etransclem20 - Mathbox for Glauco Siliprandi

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Theorem etransclem20 43263

Description: 𝐻 is smooth. (Contributed by Glauco Siliprandi, 5-Apr-2020.)

**Hypotheses**
Ref	Expression
etransclem20.s	⊢ (𝜑 → 𝑆 ∈ {ℝ, ℂ})
etransclem20.x	⊢ (𝜑 → 𝑋 ∈ ((TopOpen‘ℂ_fld) ↾_t 𝑆))
etransclem20.p	⊢ (𝜑 → 𝑃 ∈ ℕ)
etransclem20.h	⊢ 𝐻 = (𝑗 ∈ (0...𝑀) ↦ (𝑥 ∈ 𝑋 ↦ ((𝑥 − 𝑗)↑if(𝑗 = 0, (𝑃 − 1), 𝑃))))
etransclem20.J	⊢ (𝜑 → 𝐽 ∈ (0...𝑀))
etransclem20.n	⊢ (𝜑 → 𝑁 ∈ ℕ₀)

**Assertion**
Ref	Expression
etransclem20	⊢ (𝜑 → ((𝑆 D^𝑛 (𝐻‘𝐽))‘𝑁):𝑋⟶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 etransclem20**
Step	Hyp	Ref	Expression
1		iftrue 4427	. . . . . 6 ⊢ (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)
2		0cnd 10673	. . . . . 6 ⊢ (if(𝐽 = 0, (𝑃 − 1), 𝑃) < 𝑁 → 0 ∈ ℂ)
3	1, 2	eqeltrd 2853	. . . . 5 ⊢ (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)))) ∈ ℂ)
4	3	adantl 486	. . . 4 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 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)))) ∈ ℂ)
5		simpr 489	. . . . . 6 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ ¬ if(𝐽 = 0, (𝑃 − 1), 𝑃) < 𝑁) → ¬ if(𝐽 = 0, (𝑃 − 1), 𝑃) < 𝑁)
6	5	iffalsed 4432	. . . . 5 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ ¬ 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))))
7		etransclem20.p	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝑃 ∈ ℕ)
8		nnm1nn0 11976	. . . . . . . . . . . . 13 ⊢ (𝑃 ∈ ℕ → (𝑃 − 1) ∈ ℕ₀)
9	7, 8	syl 17	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝑃 − 1) ∈ ℕ₀)
10	7	nnnn0d 11995	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑃 ∈ ℕ₀)
11	9, 10	ifcld 4467	. . . . . . . . . . 11 ⊢ (𝜑 → if(𝐽 = 0, (𝑃 − 1), 𝑃) ∈ ℕ₀)
12	11	faccld 13695	. . . . . . . . . 10 ⊢ (𝜑 → (!‘if(𝐽 = 0, (𝑃 − 1), 𝑃)) ∈ ℕ)
13	12	nncnd 11691	. . . . . . . . 9 ⊢ (𝜑 → (!‘if(𝐽 = 0, (𝑃 − 1), 𝑃)) ∈ ℂ)
14	13	adantr 485	. . . . . . . 8 ⊢ ((𝜑 ∧ ¬ if(𝐽 = 0, (𝑃 − 1), 𝑃) < 𝑁) → (!‘if(𝐽 = 0, (𝑃 − 1), 𝑃)) ∈ ℂ)
15	11	nn0zd 12125	. . . . . . . . . . . . 13 ⊢ (𝜑 → if(𝐽 = 0, (𝑃 − 1), 𝑃) ∈ ℤ)
16		etransclem20.n	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝑁 ∈ ℕ₀)
17	16	nn0zd 12125	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝑁 ∈ ℤ)
18	15, 17	zsubcld 12132	. . . . . . . . . . . 12 ⊢ (𝜑 → (if(𝐽 = 0, (𝑃 − 1), 𝑃) − 𝑁) ∈ ℤ)
19	18	adantr 485	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ ¬ if(𝐽 = 0, (𝑃 − 1), 𝑃) < 𝑁) → (if(𝐽 = 0, (𝑃 − 1), 𝑃) − 𝑁) ∈ ℤ)
20	16	nn0red 11996	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝑁 ∈ ℝ)
21	20	adantr 485	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ ¬ if(𝐽 = 0, (𝑃 − 1), 𝑃) < 𝑁) → 𝑁 ∈ ℝ)
22	11	nn0red 11996	. . . . . . . . . . . . . 14 ⊢ (𝜑 → if(𝐽 = 0, (𝑃 − 1), 𝑃) ∈ ℝ)
23	22	adantr 485	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ ¬ if(𝐽 = 0, (𝑃 − 1), 𝑃) < 𝑁) → if(𝐽 = 0, (𝑃 − 1), 𝑃) ∈ ℝ)
24		simpr 489	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ ¬ if(𝐽 = 0, (𝑃 − 1), 𝑃) < 𝑁) → ¬ if(𝐽 = 0, (𝑃 − 1), 𝑃) < 𝑁)
25	21, 23, 24	nltled 10829	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ ¬ if(𝐽 = 0, (𝑃 − 1), 𝑃) < 𝑁) → 𝑁 ≤ if(𝐽 = 0, (𝑃 − 1), 𝑃))
26	23, 21	subge0d 11269	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ ¬ if(𝐽 = 0, (𝑃 − 1), 𝑃) < 𝑁) → (0 ≤ (if(𝐽 = 0, (𝑃 − 1), 𝑃) − 𝑁) ↔ 𝑁 ≤ if(𝐽 = 0, (𝑃 − 1), 𝑃)))
