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Theorem cvgcmp 15802
Description: A comparison test for convergence of a real infinite series. Exercise 3 of [Gleason] p. 182. (Contributed by NM, 1-May-2005.) (Revised by Mario Carneiro, 24-Mar-2014.)
Hypotheses
Ref Expression
cvgcmp.1 𝑍 = (β„€β‰₯β€˜π‘€)
cvgcmp.2 (πœ‘ β†’ 𝑁 ∈ 𝑍)
cvgcmp.3 ((πœ‘ ∧ π‘˜ ∈ 𝑍) β†’ (πΉβ€˜π‘˜) ∈ ℝ)
cvgcmp.4 ((πœ‘ ∧ π‘˜ ∈ 𝑍) β†’ (πΊβ€˜π‘˜) ∈ ℝ)
cvgcmp.5 (πœ‘ β†’ seq𝑀( + , 𝐹) ∈ dom ⇝ )
cvgcmp.6 ((πœ‘ ∧ π‘˜ ∈ (β„€β‰₯β€˜π‘)) β†’ 0 ≀ (πΊβ€˜π‘˜))
cvgcmp.7 ((πœ‘ ∧ π‘˜ ∈ (β„€β‰₯β€˜π‘)) β†’ (πΊβ€˜π‘˜) ≀ (πΉβ€˜π‘˜))
Assertion
Ref Expression
cvgcmp (πœ‘ β†’ seq𝑀( + , 𝐺) ∈ dom ⇝ )
Distinct variable groups:   π‘˜,𝐹   π‘˜,𝐺   πœ‘,π‘˜   π‘˜,𝑀   π‘˜,𝑁   π‘˜,𝑍

Proof of Theorem cvgcmp
Dummy variables 𝑛 π‘š π‘₯ are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 cvgcmp.1 . 2 𝑍 = (β„€β‰₯β€˜π‘€)
2 seqex 14008 . . 3 seq𝑀( + , 𝐺) ∈ V
32a1i 11 . 2 (πœ‘ β†’ seq𝑀( + , 𝐺) ∈ V)
4 cvgcmp.2 . . . . . . . 8 (πœ‘ β†’ 𝑁 ∈ 𝑍)
54, 1eleqtrdi 2839 . . . . . . 7 (πœ‘ β†’ 𝑁 ∈ (β„€β‰₯β€˜π‘€))
6 eluzel2 12865 . . . . . . 7 (𝑁 ∈ (β„€β‰₯β€˜π‘€) β†’ 𝑀 ∈ β„€)
75, 6syl 17 . . . . . 6 (πœ‘ β†’ 𝑀 ∈ β„€)
8 cvgcmp.5 . . . . . 6 (πœ‘ β†’ seq𝑀( + , 𝐹) ∈ dom ⇝ )
91climcau 15657 . . . . . 6 ((𝑀 ∈ β„€ ∧ seq𝑀( + , 𝐹) ∈ dom ⇝ ) β†’ βˆ€π‘₯ ∈ ℝ+ βˆƒπ‘š ∈ 𝑍 βˆ€π‘› ∈ (β„€β‰₯β€˜π‘š)(absβ€˜((seq𝑀( + , 𝐹)β€˜π‘›) βˆ’ (seq𝑀( + , 𝐹)β€˜π‘š))) < π‘₯)
107, 8, 9syl2anc 582 . . . . 5 (πœ‘ β†’ βˆ€π‘₯ ∈ ℝ+ βˆƒπ‘š ∈ 𝑍 βˆ€π‘› ∈ (β„€β‰₯β€˜π‘š)(absβ€˜((seq𝑀( + , 𝐹)β€˜π‘›) βˆ’ (seq𝑀( + , 𝐹)β€˜π‘š))) < π‘₯)
11 cvgcmp.3 . . . . . . . . . . 11 ((πœ‘ ∧ π‘˜ ∈ 𝑍) β†’ (πΉβ€˜π‘˜) ∈ ℝ)
121, 7, 11serfre 14036 . . . . . . . . . 10 (πœ‘ β†’ seq𝑀( + , 𝐹):π‘βŸΆβ„)
1312ffvelcdmda 7099 . . . . . . . . 9 ((πœ‘ ∧ 𝑛 ∈ 𝑍) β†’ (seq𝑀( + , 𝐹)β€˜π‘›) ∈ ℝ)
1413recnd 11280 . . . . . . . 8 ((πœ‘ ∧ 𝑛 ∈ 𝑍) β†’ (seq𝑀( + , 𝐹)β€˜π‘›) ∈ β„‚)
1514ralrimiva 3143 . . . . . . 7 (πœ‘ β†’ βˆ€π‘› ∈ 𝑍 (seq𝑀( + , 𝐹)β€˜π‘›) ∈ β„‚)
161r19.29uz 15337 . . . . . . . 8 ((βˆ€π‘› ∈ 𝑍 (seq𝑀( + , 𝐹)β€˜π‘›) ∈ β„‚ ∧ βˆƒπ‘š ∈ 𝑍 βˆ€π‘› ∈ (β„€β‰₯β€˜π‘š)(absβ€˜((seq𝑀( + , 𝐹)β€˜π‘›) βˆ’ (seq𝑀( + , 𝐹)β€˜π‘š))) < π‘₯) β†’ βˆƒπ‘š ∈ 𝑍 βˆ€π‘› ∈ (β„€β‰₯β€˜π‘š)((seq𝑀( + , 𝐹)β€˜π‘›) ∈ β„‚ ∧ (absβ€˜((seq𝑀( + , 𝐹)β€˜π‘›) βˆ’ (seq𝑀( + , 𝐹)β€˜π‘š))) < π‘₯))
1716ex 411 . . . . . . 7 (βˆ€π‘› ∈ 𝑍 (seq𝑀( + , 𝐹)β€˜π‘›) ∈ β„‚ β†’ (βˆƒπ‘š ∈ 𝑍 βˆ€π‘› ∈ (β„€β‰₯β€˜π‘š)(absβ€˜((seq𝑀( + , 𝐹)β€˜π‘›) βˆ’ (seq𝑀( + , 𝐹)β€˜π‘š))) < π‘₯ β†’ βˆƒπ‘š ∈ 𝑍 βˆ€π‘› ∈ (β„€β‰₯β€˜π‘š)((seq𝑀( + , 𝐹)β€˜π‘›) ∈ β„‚ ∧ (absβ€˜((seq𝑀( + , 𝐹)β€˜π‘›) βˆ’ (seq𝑀( + , 𝐹)β€˜π‘š))) < π‘₯)))
1815, 17syl 17 . . . . . 6 (πœ‘ β†’ (βˆƒπ‘š ∈ 𝑍 βˆ€π‘› ∈ (β„€β‰₯β€˜π‘š)(absβ€˜((seq𝑀( + , 𝐹)β€˜π‘›) βˆ’ (seq𝑀( + , 𝐹)β€˜π‘š))) < π‘₯ β†’ βˆƒπ‘š ∈ 𝑍 βˆ€π‘› ∈ (β„€β‰₯β€˜π‘š)((seq𝑀( + , 𝐹)β€˜π‘›) ∈ β„‚ ∧ (absβ€˜((seq𝑀( + , 𝐹)β€˜π‘›) βˆ’ (seq𝑀( + , 𝐹)β€˜π‘š))) < π‘₯)))
1918ralimdv 3166 . . . . 5 (πœ‘ β†’ (βˆ€π‘₯ ∈ ℝ+ βˆƒπ‘š ∈ 𝑍 βˆ€π‘› ∈ (β„€β‰₯β€˜π‘š)(absβ€˜((seq𝑀( + , 𝐹)β€˜π‘›) βˆ’ (seq𝑀( + , 𝐹)β€˜π‘š))) < π‘₯ β†’ βˆ€π‘₯ ∈ ℝ+ βˆƒπ‘š ∈ 𝑍 βˆ€π‘› ∈ (β„€β‰₯β€˜π‘š)((seq𝑀( + , 𝐹)β€˜π‘›) ∈ β„‚ ∧ (absβ€˜((seq𝑀( + , 𝐹)β€˜π‘›) βˆ’ (seq𝑀( + , 𝐹)β€˜π‘š))) < π‘₯)))
2010, 19mpd 15 . . . 4 (πœ‘ β†’ βˆ€π‘₯ ∈ ℝ+ βˆƒπ‘š ∈ 𝑍 βˆ€π‘› ∈ (β„€β‰₯β€˜π‘š)((seq𝑀( + , 𝐹)β€˜π‘›) ∈ β„‚ ∧ (absβ€˜((seq𝑀( + , 𝐹)β€˜π‘›) βˆ’ (seq𝑀( + , 𝐹)β€˜π‘š))) < π‘₯))
211uztrn2 12879 . . . . . . . . . . 11 ((𝑁 ∈ 𝑍 ∧ 𝑛 ∈ (β„€β‰₯β€˜π‘)) β†’ 𝑛 ∈ 𝑍)
224, 21sylan 578 . . . . . . . . . 10 ((πœ‘ ∧ 𝑛 ∈ (β„€β‰₯β€˜π‘)) β†’ 𝑛 ∈ 𝑍)
23 cvgcmp.4 . . . . . . . . . . . . 13 ((πœ‘ ∧ π‘˜ ∈ 𝑍) β†’ (πΊβ€˜π‘˜) ∈ ℝ)
