aannenlem3 - Metamath Proof Explorer

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Theorem aannenlem3 24085

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 24084	. . . . 5 ⊢ 𝔸 = ∪ ran 𝐻
3		omelon 8543	. . . . . . . . . 10 ⊢ ω ∈ On
4		nn0ennn 12778	. . . . . . . . . . . 12 ⊢ ℕ₀ ≈ ℕ
5		nnenom 12779	. . . . . . . . . . . 12 ⊢ ℕ ≈ ω
6	4, 5	entri 8010	. . . . . . . . . . 11 ⊢ ℕ₀ ≈ ω
7	6	ensymi 8006	. . . . . . . . . 10 ⊢ ω ≈ ℕ₀
8		isnumi 8772	. . . . . . . . . 10 ⊢ ((ω ∈ On ∧ ω ≈ ℕ₀) → ℕ₀ ∈ dom card)
9	3, 7, 8	mp2an 708	. . . . . . . . 9 ⊢ ℕ₀ ∈ dom card
10		cnex 10017	. . . . . . . . . . . 12 ⊢ ℂ ∈ V
11	10	rabex 4813	. . . . . . . . . . 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 6022	. . . . . . . . . 10 ⊢ 𝐻 Fn ℕ₀
13		dffn4 6121	. . . . . . . . . 10 ⊢ (𝐻 Fn ℕ₀ ↔ 𝐻:ℕ₀–onto→ran 𝐻)
14	12, 13	mpbi 220	. . . . . . . . 9 ⊢ 𝐻:ℕ₀–onto→ran 𝐻
15		fodomnum 8880	. . . . . . . . 9 ⊢ (ℕ₀ ∈ dom card → (𝐻:ℕ₀–onto→ran 𝐻 → ran 𝐻 ≼ ℕ₀))
16	9, 14, 15	mp2 9	. . . . . . . 8 ⊢ ran 𝐻 ≼ ℕ₀
17		domentr 8015	. . . . . . . 8 ⊢ ((ran 𝐻 ≼ ℕ₀ ∧ ℕ₀ ≈ ω) → ran 𝐻 ≼ ω)
18	16, 6, 17	mp2an 708	. . . . . . 7 ⊢ ran 𝐻 ≼ ω
19	18	a1i 11	. . . . . 6 ⊢ (𝑓 Or ℂ → ran 𝐻 ≼ ω)
20		fvelrnb 6243	. . . . . . . . . 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 24083	. . . . . . . . . . 11 ⊢ (𝑔 ∈ ℕ₀ → (𝐻‘𝑔) ∈ Fin)
23		eleq1 2689	. . . . . . . . . . 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 3027	. . . . . . . . 9 ⊢ (∃𝑔 ∈ ℕ₀ (𝐻‘𝑔) = 𝑓 → 𝑓 ∈ Fin)
26	21, 25	sylbi 207	. . . . . . . 8 ⊢ (𝑓 ∈ ran 𝐻 → 𝑓 ∈ Fin)
27	26	ssriv 3607	. . . . . . 7 ⊢ ran 𝐻 ⊆ Fin
28	27	a1i 11	. . . . . 6 ⊢ (𝑓 Or ℂ → ran 𝐻 ⊆ Fin)
29		aasscn 24073	. . . . . . . 8 ⊢ 𝔸 ⊆ ℂ
30	2, 29	eqsstr3i 3636	. . . . . . 7 ⊢ ∪ ran 𝐻 ⊆ ℂ
31		soss 5053	. . . . . . 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 8937	. . . . . 6 ⊢ ((ran 𝐻 ≼ ω ∧ ran 𝐻 ⊆ Fin ∧ 𝑓 Or ∪ ran 𝐻) → ∪ ran 𝐻 ≼ ω)
34	19, 28, 32, 33	syl3anc 1326	. . . . 5 ⊢ (𝑓 Or ℂ → ∪ ran 𝐻 ≼ ω)
35	2, 34	syl5eqbr 4688	. . . 4 ⊢ (𝑓 Or ℂ → 𝔸 ≼ ω)
36		cnso 14976	. . . 4 ⊢ ∃𝑓 𝑓 Or ℂ
37	35, 36	exlimiiv 1859	. . 3 ⊢ 𝔸 ≼ ω
38	5	ensymi 8006	. . 3 ⊢ ω ≈ ℕ
39		domentr 8015	. . 3 ⊢ ((𝔸 ≼ ω ∧ ω ≈ ℕ) → 𝔸 ≼ ℕ)
40	37, 38, 39	mp2an 708	. 2 ⊢ 𝔸 ≼ ℕ
41	10, 29	ssexi 4803	. . 3 ⊢ 𝔸 ∈ V
42		nnssq 11797	. . . 4 ⊢ ℕ ⊆ ℚ
43		qssaa 24079	. . . 4 ⊢ ℚ ⊆ 𝔸
44	42, 43	sstri 3612	. . 3 ⊢ ℕ ⊆ 𝔸
45		ssdomg 8001	. . 3 ⊢ (𝔸 ∈ V → (ℕ ⊆ 𝔸 → ℕ ≼ 𝔸))
46	41, 44, 45	mp2 9	. 2 ⊢ ℕ ≼ 𝔸
47		sbth 8080	. 2 ⊢ ((𝔸 ≼ ℕ ∧ ℕ ≼ 𝔸) → 𝔸 ≈ ℕ)
48	40, 46, 47	mp2an 708	1 ⊢ 𝔸 ≈ ℕ

