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Theorem cvgcmp 15708
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 13915 . . 3 seq𝑀( + , 𝐺) ∈ V
32a1i 11 . 2 (πœ‘ β†’ seq𝑀( + , 𝐺) ∈ V)
4 cvgcmp.2 . . . . . . . 8 (πœ‘ β†’ 𝑁 ∈ 𝑍)
54, 1eleqtrdi 2848 . . . . . . 7 (πœ‘ β†’ 𝑁 ∈ (β„€β‰₯β€˜π‘€))
6 eluzel2 12775 . . . . . . 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 15562 . . . . . 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 585 . . . . 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 13944 . . . . . . . . . 10 (πœ‘ β†’ seq𝑀( + , 𝐹):π‘βŸΆβ„)
1312ffvelcdmda 7040 . . . . . . . . 9 ((πœ‘ ∧ 𝑛 ∈ 𝑍) β†’ (seq𝑀( + , 𝐹)β€˜π‘›) ∈ ℝ)
1413recnd 11190 . . . . . . . 8 ((πœ‘ ∧ 𝑛 ∈ 𝑍) β†’ (seq𝑀( + , 𝐹)β€˜π‘›) ∈ β„‚)
1514ralrimiva 3144 . . . . . . 7 (πœ‘ β†’ βˆ€π‘› ∈ 𝑍 (seq𝑀( + , 𝐹)β€˜π‘›) ∈ β„‚)
161r19.29uz 15242 . . . . . . . 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 414 . . . . . . 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 3167 . . . . 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 12789 . . . . . . . . . . 11 ((𝑁 ∈ 𝑍 ∧ 𝑛 ∈ (β„€β‰₯β€˜π‘)) β†’ 𝑛 ∈ 𝑍)
224, 21sylan 581 . . . . . . . . . 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 13944 . . . . . . . . . . . 12 (πœ‘ β†’ seq𝑀( + , 𝐺):π‘βŸΆβ„)
2524ffvelcdmda 7040 . . . . . . . . . . 11 ((πœ‘ ∧ 𝑛 ∈ 𝑍) β†’ (seq𝑀( + , 𝐺)β€˜π‘›) ∈ ℝ)
2625recnd 11190 . . . . . . . . . 10 ((πœ‘ ∧ 𝑛 ∈ 𝑍) β†’ (seq𝑀( + , 𝐺)β€˜π‘›) ∈ β„‚)
2722, 26syldan 592 . . . . . . . . 9 ((πœ‘ ∧ 𝑛 ∈ (β„€β‰₯β€˜π‘)) β†’ (seq𝑀( + , 𝐺)β€˜π‘›) ∈ β„‚)
2827ralrimiva 3144 . . . . . . . 8 (πœ‘ β†’ βˆ€π‘› ∈ (β„€β‰₯β€˜π‘)(seq𝑀( + , 𝐺)β€˜π‘›) ∈ β„‚)
2928adantr 482 . . . . . . 7 ((πœ‘ ∧ π‘₯ ∈ ℝ+) β†’ βˆ€π‘› ∈ (β„€β‰₯β€˜π‘)(seq𝑀( + , 𝐺)β€˜π‘›) ∈ β„‚)
30 simpll 766 . . . . . . . . . . . . . . . 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 770 . . . . . . . . . . . . . . . 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 12789 . . . . . . . . . . . . . . . 16 ((𝑁 ∈ 𝑍 ∧ π‘š ∈ (β„€β‰₯β€˜π‘)) β†’ π‘š ∈ 𝑍)
3532, 33, 34syl2anc 585 . . . . . . . . . . . . . . 15 (((πœ‘ ∧ π‘₯ ∈ ℝ+) ∧ (π‘š ∈ (β„€β‰₯β€˜π‘) ∧ 𝑛 ∈ (β„€β‰₯β€˜π‘š))) β†’ π‘š ∈ 𝑍)
3631, 35ffvelcdmd 7041 . . . . . . . . . . . . . 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 2737 . . . . . . . . . . . . . . . . . 18 (β„€β‰₯β€˜π‘) = (β„€β‰₯β€˜π‘)
3837uztrn2 12789 . . . . . . . . . . . . . . . . 17 ((π‘š ∈ (β„€β‰₯β€˜π‘) ∧ 𝑛 ∈ (β„€β‰₯β€˜π‘š)) β†’ 𝑛 ∈ (β„€β‰₯β€˜π‘))
3938adantl 483 . . . . . . . . . . . . . . . 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 585 . . . . . . . . . . . . . . 15 (((πœ‘ ∧ π‘₯ ∈ ℝ+) ∧ (π‘š ∈ (β„€β‰₯β€˜π‘) ∧ 𝑛 ∈ (β„€β‰₯β€˜π‘š))) β†’ 𝑛 ∈ 𝑍)
4130, 40, 13syl2anc 585 . . . . . . . . . . . . . 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 585 . . . . . . . . . . . . . . 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 7041 . . . . . . . . . . . . . . 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 11590 . . . . . . . . . . . . . 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 2848 . . . . . . . . . . . . . . . . . 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 772 . . . . . . . . . . . . . . . . . 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 13444 . . . . . . . . . . . . . . . . . . . 20 (π‘˜ ∈ (𝑀...𝑛) β†’ π‘˜ ∈ (β„€β‰₯β€˜π‘€))
4948, 1eleqtrrdi 2849 . . . . . . . . . . . . . . . . . . 19 (π‘˜ ∈ (𝑀...𝑛) β†’ π‘˜ ∈ 𝑍)
50 fveq2 6847 . . . . . . . . . . . . . . . . . . . . . . 23 (π‘š = π‘˜ β†’ (πΉβ€˜π‘š) = (πΉβ€˜π‘˜))
