etransclem15 - Mathbox for Glauco Siliprandi

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Theorem etransclem15 43790

Description: Value of the term 𝑇, when 𝐽 = 0 and (𝐶‘0) = 𝑃 − 1 (Contributed by Glauco Siliprandi, 5-Apr-2020.)

**Hypotheses**
Ref	Expression
etransclem15.p	⊢ (𝜑 → 𝑃 ∈ ℕ)
etransclem15.m	⊢ (𝜑 → 𝑀 ∈ ℕ₀)
etransclem15.n	⊢ (𝜑 → 𝑁 ∈ ℕ₀)
etransclem15.c	⊢ (𝜑 → 𝐶:(0...𝑀)⟶(0...𝑁))
etransclem15.t	⊢ 𝑇 = (((!‘𝑁) / ∏𝑗 ∈ (0...𝑀)(!‘(𝐶‘𝑗))) · (if((𝑃 − 1) < (𝐶‘0), 0, (((!‘(𝑃 − 1)) / (!‘((𝑃 − 1) − (𝐶‘0)))) · (𝐽↑((𝑃 − 1) − (𝐶‘0))))) · ∏𝑗 ∈ (1...𝑀)if(𝑃 < (𝐶‘𝑗), 0, (((!‘𝑃) / (!‘(𝑃 − (𝐶‘𝑗)))) · ((𝐽 − 𝑗)↑(𝑃 − (𝐶‘𝑗)))))))
etransclem15.j	⊢ (𝜑 → 𝐽 = 0)
etransclem15.cpm1	⊢ (𝜑 → (𝐶‘0) ≠ (𝑃 − 1))

**Assertion**
Ref	Expression
etransclem15	⊢ (𝜑 → 𝑇 = 0)

Distinct variable groups: 𝑗,𝑀 𝜑,𝑗 Allowed substitution hints: 𝐶(𝑗) 𝑃(𝑗) 𝑇(𝑗) 𝐽(𝑗) 𝑁(𝑗)

