etransclem22 - Mathbox for Glauco Siliprandi

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Theorem etransclem22 47085

Description: The 𝑁-th derivative of 𝐻 is continuous. (Contributed by Glauco Siliprandi, 5-Apr-2020.)

**Hypotheses**
Ref	Expression
etransclem22.s	⊢ (𝜑 → 𝑆 ∈ {ℝ, ℂ})
etransclem22.x	⊢ (𝜑 → 𝑋 ∈ ((TopOpen‘ℂ_fld) ↾_t 𝑆))
etransclem22.p	⊢ (𝜑 → 𝑃 ∈ ℕ)
etransclem22.h	⊢ 𝐻 = (𝑗 ∈ (0...𝑀) ↦ (𝑥 ∈ 𝑋 ↦ ((𝑥 − 𝑗)↑if(𝑗 = 0, (𝑃 − 1), 𝑃))))
etransclem22.J	⊢ (𝜑 → 𝐽 ∈ (0...𝑀))
etransclem22.n	⊢ (𝜑 → 𝑁 ∈ ℕ₀)

**Assertion**
Ref	Expression
etransclem22	⊢ (𝜑 → ((𝑆 D^𝑛 (𝐻‘𝐽))‘𝑁) ∈ (𝑋–cn→ℂ))

Distinct variable groups: 𝑗,𝐽,𝑥 𝑗,𝑀,𝑥 𝑥,𝑁 𝑃,𝑗,𝑥 𝑥,𝑆 𝑗,𝑋,𝑥 𝜑,𝑗,𝑥 Allowed substitution hints: 𝑆(𝑗) 𝐻(𝑥, 𝑗) 𝑁(𝑗)