51 fveq2 6847 . . . . . . . . . . . . . . . . . . . . . . 23 (π‘š = π‘˜ β†’ (πΊβ€˜π‘š) = (πΊβ€˜π‘˜))
5250, 51oveq12d 7380 . . . . . . . . . . . . . . . . . . . . . 22 (π‘š = π‘˜ β†’ ((πΉβ€˜π‘š) βˆ’ (πΊβ€˜π‘š)) = ((πΉβ€˜π‘˜) βˆ’ (πΊβ€˜π‘˜)))
53 eqid 2737 . . . . . . . . . . . . . . . . . . . . . 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 7395 . . . . . . . . . . . . . . . . . . . . . 22 ((πΉβ€˜π‘˜) βˆ’ (πΊβ€˜π‘˜)) ∈ V
5552, 53, 54fvmpt 6953 . . . . . . . . . . . . . . . . . . . . 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 483 . . . . . . . . . . . . . . . . . . . 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 11590 . . . . . . . . . . . . . . . . . . . 20 ((πœ‘ ∧ π‘˜ ∈ 𝑍) β†’ ((πΉβ€˜π‘˜) βˆ’ (πΊβ€˜π‘˜)) ∈ ℝ)
5856, 57eqeltrd 2838 . . . . . . . . . . . . . . . . . . 19 ((πœ‘ ∧ π‘˜ ∈ 𝑍) β†’ ((π‘š ∈ 𝑍 ↦ ((πΉβ€˜π‘š) βˆ’ (πΊβ€˜π‘š)))β€˜π‘˜) ∈ ℝ)
5930, 49, 58syl2an 597 . . . . . . . . . . . . . . . . . 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 13444 . . . . . . . . . . . . . . . . . . 19 (π‘˜ ∈ ((π‘š + 1)...𝑛) β†’ π‘˜ ∈ (β„€β‰₯β€˜(π‘š + 1)))
61 peano2uz 12833 . . . . . . . . . . . . . . . . . . . . . 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 12789 . . . . . . . . . . . . . . . . . . . . 21 (((π‘š + 1) ∈ (β„€β‰₯β€˜π‘) ∧ π‘˜ ∈ (β„€β‰₯β€˜(π‘š + 1))) β†’ π‘˜ ∈ (β„€β‰₯β€˜π‘))
6462, 63sylan 581 . . . . . . . . . . . . . . . . . . . 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 12789 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝑁 ∈ 𝑍 ∧ π‘˜ ∈ (β„€β‰₯β€˜π‘)) β†’ π‘˜ ∈ 𝑍)
674, 66sylan 581 . . . . . . . . . . . . . . . . . . . . . . 23 ((πœ‘ ∧ π‘˜ ∈ (β„€β‰₯β€˜π‘)) β†’ π‘˜ ∈ 𝑍)
6811, 23subge0d 11752 . . . . . . . . . . . . . . . . . . . . . . 23 ((πœ‘ ∧ π‘˜ ∈ 𝑍) β†’ (0 ≀ ((πΉβ€˜π‘˜) βˆ’ (πΊβ€˜π‘˜)) ↔ (πΊβ€˜π‘˜) ≀ (πΉβ€˜π‘˜)))
6967, 68syldan 592 . . . . . . . . . . . . . . . . . . . . . 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 257 . . . . . . . . . . . . . . . . . . . . 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 5138 . . . . . . . . . . . . . . . . . . . 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 684 . . . . . . . . . . . . . . . . . . 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 594 . . . . . . . . . . . . . . . . . 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 13947 . . . . . . . . . . . . . . . . 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 13444 . . . . . . . . . . . . . . . . . . . 20 (π‘˜ ∈ (𝑀...π‘š) β†’ π‘˜ ∈ (β„€β‰₯β€˜π‘€))
7776, 1eleqtrrdi 2849 . . . . . . . . . . . . . . . . . . 19 (π‘˜ ∈ (𝑀...π‘š) β†’ π‘˜ ∈ 𝑍)
7811recnd 11190 . . . . . . . . . . . . . . . . . . 19 ((πœ‘ ∧ π‘˜ ∈ 𝑍) β†’ (πΉβ€˜π‘˜) ∈ β„‚)
7930, 77, 78syl2an 597 . . . . . . . . . . . . . . . . . 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 11190 . . . . . . . . . . . . . . . . . . 19 ((πœ‘ ∧ π‘˜ ∈ 𝑍) β†’ (πΊβ€˜π‘˜) ∈ β„‚)
8130, 77, 80syl2an 597 . . . . . . . . . . . . . . . . . 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 597 . . . . . . . . . . . . . . . . . 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 13958 . . . . . . . . . . . . . . . . 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 2848 . . . . . . . . . . . . . . . . . 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 597 . . . . . . . . . . . . . . . . . 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 597 . . . . . . . . . . . . . . . . . 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 597 . . . . . . . . . . . . . . . . . 