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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 24920 | . . . . 5 ⊢ 𝔸 = ∪ ran 𝐻 |
3 | omelon 9111 | . . . . . . . . 9 ⊢ ω ∈ On | |
4 | nn0ennn 13350 | . . . . . . . . . . 11 ⊢ ℕ0 ≈ ℕ | |
5 | nnenom 13351 | . . . . . . . . . . 11 ⊢ ℕ ≈ ω | |
6 | 4, 5 | entri 8565 | . . . . . . . . . 10 ⊢ ℕ0 ≈ ω |
7 | 6 | ensymi 8561 | . . . . . . . . 9 ⊢ ω ≈ ℕ0 |
8 | isnumi 9377 | . . . . . . . . 9 ⊢ ((ω ∈ On ∧ ω ≈ ℕ0) → ℕ0 ∈ dom card) | |
9 | 3, 7, 8 | mp2an 690 | . . . . . . . 8 ⊢ ℕ0 ∈ dom card |
10 | cnex 10620 | . . . . . . . . . . 11 ⊢ ℂ ∈ V | |
11 | 10 | rabex 5237 | . . . . . . . . . 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 6493 | . . . . . . . . 9 ⊢ 𝐻 Fn ℕ0 |
13 | dffn4 6598 | . . . . . . . . 9 ⊢ (𝐻 Fn ℕ0 ↔ 𝐻:ℕ0–onto→ran 𝐻) | |
14 | 12, 13 | mpbi 232 | . . . . . . . 8 ⊢ 𝐻:ℕ0–onto→ran 𝐻 |
15 | fodomnum 9485 | . . . . . . . 8 ⊢ (ℕ0 ∈ dom card → (𝐻:ℕ0–onto→ran 𝐻 → ran 𝐻 ≼ ℕ0)) | |
16 | 9, 14, 15 | mp2 9 | . . . . . . 7 ⊢ ran 𝐻 ≼ ℕ0 |
17 | domentr 8570 | . . . . . . 7 ⊢ ((ran 𝐻 ≼ ℕ0 ∧ ℕ0 ≈ ω) → ran 𝐻 ≼ ω) | |
18 | 16, 6, 17 | mp2an 690 | . . . . . 6 ⊢ ran 𝐻 ≼ ω |
19 | fvelrnb 6728 | . . . . . . . . 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 24919 | . . . . . . . . . 10 ⊢ (𝑔 ∈ ℕ0 → (𝐻‘𝑔) ∈ Fin) |
22 | eleq1 2902 | . . . . . . . . . 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 3282 | . . . . . . . 8 ⊢ (∃𝑔 ∈ ℕ0 (𝐻‘𝑔) = 𝑓 → 𝑓 ∈ Fin) |
25 | 20, 24 | sylbi 219 | . . . . . . 7 ⊢ (𝑓 ∈ ran 𝐻 → 𝑓 ∈ Fin) |
26 | 25 | ssriv 3973 | . . . . . 6 ⊢ ran 𝐻 ⊆ Fin |
27 | aasscn 24909 | . . . . . . . 8 ⊢ 𝔸 ⊆ ℂ | |
28 | 2, 27 | eqsstrri 4004 | . . . . . . 7 ⊢ ∪ ran 𝐻 ⊆ ℂ |
29 | soss 5495 | . . . . . . 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 9542 | . . . . . 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 5103 | . . . 4 ⊢ (𝑓 Or ℂ → 𝔸 ≼ ω) |
34 | cnso 15602 | . . . 4 ⊢ ∃𝑓 𝑓 Or ℂ | |
35 | 33, 34 | exlimiiv 1932 | . . 3 ⊢ 𝔸 ≼ ω |
36 | 5 | ensymi 8561 | . . 3 ⊢ ω ≈ ℕ |
37 | domentr 8570 | . . 3 ⊢ ((𝔸 ≼ ω ∧ ω ≈ ℕ) → 𝔸 ≼ ℕ) | |
38 | 35, 36, 37 | mp2an 690 | . 2 ⊢ 𝔸 ≼ ℕ |
39 | 10, 27 | ssexi 5228 | . . 3 ⊢ 𝔸 ∈ V |
40 | nnssq 12360 | . . . 4 ⊢ ℕ ⊆ ℚ | |
41 | qssaa 24915 | . . . 4 ⊢ ℚ ⊆ 𝔸 | |
42 | 40, 41 | sstri 3978 | . . 3 ⊢ ℕ ⊆ 𝔸 |
43 | ssdomg 8557 | . . 3 ⊢ (𝔸 ∈ V → (ℕ ⊆ 𝔸 → ℕ ≼ 𝔸)) | |
44 | 39, 42, 43 | mp2 9 | . 2 ⊢ ℕ ≼ 𝔸 |