Proof of Theorem etransclem22 Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		etransclem22.s	. . 3 ⊢ (𝜑 → 𝑆 ∈ {ℝ, ℂ})
2		etransclem22.x	. . 3 ⊢ (𝜑 → 𝑋 ∈ ((TopOpen‘ℂ_fld) ↾_t 𝑆))
3		etransclem22.p	. . 3 ⊢ (𝜑 → 𝑃 ∈ ℕ)
4		etransclem22.h	. . 3 ⊢ 𝐻 = (𝑗 ∈ (0...𝑀) ↦ (𝑥 ∈ 𝑋 ↦ ((𝑥 − 𝑗)↑if(𝑗 = 0, (𝑃 − 1), 𝑃))))
5		etransclem22.J	. . 3 ⊢ (𝜑 → 𝐽 ∈ (0...𝑀))
6		etransclem22.n	. . 3 ⊢ (𝜑 → 𝑁 ∈ ℕ₀)
7	1, 2, 3, 4, 5, 6	etransclem17 47080	. 2 ⊢ (𝜑 → ((𝑆 D^𝑛 (𝐻‘𝐽))‘𝑁) = (𝑥 ∈ 𝑋 ↦ if(if(𝐽 = 0, (𝑃 − 1), 𝑃) < 𝑁, 0, (((!‘if(𝐽 = 0, (𝑃 − 1), 𝑃)) / (!‘(if(𝐽 = 0, (𝑃 − 1), 𝑃) − 𝑁))) · ((𝑥 − 𝐽)↑(if(𝐽 = 0, (𝑃 − 1), 𝑃) − 𝑁))))))
8		simpr 490	. . . . . 6 ⊢ ((𝜑 ∧ if(𝐽 = 0, (𝑃 − 1), 𝑃) < 𝑁) → if(𝐽 = 0, (𝑃 − 1), 𝑃) < 𝑁)
9	8	iftrued 4490	. . . . 5 ⊢ ((𝜑 ∧ if(𝐽 = 0, (𝑃 − 1), 𝑃) < 𝑁) → if(if(𝐽 = 0, (𝑃 − 1), 𝑃) < 𝑁, 0, (((!‘if(𝐽 = 0, (𝑃 − 1), 𝑃)) / (!‘(if(𝐽 = 0, (𝑃 − 1), 𝑃) − 𝑁))) · ((𝑥 − 𝐽)↑(if(𝐽 = 0, (𝑃 − 1), 𝑃) − 𝑁)))) = 0)
10	9	mpteq2dv 5199	. . . 4 ⊢ ((𝜑 ∧ if(𝐽 = 0, (𝑃 − 1), 𝑃) < 𝑁) → (𝑥 ∈ 𝑋 ↦ if(if(𝐽 = 0, (𝑃 − 1), 𝑃) < 𝑁, 0, (((!‘if(𝐽 = 0, (𝑃 − 1), 𝑃)) / (!‘(if(𝐽 = 0, (𝑃 − 1), 𝑃) − 𝑁))) · ((𝑥 − 𝐽)↑(if(𝐽 = 0, (𝑃 − 1), 𝑃) − 𝑁))))) = (𝑥 ∈ 𝑋 ↦ 0))
11	1, 2	dvdmsscn 46765	. . . . . 6 ⊢ (𝜑 → 𝑋 ⊆ ℂ)
12		0cnd 11224	. . . . . 6 ⊢ (𝜑 → 0 ∈ ℂ)
13		ssid 3953	. . . . . . 7 ⊢ ℂ ⊆ ℂ
14	13	a1i 11	. . . . . 6 ⊢ (𝜑 → ℂ ⊆ ℂ)
15	11, 12, 14	constcncfg 46701	. . . . 5 ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ 0) ∈ (𝑋–cn→ℂ))
16	15	adantr 486	. . . 4 ⊢ ((𝜑 ∧ if(𝐽 = 0, (𝑃 − 1), 𝑃) < 𝑁) → (𝑥 ∈ 𝑋 ↦ 0) ∈ (𝑋–cn→ℂ))
17	10, 16	eqeltrd 2860	. . 3 ⊢ ((𝜑 ∧ if(𝐽 = 0, (𝑃 − 1), 𝑃) < 𝑁) → (𝑥 ∈ 𝑋 ↦ if(if(𝐽 = 0, (𝑃 − 1), 𝑃) < 𝑁, 0, (((!‘if(𝐽 = 0, (𝑃 − 1), 𝑃)) / (!‘(if(𝐽 = 0, (𝑃 − 1), 𝑃) − 𝑁))) · ((𝑥 − 𝐽)↑(if(𝐽 = 0, (𝑃 − 1), 𝑃) − 𝑁))))) ∈ (𝑋–cn→ℂ))
18		simpr 490	. . . . . 6 ⊢ ((𝜑 ∧ ¬ if(𝐽 = 0, (𝑃 − 1), 𝑃) < 𝑁) → ¬ if(𝐽 = 0, (𝑃 − 1), 𝑃) < 𝑁)
19	18	iffalsed 4493	. . . . 5 ⊢ ((𝜑 ∧ ¬ if(𝐽 = 0, (𝑃 − 1), 𝑃) < 𝑁) → if(if(𝐽 = 0, (𝑃 − 1), 𝑃) < 𝑁, 0, (((!‘if(𝐽 = 0, (𝑃 − 1), 𝑃)) / (!‘(if(𝐽 = 0, (𝑃 − 1), 𝑃) − 𝑁))) · ((𝑥 − 𝐽)↑(if(𝐽 = 0, (𝑃 − 1), 𝑃) − 𝑁)))) = (((!‘if(𝐽 = 0, (𝑃 − 1), 𝑃)) / (!‘(if(𝐽 = 0, (𝑃 − 1), 𝑃) − 𝑁))) · ((𝑥 − 𝐽)↑(if(𝐽 = 0, (𝑃 − 1), 𝑃) − 𝑁))))
20	19	mpteq2dv 5199	. . . 4 ⊢ ((𝜑 ∧ ¬ if(𝐽 = 0, (𝑃 − 1), 𝑃) < 𝑁) → (𝑥 ∈ 𝑋 ↦ if(if(𝐽 = 0, (𝑃 − 1), 𝑃) < 𝑁, 0, (((!‘if(𝐽 = 0, (𝑃 − 1), 𝑃)) / (!‘(if(𝐽 = 0, (𝑃 − 1), 𝑃) − 𝑁))) · ((𝑥 − 𝐽)↑(if(𝐽 = 0, (𝑃 − 1), 𝑃) − 𝑁))))) = (𝑥 ∈ 𝑋 ↦ (((!‘if(𝐽 = 0, (𝑃 − 1), 𝑃)) / (!‘(if(𝐽 = 0, (𝑃 − 1), 𝑃) − 𝑁))) · ((𝑥 − 𝐽)↑(if(𝐽 = 0, (𝑃 − 1), 𝑃) − 𝑁)))))
