aannenlem3 - Metamath Proof Explorer

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Theorem aannenlem3 24255

Description: The algebraic numbers are countable. (Contributed by Stefan O'Rear, 16-Nov-2014.)

**Hypothesis**
Ref	Expression
aannenlem.a	⊢ 𝐻 = (𝑎 ∈ ℕ₀ ↦ {𝑏 ∈ ℂ ∣ ∃𝑐 ∈ {𝑑 ∈ (Poly‘ℤ) ∣ (𝑑 ≠ 0_𝑝 ∧ (deg‘𝑑) ≤ 𝑎 ∧ ∀𝑒 ∈ ℕ₀ (abs‘((coeff‘𝑑)‘𝑒)) ≤ 𝑎)} (𝑐‘𝑏) = 0})

**Assertion**
Ref	Expression
aannenlem3	⊢ 𝔸 ≈ ℕ

Distinct variable group: 𝑎,𝑏,𝑐,𝑑,𝑒 Allowed substitution hints: 𝐻(𝑒,𝑎,𝑏,𝑐,𝑑)

Proof of Theorem aannenlem3 Dummy variables 𝑓 𝑔 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		aannenlem.a	. . . . . 6 ⊢ 𝐻 = (𝑎 ∈ ℕ₀ ↦ {𝑏 ∈ ℂ ∣ ∃𝑐 ∈ {𝑑 ∈ (Poly‘ℤ) ∣ (𝑑 ≠ 0_𝑝 ∧ (deg‘𝑑) ≤ 𝑎 ∧ ∀𝑒 ∈ ℕ₀ (abs‘((coeff‘𝑑)‘𝑒)) ≤ 𝑎)} (𝑐‘𝑏) = 0})
2	1	aannenlem2 24254	. . . . 5 ⊢ 𝔸 = ∪ ran 𝐻
3		omelon 8704	. . . . . . . . . 10 ⊢ ω ∈ On
4		nn0ennn 12943	. . . . . . . . . . . 12 ⊢ ℕ₀ ≈ ℕ
5		nnenom 12944	. . . . . . . . . . . 12 ⊢ ℕ ≈ ω
6	4, 5	entri 8163	. . . . . . . . . . 11 ⊢ ℕ₀ ≈ ω
7	6	ensymi 8159	. . . . . . . . . 10 ⊢ ω ≈ ℕ₀
8		isnumi 8933	. . . . . . . . . 10 ⊢ ((ω ∈ On ∧ ω ≈ ℕ₀) → ℕ₀ ∈ dom card)
9	3, 7, 8	mp2an 710	. . . . . . . . 9 ⊢ ℕ₀ ∈ dom card
10		cnex 10180	. . . . . . . . . . . 12 ⊢ ℂ ∈ V
11	10	rabex 4952	. . . . . . . . . . 11 ⊢ {𝑏 ∈ ℂ ∣ ∃𝑐 ∈ {𝑑 ∈ (Poly‘ℤ) ∣ (𝑑 ≠ 0_𝑝 ∧ (deg‘𝑑) ≤ 𝑎 ∧ ∀𝑒 ∈ ℕ₀ (abs‘((coeff‘𝑑)‘𝑒)) ≤ 𝑎)} (𝑐‘𝑏) = 0} ∈ V
12	11, 1	fnmpti 6171	. . . . . . . . . 10 ⊢ 𝐻 Fn ℕ₀
13		dffn4 6270	. . . . . . . . . 10 ⊢ (𝐻 Fn ℕ₀ ↔ 𝐻:ℕ₀–onto→ran 𝐻)
14	12, 13	mpbi 220	. . . . . . . . 9 ⊢ 𝐻:ℕ₀–onto→ran 𝐻
15		fodomnum 9041	. . . . . . . . 9 ⊢ (ℕ₀ ∈ dom card → (𝐻:ℕ₀–onto→ran 𝐻 → ran 𝐻 ≼ ℕ₀))
16	9, 14, 15	mp2 9	. . . . . . . 8 ⊢ ran 𝐻 ≼ ℕ₀