45 | sbth 8639 | . 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 1537 ∈ wcel 2114 ≠ wne 3018 ∀wral 3140 ∃wrex 3141 {crab 3144 Vcvv 3496 ⊆ wss 3938 ∪ cuni 4840 class class class wbr 5068 ↦ cmpt 5148 Or wor 5475 dom cdm 5557 ran crn 5558 Oncon0 6193 Fn wfn 6352 –onto→wfo 6355 ‘cfv 6357 ωcom 7582 ≈ cen 8508 ≼ cdom 8509 Fincfn 8511 cardccrd 9366 ℂcc 10537 0cc0 10539 ≤ cle 10678 ℕcn 11640 ℕ0cn0 11900 ℤcz 11984 ℚcq 12351 abscabs 14595 0𝑝c0p 24272 Polycply 24776 coeffccoe 24778 degcdgr 24779 𝔸caa 24905 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-rep 5192 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 ax-inf2 9106 ax-cnex 10595 ax-resscn 10596 ax-1cn 10597 ax-icn 10598 ax-addcl 10599 ax-addrcl 10600 ax-mulcl 10601 ax-mulrcl 10602 ax-mulcom 10603 ax-addass 10604 ax-mulass 10605 ax-distr 10606 ax-i2m1 10607 ax-1ne0 10608 ax-1rid 10609 ax-rnegex 10610 ax-rrecex 10611 ax-cnre 10612 ax-pre-lttri 10613 ax-pre-lttrn 10614 ax-pre-ltadd 10615 ax-pre-mulgt0 10616 ax-pre-sup 10617 ax-addf 10618 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-fal 1550 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-nel 3126 df-ral 3145 df-rex 3146 df-reu 3147 df-rmo 3148 df-rab 3149 df-v 3498 df-sbc 3775 df-csb 3886 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-pss 3956 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-tp 4574 df-op 4576 df-uni 4841 df-int 4879 df-iun 4923 df-br 5069 df-opab 5131 df-mpt 5149 df-tr 5175 df-id 5462 df-eprel 5467 df-po 5476 df-so 5477 df-fr 5516 df-se 5517 df-we 5518 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-pred 6150 df-ord 6196 df-on 6197 df-lim 6198 df-suc 6199 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-f1 6362 df-fo 6363 df-f1o 6364 df-fv 6365 df-isom 6366 df-riota 7116 df-ov 7161 df-oprab 7162 df-mpo 7163 df-of 7411 df-om 7583 df-1st 7691 df-2nd 7692 df-wrecs 7949 df-recs 8010 df-rdg 8048 df-1o 8104 df-2o 8105 df-oadd 8108 df-omul 8109 df-er 8291 df-map 8410 df-pm 8411 df-en 8512 df-dom 8513 df-sdom 8514 df-fin 8515 df-sup 8908 df-inf 8909 df-oi 8976 df-dju 9332 df-card 9370 df-acn 9373 df-pnf 10679 df-mnf 10680 df-xr 10681 df-ltxr 10682 df-le 10683 df-sub 10874 df-neg 10875 df-div 11300 df-nn 11641 df-2 11703 df-3 11704 df-n0 11901 df-xnn0 11971 df-z 11985 df-uz 12247 df-q 12352 df-rp 12393 df-ico 12747 df-icc 12748 df-fz 12896 df-fzo 13037 df-fl 13165 df-mod 13241 df-seq 13373 df-exp 13433 df-hash 13694 df-cj 14460 df-re 14461 df-im 14462 df-sqrt 14596 df-abs 14597 df-limsup 14830 df-clim 14847 df-rlim 14848 df-sum 15045 df-0p 24273 df-ply 24780 df-idp 24781 df-coe 24782 df-dgr 24783 df-quot 24882 df-aa 24906 |
This theorem is referenced by: aannen 24922 |
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