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Theorem iprodefisumlem 34433
Description: Lemma for iprodefisum 34434. (Contributed by Scott Fenton, 11-Feb-2018.)
Hypotheses
Ref Expression
iprodefisumlem.1 𝑍 = (β„€β‰₯β€˜π‘€)
iprodefisumlem.2 (πœ‘ β†’ 𝑀 ∈ β„€)
iprodefisumlem.3 (πœ‘ β†’ 𝐹:π‘βŸΆβ„‚)
Assertion
Ref Expression
iprodefisumlem (πœ‘ β†’ seq𝑀( Β· , (exp ∘ 𝐹)) = (exp ∘ seq𝑀( + , 𝐹)))

Proof of Theorem iprodefisumlem
Dummy variables 𝑗 π‘˜ 𝑛 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 iprodefisumlem.1 . . . 4 𝑍 = (β„€β‰₯β€˜π‘€)
2 iprodefisumlem.2 . . . 4 (πœ‘ β†’ 𝑀 ∈ β„€)
3 iprodefisumlem.3 . . . . . 6 (πœ‘ β†’ 𝐹:π‘βŸΆβ„‚)
4 fvco3 6960 . . . . . 6 ((𝐹:π‘βŸΆβ„‚ ∧ π‘˜ ∈ 𝑍) β†’ ((exp ∘ 𝐹)β€˜π‘˜) = (expβ€˜(πΉβ€˜π‘˜)))
53, 4sylan 580 . . . . 5 ((πœ‘ ∧ π‘˜ ∈ 𝑍) β†’ ((exp ∘ 𝐹)β€˜π‘˜) = (expβ€˜(πΉβ€˜π‘˜)))
63ffvelcdmda 7055 . . . . . 6 ((πœ‘ ∧ π‘˜ ∈ 𝑍) β†’ (πΉβ€˜π‘˜) ∈ β„‚)
7 efcl 15991 . . . . . 6 ((πΉβ€˜π‘˜) ∈ β„‚ β†’ (expβ€˜(πΉβ€˜π‘˜)) ∈ β„‚)
86, 7syl 17 . . . . 5 ((πœ‘ ∧ π‘˜ ∈ 𝑍) β†’ (expβ€˜(πΉβ€˜π‘˜)) ∈ β„‚)
95, 8eqeltrd 2832 . . . 4 ((πœ‘ ∧ π‘˜ ∈ 𝑍) β†’ ((exp ∘ 𝐹)β€˜π‘˜) ∈ β„‚)
101, 2, 9prodf 15798 . . 3 (πœ‘ β†’ seq𝑀( Β· , (exp ∘ 𝐹)):π‘βŸΆβ„‚)
1110ffnd 6689 . 2 (πœ‘ β†’ seq𝑀( Β· , (exp ∘ 𝐹)) Fn 𝑍)
12 eff 15990 . . . 4 exp:β„‚βŸΆβ„‚
13 ffn 6688 . . . 4 (exp:β„‚βŸΆβ„‚ β†’ exp Fn β„‚)
1412, 13ax-mp 5 . . 3 exp Fn β„‚
151, 2, 6serf 13961 . . 3 (πœ‘ β†’ seq𝑀( + , 𝐹):π‘βŸΆβ„‚)
16 fnfco 6727 . . 3 ((exp Fn β„‚ ∧ seq𝑀( + , 𝐹):π‘βŸΆβ„‚) β†’ (exp ∘ seq𝑀( + , 𝐹)) Fn 𝑍)
1714, 15, 16sylancr 587 . 2 (πœ‘ β†’ (exp ∘ seq𝑀( + , 𝐹)) Fn 𝑍)
18 fveq2 6862 . . . . . . . 8 (𝑗 = 𝑀 β†’ (seq𝑀( Β· , (exp ∘ 𝐹))β€˜π‘—) = (seq𝑀( Β· , (exp ∘ 𝐹))β€˜π‘€))
19 2fveq3 6867 . . . . . . . 8 (𝑗 = 𝑀 β†’ (expβ€˜(seq𝑀( + , 𝐹)β€˜π‘—)) = (expβ€˜(seq𝑀( + , 𝐹)β€˜π‘€)))
