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Theorem clim2prod 15840
Description: The limit of an infinite product with an initial segment added. (Contributed by Scott Fenton, 18-Dec-2017.)
Hypotheses
Ref Expression
clim2prod.1 𝑍 = (β„€β‰₯β€˜π‘€)
clim2prod.2 (πœ‘ β†’ 𝑁 ∈ 𝑍)
clim2prod.3 ((πœ‘ ∧ π‘˜ ∈ 𝑍) β†’ (πΉβ€˜π‘˜) ∈ β„‚)
clim2prod.4 (πœ‘ β†’ seq(𝑁 + 1)( Β· , 𝐹) ⇝ 𝐴)
Assertion
Ref Expression
clim2prod (πœ‘ β†’ seq𝑀( Β· , 𝐹) ⇝ ((seq𝑀( Β· , 𝐹)β€˜π‘) Β· 𝐴))
Distinct variable groups:   𝐴,π‘˜   π‘˜,𝐹   πœ‘,π‘˜   π‘˜,𝑀   π‘˜,𝑁   π‘˜,𝑍

Proof of Theorem clim2prod
Dummy variables 𝑛 π‘₯ are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqid 2726 . 2 (β„€β‰₯β€˜(𝑁 + 1)) = (β„€β‰₯β€˜(𝑁 + 1))
2 clim2prod.1 . . . . 5 𝑍 = (β„€β‰₯β€˜π‘€)
3 uzssz 12847 . . . . 5 (β„€β‰₯β€˜π‘€) βŠ† β„€
42, 3eqsstri 4011 . . . 4 𝑍 βŠ† β„€
5 clim2prod.2 . . . 4 (πœ‘ β†’ 𝑁 ∈ 𝑍)
64, 5sselid 3975 . . 3 (πœ‘ β†’ 𝑁 ∈ β„€)
76peano2zd 12673 . 2 (πœ‘ β†’ (𝑁 + 1) ∈ β„€)
8 clim2prod.4 . 2 (πœ‘ β†’ seq(𝑁 + 1)( Β· , 𝐹) ⇝ 𝐴)
95, 2eleqtrdi 2837 . . . . 5 (πœ‘ β†’ 𝑁 ∈ (β„€β‰₯β€˜π‘€))
10 eluzel2 12831 . . . . 5 (𝑁 ∈ (β„€β‰₯β€˜π‘€) β†’ 𝑀 ∈ β„€)
119, 10syl 17 . . . 4 (πœ‘ β†’ 𝑀 ∈ β„€)
12 clim2prod.3 . . . 4 ((πœ‘ ∧ π‘˜ ∈ 𝑍) β†’ (πΉβ€˜π‘˜) ∈ β„‚)
132, 11, 12prodf 15839 . . 3 (πœ‘ β†’ seq𝑀( Β· , 𝐹):π‘βŸΆβ„‚)
1413, 5ffvelcdmd 7081 . 2 (πœ‘ β†’ (seq𝑀( Β· , 𝐹)β€˜π‘) ∈ β„‚)
15 seqex 13974 . . 3 seq𝑀( Β· , 𝐹) ∈ V
1615a1i 11 . 2 (πœ‘ β†’ seq𝑀( Β· , 𝐹) ∈ V)
17 peano2uz 12889 . . . . . . . 8 (𝑁 ∈ (β„€β‰₯β€˜π‘€) β†’ (𝑁 + 1) ∈ (β„€β‰₯β€˜π‘€))
18 uzss 12849 . . . . . . . 8 ((𝑁 + 1) ∈ (β„€β‰₯β€˜π‘€) β†’ (β„€β‰₯β€˜(𝑁 + 1)) βŠ† (β„€β‰₯β€˜π‘€))
199, 17, 183syl 18 . . . . . . 7 (πœ‘ β†’ (β„€β‰₯β€˜(𝑁 + 1)) βŠ† (β„€β‰₯β€˜π‘€))