241, 7, 23serfre 14036 . . . . . . . . . . . 12 (πœ‘ β†’ seq𝑀( + , 𝐺):π‘βŸΆβ„)
2524ffvelcdmda 7099 . . . . . . . . . . 11 ((πœ‘ ∧ 𝑛 ∈ 𝑍) β†’ (seq𝑀( + , 𝐺)β€˜π‘›) ∈ ℝ)
2625recnd 11280 . . . . . . . . . 10 ((πœ‘ ∧ 𝑛 ∈ 𝑍) β†’ (seq𝑀( + , 𝐺)β€˜π‘›) ∈ β„‚)
2722, 26syldan 589 . . . . . . . . 9 ((πœ‘ ∧ 𝑛 ∈ (β„€β‰₯β€˜π‘)) β†’ (seq𝑀( + , 𝐺)β€˜π‘›) ∈ β„‚)
2827ralrimiva 3143 . . . . . . . 8 (πœ‘ β†’ βˆ€π‘› ∈ (β„€β‰₯β€˜π‘)(seq𝑀( + , 𝐺)β€˜π‘›) ∈ β„‚)
2928adantr 479 . . . . . . 7 ((πœ‘ ∧ π‘₯ ∈ ℝ+) β†’ βˆ€π‘› ∈ (β„€β‰₯β€˜π‘)(seq𝑀( + , 𝐺)β€˜π‘›) ∈ β„‚)
30 simpll 765 . . . . . . . . . . . . . . . 16 (((πœ‘ ∧ π‘₯ ∈ ℝ+) ∧ (π‘š ∈ (β„€β‰₯β€˜π‘) ∧ 𝑛 ∈ (β„€β‰₯β€˜π‘š))) β†’ πœ‘)
3130, 12syl 17 . . . . . . . . . . . . . . 15 (((πœ‘ ∧ π‘₯ ∈ ℝ+) ∧ (π‘š ∈ (β„€β‰₯β€˜π‘) ∧ 𝑛 ∈ (β„€β‰₯β€˜π‘š))) β†’ seq𝑀( + , 𝐹):π‘βŸΆβ„)
3230, 4syl 17 . . . . . . . . . . . . . . . 16 (((πœ‘ ∧ π‘₯ ∈ ℝ+) ∧ (π‘š ∈ (β„€β‰₯β€˜π‘) ∧ 𝑛 ∈ (β„€β‰₯β€˜π‘š))) β†’ 𝑁 ∈ 𝑍)
33 simprl 769 . . . . . . . . . . . . . . . 16 (((πœ‘ ∧ π‘₯ ∈ ℝ+) ∧ (π‘š ∈ (β„€β‰₯β€˜π‘) ∧ 𝑛 ∈ (β„€β‰₯β€˜π‘š))) β†’ π‘š ∈ (β„€β‰₯β€˜π‘))
341uztrn2 12879 . . . . . . . . . . . . . . . 16 ((𝑁 ∈ 𝑍 ∧ π‘š ∈ (β„€β‰₯β€˜π‘)) β†’ π‘š ∈ 𝑍)
3532, 33, 34syl2anc 582 . . . . . . . . . . . . . . 15 (((πœ‘ ∧ π‘₯ ∈ ℝ+) ∧ (π‘š ∈ (β„€β‰₯β€˜π‘) ∧ 𝑛 ∈ (β„€β‰₯β€˜π‘š))) β†’ π‘š ∈ 𝑍)
3631, 35ffvelcdmd 7100 . . . . . . . . . . . . . 14 (((πœ‘ ∧ π‘₯ ∈ ℝ+) ∧ (π‘š ∈ (β„€β‰₯β€˜π‘) ∧ 𝑛 ∈ (β„€β‰₯β€˜π‘š))) β†’ (seq𝑀( + , 𝐹)β€˜π‘š) ∈ ℝ)
37 eqid 2728 . . . . . . . . . . . . . . . . . 18 (β„€β‰₯β€˜π‘) = (β„€β‰₯β€˜π‘)
3837uztrn2 12879 . . . . . . . . . . . . . . . . 17 ((π‘š ∈ (β„€β‰₯β€˜π‘) ∧ 𝑛 ∈ (β„€β‰₯β€˜π‘š)) β†’ 𝑛 ∈ (β„€β‰₯β€˜π‘))
3938adantl 480 . . . . . . . . . . . . . . . 16 (((πœ‘ ∧ π‘₯ ∈ ℝ+) ∧ (π‘š ∈ (β„€β‰₯β€˜π‘) ∧ 𝑛 ∈ (β„€β‰₯β€˜π‘š))) β†’ 𝑛 ∈ (β„€β‰₯β€˜π‘))
4032, 39, 21syl2anc 582 . . . . . . . . . . . . . . 15 (((πœ‘ ∧ π‘₯ ∈ ℝ+) ∧ (π‘š ∈ (β„€β‰₯β€˜π‘) ∧ 𝑛 ∈ (β„€β‰₯β€˜π‘š))) β†’ 𝑛 ∈ 𝑍)
4130, 40, 13syl2anc 582 . . . . . . . . . . . . . 14 (((πœ‘ ∧ π‘₯ ∈ ℝ+) ∧ (π‘š ∈ (β„€β‰₯β€˜π‘) ∧ 𝑛 ∈ (β„€β‰₯β€˜π‘š))) β†’ (seq𝑀( + , 𝐹)β€˜π‘›) ∈ ℝ)
4230, 40, 25syl2anc 582 . . . . . . . . . . . . . . 15 (((πœ‘ ∧ π‘₯ ∈ ℝ+) ∧ (π‘š ∈ (β„€β‰₯β€˜π‘) ∧ 𝑛 ∈ (β„€β‰₯β€˜π‘š))) β†’ (seq𝑀( + , 𝐺)β€˜π‘›) ∈ ℝ)
4330, 24syl 17 . . . . . . . . . . . . . . . 16 (((πœ‘ ∧ π‘₯ ∈ ℝ+) ∧ (π‘š ∈ (β„€β‰₯β€˜π‘) ∧ 𝑛 ∈ (β„€β‰₯β€˜π‘š))) β†’ seq𝑀( + , 𝐺):π‘βŸΆβ„)
4443, 35ffvelcdmd 7100 . . . . . . . . . . . . . . 15 (((πœ‘ ∧ π‘₯ ∈ ℝ+) ∧ (π‘š ∈ (β„€β‰₯β€˜π‘) ∧ 𝑛 ∈ (β„€β‰₯β€˜π‘š))) β†’ (seq𝑀( + , 𝐺)β€˜π‘š) ∈ ℝ)
4542, 44resubcld 11680 . . . . . . . . . . . . . 14 (((πœ‘ ∧ π‘₯ ∈ ℝ+) ∧ (π‘š ∈ (β„€β‰₯β€˜π‘) ∧ 𝑛 ∈ (β„€β‰₯β€˜π‘š))) β†’ ((seq𝑀( + , 𝐺)β€˜π‘›) βˆ’ (seq𝑀( + , 𝐺)β€˜π‘š)) ∈ ℝ)
4635, 1eleqtrdi 2839 . . . . . . . . . . . . . . . . . 18 (((πœ‘ ∧ π‘₯ ∈ ℝ+) ∧ (π‘š ∈ (β„€β‰₯β€˜π‘) ∧ 𝑛 ∈ (β„€β‰₯β€˜π‘š))) β†’ π‘š ∈ (β„€β‰₯β€˜π‘€))
47 simprr 771 . . . . . . . . . . . . . . . . . 18 (((πœ‘ ∧ π‘₯ ∈ ℝ+) ∧ (π‘š ∈ (β„€β‰₯β€˜π‘) ∧ 𝑛 ∈ (β„€β‰₯β€˜π‘š))) β†’ 𝑛 ∈ (β„€β‰₯β€˜π‘š))
48 elfzuz 13537 . . . . . . . . . . . . . . . . . . . 20 (π‘˜ ∈ (𝑀...𝑛) β†’ π‘˜ ∈ (β„€β‰₯β€˜π‘€))
4948, 1eleqtrrdi 2840 . . . . . . . . . . . . . . . . . . 19 (π‘˜ ∈ (𝑀...𝑛) β†’ π‘˜ ∈ 𝑍)
50 fveq2 6902 . . . . . . . . . . . . . . . . . . . . . . 23 (π‘š = π‘˜ β†’ (πΉβ€˜π‘š) = (πΉβ€˜π‘˜))
51 fveq2 6902 . . . . . . . . . . . . . . . . . . . . . . 23 (π‘š = π‘˜ β†’ (πΊβ€˜π‘š) = (πΊβ€˜π‘˜))
5250, 51oveq12d 7444 . . . . . . . . . . . . . . . . . . . . . 22 (π‘š = π‘˜ β†’ ((πΉβ€˜π‘š) βˆ’ (πΊβ€˜π‘š)) = ((πΉβ€˜π‘˜) βˆ’ (πΊβ€˜π‘˜)))
