etransclem15 - Mathbox for Glauco Siliprandi

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Theorem etransclem15 46230

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 4482	. . . . . . 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 4485	. . . . . . . 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 7364	. . . . . . . . . 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 12498	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝑃 ∈ ℤ)
12		1zzd 12506	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 1 ∈ ℤ)
13	11, 12	zsubcld 12585	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝑃 − 1) ∈ ℤ)
14		etransclem15.c	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝐶:(0...𝑀)⟶(0...𝑁))
15		etransclem15.m	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → 𝑀 ∈ ℕ₀)
16		nn0uz 12777	. . . . . . . . . . . . . . . . 17 ⊢ ℕ₀ = (ℤ_≥‘0)
17	15, 16	eleqtrdi 2838	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 𝑀 ∈ (ℤ_≥‘0))
18		eluzfz1 13434	. . . . . . . . . . . . . . . 16 ⊢ (𝑀 ∈ (ℤ_≥‘0) → 0 ∈ (0...𝑀))
19	17, 18	syl 17	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 0 ∈ (0...𝑀))
20	14, 19	ffvelcdmd 7019	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (𝐶‘0) ∈ (0...𝑁))
21	20	elfzelzd 13428	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝐶‘0) ∈ ℤ)
22	13, 21	zsubcld 12585	. . . . . . . . . . . 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 12580	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (𝐶‘0) ∈ ℝ)
25	24	adantr 480	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ ¬ (𝑃 − 1) < (𝐶‘0)) → (𝐶‘0) ∈ ℝ)
26	13	zred 12580	. . . . . . . . . . . . . 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 11266	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ ¬ (𝑃 − 1) < (𝐶‘0)) → (𝐶‘0) ≤ (𝑃 − 1))
30		etransclem15.cpm1	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (𝐶‘0) ≠ (𝑃 − 1))
31	30	necomd 2980	. . . . . . . . . . . . . 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 11270	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ ¬ (𝑃 − 1) < (𝐶‘0)) → (𝐶‘0) < (𝑃 − 1))
34	25, 27	posdifd 11707	. . . . . . . . . . . 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 12481	. . . . . . . . . . 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 14046	. . . . . . . . 9 ⊢ ((𝜑 ∧ ¬ (𝑃 − 1) < (𝐶‘0)) → (0↑((𝑃 − 1) − (𝐶‘0))) = 0)
39	9, 38	eqtrd 2764	. . . . . . . 8 ⊢ ((𝜑 ∧ ¬ (𝑃 − 1) < (𝐶‘0)) → (𝐽↑((𝑃 − 1) − (𝐶‘0))) = 0)
40	39	oveq2d 7365	. . . . . . 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 12425	. . . . . . . . . . . . 13 ⊢ (𝑃 ∈ ℕ → (𝑃 − 1) ∈ ℕ₀)