2019, 2sseqtrrdi 4028 . . . . . 6 (πœ‘ β†’ (β„€β‰₯β€˜(𝑁 + 1)) βŠ† 𝑍)
2120sselda 3977 . . . . 5 ((πœ‘ ∧ π‘˜ ∈ (β„€β‰₯β€˜(𝑁 + 1))) β†’ π‘˜ ∈ 𝑍)
2221, 12syldan 590 . . . 4 ((πœ‘ ∧ π‘˜ ∈ (β„€β‰₯β€˜(𝑁 + 1))) β†’ (πΉβ€˜π‘˜) ∈ β„‚)
231, 7, 22prodf 15839 . . 3 (πœ‘ β†’ seq(𝑁 + 1)( Β· , 𝐹):(β„€β‰₯β€˜(𝑁 + 1))βŸΆβ„‚)
2423ffvelcdmda 7080 . 2 ((πœ‘ ∧ π‘˜ ∈ (β„€β‰₯β€˜(𝑁 + 1))) β†’ (seq(𝑁 + 1)( Β· , 𝐹)β€˜π‘˜) ∈ β„‚)
25 fveq2 6885 . . . . . 6 (π‘₯ = (𝑁 + 1) β†’ (seq𝑀( Β· , 𝐹)β€˜π‘₯) = (seq𝑀( Β· , 𝐹)β€˜(𝑁 + 1)))
26 fveq2 6885 . . . . . . 7 (π‘₯ = (𝑁 + 1) β†’ (seq(𝑁 + 1)( Β· , 𝐹)β€˜π‘₯) = (seq(𝑁 + 1)( Β· , 𝐹)β€˜(𝑁 + 1)))
2726oveq2d 7421 . . . . . 6 (π‘₯ = (𝑁 + 1) β†’ ((seq𝑀( Β· , 𝐹)β€˜π‘) Β· (seq(𝑁 + 1)( Β· , 𝐹)β€˜π‘₯)) = ((seq𝑀( Β· , 𝐹)β€˜π‘) Β· (seq(𝑁 + 1)( Β· , 𝐹)β€˜(𝑁 + 1))))
2825, 27eqeq12d 2742 . . . . 5 (π‘₯ = (𝑁 + 1) β†’ ((seq𝑀( Β· , 𝐹)β€˜π‘₯) = ((seq𝑀( Β· , 𝐹)β€˜π‘) Β· (seq(𝑁 + 1)( Β· , 𝐹)β€˜π‘₯)) ↔ (seq𝑀( Β· , 𝐹)β€˜(𝑁 + 1)) = ((seq𝑀( Β· , 𝐹)β€˜π‘) Β· (seq(𝑁 + 1)( Β· , 𝐹)β€˜(𝑁 + 1)))))
2928imbi2d 340 . . . 4 (π‘₯ = (𝑁 + 1) β†’ ((πœ‘ β†’ (seq𝑀( Β· , 𝐹)β€˜π‘₯) = ((seq𝑀( Β· , 𝐹)β€˜π‘) Β· (seq(𝑁 + 1)( Β· , 𝐹)β€˜π‘₯))) ↔ (πœ‘ β†’ (seq𝑀( Β· , 𝐹)β€˜(𝑁 + 1)) = ((seq𝑀( Β· , 𝐹)β€˜π‘) Β· (seq(𝑁 + 1)( Β· , 𝐹)β€˜(𝑁 + 1))))))
30 fveq2 6885 . . . . . 6 (π‘₯ = 𝑛 β†’ (seq𝑀( Β· , 𝐹)β€˜π‘₯) = (seq𝑀( Β· , 𝐹)β€˜π‘›))
31 fveq2 6885 . . . . . . 7 (π‘₯ = 𝑛 β†’ (seq(𝑁 + 1)( Β· , 𝐹)β€˜π‘₯) = (seq(𝑁 + 1)( Β· , 𝐹)β€˜π‘›))
3231oveq2d 7421 . . . . . 6 (π‘₯ = 𝑛 β†’ ((seq𝑀( Β· , 𝐹)β€˜π‘) Β· (seq(𝑁 + 1)( Β· , 𝐹)β€˜π‘₯)) = ((seq𝑀( Β· , 𝐹)β€˜π‘) Β· (seq(𝑁 + 1)( Β· , 𝐹)β€˜π‘›)))
3330, 32eqeq12d 2742 . . . . 5 (π‘₯ = 𝑛 β†’ ((seq𝑀( Β· , 𝐹)β€˜π‘₯) = ((seq𝑀( Β· , 𝐹)β€˜π‘) Β· (seq(𝑁 + 1)( Β· , 𝐹)β€˜π‘₯)) ↔ (seq𝑀( Β· , 𝐹)β€˜π‘›) = ((seq𝑀( Β· , 𝐹)β€˜π‘) Β· (seq(𝑁 + 1)( Β· , 𝐹)β€˜π‘›))))
