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Theorem iprodefisumlem 33046
Description: Lemma for iprodefisum 33047. (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 6742 . . . . . 6 ((𝐹:𝑍⟶ℂ ∧ 𝑘𝑍) → ((exp ∘ 𝐹)‘𝑘) = (exp‘(𝐹𝑘)))
53, 4sylan 583 . . . . 5 ((𝜑𝑘𝑍) → ((exp ∘ 𝐹)‘𝑘) = (exp‘(𝐹𝑘)))
63ffvelrnda 6833 . . . . . 6 ((𝜑𝑘𝑍) → (𝐹𝑘) ∈ ℂ)
7 efcl 15427 . . . . . 6 ((𝐹𝑘) ∈ ℂ → (exp‘(𝐹𝑘)) ∈ ℂ)
86, 7syl 17 . . . . 5 ((𝜑𝑘𝑍) → (exp‘(𝐹𝑘)) ∈ ℂ)
95, 8eqeltrd 2914 . . . 4 ((𝜑𝑘𝑍) → ((exp ∘ 𝐹)‘𝑘) ∈ ℂ)
101, 2, 9prodf 15234 . . 3 (𝜑 → seq𝑀( · , (exp ∘ 𝐹)):𝑍⟶ℂ)
1110ffnd 6495 . 2 (𝜑 → seq𝑀( · , (exp ∘ 𝐹)) Fn 𝑍)
12 eff 15426 . . . 4 exp:ℂ⟶ℂ
13 ffn 6494 . . . 4 (exp:ℂ⟶ℂ → exp Fn ℂ)
1412, 13ax-mp 5 . . 3 exp Fn ℂ
151, 2, 6serf 13394 . . 3 (𝜑 → seq𝑀( + , 𝐹):𝑍⟶ℂ)
16 fnfco 6524 . . 3 ((exp Fn ℂ ∧ seq𝑀( + , 𝐹):𝑍⟶ℂ) → (exp ∘ seq𝑀( + , 𝐹)) Fn 𝑍)
1714, 15, 16sylancr 590 . 2 (𝜑 → (exp ∘ seq𝑀( + , 𝐹)) Fn 𝑍)
18 fveq2 6652 . . . . . . . 8 (𝑗 = 𝑀 → (seq𝑀( · , (exp ∘ 𝐹))‘𝑗) = (seq𝑀( · , (exp ∘ 𝐹))‘𝑀))
19 2fveq3 6657 . . . . . . . 8 (𝑗 = 𝑀 → (exp‘(seq𝑀( + , 𝐹)‘𝑗)) = (exp‘(seq𝑀( + , 𝐹)‘𝑀)))
2018, 19eqeq12d 2838 . . . . . . 7 (𝑗 = 𝑀 → ((seq𝑀( · , (exp ∘ 𝐹))‘𝑗) = (exp‘(seq𝑀( + , 𝐹)‘𝑗)) ↔ (seq𝑀( · , (exp ∘ 𝐹))‘𝑀) = (exp‘(seq𝑀( + , 𝐹)‘𝑀))))
2120imbi2d 344 . . . . . 6 (𝑗 = 𝑀 → ((𝜑 → (seq𝑀( · , (exp ∘ 𝐹))‘𝑗) = (exp‘(seq𝑀( + , 𝐹)‘𝑗))) ↔ (𝜑 → (seq𝑀( · , (exp ∘ 𝐹))‘𝑀) = (exp‘(seq𝑀( + , 𝐹)‘𝑀)))))
22 fveq2 6652 . . . . . . . 8 (𝑗 = 𝑛 → (seq𝑀( · , (exp ∘ 𝐹))‘𝑗) = (seq𝑀( · , (exp ∘ 𝐹))‘𝑛))
23 2fveq3 6657 . . . . . . . 8 (𝑗 = 𝑛 → (exp‘(seq𝑀( + , 𝐹)‘𝑗)) = (exp‘(seq𝑀( + , 𝐹)‘𝑛)))
2422, 23eqeq12d 2838 . . . . . . 7 (𝑗 = 𝑛 → ((seq𝑀( · , (exp ∘ 𝐹))‘𝑗) = (exp‘(seq𝑀( + , 𝐹)‘𝑗)) ↔ (seq𝑀( · , (exp ∘ 𝐹))‘𝑛) = (exp‘(seq𝑀( + , 𝐹)‘𝑛))))
2524imbi2d 344 . . . . . 6 (𝑗 = 𝑛 → ((𝜑 → (seq𝑀( · , (exp ∘ 𝐹))‘𝑗) = (exp‘(seq𝑀( + , 𝐹)‘𝑗))) ↔ (𝜑 → (seq𝑀( · , (exp ∘ 𝐹))‘𝑛) = (exp‘(seq𝑀( + , 𝐹)‘𝑛)))))