2018, 19eqeq12d 2747 . . . . . . 7 (𝑗 = 𝑀 β†’ ((seq𝑀( Β· , (exp ∘ 𝐹))β€˜π‘—) = (expβ€˜(seq𝑀( + , 𝐹)β€˜π‘—)) ↔ (seq𝑀( Β· , (exp ∘ 𝐹))β€˜π‘€) = (expβ€˜(seq𝑀( + , 𝐹)β€˜π‘€))))
2120imbi2d 340 . . . . . 6 (𝑗 = 𝑀 β†’ ((πœ‘ β†’ (seq𝑀( Β· , (exp ∘ 𝐹))β€˜π‘—) = (expβ€˜(seq𝑀( + , 𝐹)β€˜π‘—))) ↔ (πœ‘ β†’ (seq𝑀( Β· , (exp ∘ 𝐹))β€˜π‘€) = (expβ€˜(seq𝑀( + , 𝐹)β€˜π‘€)))))
22 fveq2 6862 . . . . . . . 8 (𝑗 = 𝑛 β†’ (seq𝑀( Β· , (exp ∘ 𝐹))β€˜π‘—) = (seq𝑀( Β· , (exp ∘ 𝐹))β€˜π‘›))
23 2fveq3 6867 . . . . . . . 8 (𝑗 = 𝑛 β†’ (expβ€˜(seq𝑀( + , 𝐹)β€˜π‘—)) = (expβ€˜(seq𝑀( + , 𝐹)β€˜π‘›)))
2422, 23eqeq12d 2747 . . . . . . 7 (𝑗 = 𝑛 β†’ ((seq𝑀( Β· , (exp ∘ 𝐹))β€˜π‘—) = (expβ€˜(seq𝑀( + , 𝐹)β€˜π‘—)) ↔ (seq𝑀( Β· , (exp ∘ 𝐹))β€˜π‘›) = (expβ€˜(seq𝑀( + , 𝐹)β€˜π‘›))))
2524imbi2d 340 . . . . . 6 (𝑗 = 𝑛 β†’ ((πœ‘ β†’ (seq𝑀( Β· , (exp ∘ 𝐹))β€˜π‘—) = (expβ€˜(seq𝑀( + , 𝐹)β€˜π‘—))) ↔ (πœ‘ β†’ (seq𝑀( Β· , (exp ∘ 𝐹))β€˜π‘›) = (expβ€˜(seq𝑀( + , 𝐹)β€˜π‘›)))))
26 fveq2 6862 . . . . . . . 8 (𝑗 = (𝑛 + 1) β†’ (seq𝑀( Β· , (exp ∘ 𝐹))β€˜π‘—) = (seq𝑀( Β· , (exp ∘ 𝐹))β€˜(𝑛 + 1)))
27 2fveq3 6867 . . . . . . . 8 (𝑗 = (𝑛 + 1) β†’ (expβ€˜(seq𝑀( + , 𝐹)β€˜π‘—)) = (expβ€˜(seq𝑀( + , 𝐹)β€˜(𝑛 + 1))))
2826, 27eqeq12d 2747 . . . . . . 7 (𝑗 = (𝑛 + 1) β†’ ((seq𝑀( Β· , (exp ∘ 𝐹))β€˜π‘—) = (expβ€˜(seq𝑀( + , 𝐹)β€˜π‘—)) ↔ (seq𝑀( Β· , (exp ∘ 𝐹))β€˜(𝑛 + 1)) = (expβ€˜(seq𝑀( + , 𝐹)β€˜(𝑛 + 1)))))
2928imbi2d 340 . . . . . 6 (𝑗 = (𝑛 + 1) β†’ ((πœ‘ β†’ (seq𝑀( Β· , (exp ∘ 𝐹))β€˜π‘—) = (expβ€˜(seq𝑀( + , 𝐹)β€˜π‘—))) ↔ (πœ‘ β†’ (seq𝑀( Β· , (exp ∘ 𝐹))β€˜(𝑛 + 1)) = (expβ€˜(seq𝑀( + , 𝐹)β€˜(𝑛 + 1))))))
30 fveq2 6862 . . . . . . . 8 (𝑗 = π‘˜ β†’ (seq𝑀( Β· , (exp ∘ 𝐹))β€˜π‘—) = (seq𝑀( Β· , (exp ∘ 𝐹))β€˜π‘˜))
31 2fveq3 6867 . . . . . . . 8 (𝑗 = π‘˜ β†’ (expβ€˜(seq𝑀( + , 𝐹)β€˜π‘—)) = (expβ€˜(seq𝑀( + , 𝐹)β€˜π‘˜)))
3230, 31eqeq12d 2747 . . . . . . 7 (𝑗 = π‘˜ β†’ ((seq𝑀( Β· , (exp ∘ 𝐹))β€˜π‘—) = (expβ€˜(seq𝑀( + , 𝐹)β€˜π‘—)) ↔ (seq𝑀( Β· , (exp ∘ 𝐹))β€˜π‘˜) = (expβ€˜(seq𝑀( + , 𝐹)β€˜π‘˜))))
