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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 24912 | . . . . 5 ⊢ 𝔸 = ∪ ran 𝐻 |
3 | omelon 9103 | . . . . . . . . 9 ⊢ ω ∈ On | |
4 | nn0ennn 13341 | . . . . . . . . . . 11 ⊢ ℕ0 ≈ ℕ | |
5 | nnenom 13342 | . . . . . . . . . . 11 ⊢ ℕ ≈ ω | |
6 | 4, 5 | entri 8557 | . . . . . . . . . 10 ⊢ ℕ0 ≈ ω |
7 | 6 | ensymi 8553 | . . . . . . . . 9 ⊢ ω ≈ ℕ0 |
8 | isnumi 9369 | . . . . . . . . 9 ⊢ ((ω ∈ On ∧ ω ≈ ℕ0) → ℕ0 ∈ dom card) | |
9 | 3, 7, 8 | mp2an 690 | . . . . . . . 8 ⊢ ℕ0 ∈ dom card |
10 | cnex 10612 | . . . . . . . . . . 11 ⊢ ℂ ∈ V | |
11 | 10 | rabex 5227 | . . . . . . . . . 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 6485 | . . . . . . . . 9 ⊢ 𝐻 Fn ℕ0 |
13 | dffn4 6590 | . . . . . . . . 9 ⊢ (𝐻 Fn ℕ0 ↔ 𝐻:ℕ0–onto→ran 𝐻) | |
14 | 12, 13 | mpbi 232 | . . . . . . . 8 ⊢ 𝐻:ℕ0–onto→ran 𝐻 |
15 | fodomnum 9477 | . . . . . . . 8 ⊢ (ℕ0 ∈ dom card → (𝐻:ℕ0–onto→ran 𝐻 → ran 𝐻 ≼ ℕ0)) | |
16 | 9, 14, 15 | mp2 9 | . . . . . . 7 ⊢ ran 𝐻 ≼ ℕ0 |
17 | domentr 8562 | . . . . . . 7 ⊢ ((ran 𝐻 ≼ ℕ0 ∧ ℕ0 ≈ ω) → ran 𝐻 ≼ ω) | |
18 | 16, 6, 17 | mp2an 690 | . . . . . 6 ⊢ ran 𝐻 ≼ ω |
19 | fvelrnb 6720 | . . . . . . . . 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 24911 | . . . . . . . . . 10 ⊢ (𝑔 ∈ ℕ0 → (𝐻‘𝑔) ∈ Fin) |
22 | eleq1 2900 | . . . . . . . . . 10 ⊢ ((𝐻‘𝑔) = 𝑓 → ((𝐻‘𝑔) ∈ Fin ↔ 𝑓 ∈ Fin)) | |
23 | 21, 22 | syl5ibcom 247 | . . . . . . . . 9 ⊢ (𝑔 ∈ ℕ0 → ((𝐻‘𝑔) = 𝑓 → 𝑓 ∈ Fin)) |
24 | 23 | rexlimiv 3280 | . . . . . . . 8 ⊢ (∃𝑔 ∈ ℕ0 (𝐻‘𝑔) = 𝑓 → 𝑓 ∈ Fin) |
25 | 20, 24 | sylbi 219 | . . . . . . 7 ⊢ (𝑓 ∈ ran 𝐻 → 𝑓 ∈ Fin) |
26 | 25 | ssriv 3970 | . . . . . 6 ⊢ ran 𝐻 ⊆ Fin |
27 | aasscn 24901 | . . . . . . . 8 ⊢ 𝔸 ⊆ ℂ | |
28 | 2, 27 | eqsstrri 4001 | . . . . . . 7 ⊢ ∪ ran 𝐻 ⊆ ℂ |
29 | soss 5487 | . . . . . . 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 9534 | . . . . . 6 ⊢ ((ran 𝐻 ≼ ω ∧ ran 𝐻 ⊆ Fin ∧ 𝑓 Or ∪ ran 𝐻) → ∪ ran 𝐻 ≼ ω) | |
32 | 18, 26, 30, 31 | mp3an12i 1461 | . . . . 5 ⊢ (𝑓 Or ℂ → ∪ ran 𝐻 ≼ ω) |
33 | 2, 32 | eqbrtrid 5093 | . . . 4 ⊢ (𝑓 Or ℂ → 𝔸 ≼ ω) |
34 | cnso 15594 | . . . 4 ⊢ ∃𝑓 𝑓 Or ℂ | |
35 | 33, 34 | exlimiiv 1928 | . . 3 ⊢ 𝔸 ≼ ω |
36 | 5 | ensymi 8553 | . . 3 ⊢ ω ≈ ℕ |
37 | domentr 8562 | . . 3 ⊢ ((𝔸 ≼ ω ∧ ω ≈ ℕ) → 𝔸 ≼ ℕ) | |
38 | 35, 36, 37 | mp2an 690 | . 2 ⊢ 𝔸 ≼ ℕ |
39 | 10, 27 | ssexi 5218 | . . 3 ⊢ 𝔸 ∈ V |
40 | nnssq 12351 | . . . 4 ⊢ ℕ ⊆ ℚ | |
41 | qssaa 24907 | . . . 4 ⊢ ℚ ⊆ 𝔸 | |
42 | 40, 41 | sstri 3975 | . . 3 ⊢ ℕ ⊆ 𝔸 |
43 | ssdomg 8549 | . . 3 ⊢ (𝔸 ∈ V → (ℕ ⊆ 𝔸 → ℕ ≼ 𝔸)) | |
44 | 39, 42, 43 | mp2 9 | . 2 ⊢ ℕ ≼ 𝔸 |
