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Theorem iprodefisumlem 35740
Description: Lemma for iprodefisum 35741. (Contributed by Scott Fenton, 11-Feb-2018.)
Hypotheses
Ref Expression
iprodefisumlem.1 𝑍 = (ℤ𝑀)
iprodefisumlem.2 (𝜑𝑀 ∈ ℤ)
iprodefisumlem.3 (𝜑𝐹: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.
StepHypRef Expression
1 iprodefisumlem.1 . . . 4 𝑍 = (ℤ𝑀)
2 iprodefisumlem.2 . . . 4 (𝜑𝑀 ∈ ℤ)
3 iprodefisumlem.3 . . . . . 6 (𝜑𝐹:𝑍⟶ℂ)
4 fvco3 7008 . . . . . 6 ((𝐹:𝑍⟶ℂ ∧ 𝑘𝑍) → ((exp ∘ 𝐹)‘𝑘) = (exp‘(𝐹𝑘)))
53, 4sylan 580 . . . . 5 ((𝜑𝑘𝑍) → ((exp ∘ 𝐹)‘𝑘) = (exp‘(𝐹𝑘)))
63ffvelcdmda 7104 . . . . . 6 ((𝜑𝑘𝑍) → (𝐹𝑘) ∈ ℂ)
7 efcl 16118 . . . . . 6 ((𝐹𝑘) ∈ ℂ → (exp‘(𝐹𝑘)) ∈ ℂ)
86, 7syl 17 . . . . 5 ((𝜑𝑘𝑍) → (exp‘(𝐹𝑘)) ∈ ℂ)
95, 8eqeltrd 2841 . . . 4 ((𝜑𝑘𝑍) → ((exp ∘ 𝐹)‘𝑘) ∈ ℂ)
101, 2, 9prodf 15923 . . 3 (𝜑 → seq𝑀( · , (exp ∘ 𝐹)):𝑍⟶ℂ)
1110ffnd 6737 . 2 (𝜑 → seq𝑀( · , (exp ∘ 𝐹)) Fn 𝑍)
12 eff 16117 . . . 4 exp:ℂ⟶ℂ
13 ffn 6736 . . . 4 (exp:ℂ⟶ℂ → exp Fn ℂ)
1412, 13ax-mp 5 . . 3 exp Fn ℂ
151, 2, 6serf 14071 . . 3 (𝜑 → seq𝑀( + , 𝐹):𝑍⟶ℂ)
16 fnfco 6773 . . 3 ((exp Fn ℂ ∧ seq𝑀( + , 𝐹):𝑍⟶ℂ) → (exp ∘ seq𝑀( + , 𝐹)) Fn 𝑍)
1714, 15, 16sylancr 587 . 2 (𝜑 → (exp ∘ seq𝑀( + , 𝐹)) Fn 𝑍)
18 fveq2 6906 . . . . . . . 8 (𝑗 = 𝑀 → (seq𝑀( · , (exp ∘ 𝐹))‘𝑗) = (seq𝑀( · , (exp ∘ 𝐹))‘𝑀))
19 2fveq3 6911 . . . . . . . 8 (𝑗 = 𝑀 → (exp‘(seq𝑀( + , 𝐹)‘𝑗)) = (exp‘(seq𝑀( + , 𝐹)‘𝑀)))
2018, 19eqeq12d 2753 . . . . . . 7 (𝑗 = 𝑀 → ((seq𝑀( · , (exp ∘ 𝐹))‘𝑗) = (exp‘(seq𝑀( + , 𝐹)‘𝑗)) ↔ (seq𝑀( · , (exp ∘ 𝐹))‘𝑀) = (exp‘(seq𝑀( + , 𝐹)‘𝑀))))
2120imbi2d 340 . . . . . 6 (𝑗 = 𝑀 → ((𝜑 → (seq𝑀( · , (exp ∘ 𝐹))‘𝑗) = (exp‘(seq𝑀( + , 𝐹)‘𝑗))) ↔ (𝜑 → (seq𝑀( · , (exp ∘ 𝐹))‘𝑀) = (exp‘(seq𝑀( + , 𝐹)‘𝑀)))))
