etransclem15 - Mathbox for Glauco Siliprandi

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Theorem etransclem15 46245

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 4511	. . . . . . 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 481	. . . . . 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 4514	. . . . . . . 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 481	. . . . . . 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 7425	. . . . . . . . . 10 ⊢ (𝜑 → (𝐽↑((𝑃 − 1) − (𝐶‘0))) = (0↑((𝑃 − 1) − (𝐶‘0))))
9	8	adantr 480	. . . . . . . . 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 12620	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝑃 ∈ ℤ)
12		1zzd 12628	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 1 ∈ ℤ)
13	11, 12	zsubcld 12707	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝑃 − 1) ∈ ℤ)
14		etransclem15.c	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝐶:(0...𝑀)⟶(0...𝑁))
15		etransclem15.m	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → 𝑀 ∈ ℕ₀)
16		nn0uz 12899	. . . . . . . . . . . . . . . . 17 ⊢ ℕ₀ = (ℤ_≥‘0)
17	15, 16	eleqtrdi 2845	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 𝑀 ∈ (ℤ_≥‘0))
18		eluzfz1 13553	. . . . . . . . . . . . . . . 16 ⊢ (𝑀 ∈ (ℤ_≥‘0) → 0 ∈ (0...𝑀))
19	17, 18	syl 17	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 0 ∈ (0...𝑀))
20	14, 19	ffvelcdmd 7080	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (𝐶‘0) ∈ (0...𝑁))
21	20	elfzelzd 13547	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝐶‘0) ∈ ℤ)
22	13, 21	zsubcld 12707	. . . . . . . . . . . 12 ⊢ (𝜑 → ((𝑃 − 1) − (𝐶‘0)) ∈ ℤ)
23	22	adantr 480	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ ¬ (𝑃 − 1) < (𝐶‘0)) → ((𝑃 − 1) − (𝐶‘0)) ∈ ℤ)
24	21	zred 12702	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (𝐶‘0) ∈ ℝ)
25	24	adantr 480	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ ¬ (𝑃 − 1) < (𝐶‘0)) → (𝐶‘0) ∈ ℝ)
26	13	zred 12702	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (𝑃 − 1) ∈ ℝ)
27	26	adantr 480	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ ¬ (𝑃 − 1) < (𝐶‘0)) → (𝑃 − 1) ∈ ℝ)
28		simpr 484	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ ¬ (𝑃 − 1) < (𝐶‘0)) → ¬ (𝑃 − 1) < (𝐶‘0))
29	25, 27, 28	nltled 11390	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ ¬ (𝑃 − 1) < (𝐶‘0)) → (𝐶‘0) ≤ (𝑃 − 1))
30		etransclem15.cpm1	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (𝐶‘0) ≠ (𝑃 − 1))
31	30	necomd 2988	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (𝑃 − 1) ≠ (𝐶‘0))
32	31	adantr 480	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ ¬ (𝑃 − 1) < (𝐶‘0)) → (𝑃 − 1) ≠ (𝐶‘0))
33	25, 27, 29, 32	leneltd 11394	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ ¬ (𝑃 − 1) < (𝐶‘0)) → (𝐶‘0) < (𝑃 − 1))
34	25, 27	posdifd 11829	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ ¬ (𝑃 − 1) < (𝐶‘0)) → ((𝐶‘0) < (𝑃 − 1) ↔ 0 < ((𝑃 − 1) − (𝐶‘0))))
35	33, 34	mpbid 232	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ ¬ (𝑃 − 1) < (𝐶‘0)) → 0 < ((𝑃 − 1) − (𝐶‘0)))
