etransclem15 - Mathbox for Glauco Siliprandi

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Theorem etransclem15 45870

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 4539	. . . . . . 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 480	. . . . . 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 4542	. . . . . . . 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 480	. . . . . . 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 7439	. . . . . . . . . 10 ⊢ (𝜑 → (𝐽↑((𝑃 − 1) − (𝐶‘0))) = (0↑((𝑃 − 1) − (𝐶‘0))))
9	8	adantr 479	. . . . . . . . 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 12637	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝑃 ∈ ℤ)
12		1zzd 12645	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 1 ∈ ℤ)
13	11, 12	zsubcld 12723	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝑃 − 1) ∈ ℤ)
14		etransclem15.c	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝐶:(0...𝑀)⟶(0...𝑁))
15		etransclem15.m	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → 𝑀 ∈ ℕ₀)
16		nn0uz 12916	. . . . . . . . . . . . . . . . 17 ⊢ ℕ₀ = (ℤ_≥‘0)
17	15, 16	eleqtrdi 2836	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 𝑀 ∈ (ℤ_≥‘0))
18		eluzfz1 13562	. . . . . . . . . . . . . . . 16 ⊢ (𝑀 ∈ (ℤ_≥‘0) → 0 ∈ (0...𝑀))
19	17, 18	syl 17	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 0 ∈ (0...𝑀))
20	14, 19	ffvelcdmd 7099	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (𝐶‘0) ∈ (0...𝑁))
21	20	elfzelzd 13556	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝐶‘0) ∈ ℤ)
22	13, 21	zsubcld 12723	. . . . . . . . . . . 12 ⊢ (𝜑 → ((𝑃 − 1) − (𝐶‘0)) ∈ ℤ)
23	22	adantr 479	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ ¬ (𝑃 − 1) < (𝐶‘0)) → ((𝑃 − 1) − (𝐶‘0)) ∈ ℤ)
24	21	zred 12718	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (𝐶‘0) ∈ ℝ)
25	24	adantr 479	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ ¬ (𝑃 − 1) < (𝐶‘0)) → (𝐶‘0) ∈ ℝ)
26	13	zred 12718	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (𝑃 − 1) ∈ ℝ)
27	26	adantr 479	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ ¬ (𝑃 − 1) < (𝐶‘0)) → (𝑃 − 1) ∈ ℝ)
28		simpr 483	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ ¬ (𝑃 − 1) < (𝐶‘0)) → ¬ (𝑃 − 1) < (𝐶‘0))
29	25, 27, 28	nltled 11414	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ ¬ (𝑃 − 1) < (𝐶‘0)) → (𝐶‘0) ≤ (𝑃 − 1))
30		etransclem15.cpm1	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (𝐶‘0) ≠ (𝑃 − 1))
31	30	necomd 2986	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (𝑃 − 1) ≠ (𝐶‘0))
32	31	adantr 479	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ ¬ (𝑃 − 1) < (𝐶‘0)) → (𝑃 − 1) ≠ (𝐶‘0))
33	25, 27, 29, 32	leneltd 11418	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ ¬ (𝑃 − 1) < (𝐶‘0)) → (𝐶‘0) < (𝑃 − 1))
34	25, 27	posdifd 11851	. . . . . . . . . . . 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 12620	. . . . . . . . . . 11 ⊢ (((𝑃 − 1) − (𝐶‘0)) ∈ ℕ ↔ (((𝑃 − 1) − (𝐶‘0)) ∈ ℤ ∧ 0 < ((𝑃 − 1) − (𝐶‘0))))
