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Theorem cvgcmp 15852
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 = (ℤ𝑀)
cvgcmp.2 (𝜑𝑁𝑍)
cvgcmp.3 ((𝜑𝑘𝑍) → (𝐹𝑘) ∈ ℝ)
cvgcmp.4 ((𝜑𝑘𝑍) → (𝐺𝑘) ∈ ℝ)
cvgcmp.5 (𝜑 → seq𝑀( + , 𝐹) ∈ dom ⇝ )
cvgcmp.6 ((𝜑𝑘 ∈ (ℤ𝑁)) → 0 ≤ (𝐺𝑘))
cvgcmp.7 ((𝜑𝑘 ∈ (ℤ𝑁)) → (𝐺𝑘) ≤ (𝐹𝑘))
Assertion
Ref Expression
cvgcmp (𝜑 → seq𝑀( + , 𝐺) ∈ dom ⇝ )
Distinct variable groups:   𝑘,𝐹   𝑘,𝐺   𝜑,𝑘   𝑘,𝑀   𝑘,𝑁   𝑘,𝑍

Proof of Theorem cvgcmp
Dummy variables 𝑛 𝑚 𝑥 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 cvgcmp.1 . 2 𝑍 = (ℤ𝑀)
2 seqex 14044 . . 3 seq𝑀( + , 𝐺) ∈ V
32a1i 11 . 2 (𝜑 → seq𝑀( + , 𝐺) ∈ V)
4 cvgcmp.2 . . . . . . . 8 (𝜑𝑁𝑍)
54, 1eleqtrdi 2851 . . . . . . 7 (𝜑𝑁 ∈ (ℤ𝑀))
6 eluzel2 12883 . . . . . . 7 (𝑁 ∈ (ℤ𝑀) → 𝑀 ∈ ℤ)
75, 6syl 17 . . . . . 6 (𝜑𝑀 ∈ ℤ)
8 cvgcmp.5 . . . . . 6 (𝜑 → seq𝑀( + , 𝐹) ∈ dom ⇝ )
91climcau 15707 . . . . . 6 ((𝑀 ∈ ℤ ∧ seq𝑀( + , 𝐹) ∈ dom ⇝ ) → ∀𝑥 ∈ ℝ+𝑚𝑍𝑛 ∈ (ℤ𝑚)(abs‘((seq𝑀( + , 𝐹)‘𝑛) − (seq𝑀( + , 𝐹)‘𝑚))) < 𝑥)
107, 8, 9syl2anc 584 . . . . 5 (𝜑 → ∀𝑥 ∈ ℝ+𝑚𝑍𝑛 ∈ (ℤ𝑚)(abs‘((seq𝑀( + , 𝐹)‘𝑛) − (seq𝑀( + , 𝐹)‘𝑚))) < 𝑥)
11 cvgcmp.3 . . . . . . . . . . 11 ((𝜑𝑘𝑍) → (𝐹𝑘) ∈ ℝ)
121, 7, 11serfre 14072 . . . . . . . . . 10 (𝜑 → seq𝑀( + , 𝐹):𝑍⟶ℝ)
1312ffvelcdmda 7104 . . . . . . . . 9 ((𝜑𝑛𝑍) → (seq𝑀( + , 𝐹)‘𝑛) ∈ ℝ)
1413recnd 11289 . . . . . . . 8 ((𝜑𝑛𝑍) → (seq𝑀( + , 𝐹)‘𝑛) ∈ ℂ)
1514ralrimiva 3146 . . . . . . 7 (𝜑 → ∀𝑛𝑍 (seq𝑀( + , 𝐹)‘𝑛) ∈ ℂ)
161r19.29uz 15389 . . . . . . . 8 ((∀𝑛𝑍 (seq𝑀( + , 𝐹)‘𝑛) ∈ ℂ ∧ ∃𝑚𝑍𝑛 ∈ (ℤ𝑚)(abs‘((seq𝑀( + , 𝐹)‘𝑛) − (seq𝑀( + , 𝐹)‘𝑚))) < 𝑥) → ∃𝑚𝑍𝑛 ∈ (ℤ𝑚)((seq𝑀( + , 𝐹)‘𝑛) ∈ ℂ ∧ (abs‘((seq𝑀( + , 𝐹)‘𝑛) − (seq𝑀( + , 𝐹)‘𝑚))) < 𝑥))
1716ex 412 . . . . . . 7 (∀𝑛𝑍 (seq𝑀( + , 𝐹)‘𝑛) ∈ ℂ → (∃𝑚𝑍𝑛 ∈ (ℤ𝑚)(abs‘((seq𝑀( + , 𝐹)‘𝑛) − (seq𝑀( + , 𝐹)‘𝑚))) < 𝑥 → ∃𝑚𝑍𝑛 ∈ (ℤ𝑚)((seq𝑀( + , 𝐹)‘𝑛) ∈ ℂ ∧ (abs‘((seq𝑀( + , 𝐹)‘𝑛) − (seq𝑀( + , 𝐹)‘𝑚))) < 𝑥)))
1815, 17syl 17 . . . . . 6 (𝜑 → (∃𝑚𝑍𝑛 ∈ (ℤ𝑚)(abs‘((seq𝑀( + , 𝐹)‘𝑛) − (seq𝑀( + , 𝐹)‘𝑚))) < 𝑥 → ∃𝑚𝑍𝑛 ∈ (ℤ𝑚)((seq𝑀( + , 𝐹)‘𝑛) ∈ ℂ ∧ (abs‘((seq𝑀( + , 𝐹)‘𝑛) − (seq𝑀( + , 𝐹)‘𝑚))) < 𝑥)))
1918ralimdv 3169 . . . . 5 (𝜑 → (∀𝑥 ∈ ℝ+𝑚𝑍𝑛 ∈ (ℤ𝑚)(abs‘((seq𝑀( + , 𝐹)‘𝑛) − (seq𝑀( + , 𝐹)‘𝑚))) < 𝑥 → ∀𝑥 ∈ ℝ+𝑚𝑍𝑛 ∈ (ℤ𝑚)((seq𝑀( + , 𝐹)‘𝑛) ∈ ℂ ∧ (abs‘((seq𝑀( + , 𝐹)‘𝑛) − (seq𝑀( + , 𝐹)‘𝑚))) < 𝑥)))
2010, 19mpd 15 . . . 4 (𝜑 → ∀𝑥 ∈ ℝ+𝑚𝑍𝑛 ∈ (ℤ𝑚)((seq𝑀( + , 𝐹)‘𝑛) ∈ ℂ ∧ (abs‘((seq𝑀( + , 𝐹)‘𝑛) − (seq𝑀( + , 𝐹)‘𝑚))) < 𝑥))
211uztrn2 12897 . . . . . . . . . . 11 ((𝑁𝑍𝑛 ∈ (ℤ𝑁)) → 𝑛𝑍)
224, 21sylan 580 . . . . . . . . . 10 ((𝜑𝑛 ∈ (ℤ𝑁)) → 𝑛𝑍)
23 cvgcmp.4 . . . . . . . . . . . . 13 ((𝜑𝑘𝑍) → (𝐺𝑘) ∈ ℝ)
241, 7, 23serfre 14072 . . . . . . . . . . . 12 (𝜑 → seq𝑀( + , 𝐺):𝑍⟶ℝ)
2524ffvelcdmda 7104 . . . . . . . . . . 11 ((𝜑𝑛𝑍) → (seq𝑀( + , 𝐺)‘𝑛) ∈ ℝ)
