iprodefisumlem - Mathbox for Scott Fenton

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Theorem iprodefisumlem 36271

Description: Lemma for iprodefisum 36272. (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 6985	. . . . . 6 ⊢ ((𝐹:𝑍⟶ℂ ∧ 𝑘 ∈ 𝑍) → ((exp ∘ 𝐹)‘𝑘) = (exp‘(𝐹‘𝑘)))
5	3, 4	sylan 592	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → ((exp ∘ 𝐹)‘𝑘) = (exp‘(𝐹‘𝑘)))
6	3	ffvelcdmda 7083	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) ∈ ℂ)
7		efcl 16160	. . . . . 6 ⊢ ((𝐹‘𝑘) ∈ ℂ → (exp‘(𝐹‘𝑘)) ∈ ℂ)
8	6, 7	syl 18	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (exp‘(𝐹‘𝑘)) ∈ ℂ)
9	5, 8	eqeltrd 2865	. . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → ((exp ∘ 𝐹)‘𝑘) ∈ ℂ)
10	1, 2, 9	prodf 15966	. . 3 ⊢ (𝜑 → seq𝑀( · , (exp ∘ 𝐹)):𝑍⟶ℂ)
11	10	ffnd 6710	. 2 ⊢ (𝜑 → seq𝑀( · , (exp ∘ 𝐹)) Fn 𝑍)
12		eff 16159	. . . 4 ⊢ exp:ℂ⟶ℂ
13		ffn 6709	. . . 4 ⊢ (exp:ℂ⟶ℂ → exp Fn ℂ)
14	12, 13	ax-mp 5	. . 3 ⊢ exp Fn ℂ
15	1, 2, 6	serf 14086	. . 3 ⊢ (𝜑 → seq𝑀( + , 𝐹):𝑍⟶ℂ)
16		fnfco 6747	. . 3 ⊢ ((exp Fn ℂ ∧ seq𝑀( + , 𝐹):𝑍⟶ℂ) → (exp ∘ seq𝑀( + , 𝐹)) Fn 𝑍)
17	14, 15, 16	sylancr 599	. 2 ⊢ (𝜑 → (exp ∘ seq𝑀( + , 𝐹)) Fn 𝑍)
18		fveq2 6885	. . . . . . . 8 ⊢ (𝑗 = 𝑀 → (seq𝑀( · , (exp ∘ 𝐹))‘𝑗) = (seq𝑀( · , (exp ∘ 𝐹))‘𝑀))
19		2fveq3 6890	. . . . . . . 8 ⊢ (𝑗 = 𝑀 → (exp‘(seq𝑀( + , 𝐹)‘𝑗)) = (exp‘(seq𝑀( + , 𝐹)‘𝑀)))
20	18, 19	eqeq12d 2781	. . . . . . 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 6885	. . . . . . . 8 ⊢ (𝑗 = 𝑛 → (seq𝑀( · , (exp ∘ 𝐹))‘𝑗) = (seq𝑀( · , (exp ∘ 𝐹))‘𝑛))
23		2fveq3 6890	. . . . . . . 8 ⊢ (𝑗 = 𝑛 → (exp‘(seq𝑀( + , 𝐹)‘𝑗)) = (exp‘(seq𝑀( + , 𝐹)‘𝑛)))
24	22, 23	eqeq12d 2781	. . . . . . 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 6885	. . . . . . . 8 ⊢ (𝑗 = (𝑛 + 1) → (seq𝑀( · , (exp ∘ 𝐹))‘𝑗) = (seq𝑀( · , (exp ∘ 𝐹))‘(𝑛 + 1)))
27		2fveq3 6890	. . . . . . . 8 ⊢ (𝑗 = (𝑛 + 1) → (exp‘(seq𝑀( + , 𝐹)‘𝑗)) = (exp‘(seq𝑀( + , 𝐹)‘(𝑛 + 1))))
28	26, 27	eqeq12d 2781	. . . . . . 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 6885	. . . . . . . 8 ⊢ (𝑗 = 𝑘 → (seq𝑀( · , (exp ∘ 𝐹))‘𝑗) = (seq𝑀( · , (exp ∘ 𝐹))‘𝑘))
31		2fveq3 6890	. . . . . . . 8 ⊢ (𝑗 = 𝑘 → (exp‘(seq𝑀( + , 𝐹)‘𝑗)) = (exp‘(seq𝑀( + , 𝐹)‘𝑘)))
32	30, 31	eqeq12d 2781	. . . . . . 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 12895	. . . . . . . . . . 11 ⊢ (𝑀 ∈ ℤ → 𝑀 ∈ (ℤ_≥‘𝑀))
