Users' Mathboxes Mathbox for Scott Fenton < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  iprodefisumlem Structured version   Visualization version   GIF version

Theorem iprodefisumlem 32220
Description: Lemma for iprodefisum 32221. (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 6535 . . . . . 6 ((𝐹:𝑍⟶ℂ ∧ 𝑘𝑍) → ((exp ∘ 𝐹)‘𝑘) = (exp‘(𝐹𝑘)))
53, 4sylan 575 . . . . 5 ((𝜑𝑘𝑍) → ((exp ∘ 𝐹)‘𝑘) = (exp‘(𝐹𝑘)))
63ffvelrnda 6623 . . . . . 6 ((𝜑𝑘𝑍) → (𝐹𝑘) ∈ ℂ)
7 efcl 15215 . . . . . 6 ((𝐹𝑘) ∈ ℂ → (exp‘(𝐹𝑘)) ∈ ℂ)
86, 7syl 17 . . . . 5 ((𝜑𝑘𝑍) → (exp‘(𝐹𝑘)) ∈ ℂ)
95, 8eqeltrd 2859 . . . 4 ((𝜑𝑘𝑍) → ((exp ∘ 𝐹)‘𝑘) ∈ ℂ)
101, 2, 9prodf 15022 . . 3 (𝜑 → seq𝑀( · , (exp ∘ 𝐹)):𝑍⟶ℂ)
1110ffnd 6292 . 2 (𝜑 → seq𝑀( · , (exp ∘ 𝐹)) Fn 𝑍)
12 eff 15214 . . . 4 exp:ℂ⟶ℂ
13 ffn 6291 . . . 4 (exp:ℂ⟶ℂ → exp Fn ℂ)
1412, 13ax-mp 5 . . 3 exp Fn ℂ
151, 2, 6serf 13147 . . 3 (𝜑 → seq𝑀( + , 𝐹):𝑍⟶ℂ)
16 fnfco 6319 . . 3 ((exp Fn ℂ ∧ seq𝑀( + , 𝐹):𝑍⟶ℂ) → (exp ∘ seq𝑀( + , 𝐹)) Fn 𝑍)
1714, 15, 16sylancr 581 . 2 (𝜑 → (exp ∘ seq𝑀( + , 𝐹)) Fn 𝑍)
18 fveq2 6446 . . . . . . . 8 (𝑗 = 𝑀 → (seq𝑀( · , (exp ∘ 𝐹))‘𝑗) = (seq𝑀( · , (exp ∘ 𝐹))‘𝑀))
19 2fveq3 6451 . . . . . . . 8 (𝑗 = 𝑀 → (exp‘(seq𝑀( + , 𝐹)‘𝑗)) = (exp‘(seq𝑀( + , 𝐹)‘𝑀)))
2018, 19eqeq12d 2793 . . . . . . 7 (𝑗 = 𝑀 → ((seq𝑀( · , (exp ∘ 𝐹))‘𝑗) = (exp‘(seq𝑀( + , 𝐹)‘𝑗)) ↔ (seq𝑀( · , (exp ∘ 𝐹))‘𝑀) = (exp‘(seq𝑀( + , 𝐹)‘𝑀))))
2120imbi2d 332 . . . . . 6 (𝑗 = 𝑀 → ((𝜑 → (seq𝑀( · , (exp ∘ 𝐹))‘𝑗) = (exp‘(seq𝑀( + , 𝐹)‘𝑗))) ↔ (𝜑 → (seq𝑀( · , (exp ∘ 𝐹))‘𝑀) = (exp‘(seq𝑀( + , 𝐹)‘𝑀)))))
22 fveq2 6446 . . . . . . . 8 (𝑗 = 𝑛 → (seq𝑀( · , (exp ∘ 𝐹))‘𝑗) = (seq𝑀( · , (exp ∘ 𝐹))‘𝑛))
23 2fveq3 6451 . . . . . . . 8 (𝑗 = 𝑛 → (exp‘(seq𝑀( + , 𝐹)‘𝑗)) = (exp‘(seq𝑀( + , 𝐹)‘𝑛)))
2422, 23eqeq12d 2793 . . . . . . 7 (𝑗 = 𝑛 → ((seq𝑀( · , (exp ∘ 𝐹))‘𝑗) = (exp‘(seq𝑀( + , 𝐹)‘𝑗)) ↔ (seq𝑀( · , (exp ∘ 𝐹))‘𝑛) = (exp‘(seq𝑀( + , 𝐹)‘𝑛))))
2524imbi2d 332 . . . . . 6 (𝑗 = 𝑛 → ((𝜑 → (seq𝑀( · , (exp ∘ 𝐹))‘𝑗) = (exp‘(seq𝑀( + , 𝐹)‘𝑗))) ↔ (𝜑 → (seq𝑀( · , (exp ∘ 𝐹))‘𝑛) = (exp‘(seq𝑀( + , 𝐹)‘𝑛)))))
