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Theorem iprodefisumlem 31612
Description: Lemma for iprodefisum 31613. (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 6273 . . . . . 6 ((𝐹:𝑍⟶ℂ ∧ 𝑘𝑍) → ((exp ∘ 𝐹)‘𝑘) = (exp‘(𝐹𝑘)))
53, 4sylan 488 . . . . 5 ((𝜑𝑘𝑍) → ((exp ∘ 𝐹)‘𝑘) = (exp‘(𝐹𝑘)))
63ffvelrnda 6357 . . . . . 6 ((𝜑𝑘𝑍) → (𝐹𝑘) ∈ ℂ)
7 efcl 14807 . . . . . 6 ((𝐹𝑘) ∈ ℂ → (exp‘(𝐹𝑘)) ∈ ℂ)
86, 7syl 17 . . . . 5 ((𝜑𝑘𝑍) → (exp‘(𝐹𝑘)) ∈ ℂ)
95, 8eqeltrd 2700 . . . 4 ((𝜑𝑘𝑍) → ((exp ∘ 𝐹)‘𝑘) ∈ ℂ)
101, 2, 9prodf 14613 . . 3 (𝜑 → seq𝑀( · , (exp ∘ 𝐹)):𝑍⟶ℂ)
11 ffn 6043 . . 3 (seq𝑀( · , (exp ∘ 𝐹)):𝑍⟶ℂ → seq𝑀( · , (exp ∘ 𝐹)) Fn 𝑍)
1210, 11syl 17 . 2 (𝜑 → seq𝑀( · , (exp ∘ 𝐹)) Fn 𝑍)
13 eff 14806 . . . 4 exp:ℂ⟶ℂ
14 ffn 6043 . . . 4 (exp:ℂ⟶ℂ → exp Fn ℂ)
1513, 14ax-mp 5 . . 3 exp Fn ℂ
161, 2, 6serf 12824 . . 3 (𝜑 → seq𝑀( + , 𝐹):𝑍⟶ℂ)
17 fnfco 6067 . . 3 ((exp Fn ℂ ∧ seq𝑀( + , 𝐹):𝑍⟶ℂ) → (exp ∘ seq𝑀( + , 𝐹)) Fn 𝑍)
1815, 16, 17sylancr 695 . 2 (𝜑 → (exp ∘ seq𝑀( + , 𝐹)) Fn 𝑍)
19 fveq2 6189 . . . . . . . 8 (𝑗 = 𝑀 → (seq𝑀( · , (exp ∘ 𝐹))‘𝑗) = (seq𝑀( · , (exp ∘ 𝐹))‘𝑀))
20 fveq2 6189 . . . . . . . . 9 (𝑗 = 𝑀 → (seq𝑀( + , 𝐹)‘𝑗) = (seq𝑀( + , 𝐹)‘𝑀))
2120fveq2d 6193 . . . . . . . 8 (𝑗 = 𝑀 → (exp‘(seq𝑀( + , 𝐹)‘𝑗)) = (exp‘(seq𝑀( + , 𝐹)‘𝑀)))
2219, 21eqeq12d 2636 . . . . . . 7 (𝑗 = 𝑀 → ((seq𝑀( · , (exp ∘ 𝐹))‘𝑗) = (exp‘(seq𝑀( + , 𝐹)‘𝑗)) ↔ (seq𝑀( · , (exp ∘ 𝐹))‘𝑀) = (exp‘(seq𝑀( + , 𝐹)‘𝑀))))
2322imbi2d 330 . . . . . 6 (𝑗 = 𝑀 → ((𝜑 → (seq𝑀( · , (exp ∘ 𝐹))‘𝑗) = (exp‘(seq𝑀( + , 𝐹)‘𝑗))) ↔ (𝜑 → (seq𝑀( · , (exp ∘ 𝐹))‘𝑀) = (exp‘(seq𝑀( + , 𝐹)‘𝑀)))))
24 fveq2 6189 . . . . . . . 8 (𝑗 = 𝑛 → (seq𝑀( · , (exp ∘ 𝐹))‘𝑗) = (seq𝑀( · , (exp ∘ 𝐹))‘𝑛))
25 fveq2 6189 . . . . . . . . 9 (𝑗 = 𝑛 → (seq𝑀( + , 𝐹)‘𝑗) = (seq𝑀( + , 𝐹)‘𝑛))
2625fveq2d 6193 . . . . . . . 8 (𝑗 = 𝑛 → (exp‘(seq𝑀( + , 𝐹)‘𝑗)) = (exp‘(seq𝑀( + , 𝐹)‘𝑛)))