3332imbi2d 340 . . . . . 6 (𝑗 = π‘˜ β†’ ((πœ‘ β†’ (seq𝑀( Β· , (exp ∘ 𝐹))β€˜π‘—) = (expβ€˜(seq𝑀( + , 𝐹)β€˜π‘—))) ↔ (πœ‘ β†’ (seq𝑀( Β· , (exp ∘ 𝐹))β€˜π‘˜) = (expβ€˜(seq𝑀( + , 𝐹)β€˜π‘˜)))))
34 uzid 12802 . . . . . . . . . . 11 (𝑀 ∈ β„€ β†’ 𝑀 ∈ (β„€β‰₯β€˜π‘€))
352, 34syl 17 . . . . . . . . . 10 (πœ‘ β†’ 𝑀 ∈ (β„€β‰₯β€˜π‘€))
3635, 1eleqtrrdi 2843 . . . . . . . . 9 (πœ‘ β†’ 𝑀 ∈ 𝑍)
37 fvco3 6960 . . . . . . . . 9 ((𝐹:π‘βŸΆβ„‚ ∧ 𝑀 ∈ 𝑍) β†’ ((exp ∘ 𝐹)β€˜π‘€) = (expβ€˜(πΉβ€˜π‘€)))
383, 36, 37syl2anc 584 . . . . . . . 8 (πœ‘ β†’ ((exp ∘ 𝐹)β€˜π‘€) = (expβ€˜(πΉβ€˜π‘€)))
39 seq1 13944 . . . . . . . . 9 (𝑀 ∈ β„€ β†’ (seq𝑀( Β· , (exp ∘ 𝐹))β€˜π‘€) = ((exp ∘ 𝐹)β€˜π‘€))
402, 39syl 17 . . . . . . . 8 (πœ‘ β†’ (seq𝑀( Β· , (exp ∘ 𝐹))β€˜π‘€) = ((exp ∘ 𝐹)β€˜π‘€))
41 seq1 13944 . . . . . . . . . 10 (𝑀 ∈ β„€ β†’ (seq𝑀( + , 𝐹)β€˜π‘€) = (πΉβ€˜π‘€))
422, 41syl 17 . . . . . . . . 9 (πœ‘ β†’ (seq𝑀( + , 𝐹)β€˜π‘€) = (πΉβ€˜π‘€))
4342fveq2d 6866 . . . . . . . 8 (πœ‘ β†’ (expβ€˜(seq𝑀( + , 𝐹)β€˜π‘€)) = (expβ€˜(πΉβ€˜π‘€)))
4438, 40, 433eqtr4d 2781 . . . . . . 7 (πœ‘ β†’ (seq𝑀( Β· , (exp ∘ 𝐹))β€˜π‘€) = (expβ€˜(seq𝑀( + , 𝐹)β€˜π‘€)))
4544a1i 11 . . . . . 6 (𝑀 ∈ β„€ β†’ (πœ‘ β†’ (seq𝑀( Β· , (exp ∘ 𝐹))β€˜π‘€) = (expβ€˜(seq𝑀( + , 𝐹)β€˜π‘€))))
46 oveq1 7384 . . . . . . . . . . 11 ((seq𝑀( Β· , (exp ∘ 𝐹))β€˜π‘›) = (expβ€˜(seq𝑀( + , 𝐹)β€˜π‘›)) β†’ ((seq𝑀( Β· , (exp ∘ 𝐹))β€˜π‘›) Β· ((exp ∘ 𝐹)β€˜(𝑛 + 1))) = ((expβ€˜(seq𝑀( + , 𝐹)β€˜π‘›)) Β· ((exp ∘ 𝐹)β€˜(𝑛 + 1))))
47463ad2ant3 1135 . . . . . . . . . 10 ((𝑛 ∈ (β„€β‰₯β€˜π‘€) ∧ πœ‘ ∧ (seq𝑀( Β· , (exp ∘ 𝐹))β€˜π‘›) = (expβ€˜(seq𝑀( + , 𝐹)β€˜π‘›))) β†’ ((seq𝑀( Β· , (exp ∘ 𝐹))β€˜π‘›) Β· ((exp ∘ 𝐹)β€˜(𝑛 + 1))) = ((expβ€˜(seq𝑀( + , 𝐹)β€˜π‘›)) Β· ((exp ∘ 𝐹)β€˜(𝑛 + 1))))
483adantl 482 . . . . . . . . . . . . . 14 ((𝑛 ∈ (β„€β‰₯β€˜π‘€) ∧ πœ‘) β†’ 𝐹:π‘βŸΆβ„‚)