26 fveq2 6446 . . . . . . . 8 (𝑗 = (𝑛 + 1) → (seq𝑀( · , (exp ∘ 𝐹))‘𝑗) = (seq𝑀( · , (exp ∘ 𝐹))‘(𝑛 + 1)))
27 2fveq3 6451 . . . . . . . 8 (𝑗 = (𝑛 + 1) → (exp‘(seq𝑀( + , 𝐹)‘𝑗)) = (exp‘(seq𝑀( + , 𝐹)‘(𝑛 + 1))))
2826, 27eqeq12d 2793 . . . . . . 7 (𝑗 = (𝑛 + 1) → ((seq𝑀( · , (exp ∘ 𝐹))‘𝑗) = (exp‘(seq𝑀( + , 𝐹)‘𝑗)) ↔ (seq𝑀( · , (exp ∘ 𝐹))‘(𝑛 + 1)) = (exp‘(seq𝑀( + , 𝐹)‘(𝑛 + 1)))))
2928imbi2d 332 . . . . . 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 6446 . . . . . . . 8 (𝑗 = 𝑘 → (seq𝑀( · , (exp ∘ 𝐹))‘𝑗) = (seq𝑀( · , (exp ∘ 𝐹))‘𝑘))
31 2fveq3 6451 . . . . . . . 8 (𝑗 = 𝑘 → (exp‘(seq𝑀( + , 𝐹)‘𝑗)) = (exp‘(seq𝑀( + , 𝐹)‘𝑘)))
3230, 31eqeq12d 2793 . . . . . . 7 (𝑗 = 𝑘 → ((seq𝑀( · , (exp ∘ 𝐹))‘𝑗) = (exp‘(seq𝑀( + , 𝐹)‘𝑗)) ↔ (seq𝑀( · , (exp ∘ 𝐹))‘𝑘) = (exp‘(seq𝑀( + , 𝐹)‘𝑘))))
3332imbi2d 332 . . . . . 6 (𝑗 = 𝑘 → ((𝜑 → (seq𝑀( · , (exp ∘ 𝐹))‘𝑗) = (exp‘(seq𝑀( + , 𝐹)‘𝑗))) ↔ (𝜑 → (seq𝑀( · , (exp ∘ 𝐹))‘𝑘) = (exp‘(seq𝑀( + , 𝐹)‘𝑘)))))
34 uzid 12007 . . . . . . . . . . 11 (𝑀 ∈ ℤ → 𝑀 ∈ (ℤ𝑀))
352, 34syl 17 . . . . . . . . . 10 (𝜑𝑀 ∈ (ℤ𝑀))
3635, 1syl6eleqr 2870 . . . . . . . . 9 (𝜑𝑀𝑍)
37 fvco3 6535 . . . . . . . . 9 ((𝐹:𝑍⟶ℂ ∧ 𝑀𝑍) → ((exp ∘ 𝐹)‘𝑀) = (exp‘(𝐹𝑀)))
383, 36, 37syl2anc 579 . . . . . . . 8 (𝜑 → ((exp ∘ 𝐹)‘𝑀) = (exp‘(𝐹𝑀)))
39 seq1 13132 . . . . . . . . 9 (𝑀 ∈ ℤ → (seq𝑀( · , (exp ∘ 𝐹))‘𝑀) = ((exp ∘ 𝐹)‘𝑀))
402, 39syl 17 . . . . . . . 8 (𝜑 → (seq𝑀( · , (exp ∘ 𝐹))‘𝑀) = ((exp ∘ 𝐹)‘𝑀))
41 seq1 13132 . . . . . . . . . 10 (𝑀 ∈ ℤ → (seq𝑀( + , 𝐹)‘𝑀) = (𝐹𝑀))
422, 41syl 17 . . . . . . . . 9 (𝜑 → (seq𝑀( + , 𝐹)‘𝑀) = (𝐹𝑀))
4342fveq2d 6450 . . . . . . . 8 (𝜑 → (exp‘(seq𝑀( + , 𝐹)‘𝑀)) = (exp‘(𝐹𝑀)))
4438, 40, 433eqtr4d 2824 . . . . . . 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 6929 . . . . . . . . . . 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 1126 . . . . . . . . . 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 475 . . . . . . . . . . . . . 14 ((𝑛 ∈ (ℤ𝑀) ∧ 𝜑) → 𝐹:𝑍⟶ℂ)
49 peano2uz 12047 . . . . . . . . . . . . . . . 16 (𝑛 ∈ (ℤ𝑀) → (𝑛 + 1) ∈ (ℤ𝑀))