53 eqid 2728 . . . . . . . . . . . . . . . . . . . . . 22 (π‘š ∈ 𝑍 ↦ ((πΉβ€˜π‘š) βˆ’ (πΊβ€˜π‘š))) = (π‘š ∈ 𝑍 ↦ ((πΉβ€˜π‘š) βˆ’ (πΊβ€˜π‘š)))
54 ovex 7459 . . . . . . . . . . . . . . . . . . . . . 22 ((πΉβ€˜π‘˜) βˆ’ (πΊβ€˜π‘˜)) ∈ V
5552, 53, 54fvmpt 7010 . . . . . . . . . . . . . . . . . . . . 21 (π‘˜ ∈ 𝑍 β†’ ((π‘š ∈ 𝑍 ↦ ((πΉβ€˜π‘š) βˆ’ (πΊβ€˜π‘š)))β€˜π‘˜) = ((πΉβ€˜π‘˜) βˆ’ (πΊβ€˜π‘˜)))
5655adantl 480 . . . . . . . . . . . . . . . . . . . 20 ((πœ‘ ∧ π‘˜ ∈ 𝑍) β†’ ((π‘š ∈ 𝑍 ↦ ((πΉβ€˜π‘š) βˆ’ (πΊβ€˜π‘š)))β€˜π‘˜) = ((πΉβ€˜π‘˜) βˆ’ (πΊβ€˜π‘˜)))
5711, 23resubcld 11680 . . . . . . . . . . . . . . . . . . . 20 ((πœ‘ ∧ π‘˜ ∈ 𝑍) β†’ ((πΉβ€˜π‘˜) βˆ’ (πΊβ€˜π‘˜)) ∈ ℝ)
5856, 57eqeltrd 2829 . . . . . . . . . . . . . . . . . . 19 ((πœ‘ ∧ π‘˜ ∈ 𝑍) β†’ ((π‘š ∈ 𝑍 ↦ ((πΉβ€˜π‘š) βˆ’ (πΊβ€˜π‘š)))β€˜π‘˜) ∈ ℝ)
5930, 49, 58syl2an 594 . . . . . . . . . . . . . . . . . 18 ((((πœ‘ ∧ π‘₯ ∈ ℝ+) ∧ (π‘š ∈ (β„€β‰₯β€˜π‘) ∧ 𝑛 ∈ (β„€β‰₯β€˜π‘š))) ∧ π‘˜ ∈ (𝑀...𝑛)) β†’ ((π‘š ∈ 𝑍 ↦ ((πΉβ€˜π‘š) βˆ’ (πΊβ€˜π‘š)))β€˜π‘˜) ∈ ℝ)
60 elfzuz 13537 . . . . . . . . . . . . . . . . . . 19 (π‘˜ ∈ ((π‘š + 1)...𝑛) β†’ π‘˜ ∈ (β„€β‰₯β€˜(π‘š + 1)))
61 peano2uz 12923 . . . . . . . . . . . . . . . . . . . . . 22 (π‘š ∈ (β„€β‰₯β€˜π‘) β†’ (π‘š + 1) ∈ (β„€β‰₯β€˜π‘))
6233, 61syl 17 . . . . . . . . . . . . . . . . . . . . 21 (((πœ‘ ∧ π‘₯ ∈ ℝ+) ∧ (π‘š ∈ (β„€β‰₯β€˜π‘) ∧ 𝑛 ∈ (β„€β‰₯β€˜π‘š))) β†’ (π‘š + 1) ∈ (β„€β‰₯β€˜π‘))
6337uztrn2 12879 . . . . . . . . . . . . . . . . . . . . 21 (((π‘š + 1) ∈ (β„€β‰₯β€˜π‘) ∧ π‘˜ ∈ (β„€β‰₯β€˜(π‘š + 1))) β†’ π‘˜ ∈ (β„€β‰₯β€˜π‘))
6462, 63sylan 578 . . . . . . . . . . . . . . . . . . . 20 ((((πœ‘ ∧ π‘₯ ∈ ℝ+) ∧ (π‘š ∈ (β„€β‰₯β€˜π‘) ∧ 𝑛 ∈ (β„€β‰₯β€˜π‘š))) ∧ π‘˜ ∈ (β„€β‰₯β€˜(π‘š + 1))) β†’ π‘˜ ∈ (β„€β‰₯β€˜π‘))
65 cvgcmp.7 . . . . . . . . . . . . . . . . . . . . . 22 ((πœ‘ ∧ π‘˜ ∈ (β„€β‰₯β€˜π‘)) β†’ (πΊβ€˜π‘˜) ≀ (πΉβ€˜π‘˜))
661uztrn2 12879 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝑁 ∈ 𝑍 ∧ π‘˜ ∈ (β„€β‰₯β€˜π‘)) β†’ π‘˜ ∈ 𝑍)
674, 66sylan 578 . . . . . . . . . . . . . . . . . . . . . . 23 ((πœ‘ ∧ π‘˜ ∈ (β„€β‰₯β€˜π‘)) β†’ π‘˜ ∈ 𝑍)
6811, 23subge0d 11842 . . . . . . . . . . . . . . . . . . . . . . 23 ((πœ‘ ∧ π‘˜ ∈ 𝑍) β†’ (0 ≀ ((πΉβ€˜π‘˜) βˆ’ (πΊβ€˜π‘˜)) ↔ (πΊβ€˜π‘˜) ≀ (πΉβ€˜π‘˜)))
6967, 68syldan 589 . . . . . . . . . . . . . . . . . . . . . 22 ((πœ‘ ∧ π‘˜ ∈ (β„€β‰₯β€˜π‘)) β†’ (0 ≀ ((πΉβ€˜π‘˜) βˆ’ (πΊβ€˜π‘˜)) ↔ (πΊβ€˜π‘˜) ≀ (πΉβ€˜π‘˜)))
7065, 69mpbird 256 . . . . . . . . . . . . . . . . . . . . 21 ((πœ‘ ∧ π‘˜ ∈ (β„€β‰₯β€˜π‘)) β†’ 0 ≀ ((πΉβ€˜π‘˜) βˆ’ (πΊβ€˜π‘˜)))
7167, 55syl 17 . . . . . . . . . . . . . . . . . . . . 21 ((πœ‘ ∧ π‘˜ ∈ (β„€β‰₯β€˜π‘)) β†’ ((π‘š ∈ 𝑍 ↦ ((πΉβ€˜π‘š) βˆ’ (πΊβ€˜π‘š)))β€˜π‘˜) = ((πΉβ€˜π‘˜) βˆ’ (πΊβ€˜π‘˜)))
7270, 71breqtrrd 5180 . . . . . . . . . . . . . . . . . . . 20 ((πœ‘ ∧ π‘˜ ∈ (β„€β‰₯β€˜π‘)) β†’ 0 ≀ ((π‘š ∈ 𝑍 ↦ ((πΉβ€˜π‘š) βˆ’ (πΊβ€˜π‘š)))β€˜π‘˜))
7330, 64, 72syl2an2r 683 . . . . . . . . . . . . . . . . . . 19 ((((πœ‘ ∧ π‘₯ ∈ ℝ+) ∧ (π‘š ∈ (β„€β‰₯β€˜π‘) ∧ 𝑛 ∈ (β„€β‰₯β€˜π‘š))) ∧ π‘˜ ∈ (β„€β‰₯β€˜(π‘š + 1))) β†’ 0 ≀ ((π‘š ∈ 𝑍 ↦ ((πΉβ€˜π‘š) βˆ’ (πΊβ€˜π‘š)))β€˜π‘˜))
7460, 73sylan2 591 . . . . . . . . . . . . . . . . . 18 ((((πœ‘ ∧ π‘₯ ∈ ℝ+) ∧ (π‘š ∈ (β„€β‰₯β€˜π‘) ∧ 𝑛 ∈ (β„€β‰₯β€˜π‘š))) ∧ π‘˜ ∈ ((π‘š + 1)...𝑛)) β†’ 0 ≀ ((π‘š ∈ 𝑍 ↦ ((πΉβ€˜π‘š) βˆ’ (πΊβ€˜π‘š)))β€˜π‘˜))
7546, 47, 59, 74sermono 14039 . . . . . . . . . . . . . . . . 17 (((πœ‘ ∧ π‘₯ ∈ ℝ+) ∧ (π‘š ∈ (β„€β‰₯β€˜π‘) ∧ 𝑛 ∈ (β„€β‰₯β€˜π‘š))) β†’ (seq𝑀( + , (π‘š ∈ 𝑍 ↦ ((πΉβ€˜π‘š) βˆ’ (πΊβ€˜π‘š))))β€˜π‘š) ≀ (seq𝑀( + , (π‘š ∈ 𝑍 ↦ ((πΉβ€˜π‘š) βˆ’ (πΊβ€˜π‘š))))β€˜π‘›))
76 elfzuz 13537 . . . . . . . . . . . . . . . . . . . 20 (π‘˜ ∈ (𝑀...π‘š) β†’ π‘˜ ∈ (β„€β‰₯β€˜π‘€))