45 | sbth 8631 | . 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 208 ∧ w3a 1083 = wceq 1533 ∈ wcel 2110 ≠ wne 3016 ∀wral 3138 ∃wrex 3139 {crab 3142 Vcvv 3494 ⊆ wss 3935 ∪ cuni 4831 class class class wbr 5058 ↦ cmpt 5138 Or wor 5467 dom cdm 5549 ran crn 5550 Oncon0 6185 Fn wfn 6344 –onto→wfo 6347 ‘cfv 6349 ωcom 7574 ≈ cen 8500 ≼ cdom 8501 Fincfn 8503 cardccrd 9358 ℂcc 10529 0cc0 10531 ≤ cle 10670 ℕcn 11632 ℕ0cn0 11891 ℤcz 11975 ℚcq 12342 abscabs 14587 0𝑝c0p 24264 Polycply 24768 coeffccoe 24770 degcdgr 24771 𝔸caa 24897 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-rep 5182 ax-sep 5195 ax-nul 5202 ax-pow 5258 ax-pr 5321 ax-un 7455 ax-inf2 9098 ax-cnex 10587 ax-resscn 10588 ax-1cn 10589 ax-icn 10590 ax-addcl 10591 ax-addrcl 10592 ax-mulcl 10593 ax-mulrcl 10594 ax-mulcom 10595 ax-addass 10596 ax-mulass 10597 ax-distr 10598 ax-i2m1 10599 ax-1ne0 10600 ax-1rid 10601 ax-rnegex 10602 ax-rrecex 10603 ax-cnre 10604 ax-pre-lttri 10605 ax-pre-lttrn 10606 ax-pre-ltadd 10607 ax-pre-mulgt0 10608 ax-pre-sup 10609 ax-addf 10610 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1536 df-fal 1546 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3496 df-sbc 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-pss 3953 df-nul 4291 df-if 4467 df-pw 4540 df-sn 4561 df-pr 4563 df-tp 4565 df-op 4567 df-uni 4832 df-int 4869 df-iun 4913 df-br 5059 df-opab 5121 df-mpt 5139 df-tr 5165 df-id 5454 df-eprel 5459 df-po 5468 df-so 5469 df-fr 5508 df-se 5509 df-we 5510 df-xp 5555 df-rel 5556 df-cnv 5557 df-co 5558 df-dm 5559 df-rn 5560 df-res 5561 df-ima 5562 df-pred 6142 df-ord 6188 df-on 6189 df-lim 6190 df-suc 6191 df-iota 6308 df-fun 6351 df-fn 6352 df-f 6353 df-f1 6354 df-fo 6355 df-f1o 6356 df-fv 6357 df-isom 6358 df-riota 7108 df-ov 7153 df-oprab 7154 df-mpo 7155 df-of 7403 df-om 7575 df-1st 7683 df-2nd 7684 df-wrecs 7941 df-recs 8002 df-rdg 8040 df-1o 8096 df-2o 8097 df-oadd 8100 df-omul 8101 df-er 8283 df-map 8402 df-pm 8403 df-en 8504 df-dom 8505 df-sdom 8506 df-fin 8507 df-sup 8900 df-inf 8901 df-oi 8968 df-dju 9324 df-card 9362 df-acn 9365 df-pnf 10671 df-mnf 10672 df-xr 10673 df-ltxr 10674 df-le 10675 df-sub 10866 df-neg 10867 df-div 11292 df-nn 11633 df-2 11694 df-3 11695 df-n0 11892 df-xnn0 11962 df-z 11976 df-uz 12238 df-q 12343 df-rp 12384 df-ico 12738 df-icc 12739 df-fz 12887 df-fzo 13028 df-fl 13156 df-mod 13232 df-seq 13364 df-exp 13424 df-hash 13685 df-cj 14452 df-re 14453 df-im 14454 df-sqrt 14588 df-abs 14589 df-limsup 14822 df-clim 14839 df-rlim 14840 df-sum 15037 df-0p 24265 df-ply 24772 df-idp 24773 df-coe 24774 df-dgr 24775 df-quot 24874 df-aa 24898 |
This theorem is referenced by: aannen 24914 |
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