2724, 26eqeq12d 2636 . . . . . . 7 (𝑗 = 𝑛 → ((seq𝑀( · , (exp ∘ 𝐹))‘𝑗) = (exp‘(seq𝑀( + , 𝐹)‘𝑗)) ↔ (seq𝑀( · , (exp ∘ 𝐹))‘𝑛) = (exp‘(seq𝑀( + , 𝐹)‘𝑛))))
2827imbi2d 330 . . . . . 6 (𝑗 = 𝑛 → ((𝜑 → (seq𝑀( · , (exp ∘ 𝐹))‘𝑗) = (exp‘(seq𝑀( + , 𝐹)‘𝑗))) ↔ (𝜑 → (seq𝑀( · , (exp ∘ 𝐹))‘𝑛) = (exp‘(seq𝑀( + , 𝐹)‘𝑛)))))
29 fveq2 6189 . . . . . . . 8 (𝑗 = (𝑛 + 1) → (seq𝑀( · , (exp ∘ 𝐹))‘𝑗) = (seq𝑀( · , (exp ∘ 𝐹))‘(𝑛 + 1)))
30 fveq2 6189 . . . . . . . . 9 (𝑗 = (𝑛 + 1) → (seq𝑀( + , 𝐹)‘𝑗) = (seq𝑀( + , 𝐹)‘(𝑛 + 1)))
3130fveq2d 6193 . . . . . . . 8 (𝑗 = (𝑛 + 1) → (exp‘(seq𝑀( + , 𝐹)‘𝑗)) = (exp‘(seq𝑀( + , 𝐹)‘(𝑛 + 1))))
3229, 31eqeq12d 2636 . . . . . . 7 (𝑗 = (𝑛 + 1) → ((seq𝑀( · , (exp ∘ 𝐹))‘𝑗) = (exp‘(seq𝑀( + , 𝐹)‘𝑗)) ↔ (seq𝑀( · , (exp ∘ 𝐹))‘(𝑛 + 1)) = (exp‘(seq𝑀( + , 𝐹)‘(𝑛 + 1)))))
3332imbi2d 330 . . . . . 6 (𝑗 = (𝑛 + 1) → ((𝜑 → (seq𝑀( · , (exp ∘ 𝐹))‘𝑗) = (exp‘(seq𝑀( + , 𝐹)‘𝑗))) ↔ (𝜑 → (seq𝑀( · , (exp ∘ 𝐹))‘(𝑛 + 1)) = (exp‘(seq𝑀( + , 𝐹)‘(𝑛 + 1))))))
34 fveq2 6189 . . . . . . . 8 (𝑗 = 𝑘 → (seq𝑀( · , (exp ∘ 𝐹))‘𝑗) = (seq𝑀( · , (exp ∘ 𝐹))‘𝑘))
35 fveq2 6189 . . . . . . . . 9 (𝑗 = 𝑘 → (seq𝑀( + , 𝐹)‘𝑗) = (seq𝑀( + , 𝐹)‘𝑘))
3635fveq2d 6193 . . . . . . . 8 (𝑗 = 𝑘 → (exp‘(seq𝑀( + , 𝐹)‘𝑗)) = (exp‘(seq𝑀( + , 𝐹)‘𝑘)))
3734, 36eqeq12d 2636 . . . . . . 7 (𝑗 = 𝑘 → ((seq𝑀( · , (exp ∘ 𝐹))‘𝑗) = (exp‘(seq𝑀( + , 𝐹)‘𝑗)) ↔ (seq𝑀( · , (exp ∘ 𝐹))‘𝑘) = (exp‘(seq𝑀( + , 𝐹)‘𝑘))))
3837imbi2d 330 . . . . . 6 (𝑗 = 𝑘 → ((𝜑 → (seq𝑀( · , (exp ∘ 𝐹))‘𝑗) = (exp‘(seq𝑀( + , 𝐹)‘𝑗))) ↔ (𝜑 → (seq𝑀( · , (exp ∘ 𝐹))‘𝑘) = (exp‘(seq𝑀( + , 𝐹)‘𝑘)))))
39 uzid 11699 . . . . . . . . . . 11 (𝑀 ∈ ℤ → 𝑀 ∈ (ℤ𝑀))
402, 39syl 17 . . . . . . . . . 10 (𝜑𝑀 ∈ (ℤ𝑀))
4140, 1syl6eleqr 2711 . . . . . . . . 9 (𝜑𝑀𝑍)
42 fvco3 6273 . . . . . . . . 9 ((𝐹:𝑍⟶ℂ ∧ 𝑀𝑍) → ((exp ∘ 𝐹)‘𝑀) = (exp‘(𝐹𝑀)))
433, 41, 42syl2anc 693 . . . . . . . 8 (𝜑 → ((exp ∘ 𝐹)‘𝑀) = (exp‘(𝐹𝑀)))
