iprodefisumlem - Mathbox for Scott Fenton

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Theorem iprodefisumlem 32965

Description: Lemma for iprodefisum 32966. (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 6753	. . . . . 6 ⊢ ((𝐹:𝑍⟶ℂ ∧ 𝑘 ∈ 𝑍) → ((exp ∘ 𝐹)‘𝑘) = (exp‘(𝐹‘𝑘)))
5	3, 4	sylan 582	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → ((exp ∘ 𝐹)‘𝑘) = (exp‘(𝐹‘𝑘)))
6	3	ffvelrnda 6844	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) ∈ ℂ)
7		efcl 15428	. . . . . 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 2911	. . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → ((exp ∘ 𝐹)‘𝑘) ∈ ℂ)
10	1, 2, 9	prodf 15235	. . 3 ⊢ (𝜑 → seq𝑀( · , (exp ∘ 𝐹)):𝑍⟶ℂ)
11	10	ffnd 6508	. 2 ⊢ (𝜑 → seq𝑀( · , (exp ∘ 𝐹)) Fn 𝑍)
12		eff 15427	. . . 4 ⊢ exp:ℂ⟶ℂ
13		ffn 6507	. . . 4 ⊢ (exp:ℂ⟶ℂ → exp Fn ℂ)
14	12, 13	ax-mp 5	. . 3 ⊢ exp Fn ℂ
15	1, 2, 6	serf 13390	. . 3 ⊢ (𝜑 → seq𝑀( + , 𝐹):𝑍⟶ℂ)
16		fnfco 6536	. . 3 ⊢ ((exp Fn ℂ ∧ seq𝑀( + , 𝐹):𝑍⟶ℂ) → (exp ∘ seq𝑀( + , 𝐹)) Fn 𝑍)
17	14, 15, 16	sylancr 589	. 2 ⊢ (𝜑 → (exp ∘ seq𝑀( + , 𝐹)) Fn 𝑍)
18		fveq2 6663	. . . . . . . 8 ⊢ (𝑗 = 𝑀 → (seq𝑀( · , (exp ∘ 𝐹))‘𝑗) = (seq𝑀( · , (exp ∘ 𝐹))‘𝑀))
19		2fveq3 6668	. . . . . . . 8 ⊢ (𝑗 = 𝑀 → (exp‘(seq𝑀( + , 𝐹)‘𝑗)) = (exp‘(seq𝑀( + , 𝐹)‘𝑀)))
20	18, 19	eqeq12d 2835	. . . . . . 7 ⊢ (𝑗 = 𝑀 → ((seq𝑀( · , (exp ∘ 𝐹))‘𝑗) = (exp‘(seq𝑀( + , 𝐹)‘𝑗)) ↔ (seq𝑀( · , (exp ∘ 𝐹))‘𝑀) = (exp‘(seq𝑀( + , 𝐹)‘𝑀))))
21	20	imbi2d 343	. . . . . 6 ⊢ (𝑗 = 𝑀 → ((𝜑 → (seq𝑀( · , (exp ∘ 𝐹))‘𝑗) = (exp‘(seq𝑀( + , 𝐹)‘𝑗))) ↔ (𝜑 → (seq𝑀( · , (exp ∘ 𝐹))‘𝑀) = (exp‘(seq𝑀( + , 𝐹)‘𝑀)))))
22		fveq2 6663	. . . . . . . 8 ⊢ (𝑗 = 𝑛 → (seq𝑀( · , (exp ∘ 𝐹))‘𝑗) = (seq𝑀( · , (exp ∘ 𝐹))‘𝑛))
23		2fveq3 6668	. . . . . . . 8 ⊢ (𝑗 = 𝑛 → (exp‘(seq𝑀( + , 𝐹)‘𝑗)) = (exp‘(seq𝑀( + , 𝐹)‘𝑛)))
24	22, 23	eqeq12d 2835	. . . . . . 7 ⊢ (𝑗 = 𝑛 → ((seq𝑀( · , (exp ∘ 𝐹))‘𝑗) = (exp‘(seq𝑀( + , 𝐹)‘𝑗)) ↔ (seq𝑀( · , (exp ∘ 𝐹))‘𝑛) = (exp‘(seq𝑀( + , 𝐹)‘𝑛))))
