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Theorem perfectlem1 26721

Description: Lemma for perfect 26723. (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 12281	. . 3 ⊢ 2 ∈ ℕ
2		perfectlem.1	. . . . 5 ⊢ (𝜑 → 𝐴 ∈ ℕ)
3	2	nnnn0d 12528	. . . 4 ⊢ (𝜑 → 𝐴 ∈ ℕ₀)
4		peano2nn0 12508	. . . 4 ⊢ (𝐴 ∈ ℕ₀ → (𝐴 + 1) ∈ ℕ₀)
5	3, 4	syl 17	. . 3 ⊢ (𝜑 → (𝐴 + 1) ∈ ℕ₀)
6		nnexpcl 14036	. . 3 ⊢ ((2 ∈ ℕ ∧ (𝐴 + 1) ∈ ℕ₀) → (2↑(𝐴 + 1)) ∈ ℕ)
7	1, 5, 6	sylancr 587	. 2 ⊢ (𝜑 → (2↑(𝐴 + 1)) ∈ ℕ)
8		2re 12282	. . . 4 ⊢ 2 ∈ ℝ
9	2	peano2nnd 12225	. . . 4 ⊢ (𝜑 → (𝐴 + 1) ∈ ℕ)
10		1lt2 12379	. . . . 5 ⊢ 1 < 2
11	10	a1i 11	. . . 4 ⊢ (𝜑 → 1 < 2)
12		expgt1 14062	. . . 4 ⊢ ((2 ∈ ℝ ∧ (𝐴 + 1) ∈ ℕ ∧ 1 < 2) → 1 < (2↑(𝐴 + 1)))
13	8, 9, 11, 12	mp3an2i 1466	. . 3 ⊢ (𝜑 → 1 < (2↑(𝐴 + 1)))
14		1nn 12219	. . . 4 ⊢ 1 ∈ ℕ
15		nnsub 12252	. . . 4 ⊢ ((1 ∈ ℕ ∧ (2↑(𝐴 + 1)) ∈ ℕ) → (1 < (2↑(𝐴 + 1)) ↔ ((2↑(𝐴 + 1)) − 1) ∈ ℕ))
16	14, 7, 15	sylancr 587	. . 3 ⊢ (𝜑 → (1 < (2↑(𝐴 + 1)) ↔ ((2↑(𝐴 + 1)) − 1) ∈ ℕ))
17	13, 16	mpbid 231	. 2 ⊢ (𝜑 → ((2↑(𝐴 + 1)) − 1) ∈ ℕ)
18	7	nnzd 12581	. . . . . . 7 ⊢ (𝜑 → (2↑(𝐴 + 1)) ∈ ℤ)
19		peano2zm 12601	. . . . . . 7 ⊢ ((2↑(𝐴 + 1)) ∈ ℤ → ((2↑(𝐴 + 1)) − 1) ∈ ℤ)
20	18, 19	syl 17	. . . . . 6 ⊢ (𝜑 → ((2↑(𝐴 + 1)) − 1) ∈ ℤ)
21		1nn0 12484	. . . . . . . 8 ⊢ 1 ∈ ℕ₀
22		perfectlem.2	. . . . . . . 8 ⊢ (𝜑 → 𝐵 ∈ ℕ)
23		sgmnncl 26640	. . . . . . . 8 ⊢ ((1 ∈ ℕ₀ ∧ 𝐵 ∈ ℕ) → (1 σ 𝐵) ∈ ℕ)
24	21, 22, 23	sylancr 587	. . . . . . 7 ⊢ (𝜑 → (1 σ 𝐵) ∈ ℕ)
25	24	nnzd 12581	. . . . . 6 ⊢ (𝜑 → (1 σ 𝐵) ∈ ℤ)
26		dvdsmul1 16217	. . . . . 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 584	. . . . 5 ⊢ (𝜑 → ((2↑(𝐴 + 1)) − 1) ∥ (((2↑(𝐴 + 1)) − 1) · (1 σ 𝐵)))
28		2cn 12283	. . . . . . . . 9 ⊢ 2 ∈ ℂ
29		expp1 14030	. . . . . . . . 9 ⊢ ((2 ∈ ℂ ∧ 𝐴 ∈ ℕ₀) → (2↑(𝐴 + 1)) = ((2↑𝐴) · 2))
30	28, 3, 29	sylancr 587	. . . . . . . 8 ⊢ (𝜑 → (2↑(𝐴 + 1)) = ((2↑𝐴) · 2))
31		nnexpcl 14036	. . . . . . . . . . 11 ⊢ ((2 ∈ ℕ ∧ 𝐴 ∈ ℕ₀) → (2↑𝐴) ∈ ℕ)
32	1, 3, 31	sylancr 587	. . . . . . . . . 10 ⊢ (𝜑 → (2↑𝐴) ∈ ℕ)
33	32	nncnd 12224	. . . . . . . . 9 ⊢ (𝜑 → (2↑𝐴) ∈ ℂ)
34		mulcom 11192	. . . . . . . . 9 ⊢ (((2↑𝐴) ∈ ℂ ∧ 2 ∈ ℂ) → ((2↑𝐴) · 2) = (2 · (2↑𝐴)))
35	33, 28, 34	sylancl 586	. . . . . . . 8 ⊢ (𝜑 → ((2↑𝐴) · 2) = (2 · (2↑𝐴)))
36	30, 35	eqtrd 2772	. . . . . . 7 ⊢ (𝜑 → (2↑(𝐴 + 1)) = (2 · (2↑𝐴)))
37	36	oveq1d 7420	. . . . . 6 ⊢ (𝜑 → ((2↑(𝐴 + 1)) · 𝐵) = ((2 · (2↑𝐴)) · 𝐵))
38	28	a1i 11	. . . . . . 7 ⊢ (𝜑 → 2 ∈ ℂ)