37	23, 35, 36	sylanbrc 581	. . . . . . . . . 10 ⊢ ((𝜑 ∧ ¬ (𝑃 − 1) < (𝐶‘0)) → ((𝑃 − 1) − (𝐶‘0)) ∈ ℕ)
38	37	0expd 14158	. . . . . . . . 9 ⊢ ((𝜑 ∧ ¬ (𝑃 − 1) < (𝐶‘0)) → (0↑((𝑃 − 1) − (𝐶‘0))) = 0)
39	9, 38	eqtrd 2766	. . . . . . . 8 ⊢ ((𝜑 ∧ ¬ (𝑃 − 1) < (𝐶‘0)) → (𝐽↑((𝑃 − 1) − (𝐶‘0))) = 0)
40	39	oveq2d 7440	. . . . . . 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 12565	. . . . . . . . . . . . 13 ⊢ (𝑃 ∈ ℕ → (𝑃 − 1) ∈ ℕ₀)
42	10, 41	syl 17	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝑃 − 1) ∈ ℕ₀)
43	42	faccld 14301	. . . . . . . . . . 11 ⊢ (𝜑 → (!‘(𝑃 − 1)) ∈ ℕ)
44	43	nncnd 12280	. . . . . . . . . 10 ⊢ (𝜑 → (!‘(𝑃 − 1)) ∈ ℂ)
45	44	adantr 479	. . . . . . . . 9 ⊢ ((𝜑 ∧ ¬ (𝑃 − 1) < (𝐶‘0)) → (!‘(𝑃 − 1)) ∈ ℂ)
46	37	nnnn0d 12584	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ ¬ (𝑃 − 1) < (𝐶‘0)) → ((𝑃 − 1) − (𝐶‘0)) ∈ ℕ₀)
47	46	faccld 14301	. . . . . . . . . 10 ⊢ ((𝜑 ∧ ¬ (𝑃 − 1) < (𝐶‘0)) → (!‘((𝑃 − 1) − (𝐶‘0))) ∈ ℕ)
48	47	nncnd 12280	. . . . . . . . 9 ⊢ ((𝜑 ∧ ¬ (𝑃 − 1) < (𝐶‘0)) → (!‘((𝑃 − 1) − (𝐶‘0))) ∈ ℂ)
49	47	nnne0d 12314	. . . . . . . . 9 ⊢ ((𝜑 ∧ ¬ (𝑃 − 1) < (𝐶‘0)) → (!‘((𝑃 − 1) − (𝐶‘0))) ≠ 0)
50	45, 48, 49	divcld 12041	. . . . . . . 8 ⊢ ((𝜑 ∧ ¬ (𝑃 − 1) < (𝐶‘0)) → ((!‘(𝑃 − 1)) / (!‘((𝑃 − 1) − (𝐶‘0)))) ∈ ℂ)
51	50	mul01d 11463	. . . . . . 7 ⊢ ((𝜑 ∧ ¬ (𝑃 − 1) < (𝐶‘0)) → (((!‘(𝑃 − 1)) / (!‘((𝑃 − 1) − (𝐶‘0)))) · 0) = 0)
52	6, 40, 51	3eqtrd 2770	. . . . . 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 811	. . . . 5 ⊢ (𝜑 → if((𝑃 − 1) < (𝐶‘0), 0, (((!‘(𝑃 − 1)) / (!‘((𝑃 − 1) − (𝐶‘0)))) · (𝐽↑((𝑃 − 1) − (𝐶‘0))))) = 0)
54	53	oveq1d 7439	. . . 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 2826	. . . . . . 7 ⊢ (𝜑 → 𝐽 ∈ (0...𝑀))
56	10, 14, 55	etransclem7 45862	. . . . . 6 ⊢ (𝜑 → ∏𝑗 ∈ (1...𝑀)if(𝑃 < (𝐶‘𝑗), 0, (((!‘𝑃) / (!‘(𝑃 − (𝐶‘𝑗)))) · ((𝐽 − 𝑗)↑(𝑃 − (𝐶‘𝑗))))) ∈ ℤ)
57	56	zcnd 12719	. . . . 5 ⊢ (𝜑 → ∏𝑗 ∈ (1...𝑀)if(𝑃 < (𝐶‘𝑗), 0, (((!‘𝑃) / (!‘(𝑃 − (𝐶‘𝑗)))) · ((𝐽 − 𝑗)↑(𝑃 − (𝐶‘𝑗))))) ∈ ℂ)
58	57	mul02d 11462	. . . 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 2766	. . 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 7440	. 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 14301	. . . . 5 ⊢ (𝜑 → (!‘𝑁) ∈ ℕ)
63	62	nncnd 12280	. . . 4 ⊢ (𝜑 → (!‘𝑁) ∈ ℂ)
64		fzfid 13993	. . . . 5 ⊢ (𝜑 → (0...𝑀) ∈ Fin)
65		fzssnn0 44932	. . . . . . . 8 ⊢ (0...𝑁) ⊆ ℕ₀
66	14	ffvelcdmda 7098	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑗 ∈ (0...𝑀)) → (𝐶‘𝑗) ∈ (0...𝑁))
67	65, 66	sselid 3977	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑗 ∈ (0...𝑀)) → (𝐶‘𝑗) ∈ ℕ₀)
68	67	faccld 14301	. . . . . 6 ⊢ ((𝜑 ∧ 𝑗 ∈ (0...𝑀)) → (!‘(𝐶‘𝑗)) ∈ ℕ)
69	68	nncnd 12280	. . . . 5 ⊢ ((𝜑 ∧ 𝑗 ∈ (0...𝑀)) → (!‘(𝐶‘𝑗)) ∈ ℂ)
70	64, 69	fprodcl 15954	. . . 4 ⊢ (𝜑 → ∏𝑗 ∈ (0...𝑀)(!‘(𝐶‘𝑗)) ∈ ℂ)
71	68	nnne0d 12314	. . . . 5 ⊢ ((𝜑 ∧ 𝑗 ∈ (0...𝑀)) → (!‘(𝐶‘𝑗)) ≠ 0)
72	64, 69, 71	fprodn0 15981	. . . 4 ⊢ (𝜑 → ∏𝑗 ∈ (0...𝑀)(!‘(𝐶‘𝑗)) ≠ 0)
73	63, 70, 72	divcld 12041	. . 3 ⊢ (𝜑 → ((!‘𝑁) / ∏𝑗 ∈ (0...𝑀)(!‘(𝐶‘𝑗))) ∈ ℂ)
74	73	mul01d 11463	. 2 ⊢ (𝜑 → (((!‘𝑁) / ∏𝑗 ∈ (0...𝑀)(!‘(𝐶‘𝑗))) · 0) = 0)
75	2, 60, 74	3eqtrd 2770	1 ⊢ (𝜑 → 𝑇 = 0)