2625recnd 11289 . . . . . . . . . 10 ((𝜑𝑛𝑍) → (seq𝑀( + , 𝐺)‘𝑛) ∈ ℂ)
2722, 26syldan 591 . . . . . . . . 9 ((𝜑𝑛 ∈ (ℤ𝑁)) → (seq𝑀( + , 𝐺)‘𝑛) ∈ ℂ)
2827ralrimiva 3146 . . . . . . . 8 (𝜑 → ∀𝑛 ∈ (ℤ𝑁)(seq𝑀( + , 𝐺)‘𝑛) ∈ ℂ)
2928adantr 480 . . . . . . 7 ((𝜑𝑥 ∈ ℝ+) → ∀𝑛 ∈ (ℤ𝑁)(seq𝑀( + , 𝐺)‘𝑛) ∈ ℂ)
30 simpll 767 . . . . . . . . . . . . . . . 16 (((𝜑𝑥 ∈ ℝ+) ∧ (𝑚 ∈ (ℤ𝑁) ∧ 𝑛 ∈ (ℤ𝑚))) → 𝜑)
3130, 12syl 17 . . . . . . . . . . . . . . 15 (((𝜑𝑥 ∈ ℝ+) ∧ (𝑚 ∈ (ℤ𝑁) ∧ 𝑛 ∈ (ℤ𝑚))) → seq𝑀( + , 𝐹):𝑍⟶ℝ)
3230, 4syl 17 . . . . . . . . . . . . . . . 16 (((𝜑𝑥 ∈ ℝ+) ∧ (𝑚 ∈ (ℤ𝑁) ∧ 𝑛 ∈ (ℤ𝑚))) → 𝑁𝑍)
33 simprl 771 . . . . . . . . . . . . . . . 16 (((𝜑𝑥 ∈ ℝ+) ∧ (𝑚 ∈ (ℤ𝑁) ∧ 𝑛 ∈ (ℤ𝑚))) → 𝑚 ∈ (ℤ𝑁))
341uztrn2 12897 . . . . . . . . . . . . . . . 16 ((𝑁𝑍𝑚 ∈ (ℤ𝑁)) → 𝑚𝑍)
3532, 33, 34syl2anc 584 . . . . . . . . . . . . . . 15 (((𝜑𝑥 ∈ ℝ+) ∧ (𝑚 ∈ (ℤ𝑁) ∧ 𝑛 ∈ (ℤ𝑚))) → 𝑚𝑍)
3631, 35ffvelcdmd 7105 . . . . . . . . . . . . . 14 (((𝜑𝑥 ∈ ℝ+) ∧ (𝑚 ∈ (ℤ𝑁) ∧ 𝑛 ∈ (ℤ𝑚))) → (seq𝑀( + , 𝐹)‘𝑚) ∈ ℝ)
37 eqid 2737 . . . . . . . . . . . . . . . . . 18 (ℤ𝑁) = (ℤ𝑁)
3837uztrn2 12897 . . . . . . . . . . . . . . . . 17 ((𝑚 ∈ (ℤ𝑁) ∧ 𝑛 ∈ (ℤ𝑚)) → 𝑛 ∈ (ℤ𝑁))
3938adantl 481 . . . . . . . . . . . . . . . 16 (((𝜑𝑥 ∈ ℝ+) ∧ (𝑚 ∈ (ℤ𝑁) ∧ 𝑛 ∈ (ℤ𝑚))) → 𝑛 ∈ (ℤ𝑁))
4032, 39, 21syl2anc 584 . . . . . . . . . . . . . . 15 (((𝜑𝑥 ∈ ℝ+) ∧ (𝑚 ∈ (ℤ𝑁) ∧ 𝑛 ∈ (ℤ𝑚))) → 𝑛𝑍)
4130, 40, 13syl2anc 584 . . . . . . . . . . . . . 14 (((𝜑𝑥 ∈ ℝ+) ∧ (𝑚 ∈ (ℤ𝑁) ∧ 𝑛 ∈ (ℤ𝑚))) → (seq𝑀( + , 𝐹)‘𝑛) ∈ ℝ)
4230, 40, 25syl2anc 584 . . . . . . . . . . . . . . 15 (((𝜑𝑥 ∈ ℝ+) ∧ (𝑚 ∈ (ℤ𝑁) ∧ 𝑛 ∈ (ℤ𝑚))) → (seq𝑀( + , 𝐺)‘𝑛) ∈ ℝ)
4330, 24syl 17 . . . . . . . . . . . . . . . 16 (((𝜑𝑥 ∈ ℝ+) ∧ (𝑚 ∈ (ℤ𝑁) ∧ 𝑛 ∈ (ℤ𝑚))) → seq𝑀( + , 𝐺):𝑍⟶ℝ)
4443, 35ffvelcdmd 7105 . . . . . . . . . . . . . . 15 (((𝜑𝑥 ∈ ℝ+) ∧ (𝑚 ∈ (ℤ𝑁) ∧ 𝑛 ∈ (ℤ𝑚))) → (seq𝑀( + , 𝐺)‘𝑚) ∈ ℝ)
4542, 44resubcld 11691 . . . . . . . . . . . . . 14 (((𝜑𝑥 ∈ ℝ+) ∧ (𝑚 ∈ (ℤ𝑁) ∧ 𝑛 ∈ (ℤ𝑚))) → ((seq𝑀( + , 𝐺)‘𝑛) − (seq𝑀( + , 𝐺)‘𝑚)) ∈ ℝ)
4635, 1eleqtrdi 2851 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑥 ∈ ℝ+) ∧ (𝑚 ∈ (ℤ𝑁) ∧ 𝑛 ∈ (ℤ𝑚))) → 𝑚 ∈ (ℤ𝑀))
47 simprr 773 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑥 ∈ ℝ+) ∧ (𝑚 ∈ (ℤ𝑁) ∧ 𝑛 ∈ (ℤ𝑚))) → 𝑛 ∈ (ℤ𝑚))
48 elfzuz 13560 . . . . . . . . . . . . . . . . . . . 20 (𝑘 ∈ (𝑀...𝑛) → 𝑘 ∈ (ℤ𝑀))
4948, 1eleqtrrdi 2852 . . . . . . . . . . . . . . . . . . 19 (𝑘 ∈ (𝑀...𝑛) → 𝑘𝑍)
50 fveq2 6906 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑚 = 𝑘 → (𝐹𝑚) = (𝐹𝑘))
51 fveq2 6906 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑚 = 𝑘 → (𝐺𝑚) = (𝐺𝑘))
5250, 51oveq12d 7449 . . . . . . . . . . . . . . . . . . . . . 22 (𝑚 = 𝑘 → ((𝐹𝑚) − (𝐺𝑚)) = ((𝐹𝑘) − (𝐺𝑘)))