25	24	imbi2d 343	. . . . . 6 ⊢ (𝑗 = 𝑛 → ((𝜑 → (seq𝑀( · , (exp ∘ 𝐹))‘𝑗) = (exp‘(seq𝑀( + , 𝐹)‘𝑗))) ↔ (𝜑 → (seq𝑀( · , (exp ∘ 𝐹))‘𝑛) = (exp‘(seq𝑀( + , 𝐹)‘𝑛)))))
26		fveq2 6663	. . . . . . . 8 ⊢ (𝑗 = (𝑛 + 1) → (seq𝑀( · , (exp ∘ 𝐹))‘𝑗) = (seq𝑀( · , (exp ∘ 𝐹))‘(𝑛 + 1)))
27		2fveq3 6668	. . . . . . . 8 ⊢ (𝑗 = (𝑛 + 1) → (exp‘(seq𝑀( + , 𝐹)‘𝑗)) = (exp‘(seq𝑀( + , 𝐹)‘(𝑛 + 1))))
28	26, 27	eqeq12d 2835	. . . . . . 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 343	. . . . . 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 6663	. . . . . . . 8 ⊢ (𝑗 = 𝑘 → (seq𝑀( · , (exp ∘ 𝐹))‘𝑗) = (seq𝑀( · , (exp ∘ 𝐹))‘𝑘))
31		2fveq3 6668	. . . . . . . 8 ⊢ (𝑗 = 𝑘 → (exp‘(seq𝑀( + , 𝐹)‘𝑗)) = (exp‘(seq𝑀( + , 𝐹)‘𝑘)))
32	30, 31	eqeq12d 2835	. . . . . . 7 ⊢ (𝑗 = 𝑘 → ((seq𝑀( · , (exp ∘ 𝐹))‘𝑗) = (exp‘(seq𝑀( + , 𝐹)‘𝑗)) ↔ (seq𝑀( · , (exp ∘ 𝐹))‘𝑘) = (exp‘(seq𝑀( + , 𝐹)‘𝑘))))
33	32	imbi2d 343	. . . . . 6 ⊢ (𝑗 = 𝑘 → ((𝜑 → (seq𝑀( · , (exp ∘ 𝐹))‘𝑗) = (exp‘(seq𝑀( + , 𝐹)‘𝑗))) ↔ (𝜑 → (seq𝑀( · , (exp ∘ 𝐹))‘𝑘) = (exp‘(seq𝑀( + , 𝐹)‘𝑘)))))
34		uzid 12250	. . . . . . . . . . 11 ⊢ (𝑀 ∈ ℤ → 𝑀 ∈ (ℤ_≥‘𝑀))
35	2, 34	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → 𝑀 ∈ (ℤ_≥‘𝑀))
36	35, 1	eleqtrrdi 2922	. . . . . . . . 9 ⊢ (𝜑 → 𝑀 ∈ 𝑍)
37		fvco3 6753	. . . . . . . . 9 ⊢ ((𝐹:𝑍⟶ℂ ∧ 𝑀 ∈ 𝑍) → ((exp ∘ 𝐹)‘𝑀) = (exp‘(𝐹‘𝑀)))
38	3, 36, 37	syl2anc 586	. . . . . . . 8 ⊢ (𝜑 → ((exp ∘ 𝐹)‘𝑀) = (exp‘(𝐹‘𝑀)))
39		seq1 13374	. . . . . . . . 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 13374	. . . . . . . . . 10 ⊢ (𝑀 ∈ ℤ → (seq𝑀( + , 𝐹)‘𝑀) = (𝐹‘𝑀))
42	2, 41	syl 17	. . . . . . . . 9 ⊢ (𝜑 → (seq𝑀( + , 𝐹)‘𝑀) = (𝐹‘𝑀))
43	42	fveq2d 6667	. . . . . . . 8 ⊢ (𝜑 → (exp‘(seq𝑀( + , 𝐹)‘𝑀)) = (exp‘(𝐹‘𝑀)))
44	38, 40, 43	3eqtr4d 2864	. . . . . . 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 7155	. . . . . . . . . . 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 1130	. . . . . . . . . 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 484	. . . . . . . . . . . . . 14 ⊢ ((𝑛 ∈ (ℤ_≥‘𝑀) ∧ 𝜑) → 𝐹:𝑍⟶ℂ)
49		peano2uz 12293	. . . . . . . . . . . . . . . 16 ⊢ (𝑛 ∈ (ℤ_≥‘𝑀) → (𝑛 + 1) ∈ (ℤ_≥‘𝑀))
