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Theorem aaliou2 26469

Description: Liouville's approximation theorem for algebraic numbers per se. (Contributed by Stefan O'Rear, 16-Nov-2014.)

**Assertion**
Ref	Expression
aaliou2	⊢ (𝐴 ∈ (𝔸 ∩ ℝ) → ∃𝑘 ∈ ℕ ∃𝑥 ∈ ℝ⁺ ∀𝑝 ∈ ℤ ∀𝑞 ∈ ℕ (𝐴 = (𝑝 / 𝑞) ∨ (𝑥 / (𝑞↑𝑘)) < (abs‘(𝐴 − (𝑝 / 𝑞)))))

Distinct variable group: 𝐴,𝑘,𝑥,𝑝,𝑞

Proof of Theorem aaliou2 Dummy variable 𝑎 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		elin 3929	. 2 ⊢ (𝐴 ∈ (𝔸 ∩ ℝ) ↔ (𝐴 ∈ 𝔸 ∧ 𝐴 ∈ ℝ))
2		elaa 26445	. . . 4 ⊢ (𝐴 ∈ 𝔸 ↔ (𝐴 ∈ ℂ ∧ ∃𝑎 ∈ ((Poly‘ℤ) ∖ {0_𝑝})(𝑎‘𝐴) = 0))
3		eldifn 4094	. . . . . . . . . . 11 ⊢ (𝑎 ∈ ((Poly‘ℤ) ∖ {0_𝑝}) → ¬ 𝑎 ∈ {0_𝑝})
4	3	3ad2ant1 1149	. . . . . . . . . 10 ⊢ ((𝑎 ∈ ((Poly‘ℤ) ∖ {0_𝑝}) ∧ (𝑎‘𝐴) = 0 ∧ 𝐴 ∈ ℝ) → ¬ 𝑎 ∈ {0_𝑝})
5		simpr 489	. . . . . . . . . . . . 13 ⊢ (((𝑎 ∈ ((Poly‘ℤ) ∖ {0_𝑝}) ∧ (𝑎‘𝐴) = 0 ∧ 𝐴 ∈ ℝ) ∧ 𝑎 = (ℂ × {(𝑎‘0)})) → 𝑎 = (ℂ × {(𝑎‘0)}))
6		fveq1 6881	. . . . . . . . . . . . . . . . 17 ⊢ (𝑎 = (ℂ × {(𝑎‘0)}) → (𝑎‘𝐴) = ((ℂ × {(𝑎‘0)})‘𝐴))
7	6	adantl 486	. . . . . . . . . . . . . . . 16 ⊢ (((𝑎 ∈ ((Poly‘ℤ) ∖ {0_𝑝}) ∧ (𝑎‘𝐴) = 0 ∧ 𝐴 ∈ ℝ) ∧ 𝑎 = (ℂ × {(𝑎‘0)})) → (𝑎‘𝐴) = ((ℂ × {(𝑎‘0)})‘𝐴))
8		simpl2 1209	. . . . . . . . . . . . . . . 16 ⊢ (((𝑎 ∈ ((Poly‘ℤ) ∖ {0_𝑝}) ∧ (𝑎‘𝐴) = 0 ∧ 𝐴 ∈ ℝ) ∧ 𝑎 = (ℂ × {(𝑎‘0)})) → (𝑎‘𝐴) = 0)
9		simpl3 1210	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑎 ∈ ((Poly‘ℤ) ∖ {0_𝑝}) ∧ (𝑎‘𝐴) = 0 ∧ 𝐴 ∈ ℝ) ∧ 𝑎 = (ℂ × {(𝑎‘0)})) → 𝐴 ∈ ℝ)
10	9	recnd 11236	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑎 ∈ ((Poly‘ℤ) ∖ {0_𝑝}) ∧ (𝑎‘𝐴) = 0 ∧ 𝐴 ∈ ℝ) ∧ 𝑎 = (ℂ × {(𝑎‘0)})) → 𝐴 ∈ ℂ)
11		fvex 6895	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑎‘0) ∈ V
12	11	fvconst2 7203	. . . . . . . . . . . . . . . . 17 ⊢ (𝐴 ∈ ℂ → ((ℂ × {(𝑎‘0)})‘𝐴) = (𝑎‘0))
