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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 26385 | . . . . 5 ⊢ 𝔸 = ∪ ran 𝐻 |
3 | omelon 9683 | . . . . . . . . 9 ⊢ ω ∈ On | |
4 | nn0ennn 14016 | . . . . . . . . . . 11 ⊢ ℕ0 ≈ ℕ | |
5 | nnenom 14017 | . . . . . . . . . . 11 ⊢ ℕ ≈ ω | |
6 | 4, 5 | entri 9046 | . . . . . . . . . 10 ⊢ ℕ0 ≈ ω |
7 | 6 | ensymi 9042 | . . . . . . . . 9 ⊢ ω ≈ ℕ0 |
8 | isnumi 9983 | . . . . . . . . 9 ⊢ ((ω ∈ On ∧ ω ≈ ℕ0) → ℕ0 ∈ dom card) | |
9 | 3, 7, 8 | mp2an 692 | . . . . . . . 8 ⊢ ℕ0 ∈ dom card |
10 | cnex 11233 | . . . . . . . . . . 11 ⊢ ℂ ∈ V | |
11 | 10 | rabex 5344 | . . . . . . . . . 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 10094 | . . . . . . . 8 ⊢ (ℕ0 ∈ dom card → (𝐻:ℕ0–onto→ran 𝐻 → ran 𝐻 ≼ ℕ0)) | |
16 | 9, 14, 15 | mp2 9 | . . . . . . 7 ⊢ ran 𝐻 ≼ ℕ0 |
17 | domentr 9051 | . . . . . . 7 ⊢ ((ran 𝐻 ≼ ℕ0 ∧ ℕ0 ≈ ω) → ran 𝐻 ≼ ω) | |
18 | 16, 6, 17 | mp2an 692 | . . . . . 6 ⊢ ran 𝐻 ≼ ω |
19 | fvelrnb 6968 | . . . . . . . . 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 26384 | . . . . . . . . . 10 ⊢ (𝑔 ∈ ℕ0 → (𝐻‘𝑔) ∈ Fin) |
22 | eleq1 2826 | . . . . . . . . . 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 3145 | . . . . . . . 8 ⊢ (∃𝑔 ∈ ℕ0 (𝐻‘𝑔) = 𝑓 → 𝑓 ∈ Fin) |
25 | 20, 24 | sylbi 217 | . . . . . . 7 ⊢ (𝑓 ∈ ran 𝐻 → 𝑓 ∈ Fin) |
26 | 25 | ssriv 3998 | . . . . . 6 ⊢ ran 𝐻 ⊆ Fin |
27 | aasscn 26374 | . . . . . . . 8 ⊢ 𝔸 ⊆ ℂ | |
28 | 2, 27 | eqsstrri 4030 | . . . . . . 7 ⊢ ∪ ran 𝐻 ⊆ ℂ |
29 | soss 5616 | . . . . . . 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 10151 | . . . . . 6 ⊢ ((ran 𝐻 ≼ ω ∧ ran 𝐻 ⊆ Fin ∧ 𝑓 Or ∪ ran 𝐻) → ∪ ran 𝐻 ≼ ω) | |
32 | 18, 26, 30, 31 | mp3an12i 1464 | . . . . 5 ⊢ (𝑓 Or ℂ → ∪ ran 𝐻 ≼ ω) |
33 | 2, 32 | eqbrtrid 5182 | . . . 4 ⊢ (𝑓 Or ℂ → 𝔸 ≼ ω) |
34 | cnso 16279 | . . . 4 ⊢ ∃𝑓 𝑓 Or ℂ | |
35 | 33, 34 | exlimiiv 1928 | . . 3 ⊢ 𝔸 ≼ ω |
36 | 5 | ensymi 9042 | . . 3 ⊢ ω ≈ ℕ |
37 | domentr 9051 | . . 3 ⊢ ((𝔸 ≼ ω ∧ ω ≈ ℕ) → 𝔸 ≼ ℕ) | |
38 | 35, 36, 37 | mp2an 692 | . 2 ⊢ 𝔸 ≼ ℕ |
39 | 10, 27 | ssexi 5327 | . . 3 ⊢ 𝔸 ∈ V |
40 | nnssq 12997 | . . . 4 ⊢ ℕ ⊆ ℚ | |
41 | qssaa 26380 | . . . 4 ⊢ ℚ ⊆ 𝔸 | |
42 | 40, 41 | sstri 4004 | . . 3 ⊢ ℕ ⊆ 𝔸 |
43 | ssdomg 9038 | . . 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 692 | 1 ⊢ 𝔸 ≈ ℕ |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 206 ∧ w3a 1086 = wceq 1536 ∈ wcel 2105 ≠ wne 2937 ∀wral 3058 ∃wrex 3067 {crab 3432 Vcvv 3477 ⊆ wss 3962 ∪ cuni 4911 class class class wbr 5147 ↦ cmpt 5230 Or wor 5595 dom cdm 5688 ran crn 5689 Oncon0 6385 Fn wfn 6557 –onto→wfo 6560 ‘cfv 6562 ωcom 7886 ≈ cen 8980 ≼ cdom 8981 Fincfn 8983 