iprodefisumlem - Mathbox for Scott Fenton

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Theorem iprodefisumlem 35734

Description: Lemma for iprodefisum 35735. (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 6963	. . . . . 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 7059	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) ∈ ℂ)
7		efcl 16055	. . . . . 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 2829	. . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → ((exp ∘ 𝐹)‘𝑘) ∈ ℂ)
10	1, 2, 9	prodf 15860	. . 3 ⊢ (𝜑 → seq𝑀( · , (exp ∘ 𝐹)):𝑍⟶ℂ)
11	10	ffnd 6692	. 2 ⊢ (𝜑 → seq𝑀( · , (exp ∘ 𝐹)) Fn 𝑍)
12		eff 16054	. . . 4 ⊢ exp:ℂ⟶ℂ
13		ffn 6691	. . . 4 ⊢ (exp:ℂ⟶ℂ → exp Fn ℂ)
14	12, 13	ax-mp 5	. . 3 ⊢ exp Fn ℂ
15	1, 2, 6	serf 14002	. . 3 ⊢ (𝜑 → seq𝑀( + , 𝐹):𝑍⟶ℂ)
16		fnfco 6728	. . 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 6861	. . . . . . . 8 ⊢ (𝑗 = 𝑀 → (seq𝑀( · , (exp ∘ 𝐹))‘𝑗) = (seq𝑀( · , (exp ∘ 𝐹))‘𝑀))
19		2fveq3 6866	. . . . . . . 8 ⊢ (𝑗 = 𝑀 → (exp‘(seq𝑀( + , 𝐹)‘𝑗)) = (exp‘(seq𝑀( + , 𝐹)‘𝑀)))
20	18, 19	eqeq12d 2746	. . . . . . 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 6861	. . . . . . . 8 ⊢ (𝑗 = 𝑛 → (seq𝑀( · , (exp ∘ 𝐹))‘𝑗) = (seq𝑀( · , (exp ∘ 𝐹))‘𝑛))
23		2fveq3 6866	. . . . . . . 8 ⊢ (𝑗 = 𝑛 → (exp‘(seq𝑀( + , 𝐹)‘𝑗)) = (exp‘(seq𝑀( + , 𝐹)‘𝑛)))
24	22, 23	eqeq12d 2746	. . . . . . 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 6861	. . . . . . . 8 ⊢ (𝑗 = (𝑛 + 1) → (seq𝑀( · , (exp ∘ 𝐹))‘𝑗) = (seq𝑀( · , (exp ∘ 𝐹))‘(𝑛 + 1)))
27		2fveq3 6866	. . . . . . . 8 ⊢ (𝑗 = (𝑛 + 1) → (exp‘(seq𝑀( + , 𝐹)‘𝑗)) = (exp‘(seq𝑀( + , 𝐹)‘(𝑛 + 1))))
28	26, 27	eqeq12d 2746	. . . . . . 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 6861	. . . . . . . 8 ⊢ (𝑗 = 𝑘 → (seq𝑀( · , (exp ∘ 𝐹))‘𝑗) = (seq𝑀( · , (exp ∘ 𝐹))‘𝑘))
31		2fveq3 6866	. . . . . . . 8 ⊢ (𝑗 = 𝑘 → (exp‘(seq𝑀( + , 𝐹)‘𝑗)) = (exp‘(seq𝑀( + , 𝐹)‘𝑘)))
32	30, 31	eqeq12d 2746	. . . . . . 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 12815	. . . . . . . . . . 11 ⊢ (𝑀 ∈ ℤ → 𝑀 ∈ (ℤ_≥‘𝑀))
35	2, 34	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → 𝑀 ∈ (ℤ_≥‘𝑀))
36	35, 1	eleqtrrdi 2840	. . . . . . . . 9 ⊢ (𝜑 → 𝑀 ∈ 𝑍)
37		fvco3 6963	. . . . . . . . 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 13986	. . . . . . . . 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 13986	. . . . . . . . . 10 ⊢ (𝑀 ∈ ℤ → (seq𝑀( + , 𝐹)‘𝑀) = (𝐹‘𝑀))
42	2, 41	syl 17	. . . . . . . . 9 ⊢ (𝜑 → (seq𝑀( + , 𝐹)‘𝑀) = (𝐹‘𝑀))
43	42	fveq2d 6865	. . . . . . . 8 ⊢ (𝜑 → (exp‘(seq𝑀( + , 𝐹)‘𝑀)) = (exp‘(𝐹‘𝑀)))
44	38, 40, 43	3eqtr4d 2775	. . . . . . 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 7397	. . . . . . . . . . 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 12867	. . . . . . . . . . . . . . . 16 ⊢ (𝑛 ∈ (ℤ_≥‘𝑀) → (𝑛 + 1) ∈ (ℤ_≥‘𝑀))
50	49, 1	eleqtrrdi 2840	. . . . . . . . . . . . . . 15 ⊢ (𝑛 ∈ (ℤ_≥‘𝑀) → (𝑛 + 1) ∈ 𝑍)
51	50	adantr 480	. . . . . . . . . . . . . 14 ⊢ ((𝑛 ∈ (ℤ_≥‘𝑀) ∧ 𝜑) → (𝑛 + 1) ∈ 𝑍)
52		fvco3 6963	. . . . . . . . . . . . . 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 7406	. . . . . . . . . . . 12 ⊢ ((𝑛 ∈ (ℤ_≥‘𝑀) ∧ 𝜑) → ((exp‘(seq𝑀( + , 𝐹)‘𝑛)) · ((exp ∘ 𝐹)‘(𝑛 + 1))) = ((exp‘(seq𝑀( + , 𝐹)‘𝑛)) · (exp‘(𝐹‘(𝑛 + 1)))))
55	15	ffvelcdmda 7059	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → (seq𝑀( + , 𝐹)‘𝑛) ∈ ℂ)
56	55	expcom 413	. . . . . . . . . . . . . . 15 ⊢ (𝑛 ∈ 𝑍 → (𝜑 → (seq𝑀( + , 𝐹)‘𝑛) ∈ ℂ))
57	1	eqcomi 2739	. . . . . . . . . . . . . . 15 ⊢ (ℤ_≥‘𝑀) = 𝑍
58	56, 57	eleq2s 2847	. . . . . . . . . . . . . 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 7060	. . . . . . . . . . . . 13 ⊢ ((𝑛 ∈ (ℤ_≥‘𝑀) ∧ 𝜑) → (𝐹‘(𝑛 + 1)) ∈ ℂ)
61		efadd 16067	. . . . . . . . . . . . 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 2768	. . . . . . . . . . 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 2765	. . . . . . . . 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 13988	. . . . . . . . . . 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 13988	. . . . . . . . . . . 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 6865	. . . . . . . . . 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 2775	. . . . . . . 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 12872	. . . . 5 ⊢ (𝑘 ∈ (ℤ_≥‘𝑀) → (𝜑 → (seq𝑀( · , (exp ∘ 𝐹))‘𝑘) = (exp‘(seq𝑀( + , 𝐹)‘𝑘))))
77	76, 1	eleq2s 2847	. . . 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 6963	. . . 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 2768	. 2 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (seq𝑀( · , (exp ∘ 𝐹))‘𝑘) = ((exp ∘ seq𝑀( + , 𝐹))‘𝑘))
82	11, 17, 81	eqfnfvd 7009	1 ⊢ (𝜑 → seq𝑀( · , (exp ∘ 𝐹)) = (exp ∘ seq𝑀( + , 𝐹)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 ∧ w3a 1086 = wceq 1540 ∈ wcel 2109 ∘ ccom 5645 Fn wfn 6509 ⟶wf 6510 ‘cfv 6514 (class class class)co 7390 ℂcc 11073 1c1 11076 + caddc 11078 · cmul 11080 ℤcz 12536 ℤ_≥cuz 12800 seqcseq 13973 expce 16034