5049, 1syl6eleqr 2870 . . . . . . . . . . . . . . 15 (𝑛 ∈ (ℤ𝑀) → (𝑛 + 1) ∈ 𝑍)
5150adantr 474 . . . . . . . . . . . . . 14 ((𝑛 ∈ (ℤ𝑀) ∧ 𝜑) → (𝑛 + 1) ∈ 𝑍)
52 fvco3 6535 . . . . . . . . . . . . . 14 ((𝐹:𝑍⟶ℂ ∧ (𝑛 + 1) ∈ 𝑍) → ((exp ∘ 𝐹)‘(𝑛 + 1)) = (exp‘(𝐹‘(𝑛 + 1))))
5348, 51, 52syl2anc 579 . . . . . . . . . . . . 13 ((𝑛 ∈ (ℤ𝑀) ∧ 𝜑) → ((exp ∘ 𝐹)‘(𝑛 + 1)) = (exp‘(𝐹‘(𝑛 + 1))))
5453oveq2d 6938 . . . . . . . . . . . 12 ((𝑛 ∈ (ℤ𝑀) ∧ 𝜑) → ((exp‘(seq𝑀( + , 𝐹)‘𝑛)) · ((exp ∘ 𝐹)‘(𝑛 + 1))) = ((exp‘(seq𝑀( + , 𝐹)‘𝑛)) · (exp‘(𝐹‘(𝑛 + 1)))))
5515ffvelrnda 6623 . . . . . . . . . . . . . . . 16 ((𝜑𝑛𝑍) → (seq𝑀( + , 𝐹)‘𝑛) ∈ ℂ)
5655expcom 404 . . . . . . . . . . . . . . 15 (𝑛𝑍 → (𝜑 → (seq𝑀( + , 𝐹)‘𝑛) ∈ ℂ))
571eqcomi 2787 . . . . . . . . . . . . . . 15 (ℤ𝑀) = 𝑍
5856, 57eleq2s 2877 . . . . . . . . . . . . . 14 (𝑛 ∈ (ℤ𝑀) → (𝜑 → (seq𝑀( + , 𝐹)‘𝑛) ∈ ℂ))
5958imp 397 . . . . . . . . . . . . 13 ((𝑛 ∈ (ℤ𝑀) ∧ 𝜑) → (seq𝑀( + , 𝐹)‘𝑛) ∈ ℂ)
6048, 51ffvelrnd 6624 . . . . . . . . . . . . 13 ((𝑛 ∈ (ℤ𝑀) ∧ 𝜑) → (𝐹‘(𝑛 + 1)) ∈ ℂ)
61 efadd 15226 . . . . . . . . . . . . 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 579 . . . . . . . . . . . 12 ((𝑛 ∈ (ℤ𝑀) ∧ 𝜑) → (exp‘((seq𝑀( + , 𝐹)‘𝑛) + (𝐹‘(𝑛 + 1)))) = ((exp‘(seq𝑀( + , 𝐹)‘𝑛)) · (exp‘(𝐹‘(𝑛 + 1)))))
6354, 62eqtr4d 2817 . . . . . . . . . . 11 ((𝑛 ∈ (ℤ𝑀) ∧ 𝜑) → ((exp‘(seq𝑀( + , 𝐹)‘𝑛)) · ((exp ∘ 𝐹)‘(𝑛 + 1))) = (exp‘((seq𝑀( + , 𝐹)‘𝑛) + (𝐹‘(𝑛 + 1)))))
64633adant3 1123 . . . . . . . . . 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 2814 . . . . . . . . 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 13134 . . . . . . . . . . 11 (𝑛 ∈ (ℤ𝑀) → (seq𝑀( · , (exp ∘ 𝐹))‘(𝑛 + 1)) = ((seq𝑀( · , (exp ∘ 𝐹))‘𝑛) · ((exp ∘ 𝐹)‘(𝑛 + 1))))
6766adantr 474 . . . . . . . . . 10 ((𝑛 ∈ (ℤ𝑀) ∧ 𝜑) → (seq𝑀( · , (exp ∘ 𝐹))‘(𝑛 + 1)) = ((seq𝑀( · , (exp ∘ 𝐹))‘𝑛) · ((exp ∘ 𝐹)‘(𝑛 + 1))))
68673adant3 1123 . . . . . . . . 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 13134 . . . . . . . . . . . 12 (𝑛 ∈ (ℤ𝑀) → (seq𝑀( + , 𝐹)‘(𝑛 + 1)) = ((seq𝑀( + , 𝐹)‘𝑛) + (𝐹‘(𝑛 + 1))))
