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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 26357 | . . . . 5 ⊢ 𝔸 = ∪ ran 𝐻 |
3 | omelon 9689 | . . . . . . . . 9 ⊢ ω ∈ On | |
4 | nn0ennn 13999 | . . . . . . . . . . 11 ⊢ ℕ0 ≈ ℕ | |
5 | nnenom 14000 | . . . . . . . . . . 11 ⊢ ℕ ≈ ω | |
6 | 4, 5 | entri 9039 | . . . . . . . . . 10 ⊢ ℕ0 ≈ ω |
7 | 6 | ensymi 9035 | . . . . . . . . 9 ⊢ ω ≈ ℕ0 |
8 | isnumi 9989 | . . . . . . . . 9 ⊢ ((ω ∈ On ∧ ω ≈ ℕ0) → ℕ0 ∈ dom card) | |
9 | 3, 7, 8 | mp2an 690 | . . . . . . . 8 ⊢ ℕ0 ∈ dom card |
10 | cnex 11239 | . . . . . . . . . . 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 6704 | . . . . . . . . 9 ⊢ 𝐻 Fn ℕ0 |
13 | dffn4 6821 | . . . . . . . . 9 ⊢ (𝐻 Fn ℕ0 ↔ 𝐻:ℕ0–onto→ran 𝐻) | |
14 | 12, 13 | mpbi 229 | . . . . . . . 8 ⊢ 𝐻:ℕ0–onto→ran 𝐻 |
15 | fodomnum 10100 | . . . . . . . 8 ⊢ (ℕ0 ∈ dom card → (𝐻:ℕ0–onto→ran 𝐻 → ran 𝐻 ≼ ℕ0)) | |
16 | 9, 14, 15 | mp2 9 | . . . . . . 7 ⊢ ran 𝐻 ≼ ℕ0 |
17 | domentr 9044 | . . . . . . 7 ⊢ ((ran 𝐻 ≼ ℕ0 ∧ ℕ0 ≈ ω) → ran 𝐻 ≼ ω) | |
18 | 16, 6, 17 | mp2an 690 | . . . . . 6 ⊢ ran 𝐻 ≼ ω |
19 | fvelrnb 6963 | . . . . . . . . 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 26356 | . . . . . . . . . 10 ⊢ (𝑔 ∈ ℕ0 → (𝐻‘𝑔) ∈ Fin) |
22 | eleq1 2814 | . . . . . . . . . 10 ⊢ ((𝐻‘𝑔) = 𝑓 → ((𝐻‘𝑔) ∈ Fin ↔ 𝑓 ∈ Fin)) | |
23 | 21, 22 | syl5ibcom 244 | . . . . . . . . 9 ⊢ (𝑔 ∈ ℕ0 → ((𝐻‘𝑔) = 𝑓 → 𝑓 ∈ Fin)) |
24 | 23 | rexlimiv 3138 | . . . . . . . 8 ⊢ (∃𝑔 ∈ ℕ0 (𝐻‘𝑔) = 𝑓 → 𝑓 ∈ Fin) |
25 | 20, 24 | sylbi 216 | . . . . . . 7 ⊢ (𝑓 ∈ ran 𝐻 → 𝑓 ∈ Fin) |
26 | 25 | ssriv 3983 | . . . . . 6 ⊢ ran 𝐻 ⊆ Fin |
27 | aasscn 26346 | . . . . . . . 8 ⊢ 𝔸 ⊆ ℂ | |
28 | 2, 27 | eqsstrri 4015 | . . . . . . 7 ⊢ ∪ ran 𝐻 ⊆ ℂ |
29 | soss 5614 | . . . . . . 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 10157 | . . . . . 6 ⊢ ((ran 𝐻 ≼ ω ∧ ran 𝐻 ⊆ Fin ∧ 𝑓 Or ∪ ran 𝐻) → ∪ ran 𝐻 ≼ ω) | |
32 | 18, 26, 30, 31 | mp3an12i 1462 | . . . . 5 ⊢ (𝑓 Or ℂ → ∪ ran 𝐻 ≼ ω) |
33 | 2, 32 | eqbrtrid 5188 | . . . 4 ⊢ (𝑓 Or ℂ → 𝔸 ≼ ω) |
34 | cnso 16249 | . . . 4 ⊢ ∃𝑓 𝑓 Or ℂ | |
35 | 33, 34 | exlimiiv 1927 | . . 3 ⊢ 𝔸 ≼ ω |
36 | 5 | ensymi 9035 | . . 3 ⊢ ω ≈ ℕ |
37 | domentr 9044 | . . 3 ⊢ ((𝔸 ≼ ω ∧ ω ≈ ℕ) → 𝔸 ≼ ℕ) | |
38 | 35, 36, 37 | mp2an 690 | . 2 ⊢ 𝔸 ≼ ℕ |
39 | 10, 27 | ssexi 5327 | . . 3 ⊢ 𝔸 ∈ V |
40 | nnssq 12994 | . . . 4 ⊢ ℕ ⊆ ℚ | |
41 | qssaa 26352 | . . . 4 ⊢ ℚ ⊆ 𝔸 | |
42 | 40, 41 | sstri 3989 | . . 3 ⊢ ℕ ⊆ 𝔸 |
43 | ssdomg 9031 | . . 3 ⊢ (𝔸 ∈ V → (ℕ ⊆ 𝔸 → ℕ ≼ 𝔸)) | |
44 | 39, 42, 43 | mp2 9 | . 2 ⊢ ℕ ≼ 𝔸 |
45 | sbth 9131 | . 2 ⊢ ((𝔸 ≼ ℕ ∧ ℕ ≼ 𝔸) → 𝔸 ≈ ℕ) | |
