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 35270
Description: Lemma for iprodefisum 35271. (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 6991 . . . . . 6 ((𝐹:π‘βŸΆβ„‚ ∧ π‘˜ ∈ 𝑍) β†’ ((exp ∘ 𝐹)β€˜π‘˜) = (expβ€˜(πΉβ€˜π‘˜)))
53, 4sylan 579 . . . . 5 ((πœ‘ ∧ π‘˜ ∈ 𝑍) β†’ ((exp ∘ 𝐹)β€˜π‘˜) = (expβ€˜(πΉβ€˜π‘˜)))
63ffvelcdmda 7088 . . . . . 6 ((πœ‘ ∧ π‘˜ ∈ 𝑍) β†’ (πΉβ€˜π‘˜) ∈ β„‚)
7 efcl 16050 . . . . . 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 2828 . . . 4 ((πœ‘ ∧ π‘˜ ∈ 𝑍) β†’ ((exp ∘ 𝐹)β€˜π‘˜) ∈ β„‚)
101, 2, 9prodf 15857 . . 3 (πœ‘ β†’ seq𝑀( Β· , (exp ∘ 𝐹)):π‘βŸΆβ„‚)
1110ffnd 6717 . 2 (πœ‘ β†’ seq𝑀( Β· , (exp ∘ 𝐹)) Fn 𝑍)
12 eff 16049 . . . 4 exp:β„‚βŸΆβ„‚
13 ffn 6716 . . . 4 (exp:β„‚βŸΆβ„‚ β†’ exp Fn β„‚)
1412, 13ax-mp 5 . . 3 exp Fn β„‚
151, 2, 6serf 14019 . . 3 (πœ‘ β†’ seq𝑀( + , 𝐹):π‘βŸΆβ„‚)
16 fnfco 6756 . . 3 ((exp Fn β„‚ ∧ seq𝑀( + , 𝐹):π‘βŸΆβ„‚) β†’ (exp ∘ seq𝑀( + , 𝐹)) Fn 𝑍)
1714, 15, 16sylancr 586 . 2 (πœ‘ β†’ (exp ∘ seq𝑀( + , 𝐹)) Fn 𝑍)
18 fveq2 6891 . . . . . . . 8 (𝑗 = 𝑀 β†’ (seq𝑀( Β· , (exp ∘ 𝐹))β€˜π‘—) = (seq𝑀( Β· , (exp ∘ 𝐹))β€˜π‘€))
19 2fveq3 6896 . . . . . . . 8 (𝑗 = 𝑀 β†’ (expβ€˜(seq𝑀( + , 𝐹)β€˜π‘—)) = (expβ€˜(seq𝑀( + , 𝐹)β€˜π‘€)))
2018, 19eqeq12d 2743 . . . . . . 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 6891 . . . . . . . 8 (𝑗 = 𝑛 β†’ (seq𝑀( Β· , (exp ∘ 𝐹))β€˜π‘—) = (seq𝑀( Β· , (exp ∘ 𝐹))β€˜π‘›))
23 2fveq3 6896 . . . . . . . 8 (𝑗 = 𝑛 β†’ (expβ€˜(seq𝑀( + , 𝐹)β€˜π‘—)) = (expβ€˜(seq𝑀( + , 𝐹)β€˜π‘›)))
2422, 23eqeq12d 2743 . . . . . . 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 6891 . . . . . . . 8 (𝑗 = (𝑛 + 1) β†’ (seq𝑀( Β· , (exp ∘ 𝐹))β€˜π‘—) = (seq𝑀( Β· , (exp ∘ 𝐹))β€˜(𝑛 + 1)))
27 2fveq3 6896 . . . . . . . 8 (𝑗 = (𝑛 + 1) β†’ (expβ€˜(seq𝑀( + , 𝐹)β€˜π‘—)) = (expβ€˜(seq𝑀( + , 𝐹)β€˜(𝑛 + 1))))
2826, 27eqeq12d 2743 . . . . . . 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 6891 . . . . . . . 8 (𝑗 = π‘˜ β†’ (seq𝑀( Β· , (exp ∘ 𝐹))β€˜π‘—) = (seq𝑀( Β· , (exp ∘ 𝐹))β€˜π‘˜))
31 2fveq3 6896 . . . . . . . 8 (𝑗 = π‘˜ β†’ (expβ€˜(seq𝑀( + , 𝐹)β€˜π‘—)) = (expβ€˜(seq𝑀( + , 𝐹)β€˜π‘˜)))
3230, 31eqeq12d 2743 . . . . . . 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 12859 . . . . . . . . . . 11 (𝑀 ∈ β„€ β†’ 𝑀 ∈ (β„€β‰₯β€˜π‘€))
