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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 26620 | . . . 4 ⊢ (𝐴 ∈ dom arctan ↔ (𝐴 ∈ ℂ ∧ (𝐴↑2) ≠ -1)) | |
2 | 1 | simplbi 497 | . . 3 ⊢ (𝐴 ∈ dom arctan → 𝐴 ∈ ℂ) |
3 | 2 | cjcld 15148 | . 2 ⊢ (𝐴 ∈ dom arctan → (∗‘𝐴) ∈ ℂ) |
4 | 2nn0 12494 | . . . 4 ⊢ 2 ∈ ℕ0 | |
5 | cjexp 15102 | . . . 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 15151 | . . . . 5 ⊢ (𝐴 ∈ dom arctan → (∗‘(∗‘(𝐴↑2))) = (𝐴↑2)) |
9 | 1 | simprbi 496 | . . . . 5 ⊢ (𝐴 ∈ dom arctan → (𝐴↑2) ≠ -1) |
10 | 8, 9 | eqnetrd 3007 | . . . 4 ⊢ (𝐴 ∈ dom arctan → (∗‘(∗‘(𝐴↑2))) ≠ -1) |
11 | fveq2 6891 | . . . . . 6 ⊢ ((∗‘(𝐴↑2)) = -1 → (∗‘(∗‘(𝐴↑2))) = (∗‘-1)) | |
12 | neg1rr 12332 | . . . . . . 7 ⊢ -1 ∈ ℝ | |
13 | cjre 15091 | . . . . . . 7 ⊢ (-1 ∈ ℝ → (∗‘-1) = -1) | |
14 | 12, 13 | ax-mp 5 | . . . . . 6 ⊢ (∗‘-1) = -1 |
15 | 11, 14 | eqtrdi 2787 | . . . . 5 ⊢ ((∗‘(𝐴↑2)) = -1 → (∗‘(∗‘(𝐴↑2))) = -1) |
16 | 15 | necon3i 2972 | . . . 4 ⊢ ((∗‘(∗‘(𝐴↑2))) ≠ -1 → (∗‘(𝐴↑2)) ≠ -1) |
17 | 10, 16 | syl 17 | . . 3 ⊢ (𝐴 ∈ dom arctan → (∗‘(𝐴↑2)) ≠ -1) |
18 | 6, 17 | eqnetrrd 3008 | . 2 ⊢ (𝐴 ∈ dom arctan → ((∗‘𝐴)↑2) ≠ -1) |
19 | atandm3 26620 | . 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 1540 ∈ wcel 2105 ≠ wne 2939 dom cdm 5676 ‘cfv 6543 (class class class)co 7412 ℂcc 11112 ℝcr 11113 1c1 11115 -cneg 11450 2c2 12272 ℕ0cn0 12477 ↑cexp 14032 ∗ccj 15048 arctancatan 26606 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2153 ax-12 2170 ax-ext 2702 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7729 ax-cnex 11170 ax-resscn 11171 ax-1cn 11172 ax-icn 11173 ax-addcl 11174 ax-addrcl 11175 ax-mulcl 11176 ax-mulrcl 11177 ax-mulcom 11178 ax-addass 11179 ax-mulass 11180 ax-distr 11181 ax-i2m1 11182 ax-1ne0 11183 ax-1rid 11184 ax-rnegex 11185 ax-rrecex 11186 ax-cnre 11187 ax-pre-lttri 11188 ax-pre-lttrn 11189 ax-pre-ltadd 11190 ax-pre-mulgt0 11191 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-nf 1785 df-sb 2067 df-mo 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ne 2940 df-nel 3046 df-ral 3061 df-rex 3070 df-rmo 3375 df-reu 3376 df-rab 3432 df-v 3475 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-pss 3967 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-iun 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5574 df-eprel 5580 df-po 5588 df-so 5589 df-fr 5631 df-we 5633 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-pred 6300 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-riota 7368 df-ov 7415 df-oprab 7416 df-mpo 7417 df-om 7860 df-2nd 7980 df-frecs 8270 df-wrecs 8301 df-recs 8375 df-rdg 8414 df-er 8707 df-en 8944 df-dom 8945 df-sdom 8946 df-pnf 11255 df-mnf 11256 df-xr 11257 df-ltxr 11258 df-le 11259 df-sub 11451 df-neg 11452 df-div 11877 df-nn 12218 df-2 12280 df-n0 12478 df-z 12564 df-uz 12828 df-seq 13972 df-exp 14033 df-cj 15051 df-re 15052 df-im 15053 df-atan 26609 |
This theorem is referenced by: atancj 26652 |
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