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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 25194 | . . . . 5 ⊢ 𝔸 = ∪ ran 𝐻 |
3 | omelon 9250 | . . . . . . . . 9 ⊢ ω ∈ On | |
4 | nn0ennn 13535 | . . . . . . . . . . 11 ⊢ ℕ0 ≈ ℕ | |
5 | nnenom 13536 | . . . . . . . . . . 11 ⊢ ℕ ≈ ω | |
6 | 4, 5 | entri 8671 | . . . . . . . . . 10 ⊢ ℕ0 ≈ ω |
7 | 6 | ensymi 8667 | . . . . . . . . 9 ⊢ ω ≈ ℕ0 |
8 | isnumi 9545 | . . . . . . . . 9 ⊢ ((ω ∈ On ∧ ω ≈ ℕ0) → ℕ0 ∈ dom card) | |
9 | 3, 7, 8 | mp2an 692 | . . . . . . . 8 ⊢ ℕ0 ∈ dom card |
10 | cnex 10793 | . . . . . . . . . . 11 ⊢ ℂ ∈ V | |
11 | 10 | rabex 5214 | . . . . . . . . . 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 6510 | . . . . . . . . 9 ⊢ 𝐻 Fn ℕ0 |
13 | dffn4 6628 | . . . . . . . . 9 ⊢ (𝐻 Fn ℕ0 ↔ 𝐻:ℕ0–onto→ran 𝐻) | |
14 | 12, 13 | mpbi 233 | . . . . . . . 8 ⊢ 𝐻:ℕ0–onto→ran 𝐻 |
15 | fodomnum 9654 | . . . . . . . 8 ⊢ (ℕ0 ∈ dom card → (𝐻:ℕ0–onto→ran 𝐻 → ran 𝐻 ≼ ℕ0)) | |
16 | 9, 14, 15 | mp2 9 | . . . . . . 7 ⊢ ran 𝐻 ≼ ℕ0 |
17 | domentr 8676 | . . . . . . 7 ⊢ ((ran 𝐻 ≼ ℕ0 ∧ ℕ0 ≈ ω) → ran 𝐻 ≼ ω) | |
18 | 16, 6, 17 | mp2an 692 | . . . . . 6 ⊢ ran 𝐻 ≼ ω |
19 | fvelrnb 6762 | . . . . . . . . 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 25193 | . . . . . . . . . 10 ⊢ (𝑔 ∈ ℕ0 → (𝐻‘𝑔) ∈ Fin) |
22 | eleq1 2821 | . . . . . . . . . 10 ⊢ ((𝐻‘𝑔) = 𝑓 → ((𝐻‘𝑔) ∈ Fin ↔ 𝑓 ∈ Fin)) | |
23 | 21, 22 | syl5ibcom 248 | . . . . . . . . 9 ⊢ (𝑔 ∈ ℕ0 → ((𝐻‘𝑔) = 𝑓 → 𝑓 ∈ Fin)) |
24 | 23 | rexlimiv 3192 | . . . . . . . 8 ⊢ (∃𝑔 ∈ ℕ0 (𝐻‘𝑔) = 𝑓 → 𝑓 ∈ Fin) |
25 | 20, 24 | sylbi 220 | . . . . . . 7 ⊢ (𝑓 ∈ ran 𝐻 → 𝑓 ∈ Fin) |
26 | 25 | ssriv 3895 | . . . . . 6 ⊢ ran 𝐻 ⊆ Fin |
27 | aasscn 25183 | . . . . . . . 8 ⊢ 𝔸 ⊆ ℂ | |
28 | 2, 27 | eqsstrri 3926 | . . . . . . 7 ⊢ ∪ ran 𝐻 ⊆ ℂ |
29 | soss 5477 | . . . . . . 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 9711 | . . . . . 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 5078 | . . . 4 ⊢ (𝑓 Or ℂ → 𝔸 ≼ ω) |
34 | cnso 15789 | . . . 4 ⊢ ∃𝑓 𝑓 Or ℂ | |
35 | 33, 34 | exlimiiv 1939 | . . 3 ⊢ 𝔸 ≼ ω |
36 | 5 | ensymi 8667 | . . 3 ⊢ ω ≈ ℕ |
37 | domentr 8676 | . . 3 ⊢ ((𝔸 ≼ ω ∧ ω ≈ ℕ) → 𝔸 ≼ ℕ) | |
38 | 35, 36, 37 | mp2an 692 | . 2 ⊢ 𝔸 ≼ ℕ |
39 | 10, 27 | ssexi 5204 | . . 3 ⊢ 𝔸 ∈ V |
40 | nnssq 12537 | . . . 4 ⊢ ℕ ⊆ ℚ | |
41 | qssaa 25189 | . . . 4 ⊢ ℚ ⊆ 𝔸 | |
42 | 40, 41 | sstri 3900 | . . 3 ⊢ ℕ ⊆ 𝔸 |
43 | ssdomg 8663 | . . 3 ⊢ (𝔸 ∈ V → (ℕ ⊆ 𝔸 → ℕ ≼ 𝔸)) | |
44 | 39, 42, 43 | mp2 9 | . 2 ⊢ ℕ ≼ 𝔸 |
