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Theorem iprodefisumlem 35720
Description: Lemma for iprodefisum 35721. (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 16115 . . . . . 6 ((𝐹𝑘) ∈ ℂ → (exp‘(𝐹𝑘)) ∈ ℂ)
86, 7syl 17 . . . . 5 ((𝜑𝑘𝑍) → (exp‘(𝐹𝑘)) ∈ ℂ)
95, 8eqeltrd 2839 . . . 4 ((𝜑𝑘𝑍) → ((exp ∘ 𝐹)‘𝑘) ∈ ℂ)
101, 2, 9prodf 15920 . . 3 (𝜑 → seq𝑀( · , (exp ∘ 𝐹)):𝑍⟶ℂ)
1110ffnd 6738 . 2 (𝜑 → seq𝑀( · , (exp ∘ 𝐹)) Fn 𝑍)
12 eff 16114 . . . 4 exp:ℂ⟶ℂ
13 ffn 6737 . . . 4 (exp:ℂ⟶ℂ → exp Fn ℂ)
1412, 13ax-mp 5 . . 3 exp Fn ℂ
151, 2, 6serf 14068 . . 3 (𝜑 → seq𝑀( + , 𝐹):𝑍⟶ℂ)
16 fnfco 6774 . . 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 6907 . . . . . . . 8 (𝑗 = 𝑀 → (seq𝑀( · , (exp ∘ 𝐹))‘𝑗) = (seq𝑀( · , (exp ∘ 𝐹))‘𝑀))
19 2fveq3 6912 . . . . . . . 8 (𝑗 = 𝑀 → (exp‘(seq𝑀( + , 𝐹)‘𝑗)) = (exp‘(seq𝑀( + , 𝐹)‘𝑀)))
2018, 19eqeq12d 2751 . . . . . . 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 6907 . . . . . . . 8 (𝑗 = 𝑛 → (seq𝑀( · , (exp ∘ 𝐹))‘𝑗) = (seq𝑀( · , (exp ∘ 𝐹))‘𝑛))
23 2fveq3 6912 . . . . . . . 8 (𝑗 = 𝑛 → (exp‘(seq𝑀( + , 𝐹)‘𝑗)) = (exp‘(seq𝑀( + , 𝐹)‘𝑛)))
2422, 23eqeq12d 2751 . . . . . . 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 6907 . . . . . . . 8 (𝑗 = (𝑛 + 1) → (seq𝑀( · , (exp ∘ 𝐹))‘𝑗) = (seq𝑀( · , (exp ∘ 𝐹))‘(𝑛 + 1)))
27 2fveq3 6912 . . . . . . . 8 (𝑗 = (𝑛 + 1) → (exp‘(seq𝑀( + , 𝐹)‘𝑗)) = (exp‘(seq𝑀( + , 𝐹)‘(𝑛 + 1))))
2826, 27eqeq12d 2751 . . . . . . 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 6907 . . . . . . . 8 (𝑗 = 𝑘 → (seq𝑀( · , (exp ∘ 𝐹))‘𝑗) = (seq𝑀( · , (exp ∘ 𝐹))‘𝑘))
31 2fveq3 6912 . . . . . . . 8 (𝑗 = 𝑘 → (exp‘(seq𝑀( + , 𝐹)‘𝑗)) = (exp‘(seq𝑀( + , 𝐹)‘𝑘)))
3230, 31eqeq12d 2751 . . . . . . 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 12891 . . . . . . . . . . 11 (𝑀 ∈ ℤ → 𝑀 ∈ (ℤ𝑀))
352, 34syl 17 . . . . . . . . . 10 (𝜑𝑀 ∈ (ℤ𝑀))
3635, 1eleqtrrdi 2850 . . . . . . . . 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 14052 . . . . . . . . 9 (𝑀 ∈ ℤ → (seq𝑀( · , (exp ∘ 𝐹))‘𝑀) = ((exp ∘ 𝐹)‘𝑀))
402, 39syl 17 . . . . . . . 8 (𝜑 → (seq𝑀( · , (exp ∘ 𝐹))‘𝑀) = ((exp ∘ 𝐹)‘𝑀))
41 seq1 14052 . . . . . . . . . 10 (𝑀 ∈ ℤ → (seq𝑀( + , 𝐹)‘𝑀) = (𝐹𝑀))
422, 41syl 17 . . . . . . . . 9 (𝜑 → (seq𝑀( + , 𝐹)‘𝑀) = (𝐹𝑀))
4342fveq2d 6911 . . . . . . . 8 (𝜑 → (exp‘(seq𝑀( + , 𝐹)‘𝑀)) = (exp‘(𝐹𝑀)))
4438, 40, 433eqtr4d 2785 . . . . . . 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 1134 . . . . . . . . . 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 12941 . . . . . . . . . . . . . . . 16 (𝑛 ∈ (ℤ𝑀) → (𝑛 + 1) ∈ (ℤ𝑀))
5049, 1eleqtrrdi 2850 . . . . . . . . . . . . . . 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 2744 . . . . . . . . . . . . . . 15 (ℤ𝑀) = 𝑍
5856, 57eleq2s 2857 . . . . . . . . . . . . . 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 16127 . . . . . . . . . . . . 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 2778 . . . . . . . . . . 11 ((𝑛 ∈ (ℤ𝑀) ∧ 𝜑) → ((exp‘(seq𝑀( + , 𝐹)‘𝑛)) · ((exp ∘ 𝐹)‘(𝑛 + 1))) = (exp‘((seq𝑀( + , 𝐹)‘𝑛) + (𝐹‘(𝑛 + 1)))))
