iprodefisumlem - Mathbox for Scott Fenton

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Theorem iprodefisumlem 35883

Description: Lemma for iprodefisum 35884. (Contributed by Scott Fenton, 11-Feb-2018.)

**Hypotheses**
Ref	Expression
iprodefisumlem.1	⊢ 𝑍 = (ℤ_≥‘𝑀)
iprodefisumlem.2	⊢ (𝜑 → 𝑀 ∈ ℤ)
iprodefisumlem.3	⊢ (𝜑 → 𝐹:𝑍⟶ℂ)

**Assertion**
Ref	Expression
iprodefisumlem	⊢ (𝜑 → seq𝑀( · , (exp ∘ 𝐹)) = (exp ∘ seq𝑀( + , 𝐹)))

Proof of Theorem iprodefisumlem Dummy variables 𝑗 𝑘 𝑛 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		iprodefisumlem.1	. . . 4 ⊢ 𝑍 = (ℤ_≥‘𝑀)
2		iprodefisumlem.2	. . . 4 ⊢ (𝜑 → 𝑀 ∈ ℤ)
3		iprodefisumlem.3	. . . . . 6 ⊢ (𝜑 → 𝐹:𝑍⟶ℂ)
4		fvco3 6931	. . . . . 6 ⊢ ((𝐹:𝑍⟶ℂ ∧ 𝑘 ∈ 𝑍) → ((exp ∘ 𝐹)‘𝑘) = (exp‘(𝐹‘𝑘)))
5	3, 4	sylan 580	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → ((exp ∘ 𝐹)‘𝑘) = (exp‘(𝐹‘𝑘)))
6	3	ffvelcdmda 7027	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) ∈ ℂ)
7		efcl 16003	. . . . . 6 ⊢ ((𝐹‘𝑘) ∈ ℂ → (exp‘(𝐹‘𝑘)) ∈ ℂ)
8	6, 7	syl 17	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (exp‘(𝐹‘𝑘)) ∈ ℂ)
9	5, 8	eqeltrd 2834	. . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → ((exp ∘ 𝐹)‘𝑘) ∈ ℂ)
10	1, 2, 9	prodf 15808	. . 3 ⊢ (𝜑 → seq𝑀( · , (exp ∘ 𝐹)):𝑍⟶ℂ)
11	10	ffnd 6661	. 2 ⊢ (𝜑 → seq𝑀( · , (exp ∘ 𝐹)) Fn 𝑍)
12		eff 16002	. . . 4 ⊢ exp:ℂ⟶ℂ
13		ffn 6660	. . . 4 ⊢ (exp:ℂ⟶ℂ → exp Fn ℂ)
14	12, 13	ax-mp 5	. . 3 ⊢ exp Fn ℂ
15	1, 2, 6	serf 13951	. . 3 ⊢ (𝜑 → seq𝑀( + , 𝐹):𝑍⟶ℂ)
16		fnfco 6697	. . 3 ⊢ ((exp Fn ℂ ∧ seq𝑀( + , 𝐹):𝑍⟶ℂ) → (exp ∘ seq𝑀( + , 𝐹)) Fn 𝑍)
17	14, 15, 16	sylancr 587	. 2 ⊢ (𝜑 → (exp ∘ seq𝑀( + , 𝐹)) Fn 𝑍)
18		fveq2 6832	. . . . . . . 8 ⊢ (𝑗 = 𝑀 → (seq𝑀( · , (exp ∘ 𝐹))‘𝑗) = (seq𝑀( · , (exp ∘ 𝐹))‘𝑀))