53 eqid 2737 . . . . . . . . . . . . . . . . . . . . . 22 (𝑚𝑍 ↦ ((𝐹𝑚) − (𝐺𝑚))) = (𝑚𝑍 ↦ ((𝐹𝑚) − (𝐺𝑚)))
54 ovex 7464 . . . . . . . . . . . . . . . . . . . . . 22 ((𝐹𝑘) − (𝐺𝑘)) ∈ V
5552, 53, 54fvmpt 7016 . . . . . . . . . . . . . . . . . . . . 21 (𝑘𝑍 → ((𝑚𝑍 ↦ ((𝐹𝑚) − (𝐺𝑚)))‘𝑘) = ((𝐹𝑘) − (𝐺𝑘)))
5655adantl 481 . . . . . . . . . . . . . . . . . . . 20 ((𝜑𝑘𝑍) → ((𝑚𝑍 ↦ ((𝐹𝑚) − (𝐺𝑚)))‘𝑘) = ((𝐹𝑘) − (𝐺𝑘)))
5711, 23resubcld 11691 . . . . . . . . . . . . . . . . . . . 20 ((𝜑𝑘𝑍) → ((𝐹𝑘) − (𝐺𝑘)) ∈ ℝ)
5856, 57eqeltrd 2841 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑘𝑍) → ((𝑚𝑍 ↦ ((𝐹𝑚) − (𝐺𝑚)))‘𝑘) ∈ ℝ)
5930, 49, 58syl2an 596 . . . . . . . . . . . . . . . . . 18 ((((𝜑𝑥 ∈ ℝ+) ∧ (𝑚 ∈ (ℤ𝑁) ∧ 𝑛 ∈ (ℤ𝑚))) ∧ 𝑘 ∈ (𝑀...𝑛)) → ((𝑚𝑍 ↦ ((𝐹𝑚) − (𝐺𝑚)))‘𝑘) ∈ ℝ)
60 elfzuz 13560 . . . . . . . . . . . . . . . . . . 19 (𝑘 ∈ ((𝑚 + 1)...𝑛) → 𝑘 ∈ (ℤ‘(𝑚 + 1)))
61 peano2uz 12943 . . . . . . . . . . . . . . . . . . . . . 22 (𝑚 ∈ (ℤ𝑁) → (𝑚 + 1) ∈ (ℤ𝑁))
6233, 61syl 17 . . . . . . . . . . . . . . . . . . . . 21 (((𝜑𝑥 ∈ ℝ+) ∧ (𝑚 ∈ (ℤ𝑁) ∧ 𝑛 ∈ (ℤ𝑚))) → (𝑚 + 1) ∈ (ℤ𝑁))
6337uztrn2 12897 . . . . . . . . . . . . . . . . . . . . 21 (((𝑚 + 1) ∈ (ℤ𝑁) ∧ 𝑘 ∈ (ℤ‘(𝑚 + 1))) → 𝑘 ∈ (ℤ𝑁))
6462, 63sylan 580 . . . . . . . . . . . . . . . . . . . 20 ((((𝜑𝑥 ∈ ℝ+) ∧ (𝑚 ∈ (ℤ𝑁) ∧ 𝑛 ∈ (ℤ𝑚))) ∧ 𝑘 ∈ (ℤ‘(𝑚 + 1))) → 𝑘 ∈ (ℤ𝑁))
65 cvgcmp.7 . . . . . . . . . . . . . . . . . . . . . 22 ((𝜑𝑘 ∈ (ℤ𝑁)) → (𝐺𝑘) ≤ (𝐹𝑘))
661uztrn2 12897 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝑁𝑍𝑘 ∈ (ℤ𝑁)) → 𝑘𝑍)
674, 66sylan 580 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝜑𝑘 ∈ (ℤ𝑁)) → 𝑘𝑍)
6811, 23subge0d 11853 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝜑𝑘𝑍) → (0 ≤ ((𝐹𝑘) − (𝐺𝑘)) ↔ (𝐺𝑘) ≤ (𝐹𝑘)))
6967, 68syldan 591 . . . . . . . . . . . . . . . . . . . . . 22 ((𝜑𝑘 ∈ (ℤ𝑁)) → (0 ≤ ((𝐹𝑘) − (𝐺𝑘)) ↔ (𝐺𝑘) ≤ (𝐹𝑘)))
7065, 69mpbird 257 . . . . . . . . . . . . . . . . . . . . 21 ((𝜑𝑘 ∈ (ℤ𝑁)) → 0 ≤ ((𝐹𝑘) − (𝐺𝑘)))
7167, 55syl 17 . . . . . . . . . . . . . . . . . . . . 21 ((𝜑𝑘 ∈ (ℤ𝑁)) → ((𝑚𝑍 ↦ ((𝐹𝑚) − (𝐺𝑚)))‘𝑘) = ((𝐹𝑘) − (𝐺𝑘)))
7270, 71breqtrrd 5171 . . . . . . . . . . . . . . . . . . . 20 ((𝜑𝑘 ∈ (ℤ𝑁)) → 0 ≤ ((𝑚𝑍 ↦ ((𝐹𝑚) − (𝐺𝑚)))‘𝑘))
7330, 64, 72syl2an2r 685 . . . . . . . . . . . . . . . . . . 19 ((((𝜑𝑥 ∈ ℝ+) ∧ (𝑚 ∈ (ℤ𝑁) ∧ 𝑛 ∈ (ℤ𝑚))) ∧ 𝑘 ∈ (ℤ‘(𝑚 + 1))) → 0 ≤ ((𝑚𝑍 ↦ ((𝐹𝑚) − (𝐺𝑚)))‘𝑘))
7460, 73sylan2 593 . . . . . . . . . . . . . . . . . 18 ((((𝜑𝑥 ∈ ℝ+) ∧ (𝑚 ∈ (ℤ𝑁) ∧ 𝑛 ∈ (ℤ𝑚))) ∧ 𝑘 ∈ ((𝑚 + 1)...𝑛)) → 0 ≤ ((𝑚𝑍 ↦ ((𝐹𝑚) − (𝐺𝑚)))‘𝑘))
7546, 47, 59, 74sermono 14075 . . . . . . . . . . . . . . . . 17 (((𝜑𝑥 ∈ ℝ+) ∧ (𝑚 ∈ (ℤ𝑁) ∧ 𝑛 ∈ (ℤ𝑚))) → (seq𝑀( + , (𝑚𝑍 ↦ ((𝐹𝑚) − (𝐺𝑚))))‘𝑚) ≤ (seq𝑀( + , (𝑚𝑍 ↦ ((𝐹𝑚) − (𝐺𝑚))))‘𝑛))