44 seq1 12809 . . . . . . . . 9 (𝑀 ∈ ℤ → (seq𝑀( · , (exp ∘ 𝐹))‘𝑀) = ((exp ∘ 𝐹)‘𝑀))
452, 44syl 17 . . . . . . . 8 (𝜑 → (seq𝑀( · , (exp ∘ 𝐹))‘𝑀) = ((exp ∘ 𝐹)‘𝑀))
46 seq1 12809 . . . . . . . . . 10 (𝑀 ∈ ℤ → (seq𝑀( + , 𝐹)‘𝑀) = (𝐹𝑀))
472, 46syl 17 . . . . . . . . 9 (𝜑 → (seq𝑀( + , 𝐹)‘𝑀) = (𝐹𝑀))
4847fveq2d 6193 . . . . . . . 8 (𝜑 → (exp‘(seq𝑀( + , 𝐹)‘𝑀)) = (exp‘(𝐹𝑀)))
4943, 45, 483eqtr4d 2665 . . . . . . 7 (𝜑 → (seq𝑀( · , (exp ∘ 𝐹))‘𝑀) = (exp‘(seq𝑀( + , 𝐹)‘𝑀)))
5049a1i 11 . . . . . 6 (𝑀 ∈ ℤ → (𝜑 → (seq𝑀( · , (exp ∘ 𝐹))‘𝑀) = (exp‘(seq𝑀( + , 𝐹)‘𝑀))))
51 oveq1 6654 . . . . . . . . . . 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))))
52513ad2ant3 1083 . . . . . . . . . 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))))
533adantl 482 . . . . . . . . . . . . . 14 ((𝑛 ∈ (ℤ𝑀) ∧ 𝜑) → 𝐹:𝑍⟶ℂ)
54 peano2uz 11738 . . . . . . . . . . . . . . . 16 (𝑛 ∈ (ℤ𝑀) → (𝑛 + 1) ∈ (ℤ𝑀))
5554, 1syl6eleqr 2711 . . . . . . . . . . . . . . 15 (𝑛 ∈ (ℤ𝑀) → (𝑛 + 1) ∈ 𝑍)
5655adantr 481 . . . . . . . . . . . . . 14 ((𝑛 ∈ (ℤ𝑀) ∧ 𝜑) → (𝑛 + 1) ∈ 𝑍)
57 fvco3 6273 . . . . . . . . . . . . . 14 ((𝐹:𝑍⟶ℂ ∧ (𝑛 + 1) ∈ 𝑍) → ((exp ∘ 𝐹)‘(𝑛 + 1)) = (exp‘(𝐹‘(𝑛 + 1))))
5853, 56, 57syl2anc 693 . . . . . . . . . . . . 13 ((𝑛 ∈ (ℤ𝑀) ∧ 𝜑) → ((exp ∘ 𝐹)‘(𝑛 + 1)) = (exp‘(𝐹‘(𝑛 + 1))))
5958oveq2d 6663 . . . . . . . . . . . 12 ((𝑛 ∈ (ℤ𝑀) ∧ 𝜑) → ((exp‘(seq𝑀( + , 𝐹)‘𝑛)) · ((exp ∘ 𝐹)‘(𝑛 + 1))) = ((exp‘(seq𝑀( + , 𝐹)‘𝑛)) · (exp‘(𝐹‘(𝑛 + 1)))))
6016ffvelrnda 6357 . . . . . . . . . . . . . . . 16 ((𝜑𝑛𝑍) → (seq𝑀( + , 𝐹)‘𝑛) ∈ ℂ)
6160expcom 451 . . . . . . . . . . . . . . 15 (𝑛𝑍 → (𝜑 → (seq𝑀( + , 𝐹)‘𝑛) ∈ ℂ))
621eqcomi 2630 . . . . . . . . . . . . . . 15 (ℤ𝑀) = 𝑍
6361, 62eleq2s 2718 . . . . . . . . . . . . . 14 (𝑛 ∈ (ℤ𝑀) → (𝜑 → (seq𝑀( + , 𝐹)‘𝑛) ∈ ℂ))
6463imp 445 . . . . . . . . . . . . 13 ((𝑛 ∈ (ℤ𝑀) ∧ 𝜑) → (seq𝑀( + , 𝐹)‘𝑛) ∈ ℂ)
6553, 56ffvelrnd 6358 . . . . . . . . . . . . 13 ((𝑛 ∈ (ℤ𝑀) ∧ 𝜑) → (𝐹‘(𝑛 + 1)) ∈ ℂ)
