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Theorem perfectlem1 27209

Description: Lemma for perfect 27211. (Contributed by Mario Carneiro, 7-Jun-2016.)

**Hypotheses**
Ref	Expression
perfectlem.1	⊢ (𝜑 → 𝐴 ∈ ℕ)
perfectlem.2	⊢ (𝜑 → 𝐵 ∈ ℕ)
perfectlem.3	⊢ (𝜑 → ¬ 2 ∥ 𝐵)
perfectlem.4	⊢ (𝜑 → (1 σ ((2↑𝐴) · 𝐵)) = (2 · ((2↑𝐴) · 𝐵)))

**Assertion**
Ref	Expression
perfectlem1	⊢ (𝜑 → ((2↑(𝐴 + 1)) ∈ ℕ ∧ ((2↑(𝐴 + 1)) − 1) ∈ ℕ ∧ (𝐵 / ((2↑(𝐴 + 1)) − 1)) ∈ ℕ))

**Proof of Theorem perfectlem1**
Step	Hyp	Ref	Expression
1		2nn 12248	. . 3 ⊢ 2 ∈ ℕ
2		perfectlem.1	. . . . 5 ⊢ (𝜑 → 𝐴 ∈ ℕ)
3	2	nnnn0d 12492	. . . 4 ⊢ (𝜑 → 𝐴 ∈ ℕ₀)
4		peano2nn0 12471	. . . 4 ⊢ (𝐴 ∈ ℕ₀ → (𝐴 + 1) ∈ ℕ₀)
5	3, 4	syl 17	. . 3 ⊢ (𝜑 → (𝐴 + 1) ∈ ℕ₀)
6		nnexpcl 14030	. . 3 ⊢ ((2 ∈ ℕ ∧ (𝐴 + 1) ∈ ℕ₀) → (2↑(𝐴 + 1)) ∈ ℕ)
7	1, 5, 6	sylancr 588	. 2 ⊢ (𝜑 → (2↑(𝐴 + 1)) ∈ ℕ)
8		2re 12249	. . . 4 ⊢ 2 ∈ ℝ
9	2	peano2nnd 12185	. . . 4 ⊢ (𝜑 → (𝐴 + 1) ∈ ℕ)
10		1lt2 12341	. . . . 5 ⊢ 1 < 2
11	10	a1i 11	. . . 4 ⊢ (𝜑 → 1 < 2)
12		expgt1 14056	. . . 4 ⊢ ((2 ∈ ℝ ∧ (𝐴 + 1) ∈ ℕ ∧ 1 < 2) → 1 < (2↑(𝐴 + 1)))
13	8, 9, 11, 12	mp3an2i 1469	. . 3 ⊢ (𝜑 → 1 < (2↑(𝐴 + 1)))
14		1nn 12179	. . . 4 ⊢ 1 ∈ ℕ
15		nnsub 12215	. . . 4 ⊢ ((1 ∈ ℕ ∧ (2↑(𝐴 + 1)) ∈ ℕ) → (1 < (2↑(𝐴 + 1)) ↔ ((2↑(𝐴 + 1)) − 1) ∈ ℕ))
16	14, 7, 15	sylancr 588	. . 3 ⊢ (𝜑 → (1 < (2↑(𝐴 + 1)) ↔ ((2↑(𝐴 + 1)) − 1) ∈ ℕ))
17	13, 16	mpbid 232	. 2 ⊢ (𝜑 → ((2↑(𝐴 + 1)) − 1) ∈ ℕ)
18	7	nnzd 12544	. . . . . . 7 ⊢ (𝜑 → (2↑(𝐴 + 1)) ∈ ℤ)
19		peano2zm 12564	. . . . . . 7 ⊢ ((2↑(𝐴 + 1)) ∈ ℤ → ((2↑(𝐴 + 1)) − 1) ∈ ℤ)
20	18, 19	syl 17	. . . . . 6 ⊢ (𝜑 → ((2↑(𝐴 + 1)) − 1) ∈ ℤ)
21		1nn0 12447	. . . . . . . 8 ⊢ 1 ∈ ℕ₀
22		perfectlem.2	. . . . . . . 8 ⊢ (𝜑 → 𝐵 ∈ ℕ)
23		sgmnncl 27127	. . . . . . . 8 ⊢ ((1 ∈ ℕ₀ ∧ 𝐵 ∈ ℕ) → (1 σ 𝐵) ∈ ℕ)
24	21, 22, 23	sylancr 588	. . . . . . 7 ⊢ (𝜑 → (1 σ 𝐵) ∈ ℕ)
25	24	nnzd 12544	. . . . . 6 ⊢ (𝜑 → (1 σ 𝐵) ∈ ℤ)
26		dvdsmul1 16240	. . . . . 6 ⊢ ((((2↑(𝐴 + 1)) − 1) ∈ ℤ ∧ (1 σ 𝐵) ∈ ℤ) → ((2↑(𝐴 + 1)) − 1) ∥ (((2↑(𝐴 + 1)) − 1) · (1 σ 𝐵)))