7069adantr 474 . . . . . . . . . . 11 ((𝑛 ∈ (ℤ𝑀) ∧ 𝜑) → (seq𝑀( + , 𝐹)‘(𝑛 + 1)) = ((seq𝑀( + , 𝐹)‘𝑛) + (𝐹‘(𝑛 + 1))))
7170fveq2d 6450 . . . . . . . . . 10 ((𝑛 ∈ (ℤ𝑀) ∧ 𝜑) → (exp‘(seq𝑀( + , 𝐹)‘(𝑛 + 1))) = (exp‘((seq𝑀( + , 𝐹)‘𝑛) + (𝐹‘(𝑛 + 1)))))
72713adant3 1123 . . . . . . . . 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 2824 . . . . . . . 8 ((𝑛 ∈ (ℤ𝑀) ∧ 𝜑 ∧ (seq𝑀( · , (exp ∘ 𝐹))‘𝑛) = (exp‘(seq𝑀( + , 𝐹)‘𝑛))) → (seq𝑀( · , (exp ∘ 𝐹))‘(𝑛 + 1)) = (exp‘(seq𝑀( + , 𝐹)‘(𝑛 + 1))))
74733exp 1109 . . . . . . 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 12052 . . . . 5 (𝑘 ∈ (ℤ𝑀) → (𝜑 → (seq𝑀( · , (exp ∘ 𝐹))‘𝑘) = (exp‘(seq𝑀( + , 𝐹)‘𝑘))))
7776, 1eleq2s 2877 . . . 4 (𝑘𝑍 → (𝜑 → (seq𝑀( · , (exp ∘ 𝐹))‘𝑘) = (exp‘(seq𝑀( + , 𝐹)‘𝑘))))
7877impcom 398 . . 3 ((𝜑𝑘𝑍) → (seq𝑀( · , (exp ∘ 𝐹))‘𝑘) = (exp‘(seq𝑀( + , 𝐹)‘𝑘)))
79 fvco3 6535 . . . 4 ((seq𝑀( + , 𝐹):𝑍⟶ℂ ∧ 𝑘𝑍) → ((exp ∘ seq𝑀( + , 𝐹))‘𝑘) = (exp‘(seq𝑀( + , 𝐹)‘𝑘)))
8015, 79sylan 575 . . 3 ((𝜑𝑘𝑍) → ((exp ∘ seq𝑀( + , 𝐹))‘𝑘) = (exp‘(seq𝑀( + , 𝐹)‘𝑘)))
8178, 80eqtr4d 2817 . 2 ((𝜑𝑘𝑍) → (seq𝑀( · , (exp ∘ 𝐹))‘𝑘) = ((exp ∘ seq𝑀( + , 𝐹))‘𝑘))
8211, 17, 81eqfnfvd 6577 1 (𝜑 → seq𝑀( · , (exp ∘ 𝐹)) = (exp ∘ seq𝑀( + , 𝐹)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 386  w3a 1071   = wceq 1601  wcel 2107  ccom 5359   Fn wfn 6130  wf 6131  cfv 6135  (class class class)co 6922  cc 10270  1c1 10273   + caddc 10275   · cmul 10277  cz 11728  cuz 11992  seqcseq 13119  expce 15194
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2055  ax-8 2109  ax-9 2116  ax-10 2135  ax-11 2150  ax-12 2163  ax-13 2334  ax-ext 2754  ax-rep 5006  ax-sep 5017  ax-nul 5025  ax-pow 5077  ax-pr 5138  ax-un 7226  ax-inf2 8835  ax-cnex 10328  ax-resscn 10329  ax-1cn 10330  ax-icn 10331  ax-addcl 10332  ax-addrcl 10333  ax-mulcl 10334  ax-mulrcl 10335  ax-mulcom 10336  ax-addass 10337  ax-mulass 10338  ax-distr 10339  ax-i2m1 10340  ax-1ne0 10341  ax-1rid 10342  ax-rnegex 10343  ax-rrecex 10344  ax-cnre 10345  ax-pre-lttri 10346  ax-pre-lttrn 10347  ax-pre-ltadd 10348  ax-pre-mulgt0 10349  ax-pre-sup 10350  ax-addf 10351  ax-mulf 10352