**Proof of Theorem etransclem15**
Step	Hyp	Ref	Expression
1		etransclem15.t	. . 3 ⊢ 𝑇 = (((!‘𝑁) / ∏𝑗 ∈ (0...𝑀)(!‘(𝐶‘𝑗))) · (if((𝑃 − 1) < (𝐶‘0), 0, (((!‘(𝑃 − 1)) / (!‘((𝑃 − 1) − (𝐶‘0)))) · (𝐽↑((𝑃 − 1) − (𝐶‘0))))) · ∏𝑗 ∈ (1...𝑀)if(𝑃 < (𝐶‘𝑗), 0, (((!‘𝑃) / (!‘(𝑃 − (𝐶‘𝑗)))) · ((𝐽 − 𝑗)↑(𝑃 − (𝐶‘𝑗)))))))
2	1	a1i 11	. 2 ⊢ (𝜑 → 𝑇 = (((!‘𝑁) / ∏𝑗 ∈ (0...𝑀)(!‘(𝐶‘𝑗))) · (if((𝑃 − 1) < (𝐶‘0), 0, (((!‘(𝑃 − 1)) / (!‘((𝑃 − 1) − (𝐶‘0)))) · (𝐽↑((𝑃 − 1) − (𝐶‘0))))) · ∏𝑗 ∈ (1...𝑀)if(𝑃 < (𝐶‘𝑗), 0, (((!‘𝑃) / (!‘(𝑃 − (𝐶‘𝑗)))) · ((𝐽 − 𝑗)↑(𝑃 − (𝐶‘𝑗))))))))
3		iftrue 4465	. . . . . . 7 ⊢ ((𝑃 − 1) < (𝐶‘0) → if((𝑃 − 1) < (𝐶‘0), 0, (((!‘(𝑃 − 1)) / (!‘((𝑃 − 1) − (𝐶‘0)))) · (𝐽↑((𝑃 − 1) − (𝐶‘0))))) = 0)
4	3	adantl 482	. . . . . 6 ⊢ ((𝜑 ∧ (𝑃 − 1) < (𝐶‘0)) → if((𝑃 − 1) < (𝐶‘0), 0, (((!‘(𝑃 − 1)) / (!‘((𝑃 − 1) − (𝐶‘0)))) · (𝐽↑((𝑃 − 1) − (𝐶‘0))))) = 0)
5		iffalse 4468	. . . . . . . 8 ⊢ (¬ (𝑃 − 1) < (𝐶‘0) → if((𝑃 − 1) < (𝐶‘0), 0, (((!‘(𝑃 − 1)) / (!‘((𝑃 − 1) − (𝐶‘0)))) · (𝐽↑((𝑃 − 1) − (𝐶‘0))))) = (((!‘(𝑃 − 1)) / (!‘((𝑃 − 1) − (𝐶‘0)))) · (𝐽↑((𝑃 − 1) − (𝐶‘0)))))
6	5	adantl 482	. . . . . . 7 ⊢ ((𝜑 ∧ ¬ (𝑃 − 1) < (𝐶‘0)) → if((𝑃 − 1) < (𝐶‘0), 0, (((!‘(𝑃 − 1)) / (!‘((𝑃 − 1) − (𝐶‘0)))) · (𝐽↑((𝑃 − 1) − (𝐶‘0))))) = (((!‘(𝑃 − 1)) / (!‘((𝑃 − 1) − (𝐶‘0)))) · (𝐽↑((𝑃 − 1) − (𝐶‘0)))))
7		etransclem15.j	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐽 = 0)
8	7	oveq1d 7290	. . . . . . . . . 10 ⊢ (𝜑 → (𝐽↑((𝑃 − 1) − (𝐶‘0))) = (0↑((𝑃 − 1) − (𝐶‘0))))
9	8	adantr 481	. . . . . . . . 9 ⊢ ((𝜑 ∧ ¬ (𝑃 − 1) < (𝐶‘0)) → (𝐽↑((𝑃 − 1) − (𝐶‘0))) = (0↑((𝑃 − 1) − (𝐶‘0))))
10		etransclem15.p	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝑃 ∈ ℕ)
11	10	nnzd 12425	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝑃 ∈ ℤ)
12		1zzd 12351	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 1 ∈ ℤ)
13	11, 12	zsubcld 12431	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝑃 − 1) ∈ ℤ)
14		etransclem15.c	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝐶:(0...𝑀)⟶(0...𝑁))
15		etransclem15.m	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → 𝑀 ∈ ℕ₀)
16		nn0uz 12620	. . . . . . . . . . . . . . . . 17 ⊢ ℕ₀ = (ℤ_≥‘0)
17	15, 16	eleqtrdi 2849	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 𝑀 ∈ (ℤ_≥‘0))
18		eluzfz1 13263	. . . . . . . . . . . . . . . 16 ⊢ (𝑀 ∈ (ℤ_≥‘0) → 0 ∈ (0...𝑀))
19	17, 18	syl 17	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 0 ∈ (0...𝑀))
20	14, 19	ffvelrnd 6962	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (𝐶‘0) ∈ (0...𝑁))
21	20	elfzelzd 13257	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝐶‘0) ∈ ℤ)
22	13, 21	zsubcld 12431	. . . . . . . . . . . 12 ⊢ (𝜑 → ((𝑃 − 1) − (𝐶‘0)) ∈ ℤ)
23	22	adantr 481	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ ¬ (𝑃 − 1) < (𝐶‘0)) → ((𝑃 − 1) − (𝐶‘0)) ∈ ℤ)