50	49, 1	eleqtrrdi 2922	. . . . . . . . . . . . . . 15 ⊢ (𝑛 ∈ (ℤ_≥‘𝑀) → (𝑛 + 1) ∈ 𝑍)
51	50	adantr 483	. . . . . . . . . . . . . 14 ⊢ ((𝑛 ∈ (ℤ_≥‘𝑀) ∧ 𝜑) → (𝑛 + 1) ∈ 𝑍)
52		fvco3 6753	. . . . . . . . . . . . . 14 ⊢ ((𝐹:𝑍⟶ℂ ∧ (𝑛 + 1) ∈ 𝑍) → ((exp ∘ 𝐹)‘(𝑛 + 1)) = (exp‘(𝐹‘(𝑛 + 1))))
53	48, 51, 52	syl2anc 586	. . . . . . . . . . . . 13 ⊢ ((𝑛 ∈ (ℤ_≥‘𝑀) ∧ 𝜑) → ((exp ∘ 𝐹)‘(𝑛 + 1)) = (exp‘(𝐹‘(𝑛 + 1))))
54	53	oveq2d 7164	. . . . . . . . . . . 12 ⊢ ((𝑛 ∈ (ℤ_≥‘𝑀) ∧ 𝜑) → ((exp‘(seq𝑀( + , 𝐹)‘𝑛)) · ((exp ∘ 𝐹)‘(𝑛 + 1))) = ((exp‘(seq𝑀( + , 𝐹)‘𝑛)) · (exp‘(𝐹‘(𝑛 + 1)))))
55	15	ffvelrnda 6844	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → (seq𝑀( + , 𝐹)‘𝑛) ∈ ℂ)
56	55	expcom 416	. . . . . . . . . . . . . . 15 ⊢ (𝑛 ∈ 𝑍 → (𝜑 → (seq𝑀( + , 𝐹)‘𝑛) ∈ ℂ))
57	1	eqcomi 2828	. . . . . . . . . . . . . . 15 ⊢ (ℤ_≥‘𝑀) = 𝑍
58	56, 57	eleq2s 2929	. . . . . . . . . . . . . 14 ⊢ (𝑛 ∈ (ℤ_≥‘𝑀) → (𝜑 → (seq𝑀( + , 𝐹)‘𝑛) ∈ ℂ))
59	58	imp 409	. . . . . . . . . . . . 13 ⊢ ((𝑛 ∈ (ℤ_≥‘𝑀) ∧ 𝜑) → (seq𝑀( + , 𝐹)‘𝑛) ∈ ℂ)
60	48, 51	ffvelrnd 6845	. . . . . . . . . . . . 13 ⊢ ((𝑛 ∈ (ℤ_≥‘𝑀) ∧ 𝜑) → (𝐹‘(𝑛 + 1)) ∈ ℂ)
61		efadd 15439	. . . . . . . . . . . . 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 586	. . . . . . . . . . . 12 ⊢ ((𝑛 ∈ (ℤ_≥‘𝑀) ∧ 𝜑) → (exp‘((seq𝑀( + , 𝐹)‘𝑛) + (𝐹‘(𝑛 + 1)))) = ((exp‘(seq𝑀( + , 𝐹)‘𝑛)) · (exp‘(𝐹‘(𝑛 + 1)))))
63	54, 62	eqtr4d 2857	. . . . . . . . . . 11 ⊢ ((𝑛 ∈ (ℤ_≥‘𝑀) ∧ 𝜑) → ((exp‘(seq𝑀( + , 𝐹)‘𝑛)) · ((exp ∘ 𝐹)‘(𝑛 + 1))) = (exp‘((seq𝑀( + , 𝐹)‘𝑛) + (𝐹‘(𝑛 + 1)))))
64	63	3adant3 1127	. . . . . . . . . 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 2854	. . . . . . . . 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 13376	. . . . . . . . . . 11 ⊢ (𝑛 ∈ (ℤ_≥‘𝑀) → (seq𝑀( · , (exp ∘ 𝐹))‘(𝑛 + 1)) = ((seq𝑀( · , (exp ∘ 𝐹))‘𝑛) · ((exp ∘ 𝐹)‘(𝑛 + 1))))
67	66	adantr 483	. . . . . . . . . 10 ⊢ ((𝑛 ∈ (ℤ_≥‘𝑀) ∧ 𝜑) → (seq𝑀( · , (exp ∘ 𝐹))‘(𝑛 + 1)) = ((seq𝑀( · , (exp ∘ 𝐹))‘𝑛) · ((exp ∘ 𝐹)‘(𝑛 + 1))))