27	20, 25, 26	syl2anc 585	. . . . 5 ⊢ (𝜑 → ((2↑(𝐴 + 1)) − 1) ∥ (((2↑(𝐴 + 1)) − 1) · (1 σ 𝐵)))
28		2cn 12250	. . . . . . . . 9 ⊢ 2 ∈ ℂ
29		expp1 14024	. . . . . . . . 9 ⊢ ((2 ∈ ℂ ∧ 𝐴 ∈ ℕ₀) → (2↑(𝐴 + 1)) = ((2↑𝐴) · 2))
30	28, 3, 29	sylancr 588	. . . . . . . 8 ⊢ (𝜑 → (2↑(𝐴 + 1)) = ((2↑𝐴) · 2))
31		nnexpcl 14030	. . . . . . . . . . 11 ⊢ ((2 ∈ ℕ ∧ 𝐴 ∈ ℕ₀) → (2↑𝐴) ∈ ℕ)
32	1, 3, 31	sylancr 588	. . . . . . . . . 10 ⊢ (𝜑 → (2↑𝐴) ∈ ℕ)
33	32	nncnd 12184	. . . . . . . . 9 ⊢ (𝜑 → (2↑𝐴) ∈ ℂ)
34		mulcom 11118	. . . . . . . . 9 ⊢ (((2↑𝐴) ∈ ℂ ∧ 2 ∈ ℂ) → ((2↑𝐴) · 2) = (2 · (2↑𝐴)))
35	33, 28, 34	sylancl 587	. . . . . . . 8 ⊢ (𝜑 → ((2↑𝐴) · 2) = (2 · (2↑𝐴)))
36	30, 35	eqtrd 2772	. . . . . . 7 ⊢ (𝜑 → (2↑(𝐴 + 1)) = (2 · (2↑𝐴)))
37	36	oveq1d 7376	. . . . . 6 ⊢ (𝜑 → ((2↑(𝐴 + 1)) · 𝐵) = ((2 · (2↑𝐴)) · 𝐵))
38	28	a1i 11	. . . . . . 7 ⊢ (𝜑 → 2 ∈ ℂ)
39	22	nncnd 12184	. . . . . . 7 ⊢ (𝜑 → 𝐵 ∈ ℂ)
40	38, 33, 39	mulassd 11162	. . . . . 6 ⊢ (𝜑 → ((2 · (2↑𝐴)) · 𝐵) = (2 · ((2↑𝐴) · 𝐵)))
41		ax-1cn 11090	. . . . . . . . 9 ⊢ 1 ∈ ℂ
42	41	a1i 11	. . . . . . . 8 ⊢ (𝜑 → 1 ∈ ℂ)
43		perfectlem.3	. . . . . . . . . 10 ⊢ (𝜑 → ¬ 2 ∥ 𝐵)
44		2prm 16655	. . . . . . . . . . 11 ⊢ 2 ∈ ℙ
45	22	nnzd 12544	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐵 ∈ ℤ)
46		coprm 16675	. . . . . . . . . . 11 ⊢ ((2 ∈ ℙ ∧ 𝐵 ∈ ℤ) → (¬ 2 ∥ 𝐵 ↔ (2 gcd 𝐵) = 1))
47	44, 45, 46	sylancr 588	. . . . . . . . . 10 ⊢ (𝜑 → (¬ 2 ∥ 𝐵 ↔ (2 gcd 𝐵) = 1))
48	43, 47	mpbid 232	. . . . . . . . 9 ⊢ (𝜑 → (2 gcd 𝐵) = 1)
49		2z 12553	. . . . . . . . . 10 ⊢ 2 ∈ ℤ
50		rpexp1i 16687	. . . . . . . . . 10 ⊢ ((2 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝐴 ∈ ℕ₀) → ((2 gcd 𝐵) = 1 → ((2↑𝐴) gcd 𝐵) = 1))
51	49, 45, 3, 50	mp3an2i 1469	. . . . . . . . 9 ⊢ (𝜑 → ((2 gcd 𝐵) = 1 → ((2↑𝐴) gcd 𝐵) = 1))
52	48, 51	mpd 15	. . . . . . . 8 ⊢ (𝜑 → ((2↑𝐴) gcd 𝐵) = 1)
53		sgmmul 27181	. . . . . . . 8 ⊢ ((1 ∈ ℂ ∧ ((2↑𝐴) ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ ((2↑𝐴) gcd 𝐵) = 1)) → (1 σ ((2↑𝐴) · 𝐵)) = ((1 σ (2↑𝐴)) · (1 σ 𝐵)))
