etransclem20 - Mathbox for Glauco Siliprandi

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Theorem etransclem20 46225

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 4490	. . . . . 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 11143	. . . . . 6 ⊢ (if(𝐽 = 0, (𝑃 − 1), 𝑃) < 𝑁 → 0 ∈ ℂ)
3	1, 2	eqeltrd 2828	. . . . 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 481	. . . 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 484	. . . . . 6 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ ¬ if(𝐽 = 0, (𝑃 − 1), 𝑃) < 𝑁) → ¬ if(𝐽 = 0, (𝑃 − 1), 𝑃) < 𝑁)
6	5	iffalsed 4495	. . . . 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 12459	. . . . . . . . . . . . 13 ⊢ (𝑃 ∈ ℕ → (𝑃 − 1) ∈ ℕ₀)
9	7, 8	syl 17	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝑃 − 1) ∈ ℕ₀)
10	7	nnnn0d 12479	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑃 ∈ ℕ₀)
11	9, 10	ifcld 4531	. . . . . . . . . . 11 ⊢ (𝜑 → if(𝐽 = 0, (𝑃 − 1), 𝑃) ∈ ℕ₀)
12	11	faccld 14225	. . . . . . . . . 10 ⊢ (𝜑 → (!‘if(𝐽 = 0, (𝑃 − 1), 𝑃)) ∈ ℕ)
13	12	nncnd 12178	. . . . . . . . 9 ⊢ (𝜑 → (!‘if(𝐽 = 0, (𝑃 − 1), 𝑃)) ∈ ℂ)
14	13	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ ¬ if(𝐽 = 0, (𝑃 − 1), 𝑃) < 𝑁) → (!‘if(𝐽 = 0, (𝑃 − 1), 𝑃)) ∈ ℂ)
15	11	nn0zd 12531	. . . . . . . . . . . . 13 ⊢ (𝜑 → if(𝐽 = 0, (𝑃 − 1), 𝑃) ∈ ℤ)
16		etransclem20.n	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝑁 ∈ ℕ₀)
17	16	nn0zd 12531	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝑁 ∈ ℤ)
18	15, 17	zsubcld 12619	. . . . . . . . . . . 12 ⊢ (𝜑 → (if(𝐽 = 0, (𝑃 − 1), 𝑃) − 𝑁) ∈ ℤ)
19	18	adantr 480	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ ¬ if(𝐽 = 0, (𝑃 − 1), 𝑃) < 𝑁) → (if(𝐽 = 0, (𝑃 − 1), 𝑃) − 𝑁) ∈ ℤ)
20	16	nn0red 12480	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝑁 ∈ ℝ)
21	20	adantr 480	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ ¬ if(𝐽 = 0, (𝑃 − 1), 𝑃) < 𝑁) → 𝑁 ∈ ℝ)
22	11	nn0red 12480	. . . . . . . . . . . . . 14 ⊢ (𝜑 → if(𝐽 = 0, (𝑃 − 1), 𝑃) ∈ ℝ)
23	22	adantr 480	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ ¬ if(𝐽 = 0, (𝑃 − 1), 𝑃) < 𝑁) → if(𝐽 = 0, (𝑃 − 1), 𝑃) ∈ ℝ)
24		simpr 484	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ ¬ if(𝐽 = 0, (𝑃 − 1), 𝑃) < 𝑁) → ¬ if(𝐽 = 0, (𝑃 − 1), 𝑃) < 𝑁)
25	21, 23, 24	nltled 11300	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ ¬ if(𝐽 = 0, (𝑃 − 1), 𝑃) < 𝑁) → 𝑁 ≤ if(𝐽 = 0, (𝑃 − 1), 𝑃))
26	23, 21	subge0d 11744	. . . . . . . . . . . 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 257	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ ¬ if(𝐽 = 0, (𝑃 − 1), 𝑃) < 𝑁) → 0 ≤ (if(𝐽 = 0, (𝑃 − 1), 𝑃) − 𝑁))
28		elnn0z 12518	. . . . . . . . . . 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 583	. . . . . . . . . 10 ⊢ ((𝜑 ∧ ¬ if(𝐽 = 0, (𝑃 − 1), 𝑃) < 𝑁) → (if(𝐽 = 0, (𝑃 − 1), 𝑃) − 𝑁) ∈ ℕ₀)
30	29	faccld 14225	. . . . . . . . 9 ⊢ ((𝜑 ∧ ¬ if(𝐽 = 0, (𝑃 − 1), 𝑃) < 𝑁) → (!‘(if(𝐽 = 0, (𝑃 − 1), 𝑃) − 𝑁)) ∈ ℕ)
31	30	nncnd 12178	. . . . . . . 8 ⊢ ((𝜑 ∧ ¬ if(𝐽 = 0, (𝑃 − 1), 𝑃) < 𝑁) → (!‘(if(𝐽 = 0, (𝑃 − 1), 𝑃) − 𝑁)) ∈ ℂ)
32	30	nnne0d 12212	. . . . . . . 8 ⊢ ((𝜑 ∧ ¬ if(𝐽 = 0, (𝑃 − 1), 𝑃) < 𝑁) → (!‘(if(𝐽 = 0, (𝑃 − 1), 𝑃) − 𝑁)) ≠ 0)
33	14, 31, 32	divcld 11934	. . . . . . 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 45907	. . . . . . . . . 10 ⊢ (𝜑 → 𝑋 ⊆ ℂ)
38	37	sselda 3943	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝑥 ∈ ℂ)
39		etransclem20.J	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐽 ∈ (0...𝑀))
40		elfzelz 13461	. . . . . . . . . . . 12 ⊢ (𝐽 ∈ (0...𝑀) → 𝐽 ∈ ℤ)
41	40	zcnd 12615	. . . . . . . . . . 11 ⊢ (𝐽 ∈ (0...𝑀) → 𝐽 ∈ ℂ)
42	39, 41	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → 𝐽 ∈ ℂ)
43	42	adantr 480	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝐽 ∈ ℂ)
44	38, 43	subcld 11509	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → (𝑥 − 𝐽) ∈ ℂ)
45	44	adantr 480	. . . . . . 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 14087	. . . . . 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 11170	. . . . 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 2828	. . . 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 812	. . 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 2729	. . 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 7068	. 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 46222	. . 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 6652	. 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 257	1 ⊢ (𝜑 → ((𝑆 D^𝑛 (𝐻‘𝐽))‘𝑁):𝑋⟶ℂ)