26 fveq2 6652 . . . . . . . 8 (𝑗 = (𝑛 + 1) → (seq𝑀( · , (exp ∘ 𝐹))‘𝑗) = (seq𝑀( · , (exp ∘ 𝐹))‘(𝑛 + 1)))
27 2fveq3 6657 . . . . . . . 8 (𝑗 = (𝑛 + 1) → (exp‘(seq𝑀( + , 𝐹)‘𝑗)) = (exp‘(seq𝑀( + , 𝐹)‘(𝑛 + 1))))
2826, 27eqeq12d 2838 . . . . . . 7 (𝑗 = (𝑛 + 1) → ((seq𝑀( · , (exp ∘ 𝐹))‘𝑗) = (exp‘(seq𝑀( + , 𝐹)‘𝑗)) ↔ (seq𝑀( · , (exp ∘ 𝐹))‘(𝑛 + 1)) = (exp‘(seq𝑀( + , 𝐹)‘(𝑛 + 1)))))
2928imbi2d 344 . . . . . 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 6652 . . . . . . . 8 (𝑗 = 𝑘 → (seq𝑀( · , (exp ∘ 𝐹))‘𝑗) = (seq𝑀( · , (exp ∘ 𝐹))‘𝑘))
31 2fveq3 6657 . . . . . . . 8 (𝑗 = 𝑘 → (exp‘(seq𝑀( + , 𝐹)‘𝑗)) = (exp‘(seq𝑀( + , 𝐹)‘𝑘)))
3230, 31eqeq12d 2838 . . . . . . 7 (𝑗 = 𝑘 → ((seq𝑀( · , (exp ∘ 𝐹))‘𝑗) = (exp‘(seq𝑀( + , 𝐹)‘𝑗)) ↔ (seq𝑀( · , (exp ∘ 𝐹))‘𝑘) = (exp‘(seq𝑀( + , 𝐹)‘𝑘))))
3332imbi2d 344 . . . . . 6 (𝑗 = 𝑘 → ((𝜑 → (seq𝑀( · , (exp ∘ 𝐹))‘𝑗) = (exp‘(seq𝑀( + , 𝐹)‘𝑗))) ↔ (𝜑 → (seq𝑀( · , (exp ∘ 𝐹))‘𝑘) = (exp‘(seq𝑀( + , 𝐹)‘𝑘)))))
34 uzid 12246 . . . . . . . . . . 11 (𝑀 ∈ ℤ → 𝑀 ∈ (ℤ𝑀))
352, 34syl 17 . . . . . . . . . 10 (𝜑𝑀 ∈ (ℤ𝑀))
3635, 1eleqtrrdi 2925 . . . . . . . . 9 (𝜑𝑀𝑍)
37 fvco3 6742 . . . . . . . . 9 ((𝐹:𝑍⟶ℂ ∧ 𝑀𝑍) → ((exp ∘ 𝐹)‘𝑀) = (exp‘(𝐹𝑀)))
383, 36, 37syl2anc 587 . . . . . . . 8 (𝜑 → ((exp ∘ 𝐹)‘𝑀) = (exp‘(𝐹𝑀)))
39 seq1 13377 . . . . . . . . 9 (𝑀 ∈ ℤ → (seq𝑀( · , (exp ∘ 𝐹))‘𝑀) = ((exp ∘ 𝐹)‘𝑀))
402, 39syl 17 . . . . . . . 8 (𝜑 → (seq𝑀( · , (exp ∘ 𝐹))‘𝑀) = ((exp ∘ 𝐹)‘𝑀))
41 seq1 13377 . . . . . . . . . 10 (𝑀 ∈ ℤ → (seq𝑀( + , 𝐹)‘𝑀) = (𝐹𝑀))
422, 41syl 17 . . . . . . . . 9 (𝜑 → (seq𝑀( + , 𝐹)‘𝑀) = (𝐹𝑀))
4342fveq2d 6656 . . . . . . . 8 (𝜑 → (exp‘(seq𝑀( + , 𝐹)‘𝑀)) = (exp‘(𝐹𝑀)))
4438, 40, 433eqtr4d 2867 . . . . . . 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 7147 . . . . . . . . . . 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 1132 . . . . . . . . . 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 485 . . . . . . . . . . . . . 14 ((𝑛 ∈ (ℤ𝑀) ∧ 𝜑) → 𝐹:𝑍⟶ℂ)
49 peano2uz 12289 . . . . . . . . . . . . . . . 16 (𝑛 ∈ (ℤ𝑀) → (𝑛 + 1) ∈ (ℤ𝑀))
