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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 24254 | . . . . 5 ⊢ 𝔸 = ∪ ran 𝐻 |
3 | omelon 8704 | . . . . . . . . . 10 ⊢ ω ∈ On | |
4 | nn0ennn 12943 | . . . . . . . . . . . 12 ⊢ ℕ0 ≈ ℕ | |
5 | nnenom 12944 | . . . . . . . . . . . 12 ⊢ ℕ ≈ ω | |
6 | 4, 5 | entri 8163 | . . . . . . . . . . 11 ⊢ ℕ0 ≈ ω |
7 | 6 | ensymi 8159 | . . . . . . . . . 10 ⊢ ω ≈ ℕ0 |
8 | isnumi 8933 | . . . . . . . . . 10 ⊢ ((ω ∈ On ∧ ω ≈ ℕ0) → ℕ0 ∈ dom card) | |
9 | 3, 7, 8 | mp2an 710 | . . . . . . . . 9 ⊢ ℕ0 ∈ dom card |
10 | cnex 10180 | . . . . . . . . . . . 12 ⊢ ℂ ∈ V | |
11 | 10 | rabex 4952 | . . . . . . . . . . 11 ⊢ {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 6171 | . . . . . . . . . 10 ⊢ 𝐻 Fn ℕ0 |
13 | dffn4 6270 | . . . . . . . . . 10 ⊢ (𝐻 Fn ℕ0 ↔ 𝐻:ℕ0–onto→ran 𝐻) | |
14 | 12, 13 | mpbi 220 | . . . . . . . . 9 ⊢ 𝐻:ℕ0–onto→ran 𝐻 |
15 | fodomnum 9041 | . . . . . . . . 9 ⊢ (ℕ0 ∈ dom card → (𝐻:ℕ0–onto→ran 𝐻 → ran 𝐻 ≼ ℕ0)) | |
16 | 9, 14, 15 | mp2 9 | . . . . . . . 8 ⊢ ran 𝐻 ≼ ℕ0 |
17 | domentr 8168 | . . . . . . . 8 ⊢ ((ran 𝐻 ≼ ℕ0 ∧ ℕ0 ≈ ω) → ran 𝐻 ≼ ω) | |
18 | 16, 6, 17 | mp2an 710 | . . . . . . 7 ⊢ ran 𝐻 ≼ ω |
19 | 18 | a1i 11 | . . . . . 6 ⊢ (𝑓 Or ℂ → ran 𝐻 ≼ ω) |
20 | fvelrnb 6393 | . . . . . . . . . 10 ⊢ (𝐻 Fn ℕ0 → (𝑓 ∈ ran 𝐻 ↔ ∃𝑔 ∈ ℕ0 (𝐻‘𝑔) = 𝑓)) | |
21 | 12, 20 | ax-mp 5 | . . . . . . . . 9 ⊢ (𝑓 ∈ ran 𝐻 ↔ ∃𝑔 ∈ ℕ0 (𝐻‘𝑔) = 𝑓) |
22 | 1 | aannenlem1 24253 | . . . . . . . . . . 11 ⊢ (𝑔 ∈ ℕ0 → (𝐻‘𝑔) ∈ Fin) |
23 | eleq1 2815 | . . . . . . . . . . 11 ⊢ ((𝐻‘𝑔) = 𝑓 → ((𝐻‘𝑔) ∈ Fin ↔ 𝑓 ∈ Fin)) | |
24 | 22, 23 | syl5ibcom 235 | . . . . . . . . . 10 ⊢ (𝑔 ∈ ℕ0 → ((𝐻‘𝑔) = 𝑓 → 𝑓 ∈ Fin)) |
25 | 24 | rexlimiv 3153 | . . . . . . . . 9 ⊢ (∃𝑔 ∈ ℕ0 (𝐻‘𝑔) = 𝑓 → 𝑓 ∈ Fin) |
26 | 21, 25 | sylbi 207 | . . . . . . . 8 ⊢ (𝑓 ∈ ran 𝐻 → 𝑓 ∈ Fin) |
27 | 26 | ssriv 3736 | . . . . . . 7 ⊢ ran 𝐻 ⊆ Fin |
28 | 27 | a1i 11 | . . . . . 6 ⊢ (𝑓 Or ℂ → ran 𝐻 ⊆ Fin) |
29 | aasscn 24243 | . . . . . . . 8 ⊢ 𝔸 ⊆ ℂ | |
30 | 2, 29 | eqsstr3i 3765 | . . . . . . 7 ⊢ ∪ ran 𝐻 ⊆ ℂ |