22 fveq2 6906 . . . . . . . 8 (𝑗 = 𝑛 → (seq𝑀( · , (exp ∘ 𝐹))‘𝑗) = (seq𝑀( · , (exp ∘ 𝐹))‘𝑛))
23 2fveq3 6911 . . . . . . . 8 (𝑗 = 𝑛 → (exp‘(seq𝑀( + , 𝐹)‘𝑗)) = (exp‘(seq𝑀( + , 𝐹)‘𝑛)))
2422, 23eqeq12d 2753 . . . . . . 7 (𝑗 = 𝑛 → ((seq𝑀( · , (exp ∘ 𝐹))‘𝑗) = (exp‘(seq𝑀( + , 𝐹)‘𝑗)) ↔ (seq𝑀( · , (exp ∘ 𝐹))‘𝑛) = (exp‘(seq𝑀( + , 𝐹)‘𝑛))))
2524imbi2d 340 . . . . . 6 (𝑗 = 𝑛 → ((𝜑 → (seq𝑀( · , (exp ∘ 𝐹))‘𝑗) = (exp‘(seq𝑀( + , 𝐹)‘𝑗))) ↔ (𝜑 → (seq𝑀( · , (exp ∘ 𝐹))‘𝑛) = (exp‘(seq𝑀( + , 𝐹)‘𝑛)))))
26 fveq2 6906 . . . . . . . 8 (𝑗 = (𝑛 + 1) → (seq𝑀( · , (exp ∘ 𝐹))‘𝑗) = (seq𝑀( · , (exp ∘ 𝐹))‘(𝑛 + 1)))
27 2fveq3 6911 . . . . . . . 8 (𝑗 = (𝑛 + 1) → (exp‘(seq𝑀( + , 𝐹)‘𝑗)) = (exp‘(seq𝑀( + , 𝐹)‘(𝑛 + 1))))
2826, 27eqeq12d 2753 . . . . . . 7 (𝑗 = (𝑛 + 1) → ((seq𝑀( · , (exp ∘ 𝐹))‘𝑗) = (exp‘(seq𝑀( + , 𝐹)‘𝑗)) ↔ (seq𝑀( · , (exp ∘ 𝐹))‘(𝑛 + 1)) = (exp‘(seq𝑀( + , 𝐹)‘(𝑛 + 1)))))
2928imbi2d 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 6906 . . . . . . . 8 (𝑗 = 𝑘 → (seq𝑀( · , (exp ∘ 𝐹))‘𝑗) = (seq𝑀( · , (exp ∘ 𝐹))‘𝑘))
31 2fveq3 6911 . . . . . . . 8 (𝑗 = 𝑘 → (exp‘(seq𝑀( + , 𝐹)‘𝑗)) = (exp‘(seq𝑀( + , 𝐹)‘𝑘)))
3230, 31eqeq12d 2753 . . . . . . 7 (𝑗 = 𝑘 → ((seq𝑀( · , (exp ∘ 𝐹))‘𝑗) = (exp‘(seq𝑀( + , 𝐹)‘𝑗)) ↔ (seq𝑀( · , (exp ∘ 𝐹))‘𝑘) = (exp‘(seq𝑀( + , 𝐹)‘𝑘))))
3332imbi2d 340 . . . . . 6 (𝑗 = 𝑘 → ((𝜑 → (seq𝑀( · , (exp ∘ 𝐹))‘𝑗) = (exp‘(seq𝑀( + , 𝐹)‘𝑗))) ↔ (𝜑 → (seq𝑀( · , (exp ∘ 𝐹))‘𝑘) = (exp‘(seq𝑀( + , 𝐹)‘𝑘)))))
34 uzid 12893 . . . . . . . . . . 11 (𝑀 ∈ ℤ → 𝑀 ∈ (ℤ𝑀))
352, 34syl 17 . . . . . . . . . 10 (𝜑𝑀 ∈ (ℤ𝑀))
3635, 1eleqtrrdi 2852 . . . . . . . . 9 (𝜑𝑀𝑍)
37 fvco3 7008 . . . . . . . . 9 ((𝐹:𝑍⟶ℂ ∧ 𝑀𝑍) → ((exp ∘ 𝐹)‘𝑀) = (exp‘(𝐹𝑀)))
383, 36, 37syl2anc 584 . . . . . . . 8 (𝜑 → ((exp ∘ 𝐹)‘𝑀) = (exp‘(𝐹𝑀)))