17		domentr 8168	. . . . . . . 8 ⊢ ((ran 𝐻 ≼ ℕ₀ ∧ ℕ₀ ≈ ω) → ran 𝐻 ≼ ω)
18	16, 6, 17	mp2an 710	. . . . . . 7 ⊢ ran 𝐻 ≼ ω
19	18	a1i 11	. . . . . 6 ⊢ (𝑓 Or ℂ → ran 𝐻 ≼ ω)
20		fvelrnb 6393	. . . . . . . . . 10 ⊢ (𝐻 Fn ℕ₀ → (𝑓 ∈ ran 𝐻 ↔ ∃𝑔 ∈ ℕ₀ (𝐻‘𝑔) = 𝑓))
21	12, 20	ax-mp 5	. . . . . . . . 9 ⊢ (𝑓 ∈ ran 𝐻 ↔ ∃𝑔 ∈ ℕ₀ (𝐻‘𝑔) = 𝑓)
22	1	aannenlem1 24253	. . . . . . . . . . 11 ⊢ (𝑔 ∈ ℕ₀ → (𝐻‘𝑔) ∈ Fin)
23		eleq1 2815	. . . . . . . . . . 11 ⊢ ((𝐻‘𝑔) = 𝑓 → ((𝐻‘𝑔) ∈ Fin ↔ 𝑓 ∈ Fin))
24	22, 23	syl5ibcom 235	. . . . . . . . . 10 ⊢ (𝑔 ∈ ℕ₀ → ((𝐻‘𝑔) = 𝑓 → 𝑓 ∈ Fin))
25	24	rexlimiv 3153	. . . . . . . . 9 ⊢ (∃𝑔 ∈ ℕ₀ (𝐻‘𝑔) = 𝑓 → 𝑓 ∈ Fin)
26	21, 25	sylbi 207	. . . . . . . 8 ⊢ (𝑓 ∈ ran 𝐻 → 𝑓 ∈ Fin)
27	26	ssriv 3736	. . . . . . 7 ⊢ ran 𝐻 ⊆ Fin
28	27	a1i 11	. . . . . 6 ⊢ (𝑓 Or ℂ → ran 𝐻 ⊆ Fin)
29		aasscn 24243	. . . . . . . 8 ⊢ 𝔸 ⊆ ℂ
30	2, 29	eqsstr3i 3765	. . . . . . 7 ⊢ ∪ ran 𝐻 ⊆ ℂ
31		soss 5193	. . . . . . 7 ⊢ (∪ ran 𝐻 ⊆ ℂ → (𝑓 Or ℂ → 𝑓 Or ∪ ran 𝐻))
32	30, 31	ax-mp 5	. . . . . 6 ⊢ (𝑓 Or ℂ → 𝑓 Or ∪ ran 𝐻)
33		iunfictbso 9098	. . . . . 6 ⊢ ((ran 𝐻 ≼ ω ∧ ran 𝐻 ⊆ Fin ∧ 𝑓 Or ∪ ran 𝐻) → ∪ ran 𝐻 ≼ ω)
34	19, 28, 32, 33	syl3anc 1463	. . . . 5 ⊢ (𝑓 Or ℂ → ∪ ran 𝐻 ≼ ω)
35	2, 34	syl5eqbr 4827	. . . 4 ⊢ (𝑓 Or ℂ → 𝔸 ≼ ω)
36		cnso 15146	. . . 4 ⊢ ∃𝑓 𝑓 Or ℂ
37	35, 36	exlimiiv 1996	. . 3 ⊢ 𝔸 ≼ ω
38	5	ensymi 8159	. . 3 ⊢ ω ≈ ℕ
39		domentr 8168	. . 3 ⊢ ((𝔸 ≼ ω ∧ ω ≈ ℕ) → 𝔸 ≼ ℕ)
40	37, 38, 39	mp2an 710	. 2 ⊢ 𝔸 ≼ ℕ
41	10, 29	ssexi 4943	. . 3 ⊢ 𝔸 ∈ V
42		nnssq 11961	. . . 4 ⊢ ℕ ⊆ ℚ
43		qssaa 24249	. . . 4 ⊢ ℚ ⊆ 𝔸
44	42, 43	sstri 3741	. . 3 ⊢ ℕ ⊆ 𝔸
45		ssdomg 8155	. . 3 ⊢ (𝔸 ∈ V → (ℕ ⊆ 𝔸 → ℕ ≼ 𝔸))
46	41, 44, 45	mp2 9	. 2 ⊢ ℕ ≼ 𝔸
47		sbth 8233	. 2 ⊢ ((𝔸 ≼ ℕ ∧ ℕ ≼ 𝔸) → 𝔸 ≈ ℕ)
48	40, 46, 47	mp2an 710	1 ⊢ 𝔸 ≈ ℕ