27	25, 26	mpbird 260	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ ¬ if(𝐽 = 0, (𝑃 − 1), 𝑃) < 𝑁) → 0 ≤ (if(𝐽 = 0, (𝑃 − 1), 𝑃) − 𝑁))
28		elnn0z 12034	. . . . . . . . . . 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)))
29	19, 27, 28	sylanbrc 587	. . . . . . . . . 10 ⊢ ((𝜑 ∧ ¬ if(𝐽 = 0, (𝑃 − 1), 𝑃) < 𝑁) → (if(𝐽 = 0, (𝑃 − 1), 𝑃) − 𝑁) ∈ ℕ₀)
30	29	faccld 13695	. . . . . . . . 9 ⊢ ((𝜑 ∧ ¬ if(𝐽 = 0, (𝑃 − 1), 𝑃) < 𝑁) → (!‘(if(𝐽 = 0, (𝑃 − 1), 𝑃) − 𝑁)) ∈ ℕ)
31	30	nncnd 11691	. . . . . . . 8 ⊢ ((𝜑 ∧ ¬ if(𝐽 = 0, (𝑃 − 1), 𝑃) < 𝑁) → (!‘(if(𝐽 = 0, (𝑃 − 1), 𝑃) − 𝑁)) ∈ ℂ)
32	30	nnne0d 11725	. . . . . . . 8 ⊢ ((𝜑 ∧ ¬ if(𝐽 = 0, (𝑃 − 1), 𝑃) < 𝑁) → (!‘(if(𝐽 = 0, (𝑃 − 1), 𝑃) − 𝑁)) ≠ 0)
33	14, 31, 32	divcld 11455	. . . . . . 7 ⊢ ((𝜑 ∧ ¬ if(𝐽 = 0, (𝑃 − 1), 𝑃) < 𝑁) → ((!‘if(𝐽 = 0, (𝑃 − 1), 𝑃)) / (!‘(if(𝐽 = 0, (𝑃 − 1), 𝑃) − 𝑁))) ∈ ℂ)
34	33	adantlr 715	. . . . . 6 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ ¬ if(𝐽 = 0, (𝑃 − 1), 𝑃) < 𝑁) → ((!‘if(𝐽 = 0, (𝑃 − 1), 𝑃)) / (!‘(if(𝐽 = 0, (𝑃 − 1), 𝑃) − 𝑁))) ∈ ℂ)
35		etransclem20.s	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑆 ∈ {ℝ, ℂ})
36		etransclem20.x	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑋 ∈ ((TopOpen‘ℂ_fld) ↾_t 𝑆))
37	35, 36	dvdmsscn 42945	. . . . . . . . . 10 ⊢ (𝜑 → 𝑋 ⊆ ℂ)
38	37	sselda 3893	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝑥 ∈ ℂ)
39		etransclem20.J	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐽 ∈ (0...𝑀))
40		elfzelz 12957	. . . . . . . . . . . 12 ⊢ (𝐽 ∈ (0...𝑀) → 𝐽 ∈ ℤ)
41	40	zcnd 12128	. . . . . . . . . . 11 ⊢ (𝐽 ∈ (0...𝑀) → 𝐽 ∈ ℂ)
42	39, 41	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → 𝐽 ∈ ℂ)
43	42	adantr 485	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝐽 ∈ ℂ)
44	38, 43	subcld 11036	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → (𝑥 − 𝐽) ∈ ℂ)
45	44	adantr 485	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ ¬ if(𝐽 = 0, (𝑃 − 1), 𝑃) < 𝑁) → (𝑥 − 𝐽) ∈ ℂ)
46	29	adantlr 715	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ ¬ if(𝐽 = 0, (𝑃 − 1), 𝑃) < 𝑁) → (if(𝐽 = 0, (𝑃 − 1), 𝑃) − 𝑁) ∈ ℕ₀)
47	45, 46	expcld 13561	. . . . . 6 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ ¬ if(𝐽 = 0, (𝑃 − 1), 𝑃) < 𝑁) → ((𝑥 − 𝐽)↑(if(𝐽 = 0, (𝑃 − 1), 𝑃) − 𝑁)) ∈ ℂ)
48	34, 47	mulcld 10700	. . . . 5 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ ¬ 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))) ∈ ℂ)
49	6, 48	eqeltrd 2853	. . . 4 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ ¬ 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)))) ∈ ℂ)
50	4, 49	pm2.61dan 813	. . 3 ⊢ ((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)))) ∈ ℂ)
51		eqid 2759	. . 3 ⊢ (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(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)))))
52	50, 51	fmptd 6870	. 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⟶ℂ)
53		etransclem20.h	. . . 4 ⊢ 𝐻 = (𝑗 ∈ (0...𝑀) ↦ (𝑥 ∈ 𝑋 ↦ ((𝑥 − 𝑗)↑if(𝑗 = 0, (𝑃 − 1), 𝑃))))
54	35, 36, 7, 53, 39, 16	etransclem17 43260	. . 3 ⊢ (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))))))
55	54	feq1d 6484	. 2 ⊢ (𝜑 → (((𝑆 D^𝑛 (𝐻‘𝐽))‘𝑁):𝑋⟶ℂ ↔ (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⟶ℂ))
56	52, 55	mpbird 260	1 ⊢ (𝜑 → ((𝑆 D^𝑛 (𝐻‘𝐽))‘𝑁):𝑋⟶ℂ)