35	2, 34	syl 18	. . . . . . . . . 10 ⊢ (𝜑 → 𝑀 ∈ (ℤ_≥‘𝑀))
36	35, 1	eleqtrrdi 2876	. . . . . . . . 9 ⊢ (𝜑 → 𝑀 ∈ 𝑍)
37		fvco3 6985	. . . . . . . . 9 ⊢ ((𝐹:𝑍⟶ℂ ∧ 𝑀 ∈ 𝑍) → ((exp ∘ 𝐹)‘𝑀) = (exp‘(𝐹‘𝑀)))
38	3, 36, 37	syl2anc 596	. . . . . . . 8 ⊢ (𝜑 → ((exp ∘ 𝐹)‘𝑀) = (exp‘(𝐹‘𝑀)))
39		seq1 14070	. . . . . . . . 9 ⊢ (𝑀 ∈ ℤ → (seq𝑀( · , (exp ∘ 𝐹))‘𝑀) = ((exp ∘ 𝐹)‘𝑀))
40	2, 39	syl 18	. . . . . . . 8 ⊢ (𝜑 → (seq𝑀( · , (exp ∘ 𝐹))‘𝑀) = ((exp ∘ 𝐹)‘𝑀))
41		seq1 14070	. . . . . . . . . 10 ⊢ (𝑀 ∈ ℤ → (seq𝑀( + , 𝐹)‘𝑀) = (𝐹‘𝑀))
42	2, 41	syl 18	. . . . . . . . 9 ⊢ (𝜑 → (seq𝑀( + , 𝐹)‘𝑀) = (𝐹‘𝑀))
43	42	fveq2d 6889	. . . . . . . 8 ⊢ (𝜑 → (exp‘(seq𝑀( + , 𝐹)‘𝑀)) = (exp‘(𝐹‘𝑀)))
44	38, 40, 43	3eqtr4d 2810	. . . . . . 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 7426	. . . . . . . . . . 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 1153	. . . . . . . . . 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 487	. . . . . . . . . . . . . 14 ⊢ ((𝑛 ∈ (ℤ_≥‘𝑀) ∧ 𝜑) → 𝐹:𝑍⟶ℂ)
49		peano2uz 12943	. . . . . . . . . . . . . . . 16 ⊢ (𝑛 ∈ (ℤ_≥‘𝑀) → (𝑛 + 1) ∈ (ℤ_≥‘𝑀))
50	49, 1	eleqtrrdi 2876	. . . . . . . . . . . . . . 15 ⊢ (𝑛 ∈ (ℤ_≥‘𝑀) → (𝑛 + 1) ∈ 𝑍)
51	50	adantr 486	. . . . . . . . . . . . . 14 ⊢ ((𝑛 ∈ (ℤ_≥‘𝑀) ∧ 𝜑) → (𝑛 + 1) ∈ 𝑍)
52		fvco3 6985	. . . . . . . . . . . . . 14 ⊢ ((𝐹:𝑍⟶ℂ ∧ (𝑛 + 1) ∈ 𝑍) → ((exp ∘ 𝐹)‘(𝑛 + 1)) = (exp‘(𝐹‘(𝑛 + 1))))
53	48, 51, 52	syl2anc 596	. . . . . . . . . . . . 13 ⊢ ((𝑛 ∈ (ℤ_≥‘𝑀) ∧ 𝜑) → ((exp ∘ 𝐹)‘(𝑛 + 1)) = (exp‘(𝐹‘(𝑛 + 1))))
54	53	oveq2d 7435	. . . . . . . . . . . 12 ⊢ ((𝑛 ∈ (ℤ_≥‘𝑀) ∧ 𝜑) → ((exp‘(seq𝑀( + , 𝐹)‘𝑛)) · ((exp ∘ 𝐹)‘(𝑛 + 1))) = ((exp‘(seq𝑀( + , 𝐹)‘𝑛)) · (exp‘(𝐹‘(𝑛 + 1)))))
55	15	ffvelcdmda 7083	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → (seq𝑀( + , 𝐹)‘𝑛) ∈ ℂ)
56	55	expcom 419	. . . . . . . . . . . . . . 15 ⊢ (𝑛 ∈ 𝑍 → (𝜑 → (seq𝑀( + , 𝐹)‘𝑛) ∈ ℂ))
57	1	eqcomi 2774	. . . . . . . . . . . . . . 15 ⊢ (ℤ_≥‘𝑀) = 𝑍
58	56, 57	eleq2s 2883	. . . . . . . . . . . . . 14 ⊢ (𝑛 ∈ (ℤ_≥‘𝑀) → (𝜑 → (seq𝑀( + , 𝐹)‘𝑛) ∈ ℂ))
59	58	imp 412	. . . . . . . . . . . . 13 ⊢ ((𝑛 ∈ (ℤ_≥‘𝑀) ∧ 𝜑) → (seq𝑀( + , 𝐹)‘𝑛) ∈ ℂ)
60	48, 51	ffvelcdmd 7084	. . . . . . . . . . . . 13 ⊢ ((𝑛 ∈ (ℤ_≥‘𝑀) ∧ 𝜑) → (𝐹‘(𝑛 + 1)) ∈ ℂ)