19		2fveq3 6837	. . . . . . . 8 ⊢ (𝑗 = 𝑀 → (exp‘(seq𝑀( + , 𝐹)‘𝑗)) = (exp‘(seq𝑀( + , 𝐹)‘𝑀)))
20	18, 19	eqeq12d 2750	. . . . . . 7 ⊢ (𝑗 = 𝑀 → ((seq𝑀( · , (exp ∘ 𝐹))‘𝑗) = (exp‘(seq𝑀( + , 𝐹)‘𝑗)) ↔ (seq𝑀( · , (exp ∘ 𝐹))‘𝑀) = (exp‘(seq𝑀( + , 𝐹)‘𝑀))))
21	20	imbi2d 340	. . . . . 6 ⊢ (𝑗 = 𝑀 → ((𝜑 → (seq𝑀( · , (exp ∘ 𝐹))‘𝑗) = (exp‘(seq𝑀( + , 𝐹)‘𝑗))) ↔ (𝜑 → (seq𝑀( · , (exp ∘ 𝐹))‘𝑀) = (exp‘(seq𝑀( + , 𝐹)‘𝑀)))))
22		fveq2 6832	. . . . . . . 8 ⊢ (𝑗 = 𝑛 → (seq𝑀( · , (exp ∘ 𝐹))‘𝑗) = (seq𝑀( · , (exp ∘ 𝐹))‘𝑛))
23		2fveq3 6837	. . . . . . . 8 ⊢ (𝑗 = 𝑛 → (exp‘(seq𝑀( + , 𝐹)‘𝑗)) = (exp‘(seq𝑀( + , 𝐹)‘𝑛)))
24	22, 23	eqeq12d 2750	. . . . . . 7 ⊢ (𝑗 = 𝑛 → ((seq𝑀( · , (exp ∘ 𝐹))‘𝑗) = (exp‘(seq𝑀( + , 𝐹)‘𝑗)) ↔ (seq𝑀( · , (exp ∘ 𝐹))‘𝑛) = (exp‘(seq𝑀( + , 𝐹)‘𝑛))))
25	24	imbi2d 340	. . . . . 6 ⊢ (𝑗 = 𝑛 → ((𝜑 → (seq𝑀( · , (exp ∘ 𝐹))‘𝑗) = (exp‘(seq𝑀( + , 𝐹)‘𝑗))) ↔ (𝜑 → (seq𝑀( · , (exp ∘ 𝐹))‘𝑛) = (exp‘(seq𝑀( + , 𝐹)‘𝑛)))))
26		fveq2 6832	. . . . . . . 8 ⊢ (𝑗 = (𝑛 + 1) → (seq𝑀( · , (exp ∘ 𝐹))‘𝑗) = (seq𝑀( · , (exp ∘ 𝐹))‘(𝑛 + 1)))
27		2fveq3 6837	. . . . . . . 8 ⊢ (𝑗 = (𝑛 + 1) → (exp‘(seq𝑀( + , 𝐹)‘𝑗)) = (exp‘(seq𝑀( + , 𝐹)‘(𝑛 + 1))))
28	26, 27	eqeq12d 2750	. . . . . . 7 ⊢ (𝑗 = (𝑛 + 1) → ((seq𝑀( · , (exp ∘ 𝐹))‘𝑗) = (exp‘(seq𝑀( + , 𝐹)‘𝑗)) ↔ (seq𝑀( · , (exp ∘ 𝐹))‘(𝑛 + 1)) = (exp‘(seq𝑀( + , 𝐹)‘(𝑛 + 1)))))
29	28	imbi2d 340	. . . . . 6 ⊢ (𝑗 = (𝑛 + 1) → ((𝜑 → (seq𝑀( · , (exp ∘ 𝐹))‘𝑗) = (exp‘(seq𝑀( + , 𝐹)‘𝑗))) ↔ (𝜑 → (seq𝑀( · , (exp ∘ 𝐹))‘(𝑛 + 1)) = (exp‘(seq𝑀( + , 𝐹)‘(𝑛 + 1))))))
30		fveq2 6832	. . . . . . . 8 ⊢ (𝑗 = 𝑘 → (seq𝑀( · , (exp ∘ 𝐹))‘𝑗) = (seq𝑀( · , (exp ∘ 𝐹))‘𝑘))
31		2fveq3 6837	. . . . . . . 8 ⊢ (𝑗 = 𝑘 → (exp‘(seq𝑀( + , 𝐹)‘𝑗)) = (exp‘(seq𝑀( + , 𝐹)‘𝑘)))
32	30, 31	eqeq12d 2750	. . . . . . 7 ⊢ (𝑗 = 𝑘 → ((seq𝑀( · , (exp ∘ 𝐹))‘𝑗) = (exp‘(seq𝑀( + , 𝐹)‘𝑗)) ↔ (seq𝑀( · , (exp ∘ 𝐹))‘𝑘) = (exp‘(seq𝑀( + , 𝐹)‘𝑘))))