39 seq1 14055 . . . . . . . . 9 (𝑀 ∈ ℤ → (seq𝑀( · , (exp ∘ 𝐹))‘𝑀) = ((exp ∘ 𝐹)‘𝑀))
402, 39syl 17 . . . . . . . 8 (𝜑 → (seq𝑀( · , (exp ∘ 𝐹))‘𝑀) = ((exp ∘ 𝐹)‘𝑀))
41 seq1 14055 . . . . . . . . . 10 (𝑀 ∈ ℤ → (seq𝑀( + , 𝐹)‘𝑀) = (𝐹𝑀))
422, 41syl 17 . . . . . . . . 9 (𝜑 → (seq𝑀( + , 𝐹)‘𝑀) = (𝐹𝑀))
4342fveq2d 6910 . . . . . . . 8 (𝜑 → (exp‘(seq𝑀( + , 𝐹)‘𝑀)) = (exp‘(𝐹𝑀)))
4438, 40, 433eqtr4d 2787 . . . . . . 7 (𝜑 → (seq𝑀( · , (exp ∘ 𝐹))‘𝑀) = (exp‘(seq𝑀( + , 𝐹)‘𝑀)))
4544a1i 11 . . . . . 6 (𝑀 ∈ ℤ → (𝜑 → (seq𝑀( · , (exp ∘ 𝐹))‘𝑀) = (exp‘(seq𝑀( + , 𝐹)‘𝑀))))
46 oveq1 7438 . . . . . . . . . . 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))))
47463ad2ant3 1136 . . . . . . . . . 10 ((𝑛 ∈ (ℤ𝑀) ∧ 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))))
483adantl 481 . . . . . . . . . . . . . 14 ((𝑛 ∈ (ℤ𝑀) ∧ 𝜑) → 𝐹:𝑍⟶ℂ)
49 peano2uz 12943 . . . . . . . . . . . . . . . 16 (𝑛 ∈ (ℤ𝑀) → (𝑛 + 1) ∈ (ℤ𝑀))
5049, 1eleqtrrdi 2852 . . . . . . . . . . . . . . 15 (𝑛 ∈ (ℤ𝑀) → (𝑛 + 1) ∈ 𝑍)
5150adantr 480 . . . . . . . . . . . . . 14 ((𝑛 ∈ (ℤ𝑀) ∧ 𝜑) → (𝑛 + 1) ∈ 𝑍)
52 fvco3 7008 . . . . . . . . . . . . . 14 ((𝐹:𝑍⟶ℂ ∧ (𝑛 + 1) ∈ 𝑍) → ((exp ∘ 𝐹)‘(𝑛 + 1)) = (exp‘(𝐹‘(𝑛 + 1))))
5348, 51, 52syl2anc 584 . . . . . . . . . . . . 13 ((𝑛 ∈ (ℤ𝑀) ∧ 𝜑) → ((exp ∘ 𝐹)‘(𝑛 + 1)) = (exp‘(𝐹‘(𝑛 + 1))))
5453oveq2d 7447 . . . . . . . . . . . 12 ((𝑛 ∈ (ℤ𝑀) ∧ 𝜑) → ((exp‘(seq𝑀( + , 𝐹)‘𝑛)) · ((exp ∘ 𝐹)‘(𝑛 + 1))) = ((exp‘(seq𝑀( + , 𝐹)‘𝑛)) · (exp‘(𝐹‘(𝑛 + 1)))))
5515ffvelcdmda 7104 . . . . . . . . . . . . . . . 16 ((𝜑𝑛𝑍) → (seq𝑀( + , 𝐹)‘𝑛) ∈ ℂ)
5655expcom 413 . . . . . . . . . . . . . . 15 (𝑛𝑍 → (𝜑 → (seq𝑀( + , 𝐹)‘𝑛) ∈ ℂ))
571eqcomi 2746 . . . . . . . . . . . . . . 15 (ℤ𝑀) = 𝑍
5856, 57eleq2s 2859 . . . . . . . . . . . . . 14 (𝑛 ∈ (ℤ𝑀) → (𝜑 → (seq𝑀( + , 𝐹)‘𝑛) ∈ ℂ))
