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Theorem aaliou2 26224

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 3927	. 2 ⊢ (𝐴 ∈ (𝔸 ∩ ℝ) ↔ (𝐴 ∈ 𝔸 ∧ 𝐴 ∈ ℝ))
2		elaa 26200	. . . 4 ⊢ (𝐴 ∈ 𝔸 ↔ (𝐴 ∈ ℂ ∧ ∃𝑎 ∈ ((Poly‘ℤ) ∖ {0_𝑝})(𝑎‘𝐴) = 0))
3		eldifn 4091	. . . . . . . . . . 11 ⊢ (𝑎 ∈ ((Poly‘ℤ) ∖ {0_𝑝}) → ¬ 𝑎 ∈ {0_𝑝})
4	3	3ad2ant1 1133	. . . . . . . . . 10 ⊢ ((𝑎 ∈ ((Poly‘ℤ) ∖ {0_𝑝}) ∧ (𝑎‘𝐴) = 0 ∧ 𝐴 ∈ ℝ) → ¬ 𝑎 ∈ {0_𝑝})
5		simpr 484	. . . . . . . . . . . . 13 ⊢ (((𝑎 ∈ ((Poly‘ℤ) ∖ {0_𝑝}) ∧ (𝑎‘𝐴) = 0 ∧ 𝐴 ∈ ℝ) ∧ 𝑎 = (ℂ × {(𝑎‘0)})) → 𝑎 = (ℂ × {(𝑎‘0)}))
6		fveq1 6839	. . . . . . . . . . . . . . . . 17 ⊢ (𝑎 = (ℂ × {(𝑎‘0)}) → (𝑎‘𝐴) = ((ℂ × {(𝑎‘0)})‘𝐴))
7	6	adantl 481	. . . . . . . . . . . . . . . 16 ⊢ (((𝑎 ∈ ((Poly‘ℤ) ∖ {0_𝑝}) ∧ (𝑎‘𝐴) = 0 ∧ 𝐴 ∈ ℝ) ∧ 𝑎 = (ℂ × {(𝑎‘0)})) → (𝑎‘𝐴) = ((ℂ × {(𝑎‘0)})‘𝐴))
8		simpl2 1193	. . . . . . . . . . . . . . . 16 ⊢ (((𝑎 ∈ ((Poly‘ℤ) ∖ {0_𝑝}) ∧ (𝑎‘𝐴) = 0 ∧ 𝐴 ∈ ℝ) ∧ 𝑎 = (ℂ × {(𝑎‘0)})) → (𝑎‘𝐴) = 0)
9		simpl3 1194	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑎 ∈ ((Poly‘ℤ) ∖ {0_𝑝}) ∧ (𝑎‘𝐴) = 0 ∧ 𝐴 ∈ ℝ) ∧ 𝑎 = (ℂ × {(𝑎‘0)})) → 𝐴 ∈ ℝ)
10	9	recnd 11178	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑎 ∈ ((Poly‘ℤ) ∖ {0_𝑝}) ∧ (𝑎‘𝐴) = 0 ∧ 𝐴 ∈ ℝ) ∧ 𝑎 = (ℂ × {(𝑎‘0)})) → 𝐴 ∈ ℂ)
11		fvex 6853	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑎‘0) ∈ V
12	11	fvconst2 7160	. . . . . . . . . . . . . . . . 17 ⊢ (𝐴 ∈ ℂ → ((ℂ × {(𝑎‘0)})‘𝐴) = (𝑎‘0))
13	10, 12	syl 17	. . . . . . . . . . . . . . . 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 2773	. . . . . . . . . . . . . . 15 ⊢ (((𝑎 ∈ ((Poly‘ℤ) ∖ {0_𝑝}) ∧ (𝑎‘𝐴) = 0 ∧ 𝐴 ∈ ℝ) ∧ 𝑎 = (ℂ × {(𝑎‘0)})) → (𝑎‘0) = 0)
15	14	sneqd 4597	. . . . . . . . . . . . . 14 ⊢ (((𝑎 ∈ ((Poly‘ℤ) ∖ {0_𝑝}) ∧ (𝑎‘𝐴) = 0 ∧ 𝐴 ∈ ℝ) ∧ 𝑎 = (ℂ × {(𝑎‘0)})) → {(𝑎‘0)} = {0})
16	15	xpeq2d 5661	. . . . . . . . . . . . 13 ⊢ (((𝑎 ∈ ((Poly‘ℤ) ∖ {0_𝑝}) ∧ (𝑎‘𝐴) = 0 ∧ 𝐴 ∈ ℝ) ∧ 𝑎 = (ℂ × {(𝑎‘0)})) → (ℂ × {(𝑎‘0)}) = (ℂ × {0}))
17	5, 16	eqtrd 2764	. . . . . . . . . . . 12 ⊢ (((𝑎 ∈ ((Poly‘ℤ) ∖ {0_𝑝}) ∧ (𝑎‘𝐴) = 0 ∧ 𝐴 ∈ ℝ) ∧ 𝑎 = (ℂ × {(𝑎‘0)})) → 𝑎 = (ℂ × {0}))
18		df-0p 25547	. . . . . . . . . . . 12 ⊢ 0_𝑝 = (ℂ × {0})
19	17, 18	eqtr4di 2782	. . . . . . . . . . 11 ⊢ (((𝑎 ∈ ((Poly‘ℤ) ∖ {0_𝑝}) ∧ (𝑎‘𝐴) = 0 ∧ 𝐴 ∈ ℝ) ∧ 𝑎 = (ℂ × {(𝑎‘0)})) → 𝑎 = 0_𝑝)