cardccrd 9972 ℂcc 11150 0cc0 11152 ≤ cle 11293 ℕcn 12263 ℕ0cn0 12523 ℤcz 12610 ℚcq 12987 abscabs 15269 0𝑝c0p 25717 Polycply 26237 coeffccoe 26239 degcdgr 26240 𝔸caa 26370 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1791 ax-4 1805 ax-5 1907 ax-6 1964 ax-7 2004 ax-8 2107 ax-9 2115 ax-10 2138 ax-11 2154 ax-12 2174 ax-ext 2705 ax-rep 5284 ax-sep 5301 ax-nul 5311 ax-pow 5370 ax-pr 5437 ax-un 7753 ax-inf2 9678 ax-cnex 11208 ax-resscn 11209 ax-1cn 11210 ax-icn 11211 ax-addcl 11212 ax-addrcl 11213 ax-mulcl 11214 ax-mulrcl 11215 ax-mulcom 11216 ax-addass 11217 ax-mulass 11218 ax-distr 11219 ax-i2m1 11220 ax-1ne0 11221 ax-1rid 11222 ax-rnegex 11223 ax-rrecex 11224 ax-cnre 11225 ax-pre-lttri 11226 ax-pre-lttrn 11227 ax-pre-ltadd 11228 ax-pre-mulgt0 11229 ax-pre-sup 11230 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1539 df-fal 1549 df-ex 1776 df-nf 1780 df-sb 2062 df-mo 2537 df-eu 2566 df-clab 2712 df-cleq 2726 df-clel 2813 df-nfc 2889 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-rmo 3377 df-reu 3378 df-rab 3433 df-v 3479 df-sbc 3791 df-csb 3908 df-dif 3965 df-un 3967 df-in 3969 df-ss 3979 df-pss 3982 df-nul 4339 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4912 df-int 4951 df-iun 4997 df-br 5148 df-opab 5210 df-mpt 5231 df-tr 5265 df-id 5582 df-eprel 5588 df-po 5596 df-so 5597 df-fr 5640 df-se 5641 df-we 5642 df-xp 5694 df-rel 5695 df-cnv 5696 df-co 5697 df-dm 5698 df-rn 5699 df-res 5700 df-ima 5701 df-pred 6322 df-ord 6388 df-on 6389 df-lim 6390 df-suc 6391 df-iota 6515 df-fun 6564 df-fn 6565 df-f 6566 df-f1 6567 df-fo 6568 df-f1o 6569 df-fv 6570 df-isom 6571 df-riota 7387 df-ov 7433 df-oprab 7434 df-mpo 7435 df-of 7696 df-om 7887 df-1st 8012 df-2nd 8013 df-frecs 8304 df-wrecs 8335 df-recs 8409 df-rdg 8448 df-1o 8504 df-2o 8505 df-oadd 8508 df-omul 8509 df-er 8743 df-map 8866 df-pm 8867 df-en 8984 df-dom 8985 df-sdom 8986 df-fin 8987 df-sup 9479 df-inf 9480 df-oi 9547 df-dju 9938 df-card 9976 df-acn 9979 df-pnf 11294 df-mnf 11295 df-xr 11296 df-ltxr 11297 df-le 11298 df-sub 11491 df-neg 11492 df-div 11918 df-nn 12264 df-2 12326 df-3 12327 df-n0 12524 df-xnn0 12597 df-z 12611 df-uz 12876 df-q 12988 df-rp 13032 df-ico 13389 df-icc 13390 df-fz 13544 df-fzo 13691 df-fl 13828 df-mod 13906 df-seq 14039 df-exp 14099 df-hash 14366 df-cj 15134 df-re 15135 df-im 15136 df-sqrt 15270 df-abs 15271 df-limsup 15503 df-clim 15520 df-rlim 15521 df-sum 15719 df-0p 25718 df-ply 26241 df-idp 26242 df-coe 26243 df-dgr 26244 df-quot 26347 df-aa 26371 |
This theorem is referenced by: aannen 26387 |
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