13	10, 12	syl 18	. . . . . . . . . . . . . . . 16 ⊢ (((𝑎 ∈ ((Poly‘ℤ) ∖ {0_𝑝}) ∧ (𝑎‘𝐴) = 0 ∧ 𝐴 ∈ ℝ) ∧ 𝑎 = (ℂ × {(𝑎‘0)})) → ((ℂ × {(𝑎‘0)})‘𝐴) = (𝑎‘0))
14	7, 8, 13	3eqtr3rd 2813	. . . . . . . . . . . . . . 15 ⊢ (((𝑎 ∈ ((Poly‘ℤ) ∖ {0_𝑝}) ∧ (𝑎‘𝐴) = 0 ∧ 𝐴 ∈ ℝ) ∧ 𝑎 = (ℂ × {(𝑎‘0)})) → (𝑎‘0) = 0)
15	14	sneqd 4606	. . . . . . . . . . . . . 14 ⊢ (((𝑎 ∈ ((Poly‘ℤ) ∖ {0_𝑝}) ∧ (𝑎‘𝐴) = 0 ∧ 𝐴 ∈ ℝ) ∧ 𝑎 = (ℂ × {(𝑎‘0)})) → {(𝑎‘0)} = {0})
16	15	xpeq2d 5692	. . . . . . . . . . . . 13 ⊢ (((𝑎 ∈ ((Poly‘ℤ) ∖ {0_𝑝}) ∧ (𝑎‘𝐴) = 0 ∧ 𝐴 ∈ ℝ) ∧ 𝑎 = (ℂ × {(𝑎‘0)})) → (ℂ × {(𝑎‘0)}) = (ℂ × {0}))
17	5, 16	eqtrd 2804	. . . . . . . . . . . 12 ⊢ (((𝑎 ∈ ((Poly‘ℤ) ∖ {0_𝑝}) ∧ (𝑎‘𝐴) = 0 ∧ 𝐴 ∈ ℝ) ∧ 𝑎 = (ℂ × {(𝑎‘0)})) → 𝑎 = (ℂ × {0}))
18		df-0p 25797	. . . . . . . . . . . 12 ⊢ 0_𝑝 = (ℂ × {0})
19	17, 18	eqtr4di 2822	. . . . . . . . . . 11 ⊢ (((𝑎 ∈ ((Poly‘ℤ) ∖ {0_𝑝}) ∧ (𝑎‘𝐴) = 0 ∧ 𝐴 ∈ ℝ) ∧ 𝑎 = (ℂ × {(𝑎‘0)})) → 𝑎 = 0_𝑝)
20		velsn 4610	. . . . . . . . . . 11 ⊢ (𝑎 ∈ {0_𝑝} ↔ 𝑎 = 0_𝑝)
21	19, 20	sylibr 237	. . . . . . . . . 10 ⊢ (((𝑎 ∈ ((Poly‘ℤ) ∖ {0_𝑝}) ∧ (𝑎‘𝐴) = 0 ∧ 𝐴 ∈ ℝ) ∧ 𝑎 = (ℂ × {(𝑎‘0)})) → 𝑎 ∈ {0_𝑝})
22	4, 21	mtand 827	. . . . . . . . 9 ⊢ ((𝑎 ∈ ((Poly‘ℤ) ∖ {0_𝑝}) ∧ (𝑎‘𝐴) = 0 ∧ 𝐴 ∈ ℝ) → ¬ 𝑎 = (ℂ × {(𝑎‘0)}))
23		eldifi 4093	. . . . . . . . . . 11 ⊢ (𝑎 ∈ ((Poly‘ℤ) ∖ {0_𝑝}) → 𝑎 ∈ (Poly‘ℤ))
24	23	3ad2ant1 1149	. . . . . . . . . 10 ⊢ ((𝑎 ∈ ((Poly‘ℤ) ∖ {0_𝑝}) ∧ (𝑎‘𝐴) = 0 ∧ 𝐴 ∈ ℝ) → 𝑎 ∈ (Poly‘ℤ))
25		0dgrb 26371	. . . . . . . . . 10 ⊢ (𝑎 ∈ (Poly‘ℤ) → ((deg‘𝑎) = 0 ↔ 𝑎 = (ℂ × {(𝑎‘0)})))
26	24, 25	syl 18	. . . . . . . . 9 ⊢ ((𝑎 ∈ ((Poly‘ℤ) ∖ {0_𝑝}) ∧ (𝑎‘𝐴) = 0 ∧ 𝐴 ∈ ℝ) → ((deg‘𝑎) = 0 ↔ 𝑎 = (ℂ × {(𝑎‘0)})))
27	22, 26	mtbird 328	. . . . . . . 8 ⊢ ((𝑎 ∈ ((Poly‘ℤ) ∖ {0_𝑝}) ∧ (𝑎‘𝐴) = 0 ∧ 𝐴 ∈ ℝ) → ¬ (deg‘𝑎) = 0)
28		dgrcl 26358	. . . . . . . . . 10 ⊢ (𝑎 ∈ (Poly‘ℤ) → (deg‘𝑎) ∈ ℕ₀)
