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Mirrors > Home > MPE Home > Th. List > aacjcl | Structured version Visualization version GIF version |
Description: The conjugate of an algebraic number is algebraic. (Contributed by Mario Carneiro, 24-Jul-2014.) |
Ref | Expression |
---|---|
aacjcl | ⊢ (𝐴 ∈ 𝔸 → (∗‘𝐴) ∈ 𝔸) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cjcl 14861 | . . . 4 ⊢ (𝐴 ∈ ℂ → (∗‘𝐴) ∈ ℂ) | |
2 | 1 | adantr 482 | . . 3 ⊢ ((𝐴 ∈ ℂ ∧ ∃𝑓 ∈ ((Poly‘ℤ) ∖ {0𝑝})(𝑓‘𝐴) = 0) → (∗‘𝐴) ∈ ℂ) |
3 | fveq2 6804 | . . . . . . 7 ⊢ ((𝑓‘𝐴) = 0 → (∗‘(𝑓‘𝐴)) = (∗‘0)) | |
4 | cj0 14914 | . . . . . . 7 ⊢ (∗‘0) = 0 | |
5 | 3, 4 | eqtrdi 2792 | . . . . . 6 ⊢ ((𝑓‘𝐴) = 0 → (∗‘(𝑓‘𝐴)) = 0) |
6 | difss 4072 | . . . . . . . . . 10 ⊢ ((Poly‘ℤ) ∖ {0𝑝}) ⊆ (Poly‘ℤ) | |
7 | zssre 12372 | . . . . . . . . . . 11 ⊢ ℤ ⊆ ℝ | |
8 | ax-resscn 10974 | . . . . . . . . . . 11 ⊢ ℝ ⊆ ℂ | |
9 | plyss 25405 | . . . . . . . . . . 11 ⊢ ((ℤ ⊆ ℝ ∧ ℝ ⊆ ℂ) → (Poly‘ℤ) ⊆ (Poly‘ℝ)) | |
10 | 7, 8, 9 | mp2an 690 | . . . . . . . . . 10 ⊢ (Poly‘ℤ) ⊆ (Poly‘ℝ) |
11 | 6, 10 | sstri 3935 | . . . . . . . . 9 ⊢ ((Poly‘ℤ) ∖ {0𝑝}) ⊆ (Poly‘ℝ) |
12 | 11 | sseli 3922 | . . . . . . . 8 ⊢ (𝑓 ∈ ((Poly‘ℤ) ∖ {0𝑝}) → 𝑓 ∈ (Poly‘ℝ)) |
13 | id 22 | . . . . . . . 8 ⊢ (𝐴 ∈ ℂ → 𝐴 ∈ ℂ) | |
14 | plyrecj 25485 | . . . . . . . 8 ⊢ ((𝑓 ∈ (Poly‘ℝ) ∧ 𝐴 ∈ ℂ) → (∗‘(𝑓‘𝐴)) = (𝑓‘(∗‘𝐴))) | |
15 | 12, 13, 14 | syl2anr 598 | . . . . . . 7 ⊢ ((𝐴 ∈ ℂ ∧ 𝑓 ∈ ((Poly‘ℤ) ∖ {0𝑝})) → (∗‘(𝑓‘𝐴)) = (𝑓‘(∗‘𝐴))) |
16 | 15 | eqeq1d 2738 | . . . . . 6 ⊢ ((𝐴 ∈ ℂ ∧ 𝑓 ∈ ((Poly‘ℤ) ∖ {0𝑝})) → ((∗‘(𝑓‘𝐴)) = 0 ↔ (𝑓‘(∗‘𝐴)) = 0)) |
17 | 5, 16 | syl5ib 244 | . . . . 5 ⊢ ((𝐴 ∈ ℂ ∧ 𝑓 ∈ ((Poly‘ℤ) ∖ {0𝑝})) → ((𝑓‘𝐴) = 0 → (𝑓‘(∗‘𝐴)) = 0)) |
18 | 17 | reximdva 3162 | . . . 4 ⊢ (𝐴 ∈ ℂ → (∃𝑓 ∈ ((Poly‘ℤ) ∖ {0𝑝})(𝑓‘𝐴) = 0 → ∃𝑓 ∈ ((Poly‘ℤ) ∖ {0𝑝})(𝑓‘(∗‘𝐴)) = 0)) |
19 | 18 | imp 408 | . . 3 ⊢ ((𝐴 ∈ ℂ ∧ ∃𝑓 ∈ ((Poly‘ℤ) ∖ {0𝑝})(𝑓‘𝐴) = 0) → ∃𝑓 ∈ ((Poly‘ℤ) ∖ {0𝑝})(𝑓‘(∗‘𝐴)) = 0) |
20 | 2, 19 | jca 513 | . 2 ⊢ ((𝐴 ∈ ℂ ∧ ∃𝑓 ∈ ((Poly‘ℤ) ∖ {0𝑝})(𝑓‘𝐴) = 0) → ((∗‘𝐴) ∈ ℂ ∧ ∃𝑓 ∈ ((Poly‘ℤ) ∖ {0𝑝})(𝑓‘(∗‘𝐴)) = 0)) |
21 | elaa 25521 | . 2 ⊢ (𝐴 ∈ 𝔸 ↔ (𝐴 ∈ ℂ ∧ ∃𝑓 ∈ ((Poly‘ℤ) ∖ {0𝑝})(𝑓‘𝐴) = 0)) | |
22 | elaa 25521 | . 2 ⊢ ((∗‘𝐴) ∈ 𝔸 ↔ ((∗‘𝐴) ∈ ℂ ∧ ∃𝑓 ∈ ((Poly‘ℤ) ∖ {0𝑝})(𝑓‘(∗‘𝐴)) = 0)) | |