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ∧ wa 400 = wceq 1539 ∈ wcel 2112 ifcif 4421 {cpr 4525 class class class wbr 5033 ↦ cmpt 5113 ⟶wf 6332 ‘cfv 6336 (class class class)co 7151 ℂcc 10574 ℝcr 10575 0cc0 10576 1c1 10577 · cmul 10581 < clt 10714 ≤ cle 10715 − cmin 10909 / cdiv 11336 ℕcn 11675 ℕ₀cn0 11935 ℤcz 12021 ...cfz 12940 ↑cexp 13480 !cfa 13684 ↾_t crest 16753 TopOpenctopn 16754 ℂ_fldccnfld 20167 D^𝑛 cdvn 24564

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1912 ax-6 1971 ax-7 2016 ax-8 2114 ax-9 2122 ax-10 2143 ax-11 2159 ax-12 2176 ax-ext 2730 ax-rep 5157 ax-sep 5170 ax-nul 5177 ax-pow 5235 ax-pr 5299 ax-un 7460 ax-inf2 9138 ax-cnex 10632 ax-resscn 10633 ax-1cn 10634 ax-icn 10635 ax-addcl 10636 ax-addrcl 10637 ax-mulcl 10638 ax-mulrcl 10639 ax-mulcom 10640 ax-addass 10641 ax-mulass 10642 ax-distr 10643 ax-i2m1 10644 ax-1ne0 10645 ax-1rid 10646 ax-rnegex 10647 ax-rrecex 10648 ax-cnre 10649 ax-pre-lttri 10650 ax-pre-lttrn 10651 ax-pre-ltadd 10652 ax-pre-mulgt0 10653 ax-pre-sup 10654 ax-addf 10655 ax-mulf 10656