7776, 1eleqtrrdi 2840 . . . . . . . . . . . . . . . . . . 19 (π‘˜ ∈ (𝑀...π‘š) β†’ π‘˜ ∈ 𝑍)
7811recnd 11280 . . . . . . . . . . . . . . . . . . 19 ((πœ‘ ∧ π‘˜ ∈ 𝑍) β†’ (πΉβ€˜π‘˜) ∈ β„‚)
7930, 77, 78syl2an 594 . . . . . . . . . . . . . . . . . 18 ((((πœ‘ ∧ π‘₯ ∈ ℝ+) ∧ (π‘š ∈ (β„€β‰₯β€˜π‘) ∧ 𝑛 ∈ (β„€β‰₯β€˜π‘š))) ∧ π‘˜ ∈ (𝑀...π‘š)) β†’ (πΉβ€˜π‘˜) ∈ β„‚)
8023recnd 11280 . . . . . . . . . . . . . . . . . . 19 ((πœ‘ ∧ π‘˜ ∈ 𝑍) β†’ (πΊβ€˜π‘˜) ∈ β„‚)
8130, 77, 80syl2an 594 . . . . . . . . . . . . . . . . . 18 ((((πœ‘ ∧ π‘₯ ∈ ℝ+) ∧ (π‘š ∈ (β„€β‰₯β€˜π‘) ∧ 𝑛 ∈ (β„€β‰₯β€˜π‘š))) ∧ π‘˜ ∈ (𝑀...π‘š)) β†’ (πΊβ€˜π‘˜) ∈ β„‚)
8230, 77, 56syl2an 594 . . . . . . . . . . . . . . . . . 18 ((((πœ‘ ∧ π‘₯ ∈ ℝ+) ∧ (π‘š ∈ (β„€β‰₯β€˜π‘) ∧ 𝑛 ∈ (β„€β‰₯β€˜π‘š))) ∧ π‘˜ ∈ (𝑀...π‘š)) β†’ ((π‘š ∈ 𝑍 ↦ ((πΉβ€˜π‘š) βˆ’ (πΊβ€˜π‘š)))β€˜π‘˜) = ((πΉβ€˜π‘˜) βˆ’ (πΊβ€˜π‘˜)))
8346, 79, 81, 82sersub 14050 . . . . . . . . . . . . . . . . 17 (((πœ‘ ∧ π‘₯ ∈ ℝ+) ∧ (π‘š ∈ (β„€β‰₯β€˜π‘) ∧ 𝑛 ∈ (β„€β‰₯β€˜π‘š))) β†’ (seq𝑀( + , (π‘š ∈ 𝑍 ↦ ((πΉβ€˜π‘š) βˆ’ (πΊβ€˜π‘š))))β€˜π‘š) = ((seq𝑀( + , 𝐹)β€˜π‘š) βˆ’ (seq𝑀( + , 𝐺)β€˜π‘š)))
8440, 1eleqtrdi 2839 . . . . . . . . . . . . . . . . . 18 (((πœ‘ ∧ π‘₯ ∈ ℝ+) ∧ (π‘š ∈ (β„€β‰₯β€˜π‘) ∧ 𝑛 ∈ (β„€β‰₯β€˜π‘š))) β†’ 𝑛 ∈ (β„€β‰₯β€˜π‘€))
8530, 49, 78syl2an 594 . . . . . . . . . . . . . . . . . 18 ((((πœ‘ ∧ π‘₯ ∈ ℝ+) ∧ (π‘š ∈ (β„€β‰₯β€˜π‘) ∧ 𝑛 ∈ (β„€β‰₯β€˜π‘š))) ∧ π‘˜ ∈ (𝑀...𝑛)) β†’ (πΉβ€˜π‘˜) ∈ β„‚)
8630, 49, 80syl2an 594 . . . . . . . . . . . . . . . . . 18 ((((πœ‘ ∧ π‘₯ ∈ ℝ+) ∧ (π‘š ∈ (β„€β‰₯β€˜π‘) ∧ 𝑛 ∈ (β„€β‰₯β€˜π‘š))) ∧ π‘˜ ∈ (𝑀...𝑛)) β†’ (πΊβ€˜π‘˜) ∈ β„‚)
8730, 49, 56syl2an 594 . . . . . . . . . . . . . . . . . 18 ((((πœ‘ ∧ π‘₯ ∈ ℝ+) ∧ (π‘š ∈ (β„€β‰₯β€˜π‘) ∧ 𝑛 ∈ (β„€β‰₯β€˜π‘š))) ∧ π‘˜ ∈ (𝑀...𝑛)) β†’ ((π‘š ∈ 𝑍 ↦ ((πΉβ€˜π‘š) βˆ’ (πΊβ€˜π‘š)))β€˜π‘˜) = ((πΉβ€˜π‘˜) βˆ’ (πΊβ€˜π‘˜)))
8884, 85, 86, 87sersub 14050 . . . . . . . . . . . . . . . . 17 (((πœ‘ ∧ π‘₯ ∈ ℝ+) ∧ (π‘š ∈ (β„€β‰₯β€˜π‘) ∧ 𝑛 ∈ (β„€β‰₯β€˜π‘š))) β†’ (seq𝑀( + , (π‘š ∈ 𝑍 ↦ ((πΉβ€˜π‘š) βˆ’ (πΊβ€˜π‘š))))β€˜π‘›) = ((seq𝑀( + , 𝐹)β€˜π‘›) βˆ’ (seq𝑀( + , 𝐺)β€˜π‘›)))
8975, 83, 883brtr3d 5183 . . . . . . . . . . . . . . . 16 (((πœ‘ ∧ π‘₯ ∈ ℝ+) ∧ (π‘š ∈ (β„€β‰₯β€˜π‘) ∧ 𝑛 ∈ (β„€β‰₯β€˜π‘š))) β†’ ((seq𝑀( + , 𝐹)β€˜π‘š) βˆ’ (seq𝑀( + , 𝐺)β€˜π‘š)) ≀ ((seq𝑀( + , 𝐹)β€˜π‘›) βˆ’ (seq𝑀( + , 𝐺)β€˜π‘›)))
9041, 42resubcld 11680 . . . . . . . . . . . . . . . . 17 (((πœ‘ ∧ π‘₯ ∈ ℝ+) ∧ (π‘š ∈ (β„€β‰₯β€˜π‘) ∧ 𝑛 ∈ (β„€β‰₯β€˜π‘š))) β†’ ((seq𝑀( + , 𝐹)β€˜π‘›) βˆ’ (seq𝑀( + , 𝐺)β€˜π‘›)) ∈ ℝ)
9136, 44, 90lesubaddd 11849 . . . . . . . . . . . . . . . 16 (((πœ‘ ∧ π‘₯ ∈ ℝ+) ∧ (π‘š ∈ (β„€β‰₯β€˜π‘) ∧ 𝑛 ∈ (β„€β‰₯β€˜π‘š))) β†’ (((seq𝑀( + , 𝐹)β€˜π‘š) βˆ’ (seq𝑀( + , 𝐺)β€˜π‘š)) ≀ ((seq𝑀( + , 𝐹)β€˜π‘›) βˆ’ (seq𝑀( + , 𝐺)β€˜π‘›)) ↔ (seq𝑀( + , 𝐹)β€˜π‘š) ≀ (((seq𝑀( + , 𝐹)β€˜π‘›) βˆ’ (seq𝑀( + , 𝐺)β€˜π‘›)) + (seq𝑀( + , 𝐺)β€˜π‘š))))
9289, 91mpbid 231 . . . . . . . . . . . . . . 15 (((πœ‘ ∧ π‘₯ ∈ ℝ+) ∧ (π‘š ∈ (β„€β‰₯β€˜π‘) ∧ 𝑛 ∈ (β„€β‰₯β€˜π‘š))) β†’ (seq𝑀( + , 𝐹)β€˜π‘š) ≀ (((seq𝑀( + , 𝐹)β€˜π‘›) βˆ’ (seq𝑀( + , 𝐺)β€˜π‘›)) + (seq𝑀( + , 𝐺)β€˜π‘š)))
9341recnd 11280 . . . . . . . . . . . . . . . 16 (((πœ‘ ∧ π‘₯ ∈ ℝ+) ∧ (π‘š ∈ (β„€β‰₯β€˜π‘) ∧ 𝑛 ∈ (β„€β‰₯β€˜π‘š))) β†’ (seq𝑀( + , 𝐹)β€˜π‘›) ∈ β„‚)
9442recnd 11280 . . . . . . . . . . . . . . . 16 (((πœ‘ ∧ π‘₯ ∈ ℝ+) ∧ (π‘š ∈ (β„€β‰₯β€˜π‘) ∧ 𝑛 ∈ (β„€β‰₯β€˜π‘š))) β†’ (seq𝑀( + , 𝐺)β€˜π‘›) ∈ β„‚)
9544recnd 11280 . . . . . . . . . . . . . . . 16 (((πœ‘ ∧ π‘₯ ∈ ℝ+) ∧ (π‘š ∈ (β„€β‰₯β€˜π‘) ∧ 𝑛 ∈ (β„€β‰₯β€˜π‘š))) β†’ (seq𝑀( + , 𝐺)β€˜π‘š) ∈ β„‚)