76 elfzuz 13560 . . . . . . . . . . . . . . . . . . . 20 (𝑘 ∈ (𝑀...𝑚) → 𝑘 ∈ (ℤ𝑀))
7776, 1eleqtrrdi 2852 . . . . . . . . . . . . . . . . . . 19 (𝑘 ∈ (𝑀...𝑚) → 𝑘𝑍)
7811recnd 11289 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑘𝑍) → (𝐹𝑘) ∈ ℂ)
7930, 77, 78syl2an 596 . . . . . . . . . . . . . . . . . 18 ((((𝜑𝑥 ∈ ℝ+) ∧ (𝑚 ∈ (ℤ𝑁) ∧ 𝑛 ∈ (ℤ𝑚))) ∧ 𝑘 ∈ (𝑀...𝑚)) → (𝐹𝑘) ∈ ℂ)
8023recnd 11289 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑘𝑍) → (𝐺𝑘) ∈ ℂ)
8130, 77, 80syl2an 596 . . . . . . . . . . . . . . . . . 18 ((((𝜑𝑥 ∈ ℝ+) ∧ (𝑚 ∈ (ℤ𝑁) ∧ 𝑛 ∈ (ℤ𝑚))) ∧ 𝑘 ∈ (𝑀...𝑚)) → (𝐺𝑘) ∈ ℂ)
8230, 77, 56syl2an 596 . . . . . . . . . . . . . . . . . 18 ((((𝜑𝑥 ∈ ℝ+) ∧ (𝑚 ∈ (ℤ𝑁) ∧ 𝑛 ∈ (ℤ𝑚))) ∧ 𝑘 ∈ (𝑀...𝑚)) → ((𝑚𝑍 ↦ ((𝐹𝑚) − (𝐺𝑚)))‘𝑘) = ((𝐹𝑘) − (𝐺𝑘)))
8346, 79, 81, 82sersub 14086 . . . . . . . . . . . . . . . . 17 (((𝜑𝑥 ∈ ℝ+) ∧ (𝑚 ∈ (ℤ𝑁) ∧ 𝑛 ∈ (ℤ𝑚))) → (seq𝑀( + , (𝑚𝑍 ↦ ((𝐹𝑚) − (𝐺𝑚))))‘𝑚) = ((seq𝑀( + , 𝐹)‘𝑚) − (seq𝑀( + , 𝐺)‘𝑚)))
8440, 1eleqtrdi 2851 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑥 ∈ ℝ+) ∧ (𝑚 ∈ (ℤ𝑁) ∧ 𝑛 ∈ (ℤ𝑚))) → 𝑛 ∈ (ℤ𝑀))
8530, 49, 78syl2an 596 . . . . . . . . . . . . . . . . . 18 ((((𝜑𝑥 ∈ ℝ+) ∧ (𝑚 ∈ (ℤ𝑁) ∧ 𝑛 ∈ (ℤ𝑚))) ∧ 𝑘 ∈ (𝑀...𝑛)) → (𝐹𝑘) ∈ ℂ)
8630, 49, 80syl2an 596 . . . . . . . . . . . . . . . . . 18 ((((𝜑𝑥 ∈ ℝ+) ∧ (𝑚 ∈ (ℤ𝑁) ∧ 𝑛 ∈ (ℤ𝑚))) ∧ 𝑘 ∈ (𝑀...𝑛)) → (𝐺𝑘) ∈ ℂ)
8730, 49, 56syl2an 596 . . . . . . . . . . . . . . . . . 18 ((((𝜑𝑥 ∈ ℝ+) ∧ (𝑚 ∈ (ℤ𝑁) ∧ 𝑛 ∈ (ℤ𝑚))) ∧ 𝑘 ∈ (𝑀...𝑛)) → ((𝑚𝑍 ↦ ((𝐹𝑚) − (𝐺𝑚)))‘𝑘) = ((𝐹𝑘) − (𝐺𝑘)))
8884, 85, 86, 87sersub 14086 . . . . . . . . . . . . . . . . 17 (((𝜑𝑥 ∈ ℝ+) ∧ (𝑚 ∈ (ℤ𝑁) ∧ 𝑛 ∈ (ℤ𝑚))) → (seq𝑀( + , (𝑚𝑍 ↦ ((𝐹𝑚) − (𝐺𝑚))))‘𝑛) = ((seq𝑀( + , 𝐹)‘𝑛) − (seq𝑀( + , 𝐺)‘𝑛)))
8975, 83, 883brtr3d 5174 . . . . . . . . . . . . . . . 16 (((𝜑𝑥 ∈ ℝ+) ∧ (𝑚 ∈ (ℤ𝑁) ∧ 𝑛 ∈ (ℤ𝑚))) → ((seq𝑀( + , 𝐹)‘𝑚) − (seq𝑀( + , 𝐺)‘𝑚)) ≤ ((seq𝑀( + , 𝐹)‘𝑛) − (seq𝑀( + , 𝐺)‘𝑛)))
9041, 42resubcld 11691 . . . . . . . . . . . . . . . . 17 (((𝜑𝑥 ∈ ℝ+) ∧ (𝑚 ∈ (ℤ𝑁) ∧ 𝑛 ∈ (ℤ𝑚))) → ((seq𝑀( + , 𝐹)‘𝑛) − (seq𝑀( + , 𝐺)‘𝑛)) ∈ ℝ)
9136, 44, 90lesubaddd 11860 . . . . . . . . . . . . . . . 16 (((𝜑𝑥 ∈ ℝ+) ∧ (𝑚 ∈ (ℤ𝑁) ∧ 𝑛 ∈ (ℤ𝑚))) → (((seq𝑀( + , 𝐹)‘𝑚) − (seq𝑀( + , 𝐺)‘𝑚)) ≤ ((seq𝑀( + , 𝐹)‘𝑛) − (seq𝑀( + , 𝐺)‘𝑛)) ↔ (seq𝑀( + , 𝐹)‘𝑚) ≤ (((seq𝑀( + , 𝐹)‘𝑛) − (seq𝑀( + , 𝐺)‘𝑛)) + (seq𝑀( + , 𝐺)‘𝑚))))
9289, 91mpbid 232 . . . . . . . . . . . . . . 15 (((𝜑𝑥 ∈ ℝ+) ∧ (𝑚 ∈ (ℤ𝑁) ∧ 𝑛 ∈ (ℤ𝑚))) → (seq𝑀( + , 𝐹)‘𝑚) ≤ (((seq𝑀( + , 𝐹)‘𝑛) − (seq𝑀( + , 𝐺)‘𝑛)) + (seq𝑀( + , 𝐺)‘𝑚)))
9341recnd 11289 . . . . . . . . . . . . . . . 16 (((𝜑𝑥 ∈ ℝ+) ∧ (𝑚 ∈ (ℤ𝑁) ∧ 𝑛 ∈ (ℤ𝑚))) → (seq𝑀( + , 𝐹)‘𝑛) ∈ ℂ)
9442recnd 11289 . . . . . . . . . . . . . . . 16 (((𝜑𝑥 ∈ ℝ+) ∧ (𝑚 ∈ (ℤ𝑁) ∧ 𝑛 ∈ (ℤ𝑚))) → (seq𝑀( + , 𝐺)‘𝑛) ∈ ℂ)