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 196 ∧ w3a 1072 = wceq 1620 ∈ wcel 2127 ≠ wne 2920 ∀wral 3038 ∃wrex 3039 {crab 3042 Vcvv 3328 ⊆ wss 3703 ∪ cuni 4576 class class class wbr 4792 ↦ cmpt 4869 Or wor 5174 dom cdm 5254 ran crn 5255 Oncon0 5872 Fn wfn 6032 –onto→wfo 6035 ‘cfv 6037 ωcom 7218 ≈ cen 8106 ≼ cdom 8107 Fincfn 8109 cardccrd 8922 ℂcc 10097 0cc0 10099 ≤ cle 10238 ℕcn 11183 ℕ₀cn0 11455 ℤcz 11540 ℚcq 11952 abscabs 14144 0_𝑝c0p 23606 Polycply 24110 coeffccoe 24112 degcdgr 24113 𝔸caa 24239

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1859 ax-4 1874 ax-5 1976 ax-6 2042 ax-7 2078 ax-8 2129 ax-9 2136 ax-10 2156 ax-11 2171 ax-12 2184 ax-13 2379 ax-ext 2728 ax-rep 4911 ax-sep 4921 ax-nul 4929 ax-pow 4980 ax-pr 5043 ax-un 7102 ax-inf2 8699 ax-cnex 10155 ax-resscn 10156 ax-1cn 10157 ax-icn 10158 ax-addcl 10159 ax-addrcl 10160 ax-mulcl 10161 ax-mulrcl 10162 ax-mulcom 10163 ax-addass 10164 ax-mulass 10165 ax-distr 10166 ax-i2m1 10167 ax-1ne0 10168 ax-1rid 10169 ax-rnegex 10170 ax-rrecex 10171 ax-cnre 10172 ax-pre-lttri 10173 ax-pre-lttrn 10174 ax-pre-ltadd 10175 ax-pre-mulgt0 10176 ax-pre-sup 10177 ax-addf 10178

This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-3or 1073 df-3an 1074 df-tru 1623 df-fal 1626 df-ex 1842 df-nf 1847 df-sb 2035 df-eu 2599 df-mo 2600 df-clab 2735 df-cleq 2741 df-clel 2744 df-nfc 2879 df-ne 2921 df-nel 3024 df-ral 3043 df-rex 3044 df-reu 3045 df-rmo 3046 df-rab 3047 df-v 3330 df-sbc 3565 df-csb 3663 df-dif 3706 df-un 3708 df-in 3710 df-ss 3717 df-pss 3719 df-nul 4047 df-if 4219 df-pw 4292 df-sn 4310 df-pr 4312 df-tp 4314 df-op 4316 df-uni 4577 df-int 4616 df-iun 4662 df-br 4793 df-opab 4853 df-mpt 4870 df-tr 4893 df-id 5162 df-eprel 5167 df-po 5175 df-so 5176 df-fr 5213 df-se 5214 df-we 5215 df-xp 5260 df-rel 5261 df-cnv 5262 df-co 5263 df-dm 5264 df-rn 5265 df-res 5266 df-ima 5267 df-pred 5829 df-ord 5875 df-on 5876 df-lim 5877 df-suc 5878 df-iota 6000 df-fun 6039 df-fn 6040 df-f 6041 df-f1 6042 df-fo 6043 df-f1o 6044 df-fv 6045 df-isom 6046 df-riota 6762 df-ov 6804 df-oprab 6805 df-mpt2 6806 df-of 7050 df-om 7219 df-1st 7321 df-2nd 7322 df-wrecs 7564 df-recs 7625 df-rdg 7663 df-1o 7717 df-2o 7718 df-oadd 7721 df-omul 7722 df-er 7899 df-map 8013 df-pm 8014 df-en 8110 df-dom 8111 df-sdom 8112 df-fin 8113 df-sup 8501 df-inf 8502 df-oi 8568 df-card 8926 df-acn 8929 df-cda 9153 df-pnf 10239 df-mnf 10240 df-xr 10241 df-ltxr 10242 df-le 10243 df-sub 10431 df-neg 10432 df-div 10848 df-nn 11184 df-2 11242 df-3 11243 df-n0 11456 df-xnn0 11527 df-z 11541 df-uz 11851 df-q 11953 df-rp 11997 df-ico 12345 df-icc 12346 df-fz 12491 df-fzo 12631 df-fl 12758 df-mod 12834 df-seq 12967 df-exp 13026 df-hash 13283 df-cj 14009 df-re 14010 df-im 14011 df-sqrt 14145 df-abs 14146 df-limsup 14372 df-clim 14389 df-rlim 14390 df-sum 14587 df-0p 23607 df-ply 24114 df-idp 24115 df-coe 24116 df-dgr 24117 df-quot 24216 df-aa 24240

This theorem is referenced by: aannen 24256

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