9693, 94, 95subsubd 11637 . . . . . . . . . . . . . . 15 (((πœ‘ ∧ π‘₯ ∈ ℝ+) ∧ (π‘š ∈ (β„€β‰₯β€˜π‘) ∧ 𝑛 ∈ (β„€β‰₯β€˜π‘š))) β†’ ((seq𝑀( + , 𝐹)β€˜π‘›) βˆ’ ((seq𝑀( + , 𝐺)β€˜π‘›) βˆ’ (seq𝑀( + , 𝐺)β€˜π‘š))) = (((seq𝑀( + , 𝐹)β€˜π‘›) βˆ’ (seq𝑀( + , 𝐺)β€˜π‘›)) + (seq𝑀( + , 𝐺)β€˜π‘š)))
9792, 96breqtrrd 5180 . . . . . . . . . . . . . 14 (((πœ‘ ∧ π‘₯ ∈ ℝ+) ∧ (π‘š ∈ (β„€β‰₯β€˜π‘) ∧ 𝑛 ∈ (β„€β‰₯β€˜π‘š))) β†’ (seq𝑀( + , 𝐹)β€˜π‘š) ≀ ((seq𝑀( + , 𝐹)β€˜π‘›) βˆ’ ((seq𝑀( + , 𝐺)β€˜π‘›) βˆ’ (seq𝑀( + , 𝐺)β€˜π‘š))))
9836, 41, 45, 97lesubd 11856 . . . . . . . . . . . . 13 (((πœ‘ ∧ π‘₯ ∈ ℝ+) ∧ (π‘š ∈ (β„€β‰₯β€˜π‘) ∧ 𝑛 ∈ (β„€β‰₯β€˜π‘š))) β†’ ((seq𝑀( + , 𝐺)β€˜π‘›) βˆ’ (seq𝑀( + , 𝐺)β€˜π‘š)) ≀ ((seq𝑀( + , 𝐹)β€˜π‘›) βˆ’ (seq𝑀( + , 𝐹)β€˜π‘š)))
9941, 36resubcld 11680 . . . . . . . . . . . . . 14 (((πœ‘ ∧ π‘₯ ∈ ℝ+) ∧ (π‘š ∈ (β„€β‰₯β€˜π‘) ∧ 𝑛 ∈ (β„€β‰₯β€˜π‘š))) β†’ ((seq𝑀( + , 𝐹)β€˜π‘›) βˆ’ (seq𝑀( + , 𝐹)β€˜π‘š)) ∈ ℝ)
100 rpre 13022 . . . . . . . . . . . . . . 15 (π‘₯ ∈ ℝ+ β†’ π‘₯ ∈ ℝ)
101100ad2antlr 725 . . . . . . . . . . . . . 14 (((πœ‘ ∧ π‘₯ ∈ ℝ+) ∧ (π‘š ∈ (β„€β‰₯β€˜π‘) ∧ 𝑛 ∈ (β„€β‰₯β€˜π‘š))) β†’ π‘₯ ∈ ℝ)
102 lelttr 11342 . . . . . . . . . . . . . 14 ((((seq𝑀( + , 𝐺)β€˜π‘›) βˆ’ (seq𝑀( + , 𝐺)β€˜π‘š)) ∈ ℝ ∧ ((seq𝑀( + , 𝐹)β€˜π‘›) βˆ’ (seq𝑀( + , 𝐹)β€˜π‘š)) ∈ ℝ ∧ π‘₯ ∈ ℝ) β†’ ((((seq𝑀( + , 𝐺)β€˜π‘›) βˆ’ (seq𝑀( + , 𝐺)β€˜π‘š)) ≀ ((seq𝑀( + , 𝐹)β€˜π‘›) βˆ’ (seq𝑀( + , 𝐹)β€˜π‘š)) ∧ ((seq𝑀( + , 𝐹)β€˜π‘›) βˆ’ (seq𝑀( + , 𝐹)β€˜π‘š)) < π‘₯) β†’ ((seq𝑀( + , 𝐺)β€˜π‘›) βˆ’ (seq𝑀( + , 𝐺)β€˜π‘š)) < π‘₯))
10345, 99, 101, 102syl3anc 1368 . . . . . . . . . . . . 13 (((πœ‘ ∧ π‘₯ ∈ ℝ+) ∧ (π‘š ∈ (β„€β‰₯β€˜π‘) ∧ 𝑛 ∈ (β„€β‰₯β€˜π‘š))) β†’ ((((seq𝑀( + , 𝐺)β€˜π‘›) βˆ’ (seq𝑀( + , 𝐺)β€˜π‘š)) ≀ ((seq𝑀( + , 𝐹)β€˜π‘›) βˆ’ (seq𝑀( + , 𝐹)β€˜π‘š)) ∧ ((seq𝑀( + , 𝐹)β€˜π‘›) βˆ’ (seq𝑀( + , 𝐹)β€˜π‘š)) < π‘₯) β†’ ((seq𝑀( + , 𝐺)β€˜π‘›) βˆ’ (seq𝑀( + , 𝐺)β€˜π‘š)) < π‘₯))
10498, 103mpand 693 . . . . . . . . . . . 12 (((πœ‘ ∧ π‘₯ ∈ ℝ+) ∧ (π‘š ∈ (β„€β‰₯β€˜π‘) ∧ 𝑛 ∈ (β„€β‰₯β€˜π‘š))) β†’ (((seq𝑀( + , 𝐹)β€˜π‘›) βˆ’ (seq𝑀( + , 𝐹)β€˜π‘š)) < π‘₯ β†’ ((seq𝑀( + , 𝐺)β€˜π‘›) βˆ’ (seq𝑀( + , 𝐺)β€˜π‘š)) < π‘₯))
10530, 49, 11syl2an 594 . . . . . . . . . . . . . . 15 ((((πœ‘ ∧ π‘₯ ∈ ℝ+) ∧ (π‘š ∈ (β„€β‰₯β€˜π‘) ∧ 𝑛 ∈ (β„€β‰₯β€˜π‘š))) ∧ π‘˜ ∈ (𝑀...𝑛)) β†’ (πΉβ€˜π‘˜) ∈ ℝ)
10660, 64sylan2 591 . . . . . . . . . . . . . . . 16 ((((πœ‘ ∧ π‘₯ ∈ ℝ+) ∧ (π‘š ∈ (β„€β‰₯β€˜π‘) ∧ 𝑛 ∈ (β„€β‰₯β€˜π‘š))) ∧ π‘˜ ∈ ((π‘š + 1)...𝑛)) β†’ π‘˜ ∈ (β„€β‰₯β€˜π‘))
107 0red 11255 . . . . . . . . . . . . . . . . 17 ((πœ‘ ∧ π‘˜ ∈ (β„€β‰₯β€˜π‘)) β†’ 0 ∈ ℝ)
10867, 23syldan 589 . . . . . . . . . . . . . . . . 17 ((πœ‘ ∧ π‘˜ ∈ (β„€β‰₯β€˜π‘)) β†’ (πΊβ€˜π‘˜) ∈ ℝ)
10967, 11syldan 589 . . . . . . . . . . . . . . . . 17 ((πœ‘ ∧ π‘˜ ∈ (β„€β‰₯β€˜π‘)) β†’ (πΉβ€˜π‘˜) ∈ ℝ)
110 cvgcmp.6 . . . . . . . . . . . . . . . . 17 ((πœ‘ ∧ π‘˜ ∈ (β„€β‰₯β€˜π‘)) β†’ 0 ≀ (πΊβ€˜π‘˜))
111107, 108, 109, 110, 65letrd 11409 . . . . . . . . . . . . . . . 16 ((πœ‘ ∧ π‘˜ ∈ (β„€β‰₯β€˜π‘)) β†’ 0 ≀ (πΉβ€˜π‘˜))
11230, 106, 111syl2an2r 683 . . . . . . . . . . . . . . 15 ((((πœ‘ ∧ π‘₯ ∈ ℝ+) ∧ (π‘š ∈ (β„€β‰₯β€˜π‘) ∧ 𝑛 ∈ (β„€β‰₯β€˜π‘š))) ∧ π‘˜ ∈ ((π‘š + 1)...𝑛)) β†’ 0 ≀ (πΉβ€˜π‘˜))
11346, 47, 105, 112sermono 14039 . . . . . . . . . . . . . 14 (((πœ‘ ∧ π‘₯ ∈ ℝ+) ∧ (π‘š ∈ (β„€β‰₯β€˜π‘) ∧ 𝑛 ∈ (β„€β‰₯β€˜π‘š))) β†’ (seq𝑀( + , 𝐹)β€˜π‘š) ≀ (seq𝑀( + , 𝐹)β€˜π‘›))
11436, 41, 113abssubge0d 15418 . . . . . . . . . . . . 13 (((πœ‘ ∧ π‘₯ ∈ ℝ+) ∧ (π‘š ∈ (β„€β‰₯β€˜π‘) ∧ 𝑛 ∈ (β„€β‰₯β€˜π‘š))) β†’ (absβ€˜((seq𝑀( + , 𝐹)β€˜π‘›) βˆ’ (seq𝑀( + , 𝐹)β€˜π‘š))) = ((seq𝑀( + , 𝐹)β€˜π‘›) βˆ’ (seq𝑀( + , 𝐹)β€˜π‘š)))