66 efadd 14818 . . . . . . . . . . . . 13 (((seq𝑀( + , 𝐹)‘𝑛) ∈ ℂ ∧ (𝐹‘(𝑛 + 1)) ∈ ℂ) → (exp‘((seq𝑀( + , 𝐹)‘𝑛) + (𝐹‘(𝑛 + 1)))) = ((exp‘(seq𝑀( + , 𝐹)‘𝑛)) · (exp‘(𝐹‘(𝑛 + 1)))))
6764, 65, 66syl2anc 693 . . . . . . . . . . . 12 ((𝑛 ∈ (ℤ𝑀) ∧ 𝜑) → (exp‘((seq𝑀( + , 𝐹)‘𝑛) + (𝐹‘(𝑛 + 1)))) = ((exp‘(seq𝑀( + , 𝐹)‘𝑛)) · (exp‘(𝐹‘(𝑛 + 1)))))
6859, 67eqtr4d 2658 . . . . . . . . . . 11 ((𝑛 ∈ (ℤ𝑀) ∧ 𝜑) → ((exp‘(seq𝑀( + , 𝐹)‘𝑛)) · ((exp ∘ 𝐹)‘(𝑛 + 1))) = (exp‘((seq𝑀( + , 𝐹)‘𝑛) + (𝐹‘(𝑛 + 1)))))
69683adant3 1080 . . . . . . . . . 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)))))
7052, 69eqtrd 2655 . . . . . . . . 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)))))
71 seqp1 12811 . . . . . . . . . . 11 (𝑛 ∈ (ℤ𝑀) → (seq𝑀( · , (exp ∘ 𝐹))‘(𝑛 + 1)) = ((seq𝑀( · , (exp ∘ 𝐹))‘𝑛) · ((exp ∘ 𝐹)‘(𝑛 + 1))))
7271adantr 481 . . . . . . . . . 10 ((𝑛 ∈ (ℤ𝑀) ∧ 𝜑) → (seq𝑀( · , (exp ∘ 𝐹))‘(𝑛 + 1)) = ((seq𝑀( · , (exp ∘ 𝐹))‘𝑛) · ((exp ∘ 𝐹)‘(𝑛 + 1))))
73723adant3 1080 . . . . . . . . 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))))
74 seqp1 12811 . . . . . . . . . . . 12 (𝑛 ∈ (ℤ𝑀) → (seq𝑀( + , 𝐹)‘(𝑛 + 1)) = ((seq𝑀( + , 𝐹)‘𝑛) + (𝐹‘(𝑛 + 1))))
7574adantr 481 . . . . . . . . . . 11 ((𝑛 ∈ (ℤ𝑀) ∧ 𝜑) → (seq𝑀( + , 𝐹)‘(𝑛 + 1)) = ((seq𝑀( + , 𝐹)‘𝑛) + (𝐹‘(𝑛 + 1))))
7675fveq2d 6193 . . . . . . . . . 10 ((𝑛 ∈ (ℤ𝑀) ∧ 𝜑) → (exp‘(seq𝑀( + , 𝐹)‘(𝑛 + 1))) = (exp‘((seq𝑀( + , 𝐹)‘𝑛) + (𝐹‘(𝑛 + 1)))))
77763adant3 1080 . . . . . . . . 9 ((𝑛 ∈ (ℤ𝑀) ∧ 𝜑 ∧ (seq𝑀( · , (exp ∘ 𝐹))‘𝑛) = (exp‘(seq𝑀( + , 𝐹)‘𝑛))) → (exp‘(seq𝑀( + , 𝐹)‘(𝑛 + 1))) = (exp‘((seq𝑀( + , 𝐹)‘𝑛) + (𝐹‘(𝑛 + 1)))))
7870, 73, 773eqtr4d 2665 . . . . . . . 8 ((𝑛 ∈ (ℤ𝑀) ∧ 𝜑 ∧ (seq𝑀( · , (exp ∘ 𝐹))‘𝑛) = (exp‘(seq𝑀( + , 𝐹)‘𝑛))) → (seq𝑀( · , (exp ∘ 𝐹))‘(𝑛 + 1)) = (exp‘(seq𝑀( + , 𝐹)‘(𝑛 + 1))))