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ∧ wa 395 = wceq 1540 ∈ wcel 2109 ifcif 4484 {cpr 4587 class class class wbr 5102 ↦ cmpt 5183 ⟶wf 6495 ‘cfv 6499 (class class class)co 7369 ℂcc 11042 ℝcr 11043 0cc0 11044 1c1 11045 · cmul 11049 < clt 11184 ≤ cle 11185 − cmin 11381 / cdiv 11811 ℕcn 12162 ℕ₀cn0 12418 ℤcz 12505 ...cfz 13444 ↑cexp 14002 !cfa 14214 ↾_t crest 17359 TopOpenctopn 17360 ℂ_fldccnfld 21240 D^𝑛 cdvn 25741

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2008 ax-8 2111 ax-9 2119 ax-10 2142 ax-11 2158 ax-12 2178 ax-ext 2701 ax-rep 5229 ax-sep 5246 ax-nul 5256 ax-pow 5315 ax-pr 5382 ax-un 7691 ax-inf2 9570 ax-cnex 11100 ax-resscn 11101 ax-1cn 11102 ax-icn 11103 ax-addcl 11104 ax-addrcl 11105 ax-mulcl 11106 ax-mulrcl 11107 ax-mulcom 11108 ax-addass 11109 ax-mulass 11110 ax-distr 11111 ax-i2m1 11112 ax-1ne0 11113 ax-1rid 11114 ax-rnegex 11115 ax-rrecex 11116 ax-cnre 11117 ax-pre-lttri 11118 ax-pre-lttrn 11119 ax-pre-ltadd 11120 ax-pre-mulgt0 11121 ax-pre-sup 11122 ax-addf 11123

