aannenlem3 - Metamath Proof Explorer

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Theorem aannenlem3 23889

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 23888	. . . . 5 ⊢ 𝔸 = ∪ ran 𝐻
3		omelon 8426	. . . . . . . . . 10 ⊢ ω ∈ On
4		nn0ennn 12640	. . . . . . . . . . . 12 ⊢ ℕ₀ ≈ ℕ
5		nnenom 12641	. . . . . . . . . . . 12 ⊢ ℕ ≈ ω
6	4, 5	entri 7896	. . . . . . . . . . 11 ⊢ ℕ₀ ≈ ω
7	6	ensymi 7892	. . . . . . . . . 10 ⊢ ω ≈ ℕ₀
8		isnumi 8655	. . . . . . . . . 10 ⊢ ((ω ∈ On ∧ ω ≈ ℕ₀) → ℕ₀ ∈ dom card)
9	3, 7, 8	mp2an 704	. . . . . . . . 9 ⊢ ℕ₀ ∈ dom card
10		cnex 9896	. . . . . . . . . . . 12 ⊢ ℂ ∈ V
11	10	rabex 4740	. . . . . . . . . . 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 5935	. . . . . . . . . 10 ⊢ 𝐻 Fn ℕ₀
13		dffn4 6034	. . . . . . . . . 10 ⊢ (𝐻 Fn ℕ₀ ↔ 𝐻:ℕ₀–onto→ran 𝐻)
14	12, 13	mpbi 219	. . . . . . . . 9 ⊢ 𝐻:ℕ₀–onto→ran 𝐻
15		fodomnum 8763	. . . . . . . . 9 ⊢ (ℕ₀ ∈ dom card → (𝐻:ℕ₀–onto→ran 𝐻 → ran 𝐻 ≼ ℕ₀))
16	9, 14, 15	mp2 9	. . . . . . . 8 ⊢ ran 𝐻 ≼ ℕ₀
17		domentr 7901	. . . . . . . 8 ⊢ ((ran 𝐻 ≼ ℕ₀ ∧ ℕ₀ ≈ ω) → ran 𝐻 ≼ ω)
18	16, 6, 17	mp2an 704	. . . . . . 7 ⊢ ran 𝐻 ≼ ω
19	18	a1i 11	. . . . . 6 ⊢ (𝑓 Or ℂ → ran 𝐻 ≼ ω)
20		fvelrnb 6153	. . . . . . . . . 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 23887	. . . . . . . . . . 11 ⊢ (𝑔 ∈ ℕ₀ → (𝐻‘𝑔) ∈ Fin)
23		eleq1 2676	. . . . . . . . . . 11 ⊢ ((𝐻‘𝑔) = 𝑓 → ((𝐻‘𝑔) ∈ Fin ↔ 𝑓 ∈ Fin))
24	22, 23	syl5ibcom 234	. . . . . . . . . 10 ⊢ (𝑔 ∈ ℕ₀ → ((𝐻‘𝑔) = 𝑓 → 𝑓 ∈ Fin))
25	24	rexlimiv 3009	. . . . . . . . 9 ⊢ (∃𝑔 ∈ ℕ₀ (𝐻‘𝑔) = 𝑓 → 𝑓 ∈ Fin)
26	21, 25	sylbi 206	. . . . . . . 8 ⊢ (𝑓 ∈ ran 𝐻 → 𝑓 ∈ Fin)
27	26	ssriv 3572	. . . . . . 7 ⊢ ran 𝐻 ⊆ Fin
28	27	a1i 11	. . . . . 6 ⊢ (𝑓 Or ℂ → ran 𝐻 ⊆ Fin)
29		aasscn 23877	. . . . . . . 8 ⊢ 𝔸 ⊆ ℂ
30	2, 29	eqsstr3i 3599	. . . . . . 7 ⊢ ∪ ran 𝐻 ⊆ ℂ
31		soss 4977	. . . . . . 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 8820	. . . . . 6 ⊢ ((ran 𝐻 ≼ ω ∧ ran 𝐻 ⊆ Fin ∧ 𝑓 Or ∪ ran 𝐻) → ∪ ran 𝐻 ≼ ω)
34	19, 28, 32, 33	syl3anc 1318	. . . . 5 ⊢ (𝑓 Or ℂ → ∪ ran 𝐻 ≼ ω)
35	2, 34	syl5eqbr 4618	. . . 4 ⊢ (𝑓 Or ℂ → 𝔸 ≼ ω)
36		cnso 14815	. . . 4 ⊢ ∃𝑓 𝑓 Or ℂ
37	35, 36	exlimiiv 1846	. . 3 ⊢ 𝔸 ≼ ω
38	5	ensymi 7892	. . 3 ⊢ ω ≈ ℕ
39		domentr 7901	. . 3 ⊢ ((𝔸 ≼ ω ∧ ω ≈ ℕ) → 𝔸 ≼ ℕ)
40	37, 38, 39	mp2an 704	. 2 ⊢ 𝔸 ≼ ℕ
41	10, 29	ssexi 4731	. . 3 ⊢ 𝔸 ∈ V
42		nnssq 11673	. . . 4 ⊢ ℕ ⊆ ℚ
43		qssaa 23883	. . . 4 ⊢ ℚ ⊆ 𝔸
44	42, 43	sstri 3577	. . 3 ⊢ ℕ ⊆ 𝔸
45		ssdomg 7887	. . 3 ⊢ (𝔸 ∈ V → (ℕ ⊆ 𝔸 → ℕ ≼ 𝔸))
46	41, 44, 45	mp2 9	. 2 ⊢ ℕ ≼ 𝔸
47		sbth 7965	. 2 ⊢ ((𝔸 ≼ ℕ ∧ ℕ ≼ 𝔸) → 𝔸 ≈ ℕ)
48	40, 46, 47	mp2an 704	1 ⊢ 𝔸 ≈ ℕ