9544recnd 11289 . . . . . . . . . . . . . . . 16 (((𝜑𝑥 ∈ ℝ+) ∧ (𝑚 ∈ (ℤ𝑁) ∧ 𝑛 ∈ (ℤ𝑚))) → (seq𝑀( + , 𝐺)‘𝑚) ∈ ℂ)
9693, 94, 95subsubd 11648 . . . . . . . . . . . . . . 15 (((𝜑𝑥 ∈ ℝ+) ∧ (𝑚 ∈ (ℤ𝑁) ∧ 𝑛 ∈ (ℤ𝑚))) → ((seq𝑀( + , 𝐹)‘𝑛) − ((seq𝑀( + , 𝐺)‘𝑛) − (seq𝑀( + , 𝐺)‘𝑚))) = (((seq𝑀( + , 𝐹)‘𝑛) − (seq𝑀( + , 𝐺)‘𝑛)) + (seq𝑀( + , 𝐺)‘𝑚)))
9792, 96breqtrrd 5171 . . . . . . . . . . . . . 14 (((𝜑𝑥 ∈ ℝ+) ∧ (𝑚 ∈ (ℤ𝑁) ∧ 𝑛 ∈ (ℤ𝑚))) → (seq𝑀( + , 𝐹)‘𝑚) ≤ ((seq𝑀( + , 𝐹)‘𝑛) − ((seq𝑀( + , 𝐺)‘𝑛) − (seq𝑀( + , 𝐺)‘𝑚))))
9836, 41, 45, 97lesubd 11867 . . . . . . . . . . . . 13 (((𝜑𝑥 ∈ ℝ+) ∧ (𝑚 ∈ (ℤ𝑁) ∧ 𝑛 ∈ (ℤ𝑚))) → ((seq𝑀( + , 𝐺)‘𝑛) − (seq𝑀( + , 𝐺)‘𝑚)) ≤ ((seq𝑀( + , 𝐹)‘𝑛) − (seq𝑀( + , 𝐹)‘𝑚)))
9941, 36resubcld 11691 . . . . . . . . . . . . . 14 (((𝜑𝑥 ∈ ℝ+) ∧ (𝑚 ∈ (ℤ𝑁) ∧ 𝑛 ∈ (ℤ𝑚))) → ((seq𝑀( + , 𝐹)‘𝑛) − (seq𝑀( + , 𝐹)‘𝑚)) ∈ ℝ)
100 rpre 13043 . . . . . . . . . . . . . . 15 (𝑥 ∈ ℝ+𝑥 ∈ ℝ)
101100ad2antlr 727 . . . . . . . . . . . . . 14 (((𝜑𝑥 ∈ ℝ+) ∧ (𝑚 ∈ (ℤ𝑁) ∧ 𝑛 ∈ (ℤ𝑚))) → 𝑥 ∈ ℝ)
102 lelttr 11351 . . . . . . . . . . . . . 14 ((((seq𝑀( + , 𝐺)‘𝑛) − (seq𝑀( + , 𝐺)‘𝑚)) ∈ ℝ ∧ ((seq𝑀( + , 𝐹)‘𝑛) − (seq𝑀( + , 𝐹)‘𝑚)) ∈ ℝ ∧ 𝑥 ∈ ℝ) → ((((seq𝑀( + , 𝐺)‘𝑛) − (seq𝑀( + , 𝐺)‘𝑚)) ≤ ((seq𝑀( + , 𝐹)‘𝑛) − (seq𝑀( + , 𝐹)‘𝑚)) ∧ ((seq𝑀( + , 𝐹)‘𝑛) − (seq𝑀( + , 𝐹)‘𝑚)) < 𝑥) → ((seq𝑀( + , 𝐺)‘𝑛) − (seq𝑀( + , 𝐺)‘𝑚)) < 𝑥))
10345, 99, 101, 102syl3anc 1373 . . . . . . . . . . . . 13 (((𝜑𝑥 ∈ ℝ+) ∧ (𝑚 ∈ (ℤ𝑁) ∧ 𝑛 ∈ (ℤ𝑚))) → ((((seq𝑀( + , 𝐺)‘𝑛) − (seq𝑀( + , 𝐺)‘𝑚)) ≤ ((seq𝑀( + , 𝐹)‘𝑛) − (seq𝑀( + , 𝐹)‘𝑚)) ∧ ((seq𝑀( + , 𝐹)‘𝑛) − (seq𝑀( + , 𝐹)‘𝑚)) < 𝑥) → ((seq𝑀( + , 𝐺)‘𝑛) − (seq𝑀( + , 𝐺)‘𝑚)) < 𝑥))
10498, 103mpand 695 . . . . . . . . . . . 12 (((𝜑𝑥 ∈ ℝ+) ∧ (𝑚 ∈ (ℤ𝑁) ∧ 𝑛 ∈ (ℤ𝑚))) → (((seq𝑀( + , 𝐹)‘𝑛) − (seq𝑀( + , 𝐹)‘𝑚)) < 𝑥 → ((seq𝑀( + , 𝐺)‘𝑛) − (seq𝑀( + , 𝐺)‘𝑚)) < 𝑥))
10530, 49, 11syl2an 596 . . . . . . . . . . . . . . 15 ((((𝜑𝑥 ∈ ℝ+) ∧ (𝑚 ∈ (ℤ𝑁) ∧ 𝑛 ∈ (ℤ𝑚))) ∧ 𝑘 ∈ (𝑀...𝑛)) → (𝐹𝑘) ∈ ℝ)
10660, 64sylan2 593 . . . . . . . . . . . . . . . 16 ((((𝜑𝑥 ∈ ℝ+) ∧ (𝑚 ∈ (ℤ𝑁) ∧ 𝑛 ∈ (ℤ𝑚))) ∧ 𝑘 ∈ ((𝑚 + 1)...𝑛)) → 𝑘 ∈ (ℤ𝑁))
107 0red 11264 . . . . . . . . . . . . . . . . 17 ((𝜑𝑘 ∈ (ℤ𝑁)) → 0 ∈ ℝ)
10867, 23syldan 591 . . . . . . . . . . . . . . . . 17 ((𝜑𝑘 ∈ (ℤ𝑁)) → (𝐺𝑘) ∈ ℝ)
10967, 11syldan 591 . . . . . . . . . . . . . . . . 17 ((𝜑𝑘 ∈ (ℤ𝑁)) → (𝐹𝑘) ∈ ℝ)
110 cvgcmp.6 . . . . . . . . . . . . . . . . 17 ((𝜑𝑘 ∈ (ℤ𝑁)) → 0 ≤ (𝐺𝑘))
111107, 108, 109, 110, 65letrd 11418 . . . . . . . . . . . . . . . 16 ((𝜑𝑘 ∈ (ℤ𝑁)) → 0 ≤ (𝐹𝑘))
11230, 106, 111syl2an2r 685 . . . . . . . . . . . . . . 15 ((((𝜑𝑥 ∈ ℝ+) ∧ (𝑚 ∈ (ℤ𝑁) ∧ 𝑛 ∈ (ℤ𝑚))) ∧ 𝑘 ∈ ((𝑚 + 1)...𝑛)) → 0 ≤ (𝐹𝑘))
11346, 47, 105, 112sermono 14075 . . . . . . . . . . . . . 14 (((𝜑𝑥 ∈ ℝ+) ∧ (𝑚 ∈ (ℤ𝑁) ∧ 𝑛 ∈ (ℤ𝑚))) → (seq𝑀( + , 𝐹)‘𝑚) ≤ (seq𝑀( + , 𝐹)‘𝑛))
