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Mirrors > Home > MPE Home > Th. List > atandmcj | Structured version Visualization version GIF version |
Description: The arctangent function distributes under conjugation. (Contributed by Mario Carneiro, 31-Mar-2015.) |
Ref | Expression |
---|---|
atandmcj | ⊢ (𝐴 ∈ dom arctan → (∗‘𝐴) ∈ dom arctan) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | atandm3 26244 | . . . 4 ⊢ (𝐴 ∈ dom arctan ↔ (𝐴 ∈ ℂ ∧ (𝐴↑2) ≠ -1)) | |
2 | 1 | simplbi 499 | . . 3 ⊢ (𝐴 ∈ dom arctan → 𝐴 ∈ ℂ) |
3 | 2 | cjcld 15088 | . 2 ⊢ (𝐴 ∈ dom arctan → (∗‘𝐴) ∈ ℂ) |
4 | 2nn0 12437 | . . . 4 ⊢ 2 ∈ ℕ0 | |
5 | cjexp 15042 | . . . 4 ⊢ ((𝐴 ∈ ℂ ∧ 2 ∈ ℕ0) → (∗‘(𝐴↑2)) = ((∗‘𝐴)↑2)) | |
6 | 2, 4, 5 | sylancl 587 | . . 3 ⊢ (𝐴 ∈ dom arctan → (∗‘(𝐴↑2)) = ((∗‘𝐴)↑2)) |
7 | 2 | sqcld 14056 | . . . . . 6 ⊢ (𝐴 ∈ dom arctan → (𝐴↑2) ∈ ℂ) |
8 | 7 | cjcjd 15091 | . . . . 5 ⊢ (𝐴 ∈ dom arctan → (∗‘(∗‘(𝐴↑2))) = (𝐴↑2)) |
9 | 1 | simprbi 498 | . . . . 5 ⊢ (𝐴 ∈ dom arctan → (𝐴↑2) ≠ -1) |
10 | 8, 9 | eqnetrd 3012 | . . . 4 ⊢ (𝐴 ∈ dom arctan → (∗‘(∗‘(𝐴↑2))) ≠ -1) |
11 | fveq2 6847 | . . . . . 6 ⊢ ((∗‘(𝐴↑2)) = -1 → (∗‘(∗‘(𝐴↑2))) = (∗‘-1)) | |
12 | neg1rr 12275 | . . . . . . 7 ⊢ -1 ∈ ℝ | |
13 | cjre 15031 | . . . . . . 7 ⊢ (-1 ∈ ℝ → (∗‘-1) = -1) | |
14 | 12, 13 | ax-mp 5 | . . . . . 6 ⊢ (∗‘-1) = -1 |
15 | 11, 14 | eqtrdi 2793 | . . . . 5 ⊢ ((∗‘(𝐴↑2)) = -1 → (∗‘(∗‘(𝐴↑2))) = -1) |
16 | 15 | necon3i 2977 | . . . 4 ⊢ ((∗‘(∗‘(𝐴↑2))) ≠ -1 → (∗‘(𝐴↑2)) ≠ -1) |
17 | 10, 16 | syl 17 | . . 3 ⊢ (𝐴 ∈ dom arctan → (∗‘(𝐴↑2)) ≠ -1) |
18 | 6, 17 | eqnetrrd 3013 | . 2 ⊢ (𝐴 ∈ dom arctan → ((∗‘𝐴)↑2) ≠ -1) |
19 | atandm3 26244 | . 2 ⊢ ((∗‘𝐴) ∈ dom arctan ↔ ((∗‘𝐴) ∈ ℂ ∧ ((∗‘𝐴)↑2) ≠ -1)) | |
20 | 3, 18, 19 | sylanbrc 584 | 1 ⊢ (𝐴 ∈ dom arctan → (∗‘𝐴) ∈ dom arctan) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1542 ∈ wcel 2107 ≠ wne 2944 dom cdm 5638 ‘cfv 6501 (class class class)co 7362 ℂcc 11056 ℝcr 11057 1c1 11059 -cneg 11393 2c2 12215 ℕ0cn0 12420 ↑cexp 13974 ∗ccj 14988 arctancatan 26230 |
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 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2708 ax-sep 5261 ax-nul 5268 ax-pow 5325 ax-pr 5389 ax-un 7677 ax-cnex 11114 ax-resscn 11115 ax-1cn 11116 ax-icn 11117 ax-addcl 11118 ax-addrcl 11119 ax-mulcl 11120 ax-mulrcl 11121 ax-mulcom 11122 ax-addass 11123 ax-mulass 11124 ax-distr 11125 ax-i2m1 11126 ax-1ne0 11127 ax-1rid 11128 ax-rnegex 11129 ax-rrecex 11130 ax-cnre 11131 ax-pre-lttri 11132 ax-pre-lttrn 11133 ax-pre-ltadd 11134 ax-pre-mulgt0 11135 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2815 df-nfc 2890 df-ne 2945 df-nel 3051 df-ral 3066 df-rex 3075 df-rmo 3356 df-reu 3357 df-rab 3411 df-v 3450 df-sbc 3745 df-csb 3861 df-dif 3918 df-un 3920 df-in 3922 df-ss 3932 df-pss 3934 df-nul 4288 df-if 4492 df-pw 4567 df-sn 4592 df-pr 4594 df-op 4598 df-uni 4871 df-iun 4961 df-br 5111 df-opab 5173 df-mpt 5194 df-tr 5228 df-id 5536 df-eprel 5542 df-po 5550 df-so 5551 df-fr 5593 df-we 5595 df-xp 5644 df-rel 5645 df-cnv 5646 df-co 5647 df-dm 5648 df-rn 5649 df-res 5650 df-ima 5651 df-pred 6258 df-ord 6325 df-on 6326 df-lim 6327 df-suc 6328 df-iota 6453 df-fun 6503 df-fn 6504 df-f 6505 df-f1 6506 df-fo 6507 df-f1o 6508 df-fv 6509 df-riota 7318 df-ov 7365 df-oprab 7366 df-mpo 7367 df-om 7808 df-2nd 7927 df-frecs 8217 df-wrecs 8248 df-recs 8322 df-rdg 8361 df-er 8655 df-en 8891 df-dom 8892 df-sdom 8893 df-pnf 11198 df-mnf 11199 df-xr 11200 df-ltxr 11201 df-le 11202 df-sub 11394 df-neg 11395 df-div 11820 df-nn 12161 df-2 12223 df-n0 12421 df-z 12507 df-uz 12771 df-seq 13914 df-exp 13975 df-cj 14991 df-re 14992 df-im 14993 df-atan 26233 |
This theorem is referenced by: atancj 26276 |
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