68	67	3adant3 1127	. . . . . . . . 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 13376	. . . . . . . . . . . 12 ⊢ (𝑛 ∈ (ℤ_≥‘𝑀) → (seq𝑀( + , 𝐹)‘(𝑛 + 1)) = ((seq𝑀( + , 𝐹)‘𝑛) + (𝐹‘(𝑛 + 1))))
70	69	adantr 483	. . . . . . . . . . 11 ⊢ ((𝑛 ∈ (ℤ_≥‘𝑀) ∧ 𝜑) → (seq𝑀( + , 𝐹)‘(𝑛 + 1)) = ((seq𝑀( + , 𝐹)‘𝑛) + (𝐹‘(𝑛 + 1))))
71	70	fveq2d 6667	. . . . . . . . . 10 ⊢ ((𝑛 ∈ (ℤ_≥‘𝑀) ∧ 𝜑) → (exp‘(seq𝑀( + , 𝐹)‘(𝑛 + 1))) = (exp‘((seq𝑀( + , 𝐹)‘𝑛) + (𝐹‘(𝑛 + 1)))))
72	71	3adant3 1127	. . . . . . . . 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 2864	. . . . . . . 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 1114	. . . . . . 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 12298	. . . . 5 ⊢ (𝑘 ∈ (ℤ_≥‘𝑀) → (𝜑 → (seq𝑀( · , (exp ∘ 𝐹))‘𝑘) = (exp‘(seq𝑀( + , 𝐹)‘𝑘))))
77	76, 1	eleq2s 2929	. . . 4 ⊢ (𝑘 ∈ 𝑍 → (𝜑 → (seq𝑀( · , (exp ∘ 𝐹))‘𝑘) = (exp‘(seq𝑀( + , 𝐹)‘𝑘))))
78	77	impcom 410	. . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (seq𝑀( · , (exp ∘ 𝐹))‘𝑘) = (exp‘(seq𝑀( + , 𝐹)‘𝑘)))
79		fvco3 6753	. . . 4 ⊢ ((seq𝑀( + , 𝐹):𝑍⟶ℂ ∧ 𝑘 ∈ 𝑍) → ((exp ∘ seq𝑀( + , 𝐹))‘𝑘) = (exp‘(seq𝑀( + , 𝐹)‘𝑘)))
80	15, 79	sylan 582	. . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → ((exp ∘ seq𝑀( + , 𝐹))‘𝑘) = (exp‘(seq𝑀( + , 𝐹)‘𝑘)))
81	78, 80	eqtr4d 2857	. 2 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (seq𝑀( · , (exp ∘ 𝐹))‘𝑘) = ((exp ∘ seq𝑀( + , 𝐹))‘𝑘))
82	11, 17, 81	eqfnfvd 6798	1 ⊢ (𝜑 → seq𝑀( · , (exp ∘ 𝐹)) = (exp ∘ seq𝑀( + , 𝐹)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 398 ∧ w3a 1082 = wceq 1531 ∈ wcel 2108 ∘ ccom 5552 Fn wfn 6343 ⟶wf 6344 ‘cfv 6348 (class class class)co 7148 ℂcc 10527 1c1 10530 + caddc 10532 · cmul 10534 ℤcz 11973 ℤ_≥cuz 12235 seqcseq 13361 expce 15407