115114breq1d 5162 . . . . . . . . . . . 12 (((πœ‘ ∧ π‘₯ ∈ ℝ+) ∧ (π‘š ∈ (β„€β‰₯β€˜π‘) ∧ 𝑛 ∈ (β„€β‰₯β€˜π‘š))) β†’ ((absβ€˜((seq𝑀( + , 𝐹)β€˜π‘›) βˆ’ (seq𝑀( + , 𝐹)β€˜π‘š))) < π‘₯ ↔ ((seq𝑀( + , 𝐹)β€˜π‘›) βˆ’ (seq𝑀( + , 𝐹)β€˜π‘š)) < π‘₯))
11630, 49, 23syl2an 594 . . . . . . . . . . . . . . 15 ((((πœ‘ ∧ π‘₯ ∈ ℝ+) ∧ (π‘š ∈ (β„€β‰₯β€˜π‘) ∧ 𝑛 ∈ (β„€β‰₯β€˜π‘š))) ∧ π‘˜ ∈ (𝑀...𝑛)) β†’ (πΊβ€˜π‘˜) ∈ ℝ)
11730, 64, 110syl2an2r 683 . . . . . . . . . . . . . . . 16 ((((πœ‘ ∧ π‘₯ ∈ ℝ+) ∧ (π‘š ∈ (β„€β‰₯β€˜π‘) ∧ 𝑛 ∈ (β„€β‰₯β€˜π‘š))) ∧ π‘˜ ∈ (β„€β‰₯β€˜(π‘š + 1))) β†’ 0 ≀ (πΊβ€˜π‘˜))
11860, 117sylan2 591 . . . . . . . . . . . . . . 15 ((((πœ‘ ∧ π‘₯ ∈ ℝ+) ∧ (π‘š ∈ (β„€β‰₯β€˜π‘) ∧ 𝑛 ∈ (β„€β‰₯β€˜π‘š))) ∧ π‘˜ ∈ ((π‘š + 1)...𝑛)) β†’ 0 ≀ (πΊβ€˜π‘˜))
11946, 47, 116, 118sermono 14039 . . . . . . . . . . . . . 14 (((πœ‘ ∧ π‘₯ ∈ ℝ+) ∧ (π‘š ∈ (β„€β‰₯β€˜π‘) ∧ 𝑛 ∈ (β„€β‰₯β€˜π‘š))) β†’ (seq𝑀( + , 𝐺)β€˜π‘š) ≀ (seq𝑀( + , 𝐺)β€˜π‘›))
12044, 42, 119abssubge0d 15418 . . . . . . . . . . . . 13 (((πœ‘ ∧ π‘₯ ∈ ℝ+) ∧ (π‘š ∈ (β„€β‰₯β€˜π‘) ∧ 𝑛 ∈ (β„€β‰₯β€˜π‘š))) β†’ (absβ€˜((seq𝑀( + , 𝐺)β€˜π‘›) βˆ’ (seq𝑀( + , 𝐺)β€˜π‘š))) = ((seq𝑀( + , 𝐺)β€˜π‘›) βˆ’ (seq𝑀( + , 𝐺)β€˜π‘š)))
121120breq1d 5162 . . . . . . . . . . . 12 (((πœ‘ ∧ π‘₯ ∈ ℝ+) ∧ (π‘š ∈ (β„€β‰₯β€˜π‘) ∧ 𝑛 ∈ (β„€β‰₯β€˜π‘š))) β†’ ((absβ€˜((seq𝑀( + , 𝐺)β€˜π‘›) βˆ’ (seq𝑀( + , 𝐺)β€˜π‘š))) < π‘₯ ↔ ((seq𝑀( + , 𝐺)β€˜π‘›) βˆ’ (seq𝑀( + , 𝐺)β€˜π‘š)) < π‘₯))
122104, 115, 1213imtr4d 293 . . . . . . . . . . 11 (((πœ‘ ∧ π‘₯ ∈ ℝ+) ∧ (π‘š ∈ (β„€β‰₯β€˜π‘) ∧ 𝑛 ∈ (β„€β‰₯β€˜π‘š))) β†’ ((absβ€˜((seq𝑀( + , 𝐹)β€˜π‘›) βˆ’ (seq𝑀( + , 𝐹)β€˜π‘š))) < π‘₯ β†’ (absβ€˜((seq𝑀( + , 𝐺)β€˜π‘›) βˆ’ (seq𝑀( + , 𝐺)β€˜π‘š))) < π‘₯))
123122anassrs 466 . . . . . . . . . 10 ((((πœ‘ ∧ π‘₯ ∈ ℝ+) ∧ π‘š ∈ (β„€β‰₯β€˜π‘)) ∧ 𝑛 ∈ (β„€β‰₯β€˜π‘š)) β†’ ((absβ€˜((seq𝑀( + , 𝐹)β€˜π‘›) βˆ’ (seq𝑀( + , 𝐹)β€˜π‘š))) < π‘₯ β†’ (absβ€˜((seq𝑀( + , 𝐺)β€˜π‘›) βˆ’ (seq𝑀( + , 𝐺)β€˜π‘š))) < π‘₯))
124123adantld 489 . . . . . . . . 9 ((((πœ‘ ∧ π‘₯ ∈ ℝ+) ∧ π‘š ∈ (β„€β‰₯β€˜π‘)) ∧ 𝑛 ∈ (β„€β‰₯β€˜π‘š)) β†’ (((seq𝑀( + , 𝐹)β€˜π‘›) ∈ β„‚ ∧ (absβ€˜((seq𝑀( + , 𝐹)β€˜π‘›) βˆ’ (seq𝑀( + , 𝐹)β€˜π‘š))) < π‘₯) β†’ (absβ€˜((seq𝑀( + , 𝐺)β€˜π‘›) βˆ’ (seq𝑀( + , 𝐺)β€˜π‘š))) < π‘₯))
125124ralimdva 3164 . . . . . . . 8 (((πœ‘ ∧ π‘₯ ∈ ℝ+) ∧ π‘š ∈ (β„€β‰₯β€˜π‘)) β†’ (βˆ€π‘› ∈ (β„€β‰₯β€˜π‘š)((seq𝑀( + , 𝐹)β€˜π‘›) ∈ β„‚ ∧ (absβ€˜((seq𝑀( + , 𝐹)β€˜π‘›) βˆ’ (seq𝑀( + , 𝐹)β€˜π‘š))) < π‘₯) β†’ βˆ€π‘› ∈ (β„€β‰₯β€˜π‘š)(absβ€˜((seq𝑀( + , 𝐺)β€˜π‘›) βˆ’ (seq𝑀( + , 𝐺)β€˜π‘š))) < π‘₯))
126125reximdva 3165 . . . . . . 7 ((πœ‘ ∧ π‘₯ ∈ ℝ+) β†’ (βˆƒπ‘š ∈ (β„€β‰₯β€˜π‘)βˆ€π‘› ∈ (β„€β‰₯β€˜π‘š)((seq𝑀( + , 𝐹)β€˜π‘›) ∈ β„‚ ∧ (absβ€˜((seq𝑀( + , 𝐹)β€˜π‘›) βˆ’ (seq𝑀( + , 𝐹)β€˜π‘š))) < π‘₯) β†’ βˆƒπ‘š ∈ (β„€β‰₯β€˜π‘)βˆ€π‘› ∈ (β„€β‰₯β€˜π‘š)(absβ€˜((seq𝑀( + , 𝐺)β€˜π‘›) βˆ’ (seq𝑀( + , 𝐺)β€˜π‘š))) < π‘₯))
12737r19.29uz 15337 . . . . . . 7 ((βˆ€π‘› ∈ (β„€β‰₯β€˜π‘)(seq𝑀( + , 𝐺)β€˜π‘›) ∈ β„‚ ∧ βˆƒπ‘š ∈ (β„€β‰₯β€˜π‘)βˆ€π‘› ∈ (β„€β‰₯β€˜π‘š)(absβ€˜((seq𝑀( + , 𝐺)β€˜π‘›) βˆ’ (seq𝑀( + , 𝐺)β€˜π‘š))) < π‘₯) β†’ βˆƒπ‘š ∈ (β„€β‰₯β€˜π‘)βˆ€π‘› ∈ (β„€β‰₯β€˜π‘š)((seq𝑀( + , 𝐺)β€˜π‘›) ∈ β„‚ ∧ (absβ€˜((seq𝑀( + , 𝐺)β€˜π‘›) βˆ’ (seq𝑀( + , 𝐺)β€˜π‘š))) < π‘₯))
12829, 126, 127syl6an 682 . . . . . 6 ((πœ‘ ∧ π‘₯ ∈ ℝ+) β†’ (βˆƒπ‘š ∈ (β„€β‰₯β€˜π‘)βˆ€π‘› ∈ (β„€β‰₯β€˜π‘š)((seq𝑀( + , 𝐹)β€˜π‘›) ∈ β„‚ ∧ (absβ€˜((seq𝑀( + , 𝐹)β€˜π‘›) βˆ’ (seq𝑀( + , 𝐹)β€˜π‘š))) < π‘₯) β†’ βˆƒπ‘š ∈ (β„€β‰₯β€˜π‘)βˆ€π‘› ∈ (β„€β‰₯β€˜π‘š)((seq𝑀( + , 𝐺)β€˜π‘›) ∈ β„‚ ∧ (absβ€˜((seq𝑀( + , 𝐺)β€˜π‘›) βˆ’ (seq𝑀( + , 𝐺)β€˜π‘š))) < π‘₯)))
