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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 15144 | . . . 4 ⊢ (𝐴 ∈ ℂ → (∗‘𝐴) ∈ ℂ) | |
| 2 | 1 | adantr 480 | . . 3 ⊢ ((𝐴 ∈ ℂ ∧ ∃𝑓 ∈ ((Poly‘ℤ) ∖ {0𝑝})(𝑓‘𝐴) = 0) → (∗‘𝐴) ∈ ℂ) |
| 3 | fveq2 6906 | . . . . . . 7 ⊢ ((𝑓‘𝐴) = 0 → (∗‘(𝑓‘𝐴)) = (∗‘0)) | |
| 4 | cj0 15197 | . . . . . . 7 ⊢ (∗‘0) = 0 | |
| 5 | 3, 4 | eqtrdi 2793 | . . . . . 6 ⊢ ((𝑓‘𝐴) = 0 → (∗‘(𝑓‘𝐴)) = 0) |
| 6 | difss 4136 | . . . . . . . . . 10 ⊢ ((Poly‘ℤ) ∖ {0𝑝}) ⊆ (Poly‘ℤ) | |
| 7 | zssre 12620 | . . . . . . . . . . 11 ⊢ ℤ ⊆ ℝ | |
| 8 | ax-resscn 11212 | . . . . . . . . . . 11 ⊢ ℝ ⊆ ℂ | |
| 9 | plyss 26238 | . . . . . . . . . . 11 ⊢ ((ℤ ⊆ ℝ ∧ ℝ ⊆ ℂ) → (Poly‘ℤ) ⊆ (Poly‘ℝ)) | |
| 10 | 7, 8, 9 | mp2an 692 | . . . . . . . . . 10 ⊢ (Poly‘ℤ) ⊆ (Poly‘ℝ) |
| 11 | 6, 10 | sstri 3993 | . . . . . . . . 9 ⊢ ((Poly‘ℤ) ∖ {0𝑝}) ⊆ (Poly‘ℝ) |
| 12 | 11 | sseli 3979 | . . . . . . . 8 ⊢ (𝑓 ∈ ((Poly‘ℤ) ∖ {0𝑝}) → 𝑓 ∈ (Poly‘ℝ)) |
| 13 | id 22 | . . . . . . . 8 ⊢ (𝐴 ∈ ℂ → 𝐴 ∈ ℂ) | |
| 14 | plyrecj 26321 | . . . . . . . 8 ⊢ ((𝑓 ∈ (Poly‘ℝ) ∧ 𝐴 ∈ ℂ) → (∗‘(𝑓‘𝐴)) = (𝑓‘(∗‘𝐴))) | |
| 15 | 12, 13, 14 | syl2anr 597 | . . . . . . 7 ⊢ ((𝐴 ∈ ℂ ∧ 𝑓 ∈ ((Poly‘ℤ) ∖ {0𝑝})) → (∗‘(𝑓‘𝐴)) = (𝑓‘(∗‘𝐴))) |
| 16 | 15 | eqeq1d 2739 | . . . . . 6 ⊢ ((𝐴 ∈ ℂ ∧ 𝑓 ∈ ((Poly‘ℤ) ∖ {0𝑝})) → ((∗‘(𝑓‘𝐴)) = 0 ↔ (𝑓‘(∗‘𝐴)) = 0)) |
| 17 | 5, 16 | imbitrid 244 | . . . . 5 ⊢ ((𝐴 ∈ ℂ ∧ 𝑓 ∈ ((Poly‘ℤ) ∖ {0𝑝})) → ((𝑓‘𝐴) = 0 → (𝑓‘(∗‘𝐴)) = 0)) |
| 18 | 17 | reximdva 3168 | . . . 4 ⊢ (𝐴 ∈ ℂ → (∃𝑓 ∈ ((Poly‘ℤ) ∖ {0𝑝})(𝑓‘𝐴) = 0 → ∃𝑓 ∈ ((Poly‘ℤ) ∖ {0𝑝})(𝑓‘(∗‘𝐴)) = 0)) |
| 19 | 18 | imp 406 | . . 3 ⊢ ((𝐴 ∈ ℂ ∧ ∃𝑓 ∈ ((Poly‘ℤ) ∖ {0𝑝})(𝑓‘𝐴) = 0) → ∃𝑓 ∈ ((Poly‘ℤ) ∖ {0𝑝})(𝑓‘(∗‘𝐴)) = 0) |
| 20 | 2, 19 | jca 511 | . 2 ⊢ ((𝐴 ∈ ℂ ∧ ∃𝑓 ∈ ((Poly‘ℤ) ∖ {0𝑝})(𝑓‘𝐴) = 0) → ((∗‘𝐴) ∈ ℂ ∧ ∃𝑓 ∈ ((Poly‘ℤ) ∖ {0𝑝})(𝑓‘(∗‘𝐴)) = 0)) |
| 21 | elaa 26358 | . 2 ⊢ (𝐴 ∈ 𝔸 ↔ (𝐴 ∈ ℂ ∧ ∃𝑓 ∈ ((Poly‘ℤ) ∖ {0𝑝})(𝑓‘𝐴) = 0)) | |
| 22 | elaa 26358 | . 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 395 = wceq 1540 ∈ wcel 2108 ∃wrex 3070 ∖ cdif 3948 ⊆ wss 3951 {csn 4626 ‘cfv 6561 ℂcc 11153 ℝcr 11154 0cc0 11155 ℤcz 12613 ∗ccj 15135 0𝑝c0p 25704 Polycply 26223 𝔸caa 26356 |
| 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 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-rep 5279 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-inf2 9681 ax-cnex 11211 ax-resscn 11212 ax-1cn 11213 ax-icn 11214 ax-addcl 11215 ax-addrcl 11216 ax-mulcl 11217 ax-mulrcl 11218 ax-mulcom 11219 ax-addass 11220 ax-mulass 11221 ax-distr 11222 ax-i2m1 11223 ax-1ne0 11224 ax-1rid 11225 ax-rnegex 11226 ax-rrecex 11227 ax-cnre 11228 ax-pre-lttri 11229 ax-pre-lttrn 11230 ax-pre-ltadd 11231 ax-pre-mulgt0 11232 ax-pre-sup 11233 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3380 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-int 4947 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-se 5638 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-pred 6321 df-ord 6387 df-on 6388 df-lim 6389 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-isom 6570 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-of 7697 df-om 7888 df-1st 8014 df-2nd 8015 df-frecs 8306 df-wrecs 8337 df-recs 8411 df-rdg 8450 df-1o 8506 df-er 8745 df-map 8868 df-pm 8869 df-en 8986 df-dom 8987 df-sdom 8988 df-fin 8989 df-sup 9482 df-inf 9483 df-oi 9550 df-card 9979 df-pnf 11297 df-mnf 11298 df-xr 11299 df-ltxr 11300 df-le 11301 df-sub 11494 df-neg 11495 df-div 11921 df-nn 12267 df-2 12329 df-3 12330 df-n0 12527 df-z 12614 df-uz 12879 df-rp 13035 df-fz 13548 df-fzo 13695 df-fl 13832 df-seq 14043 df-exp 14103 df-hash 14370 df-cj 15138 df-re 15139 df-im 15140 df-sqrt 15274 df-abs 15275 df-clim 15524 df-rlim 15525 df-sum 15723 df-0p 25705 df-ply 26227 df-coe 26229 df-dgr 26230 df-aa 26357 |
| This theorem is referenced by: (None) |
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