24	21	zred 12426	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (𝐶‘0) ∈ ℝ)
25	24	adantr 481	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ ¬ (𝑃 − 1) < (𝐶‘0)) → (𝐶‘0) ∈ ℝ)
26	13	zred 12426	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (𝑃 − 1) ∈ ℝ)
27	26	adantr 481	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ ¬ (𝑃 − 1) < (𝐶‘0)) → (𝑃 − 1) ∈ ℝ)
28		simpr 485	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ ¬ (𝑃 − 1) < (𝐶‘0)) → ¬ (𝑃 − 1) < (𝐶‘0))
29	25, 27, 28	nltled 11125	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ ¬ (𝑃 − 1) < (𝐶‘0)) → (𝐶‘0) ≤ (𝑃 − 1))
30		etransclem15.cpm1	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (𝐶‘0) ≠ (𝑃 − 1))
31	30	necomd 2999	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (𝑃 − 1) ≠ (𝐶‘0))
32	31	adantr 481	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ ¬ (𝑃 − 1) < (𝐶‘0)) → (𝑃 − 1) ≠ (𝐶‘0))
33	25, 27, 29, 32	leneltd 11129	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ ¬ (𝑃 − 1) < (𝐶‘0)) → (𝐶‘0) < (𝑃 − 1))
34	25, 27	posdifd 11562	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ ¬ (𝑃 − 1) < (𝐶‘0)) → ((𝐶‘0) < (𝑃 − 1) ↔ 0 < ((𝑃 − 1) − (𝐶‘0))))
35	33, 34	mpbid 231	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ ¬ (𝑃 − 1) < (𝐶‘0)) → 0 < ((𝑃 − 1) − (𝐶‘0)))
36		elnnz 12329	. . . . . . . . . . 11 ⊢ (((𝑃 − 1) − (𝐶‘0)) ∈ ℕ ↔ (((𝑃 − 1) − (𝐶‘0)) ∈ ℤ ∧ 0 < ((𝑃 − 1) − (𝐶‘0))))
37	23, 35, 36	sylanbrc 583	. . . . . . . . . 10 ⊢ ((𝜑 ∧ ¬ (𝑃 − 1) < (𝐶‘0)) → ((𝑃 − 1) − (𝐶‘0)) ∈ ℕ)
38	37	0expd 13857	. . . . . . . . 9 ⊢ ((𝜑 ∧ ¬ (𝑃 − 1) < (𝐶‘0)) → (0↑((𝑃 − 1) − (𝐶‘0))) = 0)
39	9, 38	eqtrd 2778	. . . . . . . 8 ⊢ ((𝜑 ∧ ¬ (𝑃 − 1) < (𝐶‘0)) → (𝐽↑((𝑃 − 1) − (𝐶‘0))) = 0)
40	39	oveq2d 7291	. . . . . . 7 ⊢ ((𝜑 ∧ ¬ (𝑃 − 1) < (𝐶‘0)) → (((!‘(𝑃 − 1)) / (!‘((𝑃 − 1) − (𝐶‘0)))) · (𝐽↑((𝑃 − 1) − (𝐶‘0)))) = (((!‘(𝑃 − 1)) / (!‘((𝑃 − 1) − (𝐶‘0)))) · 0))
41		nnm1nn0 12274	. . . . . . . . . . . . 13 ⊢ (𝑃 ∈ ℕ → (𝑃 − 1) ∈ ℕ₀)
42	10, 41	syl 17	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝑃 − 1) ∈ ℕ₀)
43	42	faccld 13998	. . . . . . . . . . 11 ⊢ (𝜑 → (!‘(𝑃 − 1)) ∈ ℕ)
44	43	nncnd 11989	. . . . . . . . . 10 ⊢ (𝜑 → (!‘(𝑃 − 1)) ∈ ℂ)
45	44	adantr 481	. . . . . . . . 9 ⊢ ((𝜑 ∧ ¬ (𝑃 − 1) < (𝐶‘0)) → (!‘(𝑃 − 1)) ∈ ℂ)
46	37	nnnn0d 12293	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ ¬ (𝑃 − 1) < (𝐶‘0)) → ((𝑃 − 1) − (𝐶‘0)) ∈ ℕ₀)
47	46	faccld 13998	. . . . . . . . . 10 ⊢ ((𝜑 ∧ ¬ (𝑃 − 1) < (𝐶‘0)) → (!‘((𝑃 − 1) − (𝐶‘0))) ∈ ℕ)
48	47	nncnd 11989	. . . . . . . . 9 ⊢ ((𝜑 ∧ ¬ (𝑃 − 1) < (𝐶‘0)) → (!‘((𝑃 − 1) − (𝐶‘0))) ∈ ℂ)
49	47	nnne0d 12023	. . . . . . . . 9 ⊢ ((𝜑 ∧ ¬ (𝑃 − 1) < (𝐶‘0)) → (!‘((𝑃 − 1) − (𝐶‘0))) ≠ 0)