39	22	nncnd 12224	. . . . . . 7 ⊢ (𝜑 → 𝐵 ∈ ℂ)
40	38, 33, 39	mulassd 11233	. . . . . 6 ⊢ (𝜑 → ((2 · (2↑𝐴)) · 𝐵) = (2 · ((2↑𝐴) · 𝐵)))
41		ax-1cn 11164	. . . . . . . . 9 ⊢ 1 ∈ ℂ
42	41	a1i 11	. . . . . . . 8 ⊢ (𝜑 → 1 ∈ ℂ)
43		perfectlem.3	. . . . . . . . . 10 ⊢ (𝜑 → ¬ 2 ∥ 𝐵)
44		2prm 16625	. . . . . . . . . . 11 ⊢ 2 ∈ ℙ
45	22	nnzd 12581	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐵 ∈ ℤ)
46		coprm 16644	. . . . . . . . . . 11 ⊢ ((2 ∈ ℙ ∧ 𝐵 ∈ ℤ) → (¬ 2 ∥ 𝐵 ↔ (2 gcd 𝐵) = 1))
47	44, 45, 46	sylancr 587	. . . . . . . . . 10 ⊢ (𝜑 → (¬ 2 ∥ 𝐵 ↔ (2 gcd 𝐵) = 1))
48	43, 47	mpbid 231	. . . . . . . . 9 ⊢ (𝜑 → (2 gcd 𝐵) = 1)
49		2z 12590	. . . . . . . . . 10 ⊢ 2 ∈ ℤ
50		rpexp1i 16656	. . . . . . . . . 10 ⊢ ((2 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝐴 ∈ ℕ₀) → ((2 gcd 𝐵) = 1 → ((2↑𝐴) gcd 𝐵) = 1))
51	49, 45, 3, 50	mp3an2i 1466	. . . . . . . . 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 26693	. . . . . . . 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 1372	. . . . . . 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 12224	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐴 ∈ ℂ)
57		pncan 11462	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ ℂ ∧ 1 ∈ ℂ) → ((𝐴 + 1) − 1) = 𝐴)
58	56, 41, 57	sylancl 586	. . . . . . . . . . 11 ⊢ (𝜑 → ((𝐴 + 1) − 1) = 𝐴)
59	58	oveq2d 7421	. . . . . . . . . 10 ⊢ (𝜑 → (2↑((𝐴 + 1) − 1)) = (2↑𝐴))
60	59	oveq2d 7421	. . . . . . . . 9 ⊢ (𝜑 → (1 σ (2↑((𝐴 + 1) − 1))) = (1 σ (2↑𝐴)))
61		1sgm2ppw 26692	. . . . . . . . . 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 7420	. . . . . . 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 5175	. . . 4 ⊢ (𝜑 → ((2↑(𝐴 + 1)) − 1) ∥ ((2↑(𝐴 + 1)) · 𝐵))
68	20, 18	gcdcomd 16451	. . . . 5 ⊢ (𝜑 → (((2↑(𝐴 + 1)) − 1) gcd (2↑(𝐴 + 1))) = ((2↑(𝐴 + 1)) gcd ((2↑(𝐴 + 1)) − 1)))
69		iddvdsexp 16219	. . . . . . . . 9 ⊢ ((2 ∈ ℤ ∧ (𝐴 + 1) ∈ ℕ) → 2 ∥ (2↑(𝐴 + 1)))
70	49, 9, 69	sylancr 587	. . . . . . . 8 ⊢ (𝜑 → 2 ∥ (2↑(𝐴 + 1)))
71		n2dvds1 16307	. . . . . . . . . 10 ⊢ ¬ 2 ∥ 1
72	49	a1i 11	. . . . . . . . . . . 12 ⊢ (𝜑 → 2 ∈ ℤ)
73		1zzd 12589	. . . . . . . . . . . 12 ⊢ (𝜑 → 1 ∈ ℤ)
74	72, 18, 73	3jca 1128	. . . . . . . . . . 11 ⊢ (𝜑 → (2 ∈ ℤ ∧ (2↑(𝐴 + 1)) ∈ ℤ ∧ 1 ∈ ℤ))
75		dvdssub2 16240	. . . . . . . . . . 11 ⊢ (((2 ∈ ℤ ∧ (2↑(𝐴 + 1)) ∈ ℤ ∧ 1 ∈ ℤ) ∧ 2 ∥ ((2↑(𝐴 + 1)) − 1)) → (2 ∥ (2↑(𝐴 + 1)) ↔ 2 ∥ 1))
76	74, 75	sylan 580	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 2 ∥ ((2↑(𝐴 + 1)) − 1)) → (2 ∥ (2↑(𝐴 + 1)) ↔ 2 ∥ 1))