21		nfv 1947	. . . . 5 ⊢ Ⅎ𝑥(𝜑 ∧ ¬ if(𝐽 = 0, (𝑃 − 1), 𝑃) < 𝑁)
22	11, 14	idcncfg 46702	. . . . . . 7 ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ 𝑥) ∈ (𝑋–cn→ℂ))
23	5	elfzelzd 13580	. . . . . . . . 9 ⊢ (𝜑 → 𝐽 ∈ ℤ)
24	23	zcnd 12727	. . . . . . . 8 ⊢ (𝜑 → 𝐽 ∈ ℂ)
25	11, 24, 14	constcncfg 46701	. . . . . . 7 ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ 𝐽) ∈ (𝑋–cn→ℂ))
26	22, 25	subcncf 25674	. . . . . 6 ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ (𝑥 − 𝐽)) ∈ (𝑋–cn→ℂ))
27	26	adantr 486	. . . . 5 ⊢ ((𝜑 ∧ ¬ if(𝐽 = 0, (𝑃 − 1), 𝑃) < 𝑁) → (𝑥 ∈ 𝑋 ↦ (𝑥 − 𝐽)) ∈ (𝑋–cn→ℂ))
28	13	a1i 11	. . . . . . 7 ⊢ ((𝜑 ∧ ¬ if(𝐽 = 0, (𝑃 − 1), 𝑃) < 𝑁) → ℂ ⊆ ℂ)
29		nnm1nn0 12570	. . . . . . . . . . . . 13 ⊢ (𝑃 ∈ ℕ → (𝑃 − 1) ∈ ℕ₀)
30	3, 29	syl 18	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝑃 − 1) ∈ ℕ₀)
31	3	nnnn0d 12590	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑃 ∈ ℕ₀)
32	30, 31	ifcld 4529	. . . . . . . . . . 11 ⊢ (𝜑 → if(𝐽 = 0, (𝑃 − 1), 𝑃) ∈ ℕ₀)
33	32	faccld 14349	. . . . . . . . . 10 ⊢ (𝜑 → (!‘if(𝐽 = 0, (𝑃 − 1), 𝑃)) ∈ ℕ)
34	33	nncnd 12274	. . . . . . . . 9 ⊢ (𝜑 → (!‘if(𝐽 = 0, (𝑃 − 1), 𝑃)) ∈ ℂ)
35	34	adantr 486	. . . . . . . 8 ⊢ ((𝜑 ∧ ¬ if(𝐽 = 0, (𝑃 − 1), 𝑃) < 𝑁) → (!‘if(𝐽 = 0, (𝑃 − 1), 𝑃)) ∈ ℂ)
36	32	nn0zd 12641	. . . . . . . . . . . . 13 ⊢ (𝜑 → if(𝐽 = 0, (𝑃 − 1), 𝑃) ∈ ℤ)
37	6	nn0zd 12641	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝑁 ∈ ℤ)
38	36, 37	zsubcld 12731	. . . . . . . . . . . 12 ⊢ (𝜑 → (if(𝐽 = 0, (𝑃 − 1), 𝑃) − 𝑁) ∈ ℤ)
39	38	adantr 486	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ ¬ if(𝐽 = 0, (𝑃 − 1), 𝑃) < 𝑁) → (if(𝐽 = 0, (𝑃 − 1), 𝑃) − 𝑁) ∈ ℤ)
40	6	nn0red 12591	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝑁 ∈ ℝ)
41	40	adantr 486	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ ¬ if(𝐽 = 0, (𝑃 − 1), 𝑃) < 𝑁) → 𝑁 ∈ ℝ)
42	32	nn0red 12591	. . . . . . . . . . . . . 14 ⊢ (𝜑 → if(𝐽 = 0, (𝑃 − 1), 𝑃) ∈ ℝ)
43	42	adantr 486	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ ¬ if(𝐽 = 0, (𝑃 − 1), 𝑃) < 𝑁) → if(𝐽 = 0, (𝑃 − 1), 𝑃) ∈ ℝ)
44	41, 43, 18	nltled 11385	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ ¬ if(𝐽 = 0, (𝑃 − 1), 𝑃) < 𝑁) → 𝑁 ≤ if(𝐽 = 0, (𝑃 − 1), 𝑃))
45	43, 41	subge0d 11829	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ ¬ if(𝐽 = 0, (𝑃 − 1), 𝑃) < 𝑁) → (0 ≤ (if(𝐽 = 0, (𝑃 − 1), 𝑃) − 𝑁) ↔ 𝑁 ≤ if(𝐽 = 0, (𝑃 − 1), 𝑃)))
46	44, 45	mpbird 260	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ ¬ if(𝐽 = 0, (𝑃 − 1), 𝑃) < 𝑁) → 0 ≤ (if(𝐽 = 0, (𝑃 − 1), 𝑃) − 𝑁))