79783exp 1263 . . . . . . 7 (𝑛 ∈ (ℤ𝑀) → (𝜑 → ((seq𝑀( · , (exp ∘ 𝐹))‘𝑛) = (exp‘(seq𝑀( + , 𝐹)‘𝑛)) → (seq𝑀( · , (exp ∘ 𝐹))‘(𝑛 + 1)) = (exp‘(seq𝑀( + , 𝐹)‘(𝑛 + 1))))))
8079a2d 29 . . . . . 6 (𝑛 ∈ (ℤ𝑀) → ((𝜑 → (seq𝑀( · , (exp ∘ 𝐹))‘𝑛) = (exp‘(seq𝑀( + , 𝐹)‘𝑛))) → (𝜑 → (seq𝑀( · , (exp ∘ 𝐹))‘(𝑛 + 1)) = (exp‘(seq𝑀( + , 𝐹)‘(𝑛 + 1))))))
8123, 28, 33, 38, 50, 80uzind4 11743 . . . . 5 (𝑘 ∈ (ℤ𝑀) → (𝜑 → (seq𝑀( · , (exp ∘ 𝐹))‘𝑘) = (exp‘(seq𝑀( + , 𝐹)‘𝑘))))
8281, 1eleq2s 2718 . . . 4 (𝑘𝑍 → (𝜑 → (seq𝑀( · , (exp ∘ 𝐹))‘𝑘) = (exp‘(seq𝑀( + , 𝐹)‘𝑘))))
8382impcom 446 . . 3 ((𝜑𝑘𝑍) → (seq𝑀( · , (exp ∘ 𝐹))‘𝑘) = (exp‘(seq𝑀( + , 𝐹)‘𝑘)))
84 fvco3 6273 . . . 4 ((seq𝑀( + , 𝐹):𝑍⟶ℂ ∧ 𝑘𝑍) → ((exp ∘ seq𝑀( + , 𝐹))‘𝑘) = (exp‘(seq𝑀( + , 𝐹)‘𝑘)))
8516, 84sylan 488 . . 3 ((𝜑𝑘𝑍) → ((exp ∘ seq𝑀( + , 𝐹))‘𝑘) = (exp‘(seq𝑀( + , 𝐹)‘𝑘)))
8683, 85eqtr4d 2658 . 2 ((𝜑𝑘𝑍) → (seq𝑀( · , (exp ∘ 𝐹))‘𝑘) = ((exp ∘ seq𝑀( + , 𝐹))‘𝑘))
8712, 18, 86eqfnfvd 6312 1 (𝜑 → seq𝑀( · , (exp ∘ 𝐹)) = (exp ∘ seq𝑀( + , 𝐹)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384  w3a 1037   = wceq 1482  wcel 1989  ccom 5116   Fn wfn 5881  wf 5882  cfv 5886  (class class class)co 6647  cc 9931  1c1 9934   + caddc 9936   · cmul 9938  cz 11374  cuz 11684  seqcseq 12796  expce 14786
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1721  ax-4 1736  ax-5 1838  ax-6 1887  ax-7 1934  ax-8 1991  ax-9 1998  ax-10 2018  ax-11 2033  ax-12 2046  ax-13 2245  ax-ext 2601  ax-rep 4769  ax-sep 4779  ax-nul 4787  ax-pow 4841  ax-pr 4904  ax-un 6946  ax-inf2 8535  ax-cnex 9989  ax-resscn 9990  ax-1cn 9991  ax-icn 9992  ax-addcl 9993  ax-addrcl 9994  ax-mulcl 9995  ax-mulrcl 9996  ax-mulcom 9997  ax-addass 9998  ax-mulass 9999  ax-distr 10000  ax-i2m1 10001  ax-1ne0 10002  ax-1rid 10003  ax-rnegex 10004  ax-rrecex 10005  ax-cnre 10006  ax-pre-lttri 10007  ax-pre-lttrn 10008  ax-pre-ltadd 10009  ax-pre-mulgt0 10010  ax-pre-sup 10011  ax-addf 10012  ax-mulf 10013
