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| Mirrors > Home > MPE Home > Th. List > aannenlem3 | Structured version Visualization version GIF version | ||
| Description: The algebraic numbers are countable. (Contributed by Stefan O'Rear, 16-Nov-2014.) |
| Ref | Expression |
|---|---|
| aannenlem.a | ⊢ 𝐻 = (𝑎 ∈ ℕ0 ↦ {𝑏 ∈ ℂ ∣ ∃𝑐 ∈ {𝑑 ∈ (Poly‘ℤ) ∣ (𝑑 ≠ 0𝑝 ∧ (deg‘𝑑) ≤ 𝑎 ∧ ∀𝑒 ∈ ℕ0 (abs‘((coeff‘𝑑)‘𝑒)) ≤ 𝑎)} (𝑐‘𝑏) = 0}) |
| Ref | Expression |
|---|---|
| aannenlem3 | ⊢ 𝔸 ≈ ℕ |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | aannenlem.a | . . . . . 6 ⊢ 𝐻 = (𝑎 ∈ ℕ0 ↦ {𝑏 ∈ ℂ ∣ ∃𝑐 ∈ {𝑑 ∈ (Poly‘ℤ) ∣ (𝑑 ≠ 0𝑝 ∧ (deg‘𝑑) ≤ 𝑎 ∧ ∀𝑒 ∈ ℕ0 (abs‘((coeff‘𝑑)‘𝑒)) ≤ 𝑎)} (𝑐‘𝑏) = 0}) | |
| 2 | 1 | aannenlem2 26371 | . . . . 5 ⊢ 𝔸 = ∪ ran 𝐻 |
| 3 | omelon 9686 | . . . . . . . . 9 ⊢ ω ∈ On | |
| 4 | nn0ennn 14020 | . . . . . . . . . . 11 ⊢ ℕ0 ≈ ℕ | |
| 5 | nnenom 14021 | . . . . . . . . . . 11 ⊢ ℕ ≈ ω | |
| 6 | 4, 5 | entri 9048 | . . . . . . . . . 10 ⊢ ℕ0 ≈ ω |
| 7 | 6 | ensymi 9044 | . . . . . . . . 9 ⊢ ω ≈ ℕ0 |
| 8 | isnumi 9986 | . . . . . . . . 9 ⊢ ((ω ∈ On ∧ ω ≈ ℕ0) → ℕ0 ∈ dom card) | |
| 9 | 3, 7, 8 | mp2an 692 | . . . . . . . 8 ⊢ ℕ0 ∈ dom card |
| 10 | cnex 11236 | . . . . . . . . . . 11 ⊢ ℂ ∈ V | |
| 11 | 10 | rabex 5339 | . . . . . . . . . 10 ⊢ {𝑏 ∈ ℂ ∣ ∃𝑐 ∈ {𝑑 ∈ (Poly‘ℤ) ∣ (𝑑 ≠ 0𝑝 ∧ (deg‘𝑑) ≤ 𝑎 ∧ ∀𝑒 ∈ ℕ0 (abs‘((coeff‘𝑑)‘𝑒)) ≤ 𝑎)} (𝑐‘𝑏) = 0} ∈ V |
| 12 | 11, 1 | fnmpti 6711 | . . . . . . . . 9 ⊢ 𝐻 Fn ℕ0 |
| 13 | dffn4 6826 | . . . . . . . . 9 ⊢ (𝐻 Fn ℕ0 ↔ 𝐻:ℕ0–onto→ran 𝐻) | |
| 14 | 12, 13 | mpbi 230 | . . . . . . . 8 ⊢ 𝐻:ℕ0–onto→ran 𝐻 |
| 15 | fodomnum 10097 | . . . . . . . 8 ⊢ (ℕ0 ∈ dom card → (𝐻:ℕ0–onto→ran 𝐻 → ran 𝐻 ≼ ℕ0)) | |
| 16 | 9, 14, 15 | mp2 9 | . . . . . . 7 ⊢ ran 𝐻 ≼ ℕ0 |
| 17 | domentr 9053 | . . . . . . 7 ⊢ ((ran 𝐻 ≼ ℕ0 ∧ ℕ0 ≈ ω) → ran 𝐻 ≼ ω) | |
| 18 | 16, 6, 17 | mp2an 692 | . . . . . 6 ⊢ ran 𝐻 ≼ ω |
| 19 | fvelrnb 6969 | . . . . . . . . 9 ⊢ (𝐻 Fn ℕ0 → (𝑓 ∈ ran 𝐻 ↔ ∃𝑔 ∈ ℕ0 (𝐻‘𝑔) = 𝑓)) | |
| 20 | 12, 19 | ax-mp 5 | . . . . . . . 8 ⊢ (𝑓 ∈ ran 𝐻 ↔ ∃𝑔 ∈ ℕ0 (𝐻‘𝑔) = 𝑓) |
| 21 | 1 | aannenlem1 26370 | . . . . . . . . . 10 ⊢ (𝑔 ∈ ℕ0 → (𝐻‘𝑔) ∈ Fin) |
| 22 | eleq1 2829 | . . . . . . . . . 10 ⊢ ((𝐻‘𝑔) = 𝑓 → ((𝐻‘𝑔) ∈ Fin ↔ 𝑓 ∈ Fin)) | |
| 23 | 21, 22 | syl5ibcom 245 | . . . . . . . . 9 ⊢ (𝑔 ∈ ℕ0 → ((𝐻‘𝑔) = 𝑓 → 𝑓 ∈ Fin)) |
| 24 | 23 | rexlimiv 3148 | . . . . . . . 8 ⊢ (∃𝑔 ∈ ℕ0 (𝐻‘𝑔) = 𝑓 → 𝑓 ∈ Fin) |