29	24, 28	syl 18	. . . . . . . . 9 ⊢ ((𝑎 ∈ ((Poly‘ℤ) ∖ {0_𝑝}) ∧ (𝑎‘𝐴) = 0 ∧ 𝐴 ∈ ℝ) → (deg‘𝑎) ∈ ℕ₀)
30		elnn0 12505	. . . . . . . . 9 ⊢ ((deg‘𝑎) ∈ ℕ₀ ↔ ((deg‘𝑎) ∈ ℕ ∨ (deg‘𝑎) = 0))
31	29, 30	sylib 221	. . . . . . . 8 ⊢ ((𝑎 ∈ ((Poly‘ℤ) ∖ {0_𝑝}) ∧ (𝑎‘𝐴) = 0 ∧ 𝐴 ∈ ℝ) → ((deg‘𝑎) ∈ ℕ ∨ (deg‘𝑎) = 0))
32		orel2 903	. . . . . . . 8 ⊢ (¬ (deg‘𝑎) = 0 → (((deg‘𝑎) ∈ ℕ ∨ (deg‘𝑎) = 0) → (deg‘𝑎) ∈ ℕ))
33	27, 31, 32	sylc 66	. . . . . . 7 ⊢ ((𝑎 ∈ ((Poly‘ℤ) ∖ {0_𝑝}) ∧ (𝑎‘𝐴) = 0 ∧ 𝐴 ∈ ℝ) → (deg‘𝑎) ∈ ℕ)
34		eqid 2769	. . . . . . . 8 ⊢ (deg‘𝑎) = (deg‘𝑎)
35		simp3 1154	. . . . . . . 8 ⊢ ((𝑎 ∈ ((Poly‘ℤ) ∖ {0_𝑝}) ∧ (𝑎‘𝐴) = 0 ∧ 𝐴 ∈ ℝ) → 𝐴 ∈ ℝ)
36		simp2 1153	. . . . . . . 8 ⊢ ((𝑎 ∈ ((Poly‘ℤ) ∖ {0_𝑝}) ∧ (𝑎‘𝐴) = 0 ∧ 𝐴 ∈ ℝ) → (𝑎‘𝐴) = 0)
37	34, 24, 33, 35, 36	aaliou 26467	. . . . . . 7 ⊢ ((𝑎 ∈ ((Poly‘ℤ) ∖ {0_𝑝}) ∧ (𝑎‘𝐴) = 0 ∧ 𝐴 ∈ ℝ) → ∃𝑥 ∈ ℝ⁺ ∀𝑝 ∈ ℤ ∀𝑞 ∈ ℕ (𝐴 = (𝑝 / 𝑞) ∨ (𝑥 / (𝑞↑(deg‘𝑎))) < (abs‘(𝐴 − (𝑝 / 𝑞)))))
38		oveq2 7419	. . . . . . . . . . . . 13 ⊢ (𝑘 = (deg‘𝑎) → (𝑞↑𝑘) = (𝑞↑(deg‘𝑎)))
39	38	oveq2d 7427	. . . . . . . . . . . 12 ⊢ (𝑘 = (deg‘𝑎) → (𝑥 / (𝑞↑𝑘)) = (𝑥 / (𝑞↑(deg‘𝑎))))
40	39	breq1d 5123	. . . . . . . . . . 11 ⊢ (𝑘 = (deg‘𝑎) → ((𝑥 / (𝑞↑𝑘)) < (abs‘(𝐴 − (𝑝 / 𝑞))) ↔ (𝑥 / (𝑞↑(deg‘𝑎))) < (abs‘(𝐴 − (𝑝 / 𝑞)))))
41	40	orbi2d 928	. . . . . . . . . 10 ⊢ (𝑘 = (deg‘𝑎) → ((𝐴 = (𝑝 / 𝑞) ∨ (𝑥 / (𝑞↑𝑘)) < (abs‘(𝐴 − (𝑝 / 𝑞)))) ↔ (𝐴 = (𝑝 / 𝑞) ∨ (𝑥 / (𝑞↑(deg‘𝑎))) < (abs‘(𝐴 − (𝑝 / 𝑞))))))
42	41	2ralbidv 3235	. . . . . . . . 9 ⊢ (𝑘 = (deg‘𝑎) → (∀𝑝 ∈ ℤ ∀𝑞 ∈ ℕ (𝐴 = (𝑝 / 𝑞) ∨ (𝑥 / (𝑞↑𝑘)) < (abs‘(𝐴 − (𝑝 / 𝑞)))) ↔ ∀𝑝 ∈ ℤ ∀𝑞 ∈ ℕ (𝐴 = (𝑝 / 𝑞) ∨ (𝑥 / (𝑞↑(deg‘𝑎))) < (abs‘(𝐴 − (𝑝 / 𝑞))))))
43	42	rexbidv 3195	. . . . . . . 8 ⊢ (𝑘 = (deg‘𝑎) → (∃𝑥 ∈ ℝ⁺ ∀𝑝 ∈ ℤ ∀𝑞 ∈ ℕ (𝐴 = (𝑝 / 𝑞) ∨ (𝑥 / (𝑞↑𝑘)) < (abs‘(𝐴 − (𝑝 / 𝑞)))) ↔ ∃𝑥 ∈ ℝ⁺ ∀𝑝 ∈ ℤ ∀𝑞 ∈ ℕ (𝐴 = (𝑝 / 𝑞) ∨ (𝑥 / (𝑞↑(deg‘𝑎))) < (abs‘(𝐴 − (𝑝 / 𝑞))))))