3433imbi2d 340 . . . 4 (π‘₯ = 𝑛 β†’ ((πœ‘ β†’ (seq𝑀( Β· , 𝐹)β€˜π‘₯) = ((seq𝑀( Β· , 𝐹)β€˜π‘) Β· (seq(𝑁 + 1)( Β· , 𝐹)β€˜π‘₯))) ↔ (πœ‘ β†’ (seq𝑀( Β· , 𝐹)β€˜π‘›) = ((seq𝑀( Β· , 𝐹)β€˜π‘) Β· (seq(𝑁 + 1)( Β· , 𝐹)β€˜π‘›)))))
35 fveq2 6885 . . . . . 6 (π‘₯ = (𝑛 + 1) β†’ (seq𝑀( Β· , 𝐹)β€˜π‘₯) = (seq𝑀( Β· , 𝐹)β€˜(𝑛 + 1)))
36 fveq2 6885 . . . . . . 7 (π‘₯ = (𝑛 + 1) β†’ (seq(𝑁 + 1)( Β· , 𝐹)β€˜π‘₯) = (seq(𝑁 + 1)( Β· , 𝐹)β€˜(𝑛 + 1)))
3736oveq2d 7421 . . . . . 6 (π‘₯ = (𝑛 + 1) β†’ ((seq𝑀( Β· , 𝐹)β€˜π‘) Β· (seq(𝑁 + 1)( Β· , 𝐹)β€˜π‘₯)) = ((seq𝑀( Β· , 𝐹)β€˜π‘) Β· (seq(𝑁 + 1)( Β· , 𝐹)β€˜(𝑛 + 1))))
3835, 37eqeq12d 2742 . . . . 5 (π‘₯ = (𝑛 + 1) β†’ ((seq𝑀( Β· , 𝐹)β€˜π‘₯) = ((seq𝑀( Β· , 𝐹)β€˜π‘) Β· (seq(𝑁 + 1)( Β· , 𝐹)β€˜π‘₯)) ↔ (seq𝑀( Β· , 𝐹)β€˜(𝑛 + 1)) = ((seq𝑀( Β· , 𝐹)β€˜π‘) Β· (seq(𝑁 + 1)( Β· , 𝐹)β€˜(𝑛 + 1)))))
3938imbi2d 340 . . . 4 (π‘₯ = (𝑛 + 1) β†’ ((πœ‘ β†’ (seq𝑀( Β· , 𝐹)β€˜π‘₯) = ((seq𝑀( Β· , 𝐹)β€˜π‘) Β· (seq(𝑁 + 1)( Β· , 𝐹)β€˜π‘₯))) ↔ (πœ‘ β†’ (seq𝑀( Β· , 𝐹)β€˜(𝑛 + 1)) = ((seq𝑀( Β· , 𝐹)β€˜π‘) Β· (seq(𝑁 + 1)( Β· , 𝐹)β€˜(𝑛 + 1))))))
40 fveq2 6885 . . . . . 6 (π‘₯ = π‘˜ β†’ (seq𝑀( Β· , 𝐹)β€˜π‘₯) = (seq𝑀( Β· , 𝐹)β€˜π‘˜))
41 fveq2 6885 . . . . . . 7 (π‘₯ = π‘˜ β†’ (seq(𝑁 + 1)( Β· , 𝐹)β€˜π‘₯) = (seq(𝑁 + 1)( Β· , 𝐹)β€˜π‘˜))
4241oveq2d 7421 . . . . . 6 (π‘₯ = π‘˜ β†’ ((seq𝑀( Β· , 𝐹)β€˜π‘) Β· (seq(𝑁 + 1)( Β· , 𝐹)β€˜π‘₯)) = ((seq𝑀( Β· , 𝐹)β€˜π‘) Β· (seq(𝑁 + 1)( Β· , 𝐹)β€˜π‘˜)))
4340, 42eqeq12d 2742 . . . . 5 (π‘₯ = π‘˜ β†’ ((seq𝑀( Β· , 𝐹)β€˜π‘₯) = ((seq𝑀( Β· , 𝐹)β€˜π‘) Β· (seq(𝑁 + 1)( Β· , 𝐹)β€˜π‘₯)) ↔ (seq𝑀( Β· , 𝐹)β€˜π‘˜) = ((seq𝑀( Β· , 𝐹)β€˜π‘) Β· (seq(𝑁 + 1)( Β· , 𝐹)β€˜π‘˜))))