31 | soss 5193 | . . . . . . 7 ⊢ (∪ ran 𝐻 ⊆ ℂ → (𝑓 Or ℂ → 𝑓 Or ∪ ran 𝐻)) | |
32 | 30, 31 | ax-mp 5 | . . . . . 6 ⊢ (𝑓 Or ℂ → 𝑓 Or ∪ ran 𝐻) |
33 | iunfictbso 9098 | . . . . . 6 ⊢ ((ran 𝐻 ≼ ω ∧ ran 𝐻 ⊆ Fin ∧ 𝑓 Or ∪ ran 𝐻) → ∪ ran 𝐻 ≼ ω) | |
34 | 19, 28, 32, 33 | syl3anc 1463 | . . . . 5 ⊢ (𝑓 Or ℂ → ∪ ran 𝐻 ≼ ω) |
35 | 2, 34 | syl5eqbr 4827 | . . . 4 ⊢ (𝑓 Or ℂ → 𝔸 ≼ ω) |
36 | cnso 15146 | . . . 4 ⊢ ∃𝑓 𝑓 Or ℂ | |
37 | 35, 36 | exlimiiv 1996 | . . 3 ⊢ 𝔸 ≼ ω |
38 | 5 | ensymi 8159 | . . 3 ⊢ ω ≈ ℕ |
39 | domentr 8168 | . . 3 ⊢ ((𝔸 ≼ ω ∧ ω ≈ ℕ) → 𝔸 ≼ ℕ) | |
40 | 37, 38, 39 | mp2an 710 | . 2 ⊢ 𝔸 ≼ ℕ |
41 | 10, 29 | ssexi 4943 | . . 3 ⊢ 𝔸 ∈ V |
42 | nnssq 11961 | . . . 4 ⊢ ℕ ⊆ ℚ | |
43 | qssaa 24249 | . . . 4 ⊢ ℚ ⊆ 𝔸 | |
44 | 42, 43 | sstri 3741 | . . 3 ⊢ ℕ ⊆ 𝔸 |
45 | ssdomg 8155 | . . 3 ⊢ (𝔸 ∈ V → (ℕ ⊆ 𝔸 → ℕ ≼ 𝔸)) | |
46 | 41, 44, 45 | mp2 9 | . 2 ⊢ ℕ ≼ 𝔸 |
47 | sbth 8233 | . 2 ⊢ ((𝔸 ≼ ℕ ∧ ℕ ≼ 𝔸) → 𝔸 ≈ ℕ) | |
48 | 40, 46, 47 | mp2an 710 | 1 ⊢ 𝔸 ≈ ℕ |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 196 ∧ w3a 1072 = wceq 1620 ∈ wcel 2127 ≠ wne 2920 ∀wral 3038 ∃wrex 3039 {crab 3042 Vcvv 3328 ⊆ wss 3703 ∪ cuni 4576 class class class wbr 4792 ↦ cmpt 4869 Or wor 5174 dom cdm 5254 ran crn 5255 Oncon0 5872 Fn wfn 6032 –onto→wfo 6035 ‘cfv 6037 ωcom 7218 ≈ cen 8106 ≼ cdom 8107 Fincfn 8109 cardccrd 8922 ℂcc 10097 0cc0 10099 ≤ cle 10238 ℕcn 11183 ℕ0cn0 11455 ℤcz 11540 ℚcq 11952 abscabs 14144 0𝑝c0p 23606 Polycply 24110 coeffccoe 24112 degcdgr 24113 𝔸caa 24239 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1859 ax-4 1874 ax-5 1976 ax-6 2042 ax-7 2078 ax-8 2129 ax-9 2136 ax-10 2156 ax-11 2171 ax-12 2184 ax-13 2379 ax-ext 2728 ax-rep 4911 ax-sep 4921 ax-nul 4929 ax-pow 4980 ax-pr 5043 ax-un 7102 ax-inf2 8699 ax-cnex 10155 ax-resscn 10156 ax-1cn 10157 ax-icn 10158 ax-addcl 10159 ax-addrcl 10160 ax-mulcl 10161 ax-mulrcl 10162 ax-mulcom 10163 ax-addass 10164 ax-mulass 10165 ax-distr 10166 ax-i2m1 10167 ax-1ne0 10168 ax-1rid 10169 ax-rnegex 10170 ax-rrecex 10171 ax-cnre 10172 ax-pre-lttri 10173 ax-pre-lttrn 10174 ax-pre-ltadd 10175 ax-pre-mulgt0 10176 ax-pre-sup 10177 ax-addf 10178 |