77	71, 76	mtbiri 326	. . . . . . . . 9 ⊢ ((𝜑 ∧ 2 ∥ ((2↑(𝐴 + 1)) − 1)) → ¬ 2 ∥ (2↑(𝐴 + 1)))
78	77	ex 413	. . . . . . . 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 16644	. . . . . . . 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 587	. . . . . . 7 ⊢ (𝜑 → (¬ 2 ∥ ((2↑(𝐴 + 1)) − 1) ↔ (2 gcd ((2↑(𝐴 + 1)) − 1)) = 1))
82	79, 81	mpbid 231	. . . . . 6 ⊢ (𝜑 → (2 gcd ((2↑(𝐴 + 1)) − 1)) = 1)
83		rpexp1i 16656	. . . . . . 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 1466	. . . . . 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 16586	. . . . 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 1371	. . . 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 697	. . 3 ⊢ (𝜑 → ((2↑(𝐴 + 1)) − 1) ∥ 𝐵)
90		nndivdvds 16202	. . . 4 ⊢ ((𝐵 ∈ ℕ ∧ ((2↑(𝐴 + 1)) − 1) ∈ ℕ) → (((2↑(𝐴 + 1)) − 1) ∥ 𝐵 ↔ (𝐵 / ((2↑(𝐴 + 1)) − 1)) ∈ ℕ))
91	22, 17, 90	syl2anc 584	. . 3 ⊢ (𝜑 → (((2↑(𝐴 + 1)) − 1) ∥ 𝐵 ↔ (𝐵 / ((2↑(𝐴 + 1)) − 1)) ∈ ℕ))
92	89, 91	mpbid 231	. 2 ⊢ (𝜑 → (𝐵 / ((2↑(𝐴 + 1)) − 1)) ∈ ℕ)
93	7, 17, 92	3jca 1128	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 205 ∧ wa 396 ∧ w3a 1087 = wceq 1541 ∈ wcel 2106 class class class wbr 5147 (class class class)co 7405 ℂcc 11104 ℝcr 11105 1c1 11107 + caddc 11109 · cmul 11111 < clt 11244 − cmin 11440 / cdiv 11867 ℕcn 12208 2c2 12263 ℕ₀cn0 12468 ℤcz 12554 ↑cexp 14023 ∥ cdvds 16193 gcd cgcd 16431 ℙcprime 16604 σ csgm 26589

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 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2703 ax-rep 5284 ax-sep 5298 ax-nul 5305 ax-pow 5362 ax-pr 5426 ax-un 7721 ax-inf2 9632 ax-cnex 11162 ax-resscn 11163 ax-1cn 11164 ax-icn 11165 ax-addcl 11166 ax-addrcl 11167 ax-mulcl 11168 ax-mulrcl 11169 ax-mulcom 11170 ax-addass 11171 ax-mulass 11172 ax-distr 11173 ax-i2m1 11174 ax-1ne0 11175 ax-1rid 11176 ax-rnegex 11177 ax-rrecex 11178 ax-cnre 11179 ax-pre-lttri 11180 ax-pre-lttrn 11181 ax-pre-ltadd 11182 ax-pre-mulgt0 11183 ax-pre-sup 11184 ax-addf 11185 ax-mulf 11186

This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3376 df-reu 3377 df-rab 3433 df-v 3476 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4322 df-if 4528 df-pw 4603 df-sn 4628 df-pr 4630 df-tp 4632 df-op 4634 df-uni 4908 df-int 4950 df-iun 4998 df-iin 4999 df-br 5148 df-opab 5210 df-mpt 5231 df-tr 5265 df-id 5573 df-eprel 5579 df-po 5587 df-so 5588 df-fr 5630 df-se 5631 df-we 5632 df-xp 