50	45, 48, 49	divcld 11751	. . . . . . . 8 ⊢ ((𝜑 ∧ ¬ (𝑃 − 1) < (𝐶‘0)) → ((!‘(𝑃 − 1)) / (!‘((𝑃 − 1) − (𝐶‘0)))) ∈ ℂ)
51	50	mul01d 11174	. . . . . . 7 ⊢ ((𝜑 ∧ ¬ (𝑃 − 1) < (𝐶‘0)) → (((!‘(𝑃 − 1)) / (!‘((𝑃 − 1) − (𝐶‘0)))) · 0) = 0)
52	6, 40, 51	3eqtrd 2782	. . . . . 6 ⊢ ((𝜑 ∧ ¬ (𝑃 − 1) < (𝐶‘0)) → if((𝑃 − 1) < (𝐶‘0), 0, (((!‘(𝑃 − 1)) / (!‘((𝑃 − 1) − (𝐶‘0)))) · (𝐽↑((𝑃 − 1) − (𝐶‘0))))) = 0)
53	4, 52	pm2.61dan 810	. . . . 5 ⊢ (𝜑 → if((𝑃 − 1) < (𝐶‘0), 0, (((!‘(𝑃 − 1)) / (!‘((𝑃 − 1) − (𝐶‘0)))) · (𝐽↑((𝑃 − 1) − (𝐶‘0))))) = 0)
54	53	oveq1d 7290	. . . 4 ⊢ (𝜑 → (if((𝑃 − 1) < (𝐶‘0), 0, (((!‘(𝑃 − 1)) / (!‘((𝑃 − 1) − (𝐶‘0)))) · (𝐽↑((𝑃 − 1) − (𝐶‘0))))) · ∏𝑗 ∈ (1...𝑀)if(𝑃 < (𝐶‘𝑗), 0, (((!‘𝑃) / (!‘(𝑃 − (𝐶‘𝑗)))) · ((𝐽 − 𝑗)↑(𝑃 − (𝐶‘𝑗)))))) = (0 · ∏𝑗 ∈ (1...𝑀)if(𝑃 < (𝐶‘𝑗), 0, (((!‘𝑃) / (!‘(𝑃 − (𝐶‘𝑗)))) · ((𝐽 − 𝑗)↑(𝑃 − (𝐶‘𝑗)))))))
55	7, 19	eqeltrd 2839	. . . . . . 7 ⊢ (𝜑 → 𝐽 ∈ (0...𝑀))
56	10, 14, 55	etransclem7 43782	. . . . . 6 ⊢ (𝜑 → ∏𝑗 ∈ (1...𝑀)if(𝑃 < (𝐶‘𝑗), 0, (((!‘𝑃) / (!‘(𝑃 − (𝐶‘𝑗)))) · ((𝐽 − 𝑗)↑(𝑃 − (𝐶‘𝑗))))) ∈ ℤ)
57	56	zcnd 12427	. . . . 5 ⊢ (𝜑 → ∏𝑗 ∈ (1...𝑀)if(𝑃 < (𝐶‘𝑗), 0, (((!‘𝑃) / (!‘(𝑃 − (𝐶‘𝑗)))) · ((𝐽 − 𝑗)↑(𝑃 − (𝐶‘𝑗))))) ∈ ℂ)
58	57	mul02d 11173	. . . 4 ⊢ (𝜑 → (0 · ∏𝑗 ∈ (1...𝑀)if(𝑃 < (𝐶‘𝑗), 0, (((!‘𝑃) / (!‘(𝑃 − (𝐶‘𝑗)))) · ((𝐽 − 𝑗)↑(𝑃 − (𝐶‘𝑗)))))) = 0)
59	54, 58	eqtrd 2778	. . 3 ⊢ (𝜑 → (if((𝑃 − 1) < (𝐶‘0), 0, (((!‘(𝑃 − 1)) / (!‘((𝑃 − 1) − (𝐶‘0)))) · (𝐽↑((𝑃 − 1) − (𝐶‘0))))) · ∏𝑗 ∈ (1...𝑀)if(𝑃 < (𝐶‘𝑗), 0, (((!‘𝑃) / (!‘(𝑃 − (𝐶‘𝑗)))) · ((𝐽 − 𝑗)↑(𝑃 − (𝐶‘𝑗)))))) = 0)
60	59	oveq2d 7291	. 2 ⊢ (𝜑 → (((!‘𝑁) / ∏𝑗 ∈ (0...𝑀)(!‘(𝐶‘𝑗))) · (if((𝑃 − 1) < (𝐶‘0), 0, (((!‘(𝑃 − 1)) / (!‘((𝑃 − 1) − (𝐶‘0)))) · (𝐽↑((𝑃 − 1) − (𝐶‘0))))) · ∏𝑗 ∈ (1...𝑀)if(𝑃 < (𝐶‘𝑗), 0, (((!‘𝑃) / (!‘(𝑃 − (𝐶‘𝑗)))) · ((𝐽 − 𝑗)↑(𝑃 − (𝐶‘𝑗))))))) = (((!‘𝑁) / ∏𝑗 ∈ (0...𝑀)(!‘(𝐶‘𝑗))) · 0))
61		etransclem15.n	. . . . . 6 ⊢ (𝜑 → 𝑁 ∈ ℕ₀)
62	61	faccld 13998	. . . . 5 ⊢ (𝜑 → (!‘𝑁) ∈ ℕ)
63	62	nncnd 11989	. . . 4 ⊢ (𝜑 → (!‘𝑁) ∈ ℂ)
64		fzfid 13693	. . . . 5 ⊢ (𝜑 → (0...𝑀) ∈ Fin)
65		fzssnn0 42856	. . . . . . . 8 ⊢ (0...𝑁) ⊆ ℕ₀
66	14	ffvelrnda 6961	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑗 ∈ (0...𝑀)) → (𝐶‘𝑗) ∈ (0...𝑁))
67	65, 66	sselid 3919	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑗 ∈ (0...𝑀)) → (𝐶‘𝑗) ∈ ℕ₀)
68	67	faccld 13998	. . . . . 6 ⊢ ((𝜑 ∧ 𝑗 ∈ (0...𝑀)) → (!‘(𝐶‘𝑗)) ∈ ℕ)
69	68	nncnd 11989	. . . . 5 ⊢ ((𝜑 ∧ 𝑗 ∈ (0...𝑀)) → (!‘(𝐶‘𝑗)) ∈ ℂ)
70	64, 69	fprodcl 15662	. . . 4 ⊢ (𝜑 → ∏𝑗 ∈ (0...𝑀)(!‘(𝐶‘𝑗)) ∈ ℂ)
71	68	nnne0d 12023	. . . . 5 ⊢ ((𝜑 ∧ 𝑗 ∈ (0...𝑀)) → (!‘(𝐶‘𝑗)) ≠ 0)
72	64, 69, 71	fprodn0 15689	. . . 4 ⊢ (𝜑 → ∏𝑗 ∈ (0...𝑀)(!‘(𝐶‘𝑗)) ≠ 0)
73	63, 70, 72	divcld 11751	. . 3 ⊢ (𝜑 → ((!‘𝑁) / ∏𝑗 ∈ (0...𝑀)(!‘(𝐶‘𝑗))) ∈ ℂ)
74	73	mul01d 11174	. 2 ⊢ (𝜑 → (((!‘𝑁) / ∏𝑗 ∈ (0...𝑀)(!‘(𝐶‘𝑗))) · 0) = 0)
75	2, 60, 74	3eqtrd 2782	1 ⊢ (𝜑 → 𝑇 = 0)