5958imp 406 . . . . . . . . . . . . 13 ((𝑛 ∈ (ℤ𝑀) ∧ 𝜑) → (seq𝑀( + , 𝐹)‘𝑛) ∈ ℂ)
6048, 51ffvelcdmd 7105 . . . . . . . . . . . . 13 ((𝑛 ∈ (ℤ𝑀) ∧ 𝜑) → (𝐹‘(𝑛 + 1)) ∈ ℂ)
61 efadd 16130 . . . . . . . . . . . . 13 (((seq𝑀( + , 𝐹)‘𝑛) ∈ ℂ ∧ (𝐹‘(𝑛 + 1)) ∈ ℂ) → (exp‘((seq𝑀( + , 𝐹)‘𝑛) + (𝐹‘(𝑛 + 1)))) = ((exp‘(seq𝑀( + , 𝐹)‘𝑛)) · (exp‘(𝐹‘(𝑛 + 1)))))
6259, 60, 61syl2anc 584 . . . . . . . . . . . 12 ((𝑛 ∈ (ℤ𝑀) ∧ 𝜑) → (exp‘((seq𝑀( + , 𝐹)‘𝑛) + (𝐹‘(𝑛 + 1)))) = ((exp‘(seq𝑀( + , 𝐹)‘𝑛)) · (exp‘(𝐹‘(𝑛 + 1)))))
6354, 62eqtr4d 2780 . . . . . . . . . . 11 ((𝑛 ∈ (ℤ𝑀) ∧ 𝜑) → ((exp‘(seq𝑀( + , 𝐹)‘𝑛)) · ((exp ∘ 𝐹)‘(𝑛 + 1))) = (exp‘((seq𝑀( + , 𝐹)‘𝑛) + (𝐹‘(𝑛 + 1)))))
64633adant3 1133 . . . . . . . . . 10 ((𝑛 ∈ (ℤ𝑀) ∧ 𝜑 ∧ (seq𝑀( · , (exp ∘ 𝐹))‘𝑛) = (exp‘(seq𝑀( + , 𝐹)‘𝑛))) → ((exp‘(seq𝑀( + , 𝐹)‘𝑛)) · ((exp ∘ 𝐹)‘(𝑛 + 1))) = (exp‘((seq𝑀( + , 𝐹)‘𝑛) + (𝐹‘(𝑛 + 1)))))
6547, 64eqtrd 2777 . . . . . . . . 9 ((𝑛 ∈ (ℤ𝑀) ∧ 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 14057 . . . . . . . . . . 11 (𝑛 ∈ (ℤ𝑀) → (seq𝑀( · , (exp ∘ 𝐹))‘(𝑛 + 1)) = ((seq𝑀( · , (exp ∘ 𝐹))‘𝑛) · ((exp ∘ 𝐹)‘(𝑛 + 1))))
6766adantr 480 . . . . . . . . . 10 ((𝑛 ∈ (ℤ𝑀) ∧ 𝜑) → (seq𝑀( · , (exp ∘ 𝐹))‘(𝑛 + 1)) = ((seq𝑀( · , (exp ∘ 𝐹))‘𝑛) · ((exp ∘ 𝐹)‘(𝑛 + 1))))
68673adant3 1133 . . . . . . . . 9 ((𝑛 ∈ (ℤ𝑀) ∧ 𝜑 ∧ (seq𝑀( · , (exp ∘ 𝐹))‘𝑛) = (exp‘(seq𝑀( + , 𝐹)‘𝑛))) → (seq𝑀( · , (exp ∘ 𝐹))‘(𝑛 + 1)) = ((seq𝑀( · , (exp ∘ 𝐹))‘𝑛) · ((exp ∘ 𝐹)‘(𝑛 + 1))))
69 seqp1 14057 . . . . . . . . . . . 12 (𝑛 ∈ (ℤ𝑀) → (seq𝑀( + , 𝐹)‘(𝑛 + 1)) = ((seq𝑀( + , 𝐹)‘𝑛) + (𝐹‘(𝑛 + 1))))
7069adantr 480 . . . . . . . . . . 11 ((𝑛 ∈ (ℤ𝑀) ∧ 𝜑) → (seq𝑀( + , 𝐹)‘(𝑛 + 1)) = ((seq𝑀( + , 𝐹)‘𝑛) + (𝐹‘(𝑛 + 1))))