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 195 ∧ w3a 1031 = wceq 1475 ∈ wcel 1977 ≠ wne 2780 ∀wral 2896 ∃wrex 2897 {crab 2900 Vcvv 3173 ⊆ wss 3540 ∪ cuni 4372 class class class wbr 4583 ↦ cmpt 4643 Or wor 4958 dom cdm 5038 ran crn 5039 Oncon0 5640 Fn wfn 5799 –onto→wfo 5802 ‘cfv 5804 ωcom 6957 ≈ cen 7838 ≼ cdom 7839 Fincfn 7841 cardccrd 8644 ℂcc 9813 0cc0 9815 ≤ cle 9954 ℕcn 10897 ℕ₀cn0 11169 ℤcz 11254 ℚcq 11664 abscabs 13822 0_𝑝c0p 23242 Polycply 23744 coeffccoe 23746 degcdgr 23747 𝔸caa 23873

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-inf2 8421 ax-cnex 9871 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-mulcom 9879 ax-addass 9880 ax-mulass 9881 ax-distr 9882 ax-i2m1 9883 ax-1ne0 9884 ax-1rid 9885 ax-rnegex 9886 ax-rrecex 9887 ax-cnre 9888 ax-pre-lttri 9889 ax-pre-lttrn 9890 ax-pre-ltadd 9891 ax-pre-mulgt0 9892 ax-pre-sup 9893 ax-addf 9894

This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-fal 1481 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-reu 2903 df-rmo 2904 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-int 4411 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-se 4998 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-isom 5813 df-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-of 6795 df-om 6958 df-1st 7059 df-2nd 7060 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-1o 7447 df-2o 7448 df-oadd 7451 df-omul 7452 df-er 7629 df-map 7746 df-pm 7747 df-en 7842 df-dom 7843 df-sdom 7844 df-fin 7845 df-sup 8231 df-inf 8232 df-oi 8298 df-card 8648 df-acn 8651 df-cda 8873 df-pnf 9955 df-mnf 9956 df-xr 9957 df-ltxr 9958 df-le 9959 df-sub 10147 df-neg 10148 df-div 10564 df-nn 10898 df-2 10956 df-3 10957 df-n0 11170 df-xnn0 11241 df-z 11255 df-uz 11564 df-q 11665 df-rp 11709 df-ico 12052 df-icc 12053 df-fz 12198 df-fzo 12335 df-fl 12455 df-mod 12531 df-seq 12664 df-exp 12723 df-hash 12980 df-cj 13687 df-re 13688 df-im 13689 df-sqrt 13823 df-abs 13824 df-limsup 14050 df-clim 14067 df-rlim 14068 df-sum 14265 df-0p 23243 df-ply 23748 df-idp 23749 df-coe 23750 df-dgr 23751 df-quot 23850 df-aa 23874

This theorem is referenced by: aannen 23890

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