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1485  df-fal 1488  df-ex 1704  df-nf 1709  df-sb 1880  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2752  df-ne 2794  df-nel 2897  df-ral 2916  df-rex 2917  df-reu 2918  df-rmo 2919  df-rab 2920  df-v 3200  df-sbc 3434  df-csb 3532  df-dif 3575  df-un 3577  df-in 3579  df-ss 3586  df-pss 3588  df-nul 3914  df-if 4085  df-pw 4158  df-sn 4176  df-pr 4178  df-tp 4180  df-op 4182  df-uni 4435  df-int 4474  df-iun 4520  df-br 4652  df-opab 4711  df-mpt 4728  df-tr 4751  df-id 5022  df-eprel 5027  df-po 5033  df-so 5034  df-fr 5071  df-se 5072  df-we 5073  df-xp 5118  df-rel 5119  df-cnv 5120  df-co 5121  df-dm 5122  df-rn 5123  df-res 5124  df-ima 5125  df-pred 5678  df-ord 5724  df-on 5725  df-lim 5726  df-suc 5727  df-iota 5849  df-fun 5888  df-fn 5889  df-f 5890  df-f1 5891  df-fo 5892  df-f1o 5893  df-fv 5894  df-isom 5895  df-riota 6608  df-ov 6650  df-oprab 6651  df-mpt2 6652  df-om 7063  df-1st 7165  df-2nd 7166  df-wrecs 7404  df-recs 7465  df-rdg 7503  df-1o 7557  df-oadd 7561  df-er 7739  df-pm 7857  df-en 7953  df-dom 7954  df-sdom 7955  df-fin 7956  df-sup 8345  df-inf 8346  df-oi 8412  df-card 8762  df-pnf 10073  df-mnf 10074  df-xr 10075  df-ltxr 10076  df-le 10077  df-sub 10265  df-neg 10266  df-div 10682  df-nn 11018  df-2 11076  df-3 11077  df-n0 11290  df-z 11375  df-uz 11685  df-rp 11830  df-ico 12178  df-fz 12324  df-fzo 12462  df-fl 12588  df-seq 12797  df-exp 12856  df-fac 13056  df-bc 13085  df-hash 13113  df-shft 13801  df-cj 13833  df-re 13834  df-im 13835  df-sqrt 13969  df-abs 13970  df-limsup 14196  df-clim 14213  df-rlim 14214  df-sum 14411  df-ef 14792
This theorem is referenced by:  iprodefisum  31613
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