etransclem22 - Mathbox for Glauco Siliprandi

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Theorem etransclem22 45703

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 45698	. 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 483	. . . . . 6 ⊢ ((𝜑 ∧ if(𝐽 = 0, (𝑃 − 1), 𝑃) < 𝑁) → if(𝐽 = 0, (𝑃 − 1), 𝑃) < 𝑁)
9	8	iftrued 4533	. . . . 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 5246	. . . 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 45383	. . . . . 6 ⊢ (𝜑 → 𝑋 ⊆ ℂ)
12		0cnd 11232	. . . . . 6 ⊢ (𝜑 → 0 ∈ ℂ)
13		ssid 3996	. . . . . . 7 ⊢ ℂ ⊆ ℂ
14	13	a1i 11	. . . . . 6 ⊢ (𝜑 → ℂ ⊆ ℂ)
15	11, 12, 14	constcncfg 45319	. . . . 5 ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ 0) ∈ (𝑋–cn→ℂ))
16	15	adantr 479	. . . 4 ⊢ ((𝜑 ∧ if(𝐽 = 0, (𝑃 − 1), 𝑃) < 𝑁) → (𝑥 ∈ 𝑋 ↦ 0) ∈ (𝑋–cn→ℂ))
17	10, 16	eqeltrd 2825	. . 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 483	. . . . . 6 ⊢ ((𝜑 ∧ ¬ if(𝐽 = 0, (𝑃 − 1), 𝑃) < 𝑁) → ¬ if(𝐽 = 0, (𝑃 − 1), 𝑃) < 𝑁)
19	18	iffalsed 4536	. . . . 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 5246	. . . 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 1909	. . . . 5 ⊢ Ⅎ𝑥(𝜑 ∧ ¬ if(𝐽 = 0, (𝑃 − 1), 𝑃) < 𝑁)
22	11, 14	idcncfg 45320	. . . . . . 7 ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ 𝑥) ∈ (𝑋–cn→ℂ))
23	5	elfzelzd 13529	. . . . . . . . 9 ⊢ (𝜑 → 𝐽 ∈ ℤ)
24	23	zcnd 12692	. . . . . . . 8 ⊢ (𝜑 → 𝐽 ∈ ℂ)
25	11, 24, 14	constcncfg 45319	. . . . . . 7 ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ 𝐽) ∈ (𝑋–cn→ℂ))
26	22, 25	subcncf 25386	. . . . . 6 ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ (𝑥 − 𝐽)) ∈ (𝑋–cn→ℂ))
27	26	adantr 479	. . . . 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 12538	. . . . . . . . . . . . 13 ⊢ (𝑃 ∈ ℕ → (𝑃 − 1) ∈ ℕ₀)
30	3, 29	syl 17	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝑃 − 1) ∈ ℕ₀)
31	3	nnnn0d 12557	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑃 ∈ ℕ₀)
32	30, 31	ifcld 4571	. . . . . . . . . . 11 ⊢ (𝜑 → if(𝐽 = 0, (𝑃 − 1), 𝑃) ∈ ℕ₀)
33	32	faccld 14270	. . . . . . . . . 10 ⊢ (𝜑 → (!‘if(𝐽 = 0, (𝑃 − 1), 𝑃)) ∈ ℕ)
34	33	nncnd 12253	. . . . . . . . 9 ⊢ (𝜑 → (!‘if(𝐽 = 0, (𝑃 − 1), 𝑃)) ∈ ℂ)
35	34	adantr 479	. . . . . . . 8 ⊢ ((𝜑 ∧ ¬ if(𝐽 = 0, (𝑃 − 1), 𝑃) < 𝑁) → (!‘if(𝐽 = 0, (𝑃 − 1), 𝑃)) ∈ ℂ)
36	32	nn0zd 12609	. . . . . . . . . . . . 13 ⊢ (𝜑 → if(𝐽 = 0, (𝑃 − 1), 𝑃) ∈ ℤ)
37	6	nn0zd 12609	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝑁 ∈ ℤ)
38	36, 37	zsubcld 12696	. . . . . . . . . . . 12 ⊢ (𝜑 → (if(𝐽 = 0, (𝑃 − 1), 𝑃) − 𝑁) ∈ ℤ)