44	43	rspcev 3590	. . . . . . 7 ⊢ (((deg‘𝑎) ∈ ℕ ∧ ∃𝑥 ∈ ℝ⁺ ∀𝑝 ∈ ℤ ∀𝑞 ∈ ℕ (𝐴 = (𝑝 / 𝑞) ∨ (𝑥 / (𝑞↑(deg‘𝑎))) < (abs‘(𝐴 − (𝑝 / 𝑞))))) → ∃𝑘 ∈ ℕ ∃𝑥 ∈ ℝ⁺ ∀𝑝 ∈ ℤ ∀𝑞 ∈ ℕ (𝐴 = (𝑝 / 𝑞) ∨ (𝑥 / (𝑞↑𝑘)) < (abs‘(𝐴 − (𝑝 / 𝑞)))))
45	33, 37, 44	syl2anc 595	. . . . . 6 ⊢ ((𝑎 ∈ ((Poly‘ℤ) ∖ {0_𝑝}) ∧ (𝑎‘𝐴) = 0 ∧ 𝐴 ∈ ℝ) → ∃𝑘 ∈ ℕ ∃𝑥 ∈ ℝ⁺ ∀𝑝 ∈ ℤ ∀𝑞 ∈ ℕ (𝐴 = (𝑝 / 𝑞) ∨ (𝑥 / (𝑞↑𝑘)) < (abs‘(𝐴 − (𝑝 / 𝑞)))))
46	45	3exp 1135	. . . . 5 ⊢ (𝑎 ∈ ((Poly‘ℤ) ∖ {0_𝑝}) → ((𝑎‘𝐴) = 0 → (𝐴 ∈ ℝ → ∃𝑘 ∈ ℕ ∃𝑥 ∈ ℝ⁺ ∀𝑝 ∈ ℤ ∀𝑞 ∈ ℕ (𝐴 = (𝑝 / 𝑞) ∨ (𝑥 / (𝑞↑𝑘)) < (abs‘(𝐴 − (𝑝 / 𝑞)))))))
47	46	rexlimiv 3165	. . . 4 ⊢ (∃𝑎 ∈ ((Poly‘ℤ) ∖ {0_𝑝})(𝑎‘𝐴) = 0 → (𝐴 ∈ ℝ → ∃𝑘 ∈ ℕ ∃𝑥 ∈ ℝ⁺ ∀𝑝 ∈ ℤ ∀𝑞 ∈ ℕ (𝐴 = (𝑝 / 𝑞) ∨ (𝑥 / (𝑞↑𝑘)) < (abs‘(𝐴 − (𝑝 / 𝑞))))))
48	2, 47	simplbiim 513	. . 3 ⊢ (𝐴 ∈ 𝔸 → (𝐴 ∈ ℝ → ∃𝑘 ∈ ℕ ∃𝑥 ∈ ℝ⁺ ∀𝑝 ∈ ℤ ∀𝑞 ∈ ℕ (𝐴 = (𝑝 / 𝑞) ∨ (𝑥 / (𝑞↑𝑘)) < (abs‘(𝐴 − (𝑝 / 𝑞))))))
49	48	imp 411	. 2 ⊢ ((𝐴 ∈ 𝔸 ∧ 𝐴 ∈ ℝ) → ∃𝑘 ∈ ℕ ∃𝑥 ∈ ℝ⁺ ∀𝑝 ∈ ℤ ∀𝑞 ∈ ℕ (𝐴 = (𝑝 / 𝑞) ∨ (𝑥 / (𝑞↑𝑘)) < (abs‘(𝐴 − (𝑝 / 𝑞)))))
50	1, 49	sylbi 220	1 ⊢ (𝐴 ∈ (𝔸 ∩ ℝ) → ∃𝑘 ∈ ℕ ∃𝑥 ∈ ℝ⁺ ∀𝑝 ∈ ℤ ∀𝑞 ∈ ℕ (𝐴 = (𝑝 / 𝑞) ∨ (𝑥 / (𝑞↑𝑘)) < (abs‘(𝐴 − (𝑝 / 𝑞)))))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 209 ∧ wa 400 ∨ wo 860 ∧ w3a 1101 = wceq 1567 ∈ wcel 2149 ∀wral 3085 ∃wrex 3095 ∖ cdif 3910 ∩ cin 3912 {csn 4594 class class class wbr 5113 × cxp 5660 ‘cfv 6537 (class class class)co 7411 ℂcc 11097 ℝcr 11098 0cc0 11099 < clt 11242 − cmin 11440 / cdiv 11870 ℕcn 12232 ℕ₀cn0 12503 ℤcz 12590 ℝ⁺crp 13015 ↑cexp 14096 abscabs 15284 0_𝑝c0p 25796 Polycply 26309 degcdgr 26312 𝔸caa 26443

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1822 ax-4 1836 ax-5 1937 ax-6 1994 ax-7 2035 ax-8 2151 ax-9 2159 ax-10 2182 ax-11 2198 ax-12 2219 ax-ext 2741 ax-rep 5242 ax-sep 5261 ax-nul 5271 ax-pow 5337 ax-pr 5405 ax-un 7733 ax-inf2 9609 ax-cnex 11155 