This theorem depends on definitions: df-bi 210 df-an 401 df-or 846 df-3or 1086 df-3an 1087 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2071 df-mo 2558 df-eu 2589 df-clab 2737 df-cleq 2751 df-clel 2831 df-nfc 2902 df-ne 2953 df-nel 3057 df-ral 3076 df-rex 3077 df-reu 3078 df-rmo 3079 df-rab 3080 df-v 3412 df-sbc 3698 df-csb 3807 df-dif 3862 df-un 3864 df-in 3866 df-ss 3876 df-pss 3878 df-nul 4227 df-if 4422 df-pw 4497 df-sn 4524 df-pr 4526 df-tp 4528 df-op 4530 df-uni 4800 df-int 4840 df-iun 4886 df-iin 4887 df-br 5034 df-opab 5096 df-mpt 5114 df-tr 5140 df-id 5431 df-eprel 5436 df-po 5444 df-so 5445 df-fr 5484 df-se 5485 df-we 5486 df-xp 5531 df-rel 5532 df-cnv 5533 df-co 5534 df-dm 5535 df-rn 5536 df-res 5537 df-ima 5538 df-pred 6127 df-ord 6173 df-on 6174 df-lim 6175 df-suc 6176 df-iota 6295 df-fun 6338 df-fn 6339 df-f 6340 df-f1 6341 df-fo 6342 df-f1o 6343 df-fv 6344 df-isom 6345 df-riota 7109 df-ov 7154 df-oprab 7155 df-mpo 7156 df-of 7406 df-om 7581 df-1st 7694 df-2nd 7695 df-supp 7837 df-wrecs 7958 df-recs 8019 df-rdg 8057 df-1o 8113 df-2o 8114 df-er 8300 df-map 8419 df-pm 8420 df-ixp 8481 df-en 8529 df-dom 8530 df-sdom 8531 df-fin 8532 df-fsupp 8868 df-fi 8909 df-sup 8940 df-inf 8941 df-oi 9008 df-card 9402 df-pnf 10716 df-mnf 10717 df-xr 10718 df-ltxr 10719 df-le 10720 df-sub 10911 df-neg 10912 df-div 11337 df-nn 11676 df-2 11738 df-3 11739 df-4 11740 df-5 11741 df-6 11742 df-7 11743 df-8 11744 df-9 11745 df-n0 11936 df-z 12022 df-dec 12139 df-uz 12284 df-q 12390 df-rp 12432 df-xneg 12549 df-xadd 12550 df-xmul 12551 df-icc 12787 df-fz 12941 df-fzo 13084 df-seq 13420 df-exp 13481 df-fac 13685 df-hash 13742 df-cj 14507 df-re 14508 df-im 14509 df-sqrt 14643 df-abs 14644 df-struct 16544 df-ndx 16545 df-slot 16546 df-base 16548 df-sets 16549 df-ress 16550 df-plusg 16637 df-mulr 16638 df-starv 16639 df-sca 16640 df-vsca 16641 df-ip 16642 df-tset 16643 df-ple 16644 df-ds 16646 df-unif 16647 df-hom 16648 df-cco 16649 df-rest 16755 df-topn 16756 df-0g 16774 df-gsum 16775 df-topgen 16776 df-pt 16777 df-prds 16780 df-xrs 16834 df-qtop 16839 df-imas 16840 df-xps 16842 df-mre 16916 df-mrc 16917 df-acs 16919 df-mgm 17919 df-sgrp 17968 df-mnd 17979 df-submnd 18024 df-mulg 18293 df-cntz 18515 df-cmn 18976 df-psmet 20159 df-xmet 20160 df-met 20161 df-bl 20162 df-mopn 20163 df-fbas 20164 df-fg 20165 df-cnfld 20168 df-top 21595 df-topon 21612 df-topsp 21634 df-bases 21647 df-cld 21720 df-ntr 21721 df-cls 21722 df-nei 21799 df-lp 21837 df-perf 21838 df-cn 21928 df-cnp 21929 df-haus 22016 df-tx 22263 df-hmeo 22456 df-fil 22547 df-fm 22639 df-flim 22640 df-flf 22641 df-xms 23023 df-ms 23024 df-tms 23025 df-cncf 23580 df-limc 24566 df-dv 24567 df-dvn 24568

This theorem is referenced by: etransclem27 43270 etransclem29 43272 etransclem31 43274 etransclem32 43275 etransclem33 43276 etransclem34 43277

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