61		efadd 16172	. . . . . . . . . . . . 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 596	. . . . . . . . . . . 12 ⊢ ((𝑛 ∈ (ℤ_≥‘𝑀) ∧ 𝜑) → (exp‘((seq𝑀( + , 𝐹)‘𝑛) + (𝐹‘(𝑛 + 1)))) = ((exp‘(seq𝑀( + , 𝐹)‘𝑛)) · (exp‘(𝐹‘(𝑛 + 1)))))
63	54, 62	eqtr4d 2803	. . . . . . . . . . 11 ⊢ ((𝑛 ∈ (ℤ_≥‘𝑀) ∧ 𝜑) → ((exp‘(seq𝑀( + , 𝐹)‘𝑛)) · ((exp ∘ 𝐹)‘(𝑛 + 1))) = (exp‘((seq𝑀( + , 𝐹)‘𝑛) + (𝐹‘(𝑛 + 1)))))
64	63	3adant3 1150	. . . . . . . . . 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 2800	. . . . . . . . 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 14072	. . . . . . . . . . 11 ⊢ (𝑛 ∈ (ℤ_≥‘𝑀) → (seq𝑀( · , (exp ∘ 𝐹))‘(𝑛 + 1)) = ((seq𝑀( · , (exp ∘ 𝐹))‘𝑛) · ((exp ∘ 𝐹)‘(𝑛 + 1))))
67	66	adantr 486	. . . . . . . . . 10 ⊢ ((𝑛 ∈ (ℤ_≥‘𝑀) ∧ 𝜑) → (seq𝑀( · , (exp ∘ 𝐹))‘(𝑛 + 1)) = ((seq𝑀( · , (exp ∘ 𝐹))‘𝑛) · ((exp ∘ 𝐹)‘(𝑛 + 1))))
68	67	3adant3 1150	. . . . . . . . 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 14072	. . . . . . . . . . . 12 ⊢ (𝑛 ∈ (ℤ_≥‘𝑀) → (seq𝑀( + , 𝐹)‘(𝑛 + 1)) = ((seq𝑀( + , 𝐹)‘𝑛) + (𝐹‘(𝑛 + 1))))
70	69	adantr 486	. . . . . . . . . . 11 ⊢ ((𝑛 ∈ (ℤ_≥‘𝑀) ∧ 𝜑) → (seq𝑀( + , 𝐹)‘(𝑛 + 1)) = ((seq𝑀( + , 𝐹)‘𝑛) + (𝐹‘(𝑛 + 1))))
71	70	fveq2d 6889	. . . . . . . . . 10 ⊢ ((𝑛 ∈ (ℤ_≥‘𝑀) ∧ 𝜑) → (exp‘(seq𝑀( + , 𝐹)‘(𝑛 + 1))) = (exp‘((seq𝑀( + , 𝐹)‘𝑛) + (𝐹‘(𝑛 + 1)))))
72	71	3adant3 1150	. . . . . . . . 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 2810	. . . . . . . 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 1137	. . . . . . 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 30	. . . . . 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 12948	. . . . 5 ⊢ (𝑘 ∈ (ℤ_≥‘𝑀) → (𝜑 → (seq𝑀( · , (exp ∘ 𝐹))‘𝑘) = (exp‘(seq𝑀( + , 𝐹)‘𝑘))))
77	76, 1	eleq2s 2883	. . . 4 ⊢ (𝑘 ∈ 𝑍 → (𝜑 → (seq𝑀( · , (exp ∘ 𝐹))‘𝑘) = (exp‘(seq𝑀( + , 𝐹)‘𝑘))))
78	77	impcom 413	. . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (seq𝑀( · , (exp ∘ 𝐹))‘𝑘) = (exp‘(seq𝑀( + , 𝐹)‘𝑘)))
79		fvco3 6985	. . . 4 ⊢ ((seq𝑀( + , 𝐹):𝑍⟶ℂ ∧ 𝑘 ∈ 𝑍) → ((exp ∘ seq𝑀( + , 𝐹))‘𝑘) = (exp‘(seq𝑀( + , 𝐹)‘𝑘)))
80	15, 79	sylan 592	. . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → ((exp ∘ seq𝑀( + , 𝐹))‘𝑘) = (exp‘(seq𝑀( + , 𝐹)‘𝑘)))
81	78, 80	eqtr4d 2803	. 2 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (seq𝑀( · , (exp ∘ 𝐹))‘𝑘) = ((exp ∘ seq𝑀( + , 𝐹))‘𝑘))
82	11, 17, 81	eqfnfvd 7032	1 ⊢ (𝜑 → seq𝑀( · , (exp ∘ 𝐹)) = (exp ∘ seq𝑀( + , 𝐹)))