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 2702 ax-rep 5237 ax-sep 5254 ax-nul 5264 ax-pow 5323 ax-pr 5390 ax-un 7714 ax-inf2 9601 ax-cnex 11131 ax-resscn 11132 ax-1cn 11133 ax-icn 11134 ax-addcl 11135 ax-addrcl 11136 ax-mulcl 11137 ax-mulrcl 11138 ax-mulcom 11139 ax-addass 11140 ax-mulass 11141 ax-distr 11142 ax-i2m1 11143 ax-1ne0 11144 ax-1rid 11145 ax-rnegex 11146 ax-rrecex 11147 ax-cnre 11148 ax-pre-lttri 11149 ax-pre-lttrn 11150 ax-pre-ltadd 11151 ax-pre-mulgt0 11152 ax-pre-sup 11153

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 2534 df-eu 2563 df-clab 2709 df-cleq 2722 df-clel 2804 df-nfc 2879 df-ne 2927 df-nel 3031 df-ral 3046 df-rex 3055 df-rmo 3356 df-reu 3357 df-rab 3409 df-v 3452 df-sbc 3757 df-csb 3866 df-dif 3920 df-un 3922 df-in 3924 df-ss 3934 df-pss 3937 df-nul 4300 df-if 4492 df-pw 4568 df-sn 4593 df-pr 4595 df-op 4599 df-uni 4875 df-int 4914 df-iun 4960 df-br 5111 df-opab 5173 df-mpt 5192 df-tr 5218 df-id 5536 df-eprel 5541 df-po 5549 df-so 5550 df-fr 5594 df-se 5595 df-we 5596 df-xp 5647 df-rel 5648 df-cnv 5649 df-co 5650 df-dm 5651 df-rn 5652 df-res 5653 df-ima 5654 df-pred 6277 df-ord 6338 df-on 6339 df-lim 6340 df-suc 6341 df-iota 6467 df-fun 6516 df-fn 6517 df-f 6518 df-f1 6519 df-fo 6520 df-f1o 6521 df-fv 6522 df-isom 6523 df-riota 7347 df-ov 7393 df-oprab 7394 df-mpo 7395 df-om 7846 df-1st 7971 df-2nd 7972 df-frecs 8263 df-wrecs 8294 df-recs 8343 df-rdg 8381 df-1o 8437 df-er 8674 df-pm 8805 df-en 8922 df-dom 8923 df-sdom 8924 df-fin 8925 df-sup 9400 df-inf 9401 df-oi 9470 df-card 9899 df-pnf 11217 df-mnf 11218 df-xr 11219 df-ltxr 11220 df-le 11221 df-sub 11414 df-neg 11415 df-div 11843 df-nn 12194 df-2 12256 df-3 12257 df-n0 12450 df-z 12537 df-uz 12801 df-rp 12959 df-ico 13319 df-fz 13476 df-fzo 13623 df-fl 13761 df-seq 13974 df-exp 14034 df-fac 14246 df-bc 14275 df-hash 14303 df-shft 15040 df-cj 15072 df-re 15073 df-im 15074 df-sqrt 15208 df-abs 15209 df-limsup 15444 df-clim 15461 df-rlim 15462 df-sum 15660 df-ef 16040

This theorem is referenced by: iprodefisum 35735

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