54	42, 32, 22, 52, 53	syl13anc 1375	. . . . . . 7 ⊢ (𝜑 → (1 σ ((2↑𝐴) · 𝐵)) = ((1 σ (2↑𝐴)) · (1 σ 𝐵)))
55		perfectlem.4	. . . . . . 7 ⊢ (𝜑 → (1 σ ((2↑𝐴) · 𝐵)) = (2 · ((2↑𝐴) · 𝐵)))
56	2	nncnd 12184	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐴 ∈ ℂ)
57		pncan 11393	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ ℂ ∧ 1 ∈ ℂ) → ((𝐴 + 1) − 1) = 𝐴)
58	56, 41, 57	sylancl 587	. . . . . . . . . . 11 ⊢ (𝜑 → ((𝐴 + 1) − 1) = 𝐴)
59	58	oveq2d 7377	. . . . . . . . . 10 ⊢ (𝜑 → (2↑((𝐴 + 1) − 1)) = (2↑𝐴))
60	59	oveq2d 7377	. . . . . . . . 9 ⊢ (𝜑 → (1 σ (2↑((𝐴 + 1) − 1))) = (1 σ (2↑𝐴)))
61		1sgm2ppw 27180	. . . . . . . . . 10 ⊢ ((𝐴 + 1) ∈ ℕ → (1 σ (2↑((𝐴 + 1) − 1))) = ((2↑(𝐴 + 1)) − 1))
62	9, 61	syl 17	. . . . . . . . 9 ⊢ (𝜑 → (1 σ (2↑((𝐴 + 1) − 1))) = ((2↑(𝐴 + 1)) − 1))
63	60, 62	eqtr3d 2774	. . . . . . . 8 ⊢ (𝜑 → (1 σ (2↑𝐴)) = ((2↑(𝐴 + 1)) − 1))
64	63	oveq1d 7376	. . . . . . 7 ⊢ (𝜑 → ((1 σ (2↑𝐴)) · (1 σ 𝐵)) = (((2↑(𝐴 + 1)) − 1) · (1 σ 𝐵)))
65	54, 55, 64	3eqtr3d 2780	. . . . . 6 ⊢ (𝜑 → (2 · ((2↑𝐴) · 𝐵)) = (((2↑(𝐴 + 1)) − 1) · (1 σ 𝐵)))
66	37, 40, 65	3eqtrd 2776	. . . . 5 ⊢ (𝜑 → ((2↑(𝐴 + 1)) · 𝐵) = (((2↑(𝐴 + 1)) − 1) · (1 σ 𝐵)))
67	27, 66	breqtrrd 5114	. . . 4 ⊢ (𝜑 → ((2↑(𝐴 + 1)) − 1) ∥ ((2↑(𝐴 + 1)) · 𝐵))
68	20, 18	gcdcomd 16477	. . . . 5 ⊢ (𝜑 → (((2↑(𝐴 + 1)) − 1) gcd (2↑(𝐴 + 1))) = ((2↑(𝐴 + 1)) gcd ((2↑(𝐴 + 1)) − 1)))
69		iddvdsexp 16242	. . . . . . . . 9 ⊢ ((2 ∈ ℤ ∧ (𝐴 + 1) ∈ ℕ) → 2 ∥ (2↑(𝐴 + 1)))
70	49, 9, 69	sylancr 588	. . . . . . . 8 ⊢ (𝜑 → 2 ∥ (2↑(𝐴 + 1)))
71		n2dvds1 16331	. . . . . . . . . 10 ⊢ ¬ 2 ∥ 1
72	49	a1i 11	. . . . . . . . . . . 12 ⊢ (𝜑 → 2 ∈ ℤ)
73		1zzd 12552	. . . . . . . . . . . 12 ⊢ (𝜑 → 1 ∈ ℤ)
74	72, 18, 73	3jca 1129	. . . . . . . . . . 11 ⊢ (𝜑 → (2 ∈ ℤ ∧ (2↑(𝐴 + 1)) ∈ ℤ ∧ 1 ∈ ℤ))
75		dvdssub2 16264	. . . . . . . . . . 11 ⊢ (((2 ∈ ℤ ∧ (2↑(𝐴 + 1)) ∈ ℤ ∧ 1 ∈ ℤ) ∧ 2 ∥ ((2↑(𝐴 + 1)) − 1)) → (2 ∥ (2↑(𝐴 + 1)) ↔ 2 ∥ 1))