11436, 41, 113abssubge0d 15470 . . . . . . . . . . . . 13 (((𝜑𝑥 ∈ ℝ+) ∧ (𝑚 ∈ (ℤ𝑁) ∧ 𝑛 ∈ (ℤ𝑚))) → (abs‘((seq𝑀( + , 𝐹)‘𝑛) − (seq𝑀( + , 𝐹)‘𝑚))) = ((seq𝑀( + , 𝐹)‘𝑛) − (seq𝑀( + , 𝐹)‘𝑚)))
115114breq1d 5153 . . . . . . . . . . . 12 (((𝜑𝑥 ∈ ℝ+) ∧ (𝑚 ∈ (ℤ𝑁) ∧ 𝑛 ∈ (ℤ𝑚))) → ((abs‘((seq𝑀( + , 𝐹)‘𝑛) − (seq𝑀( + , 𝐹)‘𝑚))) < 𝑥 ↔ ((seq𝑀( + , 𝐹)‘𝑛) − (seq𝑀( + , 𝐹)‘𝑚)) < 𝑥))
11630, 49, 23syl2an 596 . . . . . . . . . . . . . . 15 ((((𝜑𝑥 ∈ ℝ+) ∧ (𝑚 ∈ (ℤ𝑁) ∧ 𝑛 ∈ (ℤ𝑚))) ∧ 𝑘 ∈ (𝑀...𝑛)) → (𝐺𝑘) ∈ ℝ)
11730, 64, 110syl2an2r 685 . . . . . . . . . . . . . . . 16 ((((𝜑𝑥 ∈ ℝ+) ∧ (𝑚 ∈ (ℤ𝑁) ∧ 𝑛 ∈ (ℤ𝑚))) ∧ 𝑘 ∈ (ℤ‘(𝑚 + 1))) → 0 ≤ (𝐺𝑘))
11860, 117sylan2 593 . . . . . . . . . . . . . . 15 ((((𝜑𝑥 ∈ ℝ+) ∧ (𝑚 ∈ (ℤ𝑁) ∧ 𝑛 ∈ (ℤ𝑚))) ∧ 𝑘 ∈ ((𝑚 + 1)...𝑛)) → 0 ≤ (𝐺𝑘))
11946, 47, 116, 118sermono 14075 . . . . . . . . . . . . . 14 (((𝜑𝑥 ∈ ℝ+) ∧ (𝑚 ∈ (ℤ𝑁) ∧ 𝑛 ∈ (ℤ𝑚))) → (seq𝑀( + , 𝐺)‘𝑚) ≤ (seq𝑀( + , 𝐺)‘𝑛))
12044, 42, 119abssubge0d 15470 . . . . . . . . . . . . 13 (((𝜑𝑥 ∈ ℝ+) ∧ (𝑚 ∈ (ℤ𝑁) ∧ 𝑛 ∈ (ℤ𝑚))) → (abs‘((seq𝑀( + , 𝐺)‘𝑛) − (seq𝑀( + , 𝐺)‘𝑚))) = ((seq𝑀( + , 𝐺)‘𝑛) − (seq𝑀( + , 𝐺)‘𝑚)))
121120breq1d 5153 . . . . . . . . . . . 12 (((𝜑𝑥 ∈ ℝ+) ∧ (𝑚 ∈ (ℤ𝑁) ∧ 𝑛 ∈ (ℤ𝑚))) → ((abs‘((seq𝑀( + , 𝐺)‘𝑛) − (seq𝑀( + , 𝐺)‘𝑚))) < 𝑥 ↔ ((seq𝑀( + , 𝐺)‘𝑛) − (seq𝑀( + , 𝐺)‘𝑚)) < 𝑥))
122104, 115, 1213imtr4d 294 . . . . . . . . . . 11 (((𝜑𝑥 ∈ ℝ+) ∧ (𝑚 ∈ (ℤ𝑁) ∧ 𝑛 ∈ (ℤ𝑚))) → ((abs‘((seq𝑀( + , 𝐹)‘𝑛) − (seq𝑀( + , 𝐹)‘𝑚))) < 𝑥 → (abs‘((seq𝑀( + , 𝐺)‘𝑛) − (seq𝑀( + , 𝐺)‘𝑚))) < 𝑥))
123122anassrs 467 . . . . . . . . . 10 ((((𝜑𝑥 ∈ ℝ+) ∧ 𝑚 ∈ (ℤ𝑁)) ∧ 𝑛 ∈ (ℤ𝑚)) → ((abs‘((seq𝑀( + , 𝐹)‘𝑛) − (seq𝑀( + , 𝐹)‘𝑚))) < 𝑥 → (abs‘((seq𝑀( + , 𝐺)‘𝑛) − (seq𝑀( + , 𝐺)‘𝑚))) < 𝑥))
124123adantld 490 . . . . . . . . 9 ((((𝜑𝑥 ∈ ℝ+) ∧ 𝑚 ∈ (ℤ𝑁)) ∧ 𝑛 ∈ (ℤ𝑚)) → (((seq𝑀( + , 𝐹)‘𝑛) ∈ ℂ ∧ (abs‘((seq𝑀( + , 𝐹)‘𝑛) − (seq𝑀( + , 𝐹)‘𝑚))) < 𝑥) → (abs‘((seq𝑀( + , 𝐺)‘𝑛) − (seq𝑀( + , 𝐺)‘𝑚))) < 𝑥))
125124ralimdva 3167 . . . . . . . 8 (((𝜑𝑥 ∈ ℝ+) ∧ 𝑚 ∈ (ℤ𝑁)) → (∀𝑛 ∈ (ℤ𝑚)((seq𝑀( + , 𝐹)‘𝑛) ∈ ℂ ∧ (abs‘((seq𝑀( + , 𝐹)‘𝑛) − (seq𝑀( + , 𝐹)‘𝑚))) < 𝑥) → ∀𝑛 ∈ (ℤ𝑚)(abs‘((seq𝑀( + , 𝐺)‘𝑛) − (seq𝑀( + , 𝐺)‘𝑚))) < 𝑥))
126125reximdva 3168 . . . . . . 7 ((𝜑𝑥 ∈ ℝ+) → (∃𝑚 ∈ (ℤ𝑁)∀𝑛 ∈ (ℤ𝑚)((seq𝑀( + , 𝐹)‘𝑛) ∈ ℂ ∧ (abs‘((seq𝑀( + , 𝐹)‘𝑛) − (seq𝑀( + , 𝐹)‘𝑚))) < 𝑥) → ∃𝑚 ∈ (ℤ𝑁)∀𝑛 ∈ (ℤ𝑚)(abs‘((seq𝑀( + , 𝐺)‘𝑛) − (seq𝑀( + , 𝐺)‘𝑚))) < 𝑥))
12737r19.29uz 15389 . . . . . . 7 ((∀𝑛 ∈ (ℤ𝑁)(seq𝑀( + , 𝐺)‘𝑛) ∈ ℂ ∧ ∃𝑚 ∈ (ℤ𝑁)∀𝑛 ∈ (ℤ𝑚)(abs‘((seq𝑀( + , 𝐺)‘𝑛) − (seq𝑀( + , 𝐺)‘𝑚))) < 𝑥) → ∃𝑚 ∈ (ℤ𝑁)∀𝑛 ∈ (ℤ𝑚)((seq𝑀( + , 𝐺)‘𝑛) ∈ ℂ ∧ (abs‘((seq𝑀( + , 𝐺)‘𝑛) − (seq𝑀( + , 𝐺)‘𝑚))) < 𝑥))
12829, 126, 127syl6an 684 . . . . . 6 ((𝜑𝑥 ∈ ℝ+) → (∃𝑚 ∈ (ℤ𝑁)∀𝑛 ∈ (ℤ𝑚)((seq𝑀( + , 𝐹)‘𝑛) ∈ ℂ ∧ (abs‘((seq𝑀( + , 𝐹)‘𝑛) − (seq𝑀( + , 𝐹)‘𝑚))) < 𝑥) → ∃𝑚 ∈ (ℤ𝑁)∀𝑛 ∈ (ℤ𝑚)((seq𝑀( + , 𝐺)‘𝑛) ∈ ℂ ∧ (abs‘((seq𝑀( + , 𝐺)‘𝑛) − (seq𝑀( + , 𝐺)‘𝑚))) < 𝑥)))