33	32	imbi2d 340	. . . . . 6 ⊢ (𝑗 = 𝑘 → ((𝜑 → (seq𝑀( · , (exp ∘ 𝐹))‘𝑗) = (exp‘(seq𝑀( + , 𝐹)‘𝑗))) ↔ (𝜑 → (seq𝑀( · , (exp ∘ 𝐹))‘𝑘) = (exp‘(seq𝑀( + , 𝐹)‘𝑘)))))
34		uzid 12764	. . . . . . . . . . 11 ⊢ (𝑀 ∈ ℤ → 𝑀 ∈ (ℤ_≥‘𝑀))
35	2, 34	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → 𝑀 ∈ (ℤ_≥‘𝑀))
36	35, 1	eleqtrrdi 2845	. . . . . . . . 9 ⊢ (𝜑 → 𝑀 ∈ 𝑍)
37		fvco3 6931	. . . . . . . . 9 ⊢ ((𝐹:𝑍⟶ℂ ∧ 𝑀 ∈ 𝑍) → ((exp ∘ 𝐹)‘𝑀) = (exp‘(𝐹‘𝑀)))
38	3, 36, 37	syl2anc 584	. . . . . . . 8 ⊢ (𝜑 → ((exp ∘ 𝐹)‘𝑀) = (exp‘(𝐹‘𝑀)))
39		seq1 13935	. . . . . . . . 9 ⊢ (𝑀 ∈ ℤ → (seq𝑀( · , (exp ∘ 𝐹))‘𝑀) = ((exp ∘ 𝐹)‘𝑀))
40	2, 39	syl 17	. . . . . . . 8 ⊢ (𝜑 → (seq𝑀( · , (exp ∘ 𝐹))‘𝑀) = ((exp ∘ 𝐹)‘𝑀))
41		seq1 13935	. . . . . . . . . 10 ⊢ (𝑀 ∈ ℤ → (seq𝑀( + , 𝐹)‘𝑀) = (𝐹‘𝑀))
42	2, 41	syl 17	. . . . . . . . 9 ⊢ (𝜑 → (seq𝑀( + , 𝐹)‘𝑀) = (𝐹‘𝑀))
43	42	fveq2d 6836	. . . . . . . 8 ⊢ (𝜑 → (exp‘(seq𝑀( + , 𝐹)‘𝑀)) = (exp‘(𝐹‘𝑀)))
44	38, 40, 43	3eqtr4d 2779	. . . . . . 7 ⊢ (𝜑 → (seq𝑀( · , (exp ∘ 𝐹))‘𝑀) = (exp‘(seq𝑀( + , 𝐹)‘𝑀)))
45	44	a1i 11	. . . . . 6 ⊢ (𝑀 ∈ ℤ → (𝜑 → (seq𝑀( · , (exp ∘ 𝐹))‘𝑀) = (exp‘(seq𝑀( + , 𝐹)‘𝑀))))
46		oveq1 7363	. . . . . . . . . . 11 ⊢ ((seq𝑀( · , (exp ∘ 𝐹))‘𝑛) = (exp‘(seq𝑀( + , 𝐹)‘𝑛)) → ((seq𝑀( · , (exp ∘ 𝐹))‘𝑛) · ((exp ∘ 𝐹)‘(𝑛 + 1))) = ((exp‘(seq𝑀( + , 𝐹)‘𝑛)) · ((exp ∘ 𝐹)‘(𝑛 + 1))))
47	46	3ad2ant3 1135	. . . . . . . . . 10 ⊢ ((𝑛 ∈ (ℤ_≥‘𝑀) ∧ 𝜑 ∧ (seq𝑀( · , (exp ∘ 𝐹))‘𝑛) = (exp‘(seq𝑀( + , 𝐹)‘𝑛))) → ((seq𝑀( · , (exp ∘ 𝐹))‘𝑛) · ((exp ∘ 𝐹)‘(𝑛 + 1))) = ((exp‘(seq𝑀( + , 𝐹)‘𝑛)) · ((exp ∘ 𝐹)‘(𝑛 + 1))))
48	3	adantl 481	. . . . . . . . . . . . . 14 ⊢ ((𝑛 ∈ (ℤ_≥‘𝑀) ∧ 𝜑) → 𝐹:𝑍⟶ℂ)
49		peano2uz 12812	. . . . . . . . . . . . . . . 16 ⊢ (𝑛 ∈ (ℤ_≥‘𝑀) → (𝑛 + 1) ∈ (ℤ_≥‘𝑀))