ax-resscn 11156 ax-1cn 11157 ax-icn 11158 ax-addcl 11159 ax-addrcl 11160 ax-mulcl 11161 ax-mulrcl 11162 ax-mulcom 11163 ax-addass 11164 ax-mulass 11165 ax-distr 11166 ax-i2m1 11167 ax-1ne0 11168 ax-1rid 11169 ax-rnegex 11170 ax-rrecex 11171 ax-cnre 11172 ax-pre-lttri 11173 ax-pre-lttrn 11174 ax-pre-ltadd 11175 ax-pre-mulgt0 11176 ax-pre-sup 11177 ax-addf 11178

This theorem depends on definitions: df-bi 210 df-an 401 df-or 861 df-3or 1102 df-3an 1103 df-tru 1570 df-fal 1580 df-ex 1807 df-nf 1811 df-sb 2098 df-mo 2573 df-eu 2603 df-clab 2748 df-cleq 2761 df-clel 2844 df-nfc 2918 df-ne 2965 df-nel 3071 df-ral 3086 df-rex 3096 df-rmo 3376 df-reu 3377 df-rab 3424 df-v 3465 df-sbc 3754 df-csb 3862 df-dif 3916 df-un 3918 df-in 3920 df-ss 3930 df-pss 3933 df-nul 4295 df-if 4493 df-pw 4569 df-sn 4595 df-pr 4597 df-tp 4599 df-op 4601 df-uni 4877 df-int 4917 df-iun 4962 df-iin 4963 df-br 5114 df-opab 5178 df-mpt 5197 df-tr 5223 df-id 5557 df-eprel 5562 df-po 5570 df-so 5571 df-fr 5615 df-se 5616 df-we 5617 df-xp 5668 df-rel 5669 df-cnv 5670 df-co 5671 df-dm 5672 df-rn 5673 df-res 5674 df-ima 5675 df-pred 6303 df-ord 6364 df-on 6365 df-lim 6366 df-suc 6367 df-iota 6493 df-fun 6539 df-fn 6540 df-f 6541 df-f1 6542 df-fo 6543 df-f1o 6544 df-fv 6545 df-isom 6546 df-riota 7368 df-ov 7414 df-oprab 7415 df-mpo 7416 df-of 7675 df-om 7862 df-1st 7985 df-2nd 7986 df-supp 8156 df-frecs 8277 df-wrecs 8308 df-recs 8357 df-rdg 8396 df-1o 8452 df-2o 8453 df-oadd 8456 df-er 8693 df-map 8825 df-pm 8826 df-ixp 8895 df-en 8943 df-dom 8944 df-sdom 8945 df-fin 8946 df-fsupp 9321 df-fi 9370 df-sup 9401 df-inf 9402 df-oi 9471 df-dju 9886 df-card 9924 df-pnf 11244 df-mnf 11245 df-xr 11246 df-ltxr 11247 df-le 11248 df-sub 11442 df-neg 11443 df-div 11871 df-nn 12233 