4443imbi2d 340 . . . 4 (π‘₯ = π‘˜ β†’ ((πœ‘ β†’ (seq𝑀( Β· , 𝐹)β€˜π‘₯) = ((seq𝑀( Β· , 𝐹)β€˜π‘) Β· (seq(𝑁 + 1)( Β· , 𝐹)β€˜π‘₯))) ↔ (πœ‘ β†’ (seq𝑀( Β· , 𝐹)β€˜π‘˜) = ((seq𝑀( Β· , 𝐹)β€˜π‘) Β· (seq(𝑁 + 1)( Β· , 𝐹)β€˜π‘˜)))))
459adantr 480 . . . . . . 7 ((πœ‘ ∧ (𝑁 + 1) ∈ β„€) β†’ 𝑁 ∈ (β„€β‰₯β€˜π‘€))
46 seqp1 13987 . . . . . . 7 (𝑁 ∈ (β„€β‰₯β€˜π‘€) β†’ (seq𝑀( Β· , 𝐹)β€˜(𝑁 + 1)) = ((seq𝑀( Β· , 𝐹)β€˜π‘) Β· (πΉβ€˜(𝑁 + 1))))
4745, 46syl 17 . . . . . 6 ((πœ‘ ∧ (𝑁 + 1) ∈ β„€) β†’ (seq𝑀( Β· , 𝐹)β€˜(𝑁 + 1)) = ((seq𝑀( Β· , 𝐹)β€˜π‘) Β· (πΉβ€˜(𝑁 + 1))))
48 seq1 13985 . . . . . . . 8 ((𝑁 + 1) ∈ β„€ β†’ (seq(𝑁 + 1)( Β· , 𝐹)β€˜(𝑁 + 1)) = (πΉβ€˜(𝑁 + 1)))
4948adantl 481 . . . . . . 7 ((πœ‘ ∧ (𝑁 + 1) ∈ β„€) β†’ (seq(𝑁 + 1)( Β· , 𝐹)β€˜(𝑁 + 1)) = (πΉβ€˜(𝑁 + 1)))
5049oveq2d 7421 . . . . . 6 ((πœ‘ ∧ (𝑁 + 1) ∈ β„€) β†’ ((seq𝑀( Β· , 𝐹)β€˜π‘) Β· (seq(𝑁 + 1)( Β· , 𝐹)β€˜(𝑁 + 1))) = ((seq𝑀( Β· , 𝐹)β€˜π‘) Β· (πΉβ€˜(𝑁 + 1))))
5147, 50eqtr4d 2769 . . . . 5 ((πœ‘ ∧ (𝑁 + 1) ∈ β„€) β†’ (seq𝑀( Β· , 𝐹)β€˜(𝑁 + 1)) = ((seq𝑀( Β· , 𝐹)β€˜π‘) Β· (seq(𝑁 + 1)( Β· , 𝐹)β€˜(𝑁 + 1))))
5251expcom 413 . . . 4 ((𝑁 + 1) ∈ β„€ β†’ (πœ‘ β†’ (seq𝑀( Β· , 𝐹)β€˜(𝑁 + 1)) = ((seq𝑀( Β· , 𝐹)β€˜π‘) Β· (seq(𝑁 + 1)( Β· , 𝐹)β€˜(𝑁 + 1)))))
5319sselda 3977 . . . . . . . . . 10 ((πœ‘ ∧ 𝑛 ∈ (β„€β‰₯β€˜(𝑁 + 1))) β†’ 𝑛 ∈ (β„€β‰₯β€˜π‘€))
54 seqp1 13987 . . . . . . . . . 10 (𝑛 ∈ (β„€β‰₯β€˜π‘€) β†’ (seq𝑀( Β· , 𝐹)β€˜(𝑛 + 1)) = ((seq𝑀( Β· , 𝐹)β€˜π‘›) Β· (πΉβ€˜(𝑛 + 1))))
5553, 54syl 17 . . . . . . . . 9 ((πœ‘ ∧ 𝑛 ∈ (β„€β‰₯β€˜(𝑁 + 1))) β†’ (seq𝑀( Β· , 𝐹)β€˜(𝑛 + 1)) = ((seq𝑀( Β· , 𝐹)β€˜π‘›) Β· (πΉβ€˜(𝑛 + 1))))
5655adantr 480 . . . . . . . 8 (((πœ‘ ∧ 𝑛 ∈ (β„€β‰₯β€˜(𝑁 + 1))) ∧ (seq𝑀( Β· , 𝐹)β€˜π‘›) = ((seq𝑀( Β· , 𝐹)β€˜π‘) Β· (seq(𝑁 + 1)( Β· , 𝐹)β€˜π‘›))) β†’ (seq𝑀( Β· , 𝐹)β€˜(𝑛 + 1)) = ((seq𝑀( Β· , 𝐹)β€˜π‘›) Β· (πΉβ€˜(𝑛 + 1))))