76	74, 75	sylan 581	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 2 ∥ ((2↑(𝐴 + 1)) − 1)) → (2 ∥ (2↑(𝐴 + 1)) ↔ 2 ∥ 1))
77	71, 76	mtbiri 327	. . . . . . . . 9 ⊢ ((𝜑 ∧ 2 ∥ ((2↑(𝐴 + 1)) − 1)) → ¬ 2 ∥ (2↑(𝐴 + 1)))
78	77	ex 412	. . . . . . . 8 ⊢ (𝜑 → (2 ∥ ((2↑(𝐴 + 1)) − 1) → ¬ 2 ∥ (2↑(𝐴 + 1))))
79	70, 78	mt2d 136	. . . . . . 7 ⊢ (𝜑 → ¬ 2 ∥ ((2↑(𝐴 + 1)) − 1))
80		coprm 16675	. . . . . . . 8 ⊢ ((2 ∈ ℙ ∧ ((2↑(𝐴 + 1)) − 1) ∈ ℤ) → (¬ 2 ∥ ((2↑(𝐴 + 1)) − 1) ↔ (2 gcd ((2↑(𝐴 + 1)) − 1)) = 1))
81	44, 20, 80	sylancr 588	. . . . . . 7 ⊢ (𝜑 → (¬ 2 ∥ ((2↑(𝐴 + 1)) − 1) ↔ (2 gcd ((2↑(𝐴 + 1)) − 1)) = 1))
82	79, 81	mpbid 232	. . . . . 6 ⊢ (𝜑 → (2 gcd ((2↑(𝐴 + 1)) − 1)) = 1)
83		rpexp1i 16687	. . . . . . 7 ⊢ ((2 ∈ ℤ ∧ ((2↑(𝐴 + 1)) − 1) ∈ ℤ ∧ (𝐴 + 1) ∈ ℕ₀) → ((2 gcd ((2↑(𝐴 + 1)) − 1)) = 1 → ((2↑(𝐴 + 1)) gcd ((2↑(𝐴 + 1)) − 1)) = 1))
84	49, 20, 5, 83	mp3an2i 1469	. . . . . 6 ⊢ (𝜑 → ((2 gcd ((2↑(𝐴 + 1)) − 1)) = 1 → ((2↑(𝐴 + 1)) gcd ((2↑(𝐴 + 1)) − 1)) = 1))
85	82, 84	mpd 15	. . . . 5 ⊢ (𝜑 → ((2↑(𝐴 + 1)) gcd ((2↑(𝐴 + 1)) − 1)) = 1)
86	68, 85	eqtrd 2772	. . . 4 ⊢ (𝜑 → (((2↑(𝐴 + 1)) − 1) gcd (2↑(𝐴 + 1))) = 1)
87		coprmdvds 16616	. . . . 5 ⊢ ((((2↑(𝐴 + 1)) − 1) ∈ ℤ ∧ (2↑(𝐴 + 1)) ∈ ℤ ∧ 𝐵 ∈ ℤ) → ((((2↑(𝐴 + 1)) − 1) ∥ ((2↑(𝐴 + 1)) · 𝐵) ∧ (((2↑(𝐴 + 1)) − 1) gcd (2↑(𝐴 + 1))) = 1) → ((2↑(𝐴 + 1)) − 1) ∥ 𝐵))
88	20, 18, 45, 87	syl3anc 1374	. . . 4 ⊢ (𝜑 → ((((2↑(𝐴 + 1)) − 1) ∥ ((2↑(𝐴 + 1)) · 𝐵) ∧ (((2↑(𝐴 + 1)) − 1) gcd (2↑(𝐴 + 1))) = 1) → ((2↑(𝐴 + 1)) − 1) ∥ 𝐵))
89	67, 86, 88	mp2and 700	. . 3 ⊢ (𝜑 → ((2↑(𝐴 + 1)) − 1) ∥ 𝐵)
90		nndivdvds 16224	. . . 4 ⊢ ((𝐵 ∈ ℕ ∧ ((2↑(𝐴 + 1)) − 1) ∈ ℕ) → (((2↑(𝐴 + 1)) − 1) ∥ 𝐵 ↔ (𝐵 / ((2↑(𝐴 + 1)) − 1)) ∈ ℕ))
91	22, 17, 90	syl2anc 585	. . 3 ⊢ (𝜑 → (((2↑(𝐴 + 1)) − 1) ∥ 𝐵 ↔ (𝐵 / ((2↑(𝐴 + 1)) − 1)) ∈ ℕ))
92	89, 91	mpbid 232	. 2 ⊢ (𝜑 → (𝐵 / ((2↑(𝐴 + 1)) − 1)) ∈ ℕ)