20		velsn 4601	. . . . . . . . . . 11 ⊢ (𝑎 ∈ {0_𝑝} ↔ 𝑎 = 0_𝑝)
21	19, 20	sylibr 234	. . . . . . . . . 10 ⊢ (((𝑎 ∈ ((Poly‘ℤ) ∖ {0_𝑝}) ∧ (𝑎‘𝐴) = 0 ∧ 𝐴 ∈ ℝ) ∧ 𝑎 = (ℂ × {(𝑎‘0)})) → 𝑎 ∈ {0_𝑝})
22	4, 21	mtand 815	. . . . . . . . 9 ⊢ ((𝑎 ∈ ((Poly‘ℤ) ∖ {0_𝑝}) ∧ (𝑎‘𝐴) = 0 ∧ 𝐴 ∈ ℝ) → ¬ 𝑎 = (ℂ × {(𝑎‘0)}))
23		eldifi 4090	. . . . . . . . . . 11 ⊢ (𝑎 ∈ ((Poly‘ℤ) ∖ {0_𝑝}) → 𝑎 ∈ (Poly‘ℤ))
24	23	3ad2ant1 1133	. . . . . . . . . 10 ⊢ ((𝑎 ∈ ((Poly‘ℤ) ∖ {0_𝑝}) ∧ (𝑎‘𝐴) = 0 ∧ 𝐴 ∈ ℝ) → 𝑎 ∈ (Poly‘ℤ))
25		0dgrb 26127	. . . . . . . . . 10 ⊢ (𝑎 ∈ (Poly‘ℤ) → ((deg‘𝑎) = 0 ↔ 𝑎 = (ℂ × {(𝑎‘0)})))
26	24, 25	syl 17	. . . . . . . . 9 ⊢ ((𝑎 ∈ ((Poly‘ℤ) ∖ {0_𝑝}) ∧ (𝑎‘𝐴) = 0 ∧ 𝐴 ∈ ℝ) → ((deg‘𝑎) = 0 ↔ 𝑎 = (ℂ × {(𝑎‘0)})))
27	22, 26	mtbird 325	. . . . . . . 8 ⊢ ((𝑎 ∈ ((Poly‘ℤ) ∖ {0_𝑝}) ∧ (𝑎‘𝐴) = 0 ∧ 𝐴 ∈ ℝ) → ¬ (deg‘𝑎) = 0)
28		dgrcl 26114	. . . . . . . . . 10 ⊢ (𝑎 ∈ (Poly‘ℤ) → (deg‘𝑎) ∈ ℕ₀)
29	24, 28	syl 17	. . . . . . . . 9 ⊢ ((𝑎 ∈ ((Poly‘ℤ) ∖ {0_𝑝}) ∧ (𝑎‘𝐴) = 0 ∧ 𝐴 ∈ ℝ) → (deg‘𝑎) ∈ ℕ₀)
30		elnn0 12420	. . . . . . . . 9 ⊢ ((deg‘𝑎) ∈ ℕ₀ ↔ ((deg‘𝑎) ∈ ℕ ∨ (deg‘𝑎) = 0))
31	29, 30	sylib 218	. . . . . . . 8 ⊢ ((𝑎 ∈ ((Poly‘ℤ) ∖ {0_𝑝}) ∧ (𝑎‘𝐴) = 0 ∧ 𝐴 ∈ ℝ) → ((deg‘𝑎) ∈ ℕ ∨ (deg‘𝑎) = 0))
32		orel2 890	. . . . . . . 8 ⊢ (¬ (deg‘𝑎) = 0 → (((deg‘𝑎) ∈ ℕ ∨ (deg‘𝑎) = 0) → (deg‘𝑎) ∈ ℕ))
33	27, 31, 32	sylc 65	. . . . . . 7 ⊢ ((𝑎 ∈ ((Poly‘ℤ) ∖ {0_𝑝}) ∧ (𝑎‘𝐴) = 0 ∧ 𝐴 ∈ ℝ) → (deg‘𝑎) ∈ ℕ)
34		eqid 2729	. . . . . . . 8 ⊢ (deg‘𝑎) = (deg‘𝑎)
35		simp3 1138	. . . . . . . 8 ⊢ ((𝑎 ∈ ((Poly‘ℤ) ∖ {0_𝑝}) ∧ (𝑎‘𝐴) = 0 ∧ 𝐴 ∈ ℝ) → 𝐴 ∈ ℝ)
36		simp2 1137	. . . . . . . 8 ⊢ ((𝑎 ∈ ((Poly‘ℤ) ∖ {0_𝑝}) ∧ (𝑎‘𝐴) = 0 ∧ 𝐴 ∈ ℝ) → (𝑎‘𝐴) = 0)
37	34, 24, 33, 35, 36	aaliou 26222	. . . . . . 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 7377	. . . . . . . . . . . . 13 ⊢ (𝑘 = (deg‘𝑎) → (𝑞↑𝑘) = (𝑞↑(deg‘𝑎)))
39	38	oveq2d 7385	. . . . . . . . . . . 12 ⊢ (𝑘 = (deg‘𝑎) → (𝑥 / (𝑞↑𝑘)) = (𝑥 / (𝑞↑(deg‘𝑎))))
40	39	breq1d 5112	. . . . . . . . . . 11 ⊢ (𝑘 = (deg‘𝑎) → ((𝑥 / (𝑞↑𝑘)) < (abs‘(𝐴 − (𝑝 / 𝑞))) ↔ (𝑥 / (𝑞↑(deg‘𝑎))) < (abs‘(𝐴 − (𝑝 / 𝑞)))))