47		elnn0z 12629	. . . . . . . . . . 11 ⊢ ((if(𝐽 = 0, (𝑃 − 1), 𝑃) − 𝑁) ∈ ℕ₀ ↔ ((if(𝐽 = 0, (𝑃 − 1), 𝑃) − 𝑁) ∈ ℤ ∧ 0 ≤ (if(𝐽 = 0, (𝑃 − 1), 𝑃) − 𝑁)))
48	39, 46, 47	sylanbrc 595	. . . . . . . . . 10 ⊢ ((𝜑 ∧ ¬ if(𝐽 = 0, (𝑃 − 1), 𝑃) < 𝑁) → (if(𝐽 = 0, (𝑃 − 1), 𝑃) − 𝑁) ∈ ℕ₀)
49	48	faccld 14349	. . . . . . . . 9 ⊢ ((𝜑 ∧ ¬ if(𝐽 = 0, (𝑃 − 1), 𝑃) < 𝑁) → (!‘(if(𝐽 = 0, (𝑃 − 1), 𝑃) − 𝑁)) ∈ ℕ)
50	49	nncnd 12274	. . . . . . . 8 ⊢ ((𝜑 ∧ ¬ if(𝐽 = 0, (𝑃 − 1), 𝑃) < 𝑁) → (!‘(if(𝐽 = 0, (𝑃 − 1), 𝑃) − 𝑁)) ∈ ℂ)
51	49	nnne0d 12311	. . . . . . . 8 ⊢ ((𝜑 ∧ ¬ if(𝐽 = 0, (𝑃 − 1), 𝑃) < 𝑁) → (!‘(if(𝐽 = 0, (𝑃 − 1), 𝑃) − 𝑁)) ≠ 0)
52	35, 50, 51	divcld 12016	. . . . . . 7 ⊢ ((𝜑 ∧ ¬ if(𝐽 = 0, (𝑃 − 1), 𝑃) < 𝑁) → ((!‘if(𝐽 = 0, (𝑃 − 1), 𝑃)) / (!‘(if(𝐽 = 0, (𝑃 − 1), 𝑃) − 𝑁))) ∈ ℂ)
53	28, 52, 28	constcncfg 46701	. . . . . 6 ⊢ ((𝜑 ∧ ¬ if(𝐽 = 0, (𝑃 − 1), 𝑃) < 𝑁) → (𝑦 ∈ ℂ ↦ ((!‘if(𝐽 = 0, (𝑃 − 1), 𝑃)) / (!‘(if(𝐽 = 0, (𝑃 − 1), 𝑃) − 𝑁)))) ∈ (ℂ–cn→ℂ))
54		expcncf 25155	. . . . . . 7 ⊢ ((if(𝐽 = 0, (𝑃 − 1), 𝑃) − 𝑁) ∈ ℕ₀ → (𝑦 ∈ ℂ ↦ (𝑦↑(if(𝐽 = 0, (𝑃 − 1), 𝑃) − 𝑁))) ∈ (ℂ–cn→ℂ))
55	48, 54	syl 18	. . . . . 6 ⊢ ((𝜑 ∧ ¬ if(𝐽 = 0, (𝑃 − 1), 𝑃) < 𝑁) → (𝑦 ∈ ℂ ↦ (𝑦↑(if(𝐽 = 0, (𝑃 − 1), 𝑃) − 𝑁))) ∈ (ℂ–cn→ℂ))
56	53, 55	mulcncf 25675	. . . . 5 ⊢ ((𝜑 ∧ ¬ if(𝐽 = 0, (𝑃 − 1), 𝑃) < 𝑁) → (𝑦 ∈ ℂ ↦ (((!‘if(𝐽 = 0, (𝑃 − 1), 𝑃)) / (!‘(if(𝐽 = 0, (𝑃 − 1), 𝑃) − 𝑁))) · (𝑦↑(if(𝐽 = 0, (𝑃 − 1), 𝑃) − 𝑁)))) ∈ (ℂ–cn→ℂ))
57		oveq1 7421	. . . . . 6 ⊢ (𝑦 = (𝑥 − 𝐽) → (𝑦↑(if(𝐽 = 0, (𝑃 − 1), 𝑃) − 𝑁)) = ((𝑥 − 𝐽)↑(if(𝐽 = 0, (𝑃 − 1), 𝑃) − 𝑁)))
58	57	oveq2d 7430	. . . . 5 ⊢ (𝑦 = (𝑥 − 𝐽) → (((!‘if(𝐽 = 0, (𝑃 − 1), 𝑃)) / (!‘(if(𝐽 = 0, (𝑃 − 1), 𝑃) − 𝑁))) · (𝑦↑(if(𝐽 = 0, (𝑃 − 1), 𝑃) − 𝑁))) = (((!‘if(𝐽 = 0, (𝑃 − 1), 𝑃)) / (!‘(if(𝐽 = 0, (𝑃 − 1), 𝑃) − 𝑁))) · ((𝑥 − 𝐽)↑(if(𝐽 = 0, (𝑃 − 1), 𝑃) − 𝑁))))
59	21, 27, 56, 28, 58	cncfcompt2 25137	. . . 4 ⊢ ((𝜑 ∧ ¬ if(𝐽 = 0, (𝑃 − 1), 𝑃) < 𝑁) → (𝑥 ∈ 𝑋 ↦ (((!‘if(𝐽 = 0, (𝑃 − 1), 𝑃)) / (!‘(if(𝐽 = 0, (𝑃 − 1), 𝑃) − 𝑁))) · ((𝑥 − 𝐽)↑(if(𝐽 = 0, (𝑃 − 1), 𝑃) − 𝑁)))) ∈ (𝑋–cn→ℂ))
60	20, 59	eqeltrd 2860	. . 3 ⊢ ((𝜑 ∧ ¬ if(𝐽 = 0, (𝑃 − 1), 𝑃) < 𝑁) → (𝑥 ∈ 𝑋 ↦ if(if(𝐽 = 0, (𝑃 − 1), 𝑃) < 𝑁, 0, (((!‘if(𝐽 = 0, (𝑃 − 1), 𝑃)) / (!‘(if(𝐽 = 0, (𝑃 − 1), 𝑃) − 𝑁))) · ((𝑥 − 𝐽)↑(if(𝐽 = 0, (𝑃 − 1), 𝑃) − 𝑁))))) ∈ (𝑋–cn→ℂ))
61	17, 60	pm2.61dan 825	. 2 ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ if(if(𝐽 = 0, (𝑃 − 1), 𝑃) < 𝑁, 0, (((!‘if(𝐽 = 0, (𝑃 − 1), 𝑃)) / (!‘(if(𝐽 = 0, (𝑃 − 1), 𝑃) − 𝑁))) · ((𝑥 − 𝐽)↑(if(𝐽 = 0, (𝑃 − 1), 𝑃) − 𝑁))))) ∈ (𝑋–cn→ℂ))
62	7, 61	eqeltrd 2860	1 ⊢ (𝜑 → ((𝑆 D^𝑛 (𝐻‘𝐽))‘𝑁) ∈ (𝑋–cn→ℂ))