64633adant3 1131 . . . . . . . . . 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 2775 . . . . . . . . 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 14054 . . . . . . . . . . 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 1131 . . . . . . . . 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 14054 . . . . . . . . . . . 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 6911 . . . . . . . . . 10 ((𝑛 ∈ (ℤ𝑀) ∧ 𝜑) → (exp‘(seq𝑀( + , 𝐹)‘(𝑛 + 1))) = (exp‘((seq𝑀( + , 𝐹)‘𝑛) + (𝐹‘(𝑛 + 1)))))
72713adant3 1131 . . . . . . . . 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 2785 . . . . . . . 8 ((𝑛 ∈ (ℤ𝑀) ∧ 𝜑 ∧ (seq𝑀( · , (exp ∘ 𝐹))‘𝑛) = (exp‘(seq𝑀( + , 𝐹)‘𝑛))) → (seq𝑀( · , (exp ∘ 𝐹))‘(𝑛 + 1)) = (exp‘(seq𝑀( + , 𝐹)‘(𝑛 + 1))))
74733exp 1118 . . . . . . 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 12946 . . . . 5 (𝑘 ∈ (ℤ𝑀) → (𝜑 → (seq𝑀( · , (exp ∘ 𝐹))‘𝑘) = (exp‘(seq𝑀( + , 𝐹)‘𝑘))))
7776, 1eleq2s 2857 . . . 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 2778 . 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 1086   = wceq 1537  wcel 2106  ccom 5693   Fn wfn 6558  wf 6559  cfv 6563  (class class class)co 7431  cc 11151  1c1 11154   + caddc 11156   · cmul 11158  cz 12611  cuz 12876  seqcseq 14039  expce 16094
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754  ax-inf2 9679  ax-cnex 11209  ax-resscn 11210  ax-1cn 11211  ax-icn 11212  ax-addcl 11213  ax-addrcl 11214  ax-mulcl 11215  ax-mulrcl 11216  ax-mulcom 11217  ax-addass 11218  ax-mulass 11219  ax-distr 11220  ax-i2m1 11221  ax-1ne0 11222  ax-1rid 11223  ax-rnegex 11224  ax-rrecex 11225  ax-cnre 11226  ax-pre-lttri 11227  ax-pre-lttrn 11228  ax-pre-ltadd 11229  ax-pre-mulgt0 11230  ax-pre-sup 11231
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-nel 3045  df-ral 3060  df-rex 3069  df-rmo 3378  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-pss 3983  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-int 4952  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5583  df-eprel 5589  df-po 5597  df-so 5598  df-fr 5641  df-se 5642  df-we 5643  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-pred 6323  df-ord 6389  df-on 6390  df-lim 6391  df-suc 6392  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-isom 6572  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-1st 8013  df-2nd 8014  df-frecs 8305  df-wrecs 8336  df-recs 8410  df-rdg 8449  df-1o 8505  df-er 8744  df-pm 8868  df-en 8985  df-dom 8986  df-sdom 8987  df-fin 8988  df-sup 9480  df-inf 9481  df-oi 9548  df-card 9977  df-pnf 11295  df-mnf 11296  df-xr 11297  df-ltxr 11298  df-le 11299  df-sub 11492  df-neg 11493  df-div 11919  df-nn 12265  df-2 12327  df-3 12328  df-n0 12525  df-z 12612  df-uz 12877  df-rp 13033  df-ico 13390  df-fz 13545  df-fzo 13692  df-fl 13829  df-seq 14040  df-exp 14100  df-fac 14310  df-bc 14339  df-hash 14367  df-shft 15103  df-cj 15135  df-re 15136  df-im 15137  df-sqrt 15271  df-abs 15272  df-limsup 15504  df-clim 15521  df-rlim 15522  df-sum 15720  df-ef 16100
This theorem is referenced by:  iprodefisum  35721
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