93	7, 17, 92	3jca 1129	1 ⊢ (𝜑 → ((2↑(𝐴 + 1)) ∈ ℕ ∧ ((2↑(𝐴 + 1)) − 1) ∈ ℕ ∧ (𝐵 / ((2↑(𝐴 + 1)) − 1)) ∈ ℕ))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 206 ∧ wa 395 ∧ w3a 1087 = wceq 1542 ∈ wcel 2114 class class class wbr 5086 (class class class)co 7361 ℂcc 11030 ℝcr 11031 1c1 11033 + caddc 11035 · cmul 11037 < clt 11173 − cmin 11371 / cdiv 11801 ℕcn 12168 2c2 12230 ℕ₀cn0 12431 ℤcz 12518 ↑cexp 14017 ∥ cdvds 16215 gcd cgcd 16457 ℙcprime 16634 σ csgm 27076

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1912 ax-6 1969 ax-7 2010 ax-8 2116 ax-9 2124 ax-10 2147 ax-11 2163 ax-12 2185 ax-ext 2709 ax-rep 5213 ax-sep 5232 ax-nul 5242 ax-pow 5303 ax-pr 5371 ax-un 7683 ax-inf2 9556 ax-cnex 11088 ax-resscn 11089 ax-1cn 11090 ax-icn 11091 ax-addcl 11092 ax-addrcl 11093 ax-mulcl 11094 ax-mulrcl 11095 ax-mulcom 11096 ax-addass 11097 ax-mulass 11098 ax-distr 11099 ax-i2m1 11100 ax-1ne0 11101 ax-1rid 11102 ax-rnegex 11103 ax-rrecex 11104 ax-cnre 11105 ax-pre-lttri 11106 ax-pre-lttrn 11107 ax-pre-ltadd 11108 ax-pre-mulgt0 11109 ax-pre-sup 11110 ax-addf 11111

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1545 df-fal 1555 df-ex 1782 df-nf 1786 df-sb 2069 df-mo 2540 df-eu 2570 df-clab 2716 df-cleq 2729 df-clel 2812 df-nfc 2886 df-ne 2934 df-nel 3038 df-ral 3053 df-rex 3063 df-rmo 3343 df-reu 3344 df-rab 3391 df-v 3432 df-sbc 3730 df-csb 3839 df-dif 3893 df-un 3895 df-in 3897 df-ss 3907 df-pss 3910 df-nul 4275 df-if 4468 df-pw 4544 df-sn 4569 df-pr 4571 df-tp 4573 df-op 4575 df-uni 4852 df-int 4891 df-iun 4936 df-iin 4937 df-br 5087 df-opab 5149 df-mpt 5168 df-tr 5194 df-id 5520 df-eprel 5525 df-po 5533 df-so 5534 df-fr 5578 df-se 5579 df-we 5580 df-xp 5631 df-rel 5632 df-cnv 5633 df-co 5634 df-dm 5635 df-rn 5636 df-res 5637 df-ima 5638 df-pred 6260 df-ord 6321 df-on 6322 df-lim 6323 df-suc 6324 df-iota 6449 df-fun 6495 df-fn 6496 df-f 6497 df-f1 6498 df-fo 6499 df-f1o 6500 df-fv 6501 df-isom 6502 df-riota 7318 df-ov 7364 df-oprab 7365 df-mpo 7366 df-of 7625 df-om 7812 df-1st 7936 df-2nd 7937 df-supp 8105 df-frecs 8225 df-wrecs 8256 df-recs 8305 df-rdg 8343 df-1o 8399 df-2o 8400 df-er 8637 df-map 