129128ralimdva 3164 . . . . 5 (πœ‘ β†’ (βˆ€π‘₯ ∈ ℝ+ βˆƒπ‘š ∈ (β„€β‰₯β€˜π‘)βˆ€π‘› ∈ (β„€β‰₯β€˜π‘š)((seq𝑀( + , 𝐹)β€˜π‘›) ∈ β„‚ ∧ (absβ€˜((seq𝑀( + , 𝐹)β€˜π‘›) βˆ’ (seq𝑀( + , 𝐹)β€˜π‘š))) < π‘₯) β†’ βˆ€π‘₯ ∈ ℝ+ βˆƒπ‘š ∈ (β„€β‰₯β€˜π‘)βˆ€π‘› ∈ (β„€β‰₯β€˜π‘š)((seq𝑀( + , 𝐺)β€˜π‘›) ∈ β„‚ ∧ (absβ€˜((seq𝑀( + , 𝐺)β€˜π‘›) βˆ’ (seq𝑀( + , 𝐺)β€˜π‘š))) < π‘₯)))
1301, 37cau4 15343 . . . . . 6 (𝑁 ∈ 𝑍 β†’ (βˆ€π‘₯ ∈ ℝ+ βˆƒπ‘š ∈ 𝑍 βˆ€π‘› ∈ (β„€β‰₯β€˜π‘š)((seq𝑀( + , 𝐹)β€˜π‘›) ∈ β„‚ ∧ (absβ€˜((seq𝑀( + , 𝐹)β€˜π‘›) βˆ’ (seq𝑀( + , 𝐹)β€˜π‘š))) < π‘₯) ↔ βˆ€π‘₯ ∈ ℝ+ βˆƒπ‘š ∈ (β„€β‰₯β€˜π‘)βˆ€π‘› ∈ (β„€β‰₯β€˜π‘š)((seq𝑀( + , 𝐹)β€˜π‘›) ∈ β„‚ ∧ (absβ€˜((seq𝑀( + , 𝐹)β€˜π‘›) βˆ’ (seq𝑀( + , 𝐹)β€˜π‘š))) < π‘₯)))
1314, 130syl 17 . . . . 5 (πœ‘ β†’ (βˆ€π‘₯ ∈ ℝ+ βˆƒπ‘š ∈ 𝑍 βˆ€π‘› ∈ (β„€β‰₯β€˜π‘š)((seq𝑀( + , 𝐹)β€˜π‘›) ∈ β„‚ ∧ (absβ€˜((seq𝑀( + , 𝐹)β€˜π‘›) βˆ’ (seq𝑀( + , 𝐹)β€˜π‘š))) < π‘₯) ↔ βˆ€π‘₯ ∈ ℝ+ βˆƒπ‘š ∈ (β„€β‰₯β€˜π‘)βˆ€π‘› ∈ (β„€β‰₯β€˜π‘š)((seq𝑀( + , 𝐹)β€˜π‘›) ∈ β„‚ ∧ (absβ€˜((seq𝑀( + , 𝐹)β€˜π‘›) βˆ’ (seq𝑀( + , 𝐹)β€˜π‘š))) < π‘₯)))
1321, 37cau4 15343 . . . . . 6 (𝑁 ∈ 𝑍 β†’ (βˆ€π‘₯ ∈ ℝ+ βˆƒπ‘š ∈ 𝑍 βˆ€π‘› ∈ (β„€β‰₯β€˜π‘š)((seq𝑀( + , 𝐺)β€˜π‘›) ∈ β„‚ ∧ (absβ€˜((seq𝑀( + , 𝐺)β€˜π‘›) βˆ’ (seq𝑀( + , 𝐺)β€˜π‘š))) < π‘₯) ↔ βˆ€π‘₯ ∈ ℝ+ βˆƒπ‘š ∈ (β„€β‰₯β€˜π‘)βˆ€π‘› ∈ (β„€β‰₯β€˜π‘š)((seq𝑀( + , 𝐺)β€˜π‘›) ∈ β„‚ ∧ (absβ€˜((seq𝑀( + , 𝐺)β€˜π‘›) βˆ’ (seq𝑀( + , 𝐺)β€˜π‘š))) < π‘₯)))
1334, 132syl 17 . . . . 5 (πœ‘ β†’ (βˆ€π‘₯ ∈ ℝ+ βˆƒπ‘š ∈ 𝑍 βˆ€π‘› ∈ (β„€β‰₯β€˜π‘š)((seq𝑀( + , 𝐺)β€˜π‘›) ∈ β„‚ ∧ (absβ€˜((seq𝑀( + , 𝐺)β€˜π‘›) βˆ’ (seq𝑀( + , 𝐺)β€˜π‘š))) < π‘₯) ↔ βˆ€π‘₯ ∈ ℝ+ βˆƒπ‘š ∈ (β„€β‰₯β€˜π‘)βˆ€π‘› ∈ (β„€β‰₯β€˜π‘š)((seq𝑀( + , 𝐺)β€˜π‘›) ∈ β„‚ ∧ (absβ€˜((seq𝑀( + , 𝐺)β€˜π‘›) βˆ’ (seq𝑀( + , 𝐺)β€˜π‘š))) < π‘₯)))
134129, 131, 1333imtr4d 293 . . . 4 (πœ‘ β†’ (βˆ€π‘₯ ∈ ℝ+ βˆƒπ‘š ∈ 𝑍 βˆ€π‘› ∈ (β„€β‰₯β€˜π‘š)((seq𝑀( + , 𝐹)β€˜π‘›) ∈ β„‚ ∧ (absβ€˜((seq𝑀( + , 𝐹)β€˜π‘›) βˆ’ (seq𝑀( + , 𝐹)β€˜π‘š))) < π‘₯) β†’ βˆ€π‘₯ ∈ ℝ+ βˆƒπ‘š ∈ 𝑍 βˆ€π‘› ∈ (β„€β‰₯β€˜π‘š)((seq𝑀( + , 𝐺)β€˜π‘›) ∈ β„‚ ∧ (absβ€˜((seq𝑀( + , 𝐺)β€˜π‘›) βˆ’ (seq𝑀( + , 𝐺)β€˜π‘š))) < π‘₯)))
13520, 134mpd 15 . . 3 (πœ‘ β†’ βˆ€π‘₯ ∈ ℝ+ βˆƒπ‘š ∈ 𝑍 βˆ€π‘› ∈ (β„€β‰₯β€˜π‘š)((seq𝑀( + , 𝐺)β€˜π‘›) ∈ β„‚ ∧ (absβ€˜((seq𝑀( + , 𝐺)β€˜π‘›) βˆ’ (seq𝑀( + , 𝐺)β€˜π‘š))) < π‘₯))
1361uztrn2 12879 . . . . . . . 8 ((π‘š ∈ 𝑍 ∧ 𝑛 ∈ (β„€β‰₯β€˜π‘š)) β†’ 𝑛 ∈ 𝑍)
137 simpr 483 . . . . . . . . 9 (((seq𝑀( + , 𝐺)β€˜π‘›) ∈ β„‚ ∧ (absβ€˜((seq𝑀( + , 𝐺)β€˜π‘›) βˆ’ (seq𝑀( + , 𝐺)β€˜π‘š))) < π‘₯) β†’ (absβ€˜((seq𝑀( + , 𝐺)β€˜π‘›) βˆ’ (seq𝑀( + , 𝐺)β€˜π‘š))) < π‘₯)
13825biantrurd 531 . . . . . . . . 9 ((πœ‘ ∧ 𝑛 ∈ 𝑍) β†’ ((absβ€˜((seq𝑀( + , 𝐺)β€˜π‘›) βˆ’ (seq𝑀( + , 𝐺)β€˜π‘š))) < π‘₯ ↔ ((seq𝑀( + , 𝐺)β€˜π‘›) ∈ ℝ ∧ (absβ€˜((seq𝑀( + , 𝐺)β€˜π‘›) βˆ’ (seq𝑀( + , 𝐺)β€˜π‘š))) < π‘₯)))
139137, 138imbitrid 243 . . . . . . . 8 ((πœ‘ ∧ 𝑛 ∈ 𝑍) β†’ (((seq𝑀( + , 𝐺)β€˜π‘›) ∈ β„‚ ∧ (absβ€˜((seq𝑀( + , 𝐺)β€˜π‘›) βˆ’ (seq𝑀( + , 𝐺)β€˜π‘š))) < π‘₯) β†’ ((seq𝑀( + , 𝐺)β€˜π‘›) ∈ ℝ ∧ (absβ€˜((seq𝑀( + , 𝐺)β€˜π‘›) βˆ’ (seq𝑀( + , 𝐺)β€˜π‘š))) < π‘₯)))
140136, 139sylan2 591 . . . . . . 7 ((πœ‘ ∧ (π‘š ∈ 𝑍 ∧ 𝑛 ∈ (β„€β‰₯β€˜π‘š))) β†’ (((seq𝑀( + , 𝐺)β€˜π‘›) ∈ β„‚ ∧ (absβ€˜((seq𝑀( + , 𝐺)β€˜π‘›) βˆ’ (seq𝑀( + , 𝐺)β€˜π‘š))) < π‘₯) β†’ ((seq𝑀( + , 𝐺)β€˜π‘›) ∈ ℝ ∧ (absβ€˜((seq𝑀( + , 𝐺)β€˜π‘›) βˆ’ (seq𝑀( + , 𝐺)β€˜π‘š))) < π‘₯)))
141140anassrs 466 . . . . . 6 (((πœ‘ ∧ π‘š ∈ 𝑍) ∧ 𝑛 ∈ (β„€β‰₯β€˜π‘š)) β†’ (((seq𝑀( + , 𝐺)β€˜π‘›) ∈ β„‚ ∧ (absβ€˜((seq𝑀( + , 𝐺)β€˜π‘›) βˆ’ (seq𝑀( + , 𝐺)β€˜π‘š))) < π‘₯) β†’ ((seq𝑀( + , 𝐺)β€˜π‘›) ∈ ℝ ∧ (absβ€˜((seq𝑀( + , 𝐺)β€˜π‘›) βˆ’ (seq𝑀( + , 𝐺)β€˜π‘š))) < π‘₯)))
142141ralimdva 3164 . . . . 5 ((πœ‘ ∧ π‘š ∈ 𝑍) β†’ (βˆ€π‘› ∈ (β„€β‰₯β€˜π‘š)((seq𝑀( + , 𝐺)β€˜π‘›) ∈ β„‚ ∧ (absβ€˜((seq𝑀( + , 𝐺)β€˜π‘›) βˆ’ (seq𝑀( + , 𝐺)β€˜π‘š))) < π‘₯) β†’ βˆ€π‘› ∈ (β„€β‰₯β€˜π‘š)((seq𝑀( + , 𝐺)β€˜π‘›) ∈ ℝ ∧ (absβ€˜((seq𝑀( + , 𝐺)β€˜π‘›) βˆ’ (seq𝑀( + , 𝐺)β€˜π‘š))) < π‘₯)))