42	10, 41	syl 17	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝑃 − 1) ∈ ℕ₀)
43	42	faccld 14191	. . . . . . . . . . 11 ⊢ (𝜑 → (!‘(𝑃 − 1)) ∈ ℕ)
44	43	nncnd 12144	. . . . . . . . . 10 ⊢ (𝜑 → (!‘(𝑃 − 1)) ∈ ℂ)
45	44	adantr 480	. . . . . . . . 9 ⊢ ((𝜑 ∧ ¬ (𝑃 − 1) < (𝐶‘0)) → (!‘(𝑃 − 1)) ∈ ℂ)
46	37	nnnn0d 12445	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ ¬ (𝑃 − 1) < (𝐶‘0)) → ((𝑃 − 1) − (𝐶‘0)) ∈ ℕ₀)
47	46	faccld 14191	. . . . . . . . . 10 ⊢ ((𝜑 ∧ ¬ (𝑃 − 1) < (𝐶‘0)) → (!‘((𝑃 − 1) − (𝐶‘0))) ∈ ℕ)
48	47	nncnd 12144	. . . . . . . . 9 ⊢ ((𝜑 ∧ ¬ (𝑃 − 1) < (𝐶‘0)) → (!‘((𝑃 − 1) − (𝐶‘0))) ∈ ℂ)
49	47	nnne0d 12178	. . . . . . . . 9 ⊢ ((𝜑 ∧ ¬ (𝑃 − 1) < (𝐶‘0)) → (!‘((𝑃 − 1) − (𝐶‘0))) ≠ 0)
50	45, 48, 49	divcld 11900	. . . . . . . 8 ⊢ ((𝜑 ∧ ¬ (𝑃 − 1) < (𝐶‘0)) → ((!‘(𝑃 − 1)) / (!‘((𝑃 − 1) − (𝐶‘0)))) ∈ ℂ)
51	50	mul01d 11315	. . . . . . 7 ⊢ ((𝜑 ∧ ¬ (𝑃 − 1) < (𝐶‘0)) → (((!‘(𝑃 − 1)) / (!‘((𝑃 − 1) − (𝐶‘0)))) · 0) = 0)
52	6, 40, 51	3eqtrd 2768	. . . . . 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 7364	. . . 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 2828	. . . . . . 7 ⊢ (𝜑 → 𝐽 ∈ (0...𝑀))
56	10, 14, 55	etransclem7 46222	. . . . . 6 ⊢ (𝜑 → ∏𝑗 ∈ (1...𝑀)if(𝑃 < (𝐶‘𝑗), 0, (((!‘𝑃) / (!‘(𝑃 − (𝐶‘𝑗)))) · ((𝐽 − 𝑗)↑(𝑃 − (𝐶‘𝑗))))) ∈ ℤ)
57	56	zcnd 12581	. . . . 5 ⊢ (𝜑 → ∏𝑗 ∈ (1...𝑀)if(𝑃 < (𝐶‘𝑗), 0, (((!‘𝑃) / (!‘(𝑃 − (𝐶‘𝑗)))) · ((𝐽 − 𝑗)↑(𝑃 − (𝐶‘𝑗))))) ∈ ℂ)
58	57	mul02d 11314	. . . 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 2764	. . 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 7365	. 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 14191	. . . . 5 ⊢ (𝜑 → (!‘𝑁) ∈ ℕ)
63	62	nncnd 12144	. . . 4 ⊢ (𝜑 → (!‘𝑁) ∈ ℂ)
64		fzfid 13880	. . . . 5 ⊢ (𝜑 → (0...𝑀) ∈ Fin)
65		fzssnn0 45298	. . . . . . . 8 ⊢ (0...𝑁) ⊆ ℕ₀
66	14	ffvelcdmda 7018	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑗 ∈ (0...𝑀)) → (𝐶‘𝑗) ∈ (0...𝑁))
67	65, 66	sselid 3933	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑗 ∈ (0...𝑀)) → (𝐶‘𝑗) ∈ ℕ₀)
68	67	faccld 14191	. . . . . 6 ⊢ ((𝜑 ∧ 𝑗 ∈ (0...𝑀)) → (!‘(𝐶‘𝑗)) ∈ ℕ)
69	68	nncnd 12144	. . . . 5 ⊢ ((𝜑 ∧ 𝑗 ∈ (0...𝑀)) → (!‘(𝐶‘𝑗)) ∈ ℂ)
70	64, 69	fprodcl 15859	. . . 4 ⊢ (𝜑 → ∏𝑗 ∈ (0...𝑀)(!‘(𝐶‘𝑗)) ∈ ℂ)
71	68	nnne0d 12178	. . . . 5 ⊢ ((𝜑 ∧ 𝑗 ∈ (0...𝑀)) → (!‘(𝐶‘𝑗)) ≠ 0)
72	64, 69, 71	fprodn0 15886	. . . 4 ⊢ (𝜑 → ∏𝑗 ∈ (0...𝑀)(!‘(𝐶‘𝑗)) ≠ 0)
73	63, 70, 72	divcld 11900	. . 3 ⊢ (𝜑 → ((!‘𝑁) / ∏𝑗 ∈ (0...𝑀)(!‘(𝐶‘𝑗))) ∈ ℂ)
74	73	mul01d 11315	. 2 ⊢ (𝜑 → (((!‘𝑁) / ∏𝑗 ∈ (0...𝑀)(!‘(𝐶‘𝑗))) · 0) = 0)
75	2, 60, 74	3eqtrd 2768	1 ⊢ (𝜑 → 𝑇 = 0)