7170fveq2d 6910 . . . . . . . . . 10 ((𝑛 ∈ (ℤ𝑀) ∧ 𝜑) → (exp‘(seq𝑀( + , 𝐹)‘(𝑛 + 1))) = (exp‘((seq𝑀( + , 𝐹)‘𝑛) + (𝐹‘(𝑛 + 1)))))
72713adant3 1133 . . . . . . . . 9 ((𝑛 ∈ (ℤ𝑀) ∧ 𝜑 ∧ (seq𝑀( · , (exp ∘ 𝐹))‘𝑛) = (exp‘(seq𝑀( + , 𝐹)‘𝑛))) → (exp‘(seq𝑀( + , 𝐹)‘(𝑛 + 1))) = (exp‘((seq𝑀( + , 𝐹)‘𝑛) + (𝐹‘(𝑛 + 1)))))
7365, 68, 723eqtr4d 2787 . . . . . . . 8 ((𝑛 ∈ (ℤ𝑀) ∧ 𝜑 ∧ (seq𝑀( · , (exp ∘ 𝐹))‘𝑛) = (exp‘(seq𝑀( + , 𝐹)‘𝑛))) → (seq𝑀( · , (exp ∘ 𝐹))‘(𝑛 + 1)) = (exp‘(seq𝑀( + , 𝐹)‘(𝑛 + 1))))
74733exp 1120 . . . . . . 7 (𝑛 ∈ (ℤ𝑀) → (𝜑 → ((seq𝑀( · , (exp ∘ 𝐹))‘𝑛) = (exp‘(seq𝑀( + , 𝐹)‘𝑛)) → (seq𝑀( · , (exp ∘ 𝐹))‘(𝑛 + 1)) = (exp‘(seq𝑀( + , 𝐹)‘(𝑛 + 1))))))
7574a2d 29 . . . . . 6 (𝑛 ∈ (ℤ𝑀) → ((𝜑 → (seq𝑀( · , (exp ∘ 𝐹))‘𝑛) = (exp‘(seq𝑀( + , 𝐹)‘𝑛))) → (𝜑 → (seq𝑀( · , (exp ∘ 𝐹))‘(𝑛 + 1)) = (exp‘(seq𝑀( + , 𝐹)‘(𝑛 + 1))))))
7621, 25, 29, 33, 45, 75uzind4 12948 . . . . 5 (𝑘 ∈ (ℤ𝑀) → (𝜑 → (seq𝑀( · , (exp ∘ 𝐹))‘𝑘) = (exp‘(seq𝑀( + , 𝐹)‘𝑘))))
7776, 1eleq2s 2859 . . . 4 (𝑘𝑍 → (𝜑 → (seq𝑀( · , (exp ∘ 𝐹))‘𝑘) = (exp‘(seq𝑀( + , 𝐹)‘𝑘))))
7877impcom 407 . . 3 ((𝜑𝑘𝑍) → (seq𝑀( · , (exp ∘ 𝐹))‘𝑘) = (exp‘(seq𝑀( + , 𝐹)‘𝑘)))
79 fvco3 7008 . . . 4 ((seq𝑀( + , 𝐹):𝑍⟶ℂ ∧ 𝑘𝑍) → ((exp ∘ seq𝑀( + , 𝐹))‘𝑘) = (exp‘(seq𝑀( + , 𝐹)‘𝑘)))
8015, 79sylan 580 . . 3 ((𝜑𝑘𝑍) → ((exp ∘ seq𝑀( + , 𝐹))‘𝑘) = (exp‘(seq𝑀( + , 𝐹)‘𝑘)))
8178, 80eqtr4d 2780 . 2 ((𝜑𝑘𝑍) → (seq𝑀( · , (exp ∘ 𝐹))‘𝑘) = ((exp ∘ seq𝑀( + , 𝐹))‘𝑘))
8211, 17, 81eqfnfvd 7054 1 (𝜑 → seq𝑀( · , (exp ∘ 𝐹)) = (exp ∘ seq𝑀( + , 𝐹)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  w3a 1087   = wceq 1540  wcel 2108  ccom 5689   Fn wfn 6556  wf 6557  cfv 6561  (class class class)co 7431  cc 11153  1c1 11156   + caddc 11158   · cmul 11160  cz 12613  cuz 12878  seqcseq 14042  expce 16097
