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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 26765 | . . . 4 ⊢ (𝐴 ∈ dom arctan ↔ (𝐴 ∈ ℂ ∧ (𝐴↑2) ≠ -1)) | |
2 | 1 | simplbi 497 | . . 3 ⊢ (𝐴 ∈ dom arctan → 𝐴 ∈ ℂ) |
3 | 2 | cjcld 15149 | . 2 ⊢ (𝐴 ∈ dom arctan → (∗‘𝐴) ∈ ℂ) |
4 | 2nn0 12493 | . . . 4 ⊢ 2 ∈ ℕ0 | |
5 | cjexp 15103 | . . . 4 ⊢ ((𝐴 ∈ ℂ ∧ 2 ∈ ℕ0) → (∗‘(𝐴↑2)) = ((∗‘𝐴)↑2)) | |
6 | 2, 4, 5 | sylancl 585 | . . 3 ⊢ (𝐴 ∈ dom arctan → (∗‘(𝐴↑2)) = ((∗‘𝐴)↑2)) |
7 | 2 | sqcld 14114 | . . . . . 6 ⊢ (𝐴 ∈ dom arctan → (𝐴↑2) ∈ ℂ) |
8 | 7 | cjcjd 15152 | . . . . 5 ⊢ (𝐴 ∈ dom arctan → (∗‘(∗‘(𝐴↑2))) = (𝐴↑2)) |
9 | 1 | simprbi 496 | . . . . 5 ⊢ (𝐴 ∈ dom arctan → (𝐴↑2) ≠ -1) |
10 | 8, 9 | eqnetrd 3002 | . . . 4 ⊢ (𝐴 ∈ dom arctan → (∗‘(∗‘(𝐴↑2))) ≠ -1) |
11 | fveq2 6885 | . . . . . 6 ⊢ ((∗‘(𝐴↑2)) = -1 → (∗‘(∗‘(𝐴↑2))) = (∗‘-1)) | |
12 | neg1rr 12331 | . . . . . . 7 ⊢ -1 ∈ ℝ | |
13 | cjre 15092 | . . . . . . 7 ⊢ (-1 ∈ ℝ → (∗‘-1) = -1) | |
14 | 12, 13 | ax-mp 5 | . . . . . 6 ⊢ (∗‘-1) = -1 |
15 | 11, 14 | eqtrdi 2782 | . . . . 5 ⊢ ((∗‘(𝐴↑2)) = -1 → (∗‘(∗‘(𝐴↑2))) = -1) |
16 | 15 | necon3i 2967 | . . . 4 ⊢ ((∗‘(∗‘(𝐴↑2))) ≠ -1 → (∗‘(𝐴↑2)) ≠ -1) |
17 | 10, 16 | syl 17 | . . 3 ⊢ (𝐴 ∈ dom arctan → (∗‘(𝐴↑2)) ≠ -1) |
18 | 6, 17 | eqnetrrd 3003 | . 2 ⊢ (𝐴 ∈ dom arctan → ((∗‘𝐴)↑2) ≠ -1) |
19 | atandm3 26765 | . 2 ⊢ ((∗‘𝐴) ∈ dom arctan ↔ ((∗‘𝐴) ∈ ℂ ∧ ((∗‘𝐴)↑2) ≠ -1)) | |
20 | 3, 18, 19 | sylanbrc 582 | 1 ⊢ (𝐴 ∈ dom arctan → (∗‘𝐴) ∈ dom arctan) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1533 ∈ wcel 2098 ≠ wne 2934 dom cdm 5669 ‘cfv 6537 (class class class)co 7405 ℂcc 11110 ℝcr 11111 1c1 11113 -cneg 11449 2c2 12271 ℕ0cn0 12476 ↑cexp 14032 ∗ccj 15049 arctancatan 26751 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2697 ax-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 ax-un 7722 ax-cnex 11168 ax-resscn 11169 ax-1cn 11170 ax-icn 11171 ax-addcl 11172 ax-addrcl 11173 ax-mulcl 11174 ax-mulrcl 11175 ax-mulcom 11176 ax-addass 11177 ax-mulass 11178 ax-distr 11179 ax-i2m1 11180 ax-1ne0 11181 ax-1rid 11182 ax-rnegex 11183 ax-rrecex 11184 ax-cnre 11185 ax-pre-lttri 11186 ax-pre-lttrn 11187 ax-pre-ltadd 11188 ax-pre-mulgt0 11189 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ne 2935 df-nel 3041 df-ral 3056 df-rex 3065 df-rmo 3370 df-reu 3371 df-rab 3427 df-v 3470 df-sbc 3773 df-csb 3889 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-pss 3962 df-nul 4318 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4903 df-iun 4992 df-br 5142 df-opab 5204 df-mpt 5225 df-tr 5259 df-id 5567 df-eprel 5573 df-po 5581 df-so 5582 df-fr 5624 df-we 5626 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-pred 6294 df-ord 6361 df-on 6362 df-lim 6363 df-suc 6364 df-iota 6489 df-fun 6539 df-fn 6540 df-f 6541 df-f1 6542 df-fo 6543 df-f1o 6544 df-fv 6545 df-riota 7361 df-ov 7408 df-oprab 7409 df-mpo 7410 df-om 7853 df-2nd 7975 df-frecs 8267 df-wrecs 8298 df-recs 8372 df-rdg 8411 df-er 8705 df-en 8942 df-dom 8943 df-sdom 8944 df-pnf 11254 df-mnf 11255 df-xr 11256 df-ltxr 11257 df-le 11258 df-sub 11450 df-neg 11451 df-div 11876 df-nn 12217 df-2 12279 df-n0 12477 df-z 12563 df-uz 12827 df-seq 13973 df-exp 14033 df-cj 15052 df-re 15053 df-im 15054 df-atan 26754 |
This theorem is referenced by: atancj 26797 |
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