36		elnnz 12603	. . . . . . . . . . 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 14162	. . . . . . . . 9 ⊢ ((𝜑 ∧ ¬ (𝑃 − 1) < (𝐶‘0)) → (0↑((𝑃 − 1) − (𝐶‘0))) = 0)
39	9, 38	eqtrd 2771	. . . . . . . 8 ⊢ ((𝜑 ∧ ¬ (𝑃 − 1) < (𝐶‘0)) → (𝐽↑((𝑃 − 1) − (𝐶‘0))) = 0)
40	39	oveq2d 7426	. . . . . . 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 12547	. . . . . . . . . . . . 13 ⊢ (𝑃 ∈ ℕ → (𝑃 − 1) ∈ ℕ₀)
42	10, 41	syl 17	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝑃 − 1) ∈ ℕ₀)
43	42	faccld 14307	. . . . . . . . . . 11 ⊢ (𝜑 → (!‘(𝑃 − 1)) ∈ ℕ)
44	43	nncnd 12261	. . . . . . . . . 10 ⊢ (𝜑 → (!‘(𝑃 − 1)) ∈ ℂ)
45	44	adantr 480	. . . . . . . . 9 ⊢ ((𝜑 ∧ ¬ (𝑃 − 1) < (𝐶‘0)) → (!‘(𝑃 − 1)) ∈ ℂ)
46	37	nnnn0d 12567	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ ¬ (𝑃 − 1) < (𝐶‘0)) → ((𝑃 − 1) − (𝐶‘0)) ∈ ℕ₀)
47	46	faccld 14307	. . . . . . . . . 10 ⊢ ((𝜑 ∧ ¬ (𝑃 − 1) < (𝐶‘0)) → (!‘((𝑃 − 1) − (𝐶‘0))) ∈ ℕ)
48	47	nncnd 12261	. . . . . . . . 9 ⊢ ((𝜑 ∧ ¬ (𝑃 − 1) < (𝐶‘0)) → (!‘((𝑃 − 1) − (𝐶‘0))) ∈ ℂ)
49	47	nnne0d 12295	. . . . . . . . 9 ⊢ ((𝜑 ∧ ¬ (𝑃 − 1) < (𝐶‘0)) → (!‘((𝑃 − 1) − (𝐶‘0))) ≠ 0)
50	45, 48, 49	divcld 12022	. . . . . . . 8 ⊢ ((𝜑 ∧ ¬ (𝑃 − 1) < (𝐶‘0)) → ((!‘(𝑃 − 1)) / (!‘((𝑃 − 1) − (𝐶‘0)))) ∈ ℂ)
51	50	mul01d 11439	. . . . . . 7 ⊢ ((𝜑 ∧ ¬ (𝑃 − 1) < (𝐶‘0)) → (((!‘(𝑃 − 1)) / (!‘((𝑃 − 1) − (𝐶‘0)))) · 0) = 0)
52	6, 40, 51	3eqtrd 2775	. . . . . 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 812	. . . . 5 ⊢ (𝜑 → if((𝑃 − 1) < (𝐶‘0), 0, (((!‘(𝑃 − 1)) / (!‘((𝑃 − 1) − (𝐶‘0)))) · (𝐽↑((𝑃 − 1) − (𝐶‘0))))) = 0)
54	53	oveq1d 7425	. . . 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 2835	. . . . . . 7 ⊢ (𝜑 → 𝐽 ∈ (0...𝑀))
56	10, 14, 55	etransclem7 46237	. . . . . 6 ⊢ (𝜑 → ∏𝑗 ∈ (1...𝑀)if(𝑃 < (𝐶‘𝑗), 0, (((!‘𝑃) / (!‘(𝑃 − (𝐶‘𝑗)))) · ((𝐽 − 𝑗)↑(𝑃 − (𝐶‘𝑗))))) ∈ ℤ)
57	56	zcnd 12703	. . . . 5 ⊢ (𝜑 → ∏𝑗 ∈ (1...𝑀)if(𝑃 < (𝐶‘𝑗), 0, (((!‘𝑃) / (!‘(𝑃 − (𝐶‘𝑗)))) · ((𝐽 − 𝑗)↑(𝑃 − (𝐶‘𝑗))))) ∈ ℂ)
58	57	mul02d 11438	. . . 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 2771	. . 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 7426	. 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 14307	. . . . 5 ⊢ (𝜑 → (!‘𝑁) ∈ ℕ)
63	62	nncnd 12261	. . . 4 ⊢ (𝜑 → (!‘𝑁) ∈ ℂ)
64		fzfid 13996	. . . . 5 ⊢ (𝜑 → (0...𝑀) ∈ Fin)
65		fzssnn0 45312	. . . . . . . 8 ⊢ (0...𝑁) ⊆ ℕ₀
66	14	ffvelcdmda 7079	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑗 ∈ (0...𝑀)) → (𝐶‘𝑗) ∈ (0...𝑁))
67	65, 66	sselid 3961	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑗 ∈ (0...𝑀)) → (𝐶‘𝑗) ∈ ℕ₀)
68	67	faccld 14307	. . . . . 6 ⊢ ((𝜑 ∧ 𝑗 ∈ (0...𝑀)) → (!‘(𝐶‘𝑗)) ∈ ℕ)
69	68	nncnd 12261	. . . . 5 ⊢ ((𝜑 ∧ 𝑗 ∈ (0...𝑀)) → (!‘(𝐶‘𝑗)) ∈ ℂ)
70	64, 69	fprodcl 15973	. . . 4 ⊢ (𝜑 → ∏𝑗 ∈ (0...𝑀)(!‘(𝐶‘𝑗)) ∈ ℂ)
71	68	nnne0d 12295	. . . . 5 ⊢ ((𝜑 ∧ 𝑗 ∈ (0...𝑀)) → (!‘(𝐶‘𝑗)) ≠ 0)
72	64, 69, 71	fprodn0 16000	. . . 4 ⊢ (𝜑 → ∏𝑗 ∈ (0...𝑀)(!‘(𝐶‘𝑗)) ≠ 0)
73	63, 70, 72	divcld 12022	. . 3 ⊢ (𝜑 → ((!‘𝑁) / ∏𝑗 ∈ (0...𝑀)(!‘(𝐶‘𝑗))) ∈ ℂ)
74	73	mul01d 11439	. 2 ⊢ (𝜑 → (((!‘𝑁) / ∏𝑗 ∈ (0...𝑀)(!‘(𝐶‘𝑗))) · 0) = 0)
75	2, 60, 74	3eqtrd 2775	1 ⊢ (𝜑 → 𝑇 = 0)