352, 34syl 17 . . . . . . . . . 10 (πœ‘ β†’ 𝑀 ∈ (β„€β‰₯β€˜π‘€))
3635, 1eleqtrrdi 2839 . . . . . . . . 9 (πœ‘ β†’ 𝑀 ∈ 𝑍)
37 fvco3 6991 . . . . . . . . 9 ((𝐹:π‘βŸΆβ„‚ ∧ 𝑀 ∈ 𝑍) β†’ ((exp ∘ 𝐹)β€˜π‘€) = (expβ€˜(πΉβ€˜π‘€)))
383, 36, 37syl2anc 583 . . . . . . . 8 (πœ‘ β†’ ((exp ∘ 𝐹)β€˜π‘€) = (expβ€˜(πΉβ€˜π‘€)))
39 seq1 14003 . . . . . . . . 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 14003 . . . . . . . . . 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 6895 . . . . . . . 8 (πœ‘ β†’ (expβ€˜(seq𝑀( + , 𝐹)β€˜π‘€)) = (expβ€˜(πΉβ€˜π‘€)))
4438, 40, 433eqtr4d 2777 . . . . . . 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 7421 . . . . . . . . . . 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 1133 . . . . . . . . . 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 481 . . . . . . . . . . . . . 14 ((𝑛 ∈ (β„€β‰₯β€˜π‘€) ∧ πœ‘) β†’ 𝐹:π‘βŸΆβ„‚)
49 peano2uz 12907 . . . . . . . . . . . . . . . 16 (𝑛 ∈ (β„€β‰₯β€˜π‘€) β†’ (𝑛 + 1) ∈ (β„€β‰₯β€˜π‘€))
5049, 1eleqtrrdi 2839 . . . . . . . . . . . . . . 15 (𝑛 ∈ (β„€β‰₯β€˜π‘€) β†’ (𝑛 + 1) ∈ 𝑍)
5150adantr 480 . . . . . . . . . . . . . 14 ((𝑛 ∈ (β„€β‰₯β€˜π‘€) ∧ πœ‘) β†’ (𝑛 + 1) ∈ 𝑍)
52 fvco3 6991 . . . . . . . . . . . . . 14 ((𝐹:π‘βŸΆβ„‚ ∧ (𝑛 + 1) ∈ 𝑍) β†’ ((exp ∘ 𝐹)β€˜(𝑛 + 1)) = (expβ€˜(πΉβ€˜(𝑛 + 1))))
5348, 51, 52syl2anc 583 . . . . . . . . . . . . 13 ((𝑛 ∈ (β„€β‰₯β€˜π‘€) ∧ πœ‘) β†’ ((exp ∘ 𝐹)β€˜(𝑛 + 1)) = (expβ€˜(πΉβ€˜(𝑛 + 1))))
5453oveq2d 7430 . . . . . . . . . . . 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 7088 . . . . . . . . . . . . . . . 16 ((πœ‘ ∧ 𝑛 ∈ 𝑍) β†’ (seq𝑀( + , 𝐹)β€˜π‘›) ∈ β„‚)
5655expcom 413 . . . . . . . . . . . . . . 15 (𝑛 ∈ 𝑍 β†’ (πœ‘ β†’ (seq𝑀( + , 𝐹)β€˜π‘›) ∈ β„‚))
571eqcomi 2736 . . . . . . . . . . . . . . 15 (β„€β‰₯β€˜π‘€) = 𝑍
5856, 57eleq2s 2846 . . . . . . . . . . . . . 14 (𝑛 ∈ (β„€β‰₯β€˜π‘€) β†’ (πœ‘ β†’ (seq𝑀( + , 𝐹)β€˜π‘›) ∈ β„‚))
5958imp 406 . . . . . . . . . . . . 13 ((𝑛 ∈ (β„€β‰₯β€˜π‘€) ∧ πœ‘) β†’ (seq𝑀( + , 𝐹)β€˜π‘›) ∈ β„‚)
6048, 51ffvelcdmd 7089 . . . . . . . . . . . . 13 ((𝑛 ∈ (β„€β‰₯β€˜π‘€) ∧ πœ‘) β†’ (πΉβ€˜(𝑛 + 1)) ∈ β„‚)