Colors of variables: wff setvar class

This proof depends on syntax axioms: → wi 4 ∧ wa 401 ∧ w3a 1103 = wceq 1570 ∈ wcel 2146 ∘ ccom 5667 Fn wfn 6535 ⟶wf 6536 ‘cfv 6540 (class class class)co 7419 ℂcc 11115 1c1 11118 + caddc 11120 · cmul 11122 ℤcz 12608 ℤ_≥cuz 12880 seqcseq 14057 expce 16139

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1828 ax-4 1842 ax-5 1943 ax-6 2000 ax-7 2041 ax-8 2148 ax-9 2156 ax-10 2179 ax-11 2195 ax-12 2216 ax-ext 2737 ax-rep 5240 ax-sep 5259 ax-nul 5271 ax-pow 5338 ax-pr 5406 ax-un 7742 ax-inf2 9617 ax-cnex 11173 ax-resscn 11174 ax-1cn 11175 ax-icn 11176 ax-addcl 11177 ax-addrcl 11178 ax-mulcl 11179 ax-mulrcl 11180 ax-mulcom 11181 ax-addass 11182 ax-mulass 11183 ax-distr 11184 ax-i2m1 11185 ax-1ne0 11186 ax-1rid 11187 ax-rnegex 11188 ax-rrecex 11189 ax-cnre 11190 ax-pre-lttri 11191 ax-pre-lttrn 11192 ax-pre-ltadd 11193 ax-pre-mulgt0 11194 ax-pre-sup 11195

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1813 df-nf 1817 df-sb 2100 df-mo 2569 df-eu 2599 df-clab 2744 df-cleq 2757 df-clel 2840 df-nfc 2914 df-ne 2961 df-nel 3067 df-ral 3082 df-rex 3092 df-rmo 3371 df-reu 3372 df-rab 3419 df-v 3459 df-sbc 3747 df-csb 3855 df-dif 3909 df-un 3911 df-in 3913 df-ss 3923 df-pss 3926 df-nul 4287 df-if 4490 df-pw 4566 df-sn 4592 df-pr 4594 df-op 4598 df-uni 4875 df-int 4915 df-iun 4960 df-br 5112 df-opab 5176 df-mpt 5195 df-tr 5221 df-id 5558 df-eprel 5563 df-po 5571 df-so 5572 df-fr 5616 df-se 5617 df-we 5618 df-xp 5669 df-rel 5670 df-cnv 5671 df-co 5672 df-dm 5673 df-rn 5674 df-res 5675 df-ima 5676 df-pred 6306 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6496 df-fun 6542 df-fn 6543 df-f 6544 df-f1 6545 df-fo 6546 df-f1o 6547 df-fv 6548 df-isom 6549 df-riota 7376 df-ov 7422 df-oprab 7423 df-mpo 7424 df-om 7869 df-1st 7992 df-2nd 7993 df-frecs 8284 df-wrecs 8315 df-recs 8364 df-rdg 8403 df-1o 8459 df-er 8700 df-pm 8833 df-en 8950 df-dom 8951 df-sdom 8952 df-fin 8953 df-sup 9409 df-inf 9410 df-oi 9479 df-card 9941 df-pnf 11262 df-mnf 11263 df-xr 11264 df-ltxr 11265 df-le 11266 df-sub 11460 df-neg 11461 df-div 11889 df-nn 12251 df-2 12320 df-3 12321 df-n0 12522 df-z 12609 df-uz 12881 df-rp 13035 df-ico 13396 df-fz 13554 df-fzo 13702 df-fl 13845 df-seq 14058 df-exp 14118 df-fac 14330 df-bc 14359 df-hash 14387 df-shft 15130 df-cj 15176 df-re 15177 df-im 15178 df-sqrt 15312 df-abs 15313 df-limsup 15548 df-clim 15565 df-rlim 15566 df-sum 15764 df-ef 16145

This theorem is used by: iprodefisum 36272

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