Colors of variables: wff setvar class

This proof depends on syntax axioms: ¬ wn 3 → wi 4 ∧ wa 401 = wceq 1570 ∈ wcel 2145 ⊆ wss 3899 ifcif 4482 {cpr 4586 class class class wbr 5103 ↦ cmpt 5186 ‘cfv 6533 (class class class)co 7414 ℂcc 11123 ℝcr 11124 0cc0 11125 1c1 11126 · cmul 11130 < clt 11268 ≤ cle 11269 − cmin 11466 / cdiv 11896 ℕcn 12258 ℕ₀cn0 12529 ℤcz 12616 ...cfz 13562 ↑cexp 14126 !cfa 14338 ↾_t crest 17506 TopOpenctopn 17507 ℂ_fldccnfld 21586 –cn→ccncf 25105 D^𝑛 cdvn 26092

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1828 ax-4 1842 ax-5 1943 ax-6 2000 ax-7 2041 ax-8 2147 ax-9 2155 ax-10 2178 ax-11 2194 ax-12 2213 ax-ext 2732 ax-rep 5232 ax-sep 5251 ax-nul 5263 ax-pow 5330 ax-pr 5398 ax-un 7737 ax-inf2 9621 ax-cnex 11181 ax-resscn 11182 ax-1cn 11183 ax-icn 11184 ax-addcl 11185 ax-addrcl 11186 ax-mulcl 11187 ax-mulrcl 11188 ax-mulcom 11189 ax-addass 11190 ax-mulass 11191 ax-distr 11192 ax-i2m1 11193 ax-1ne0 11194 ax-1rid 11195 ax-rnegex 11196 ax-rrecex 11197 ax-cnre 11198 ax-pre-lttri 11199 ax-pre-lttrn 11200 ax-pre-ltadd 11201 ax-pre-mulgt0 11202 ax-pre-sup 11203 ax-addf 11204