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ∧ wa 396 = wceq 1539 ∈ wcel 2106 ≠ wne 2943 ifcif 4459 class class class wbr 5074 ⟶wf 6429 ‘cfv 6433 (class class class)co 7275 ℂcc 10869 ℝcr 10870 0cc0 10871 1c1 10872 · cmul 10876 < clt 11009 − cmin 11205 / cdiv 11632 ℕcn 11973 ℕ₀cn0 12233 ℤcz 12319 ℤ_≥cuz 12582 ...cfz 13239 ↑cexp 13782 !cfa 13987 ∏cprod 15615

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 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2709 ax-rep 5209 ax-sep 5223 ax-nul 5230 ax-pow 5288 ax-pr 5352 ax-un 7588 ax-inf2 9399 ax-cnex 10927 ax-resscn 10928 ax-1cn 10929 ax-icn 10930 ax-addcl 10931 ax-addrcl 10932 ax-mulcl 10933 ax-mulrcl 10934 ax-mulcom 10935 ax-addass 10936 ax-mulass 10937 ax-distr 10938 ax-i2m1 10939 ax-1ne0 10940 ax-1rid 10941 ax-rnegex 10942 ax-rrecex 10943 ax-cnre 10944 ax-pre-lttri 10945 ax-pre-lttrn 10946 ax-pre-ltadd 10947 ax-pre-mulgt0 10948 ax-pre-sup 10949