49 peano2uz 12850 . . . . . . . . . . . . . . . 16 (𝑛 ∈ (β„€β‰₯β€˜π‘€) β†’ (𝑛 + 1) ∈ (β„€β‰₯β€˜π‘€))
5049, 1eleqtrrdi 2843 . . . . . . . . . . . . . . 15 (𝑛 ∈ (β„€β‰₯β€˜π‘€) β†’ (𝑛 + 1) ∈ 𝑍)
5150adantr 481 . . . . . . . . . . . . . 14 ((𝑛 ∈ (β„€β‰₯β€˜π‘€) ∧ πœ‘) β†’ (𝑛 + 1) ∈ 𝑍)
52 fvco3 6960 . . . . . . . . . . . . . 14 ((𝐹:π‘βŸΆβ„‚ ∧ (𝑛 + 1) ∈ 𝑍) β†’ ((exp ∘ 𝐹)β€˜(𝑛 + 1)) = (expβ€˜(πΉβ€˜(𝑛 + 1))))
5348, 51, 52syl2anc 584 . . . . . . . . . . . . 13 ((𝑛 ∈ (β„€β‰₯β€˜π‘€) ∧ πœ‘) β†’ ((exp ∘ 𝐹)β€˜(𝑛 + 1)) = (expβ€˜(πΉβ€˜(𝑛 + 1))))
5453oveq2d 7393 . . . . . . . . . . . 12 ((𝑛 ∈ (β„€β‰₯β€˜π‘€) ∧ πœ‘) β†’ ((expβ€˜(seq𝑀( + , 𝐹)β€˜π‘›)) Β· ((exp ∘ 𝐹)β€˜(𝑛 + 1))) = ((expβ€˜(seq𝑀( + , 𝐹)β€˜π‘›)) Β· (expβ€˜(πΉβ€˜(𝑛 + 1)))))
5515ffvelcdmda 7055 . . . . . . . . . . . . . . . 16 ((πœ‘ ∧ 𝑛 ∈ 𝑍) β†’ (seq𝑀( + , 𝐹)β€˜π‘›) ∈ β„‚)
5655expcom 414 . . . . . . . . . . . . . . 15 (𝑛 ∈ 𝑍 β†’ (πœ‘ β†’ (seq𝑀( + , 𝐹)β€˜π‘›) ∈ β„‚))
571eqcomi 2740 . . . . . . . . . . . . . . 15 (β„€β‰₯β€˜π‘€) = 𝑍
5856, 57eleq2s 2850 . . . . . . . . . . . . . 14 (𝑛 ∈ (β„€β‰₯β€˜π‘€) β†’ (πœ‘ β†’ (seq𝑀( + , 𝐹)β€˜π‘›) ∈ β„‚))
5958imp 407 . . . . . . . . . . . . 13 ((𝑛 ∈ (β„€β‰₯β€˜π‘€) ∧ πœ‘) β†’ (seq𝑀( + , 𝐹)β€˜π‘›) ∈ β„‚)
6048, 51ffvelcdmd 7056 . . . . . . . . . . . . 13 ((𝑛 ∈ (β„€β‰₯β€˜π‘€) ∧ πœ‘) β†’ (πΉβ€˜(𝑛 + 1)) ∈ β„‚)
61 efadd 16002 . . . . . . . . . . . . 13 (((seq𝑀( + , 𝐹)β€˜π‘›) ∈ β„‚ ∧ (πΉβ€˜(𝑛 + 1)) ∈ β„‚) β†’ (expβ€˜((seq𝑀( + , 𝐹)β€˜π‘›) + (πΉβ€˜(𝑛 + 1)))) = ((expβ€˜(seq𝑀( + , 𝐹)β€˜π‘›)) Β· (expβ€˜(πΉβ€˜(𝑛 + 1)))))
6259, 60, 61syl2anc 584 . . . . . . . . . . . 12 ((𝑛 ∈ (β„€β‰₯β€˜π‘€) ∧ πœ‘) β†’ (expβ€˜((seq𝑀( + , 𝐹)β€˜π‘›) + (πΉβ€˜(𝑛 + 1)))) = ((expβ€˜(seq𝑀( + , 𝐹)β€˜π‘›)) Β· (expβ€˜(πΉβ€˜(𝑛 + 1)))))
6354, 62eqtr4d 2774 . . . . . . . . . . 11 ((𝑛 ∈ (β„€β‰₯β€˜π‘€) ∧ πœ‘) β†’ ((expβ€˜(seq𝑀( + , 𝐹)β€˜π‘›)) Β· ((exp ∘ 𝐹)β€˜(𝑛 + 1))) = (expβ€˜((seq𝑀( + , 𝐹)β€˜π‘›) + (πΉβ€˜(𝑛 + 1)))))