50	49, 1	eleqtrrdi 2845	. . . . . . . . . . . . . . 15 ⊢ (𝑛 ∈ (ℤ_≥‘𝑀) → (𝑛 + 1) ∈ 𝑍)
51	50	adantr 480	. . . . . . . . . . . . . 14 ⊢ ((𝑛 ∈ (ℤ_≥‘𝑀) ∧ 𝜑) → (𝑛 + 1) ∈ 𝑍)
52		fvco3 6931	. . . . . . . . . . . . . 14 ⊢ ((𝐹:𝑍⟶ℂ ∧ (𝑛 + 1) ∈ 𝑍) → ((exp ∘ 𝐹)‘(𝑛 + 1)) = (exp‘(𝐹‘(𝑛 + 1))))
53	48, 51, 52	syl2anc 584	. . . . . . . . . . . . 13 ⊢ ((𝑛 ∈ (ℤ_≥‘𝑀) ∧ 𝜑) → ((exp ∘ 𝐹)‘(𝑛 + 1)) = (exp‘(𝐹‘(𝑛 + 1))))
54	53	oveq2d 7372	. . . . . . . . . . . 12 ⊢ ((𝑛 ∈ (ℤ_≥‘𝑀) ∧ 𝜑) → ((exp‘(seq𝑀( + , 𝐹)‘𝑛)) · ((exp ∘ 𝐹)‘(𝑛 + 1))) = ((exp‘(seq𝑀( + , 𝐹)‘𝑛)) · (exp‘(𝐹‘(𝑛 + 1)))))
55	15	ffvelcdmda 7027	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → (seq𝑀( + , 𝐹)‘𝑛) ∈ ℂ)
56	55	expcom 413	. . . . . . . . . . . . . . 15 ⊢ (𝑛 ∈ 𝑍 → (𝜑 → (seq𝑀( + , 𝐹)‘𝑛) ∈ ℂ))
57	1	eqcomi 2743	. . . . . . . . . . . . . . 15 ⊢ (ℤ_≥‘𝑀) = 𝑍
58	56, 57	eleq2s 2852	. . . . . . . . . . . . . 14 ⊢ (𝑛 ∈ (ℤ_≥‘𝑀) → (𝜑 → (seq𝑀( + , 𝐹)‘𝑛) ∈ ℂ))
59	58	imp 406	. . . . . . . . . . . . 13 ⊢ ((𝑛 ∈ (ℤ_≥‘𝑀) ∧ 𝜑) → (seq𝑀( + , 𝐹)‘𝑛) ∈ ℂ)
60	48, 51	ffvelcdmd 7028	. . . . . . . . . . . . 13 ⊢ ((𝑛 ∈ (ℤ_≥‘𝑀) ∧ 𝜑) → (𝐹‘(𝑛 + 1)) ∈ ℂ)
61		efadd 16015	. . . . . . . . . . . . 13 ⊢ (((seq𝑀( + , 𝐹)‘𝑛) ∈ ℂ ∧ (𝐹‘(𝑛 + 1)) ∈ ℂ) → (exp‘((seq𝑀( + , 𝐹)‘𝑛) + (𝐹‘(𝑛 + 1)))) = ((exp‘(seq𝑀( + , 𝐹)‘𝑛)) · (exp‘(𝐹‘(𝑛 + 1)))))
62	59, 60, 61	syl2anc 584	. . . . . . . . . . . 12 ⊢ ((𝑛 ∈ (ℤ_≥‘𝑀) ∧ 𝜑) → (exp‘((seq𝑀( + , 𝐹)‘𝑛) + (𝐹‘(𝑛 + 1)))) = ((exp‘(seq𝑀( + , 𝐹)‘𝑛)) · (exp‘(𝐹‘(𝑛 + 1)))))
63	54, 62	eqtr4d 2772	. . . . . . . . . . 11 ⊢ ((𝑛 ∈ (ℤ_≥‘𝑀) ∧ 𝜑) → ((exp‘(seq𝑀( + , 𝐹)‘𝑛)) · ((exp ∘ 𝐹)‘(𝑛 + 1))) = (exp‘((seq𝑀( + , 𝐹)‘𝑛) + (𝐹‘(𝑛 + 1)))))
64	63	3adant3 1132	. . . . . . . . . 10 ⊢ ((𝑛 ∈ (ℤ_≥‘𝑀) ∧ 𝜑 ∧ (seq𝑀( · , (exp ∘ 𝐹))‘𝑛) = (exp‘(seq𝑀( + , 𝐹)‘𝑛))) → ((exp‘(seq𝑀( + , 𝐹)‘𝑛)) · ((exp ∘ 𝐹)‘(𝑛 + 1))) = (exp‘((seq𝑀( + , 𝐹)‘𝑛) + (𝐹‘(𝑛 + 1)))))