46 | 38, 44, 45 | mp2an 690 | 1 ⊢ 𝔸 ≈ ℕ |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ w3a 1084 = wceq 1534 ∈ wcel 2099 ≠ wne 2930 ∀wral 3051 ∃wrex 3060 {crab 3419 Vcvv 3462 ⊆ wss 3947 ∪ cuni 4913 class class class wbr 5153 ↦ cmpt 5236 Or wor 5593 dom cdm 5682 ran crn 5683 Oncon0 6376 Fn wfn 6549 –onto→wfo 6552 ‘cfv 6554 ωcom 7876 ≈ cen 8971 ≼ cdom 8972 Fincfn 8974 cardccrd 9978 ℂcc 11156 0cc0 11158 ≤ cle 11299 ℕcn 12264 ℕ0cn0 12524 ℤcz 12610 ℚcq 12984 abscabs 15239 0𝑝c0p 25689 Polycply 26211 coeffccoe 26213 degcdgr 26214 𝔸caa 26342 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2697 ax-rep 5290 ax-sep 5304 ax-nul 5311 ax-pow 5369 ax-pr 5433 ax-un 7746 ax-inf2 9684 ax-cnex 11214 ax-resscn 11215 ax-1cn 11216 ax-icn 11217 ax-addcl 11218 ax-addrcl 11219 ax-mulcl 11220 ax-mulrcl 11221 ax-mulcom 11222 ax-addass 11223 ax-mulass 11224 ax-distr 11225 ax-i2m1 11226 ax-1ne0 11227 ax-1rid 11228 ax-rnegex 11229 ax-rrecex 11230 ax-cnre 11231 ax-pre-lttri 11232 ax-pre-lttrn 11233 ax-pre-ltadd 11234 ax-pre-mulgt0 11235 ax-pre-sup 11236 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2704 df-cleq 2718 df-clel 2803 df-nfc 2878 df-ne 2931 df-nel 3037 df-ral 3052 df-rex 3061 df-rmo 3364 df-reu 3365 df-rab 3420 df-v 3464 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3967 df-nul 4326 df-if 4534 df-pw 4609 df-sn 4634 df-pr 4636 df-op 4640 df-uni 4914 df-int 4955 df-iun 5003 df-br 5154 df-opab 5216 df-mpt 5237 df-tr 5271 df-id 5580 df-eprel 5586 df-po 5594 df-so 5595 df-fr 5637 df-se 5638 df-we 5639 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-rn 5693 df-res 5694 df-ima 5695 df-pred 6312 df-ord 6379 df-on 6380 df-lim 6381 df-suc 6382 df-iota 6506 df-fun 6556 df-fn 6557 df-f 6558 df-f1 6559 df-fo 6560 df-f1o 6561 df-fv 6562 df-isom 6563 df-riota 7380 df-ov 7427 df-oprab 7428 df-mpo 7429 df-of 7690 df-om 7877 df-1st 8003 df-2nd 8004 df-frecs 8296 df-wrecs 8327 df-recs 8401 df-rdg 8440 df-1o 8496 df-2o 8497 df-oadd 8500 df-omul 8501 df-er 8734 df-map 8857 df-pm 8858 df-en 8975 df-dom 8976 df-sdom 8977 df-fin 8978 df-sup 9485 df-inf 9486 df-oi 9553 df-dju 9944 df-card 9982 df-acn 9985 df-pnf 11300 df-mnf 11301 df-xr 11302 df-ltxr 11303 df-le 11304 df-sub 11496 df-neg 11497 df-div 11922 df-nn 12265 df-2 12327 df-3 12328 df-n0 12525 df-xnn0 12597 df-z 12611 df-uz 12875 df-q 12985 df-rp 13029 df-ico 13384 df-icc 13385 df-fz 13539 df-fzo 13682 df-fl 13812 df-mod 13890 df-seq 14022 df-exp 14082 df-hash 14348 df-cj 15104 df-re 15105 df-im 15106 df-sqrt 15240 df-abs 15241 df-limsup 15473 df-clim 15490 df-rlim 15491 df-sum 15691 df-0p 25690 df-ply 26215 df-idp 26216 df-coe 26217 df-dgr 26218 df-quot 26319 df-aa 26343 |
This theorem is referenced by: aannen 26359 |
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