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 1905 ax-6 1964 ax-7 2009 ax-8 2110 ax-9 2118 ax-10 2139 ax-11 2154 ax-12 2170 ax-ext 2791 ax-rep 5181 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7453 ax-inf2 9096 ax-cnex 10585 ax-resscn 10586 ax-1cn 10587 ax-icn 10588 ax-addcl 10589 ax-addrcl 10590 ax-mulcl 10591 ax-mulrcl 10592 ax-mulcom 10593 ax-addass 10594 ax-mulass 10595 ax-distr 10596 ax-i2m1 10597 ax-1ne0 10598 ax-1rid 10599 ax-rnegex 10600 ax-rrecex 10601 ax-cnre 10602 ax-pre-lttri 10603 ax-pre-lttrn 10604 ax-pre-ltadd 10605 ax-pre-mulgt0 10606 ax-pre-sup 10607 ax-addf 10608 ax-mulf 10609

This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1083 df-3an 1084 df-tru 1534 df-fal 1544 df-ex 1775 df-nf 1779 df-sb 2064 df-mo 2616 df-eu 2648 df-clab 2798 df-cleq 2812 df-clel 2891 df-nfc 2961 df-ne 3015 df-nel 3122 df-ral 3141 df-rex 3142 df-reu 3143 df-rmo 3144 df-rab 3145 df-v 3495 df-sbc 3771 df-csb 3882 df-dif 3937 df-un 3939 df-in 3941 df-ss 3950 df-pss 3952 df-nul 4290 df-if 4466 df-pw 4539 df-sn 4560 df-pr 4562 df-tp 4564 df-op 4566 df-uni 4831 df-int 4868 df-iun 4912 df-br 5058 df-opab 5120 df-mpt 5138 df-tr 5164 df-id 5453 df-eprel 5458 df-po 5467 df-so 5468 df-fr 5507 df-se 5508 df-we 5509 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-pred 6141 df-ord 6187 df-on 6188 df-lim 6189 df-suc 6190 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-isom 6357 df-riota 7106 df-ov 7151 df-oprab 7152 df-mpo 7153 df-om 7573 df-1st 7681 df-2nd 7682 df-wrecs 7939 df-recs 8000 df-rdg 8038 df-1o 8094 df-oadd 8098 df-er 8281 df-pm 8401 df-en 8502 df-dom 8503 df-sdom 8504 df-fin 8505 df-sup 8898 df-inf 8899 df-oi 8966 df-card 9360 df-pnf 10669 df-mnf 10670 df-xr 10671 df-ltxr 10672 df-le 10673 df-sub 10864 df-neg 10865 df-div 11290 df-nn 11631 df-2 11692 df-3 11693 df-n0 11890 df-z 11974 df-uz 12236 df-rp 12382 df-ico 12736 df-fz 12885 df-fzo 13026 df-fl 13154 df-seq 13362 df-exp 13422 df-fac 13626 df-bc 13655 df-hash 13683 df-shft 14418 df-cj 14450 df-re 14451 df-im 14452 df-sqrt 14586 df-abs 14587 df-limsup 14820 df-clim 14837 df-rlim 14838 df-sum 15035 df-ef 15413

This theorem is referenced by: iprodefisum 32966

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