| 25 | 20, 24 | sylbi 217 | . . . . . . 7 ⊢ (𝑓 ∈ ran 𝐻 → 𝑓 ∈ Fin) |
| 26 | 25 | ssriv 3987 | . . . . . 6 ⊢ ran 𝐻 ⊆ Fin |
| 27 | aasscn 26360 | . . . . . . . 8 ⊢ 𝔸 ⊆ ℂ | |
| 28 | 2, 27 | eqsstrri 4031 | . . . . . . 7 ⊢ ∪ ran 𝐻 ⊆ ℂ |
| 29 | soss 5612 | . . . . . . 7 ⊢ (∪ ran 𝐻 ⊆ ℂ → (𝑓 Or ℂ → 𝑓 Or ∪ ran 𝐻)) | |
| 30 | 28, 29 | ax-mp 5 | . . . . . 6 ⊢ (𝑓 Or ℂ → 𝑓 Or ∪ ran 𝐻) |
| 31 | iunfictbso 10154 | . . . . . 6 ⊢ ((ran 𝐻 ≼ ω ∧ ran 𝐻 ⊆ Fin ∧ 𝑓 Or ∪ ran 𝐻) → ∪ ran 𝐻 ≼ ω) | |
| 32 | 18, 26, 30, 31 | mp3an12i 1467 | . . . . 5 ⊢ (𝑓 Or ℂ → ∪ ran 𝐻 ≼ ω) |
| 33 | 2, 32 | eqbrtrid 5178 | . . . 4 ⊢ (𝑓 Or ℂ → 𝔸 ≼ ω) |
| 34 | cnso 16283 | . . . 4 ⊢ ∃𝑓 𝑓 Or ℂ | |
| 35 | 33, 34 | exlimiiv 1931 | . . 3 ⊢ 𝔸 ≼ ω |
| 36 | 5 | ensymi 9044 | . . 3 ⊢ ω ≈ ℕ |
| 37 | domentr 9053 | . . 3 ⊢ ((𝔸 ≼ ω ∧ ω ≈ ℕ) → 𝔸 ≼ ℕ) | |
| 38 | 35, 36, 37 | mp2an 692 | . 2 ⊢ 𝔸 ≼ ℕ |
| 39 | 10, 27 | ssexi 5322 | . . 3 ⊢ 𝔸 ∈ V |
| 40 | nnssq 13000 | . . . 4 ⊢ ℕ ⊆ ℚ | |
| 41 | qssaa 26366 | . . . 4 ⊢ ℚ ⊆ 𝔸 | |
| 42 | 40, 41 | sstri 3993 | . . 3 ⊢ ℕ ⊆ 𝔸 |
| 43 | ssdomg 9040 | . . 3 ⊢ (𝔸 ∈ V → (ℕ ⊆ 𝔸 → ℕ ≼ 𝔸)) | |
| 44 | 39, 42, 43 | mp2 9 | . 2 ⊢ ℕ ≼ 𝔸 |
| 45 | sbth 9133 | . 2 ⊢ ((𝔸 ≼ ℕ ∧ ℕ ≼ 𝔸) → 𝔸 ≈ ℕ) | |
| 46 | 38, 44, 45 | mp2an 692 | 1 ⊢ 𝔸 ≈ ℕ |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ↔ wb 206 ∧ w3a 1087 = wceq 1540 ∈ wcel 2108 ≠ wne 2940 ∀wral 3061 ∃wrex 3070 {crab 3436 Vcvv 3480 ⊆ wss 3951 ∪ cuni 4907 class class class wbr 5143 ↦ cmpt 5225 Or wor 5591 dom cdm 5685 ran crn 5686 Oncon0 6384 Fn wfn 6556 –onto→wfo 6559 ‘cfv 6561 ωcom 7887 ≈ cen 8982 ≼ cdom 8983 Fincfn 8985 cardccrd 9975 ℂcc 11153 0cc0 11155 ≤ cle 11296 ℕcn 12266 ℕ0cn0 12526 ℤcz 12613 ℚcq 12990 abscabs 15273 0𝑝c0p 25704 Polycply 26223 coeffccoe 26225 degcdgr 26226 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-2o 8507 df-oadd 8510 df-omul 8511 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-dju 9941 df-card 9979 df-acn 9982 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-xnn0 12600 df-z 12614 df-uz 12879 df-q 12991 df-rp 13035 df-ico 13393 df-icc 13394 df-fz 13548 df-fzo 13695 df-fl 13832 df-mod 13910 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-limsup 15507 df-clim 15524 df-rlim 15525 df-sum 15723 df-0p 25705 df-ply 26227 df-idp 26228 df-coe 26229 df-dgr 26230 df-quot 26333 df-aa 26357 |
| This theorem is referenced by: aannen 26373 |
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