143142reximdva 3165 . . . 4 (πœ‘ β†’ (βˆƒπ‘š ∈ 𝑍 βˆ€π‘› ∈ (β„€β‰₯β€˜π‘š)((seq𝑀( + , 𝐺)β€˜π‘›) ∈ β„‚ ∧ (absβ€˜((seq𝑀( + , 𝐺)β€˜π‘›) βˆ’ (seq𝑀( + , 𝐺)β€˜π‘š))) < π‘₯) β†’ βˆƒπ‘š ∈ 𝑍 βˆ€π‘› ∈ (β„€β‰₯β€˜π‘š)((seq𝑀( + , 𝐺)β€˜π‘›) ∈ ℝ ∧ (absβ€˜((seq𝑀( + , 𝐺)β€˜π‘›) βˆ’ (seq𝑀( + , 𝐺)β€˜π‘š))) < π‘₯)))
144143ralimdv 3166 . . 3 (πœ‘ β†’ (βˆ€π‘₯ ∈ ℝ+ βˆƒπ‘š ∈ 𝑍 βˆ€π‘› ∈ (β„€β‰₯β€˜π‘š)((seq𝑀( + , 𝐺)β€˜π‘›) ∈ β„‚ ∧ (absβ€˜((seq𝑀( + , 𝐺)β€˜π‘›) βˆ’ (seq𝑀( + , 𝐺)β€˜π‘š))) < π‘₯) β†’ βˆ€π‘₯ ∈ ℝ+ βˆƒπ‘š ∈ 𝑍 βˆ€π‘› ∈ (β„€β‰₯β€˜π‘š)((seq𝑀( + , 𝐺)β€˜π‘›) ∈ ℝ ∧ (absβ€˜((seq𝑀( + , 𝐺)β€˜π‘›) βˆ’ (seq𝑀( + , 𝐺)β€˜π‘š))) < π‘₯)))
145135, 144mpd 15 . 2 (πœ‘ β†’ βˆ€π‘₯ ∈ ℝ+ βˆƒπ‘š ∈ 𝑍 βˆ€π‘› ∈ (β„€β‰₯β€˜π‘š)((seq𝑀( + , 𝐺)β€˜π‘›) ∈ ℝ ∧ (absβ€˜((seq𝑀( + , 𝐺)β€˜π‘›) βˆ’ (seq𝑀( + , 𝐺)β€˜π‘š))) < π‘₯))
1461, 3, 145caurcvg2 15664 1 (πœ‘ β†’ seq𝑀( + , 𝐺) ∈ dom ⇝ )
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ↔ wb 205   ∧ wa 394   = wceq 1533   ∈ wcel 2098  βˆ€wral 3058  βˆƒwrex 3067  Vcvv 3473   class class class wbr 5152   ↦ cmpt 5235  dom cdm 5682  βŸΆwf 6549  β€˜cfv 6553  (class class class)co 7426  β„‚cc 11144  β„cr 11145  0cc0 11146  1c1 11147   + caddc 11149   < clt 11286   ≀ cle 11287   βˆ’ cmin 11482  β„€cz 12596  β„€β‰₯cuz 12860  β„+crp 13014  ...cfz 13524  seqcseq 14006  abscabs 15221   ⇝ cli 15468
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 2166  ax-ext 2699  ax-rep 5289  ax-sep 5303  ax-nul 5310  ax-pow 5369  ax-pr 5433  ax-un 7746  ax-inf2 9672  ax-cnex 11202  ax-resscn 11203  ax-1cn 11204  ax-icn 11205  ax-addcl 11206  ax-addrcl 11207  ax-mulcl 11208  ax-mulrcl 11209  ax-mulcom 11210  ax-addass 11211  ax-mulass 11212  ax-distr 11213  ax-i2m1 11214  ax-1ne0 11215  ax-1rid 11216  ax-rnegex 11217  ax-rrecex 11218  ax-cnre 11219  ax-pre-lttri 11220  ax-pre-lttrn 11221  ax-pre-ltadd 11222  ax-pre-mulgt0 11223  ax-pre-sup 11224
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2529  df-eu 2558  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2938  df-nel 3044  df-ral 3059  df-rex 3068  df-rmo 3374  df-reu 3375  df-rab 3431  df-v 3475  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-pss 3968  df-nul 4327  df-if 4533  df-pw 4608  df-sn 4633  df-pr 4635  df-op 4639  df-uni 4913  df-iun 5002  df-br 5153  df-opab 5215  df-mpt 5236  df-tr 5270  df-id 5580  df-eprel 5586  df-po 5594  df-so 5595  df-fr 5637  df-we 5639  df-xp 5688  df-rel 5689  df-cnv 5690  df-co 5691  df-dm 5692  df-rn 5693  df-res 5694  df-ima 5695  df-pred 6310  df-ord 6377  df-on 6378  df-lim 6379  df-suc 6380  df-iota 6505  df-fun 6555  df-fn 6556  df-f 6557  df-f1 6558  df-fo 6559  df-f1o 6560  df-fv 6561  df-riota 7382  df-ov 7429  df-oprab 7430  df-mpo 7431  df-om 7877  df-1st 7999  df-2nd 8000  df-frecs 8293  df-wrecs 8324  df-recs 8398  df-rdg 8437  df-er 8731  df-pm 8854  df-en 8971  df-dom 8972  df-sdom 8973  df-sup 9473  df-inf 9474  df-pnf 11288  df-mnf 11289  df-xr 11290  df-ltxr 11291  df-le 11292  df-sub 11484  df-neg 11485  df-div 11910  df-nn 12251  df-2 12313  df-3 12314  df-n0 12511  df-z 12597  df-uz 12861  df-rp 13015  df-ico 13370  df-fz 13525  df-fzo 13668  df-fl 13797  df-seq 14007  df-exp 14067  df-cj 15086  df-re 15087  df-im 15088  df-sqrt 15222  df-abs 15223  df-limsup 15455  df-clim 15472  df-rlim 15473
This theorem is referenced by:  cvgcmpce  15804  rpnnen2lem5  16202  aaliou3lem3  26299
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