5681 df-rel 5682 df-cnv 5683 df-co 5684 df-dm 5685 df-rn 5686 df-res 5687 df-ima 5688 df-pred 6297 df-ord 6364 df-on 6365 df-lim 6366 df-suc 6367 df-iota 6492 df-fun 6542 df-fn 6543 df-f 6544 df-f1 6545 df-fo 6546 df-f1o 6547 df-fv 6548 df-isom 6549 df-riota 7361 df-ov 7408 df-oprab 7409 df-mpo 7410 df-of 7666 df-om 7852 df-1st 7971 df-2nd 7972 df-supp 8143 df-frecs 8262 df-wrecs 8293 df-recs 8367 df-rdg 8406 df-1o 8462 df-2o 8463 df-er 8699 df-map 8818 df-pm 8819 df-ixp 8888 df-en 8936 df-dom 8937 df-sdom 8938 df-fin 8939 df-fsupp 9358 df-fi 9402 df-sup 9433 df-inf 9434 df-oi 9501 df-card 9930 df-pnf 11246 df-mnf 11247 df-xr 11248 df-ltxr 11249 df-le 11250 df-sub 11442 df-neg 11443 df-div 11868 df-nn 12209 df-2 12271 df-3 12272 df-4 12273 df-5 12274 df-6 12275 df-7 12276 df-8 12277 df-9 12278 df-n0 12469 df-z 12555 df-dec 12674 df-uz 12819 df-q 12929 df-rp 12971 df-xneg 13088 df-xadd 13089 df-xmul 13090 df-ioo 13324 df-ioc 13325 df-ico 13326 df-icc 13327 df-fz 13481 df-fzo 13624 df-fl 13753 df-mod 13831 df-seq 13963 df-exp 14024 df-fac 14230 df-bc 14259 df-hash 14287 df-shft 15010 df-cj 15042 df-re 15043 df-im 15044 df-sqrt 15178 df-abs 15179 df-limsup 15411 df-clim 15428 df-rlim 15429 df-sum 15629 df-ef 16007 df-sin 16009 df-cos 16010 df-pi 16012 df-dvds 16194 df-gcd 16432 df-prm 16605 df-pc 16766 df-struct 17076 df-sets 17093 df-slot 17111 df-ndx 17123 df-base 17141 df-ress 17170 df-plusg 17206 df-mulr 17207 df-starv 17208 df-sca 17209 df-vsca 17210 df-ip 17211 df-tset 17212 df-ple 17213 df-ds 17215 df-unif 17216 df-hom 17217 df-cco 17218 df-rest 17364 df-topn 17365 df-0g 17383 df-gsum 17384 df-topgen 17385 df-pt 17386 df-prds 17389 df-xrs 17444 df-qtop 17449 df-imas 17450 df-xps 17452 df-mre 17526 df-mrc 17527 df-acs 17529 df-mgm 18557 df-sgrp 18606 df-mnd 18622 df-submnd 18668 df-mulg 18945 df-cntz 19175 df-cmn 19644 df-psmet 20928 df-xmet 20929 df-met 20930 df-bl 20931 df-mopn 20932 df-fbas 20933 df-fg 20934 df-cnfld 20937 df-top 22387 df-topon 22404 df-topsp 22426 df-bases 22440 df-cld 22514 df-ntr 22515 df-cls 22516 df-nei 22593 df-lp 22631 df-perf 22632 df-cn 22722 df-cnp 22723 df-haus 22810 df-tx 23057 df-hmeo 23250 df-fil 23341 df-fm 23433 df-flim 23434 df-flf 23435 df-xms 23817 df-ms 23818 df-tms 23819 df-cncf 24385 df-limc 25374 df-dv 25375 df-log 26056 df-cxp 26057 df-sgm 26595

This theorem is referenced by: perfectlem2 26722

W3C validator