57 oveq1 7412 . . . . . . . . 9 ((seq𝑀( Β· , 𝐹)β€˜π‘›) = ((seq𝑀( Β· , 𝐹)β€˜π‘) Β· (seq(𝑁 + 1)( Β· , 𝐹)β€˜π‘›)) β†’ ((seq𝑀( Β· , 𝐹)β€˜π‘›) Β· (πΉβ€˜(𝑛 + 1))) = (((seq𝑀( Β· , 𝐹)β€˜π‘) Β· (seq(𝑁 + 1)( Β· , 𝐹)β€˜π‘›)) Β· (πΉβ€˜(𝑛 + 1))))
5857adantl 481 . . . . . . . 8 (((πœ‘ ∧ 𝑛 ∈ (β„€β‰₯β€˜(𝑁 + 1))) ∧ (seq𝑀( Β· , 𝐹)β€˜π‘›) = ((seq𝑀( Β· , 𝐹)β€˜π‘) Β· (seq(𝑁 + 1)( Β· , 𝐹)β€˜π‘›))) β†’ ((seq𝑀( Β· , 𝐹)β€˜π‘›) Β· (πΉβ€˜(𝑛 + 1))) = (((seq𝑀( Β· , 𝐹)β€˜π‘) Β· (seq(𝑁 + 1)( Β· , 𝐹)β€˜π‘›)) Β· (πΉβ€˜(𝑛 + 1))))
5914adantr 480 . . . . . . . . . . 11 ((πœ‘ ∧ 𝑛 ∈ (β„€β‰₯β€˜(𝑁 + 1))) β†’ (seq𝑀( Β· , 𝐹)β€˜π‘) ∈ β„‚)
6023ffvelcdmda 7080 . . . . . . . . . . 11 ((πœ‘ ∧ 𝑛 ∈ (β„€β‰₯β€˜(𝑁 + 1))) β†’ (seq(𝑁 + 1)( Β· , 𝐹)β€˜π‘›) ∈ β„‚)
61 peano2uz 12889 . . . . . . . . . . . . . 14 (𝑛 ∈ (β„€β‰₯β€˜π‘€) β†’ (𝑛 + 1) ∈ (β„€β‰₯β€˜π‘€))
6261, 2eleqtrrdi 2838 . . . . . . . . . . . . 13 (𝑛 ∈ (β„€β‰₯β€˜π‘€) β†’ (𝑛 + 1) ∈ 𝑍)
6353, 62syl 17 . . . . . . . . . . . 12 ((πœ‘ ∧ 𝑛 ∈ (β„€β‰₯β€˜(𝑁 + 1))) β†’ (𝑛 + 1) ∈ 𝑍)
6412ralrimiva 3140 . . . . . . . . . . . . 13 (πœ‘ β†’ βˆ€π‘˜ ∈ 𝑍 (πΉβ€˜π‘˜) ∈ β„‚)
65 fveq2 6885 . . . . . . . . . . . . . . 15 (π‘˜ = (𝑛 + 1) β†’ (πΉβ€˜π‘˜) = (πΉβ€˜(𝑛 + 1)))
6665eleq1d 2812 . . . . . . . . . . . . . 14 (π‘˜ = (𝑛 + 1) β†’ ((πΉβ€˜π‘˜) ∈ β„‚ ↔ (πΉβ€˜(𝑛 + 1)) ∈ β„‚))
6766rspcv 3602 . . . . . . . . . . . . 13 ((𝑛 + 1) ∈ 𝑍 β†’ (βˆ€π‘˜ ∈ 𝑍 (πΉβ€˜π‘˜) ∈ β„‚ β†’ (πΉβ€˜(𝑛 + 1)) ∈ β„‚))
6864, 67mpan9 506 . . . . . . . . . . . 12 ((πœ‘ ∧ (𝑛 + 1) ∈ 𝑍) β†’ (πΉβ€˜(𝑛 + 1)) ∈ β„‚)
6963, 68syldan 590 . . . . . . . . . . 11 ((πœ‘ ∧ 𝑛 ∈ (β„€β‰₯β€˜(𝑁 + 1))) β†’ (πΉβ€˜(𝑛 + 1)) ∈ β„‚)