65	47, 64	eqtrd 2769	. . . . . . . . 9 ⊢ ((𝑛 ∈ (ℤ_≥‘𝑀) ∧ 𝜑 ∧ (seq𝑀( · , (exp ∘ 𝐹))‘𝑛) = (exp‘(seq𝑀( + , 𝐹)‘𝑛))) → ((seq𝑀( · , (exp ∘ 𝐹))‘𝑛) · ((exp ∘ 𝐹)‘(𝑛 + 1))) = (exp‘((seq𝑀( + , 𝐹)‘𝑛) + (𝐹‘(𝑛 + 1)))))
66		seqp1 13937	. . . . . . . . . . 11 ⊢ (𝑛 ∈ (ℤ_≥‘𝑀) → (seq𝑀( · , (exp ∘ 𝐹))‘(𝑛 + 1)) = ((seq𝑀( · , (exp ∘ 𝐹))‘𝑛) · ((exp ∘ 𝐹)‘(𝑛 + 1))))
67	66	adantr 480	. . . . . . . . . 10 ⊢ ((𝑛 ∈ (ℤ_≥‘𝑀) ∧ 𝜑) → (seq𝑀( · , (exp ∘ 𝐹))‘(𝑛 + 1)) = ((seq𝑀( · , (exp ∘ 𝐹))‘𝑛) · ((exp ∘ 𝐹)‘(𝑛 + 1))))
68	67	3adant3 1132	. . . . . . . . 9 ⊢ ((𝑛 ∈ (ℤ_≥‘𝑀) ∧ 𝜑 ∧ (seq𝑀( · , (exp ∘ 𝐹))‘𝑛) = (exp‘(seq𝑀( + , 𝐹)‘𝑛))) → (seq𝑀( · , (exp ∘ 𝐹))‘(𝑛 + 1)) = ((seq𝑀( · , (exp ∘ 𝐹))‘𝑛) · ((exp ∘ 𝐹)‘(𝑛 + 1))))
69		seqp1 13937	. . . . . . . . . . . 12 ⊢ (𝑛 ∈ (ℤ_≥‘𝑀) → (seq𝑀( + , 𝐹)‘(𝑛 + 1)) = ((seq𝑀( + , 𝐹)‘𝑛) + (𝐹‘(𝑛 + 1))))
70	69	adantr 480	. . . . . . . . . . 11 ⊢ ((𝑛 ∈ (ℤ_≥‘𝑀) ∧ 𝜑) → (seq𝑀( + , 𝐹)‘(𝑛 + 1)) = ((seq𝑀( + , 𝐹)‘𝑛) + (𝐹‘(𝑛 + 1))))
71	70	fveq2d 6836	. . . . . . . . . 10 ⊢ ((𝑛 ∈ (ℤ_≥‘𝑀) ∧ 𝜑) → (exp‘(seq𝑀( + , 𝐹)‘(𝑛 + 1))) = (exp‘((seq𝑀( + , 𝐹)‘𝑛) + (𝐹‘(𝑛 + 1)))))
72	71	3adant3 1132	. . . . . . . . 9 ⊢ ((𝑛 ∈ (ℤ_≥‘𝑀) ∧ 𝜑 ∧ (seq𝑀( · , (exp ∘ 𝐹))‘𝑛) = (exp‘(seq𝑀( + , 𝐹)‘𝑛))) → (exp‘(seq𝑀( + , 𝐹)‘(𝑛 + 1))) = (exp‘((seq𝑀( + , 𝐹)‘𝑛) + (𝐹‘(𝑛 + 1)))))
73	65, 68, 72	3eqtr4d 2779	. . . . . . . 8 ⊢ ((𝑛 ∈ (ℤ_≥‘𝑀) ∧ 𝜑 ∧ (seq𝑀( · , (exp ∘ 𝐹))‘𝑛) = (exp‘(seq𝑀( + , 𝐹)‘𝑛))) → (seq𝑀( · , (exp ∘ 𝐹))‘(𝑛 + 1)) = (exp‘(seq𝑀( + , 𝐹)‘(𝑛 + 1))))
74	73	3exp 1119	. . . . . . 7 ⊢ (𝑛 ∈ (ℤ_≥‘𝑀) → (𝜑 → ((seq𝑀( · , (exp ∘ 𝐹))‘𝑛) = (exp‘(seq𝑀( + , 𝐹)‘𝑛)) → (seq𝑀( · , (exp ∘ 𝐹))‘(𝑛 + 1)) = (exp‘(seq𝑀( + , 𝐹)‘(𝑛 + 1))))))