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 196 ∧ w3a 1037 = wceq 1483 ∈ wcel 1990 ≠ wne 2794 ∀wral 2912 ∃wrex 2913 {crab 2916 Vcvv 3200 ⊆ wss 3574 ∪ cuni 4436 class class class wbr 4653 ↦ cmpt 4729 Or wor 5034 dom cdm 5114 ran crn 5115 Oncon0 5723 Fn wfn 5883 –onto→wfo 5886 ‘cfv 5888 ωcom 7065 ≈ cen 7952 ≼ cdom 7953 Fincfn 7955 cardccrd 8761 ℂcc 9934 0cc0 9936 ≤ cle 10075 ℕcn 11020 ℕ₀cn0 11292 ℤcz 11377 ℚcq 11788 abscabs 13974 0_𝑝c0p 23436 Polycply 23940 coeffccoe 23942 degcdgr 23943 𝔸caa 24069

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-rep 4771 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-inf2 8538 ax-cnex 9992 ax-resscn 9993 ax-1cn 9994 ax-icn 9995 ax-addcl 9996 ax-addrcl 9997 ax-mulcl 9998 ax-mulrcl 9999 ax-mulcom 10000 ax-addass 10001 ax-mulass 10002 ax-distr 10003 ax-i2m1 10004 ax-1ne0 10005 ax-1rid 10006 ax-rnegex 10007 ax-rrecex 10008 ax-cnre 10009 ax-pre-lttri 10010 ax-pre-lttrn 10011 ax-pre-ltadd 10012 ax-pre-mulgt0 10013 ax-pre-sup 10014 ax-addf 10015

This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-fal 1489 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-nel 2898 df-ral 2917 df-rex 2918 df-reu 2919 df-rmo 2920 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-int 4476 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-se 5074 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-pred 5680 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-isom 5897 df-riota 6611 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-of 6897 df-om 7066 df-1st 7168 df-2nd 7169 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-1o 7560 df-2o 7561 df-oadd 7564 df-omul 7565 df-er 7742 df-map 7859 df-pm 7860 df-en 7956 df-dom 7957 df-sdom 7958 df-fin 7959 df-sup 8348 df-inf 8349 df-oi 8415 df-card 8765 df-acn 8768 df-cda 8990 df-pnf 10076 df-mnf 10077 df-xr 10078 df-ltxr 10079 df-le 10080 df-sub 10268 df-neg 10269 df-div 10685 df-nn 11021 df-2 11079 df-3 11080 df-n0 11293 df-xnn0 11364 df-z 11378 df-uz 11688 df-q 11789 df-rp 11833 df-ico 12181 df-icc 12182 df-fz 12327 df-fzo 12466 df-fl 12593 df-mod 12669 df-seq 12802 df-exp 12861 df-hash 13118 df-cj 13839 df-re 13840 df-im 13841 df-sqrt 13975 df-abs 13976 df-limsup 14202 df-clim 14219 df-rlim 14220 df-sum 14417 df-0p 23437 df-ply 23944 df-idp 23945 df-coe 23946 df-dgr 23947 df-quot 24046 df-aa 24070

This theorem is referenced by: aannen 24086

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