41	40	orbi2d 915	. . . . . . . . . 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 3199	. . . . . . . . 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 3157	. . . . . . . 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 3585	. . . . . . 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 584	. . . . . 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 1119	. . . . 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 3127	. . . 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 504	. . 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 406	. 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 217	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 206 ∧ wa 395 ∨ wo 847 ∧ w3a 1086 = wceq 1540 ∈ wcel 2109 ∀wral 3044 ∃wrex 3053 ∖ cdif 3908 ∩ cin 3910 {csn 4585 class class class wbr 5102 × cxp 5629 ‘cfv 6499 (class class class)co 7369 ℂcc 11042 ℝcr 11043 0cc0 11044 < clt 11184 − cmin 11381 / cdiv 11811 ℕcn 12162 ℕ₀cn0 12418 ℤcz 12505 ℝ⁺crp 12927 ↑cexp 14002 abscabs 15176 0_𝑝c0p 25546 Polycply 26065 degcdgr 26068 𝔸caa 26198

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2008 ax-8 2111 ax-9 2119 ax-10 2142 ax-11 2158 ax-12 2178 ax-ext 2701 ax-rep 5229 ax-sep 5246 ax-nul 5256 ax-pow 5315 ax-pr 5382 ax-un 7691 ax-inf2 9570 ax-cnex 11100 ax-resscn 11101 ax-1cn 11102 ax-icn 11103 ax-addcl 11104 ax-addrcl 11105 ax-mulcl 11106 ax-mulrcl 11107 ax-mulcom 11108 ax-addass 11109 ax-mulass 11110 ax-distr 11111 ax-i2m1 11112 ax-1ne0 11113 ax-1rid 11114 ax-rnegex 11115 ax-rrecex 11116 ax-cnre 11117 ax-pre-lttri 11118 ax-pre-lttrn 11119 ax-pre-ltadd 11120 ax-pre-mulgt0 11121 ax-pre-sup 11122 ax-addf 11123

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2533 df-eu 2562 df-clab 2708 df-cleq 2721 df-clel 2803 df-nfc 2878 df-ne 2926 df-nel 3030 df-ral 3045 df-rex 3054 df-rmo 3351 df-reu 3352 df-rab 3403 df-v 3446 df-sbc 3751 df-csb 3860 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-pss 3931 df-nul 4293 df-if 4485 df-pw 4561 df-sn 4586 df-pr 4588 df-tp 4590 df-op 4592 df-uni 4868 df-int 4907 df-iun 4953 df-iin 4954 df-br 5103 df-opab 5165 df-mpt 5184 df-tr 5210 df-id 5526 df-eprel 5531 df-po 5539 df-so 5540 df-fr 5584 df-se 5585 df-we 5586 df-xp 5637 df-rel 5638 df-cnv 5639 df-co 5640 df-dm 5641 df-rn 5642 df-res 5643 df-ima 5644 df-pred 6262 df-ord 6323 df-on 6324 df-lim 6325 df-suc 6326 df-iota 6452 df-fun 6501 df-fn 6502 df-f 6503 df-f1 6504 df-fo 6505 df-f1o 6506 df-fv 6507 df-isom 6508 df-riota 7326 df-ov 7372 df-oprab 7373 df-mpo 7374 