-nul 4339 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4912 df-iun 4997 df-br 5148 df-opab 5210 df-mpt 5231 df-tr 5265 df-id 5582 df-eprel 5588 df-po 5596 df-so 5597 df-fr 5640 df-we 5642 df-xp 5694 df-rel 5695 df-cnv 5696 df-co 5697 df-dm 5698 df-rn 5699 df-res 5700 df-ima 5701 df-pred 6322 df-ord 6388 df-on 6389 df-lim 6390 df-suc 6391 df-iota 6515 df-fun 6564 df-fn 6565 df-f 6566 df-f1 6567 df-fo 6568 df-f1o 6569 df-fv 6570 df-riota 7387 df-ov 7433 df-oprab 7434 df-mpo 7435 df-om 7887 df-2nd 8013 df-frecs 8304 df-wrecs 8335 df-recs 8409 df-rdg 8448 df-er 8743 df-en 8984 df-dom 8985 df-sdom 8986 df-pnf 11294 df-mnf 11295 df-xr 11296 df-ltxr 11297 df-le 11298 df-sub 11491 df-neg 11492 df-nn 12264 df-2 12326 df-n0 12524 df-z 12611 df-uz 12876 df-seq 14039 df-exp 14099 df-atan 26924 |
This theorem is referenced by: efiatan2 26974 cosatan 26978 cosatanne0 26979 atansssdm 26990 dvatan 26992 |
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