39	38	adantr 479	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ ¬ if(𝐽 = 0, (𝑃 − 1), 𝑃) < 𝑁) → (if(𝐽 = 0, (𝑃 − 1), 𝑃) − 𝑁) ∈ ℤ)
40	6	nn0red 12558	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝑁 ∈ ℝ)
41	40	adantr 479	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ ¬ if(𝐽 = 0, (𝑃 − 1), 𝑃) < 𝑁) → 𝑁 ∈ ℝ)
42	32	nn0red 12558	. . . . . . . . . . . . . 14 ⊢ (𝜑 → if(𝐽 = 0, (𝑃 − 1), 𝑃) ∈ ℝ)
43	42	adantr 479	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ ¬ if(𝐽 = 0, (𝑃 − 1), 𝑃) < 𝑁) → if(𝐽 = 0, (𝑃 − 1), 𝑃) ∈ ℝ)
44	41, 43, 18	nltled 11389	. . . . . . . . . . . 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 256	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ ¬ if(𝐽 = 0, (𝑃 − 1), 𝑃) < 𝑁) → 0 ≤ (if(𝐽 = 0, (𝑃 − 1), 𝑃) − 𝑁))
47		elnn0z 12596	. . . . . . . . . . 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 581	. . . . . . . . . 10 ⊢ ((𝜑 ∧ ¬ if(𝐽 = 0, (𝑃 − 1), 𝑃) < 𝑁) → (if(𝐽 = 0, (𝑃 − 1), 𝑃) − 𝑁) ∈ ℕ₀)
49	48	faccld 14270	. . . . . . . . 9 ⊢ ((𝜑 ∧ ¬ if(𝐽 = 0, (𝑃 − 1), 𝑃) < 𝑁) → (!‘(if(𝐽 = 0, (𝑃 − 1), 𝑃) − 𝑁)) ∈ ℕ)
50	49	nncnd 12253	. . . . . . . 8 ⊢ ((𝜑 ∧ ¬ if(𝐽 = 0, (𝑃 − 1), 𝑃) < 𝑁) → (!‘(if(𝐽 = 0, (𝑃 − 1), 𝑃) − 𝑁)) ∈ ℂ)
51	49	nnne0d 12287	. . . . . . . 8 ⊢ ((𝜑 ∧ ¬ if(𝐽 = 0, (𝑃 − 1), 𝑃) < 𝑁) → (!‘(if(𝐽 = 0, (𝑃 − 1), 𝑃) − 𝑁)) ≠ 0)
52	35, 50, 51	divcld 12015	. . . . . . 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 45319	. . . . . 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 24860	. . . . . . 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 17	. . . . . 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 25387	. . . . 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 7420	. . . . . 6 ⊢ (𝑦 = (𝑥 − 𝐽) → (𝑦↑(if(𝐽 = 0, (𝑃 − 1), 𝑃) − 𝑁)) = ((𝑥 − 𝐽)↑(if(𝐽 = 0, (𝑃 − 1), 𝑃) − 𝑁)))
58	57	oveq2d 7429	. . . . 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 24841	. . . 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 2825	. . 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 811	. 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 2825	1 ⊢ (𝜑 → ((𝑆 D^𝑛 (𝐻‘𝐽))‘𝑁) ∈ (𝑋–cn→ℂ))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ∧ wa 394 = wceq 1533 ∈ wcel 2098 ⊆ wss 3941 ifcif 4525 {cpr 4627 class class class wbr 5144 ↦ cmpt 5227 ‘cfv 6543 (class class class)co 7413 ℂcc 11131 ℝcr 11132 0cc0 11133 1c1 11134 · cmul 11138 < clt 11273 ≤ cle 11274 − cmin 11469 / cdiv 11896 ℕcn 12237 ℕ₀cn0 12497 ℤcz 12583 ...cfz 13511 ↑cexp 14053 !cfa 14259 ↾_t crest 17396 TopOpenctopn 17397 ℂ_fldccnfld 21278 –cn→ccncf 24809 D^𝑛 cdvn 25806