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1813 df-nf 1817 df-sb 2100 df-mo 2564 df-eu 2594 df-clab 2739 df-cleq 2752 df-clel 2835 df-nfc 2909 df-ne 2956 df-nel 3062 df-ral 3077 df-rex 3087 df-rmo 3365 df-reu 3366 df-rab 3413 df-v 3452 df-sbc 3740 df-csb 3848 df-dif 3902 df-un 3904 df-in 3906 df-ss 3916 df-pss 3919 df-nul 4280 df-if 4483 df-pw 4559 df-sn 4585 df-pr 4587 df-tp 4589 df-op 4591 df-uni 4868 df-int 4908 df-iun 4953 df-iin 4954 df-br 5104 df-opab 5168 df-mpt 5187 df-tr 5213 df-id 5550 df-eprel 5555 df-po 5563 df-so 5564 df-fr 5608 df-se 5609 df-we 5610 df-xp 5661 df-rel 5662 df-cnv 5663 df-co 5664 df-dm 5665 df-rn 5666 df-res 5667 df-ima 5668 df-pred 6299 df-ord 6360 df-on 6361 df-lim 6362 df-suc 6363 df-iota 6489 df-fun 6535 df-fn 6536 df-f 6537 df-f1 6538 df-fo 6539 df-f1o 6540 df-fv 6541 df-isom 6542 df-riota 7371 df-ov 7417 df-oprab 7418 df-mpo 7419 df-of 7679 df-om 7864 df-1st 7987 df-2nd 7988 df-supp 8160 df-frecs 8281 df-wrecs 8312 df-recs 8361 df-rdg 8400 df-1o 8456 df-2o 8457 df-er 8697 df-map 8829 df-pm 8830 df-ixp 8906 df-en 8954 df-dom 8955 df-sdom 8956 df-fin 8957 df-fsupp 9333 df-fi 9382 df-sup 9413 df-inf 9414 df-oi 9483 df-card 9945 df-pnf 11270 df-mnf 11271 df-xr 11272 df-ltxr 11273 df-le 11274 df-sub 11468 df-neg 11469 df-div 11897 df-nn 12259 df-2 12328 df-3 12329 df-4 12330 df-5 12331 df-6 12332 df-7 12333 df-8 12334 df-9 12335 df-n0 12530 df-z 12617 df-dec 12738 df-uz 12889 df-q 12999 df-rp 13044 df-xneg 13164 df-xadd 13165 df-xmul 13166 df-icc 13406 df-fz 13563 df-fzo 13711 df-seq 14067 df-exp 14127 df-fac 14339 df-hash 14396 df-cj 15187 df-re 15188 df-im 15189 df-sqrt 15323 df-abs 15324 df-struct 17240 df-sets 17257 df-slot 17275 df-ndx 17287 df-base 17303 df-ress 17324 df-plusg 17356 df-mulr 17357 df-starv 17358 df-sca 17359 df-vsca 17360 df-ip 17361 df-tset 17362 df-ple 17363 df-ds 17365 df-unif 17366 df-hom 17367 df-cco 17368 df-rest 17508 df-topn 17509 df-0g 17527 df-gsum 17528 df-topgen 17529 df-pt 17530 df-prds 17533 df-xrs 17589 df-qtop 17594 df-imas 17595 df-xps 17597 df-mre 17671 df-mrc 17672 df-acs 17674 df-mgm 18731 df-sgrp 18822 df-mnd 18838 df-submnd 18893 df-mulg 19192 df-cntz 19445 df-cmn 19910 df-psmet 21578 df-xmet 21579 df-met 21580 df-bl 21581 df-mopn 21582 df-fbas 21583 df-fg 21584 df-cnfld 21587 df-top 23120 df-topon 23137 df-topsp 23159 df-bases 23172 df-cld 23245 df-ntr 23246 df-cls 23247 df-nei 23324 df-lp 23362 df-perf 23363 df-cn 23453 df-cnp 23454 df-haus 23541 df-tx 23789 df-hmeo 23982 df-fil 24073 df-fm 24165 df-flim 24166 df-flf 24167 df-xms 24547 df-ms 24548 df-tms 24549 df-cncf 25107 df-limc 26094 df-dv 26095 df-dvn 26096

This theorem is used by: etransclem34 47097

W3C validator