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-3or 1072  df-3an 1073  df-tru 1605  df-fal 1615  df-ex 1824  df-nf 1828  df-sb 2012  df-mo 2551  df-eu 2587  df-clab 2764  df-cleq 2770  df-clel 2774  df-nfc 2921  df-ne 2970  df-nel 3076  df-ral 3095  df-rex 3096  df-reu 3097  df-rmo 3098  df-rab 3099  df-v 3400  df-sbc 3653  df-csb 3752  df-dif 3795  df-un 3797  df-in 3799  df-ss 3806  df-pss 3808  df-nul 4142  df-if 4308  df-pw 4381  df-sn 4399  df-pr 4401  df-tp 4403  df-op 4405  df-uni 4672  df-int 4711  df-iun 4755  df-br 4887  df-opab 4949  df-mpt 4966  df-tr 4988  df-id 5261  df-eprel 5266  df-po 5274  df-so 5275  df-fr 5314  df-se 5315  df-we 5316  df-xp 5361  df-rel 5362  df-cnv 5363  df-co 5364  df-dm 5365  df-rn 5366  df-res 5367  df-ima 5368  df-pred 5933  df-ord 5979  df-on 5980  df-lim 5981  df-suc 5982  df-iota 6099  df-fun 6137  df-fn 6138  df-f 6139  df-f1 6140  df-fo 6141  df-f1o 6142  df-fv 6143  df-isom 6144  df-riota 6883  df-ov 6925  df-oprab 6926  df-mpt2 6927  df-om 7344  df-1st 7445  df-2nd 7446  df-wrecs 7689  df-recs 7751  df-rdg 7789  df-1o 7843  df-oadd 7847  df-er 8026  df-pm 8143  df-en 8242  df-dom 8243  df-sdom 8244  df-fin 8245  df-sup 8636  df-inf 8637  df-oi 8704  df-card 9098  df-pnf 10413  df-mnf 10414  df-xr 10415  df-ltxr 10416  df-le 10417  df-sub 10608  df-neg 10609  df-div 11033  df-nn 11375  df-2 11438  df-3 11439  df-n0 11643  df-z 11729  df-uz 11993  df-rp 12138  df-ico 12493  df-fz 12644  df-fzo 12785  df-fl 12912  df-seq 13120  df-exp 13179  df-fac 13379  df-bc 13408  df-hash 13436  df-shft 14214  df-cj 14246  df-re 14247  df-im 14248  df-sqrt 14382  df-abs 14383  df-limsup 14610  df-clim 14627  df-rlim 14628  df-sum 14825  df-ef 15200
This theorem is referenced by:  iprodefisum  32221
  Copyright terms: Public domain W3C validator