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2696 ax-rep 5281 ax-sep 5295 ax-nul 5302 ax-pow 5360 ax-pr 5424 ax-un 7735 ax-inf2 9659 ax-cnex 11189 ax-resscn 11190 ax-1cn 11191 ax-icn 11192 ax-addcl 11193 ax-addrcl 11194 ax-mulcl 11195 ax-mulrcl 11196 ax-mulcom 11197 ax-addass 11198 ax-mulass 11199 ax-distr 11200 ax-i2m1 11201 ax-1ne0 11202 ax-1rid 11203 ax-rnegex 11204 ax-rrecex 11205 ax-cnre 11206 ax-pre-lttri 11207 ax-pre-lttrn 11208 ax-pre-ltadd 11209 ax-pre-mulgt0 11210 ax-pre-sup 11211 ax-addf 11212

This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2931 df-nel 3037 df-ral 3052 df-rex 3061 df-rmo 3364 df-reu 3365 df-rab 3420 df-v 3465 df-sbc 3771 df-csb 3887 df-dif 3944 df-un 3946 df-in 3948 df-ss 3958 df-pss 3961 df-nul 4320 df-if 4526 df-pw 4601 df-sn 4626 df-pr 4628 df-tp 4630 df-op 4632 df-uni 4905 df-int 4946 df-iun 4994 df-iin 4995 df-br 5145 df-opab 5207 df-mpt 5228 df-tr 5262 df-id 5571 df-eprel 5577 df-po 5585 df-so 5586 df-fr 5628 df-se 5629 df-we 5630 df-xp 5679 df-rel 5680 df-cnv 5681 df-co 5682 df-dm 5683 df-rn 5684 df-res 5685 df-ima 5686 df-pred 6301 df-ord 6368 df-on 6369 df-lim 6370 df-suc 6371 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-isom 6552 df-riota 7369 df-ov 7416 df-oprab 7417 df-mpo 7418 df-of 7679 df-om 7866 df-1st 7987 df-2nd 7988 df-supp 8159 df-frecs 8280 df-wrecs 8311 df-recs 8385 df-rdg 8424 df-1o 8480 df-2o 8481 df-er 8718 df-map 8840 df-pm 8841 df-ixp 8910 df-en 8958 df-dom 8959 df-sdom 8960 df-fin 8961 df-fsupp 9381 df-fi 9429 df-sup 9460 df-inf 9461 df-oi 9528 df-card 9957 df-pnf 11275 df-mnf 11276 df-xr 11277 df-ltxr 11278 df-le 11279 df-sub 11471 df-neg 11472 df-div 11897 df-nn 12238 df-2 12300 df-3 12301 df-4 12302 df-5 12303 df-6 12304 df-7 12305 df-8 12306 df-9 12307 df-n0 12498 df-z 12584 df-dec 12703 df-uz 12848 df-q 12958 df-rp 13002 df-xneg 13119 df-xadd 13120 df-xmul 13121 df-icc 13358 df-fz 13512 df-fzo 13655 df-seq 13994 df-exp 14054 df-fac 14260 df-hash 14317 df-cj 15073 df-re 15074 df-im 15075 df-sqrt 15209 df-abs 15210 df-struct 17110 df-sets 17127 df-slot 17145 df-ndx 17157 df-base 17175 df-ress 17204 df-plusg 17240 df-mulr 17241 df-starv 17242 df-sca 17243 df-vsca 17244 df-ip 17245 df-tset 17246 df-ple 17247 df-ds 17249 df-unif 17250 df-hom 17251 df-cco 17252 df-rest 17398 df-topn 17399 df-0g 17417 df-gsum 17418 df-topgen 17419 df-pt 17420 df-prds 17423 df-xrs 17478 df-qtop 17483 df-imas 17484 df-xps 17486 df-mre 17560 df-mrc 17561 df-acs 17563 df-mgm 18594 df-sgrp 18673 df-mnd 18689 df-submnd 18735 df-mulg 19023 df-cntz 19267 df-cmn 19736 df-psmet 21270 df-xmet 21271 df-met 21272 df-bl 21273 df-mopn 21274 df-fbas 21275 df-fg 21276 df-cnfld 21279 df-top 22809 df-topon 22826 df-topsp 22848 df-bases 22862 df-cld 22936 df-ntr 22937 df-cls 22938 df-nei 23015 df-lp 23053 df-perf 23054 df-cn 23144 df-cnp 23145 df-haus 23232 df-tx 23479 df-hmeo 23672 df-fil 23763 df-fm 23855 df-flim 23856 df-flf 23857 df-xms 24239 df-ms 24240 df-tms 24241 df-cncf 24811 df-limc 25808 df-dv 25809 df-dvn 25810

This theorem is referenced by: etransclem34 45715

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