64633adant3 1132 . . . . . . . . . 10 ((𝑛 ∈ (β„€β‰₯β€˜π‘€) ∧ πœ‘ ∧ (seq𝑀( Β· , (exp ∘ 𝐹))β€˜π‘›) = (expβ€˜(seq𝑀( + , 𝐹)β€˜π‘›))) β†’ ((expβ€˜(seq𝑀( + , 𝐹)β€˜π‘›)) Β· ((exp ∘ 𝐹)β€˜(𝑛 + 1))) = (expβ€˜((seq𝑀( + , 𝐹)β€˜π‘›) + (πΉβ€˜(𝑛 + 1)))))
6547, 64eqtrd 2771 . . . . . . . . 9 ((𝑛 ∈ (β„€β‰₯β€˜π‘€) ∧ πœ‘ ∧ (seq𝑀( Β· , (exp ∘ 𝐹))β€˜π‘›) = (expβ€˜(seq𝑀( + , 𝐹)β€˜π‘›))) β†’ ((seq𝑀( Β· , (exp ∘ 𝐹))β€˜π‘›) Β· ((exp ∘ 𝐹)β€˜(𝑛 + 1))) = (expβ€˜((seq𝑀( + , 𝐹)β€˜π‘›) + (πΉβ€˜(𝑛 + 1)))))
66 seqp1 13946 . . . . . . . . . . 11 (𝑛 ∈ (β„€β‰₯β€˜π‘€) β†’ (seq𝑀( Β· , (exp ∘ 𝐹))β€˜(𝑛 + 1)) = ((seq𝑀( Β· , (exp ∘ 𝐹))β€˜π‘›) Β· ((exp ∘ 𝐹)β€˜(𝑛 + 1))))
6766adantr 481 . . . . . . . . . 10 ((𝑛 ∈ (β„€β‰₯β€˜π‘€) ∧ πœ‘) β†’ (seq𝑀( Β· , (exp ∘ 𝐹))β€˜(𝑛 + 1)) = ((seq𝑀( Β· , (exp ∘ 𝐹))β€˜π‘›) Β· ((exp ∘ 𝐹)β€˜(𝑛 + 1))))
68673adant3 1132 . . . . . . . . 9 ((𝑛 ∈ (β„€β‰₯β€˜π‘€) ∧ πœ‘ ∧ (seq𝑀( Β· , (exp ∘ 𝐹))β€˜π‘›) = (expβ€˜(seq𝑀( + , 𝐹)β€˜π‘›))) β†’ (seq𝑀( Β· , (exp ∘ 𝐹))β€˜(𝑛 + 1)) = ((seq𝑀( Β· , (exp ∘ 𝐹))β€˜π‘›) Β· ((exp ∘ 𝐹)β€˜(𝑛 + 1))))
69 seqp1 13946 . . . . . . . . . . . 12 (𝑛 ∈ (β„€β‰₯β€˜π‘€) β†’ (seq𝑀( + , 𝐹)β€˜(𝑛 + 1)) = ((seq𝑀( + , 𝐹)β€˜π‘›) + (πΉβ€˜(𝑛 + 1))))
7069adantr 481 . . . . . . . . . . 11 ((𝑛 ∈ (β„€β‰₯β€˜π‘€) ∧ πœ‘) β†’ (seq𝑀( + , 𝐹)β€˜(𝑛 + 1)) = ((seq𝑀( + , 𝐹)β€˜π‘›) + (πΉβ€˜(𝑛 + 1))))
7170fveq2d 6866 . . . . . . . . . 10 ((𝑛 ∈ (β„€β‰₯β€˜π‘€) ∧ πœ‘) β†’ (expβ€˜(seq𝑀( + , 𝐹)β€˜(𝑛 + 1))) = (expβ€˜((seq𝑀( + , 𝐹)β€˜π‘›) + (πΉβ€˜(𝑛 + 1)))))
72713adant3 1132 . . . . . . . . 9 ((𝑛 ∈ (β„€β‰₯β€˜π‘€) ∧ πœ‘ ∧ (seq𝑀( Β· , (exp ∘ 𝐹))β€˜π‘›) = (expβ€˜(seq𝑀( + , 𝐹)β€˜π‘›))) β†’ (expβ€˜(seq𝑀( + , 𝐹)β€˜(𝑛 + 1))) = (expβ€˜((seq𝑀( + , 𝐹)β€˜π‘›) + (πΉβ€˜(𝑛 + 1)))))
7365, 68, 723eqtr4d 2781 . . . . . . . 8 ((𝑛 ∈ (β„€β‰₯β€˜π‘€) ∧ πœ‘ ∧ (seq𝑀( Β· , (exp ∘ 𝐹))β€˜π‘›) = (expβ€˜(seq𝑀( + , 𝐹)β€˜π‘›))) β†’ (seq𝑀( Β· , (exp ∘ 𝐹))β€˜(𝑛 + 1)) = (expβ€˜(seq𝑀( + , 𝐹)β€˜(𝑛 + 1))))