45 | sbth 8755 | . 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 209 ∧ w3a 1089 = wceq 1543 ∈ wcel 2110 ≠ wne 2935 ∀wral 3054 ∃wrex 3055 {crab 3058 Vcvv 3401 ⊆ wss 3857 ∪ cuni 4809 class class class wbr 5043 ↦ cmpt 5124 Or wor 5456 dom cdm 5540 ran crn 5541 Oncon0 6202 Fn wfn 6364 –onto→wfo 6367 ‘cfv 6369 ωcom 7633 ≈ cen 8612 ≼ cdom 8613 Fincfn 8615 cardccrd 9534 ℂcc 10710 0cc0 10712 ≤ cle 10851 ℕcn 11813 ℕ0cn0 12073 ℤcz 12159 ℚcq 12527 abscabs 14780 0𝑝c0p 24538 Polycply 25050 coeffccoe 25052 degcdgr 25053 𝔸caa 25179 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2706 ax-rep 5168 ax-sep 5181 ax-nul 5188 ax-pow 5247 ax-pr 5311 ax-un 7512 ax-inf2 9245 ax-cnex 10768 ax-resscn 10769 ax-1cn 10770 ax-icn 10771 ax-addcl 10772 ax-addrcl 10773 ax-mulcl 10774 ax-mulrcl 10775 ax-mulcom 10776 ax-addass 10777 ax-mulass 10778 ax-distr 10779 ax-i2m1 10780 ax-1ne0 10781 ax-1rid 10782 ax-rnegex 10783 ax-rrecex 10784 ax-cnre 10785 ax-pre-lttri 10786 ax-pre-lttrn 10787 ax-pre-ltadd 10788 ax-pre-mulgt0 10789 ax-pre-sup 10790 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3or 1090 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2071 df-mo 2537 df-eu 2566 df-clab 2713 df-cleq 2726 df-clel 2812 df-nfc 2882 df-ne 2936 df-nel 3040 df-ral 3059 df-rex 3060 df-reu 3061 df-rmo 3062 df-rab 3063 df-v 3403 df-sbc 3688 df-csb 3803 df-dif 3860 df-un 3862 df-in 3864 df-ss 3874 df-pss 3876 df-nul 4228 df-if 4430 df-pw 4505 df-sn 4532 df-pr 4534 df-tp 4536 df-op 4538 df-uni 4810 df-int 4850 df-iun 4896 df-br 5044 df-opab 5106 df-mpt 5125 df-tr 5151 df-id 5444 df-eprel 5449 df-po 5457 df-so 5458 df-fr 5498 df-se 5499 df-we 5500 df-xp 5546 df-rel 5547 df-cnv 5548 df-co 5549 df-dm 5550 df-rn 5551 df-res 5552 df-ima 5553 df-pred 6149 df-ord 6205 df-on 6206 df-lim 6207 df-suc 6208 df-iota 6327 df-fun 6371 df-fn 6372 df-f 6373 df-f1 6374 df-fo 6375 df-f1o 6376 df-fv 6377 df-isom 6378 df-riota 7159 df-ov 7205 df-oprab 7206 df-mpo 7207 df-of 7458 df-om 7634 df-1st 7750 df-2nd 7751 df-wrecs 8036 df-recs 8097 df-rdg 8135 df-1o 8191 df-2o 8192 df-oadd 8195 df-omul 8196 df-er 8380 df-map 8499 df-pm 8500 df-en 8616 df-dom 8617 df-sdom 8618 df-fin 8619 df-sup 9047 df-inf 9048 df-oi 9115 df-dju 9500 df-card 9538 df-acn 9541 df-pnf 10852 df-mnf 10853 df-xr 10854 df-ltxr 10855 df-le 10856 df-sub 11047 df-neg 11048 df-div 11473 df-nn 11814 df-2 11876 df-3 11877 df-n0 12074 df-xnn0 12146 df-z 12160 df-uz 12422 df-q 12528 df-rp 12570 df-ico 12924 df-icc 12925 df-fz 13079 df-fzo 13222 df-fl 13350 df-mod 13426 df-seq 13558 df-exp 13619 df-hash 13880 df-cj 14645 df-re 14646 df-im 14647 df-sqrt 14781 df-abs 14782 df-limsup 15015 df-clim 15032 df-rlim 15033 df-sum 15233 df-0p 24539 df-ply 25054 df-idp 25055 df-coe 25056 df-dgr 25057 df-quot 25156 df-aa 25180 |
This theorem is referenced by: aannen 25196 |
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