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2533 df-eu 2562 df-clab 2708 df-cleq 2721 df-clel 2803 df-nfc 2878 df-ne 2926 df-nel 3030 df-ral 3045 df-rex 3054 df-rmo 3351 df-reu 3352 df-rab 3403 df-v 3446 df-sbc 3751 df-csb 3860 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-pss 3931 df-nul 4293 df-if 4485 df-pw 4561 df-sn 4586 df-pr 4588 df-tp 4590 df-op 4592 df-uni 4868 df-int 4907 df-iun 4953 df-iin 4954 df-br 5103 df-opab 5165 df-mpt 5184 df-tr 5210 df-id 5526 df-eprel 5531 df-po 5539 df-so 5540 df-fr 5584 df-se 5585 df-we 5586 df-xp 5637 df-rel 5638 df-cnv 5639 df-co 5640 df-dm 5641 df-rn 5642 df-res 5643 df-ima 5644 df-pred 6262 df-ord 6323 df-on 6324 df-lim 6325 df-suc 6326 df-iota 6452 df-fun 6501 df-fn 6502 df-f 6503 df-f1 6504 df-fo 6505 df-f1o 6506 df-fv 6507 df-isom 6508 df-riota 7326 df-ov 7372 df-oprab 7373 df-mpo 7374 df-of 7633 df-om 7823 df-1st 7947 df-2nd 7948 df-supp 8117 df-frecs 8237 df-wrecs 8268 df-recs 8317 df-rdg 8355 df-1o 8411 df-2o 8412 df-er 8648 df-map 8778 df-pm 8779 df-ixp 8848 df-en 8896 df-dom 8897 df-sdom 8898 df-fin 8899 df-fsupp 9289 df-fi 9338 df-sup 9369 df-inf 9370 df-oi 9439 df-card 9868 df-pnf 11186 df-mnf 11187 df-xr 11188 df-ltxr 11189 df-le 11190 df-sub 11383 df-neg 11384 df-div 11812 df-nn 12163 df-2 12225 df-3 12226 df-4 12227 df-5 12228 df-6 12229 df-7 12230 df-8 12231 df-9 12232 df-n0 12419 df-z 12506 df-dec 12626 df-uz 12770 df-q 12884 df-rp 12928 df-xneg 13048 df-xadd 13049 df-xmul 13050 df-icc 13289 df-fz 13445 df-fzo 13592 df-seq 13943 df-exp 14003 df-fac 14215 df-hash 14272 df-cj 15041 df-re 15042 df-im 15043 df-sqrt 15177 df-abs 15178 df-struct 17093 df-sets 17110 df-slot 17128 df-ndx 17140 df-base 17156 df-ress 17177 df-plusg 17209 df-mulr 17210 df-starv 17211 df-sca 17212 df-vsca 17213 df-ip 17214 df-tset 17215 df-ple 17216 df-ds 17218 df-unif 17219 df-hom 17220 df-cco 17221 df-rest 17361 df-topn 17362 df-0g 17380 df-gsum 17381 df-topgen 17382 df-pt 17383 df-prds 17386 df-xrs 17441 df-qtop 17446 df-imas 17447 df-xps 17449 df-mre 17523 df-mrc 17524 df-acs 17526 df-mgm 18543 df-sgrp 18622 df-mnd 18638 df-submnd 18687 df-mulg 18976 df-cntz 19225 df-cmn 19688 df-psmet 21232 df-xmet 21233 df-met 21234 df-bl 21235 df-mopn 21236 df-fbas 21237 df-fg 21238 df-cnfld 21241 df-top 22757 df-topon 22774 df-topsp 22796 df-bases 22809 df-cld 22882 df-ntr 22883 df-cls 22884 df-nei 22961 df-lp 22999 df-perf 23000 df-cn 23090 df-cnp 23091 df-haus 23178 df-tx 23425 df-hmeo 23618 df-fil 23709 df-fm 23801 df-flim 23802 df-flf 23803 df-xms 24184 df-ms 24185 df-tms 24186 df-cncf 24747 df-limc 25743 df-dv 25744 df-dvn 25745

This theorem is referenced by: etransclem27 46232 etransclem29 46234 etransclem31 46236 etransclem32 46237 etransclem33 46238 etransclem34 46239

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