75	74	a2d 29	. . . . . 6 ⊢ (𝑛 ∈ (ℤ_≥‘𝑀) → ((𝜑 → (seq𝑀( · , (exp ∘ 𝐹))‘𝑛) = (exp‘(seq𝑀( + , 𝐹)‘𝑛))) → (𝜑 → (seq𝑀( · , (exp ∘ 𝐹))‘(𝑛 + 1)) = (exp‘(seq𝑀( + , 𝐹)‘(𝑛 + 1))))))
76	21, 25, 29, 33, 45, 75	uzind4 12817	. . . . 5 ⊢ (𝑘 ∈ (ℤ_≥‘𝑀) → (𝜑 → (seq𝑀( · , (exp ∘ 𝐹))‘𝑘) = (exp‘(seq𝑀( + , 𝐹)‘𝑘))))
77	76, 1	eleq2s 2852	. . . 4 ⊢ (𝑘 ∈ 𝑍 → (𝜑 → (seq𝑀( · , (exp ∘ 𝐹))‘𝑘) = (exp‘(seq𝑀( + , 𝐹)‘𝑘))))
78	77	impcom 407	. . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (seq𝑀( · , (exp ∘ 𝐹))‘𝑘) = (exp‘(seq𝑀( + , 𝐹)‘𝑘)))
79		fvco3 6931	. . . 4 ⊢ ((seq𝑀( + , 𝐹):𝑍⟶ℂ ∧ 𝑘 ∈ 𝑍) → ((exp ∘ seq𝑀( + , 𝐹))‘𝑘) = (exp‘(seq𝑀( + , 𝐹)‘𝑘)))
80	15, 79	sylan 580	. . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → ((exp ∘ seq𝑀( + , 𝐹))‘𝑘) = (exp‘(seq𝑀( + , 𝐹)‘𝑘)))
81	78, 80	eqtr4d 2772	. 2 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (seq𝑀( · , (exp ∘ 𝐹))‘𝑘) = ((exp ∘ seq𝑀( + , 𝐹))‘𝑘))
82	11, 17, 81	eqfnfvd 6977	1 ⊢ (𝜑 → seq𝑀( · , (exp ∘ 𝐹)) = (exp ∘ seq𝑀( + , 𝐹)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 ∧ w3a 1086 = wceq 1541 ∈ wcel 2113 ∘ ccom 5626 Fn wfn 6485 ⟶wf 6486 ‘cfv 6490 (class class class)co 7356 ℂcc 11022 1c1 11025 + caddc 11027 · cmul 11029 ℤcz 12486 ℤ_≥cuz 12749 seqcseq 13922 expce 15982

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1968 ax-7 2009 ax-8 2115 ax-9 2123 ax-10 2146 ax-11 2162 ax-12 2182 ax-ext 2706 ax-rep 5222 ax-sep 5239 ax-nul 5249 ax-pow 5308 ax-pr 5375 ax-un 7678 ax-inf2 9548 ax-cnex 11080 ax-resscn 11081 ax-1cn 11082 ax-icn 11083 ax-addcl 11084 ax-addrcl 11085 ax-mulcl 11086 ax-mulrcl 11087 ax-mulcom 11088 ax-addass 11089 ax-mulass 11090 ax-distr 11091 ax-i2m1 11092 ax-1ne0 11093 ax-1rid 11094 ax-rnegex 11095 ax-rrecex 11096 ax-cnre 11097 ax-pre-lttri 11098 ax-pre-lttrn 11099 ax-pre-ltadd 11100 ax-pre-mulgt0 11101 ax-pre-sup 11102