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 13958 . . . . . . . . . . . . . . . . 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 5141 . . . . . . . . . . . . . . . 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 11590 . . . . . . . . . . . . . . . . 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 11759 . . . . . . . . . . . . . . . 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 11190 . . . . . . . . . . . . . . . 16 (((πœ‘ ∧ π‘₯ ∈ ℝ+) ∧ (π‘š ∈ (β„€β‰₯β€˜π‘) ∧ 𝑛 ∈ (β„€β‰₯β€˜π‘š))) β†’ (seq𝑀( + , 𝐹)β€˜π‘›) ∈ β„‚)
9442recnd 11190 . . . . . . . . . . . . . . . 16 (((πœ‘ ∧ π‘₯ ∈ ℝ+) ∧ (π‘š ∈ (β„€β‰₯β€˜π‘) ∧ 𝑛 ∈ (β„€β‰₯β€˜π‘š))) β†’ (seq𝑀( + , 𝐺)β€˜π‘›) ∈ β„‚)
9544recnd 11190 . . . . . . . . . . . . . . . 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 11547 . . . . . . . . . . . . . . 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 5138 . . . . . . . . . . . . . 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 11766 . . . . . . . . . . . . 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 11590 . . . . . . . . . . . . . 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 12930 . . . . . . . . . . . . . . 15 (π‘₯ ∈ ℝ+ β†’ π‘₯ ∈ ℝ)
101100ad2antlr 726 . . . . . . . . . . . . . 14 (((πœ‘ ∧ π‘₯ ∈ ℝ+) ∧ (π‘š ∈ (β„€β‰₯β€˜π‘) ∧ 𝑛 ∈ (β„€β‰₯β€˜π‘š))) β†’ π‘₯ ∈ ℝ)
102 lelttr 11252 . . . . . . . . . . . . . 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 1372 . . . . . . . . . . . . 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 694 . . . . . . . . . . . 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 597 . . . . . . . . . . . . . . 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 594 . . . . . . . . . . . . . . . 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 11165 . . . . . . . . . . . . . . . . 17 ((πœ‘ ∧ π‘˜ ∈ (β„€β‰₯β€˜π‘)) β†’ 0 ∈ ℝ)
10867, 23syldan 592 . . . . . . . . . . . . . . . . 17 ((πœ‘ ∧ π‘˜ ∈ (β„€β‰₯β€˜π‘)) β†’ (πΊβ€˜π‘˜) ∈ ℝ)
10967, 11syldan 592 . . . . . . . . . . . . . . . . 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 11319 . . . . . . . . . . . . . . . 16 ((πœ‘ ∧ π‘˜ ∈ (β„€β‰₯β€˜π‘)) β†’ 0 ≀ (πΉβ€˜π‘˜))
11230, 106, 111syl2an2r 684 . . . . . . . . . . . . . . 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 13947 . . . . . . . . . . . . . 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 15323 . . . . . . . . . . . . 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 5120 . . . . . . . . . . . 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 597 . . . . . . . . . . . . . . 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 684 . . . . . . . . . . . . . . . 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 594 . . . . . . . . . . . . . . 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 13947 . . . . . . . . . . . . . 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 15323 . . . . . . . . . . . . 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 5120 . . . . . . . . . . . 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 294 . . . . . . . . . . 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 469 . . . . . . . . . 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 492 . . . . . . . . 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 3165 . . . . . . . 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 3166 . . . . . . 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 15242 . . . . . . 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 683 . . . . . 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 3165 . . . . 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 15248 . . . . . 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 15248 . . . . . 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 294 . . . 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 12789 . . . . . . . 8 ((π‘š ∈ 𝑍 ∧ 𝑛 ∈ (β„€β‰₯β€˜π‘š)) β†’ 𝑛 ∈ 𝑍)