8769 df-pm 8770 df-ixp 8840 df-en 8888 df-dom 8889 df-sdom 8890 df-fin 8891 df-fsupp 9269 df-fi 9318 df-sup 9349 df-inf 9350 df-oi 9419 df-card 9857 df-pnf 11175 df-mnf 11176 df-xr 11177 df-ltxr 11178 df-le 11179 df-sub 11373 df-neg 11374 df-div 11802 df-nn 12169 df-2 12238 df-3 12239 df-4 12240 df-5 12241 df-6 12242 df-7 12243 df-8 12244 df-9 12245 df-n0 12432 df-z 12519 df-dec 12639 df-uz 12783 df-q 12893 df-rp 12937 df-xneg 13057 df-xadd 13058 df-xmul 13059 df-ioo 13296 df-ioc 13297 df-ico 13298 df-icc 13299 df-fz 13456 df-fzo 13603 df-fl 13745 df-mod 13823 df-seq 13958 df-exp 14018 df-fac 14230 df-bc 14259 df-hash 14287 df-shft 15023 df-cj 15055 df-re 15056 df-im 15057 df-sqrt 15191 df-abs 15192 df-limsup 15427 df-clim 15444 df-rlim 15445 df-sum 15643 df-ef 16026 df-sin 16028 df-cos 16029 df-pi 16031 df-dvds 16216 df-gcd 16458 df-prm 16635 df-pc 16802 df-struct 17111 df-sets 17128 df-slot 17146 df-ndx 17158 df-base 17174 df-ress 17195 df-plusg 17227 df-mulr 17228 df-starv 17229 df-sca 17230 df-vsca 17231 df-ip 17232 df-tset 17233 df-ple 17234 df-ds 17236 df-unif 17237 df-hom 17238 df-cco 17239 df-rest 17379 df-topn 17380 df-0g 17398 df-gsum 17399 df-topgen 17400 df-pt 17401 df-prds 17404 df-xrs 17460 df-qtop 17465 df-imas 17466 df-xps 17468 df-mre 17542 df-mrc 17543 df-acs 17545 df-mgm 18602 df-sgrp 18681 df-mnd 18697 df-submnd 18746 df-mulg 19038 df-cntz 19286 df-cmn 19751 df-psmet 21339 df-xmet 21340 df-met 21341 df-bl 21342 df-mopn 21343 df-fbas 21344 df-fg 21345 df-cnfld 21348 df-top 22872 df-topon 22889 df-topsp 22911 df-bases 22924 df-cld 22997 df-ntr 22998 df-cls 22999 df-nei 23076 df-lp 23114 df-perf 23115 df-cn 23205 df-cnp 23206 df-haus 23293 df-tx 23540 df-hmeo 23733 df-fil 23824 df-fm 23916 df-flim 23917 df-flf 23918 df-xms 24298 df-ms 24299 df-tms 24300 df-cncf 24858 df-limc 25846 df-dv 25847 df-log 26536 df-cxp 26537 df-sgm 27082

This theorem is referenced by: perfectlem2 27210

W3C validator