This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-3or 1073 df-3an 1074 df-tru 1623 df-fal 1626 df-ex 1842 df-nf 1847 df-sb 2035 df-eu 2599 df-mo 2600 df-clab 2735 df-cleq 2741 df-clel 2744 df-nfc 2879 df-ne 2921 df-nel 3024 df-ral 3043 df-rex 3044 df-reu 3045 df-rmo 3046 df-rab 3047 df-v 3330 df-sbc 3565 df-csb 3663 df-dif 3706 df-un 3708 df-in 3710 df-ss 3717 df-pss 3719 df-nul 4047 df-if 4219 df-pw 4292 df-sn 4310 df-pr 4312 df-tp 4314 df-op 4316 df-uni 4577 df-int 4616 df-iun 4662 df-br 4793 df-opab 4853 df-mpt 4870 df-tr 4893 df-id 5162 df-eprel 5167 df-po 5175 df-so 5176 df-fr 5213 df-se 5214 df-we 5215 df-xp 5260 df-rel 5261 df-cnv 5262 df-co 5263 df-dm 5264 df-rn 5265 df-res 5266 df-ima 5267 df-pred 5829 df-ord 5875 df-on 5876 df-lim 5877 df-suc 5878 df-iota 6000 df-fun 6039 df-fn 6040 df-f 6041 df-f1 6042 df-fo 6043 df-f1o 6044 df-fv 6045 df-isom 6046 df-riota 6762 df-ov 6804 df-oprab 6805 df-mpt2 6806 df-of 7050 df-om 7219 df-1st 7321 df-2nd 7322 df-wrecs 7564 df-recs 7625 df-rdg 7663 df-1o 7717 df-2o 7718 df-oadd 7721 df-omul 7722 df-er 7899 df-map 8013 df-pm 8014 df-en 8110 df-dom 8111 df-sdom 8112 df-fin 8113 df-sup 8501 df-inf 8502 df-oi 8568 df-card 8926 df-acn 8929 df-cda 9153 df-pnf 10239 df-mnf 10240 df-xr 10241 df-ltxr 10242 df-le 10243 df-sub 10431 df-neg 10432 df-div 10848 df-nn 11184 df-2 11242 df-3 11243 df-n0 11456 df-xnn0 11527 df-z 11541 df-uz 11851 df-q 11953 df-rp 11997 df-ico 12345 df-icc 12346 df-fz 12491 df-fzo 12631 df-fl 12758 df-mod 12834 df-seq 12967 df-exp 13026 df-hash 13283 df-cj 14009 df-re 14010 df-im 14011 df-sqrt 14145 df-abs 14146 df-limsup 14372 df-clim 14389 df-rlim 14390 df-sum 14587 df-0p 23607 df-ply 24114 df-idp 24115 df-coe 24116 df-dgr 24117 df-quot 24216 df-aa 24240 |
This theorem is referenced by: aannen 24256 |
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