129128ralimdva 3167 . . . . 5 (𝜑 → (∀𝑥 ∈ ℝ+𝑚 ∈ (ℤ𝑁)∀𝑛 ∈ (ℤ𝑚)((seq𝑀( + , 𝐹)‘𝑛) ∈ ℂ ∧ (abs‘((seq𝑀( + , 𝐹)‘𝑛) − (seq𝑀( + , 𝐹)‘𝑚))) < 𝑥) → ∀𝑥 ∈ ℝ+𝑚 ∈ (ℤ𝑁)∀𝑛 ∈ (ℤ𝑚)((seq𝑀( + , 𝐺)‘𝑛) ∈ ℂ ∧ (abs‘((seq𝑀( + , 𝐺)‘𝑛) − (seq𝑀( + , 𝐺)‘𝑚))) < 𝑥)))
1301, 37cau4 15395 . . . . . 6 (𝑁𝑍 → (∀𝑥 ∈ ℝ+𝑚𝑍𝑛 ∈ (ℤ𝑚)((seq𝑀( + , 𝐹)‘𝑛) ∈ ℂ ∧ (abs‘((seq𝑀( + , 𝐹)‘𝑛) − (seq𝑀( + , 𝐹)‘𝑚))) < 𝑥) ↔ ∀𝑥 ∈ ℝ+𝑚 ∈ (ℤ𝑁)∀𝑛 ∈ (ℤ𝑚)((seq𝑀( + , 𝐹)‘𝑛) ∈ ℂ ∧ (abs‘((seq𝑀( + , 𝐹)‘𝑛) − (seq𝑀( + , 𝐹)‘𝑚))) < 𝑥)))
1314, 130syl 17 . . . . 5 (𝜑 → (∀𝑥 ∈ ℝ+𝑚𝑍𝑛 ∈ (ℤ𝑚)((seq𝑀( + , 𝐹)‘𝑛) ∈ ℂ ∧ (abs‘((seq𝑀( + , 𝐹)‘𝑛) − (seq𝑀( + , 𝐹)‘𝑚))) < 𝑥) ↔ ∀𝑥 ∈ ℝ+𝑚 ∈ (ℤ𝑁)∀𝑛 ∈ (ℤ𝑚)((seq𝑀( + , 𝐹)‘𝑛) ∈ ℂ ∧ (abs‘((seq𝑀( + , 𝐹)‘𝑛) − (seq𝑀( + , 𝐹)‘𝑚))) < 𝑥)))
1321, 37cau4 15395 . . . . . 6 (𝑁𝑍 → (∀𝑥 ∈ ℝ+𝑚𝑍𝑛 ∈ (ℤ𝑚)((seq𝑀( + , 𝐺)‘𝑛) ∈ ℂ ∧ (abs‘((seq𝑀( + , 𝐺)‘𝑛) − (seq𝑀( + , 𝐺)‘𝑚))) < 𝑥) ↔ ∀𝑥 ∈ ℝ+𝑚 ∈ (ℤ𝑁)∀𝑛 ∈ (ℤ𝑚)((seq𝑀( + , 𝐺)‘𝑛) ∈ ℂ ∧ (abs‘((seq𝑀( + , 𝐺)‘𝑛) − (seq𝑀( + , 𝐺)‘𝑚))) < 𝑥)))
1334, 132syl 17 . . . . 5 (𝜑 → (∀𝑥 ∈ ℝ+𝑚𝑍𝑛 ∈ (ℤ𝑚)((seq𝑀( + , 𝐺)‘𝑛) ∈ ℂ ∧ (abs‘((seq𝑀( + , 𝐺)‘𝑛) − (seq𝑀( + , 𝐺)‘𝑚))) < 𝑥) ↔ ∀𝑥 ∈ ℝ+𝑚 ∈ (ℤ𝑁)∀𝑛 ∈ (ℤ𝑚)((seq𝑀( + , 𝐺)‘𝑛) ∈ ℂ ∧ (abs‘((seq𝑀( + , 𝐺)‘𝑛) − (seq𝑀( + , 𝐺)‘𝑚))) < 𝑥)))
134129, 131, 1333imtr4d 294 . . . 4 (𝜑 → (∀𝑥 ∈ ℝ+𝑚𝑍𝑛 ∈ (ℤ𝑚)((seq𝑀( + , 𝐹)‘𝑛) ∈ ℂ ∧ (abs‘((seq𝑀( + , 𝐹)‘𝑛) − (seq𝑀( + , 𝐹)‘𝑚))) < 𝑥) → ∀𝑥 ∈ ℝ+𝑚𝑍𝑛 ∈ (ℤ𝑚)((seq𝑀( + , 𝐺)‘𝑛) ∈ ℂ ∧ (abs‘((seq𝑀( + , 𝐺)‘𝑛) − (seq𝑀( + , 𝐺)‘𝑚))) < 𝑥)))
13520, 134mpd 15 . . 3 (𝜑 → ∀𝑥 ∈ ℝ+𝑚𝑍𝑛 ∈ (ℤ𝑚)((seq𝑀( + , 𝐺)‘𝑛) ∈ ℂ ∧ (abs‘((seq𝑀( + , 𝐺)‘𝑛) − (seq𝑀( + , 𝐺)‘𝑚))) < 𝑥))
1361uztrn2 12897 . . . . . . . 8 ((𝑚𝑍𝑛 ∈ (ℤ𝑚)) → 𝑛𝑍)
137 simpr 484 . . . . . . . . 9 (((seq𝑀( + , 𝐺)‘𝑛) ∈ ℂ ∧ (abs‘((seq𝑀( + , 𝐺)‘𝑛) − (seq𝑀( + , 𝐺)‘𝑚))) < 𝑥) → (abs‘((seq𝑀( + , 𝐺)‘𝑛) − (seq𝑀( + , 𝐺)‘𝑚))) < 𝑥)
13825biantrurd 532 . . . . . . . . 9 ((𝜑𝑛𝑍) → ((abs‘((seq𝑀( + , 𝐺)‘𝑛) − (seq𝑀( + , 𝐺)‘𝑚))) < 𝑥 ↔ ((seq𝑀( + , 𝐺)‘𝑛) ∈ ℝ ∧ (abs‘((seq𝑀( + , 𝐺)‘𝑛) − (seq𝑀( + , 𝐺)‘𝑚))) < 𝑥)))
139137, 138imbitrid 244 . . . . . . . 8 ((𝜑𝑛𝑍) → (((seq𝑀( + , 𝐺)‘𝑛) ∈ ℂ ∧ (abs‘((seq𝑀( + , 𝐺)‘𝑛) − (seq𝑀( + , 𝐺)‘𝑚))) < 𝑥) → ((seq𝑀( + , 𝐺)‘𝑛) ∈ ℝ ∧ (abs‘((seq𝑀( + , 𝐺)‘𝑛) − (seq𝑀( + , 𝐺)‘𝑚))) < 𝑥)))
140136, 139sylan2 593 . . . . . . 7 ((𝜑 ∧ (𝑚𝑍𝑛 ∈ (ℤ𝑚))) → (((seq𝑀( + , 𝐺)‘𝑛) ∈ ℂ ∧ (abs‘((seq𝑀( + , 𝐺)‘𝑛) − (seq𝑀( + , 𝐺)‘𝑚))) < 𝑥) → ((seq𝑀( + , 𝐺)‘𝑛) ∈ ℝ ∧ (abs‘((seq𝑀( + , 𝐺)‘𝑛) − (seq𝑀( + , 𝐺)‘𝑚))) < 𝑥)))
141140anassrs 467 . . . . . 6 (((𝜑𝑚𝑍) ∧ 𝑛 ∈ (ℤ𝑚)) → (((seq𝑀( + , 𝐺)‘𝑛) ∈ ℂ ∧ (abs‘((seq𝑀( + , 𝐺)‘𝑛) − (seq𝑀( + , 𝐺)‘𝑚))) < 𝑥) → ((seq𝑀( + , 𝐺)‘𝑛) ∈ ℝ ∧ (abs‘((seq𝑀( + , 𝐺)‘𝑛) − (seq𝑀( + , 𝐺)‘𝑚))) < 𝑥)))