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 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755  ax-inf2 9681  ax-cnex 11211  ax-resscn 11212  ax-1cn 11213  ax-icn 11214  ax-addcl 11215  ax-addrcl 11216  ax-mulcl 11217  ax-mulrcl 11218  ax-mulcom 11219  ax-addass 11220  ax-mulass 11221  ax-distr 11222  ax-i2m1 11223  ax-1ne0 11224  ax-1rid 11225  ax-rnegex 11226  ax-rrecex 11227  ax-cnre 11228  ax-pre-lttri 11229  ax-pre-lttrn 11230  ax-pre-ltadd 11231  ax-pre-mulgt0 11232  ax-pre-sup 11233
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-rmo 3380  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-int 4947  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-se 5638  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-pred 6321  df-ord 6387  df-on 6388  df-lim 6389  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-isom 6570  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-1st 8014  df-2nd 8015  df-frecs 8306  df-wrecs 8337  df-recs 8411  df-rdg 8450  df-1o 8506  df-er 8745  df-pm 8869  df-en 8986  df-dom 8987  df-sdom 8988  df-fin 8989  df-sup 9482  df-inf 9483  df-oi 9550  df-card 9979  df-pnf 11297  df-mnf 11298  df-xr 11299  df-ltxr 11300  df-le 11301  df-sub 11494  df-neg 11495  df-div 11921  df-nn 12267  df-2 12329  df-3 12330  df-n0 12527  df-z 12614  df-uz 12879  df-rp 13035  df-ico 13393  df-fz 13548  df-fzo 13695  df-fl 13832  df-seq 14043  df-exp 14103  df-fac 14313  df-bc 14342  df-hash 14370  df-shft 15106  df-cj 15138  df-re 15139  df-im 15140  df-sqrt 15274  df-abs 15275  df-limsup 15507  df-clim 15524  df-rlim 15525  df-sum 15723  df-ef 16103
This theorem is referenced by:  iprodefisum  35741
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