Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > atanre | Structured version Visualization version GIF version |
Description: A real number is in the domain of the arctangent function. (Contributed by Mario Carneiro, 31-Mar-2015.) |
Ref | Expression |
---|---|
atanre | ⊢ (𝐴 ∈ ℝ → 𝐴 ∈ dom arctan) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | recn 10971 | . 2 ⊢ (𝐴 ∈ ℝ → 𝐴 ∈ ℂ) | |
2 | neg1rr 12098 | . . . 4 ⊢ -1 ∈ ℝ | |
3 | 2 | a1i 11 | . . 3 ⊢ (𝐴 ∈ ℝ → -1 ∈ ℝ) |
4 | 0red 10988 | . . . 4 ⊢ (𝐴 ∈ ℝ → 0 ∈ ℝ) | |
5 | resqcl 13854 | . . . 4 ⊢ (𝐴 ∈ ℝ → (𝐴↑2) ∈ ℝ) | |
6 | neg1lt0 12100 | . . . . 5 ⊢ -1 < 0 | |
7 | 6 | a1i 11 | . . . 4 ⊢ (𝐴 ∈ ℝ → -1 < 0) |
8 | sqge0 13865 | . . . 4 ⊢ (𝐴 ∈ ℝ → 0 ≤ (𝐴↑2)) | |
9 | 3, 4, 5, 7, 8 | ltletrd 11145 | . . 3 ⊢ (𝐴 ∈ ℝ → -1 < (𝐴↑2)) |
10 | 3, 9 | gtned 11120 | . 2 ⊢ (𝐴 ∈ ℝ → (𝐴↑2) ≠ -1) |
11 | atandm3 26038 | . 2 ⊢ (𝐴 ∈ dom arctan ↔ (𝐴 ∈ ℂ ∧ (𝐴↑2) ≠ -1)) | |
12 | 1, 10, 11 | sylanbrc 583 | 1 ⊢ (𝐴 ∈ ℝ → 𝐴 ∈ dom arctan) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2106 ≠ wne 2943 class class class wbr 5073 dom cdm 5584 (class class class)co 7267 ℂcc 10879 ℝcr 10880 0cc0 10881 1c1 10882 < clt 11019 -cneg 11216 2c2 12038 ↑cexp 13792 arctancatan 26024 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2709 ax-sep 5221 ax-nul 5228 ax-pow 5286 ax-pr 5350 ax-un 7578 ax-cnex 10937 ax-resscn 10938 ax-1cn 10939 ax-icn 10940 ax-addcl 10941 ax-addrcl 10942 ax-mulcl 10943 ax-mulrcl 10944 ax-mulcom 10945 ax-addass 10946 ax-mulass 10947 ax-distr 10948 ax-i2m1 10949 ax-1ne0 10950 ax-1rid 10951 ax-rnegex 10952 ax-rrecex 10953 ax-cnre 10954 ax-pre-lttri 10955 ax-pre-lttrn 10956 ax-pre-ltadd 10957 ax-pre-mulgt0 10958 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2068 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2889 df-ne 2944 df-nel 3050 df-ral 3069 df-rex 3070 df-reu 3071 df-rab 3073 df-v 3431 df-sbc 3716 df-csb 3832 df-dif 3889 df-un 3891 df-in 3893 df-ss 3903 df-pss 3905 df-nul 4257 df-if 4460 df-pw 4535 df-sn 4562 df-pr 4564 df-op 4568 df-uni 4840 df-iun 4926 df-br 5074 df-opab 5136 df-mpt 5157 df-tr 5191 df-id 5484 df-eprel 5490 df-po 5498 df-so 5499 df-fr 5539 df-we 5541 df-xp 5590 df-rel 5591 df-cnv 5592 df-co 5593 df-dm 5594 df-rn 5595 df-res 5596 df-ima 5597 df-pred 6195 df-ord 6262 df-on 6263 df-lim 6264 df-suc 6265 df-iota 6384 df-fun 6428 df-fn 6429 df-f 6430 df-f1 6431 df-fo 6432 df-f1o 6433 df-fv 6434 df-riota 7224 df-ov 7270 df-oprab 7271 df-mpo 7272 df-om 7703 df-2nd 7821 df-frecs 8084 df-wrecs 8115 df-recs 8189 df-rdg 8228 df-er 8485 df-en 8721 df-dom 8722 df-sdom 8723 df-pnf 11021 df-mnf 11022 df-xr 11023 df-ltxr 11024 df-le 11025 df-sub 11217 df-neg 11218 df-nn 11984 df-2 12046 df-n0 12244 df-z 12330 df-uz 12593 df-seq 13732 df-exp 13793 df-atan 26027 |
This theorem is referenced by: atan0 26068 atanrecl 26071 atanbndlem 26085 atanbnd 26086 atanord 26087 |
Copyright terms: Public domain | W3C validator |