5049, 1eleqtrrdi 2925 . . . . . . . . . . . . . . 15 (𝑛 ∈ (ℤ𝑀) → (𝑛 + 1) ∈ 𝑍)
5150adantr 484 . . . . . . . . . . . . . 14 ((𝑛 ∈ (ℤ𝑀) ∧ 𝜑) → (𝑛 + 1) ∈ 𝑍)
52 fvco3 6742 . . . . . . . . . . . . . 14 ((𝐹:𝑍⟶ℂ ∧ (𝑛 + 1) ∈ 𝑍) → ((exp ∘ 𝐹)‘(𝑛 + 1)) = (exp‘(𝐹‘(𝑛 + 1))))
5348, 51, 52syl2anc 587 . . . . . . . . . . . . 13 ((𝑛 ∈ (ℤ𝑀) ∧ 𝜑) → ((exp ∘ 𝐹)‘(𝑛 + 1)) = (exp‘(𝐹‘(𝑛 + 1))))
5453oveq2d 7156 . . . . . . . . . . . 12 ((𝑛 ∈ (ℤ𝑀) ∧ 𝜑) → ((exp‘(seq𝑀( + , 𝐹)‘𝑛)) · ((exp ∘ 𝐹)‘(𝑛 + 1))) = ((exp‘(seq𝑀( + , 𝐹)‘𝑛)) · (exp‘(𝐹‘(𝑛 + 1)))))
5515ffvelrnda 6833 . . . . . . . . . . . . . . . 16 ((𝜑𝑛𝑍) → (seq𝑀( + , 𝐹)‘𝑛) ∈ ℂ)
5655expcom 417 . . . . . . . . . . . . . . 15 (𝑛𝑍 → (𝜑 → (seq𝑀( + , 𝐹)‘𝑛) ∈ ℂ))
571eqcomi 2831 . . . . . . . . . . . . . . 15 (ℤ𝑀) = 𝑍
5856, 57eleq2s 2932 . . . . . . . . . . . . . 14 (𝑛 ∈ (ℤ𝑀) → (𝜑 → (seq𝑀( + , 𝐹)‘𝑛) ∈ ℂ))
5958imp 410 . . . . . . . . . . . . 13 ((𝑛 ∈ (ℤ𝑀) ∧ 𝜑) → (seq𝑀( + , 𝐹)‘𝑛) ∈ ℂ)
6048, 51ffvelrnd 6834 . . . . . . . . . . . . 13 ((𝑛 ∈ (ℤ𝑀) ∧ 𝜑) → (𝐹‘(𝑛 + 1)) ∈ ℂ)
61 efadd 15438 . . . . . . . . . . . . 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 587 . . . . . . . . . . . 12 ((𝑛 ∈ (ℤ𝑀) ∧ 𝜑) → (exp‘((seq𝑀( + , 𝐹)‘𝑛) + (𝐹‘(𝑛 + 1)))) = ((exp‘(seq𝑀( + , 𝐹)‘𝑛)) · (exp‘(𝐹‘(𝑛 + 1)))))
6354, 62eqtr4d 2860 . . . . . . . . . . 11 ((𝑛 ∈ (ℤ𝑀) ∧ 𝜑) → ((exp‘(seq𝑀( + , 𝐹)‘𝑛)) · ((exp ∘ 𝐹)‘(𝑛 + 1))) = (exp‘((seq𝑀( + , 𝐹)‘𝑛) + (𝐹‘(𝑛 + 1)))))
64633adant3 1129 . . . . . . . . . 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 2857 . . . . . . . . 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 13379 . . . . . . . . . . 11 (𝑛 ∈ (ℤ𝑀) → (seq𝑀( · , (exp ∘ 𝐹))‘(𝑛 + 1)) = ((seq𝑀( · , (exp ∘ 𝐹))‘𝑛) · ((exp ∘ 𝐹)‘(𝑛 + 1))))
6766adantr 484 . . . . . . . . . 10 ((𝑛 ∈ (ℤ𝑀) ∧ 𝜑) → (seq𝑀( · , (exp ∘ 𝐹))‘(𝑛 + 1)) = ((seq𝑀( · , (exp ∘ 𝐹))‘𝑛) · ((exp ∘ 𝐹)‘(𝑛 + 1))))
68673adant3 1129 . . . . . . . . 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 13379 . . . . . . . . . . . 12 (𝑛 ∈ (ℤ𝑀) → (seq𝑀( + , 𝐹)‘(𝑛 + 1)) = ((seq𝑀( + , 𝐹)‘𝑛) + (𝐹‘(𝑛 + 1))))