df-of 7633 df-om 7823 df-1st 7947 df-2nd 7948 df-supp 8117 df-frecs 8237 df-wrecs 8268 df-recs 8317 df-rdg 8355 df-1o 8411 df-2o 8412 df-oadd 8415 df-er 8648 df-map 8778 df-pm 8779 df-ixp 8848 df-en 8896 df-dom 8897 df-sdom 8898 df-fin 8899 df-fsupp 9289 df-fi 9338 df-sup 9369 df-inf 9370 df-oi 9439 df-dju 9830 df-card 9868 df-pnf 11186 df-mnf 11187 df-xr 11188 df-ltxr 11189 df-le 11190 df-sub 11383 df-neg 11384 df-div 11812 df-nn 12163 df-2 12225 df-3 12226 df-4 12227 df-5 12228 df-6 12229 df-7 12230 df-8 12231 df-9 12232 df-n0 12419 df-xnn0 12492 df-z 12506 df-dec 12626 df-uz 12770 df-q 12884 df-rp 12928 df-xneg 13048 df-xadd 13049 df-xmul 13050 df-ioo 13286 df-ico 13288 df-icc 13289 df-fz 13445 df-fzo 13592 df-fl 13730 df-seq 13943 df-exp 14003 df-hash 14272 df-cj 15041 df-re 15042 df-im 15043 df-sqrt 15177 df-abs 15178 df-clim 15430 df-rlim 15431 df-sum 15629 df-struct 17093 df-sets 17110 df-slot 17128 df-ndx 17140 df-base 17156 df-ress 17177 df-plusg 17209 df-mulr 17210 df-starv 17211 df-sca 17212 df-vsca 17213 df-ip 17214 df-tset 17215 df-ple 17216 df-ds 17218 df-unif 17219 df-hom 17220 df-cco 17221 df-rest 17361 df-topn 17362 df-0g 17380 df-gsum 17381 df-topgen 17382 df-pt 17383 df-prds 17386 df-xrs 17441 df-qtop 17446 df-imas 17447 df-xps 17449 df-mre 17523 df-mrc 17524 df-acs 17526 df-mgm 18543 df-sgrp 18622 df-mnd 18638 df-submnd 18687 df-grp 18844 df-minusg 18845 df-mulg 18976 df-subg 19031 df-cntz 19225 df-cmn 19688 df-abl 19689 df-mgp 20026 df-rng 20038 df-ur 20067 df-ring 20120 df-cring 20121 df-subrng 20431 df-subrg 20455 df-psmet 21232 df-xmet 21233 df-met 21234 df-bl 21235 df-mopn 21236 df-fbas 21237 df-fg 21238 df-cnfld 21241 df-top 22757 df-topon 22774 df-topsp 22796 df-bases 22809 df-cld 22882 df-ntr 22883 df-cls 22884 df-nei 22961 df-lp 22999 df-perf 23000 df-cn 23090 df-cnp 23091 df-haus 23178 df-cmp 23250 df-tx 23425 df-hmeo 23618 df-fil 23709 df-fm 23801 df-flim 23802 df-flf 23803 df-xms 24184 df-ms 24185 df-tms 24186 df-cncf 24747 df-0p 25547 df-limc 25743 df-dv 25744 df-dvn 25745 df-cpn 25746 df-ply 26069 df-idp 26070 df-coe 26071 df-dgr 26072 df-quot 26175 df-aa 26199

This theorem is referenced by: aaliou2b 26225

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