61 efadd 16062 . . . . . . . . . . . . 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 583 . . . . . . . . . . . 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 2770 . . . . . . . . . . 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 1130 . . . . . . . . . 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 2767 . . . . . . . . 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 14005 . . . . . . . . . . 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 480 . . . . . . . . . 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 1130 . . . . . . . . 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 14005 . . . . . . . . . . . 12 (𝑛 ∈ (β„€β‰₯β€˜π‘€) β†’ (seq𝑀( + , 𝐹)β€˜(𝑛 + 1)) = ((seq𝑀( + , 𝐹)β€˜π‘›) + (πΉβ€˜(𝑛 + 1))))
7069adantr 480 . . . . . . . . . . 11 ((𝑛 ∈ (β„€β‰₯β€˜π‘€) ∧ πœ‘) β†’ (seq𝑀( + , 𝐹)β€˜(𝑛 + 1)) = ((seq𝑀( + , 𝐹)β€˜π‘›) + (πΉβ€˜(𝑛 + 1))))
7170fveq2d 6895 . . . . . . . . . 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 1130 . . . . . . . . 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 2777 . . . . . . . 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 1117 . . . . . . 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 12912 . . . . 5 (π‘˜ ∈ (β„€β‰₯β€˜π‘€) β†’ (πœ‘ β†’ (seq𝑀( Β· , (exp ∘ 𝐹))β€˜π‘˜) = (expβ€˜(seq𝑀( + , 𝐹)β€˜π‘˜))))
7776, 1eleq2s 2846 . . . 4 (π‘˜ ∈ 𝑍 β†’ (πœ‘ β†’ (seq𝑀( Β· , (exp ∘ 𝐹))β€˜π‘˜) = (expβ€˜(seq𝑀( + , 𝐹)β€˜π‘˜))))
7877impcom 407 . . 3 ((πœ‘ ∧ π‘˜ ∈ 𝑍) β†’ (seq𝑀( Β· , (exp ∘ 𝐹))β€˜π‘˜) = (expβ€˜(seq𝑀( + , 𝐹)β€˜π‘˜)))
79 fvco3 6991 . . . 4 ((seq𝑀( + , 𝐹):π‘βŸΆβ„‚ ∧ π‘˜ ∈ 𝑍) β†’ ((exp ∘ seq𝑀( + , 𝐹))β€˜π‘˜) = (expβ€˜(seq𝑀( + , 𝐹)β€˜π‘˜)))
8015, 79sylan 579 . . 3 ((πœ‘ ∧ π‘˜ ∈ 𝑍) β†’ ((exp ∘ seq𝑀( + , 𝐹))β€˜π‘˜) = (expβ€˜(seq𝑀( + , 𝐹)β€˜π‘˜)))
8178, 80eqtr4d 2770 . 2 ((πœ‘ ∧ π‘˜ ∈ 𝑍) β†’ (seq𝑀( Β· , (exp ∘ 𝐹))β€˜π‘˜) = ((exp ∘ seq𝑀( + , 𝐹))β€˜π‘˜))
8211, 17, 81eqfnfvd 7037 1 (πœ‘ β†’ seq𝑀( Β· , (exp ∘ 𝐹)) = (exp ∘ seq𝑀( + , 𝐹)))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ wa 395   ∧ w3a 1085   = wceq 1534   ∈ wcel 2099   ∘ ccom 5676   Fn wfn 6537  βŸΆwf 6538  β€˜cfv 6542  (class class class)co 7414  β„‚cc 11128  1c1 11131   + caddc 11133   Β· cmul 11135  β„€cz 12580  β„€β‰₯cuz 12844  seqcseq 13990  expce 16029