df-2 12302 df-3 12303 df-4 12304 df-5 12305 df-6 12306 df-7 12307 df-8 12308 df-9 12309 df-n0 12504 df-xnn0 12577 df-z 12591 df-dec 12711 df-uz 12862 df-q 12972 df-rp 13016 df-xneg 13136 df-xadd 13137 df-xmul 13138 df-ioo 13375 df-ico 13377 df-icc 13378 df-fz 13535 df-fzo 13682 df-fl 13824 df-seq 14037 df-exp 14097 df-hash 14366 df-cj 15149 df-re 15150 df-im 15151 df-sqrt 15285 df-abs 15286 df-clim 15538 df-rlim 15539 df-sum 15737 df-struct 17206 df-sets 17223 df-slot 17241 df-ndx 17253 df-base 17269 df-ress 17290 df-plusg 17322 df-mulr 17323 df-starv 17324 df-sca 17325 df-vsca 17326 df-ip 17327 df-tset 17328 df-ple 17329 df-ds 17331 df-unif 17332 df-hom 17333 df-cco 17334 df-rest 17474 df-topn 17475 df-0g 17493 df-gsum 17494 df-topgen 17495 df-pt 17496 df-prds 17499 df-xrs 17555 df-qtop 17560 df-imas 17561 df-xps 17563 df-mre 17637 df-mrc 17638 df-acs 17640 df-mgm 18697 df-sgrp 18776 df-mnd 18792 df-submnd 18841 df-grp 19002 df-minusg 19003 df-mulg 19133 df-subg 19188 df-cntz 19386 df-cmn 19851 df-abl 19852 df-mgp 20216 df-rng 20230 df-ur 20263 df-ring 20316 df-cring 20317 df-subrng 20630 df-subrg 20654 df-psmet 21482 df-xmet 21483 df-met 21484 df-bl 21485 df-mopn 21486 df-fbas 21487 df-fg 21488 df-cnfld 21491 df-top 23019 df-topon 23036 df-topsp 23058 df-bases 23071 df-cld 23144 df-ntr 23145 df-cls 23146 df-nei 23223 df-lp 23261 df-perf 23262 df-cn 23352 df-cnp 23353 df-haus 23440 df-cmp 23512 df-tx 23687 df-hmeo 23880 df-fil 23971 df-fm 24063 df-flim 24064 df-flf 24065 df-xms 24445 df-ms 24446 df-tms 24447 df-cncf 25005 df-0p 25797 df-limc 25993 df-dv 25994 df-dvn 25995 df-cpn 25996 df-ply 26313 df-idp 26314 df-coe 26315 df-dgr 26316 df-quot 26420 df-aa 26444

This theorem is referenced by: aaliou2b 26470

W3C validator