23 | 20, 21, 22 | 3imtr4i 292 | 1 ⊢ (𝐴 ∈ 𝔸 → (∗‘𝐴) ∈ 𝔸) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 397 = wceq 1539 ∈ wcel 2104 ∃wrex 3071 ∖ cdif 3889 ⊆ wss 3892 {csn 4565 ‘cfv 6458 ℂcc 10915 ℝcr 10916 0cc0 10917 ℤcz 12365 ∗ccj 14852 0𝑝c0p 24878 Polycply 25390 𝔸caa 25519 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1911 ax-6 1969 ax-7 2009 ax-8 2106 ax-9 2114 ax-10 2135 ax-11 2152 ax-12 2169 ax-ext 2707 ax-rep 5218 ax-sep 5232 ax-nul 5239 ax-pow 5297 ax-pr 5361 ax-un 7620 ax-inf2 9443 ax-cnex 10973 ax-resscn 10974 ax-1cn 10975 ax-icn 10976 ax-addcl 10977 ax-addrcl 10978 ax-mulcl 10979 ax-mulrcl 10980 ax-mulcom 10981 ax-addass 10982 ax-mulass 10983 ax-distr 10984 ax-i2m1 10985 ax-1ne0 10986 ax-1rid 10987 ax-rnegex 10988 ax-rrecex 10989 ax-cnre 10990 ax-pre-lttri 10991 ax-pre-lttrn 10992 ax-pre-ltadd 10993 ax-pre-mulgt0 10994 ax-pre-sup 10995 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 846 df-3or 1088 df-3an 1089 df-tru 1542 df-fal 1552 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2887 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-rmo 3285 df-reu 3286 df-rab 3287 df-v 3439 df-sbc 3722 df-csb 3838 df-dif 3895 df-un 3897 df-in 3899 df-ss 3909 df-pss 3911 df-nul 4263 df-if 4466 df-pw 4541 df-sn 4566 df-pr 4568 df-op 4572 df-uni 4845 df-int 4887 df-iun 4933 df-br 5082 df-opab 5144 df-mpt 5165 df-tr 5199 df-id 5500 df-eprel 5506 df-po 5514 df-so 5515 df-fr 5555 df-se 5556 df-we 5557 df-xp 5606 df-rel 5607 df-cnv 5608 df-co 5609 df-dm 5610 df-rn 5611 df-res 5612 df-ima 5613 df-pred 6217 df-ord 6284 df-on 6285 df-lim 6286 df-suc 6287 df-iota 6410 df-fun 6460 df-fn 6461 df-f 6462 df-f1 6463 df-fo 6464 df-f1o 6465 df-fv 6466 df-isom 6467 df-riota 7264 df-ov 7310 df-oprab 7311 df-mpo 7312 df-of 7565 df-om 7745 df-1st 7863 df-2nd 7864 df-frecs 8128 df-wrecs 8159 df-recs 8233 df-rdg 8272 df-1o 8328 df-er 8529 df-map 8648 df-pm 8649 df-en 8765 df-dom 8766 df-sdom 8767 df-fin 8768 df-sup 9245 df-inf 9246 df-oi 9313 df-card 9741 df-pnf 11057 df-mnf 11058 df-xr 11059 df-ltxr 11060 df-le 11061 df-sub 11253 df-neg 11254 df-div 11679 df-nn 12020 df-2 12082 df-3 12083 df-n0 12280 df-z 12366 df-uz 12629 df-rp 12777 df-fz 13286 df-fzo 13429 df-fl 13558 df-seq 13768 df-exp 13829 df-hash 14091 df-cj 14855 df-re 14856 df-im 14857 df-sqrt 14991 df-abs 14992 df-clim 15242 df-rlim 15243 df-sum 15443 df-0p 24879 df-ply 25394 df-coe 25396 df-dgr 25397 df-aa 25520 |
This theorem is referenced by: (None) |
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