74733exp 1119 . . . . . . 7 (𝑛 ∈ (β„€β‰₯β€˜π‘€) β†’ (πœ‘ β†’ ((seq𝑀( Β· , (exp ∘ 𝐹))β€˜π‘›) = (expβ€˜(seq𝑀( + , 𝐹)β€˜π‘›)) β†’ (seq𝑀( Β· , (exp ∘ 𝐹))β€˜(𝑛 + 1)) = (expβ€˜(seq𝑀( + , 𝐹)β€˜(𝑛 + 1))))))
7574a2d 29 . . . . . 6 (𝑛 ∈ (β„€β‰₯β€˜π‘€) β†’ ((πœ‘ β†’ (seq𝑀( Β· , (exp ∘ 𝐹))β€˜π‘›) = (expβ€˜(seq𝑀( + , 𝐹)β€˜π‘›))) β†’ (πœ‘ β†’ (seq𝑀( Β· , (exp ∘ 𝐹))β€˜(𝑛 + 1)) = (expβ€˜(seq𝑀( + , 𝐹)β€˜(𝑛 + 1))))))
7621, 25, 29, 33, 45, 75uzind4 12855 . . . . 5 (π‘˜ ∈ (β„€β‰₯β€˜π‘€) β†’ (πœ‘ β†’ (seq𝑀( Β· , (exp ∘ 𝐹))β€˜π‘˜) = (expβ€˜(seq𝑀( + , 𝐹)β€˜π‘˜))))
7776, 1eleq2s 2850 . . . 4 (π‘˜ ∈ 𝑍 β†’ (πœ‘ β†’ (seq𝑀( Β· , (exp ∘ 𝐹))β€˜π‘˜) = (expβ€˜(seq𝑀( + , 𝐹)β€˜π‘˜))))
7877impcom 408 . . 3 ((πœ‘ ∧ π‘˜ ∈ 𝑍) β†’ (seq𝑀( Β· , (exp ∘ 𝐹))β€˜π‘˜) = (expβ€˜(seq𝑀( + , 𝐹)β€˜π‘˜)))
79 fvco3 6960 . . . 4 ((seq𝑀( + , 𝐹):π‘βŸΆβ„‚ ∧ π‘˜ ∈ 𝑍) β†’ ((exp ∘ seq𝑀( + , 𝐹))β€˜π‘˜) = (expβ€˜(seq𝑀( + , 𝐹)β€˜π‘˜)))
8015, 79sylan 580 . . 3 ((πœ‘ ∧ π‘˜ ∈ 𝑍) β†’ ((exp ∘ seq𝑀( + , 𝐹))β€˜π‘˜) = (expβ€˜(seq𝑀( + , 𝐹)β€˜π‘˜)))
8178, 80eqtr4d 2774 . 2 ((πœ‘ ∧ π‘˜ ∈ 𝑍) β†’ (seq𝑀( Β· , (exp ∘ 𝐹))β€˜π‘˜) = ((exp ∘ seq𝑀( + , 𝐹))β€˜π‘˜))
8211, 17, 81eqfnfvd 7005 1 (πœ‘ β†’ seq𝑀( Β· , (exp ∘ 𝐹)) = (exp ∘ seq𝑀( + , 𝐹)))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ wa 396   ∧ w3a 1087   = wceq 1541   ∈ wcel 2106   ∘ ccom 5657   Fn wfn 6511  βŸΆwf 6512  β€˜cfv 6516  (class class class)co 7377  β„‚cc 11073  1c1 11076   + caddc 11078   Β· cmul 11080  β„€cz 12523  β„€β‰₯cuz 12787  seqcseq 13931  expce 15970
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2702  ax-rep 5262  ax-sep 5276  ax-nul 5283  ax-pow 5340  ax-pr 5404  ax-un 7692  ax-inf2 9601  ax-cnex 11131  ax-resscn 11132  ax-1cn 11133  ax-icn 11134  ax-addcl 11135  ax-addrcl 11136  ax-mulcl 11137  ax-mulrcl 11138  ax-mulcom 11139  ax-addass 11140  ax-mulass 11141  ax-distr 11142  ax-i2m1 11143  ax-1ne0 11144  ax-1rid 11145  ax-rnegex 11146  ax-rrecex 11147  ax-cnre 11148  ax-pre-lttri 11149  ax-pre-lttrn 11150  ax-pre-ltadd 11151  ax-pre-mulgt0 11152  ax-pre-sup 11153