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ∧ wa 395 = wceq 1540 ∈ wcel 2109 ≠ wne 2933 ifcif 4505 class class class wbr 5124 ⟶wf 6532 ‘cfv 6536 (class class class)co 7410 ℂcc 11132 ℝcr 11133 0cc0 11134 1c1 11135 · cmul 11139 < clt 11274 − cmin 11471 / cdiv 11899 ℕcn 12245 ℕ₀cn0 12506 ℤcz 12593 ℤ_≥cuz 12857 ...cfz 13529 ↑cexp 14084 !cfa 14296 ∏cprod 15924

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 2708 ax-rep 5254 ax-sep 5271 ax-nul 5281 ax-pow 5340 ax-pr 5407 ax-un 7734 ax-inf2 9660 ax-cnex 11190 ax-resscn 11191 ax-1cn 11192 ax-icn 11193 ax-addcl 11194 ax-addrcl 11195 ax-mulcl 11196 ax-mulrcl 11197 ax-mulcom 11198 ax-addass 11199 ax-mulass 11200 ax-distr 11201 ax-i2m1 11202 ax-1ne0 11203 ax-1rid 11204 ax-rnegex 11205 ax-rrecex 11206 ax-cnre 11207 ax-pre-lttri 11208 ax-pre-lttrn 11209 ax-pre-ltadd 11210 ax-pre-mulgt0 11211 ax-pre-sup 11212

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 2540 df-eu 2569 df-clab 2715 df-cleq 2728 df-clel 2810 df-nfc 2886 df-ne 2934 df-nel 3038 df-ral 3053 df-rex 3062 df-rmo 3364 df-reu 3365 df-rab 3421 df-v 3466 df-sbc 3771 df-csb 3880 df-dif 3934 df-un 3936 df-in 3938 df-ss 3948 df-pss 3951 df-nul 4314 df-if 4506 df-pw 4582 df-sn 4607 df-pr 4609 df-op 4613 df-uni 4889 df-int 4928 df-iun 4974 df-br 5125 df-opab 5187 df-mpt 5207 df-tr 5235 df-id 5553 df-eprel 5558 df-po 5566 df-so 5567 df-fr 5611 df-se 5612 df-we 5613 df-xp 5665 df-rel 5666 df-cnv 5667 df-co 5668 df-dm 5669 df-rn 5670 df-res 5671 df-ima 5672 df-pred 6295 df-ord 6360 df-on 6361 df-lim 6362 df-suc 6363 df-iota 6489 df-fun 6538 df-fn 6539 df-f 6540 df-f1 6541 df-fo 6542 df-f1o 6543 df-fv 6544 df-isom 6545 df-riota 7367 df-ov 7413 df-oprab 7414 df-mpo 7415 df-om 7867 df-1st 7993 df-2nd 7994 df-frecs 8285 df-wrecs 8316 df-recs 8390 df-rdg 8429 df-1o 8485 df-er 8724 df-en 8965 df-dom 8966 df-sdom 8967 df-fin 8968 df-sup 9459 df-oi 9529 df-card 9958 df-pnf 11276 df-mnf 11277 df-xr 11278 df-ltxr 11279 df-le 11280 df-sub 11473 df-neg 11474 df-div 11900 df-nn 12246 df-2 12308 df-3 12309 df-n0 12507 df-z 12594 df-uz 12858 df-rp 13014 df-fz 13530 df-fzo 13677 df-seq 14025 df-exp 14085 df-fac 14297 df-bc 14326 df-hash 14354 df-cj 15123 df-re 15124 df-im 15125 df-sqrt 15259 df-abs 15260 df-clim 15509 df-prod 15925

This theorem is referenced by: etransclem28 46258

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