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ∧ wa 395 = wceq 1540 ∈ wcel 2109 ≠ wne 2925 ifcif 4476 class class class wbr 5092 ⟶wf 6478 ‘cfv 6482 (class class class)co 7349 ℂcc 11007 ℝcr 11008 0cc0 11009 1c1 11010 · cmul 11014 < clt 11149 − cmin 11347 / cdiv 11777 ℕcn 12128 ℕ₀cn0 12384 ℤcz 12471 ℤ_≥cuz 12735 ...cfz 13410 ↑cexp 13968 !cfa 14180 ∏cprod 15810

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 5218 ax-sep 5235 ax-nul 5245 ax-pow 5304 ax-pr 5371 ax-un 7671 ax-inf2 9537 ax-cnex 11065 ax-resscn 11066 ax-1cn 11067 ax-icn 11068 ax-addcl 11069 ax-addrcl 11070 ax-mulcl 11071 ax-mulrcl 11072 ax-mulcom 11073 ax-addass 11074 ax-mulass 11075 ax-distr 11076 ax-i2m1 11077 ax-1ne0 11078 ax-1rid 11079 ax-rnegex 11080 ax-rrecex 11081 ax-cnre 11082 ax-pre-lttri 11083 ax-pre-lttrn 11084 ax-pre-ltadd 11085 ax-pre-mulgt0 11086 ax-pre-sup 11087

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 3343 df-reu 3344 df-rab 3395 df-v 3438 df-sbc 3743 df-csb 3852 df-dif 3906 df-un 3908 df-in 3910 df-ss 3920 df-pss 3923 df-nul 4285 df-if 4477 df-pw 4553 df-sn 4578 df-pr 4580 df-op 4584 df-uni 4859 df-int 4897 df-iun 4943 df-br 5093 df-opab 5155 df-mpt 5174 df-tr 5200 df-id 5514 df-eprel 5519 df-po 5527 df-so 5528 df-fr 5572 df-se 5573 df-we 5574 df-xp 5625 df-rel 5626 df-cnv 5627 df-co 5628 df-dm 5629 df-rn 5630 df-res 5631 df-ima 5632 df-pred 6249 df-ord 6310 df-on 6311 df-lim 6312 df-suc 6313 df-iota 6438 df-fun 6484 df-fn 6485 df-f 6486 df-f1 6487 df-fo 6488 df-f1o 6489 df-fv 6490 df-isom 6491 df-riota 7306 df-ov 7352 df-oprab 7353 df-mpo 7354 df-om 7800 df-1st 7924 df-2nd 7925 df-frecs 8214 df-wrecs 8245 df-recs 8294 df-rdg 8332 df-1o 8388 df-er 8625 df-en 8873 df-dom 8874 df-sdom 8875 df-fin 8876 df-sup 9332 df-oi 9402 df-card 9835 df-pnf 11151 df-mnf 11152 df-xr 11153 df-ltxr 11154 df-le 11155 df-sub 11349 df-neg 11350 df-div 11778 df-nn 12129 df-2 12191 df-3 12192 df-n0 12385 df-z 12472 df-uz 12736 df-rp 12894 df-fz 13411 df-fzo 13558 df-seq 13909 df-exp 13969 df-fac 14181 df-bc 14210 df-hash 14238 df-cj 15006 df-re 15007 df-im 15008 df-sqrt 15142 df-abs 15143 df-clim 15395 df-prod 15811

This theorem is referenced by: etransclem28 46243

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