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1544 df-fal 1554 df-ex 1781 df-nf 1785 df-sb 2068 df-mo 2537 df-eu 2567 df-clab 2713 df-cleq 2726 df-clel 2809 df-nfc 2883 df-ne 2931 df-nel 3035 df-ral 3050 df-rex 3059 df-rmo 3348 df-reu 3349 df-rab 3398 df-v 3440 df-sbc 3739 df-csb 3848 df-dif 3902 df-un 3904 df-in 3906 df-ss 3916 df-pss 3919 df-nul 4284 df-if 4478 df-pw 4554 df-sn 4579 df-pr 4581 df-op 4585 df-uni 4862 df-int 4901 df-iun 4946 df-br 5097 df-opab 5159 df-mpt 5178 df-tr 5204 df-id 5517 df-eprel 5522 df-po 5530 df-so 5531 df-fr 5575 df-se 5576 df-we 5577 df-xp 5628 df-rel 5629 df-cnv 5630 df-co 5631 df-dm 5632 df-rn 5633 df-res 5634 df-ima 5635 df-pred 6257 df-ord 6318 df-on 6319 df-lim 6320 df-suc 6321 df-iota 6446 df-fun 6492 df-fn 6493 df-f 6494 df-f1 6495 df-fo 6496 df-f1o 6497 df-fv 6498 df-isom 6499 df-riota 7313 df-ov 7359 df-oprab 7360 df-mpo 7361 df-om 7807 df-1st 7931 df-2nd 7932 df-frecs 8221 df-wrecs 8252 df-recs 8301 df-rdg 8339 df-1o 8395 df-er 8633 df-pm 8764 df-en 8882 df-dom 8883 df-sdom 8884 df-fin 8885 df-sup 9343 df-inf 9344 df-oi 9413 df-card 9849 df-pnf 11166 df-mnf 11167 df-xr 11168 df-ltxr 11169 df-le 11170 df-sub 11364 df-neg 11365 df-div 11793 df-nn 12144 df-2 12206 df-3 12207 df-n0 12400 df-z 12487 df-uz 12750 df-rp 12904 df-ico 13265 df-fz 13422 df-fzo 13569 df-fl 13710 df-seq 13923 df-exp 13983 df-fac 14195 df-bc 14224 df-hash 14252 df-shft 14988 df-cj 15020 df-re 15021 df-im 15022 df-sqrt 15156 df-abs 15157 df-limsup 15392 df-clim 15409 df-rlim 15410 df-sum 15608 df-ef 15988

This theorem is referenced by: iprodefisum 35884

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