7059, 60, 69mulassd 11241 . . . . . . . . . 10 ((πœ‘ ∧ 𝑛 ∈ (β„€β‰₯β€˜(𝑁 + 1))) β†’ (((seq𝑀( Β· , 𝐹)β€˜π‘) Β· (seq(𝑁 + 1)( Β· , 𝐹)β€˜π‘›)) Β· (πΉβ€˜(𝑛 + 1))) = ((seq𝑀( Β· , 𝐹)β€˜π‘) Β· ((seq(𝑁 + 1)( Β· , 𝐹)β€˜π‘›) Β· (πΉβ€˜(𝑛 + 1)))))
7170adantr 480 . . . . . . . . 9 (((πœ‘ ∧ 𝑛 ∈ (β„€β‰₯β€˜(𝑁 + 1))) ∧ (seq𝑀( Β· , 𝐹)β€˜π‘›) = ((seq𝑀( Β· , 𝐹)β€˜π‘) Β· (seq(𝑁 + 1)( Β· , 𝐹)β€˜π‘›))) β†’ (((seq𝑀( Β· , 𝐹)β€˜π‘) Β· (seq(𝑁 + 1)( Β· , 𝐹)β€˜π‘›)) Β· (πΉβ€˜(𝑛 + 1))) = ((seq𝑀( Β· , 𝐹)β€˜π‘) Β· ((seq(𝑁 + 1)( Β· , 𝐹)β€˜π‘›) Β· (πΉβ€˜(𝑛 + 1)))))
72 seqp1 13987 . . . . . . . . . . . 12 (𝑛 ∈ (β„€β‰₯β€˜(𝑁 + 1)) β†’ (seq(𝑁 + 1)( Β· , 𝐹)β€˜(𝑛 + 1)) = ((seq(𝑁 + 1)( Β· , 𝐹)β€˜π‘›) Β· (πΉβ€˜(𝑛 + 1))))
7372adantl 481 . . . . . . . . . . 11 ((πœ‘ ∧ 𝑛 ∈ (β„€β‰₯β€˜(𝑁 + 1))) β†’ (seq(𝑁 + 1)( Β· , 𝐹)β€˜(𝑛 + 1)) = ((seq(𝑁 + 1)( Β· , 𝐹)β€˜π‘›) Β· (πΉβ€˜(𝑛 + 1))))
7473oveq2d 7421 . . . . . . . . . 10 ((πœ‘ ∧ 𝑛 ∈ (β„€β‰₯β€˜(𝑁 + 1))) β†’ ((seq𝑀( Β· , 𝐹)β€˜π‘) Β· (seq(𝑁 + 1)( Β· , 𝐹)β€˜(𝑛 + 1))) = ((seq𝑀( Β· , 𝐹)β€˜π‘) Β· ((seq(𝑁 + 1)( Β· , 𝐹)β€˜π‘›) Β· (πΉβ€˜(𝑛 + 1)))))
7574adantr 480 . . . . . . . . 9 (((πœ‘ ∧ 𝑛 ∈ (β„€β‰₯β€˜(𝑁 + 1))) ∧ (seq𝑀( Β· , 𝐹)β€˜π‘›) = ((seq𝑀( Β· , 𝐹)β€˜π‘) Β· (seq(𝑁 + 1)( Β· , 𝐹)β€˜π‘›))) β†’ ((seq𝑀( Β· , 𝐹)β€˜π‘) Β· (seq(𝑁 + 1)( Β· , 𝐹)β€˜(𝑛 + 1))) = ((seq𝑀( Β· , 𝐹)β€˜π‘) Β· ((seq(𝑁 + 1)( Β· , 𝐹)β€˜π‘›) Β· (πΉβ€˜(𝑛 + 1)))))
7671, 75eqtr4d 2769 . . . . . . . 8 (((πœ‘ ∧ 𝑛 ∈ (β„€β‰₯β€˜(𝑁 + 1))) ∧ (seq𝑀( Β· , 𝐹)β€˜π‘›) = ((seq𝑀( Β· , 𝐹)β€˜π‘) Β· (seq(𝑁 + 1)( Β· , 𝐹)β€˜π‘›))) β†’ (((seq𝑀( Β· , 𝐹)β€˜π‘) Β· (seq(𝑁 + 1)( Β· , 𝐹)β€˜π‘›)) Β· (πΉβ€˜(𝑛 + 1))) = ((seq𝑀( Β· , 𝐹)β€˜π‘) Β· (seq(𝑁 + 1)( Β· , 𝐹)β€˜(𝑛 + 1))))