This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2068 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2889 df-ne 2944 df-nel 3050 df-ral 3069 df-rex 3070 df-rmo 3071 df-reu 3072 df-rab 3073 df-v 3434 df-sbc 3717 df-csb 3833 df-dif 3890 df-un 3892 df-in 3894 df-ss 3904 df-pss 3906 df-nul 4257 df-if 4460 df-pw 4535 df-sn 4562 df-pr 4564 df-op 4568 df-uni 4840 df-int 4880 df-iun 4926 df-br 5075 df-opab 5137 df-mpt 5158 df-tr 5192 df-id 5489 df-eprel 5495 df-po 5503 df-so 5504 df-fr 5544 df-se 5545 df-we 5546 df-xp 5595 df-rel 5596 df-cnv 5597 df-co 5598 df-dm 5599 df-rn 5600 df-res 5601 df-ima 5602 df-pred 6202 df-ord 6269 df-on 6270 df-lim 6271 df-suc 6272 df-iota 6391 df-fun 6435 df-fn 6436 df-f 6437 df-f1 6438 df-fo 6439 df-f1o 6440 df-fv 6441 df-isom 6442 df-riota 7232 df-ov 7278 df-oprab 7279 df-mpo 7280 df-om 7713 df-1st 7831 df-2nd 7832 df-frecs 8097 df-wrecs 8128 df-recs 8202 df-rdg 8241 df-1o 8297 df-er 8498 df-en 8734 df-dom 8735 df-sdom 8736 df-fin 8737 df-sup 9201 df-oi 9269 df-card 9697 df-pnf 11011 df-mnf 11012 df-xr 11013 df-ltxr 11014 df-le 11015 df-sub 11207 df-neg 11208 df-div 11633 df-nn 11974 df-2 12036 df-3 12037 df-n0 12234 df-z 12320 df-uz 12583 df-rp 12731 df-fz 13240 df-fzo 13383 df-seq 13722 df-exp 13783 df-fac 13988 df-bc 14017 df-hash 14045 df-cj 14810 df-re 14811 df-im 14812 df-sqrt 14946 df-abs 14947 df-clim 15197 df-prod 15616

This theorem is referenced by: etransclem28 43803

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