142141ralimdva 3167 . . . . 5 ((𝜑𝑚𝑍) → (∀𝑛 ∈ (ℤ𝑚)((seq𝑀( + , 𝐺)‘𝑛) ∈ ℂ ∧ (abs‘((seq𝑀( + , 𝐺)‘𝑛) − (seq𝑀( + , 𝐺)‘𝑚))) < 𝑥) → ∀𝑛 ∈ (ℤ𝑚)((seq𝑀( + , 𝐺)‘𝑛) ∈ ℝ ∧ (abs‘((seq𝑀( + , 𝐺)‘𝑛) − (seq𝑀( + , 𝐺)‘𝑚))) < 𝑥)))
143142reximdva 3168 . . . 4 (𝜑 → (∃𝑚𝑍𝑛 ∈ (ℤ𝑚)((seq𝑀( + , 𝐺)‘𝑛) ∈ ℂ ∧ (abs‘((seq𝑀( + , 𝐺)‘𝑛) − (seq𝑀( + , 𝐺)‘𝑚))) < 𝑥) → ∃𝑚𝑍𝑛 ∈ (ℤ𝑚)((seq𝑀( + , 𝐺)‘𝑛) ∈ ℝ ∧ (abs‘((seq𝑀( + , 𝐺)‘𝑛) − (seq𝑀( + , 𝐺)‘𝑚))) < 𝑥)))
144143ralimdv 3169 . . 3 (𝜑 → (∀𝑥 ∈ ℝ+𝑚𝑍𝑛 ∈ (ℤ𝑚)((seq𝑀( + , 𝐺)‘𝑛) ∈ ℂ ∧ (abs‘((seq𝑀( + , 𝐺)‘𝑛) − (seq𝑀( + , 𝐺)‘𝑚))) < 𝑥) → ∀𝑥 ∈ ℝ+𝑚𝑍𝑛 ∈ (ℤ𝑚)((seq𝑀( + , 𝐺)‘𝑛) ∈ ℝ ∧ (abs‘((seq𝑀( + , 𝐺)‘𝑛) − (seq𝑀( + , 𝐺)‘𝑚))) < 𝑥)))
145135, 144mpd 15 . 2 (𝜑 → ∀𝑥 ∈ ℝ+𝑚𝑍𝑛 ∈ (ℤ𝑚)((seq𝑀( + , 𝐺)‘𝑛) ∈ ℝ ∧ (abs‘((seq𝑀( + , 𝐺)‘𝑛) − (seq𝑀( + , 𝐺)‘𝑚))) < 𝑥))
1461, 3, 145caurcvg2 15714 1 (𝜑 → seq𝑀( + , 𝐺) ∈ dom ⇝ )
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395   = wceq 1540  wcel 2108  wral 3061  wrex 3070  Vcvv 3480   class class class wbr 5143  cmpt 5225  dom cdm 5685  wf 6557  cfv 6561  (class class class)co 7431  cc 11153  cr 11154  0cc0 11155  1c1 11156   + caddc 11158   < clt 11295  cle 11296  cmin 11492  cz 12613  cuz 12878  +crp 13034  ...cfz 13547  seqcseq 14042  abscabs 15273  cli 15520
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755  ax-inf2 9681  ax-cnex 11211  ax-resscn 11212  ax-1cn 11213  ax-icn 11214  ax-addcl 11215  ax-addrcl 11216  ax-mulcl 11217  ax-mulrcl 11218  ax-mulcom 11219  ax-addass 11220  ax-mulass 11221  ax-distr 11222  ax-i2m1 11223  ax-1ne0 11224  ax-1rid 11225  ax-rnegex 11226  ax-rrecex 11227  ax-cnre 11228  ax-pre-lttri 11229  ax-pre-lttrn 11230  ax-pre-ltadd 11231  ax-pre-mulgt0 11232  ax-pre-sup 11233
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-rmo 3380  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-pred 6321  df-ord 6387  df-on 6388  df-lim 6389  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-1st 8014  df-2nd 8015  df-frecs 8306  df-wrecs 8337  df-recs 8411  df-rdg 8450  df-er 8745  df-pm 8869  df-en 8986  df-dom 8987  df-sdom 8988  df-sup 9482  df-inf 9483  df-pnf 11297  df-mnf 11298  df-xr 11299  df-ltxr 11300  df-le 11301  df-sub 11494  df-neg 11495  df-div 11921  df-nn 12267  df-2 12329  df-3 12330  df-n0 12527  df-z 12614  df-uz 12879  df-rp 13035  df-ico 13393  df-fz 13548  df-fzo 13695  df-fl 13832  df-seq 14043  df-exp 14103  df-cj 15138  df-re 15139  df-im 15140  df-sqrt 15274  df-abs 15275  df-limsup 15507  df-clim 15524  df-rlim 15525
This theorem is referenced by:  cvgcmpce  15854  rpnnen2lem5  16254  aaliou3lem3  26386
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