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-nel 3046  df-ral 3061  df-rex 3070  df-rmo 3364  df-reu 3365  df-rab 3419  df-v 3461  df-sbc 3758  df-csb 3874  df-dif 3931  df-un 3933  df-in 3935  df-ss 3945  df-pss 3947  df-nul 4303  df-if 4507  df-pw 4582  df-sn 4607  df-pr 4609  df-op 4613  df-uni 4886  df-int 4928  df-iun 4976  df-br 5126  df-opab 5188  df-mpt 5209  df-tr 5243  df-id 5551  df-eprel 5557  df-po 5565  df-so 5566  df-fr 5608  df-se 5609  df-we 5610  df-xp 5659  df-rel 5660  df-cnv 5661  df-co 5662  df-dm 5663  df-rn 5664  df-res 5665  df-ima 5666  df-pred 6273  df-ord 6340  df-on 6341  df-lim 6342  df-suc 6343  df-iota 6468  df-fun 6518  df-fn 6519  df-f 6520  df-f1 6521  df-fo 6522  df-f1o 6523  df-fv 6524  df-isom 6525  df-riota 7333  df-ov 7380  df-oprab 7381  df-mpo 7382  df-om 7823  df-1st 7941  df-2nd 7942  df-frecs 8232  df-wrecs 8263  df-recs 8337  df-rdg 8376  df-1o 8432  df-er 8670  df-pm 8790  df-en 8906  df-dom 8907  df-sdom 8908  df-fin 8909  df-sup 9402  df-inf 9403  df-oi 9470  df-card 9899  df-pnf 11215  df-mnf 11216  df-xr 11217  df-ltxr 11218  df-le 11219  df-sub 11411  df-neg 11412  df-div 11837  df-nn 12178  df-2 12240  df-3 12241  df-n0 12438  df-z 12524  df-uz 12788  df-rp 12940  df-ico 13295  df-fz 13450  df-fzo 13593  df-fl 13722  df-seq 13932  df-exp 13993  df-fac 14199  df-bc 14228  df-hash 14256  df-shft 14979  df-cj 15011  df-re 15012  df-im 15013  df-sqrt 15147  df-abs 15148  df-limsup 15380  df-clim 15397  df-rlim 15398  df-sum 15598  df-ef 15976
This theorem is referenced by:  iprodefisum  34434
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