137 simpr 486 . . . . . . . . 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 534 . . . . . . . . 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 594 . . . . . . 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 469 . . . . . 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 3165 . . . . 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 3166 . . . 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 3167 . . 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 15569 1 (πœ‘ β†’ seq𝑀( + , 𝐺) ∈ dom ⇝ )
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ↔ wb 205   ∧ wa 397   = wceq 1542   ∈ wcel 2107  βˆ€wral 3065  βˆƒwrex 3074  Vcvv 3448   class class class wbr 5110   ↦ cmpt 5193  dom cdm 5638  βŸΆwf 6497  β€˜cfv 6501  (class class class)co 7362  β„‚cc 11056  β„cr 11057  0cc0 11058  1c1 11059   + caddc 11061   < clt 11196   ≀ cle 11197   βˆ’ cmin 11392  β„€cz 12506  β„€β‰₯cuz 12770  β„+crp 12922  ...cfz 13431  seqcseq 13913  abscabs 15126   ⇝ cli 15373
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2708  ax-rep 5247  ax-sep 5261  ax-nul 5268  ax-pow 5325  ax-pr 5389  ax-un 7677  ax-inf2 9584  ax-cnex 11114  ax-resscn 11115  ax-1cn 11116  ax-icn 11117  ax-addcl 11118  ax-addrcl 11119  ax-mulcl 11120  ax-mulrcl 11121  ax-mulcom 11122  ax-addass 11123  ax-mulass 11124  ax-distr 11125  ax-i2m1 11126  ax-1ne0 11127  ax-1rid 11128  ax-rnegex 11129  ax-rrecex 11130  ax-cnre 11131  ax-pre-lttri 11132  ax-pre-lttrn 11133  ax-pre-ltadd 11134  ax-pre-mulgt0 11135  ax-pre-sup 11136
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2890  df-ne 2945  df-nel 3051  df-ral 3066  df-rex 3075  df-rmo 3356  df-reu 3357  df-rab 3411  df-v 3450  df-sbc 3745  df-csb 3861  df-dif 3918  df-un 3920  df-in 3922  df-ss 3932  df-pss 3934  df-nul 4288  df-if 4492  df-pw 4567  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4871  df-iun 4961  df-br 5111  df-opab 5173  df-mpt 5194  df-tr 5228  df-id 5536  df-eprel 5542  df-po 5550  df-so 5551  df-fr 5593  df-we 5595  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651  df-pred 6258  df-ord 6325  df-on 6326  df-lim 6327  df-suc 6328  df-iota 6453  df-fun 6503  df-fn 6504  df-f 6505  df-f1 6506  df-fo 6507  df-f1o 6508  df-fv 6509  df-riota 7318  df-ov 7365  df-oprab 7366  df-mpo 7367  df-om 7808  df-1st 7926  df-2nd 7927  df-frecs 8217  df-wrecs 8248  df-recs 8322  df-rdg 8361  df-er 8655  df-pm 8775  df-en 8891  df-dom 8892  df-sdom 8893  df-sup 9385  df-inf 9386  df-pnf 11198  df-mnf 11199  df-xr 11200  df-ltxr 11201  df-le 11202  df-sub 11394  df-neg 11395  df-div 11820  df-nn 12161  df-2 12223  df-3 12224  df-n0 12421  df-z 12507  df-uz 12771  df-rp 12923  df-ico 13277  df-fz 13432  df-fzo 13575  df-fl 13704  df-seq 13914  df-exp 13975  df-cj 14991  df-re 14992  df-im 14993  df-sqrt 15127  df-abs 15128  df-limsup 15360  df-clim 15377  df-rlim 15378
This theorem is referenced by:  cvgcmpce  15710  rpnnen2lem5  16107  aaliou3lem3  25720
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