7756, 58, 763eqtrd 2770 . . . . . . 7 (((πœ‘ ∧ 𝑛 ∈ (β„€β‰₯β€˜(𝑁 + 1))) ∧ (seq𝑀( Β· , 𝐹)β€˜π‘›) = ((seq𝑀( Β· , 𝐹)β€˜π‘) Β· (seq(𝑁 + 1)( Β· , 𝐹)β€˜π‘›))) β†’ (seq𝑀( Β· , 𝐹)β€˜(𝑛 + 1)) = ((seq𝑀( Β· , 𝐹)β€˜π‘) Β· (seq(𝑁 + 1)( Β· , 𝐹)β€˜(𝑛 + 1))))
7877exp31 419 . . . . . 6 (πœ‘ β†’ (𝑛 ∈ (β„€β‰₯β€˜(𝑁 + 1)) β†’ ((seq𝑀( Β· , 𝐹)β€˜π‘›) = ((seq𝑀( Β· , 𝐹)β€˜π‘) Β· (seq(𝑁 + 1)( Β· , 𝐹)β€˜π‘›)) β†’ (seq𝑀( Β· , 𝐹)β€˜(𝑛 + 1)) = ((seq𝑀( Β· , 𝐹)β€˜π‘) Β· (seq(𝑁 + 1)( Β· , 𝐹)β€˜(𝑛 + 1))))))
7978com12 32 . . . . 5 (𝑛 ∈ (β„€β‰₯β€˜(𝑁 + 1)) β†’ (πœ‘ β†’ ((seq𝑀( Β· , 𝐹)β€˜π‘›) = ((seq𝑀( Β· , 𝐹)β€˜π‘) Β· (seq(𝑁 + 1)( Β· , 𝐹)β€˜π‘›)) β†’ (seq𝑀( Β· , 𝐹)β€˜(𝑛 + 1)) = ((seq𝑀( Β· , 𝐹)β€˜π‘) Β· (seq(𝑁 + 1)( Β· , 𝐹)β€˜(𝑛 + 1))))))
8079a2d 29 . . . 4 (𝑛 ∈ (β„€β‰₯β€˜(𝑁 + 1)) β†’ ((πœ‘ β†’ (seq𝑀( Β· , 𝐹)β€˜π‘›) = ((seq𝑀( Β· , 𝐹)β€˜π‘) Β· (seq(𝑁 + 1)( Β· , 𝐹)β€˜π‘›))) β†’ (πœ‘ β†’ (seq𝑀( Β· , 𝐹)β€˜(𝑛 + 1)) = ((seq𝑀( Β· , 𝐹)β€˜π‘) Β· (seq(𝑁 + 1)( Β· , 𝐹)β€˜(𝑛 + 1))))))
8129, 34, 39, 44, 52, 80uzind4 12894 . . 3 (π‘˜ ∈ (β„€β‰₯β€˜(𝑁 + 1)) β†’ (πœ‘ β†’ (seq𝑀( Β· , 𝐹)β€˜π‘˜) = ((seq𝑀( Β· , 𝐹)β€˜π‘) Β· (seq(𝑁 + 1)( Β· , 𝐹)β€˜π‘˜))))
8281impcom 407 . 2 ((πœ‘ ∧ π‘˜ ∈ (β„€β‰₯β€˜(𝑁 + 1))) β†’ (seq𝑀( Β· , 𝐹)β€˜π‘˜) = ((seq𝑀( Β· , 𝐹)β€˜π‘) Β· (seq(𝑁 + 1)( Β· , 𝐹)β€˜π‘˜)))
831, 7, 8, 14, 16, 24, 82climmulc2 15587 1 (πœ‘ β†’ seq𝑀( Β· , 𝐹) ⇝ ((seq𝑀( Β· , 𝐹)β€˜π‘) Β· 𝐴))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ wa 395   = wceq 1533   ∈ wcel 2098  βˆ€wral 3055  Vcvv 3468   βŠ† wss 3943   class class class wbr 5141  β€˜cfv 6537  (class class class)co 7405  β„‚cc 11110  1c1 11113   + caddc 11115   Β· cmul 11117  β„€cz 12562  β„€β‰₯cuz 12826  seqcseq 13972   ⇝ cli 15434