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2164  ax-ext 2698  ax-rep 5279  ax-sep 5293  ax-nul 5300  ax-pow 5359  ax-pr 5423  ax-un 7734  ax-inf2 9656  ax-cnex 11186  ax-resscn 11187  ax-1cn 11188  ax-icn 11189  ax-addcl 11190  ax-addrcl 11191  ax-mulcl 11192  ax-mulrcl 11193  ax-mulcom 11194  ax-addass 11195  ax-mulass 11196  ax-distr 11197  ax-i2m1 11198  ax-1ne0 11199  ax-1rid 11200  ax-rnegex 11201  ax-rrecex 11202  ax-cnre 11203  ax-pre-lttri 11204  ax-pre-lttrn 11205  ax-pre-ltadd 11206  ax-pre-mulgt0 11207  ax-pre-sup 11208
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3or 1086  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2705  df-cleq 2719  df-clel 2805  df-nfc 2880  df-ne 2936  df-nel 3042  df-ral 3057  df-rex 3066  df-rmo 3371  df-reu 3372  df-rab 3428  df-v 3471  df-sbc 3775  df-csb 3890  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-pss 3963  df-nul 4319  df-if 4525  df-pw 4600  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4904  df-int 4945  df-iun 4993  df-br 5143  df-opab 5205  df-mpt 5226  df-tr 5260  df-id 5570  df-eprel 5576  df-po 5584  df-so 5585  df-fr 5627  df-se 5628  df-we 5629  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-rn 5683  df-res 5684  df-ima 5685  df-pred 6299  df-ord 6366  df-on 6367  df-lim 6368  df-suc 6369  df-iota 6494  df-fun 6544  df-fn 6545  df-f 6546  df-f1 6547  df-fo 6548  df-f1o 6549  df-fv 6550  df-isom 6551  df-riota 7370  df-ov 7417  df-oprab 7418  df-mpo 7419  df-om 7865  df-1st 7987  df-2nd 7988  df-frecs 8280  df-wrecs 8311  df-recs 8385  df-rdg 8424  df-1o 8480  df-er 8718  df-pm 8839  df-en 8956  df-dom 8957  df-sdom 8958  df-fin 8959  df-sup 9457  df-inf 9458  df-oi 9525  df-card 9954  df-pnf 11272  df-mnf 11273  df-xr 11274  df-ltxr 11275  df-le 11276  df-sub 11468  df-neg 11469  df-div 11894  df-nn 12235  df-2 12297  df-3 12298  df-n0 12495  df-z 12581  df-uz 12845  df-rp 12999  df-ico 13354  df-fz 13509  df-fzo 13652  df-fl 13781  df-seq 13991  df-exp 14051  df-fac 14257  df-bc 14286  df-hash 14314  df-shft 15038  df-cj 15070  df-re 15071  df-im 15072  df-sqrt 15206  df-abs 15207  df-limsup 15439  df-clim 15456  df-rlim 15457  df-sum 15657  df-ef 16035
This theorem is referenced by:  iprodefisum  35271
  Copyright terms: Public domain W3C validator