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ∧ wa 394 = wceq 1534 ∈ wcel 2099 ≠ wne 2930 ifcif 4533 class class class wbr 5153 ⟶wf 6550 ‘cfv 6554 (class class class)co 7424 ℂcc 11156 ℝcr 11157 0cc0 11158 1c1 11159 · cmul 11163 < clt 11298 − cmin 11494 / cdiv 11921 ℕcn 12264 ℕ₀cn0 12524 ℤcz 12610 ℤ_≥cuz 12874 ...cfz 13538 ↑cexp 14081 !cfa 14290 ∏cprod 15907

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2697 ax-rep 5290 ax-sep 5304 ax-nul 5311 ax-pow 5369 ax-pr 5433 ax-un 7746 ax-inf2 9684 ax-cnex 11214 ax-resscn 11215 ax-1cn 11216 ax-icn 11217 ax-addcl 11218 ax-addrcl 11219 ax-mulcl 11220 ax-mulrcl 11221 ax-mulcom 11222 ax-addass 11223 ax-mulass 11224 ax-distr 11225 ax-i2m1 11226 ax-1ne0 11227 ax-1rid 11228 ax-rnegex 11229 ax-rrecex 11230 ax-cnre 11231 ax-pre-lttri 11232 ax-pre-lttrn 11233 ax-pre-ltadd 11234 ax-pre-mulgt0 11235 ax-pre-sup 11236

This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2704 df-cleq 2718 df-clel 2803 df-nfc 2878 df-ne 2931 df-nel 3037 df-ral 3052 df-rex 3061 df-rmo 3364 df-reu 3365 df-rab 3420 df-v 3464 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3967 df-nul 4326 df-if 4534 df-pw 4609 df-sn 4634 df-pr 4636 df-op 4640 df-uni 4914 df-int 4955 df-iun 5003 df-br 5154 df-opab 5216 df-mpt 5237 df-tr 5271 df-id 5580 df-eprel 5586 df-po 5594 df-so 5595 df-fr 5637 df-se 5638 df-we 5639 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-rn 5693 df-res 5694 df-ima 5695 df-pred 6312 df-ord 6379 df-on 6380 df-lim 6381 df-suc 6382 df-iota 6506 df-fun 6556 df-fn 6557 df-f 6558 df-f1 6559 df-fo 6560 df-f1o 6561 df-fv 6562 df-isom 6563 df-riota 7380 df-ov 7427 df-oprab 7428 df-mpo 7429 df-om 7877 df-1st 8003 df-2nd 8004 df-frecs 8296 df-wrecs 8327 df-recs 8401 df-rdg 8440 df-1o 8496 df-er 8734 df-en 8975 df-dom 8976 df-sdom 8977 df-fin 8978 df-sup 9485 df-oi 9553 df-card 9982 df-pnf 11300 df-mnf 11301 df-xr 11302 df-ltxr 11303 df-le 11304 df-sub 11496 df-neg 11497 df-div 11922 df-nn 12265 df-2 12327 df-3 12328 df-n0 12525 df-z 12611 df-uz 12875 df-rp 13029 df-fz 13539 df-fzo 13682 df-seq 14022 df-exp 14082 df-fac 14291 df-bc 14320 df-hash 14348 df-cj 15104 df-re 15105 df-im 15106 df-sqrt 15240 df-abs 15241 df-clim 15490 df-prod 15908

This theorem is referenced by: etransclem28 45883

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