7069adantr 484 . . . . . . . . . . 11 ((𝑛 ∈ (ℤ𝑀) ∧ 𝜑) → (seq𝑀( + , 𝐹)‘(𝑛 + 1)) = ((seq𝑀( + , 𝐹)‘𝑛) + (𝐹‘(𝑛 + 1))))
7170fveq2d 6656 . . . . . . . . . 10 ((𝑛 ∈ (ℤ𝑀) ∧ 𝜑) → (exp‘(seq𝑀( + , 𝐹)‘(𝑛 + 1))) = (exp‘((seq𝑀( + , 𝐹)‘𝑛) + (𝐹‘(𝑛 + 1)))))
72713adant3 1129 . . . . . . . . 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 2867 . . . . . . . 8 ((𝑛 ∈ (ℤ𝑀) ∧ 𝜑 ∧ (seq𝑀( · , (exp ∘ 𝐹))‘𝑛) = (exp‘(seq𝑀( + , 𝐹)‘𝑛))) → (seq𝑀( · , (exp ∘ 𝐹))‘(𝑛 + 1)) = (exp‘(seq𝑀( + , 𝐹)‘(𝑛 + 1))))
74733exp 1116 . . . . . . 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 12294 . . . . 5 (𝑘 ∈ (ℤ𝑀) → (𝜑 → (seq𝑀( · , (exp ∘ 𝐹))‘𝑘) = (exp‘(seq𝑀( + , 𝐹)‘𝑘))))
7776, 1eleq2s 2932 . . . 4 (𝑘𝑍 → (𝜑 → (seq𝑀( · , (exp ∘ 𝐹))‘𝑘) = (exp‘(seq𝑀( + , 𝐹)‘𝑘))))
7877impcom 411 . . 3 ((𝜑𝑘𝑍) → (seq𝑀( · , (exp ∘ 𝐹))‘𝑘) = (exp‘(seq𝑀( + , 𝐹)‘𝑘)))
79 fvco3 6742 . . . 4 ((seq𝑀( + , 𝐹):𝑍⟶ℂ ∧ 𝑘𝑍) → ((exp ∘ seq𝑀( + , 𝐹))‘𝑘) = (exp‘(seq𝑀( + , 𝐹)‘𝑘)))
8015, 79sylan 583 . . 3 ((𝜑𝑘𝑍) → ((exp ∘ seq𝑀( + , 𝐹))‘𝑘) = (exp‘(seq𝑀( + , 𝐹)‘𝑘)))
8178, 80eqtr4d 2860 . 2 ((𝜑𝑘𝑍) → (seq𝑀( · , (exp ∘ 𝐹))‘𝑘) = ((exp ∘ seq𝑀( + , 𝐹))‘𝑘))
8211, 17, 81eqfnfvd 6787 1 (𝜑 → seq𝑀( · , (exp ∘ 𝐹)) = (exp ∘ seq𝑀( + , 𝐹)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 399  w3a 1084   = wceq 1538  wcel 2114  ccom 5536   Fn wfn 6329  wf 6330  cfv 6334  (class class class)co 7140  cc 10524  1c1 10527   + caddc 10529   · cmul 10531  cz 11969  cuz 12231  seqcseq 13364  expce 15406
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2178  ax-ext 2794  ax-rep 5166  ax-sep 5179  ax-nul 5186  ax-pow 5243  ax-pr 5307  ax-un 7446  ax-inf2 9092  ax-cnex 10582  ax-resscn 10583  ax-1cn 10584  ax-icn 10585  ax-addcl 10586  ax-addrcl 10587  ax-mulcl 10588  ax-mulrcl 10589  ax-mulcom 10590  ax-addass 10591  ax-mulass 10592  ax-distr 10593  ax-i2m1 10594  ax-1ne0 10595  ax-1rid 10596  ax-rnegex 10597  ax-rrecex 10598  ax-cnre 10599  ax-pre-lttri 10600  ax-pre-lttrn 10601  ax-pre-ltadd 10602  ax-pre-mulgt0 10603  ax-pre-sup 10604  ax-addf 10605  ax-mulf 10606