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2697  ax-rep 5278  ax-sep 5292  ax-nul 5299  ax-pow 5356  ax-pr 5420  ax-un 7722  ax-inf2 9638  ax-cnex 11168  ax-resscn 11169  ax-1cn 11170  ax-icn 11171  ax-addcl 11172  ax-addrcl 11173  ax-mulcl 11174  ax-mulrcl 11175  ax-mulcom 11176  ax-addass 11177  ax-mulass 11178  ax-distr 11179  ax-i2m1 11180  ax-1ne0 11181  ax-1rid 11182  ax-rnegex 11183  ax-rrecex 11184  ax-cnre 11185  ax-pre-lttri 11186  ax-pre-lttrn 11187  ax-pre-ltadd 11188  ax-pre-mulgt0 11189  ax-pre-sup 11190
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ne 2935  df-nel 3041  df-ral 3056  df-rex 3065  df-rmo 3370  df-reu 3371  df-rab 3427  df-v 3470  df-sbc 3773  df-csb 3889  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-pss 3962  df-nul 4318  df-if 4524  df-pw 4599  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-iun 4992  df-br 5142  df-opab 5204  df-mpt 5225  df-tr 5259  df-id 5567  df-eprel 5573  df-po 5581  df-so 5582  df-fr 5624  df-we 5626  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-res 5681  df-ima 5682  df-pred 6294  df-ord 6361  df-on 6362  df-lim 6363  df-suc 6364  df-iota 6489  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-riota 7361  df-ov 7408  df-oprab 7409  df-mpo 7410  df-om 7853  df-1st 7974  df-2nd 7975  df-frecs 8267  df-wrecs 8298  df-recs 8372  df-rdg 8411  df-er 8705  df-en 8942  df-dom 8943  df-sdom 8944  df-sup 9439  df-pnf 11254  df-mnf 11255  df-xr 11256  df-ltxr 11257  df-le 11258  df-sub 11450  df-neg 11451  df-div 11876  df-nn 12217  df-2 12279  df-3 12280  df-n0 12477  df-z 12563  df-uz 12827  df-rp 12981  df-fz 13491  df-seq 13973  df-exp 14033  df-cj 15052  df-re 15053  df-im 15054  df-sqrt 15188  df-abs 15189  df-clim 15438
This theorem is referenced by:  ntrivcvg  15849
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