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2622  df-eu 2653  df-clab 2801  df-cleq 2815  df-clel 2894  df-nfc 2962  df-ne 3012  df-nel 3116  df-ral 3135  df-rex 3136  df-reu 3137  df-rmo 3138  df-rab 3139  df-v 3471  df-sbc 3748  df-csb 3856  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-pss 3927  df-nul 4266  df-if 4440  df-pw 4513  df-sn 4540  df-pr 4542  df-tp 4544  df-op 4546  df-uni 4814  df-int 4852  df-iun 4896  df-br 5043  df-opab 5105  df-mpt 5123  df-tr 5149  df-id 5437  df-eprel 5442  df-po 5451  df-so 5452  df-fr 5491  df-se 5492  df-we 5493  df-xp 5538  df-rel 5539  df-cnv 5540  df-co 5541  df-dm 5542  df-rn 5543  df-res 5544  df-ima 5545  df-pred 6126  df-ord 6172  df-on 6173  df-lim 6174  df-suc 6175  df-iota 6293  df-fun 6336  df-fn 6337  df-f 6338  df-f1 6339  df-fo 6340  df-f1o 6341  df-fv 6342  df-isom 6343  df-riota 7098  df-ov 7143  df-oprab 7144  df-mpo 7145  df-om 7566  df-1st 7675  df-2nd 7676  df-wrecs 7934  df-recs 7995  df-rdg 8033  df-1o 8089  df-oadd 8093  df-er 8276  df-pm 8396  df-en 8497  df-dom 8498  df-sdom 8499  df-fin 8500  df-sup 8894  df-inf 8895  df-oi 8962  df-card 9356  df-pnf 10666  df-mnf 10667  df-xr 10668  df-ltxr 10669  df-le 10670  df-sub 10861  df-neg 10862  df-div 11287  df-nn 11626  df-2 11688  df-3 11689  df-n0 11886  df-z 11970  df-uz 12232  df-rp 12378  df-ico 12732  df-fz 12886  df-fzo 13029  df-fl 13157  df-seq 13365  df-exp 13426  df-fac 13630  df-bc 13659  df-hash 13687  df-shft 14417  df-cj 14449  df-re 14450  df-im 14451  df-sqrt 14585  df-abs 14586  df-limsup 14819  df-clim 14836  df-rlim 14837  df-sum 15034  df-ef 15412
This theorem is referenced by:  iprodefisum  33047
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