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Theorem cvgcmp 15166
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 13366 . . 3 seq𝑀( + , 𝐺) ∈ V
32a1i 11 . 2 (𝜑 → seq𝑀( + , 𝐺) ∈ V)
4 cvgcmp.2 . . . . . . . 8 (𝜑𝑁𝑍)
54, 1syl6eleq 2928 . . . . . . 7 (𝜑𝑁 ∈ (ℤ𝑀))
6 eluzel2 12242 . . . . . . 7 (𝑁 ∈ (ℤ𝑀) → 𝑀 ∈ ℤ)
75, 6syl 17 . . . . . 6 (𝜑𝑀 ∈ ℤ)
8 cvgcmp.5 . . . . . 6 (𝜑 → seq𝑀( + , 𝐹) ∈ dom ⇝ )
91climcau 15022 . . . . . 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 13394 . . . . . . . . . 10 (𝜑 → seq𝑀( + , 𝐹):𝑍⟶ℝ)
1312ffvelrnda 6849 . . . . . . . . 9 ((𝜑𝑛𝑍) → (seq𝑀( + , 𝐹)‘𝑛) ∈ ℝ)
1413recnd 10663 . . . . . . . 8 ((𝜑𝑛𝑍) → (seq𝑀( + , 𝐹)‘𝑛) ∈ ℂ)
1514ralrimiva 3187 . . . . . . 7 (𝜑 → ∀𝑛𝑍 (seq𝑀( + , 𝐹)‘𝑛) ∈ ℂ)
161r19.29uz 14705 . . . . . . . 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 413 . . . . . . 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 3183 . . . . 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 12256 . . . . . . . . . . 11 ((𝑁𝑍𝑛 ∈ (ℤ𝑁)) → 𝑛𝑍)
224, 21sylan 580 . . . . . . . . . 10 ((𝜑𝑛 ∈ (ℤ𝑁)) → 𝑛𝑍)
23 cvgcmp.4 . . . . . . . . . . . . 13 ((𝜑𝑘𝑍) → (𝐺𝑘) ∈ ℝ)
241, 7, 23serfre 13394 . . . . . . . . . . . 12 (𝜑 → seq𝑀( + , 𝐺):𝑍⟶ℝ)
2524ffvelrnda 6849 . . . . . . . . . . 11 ((𝜑𝑛𝑍) → (seq𝑀( + , 𝐺)‘𝑛) ∈ ℝ)
2625recnd 10663 . . . . . . . . . 10 ((𝜑𝑛𝑍) → (seq𝑀( + , 𝐺)‘𝑛) ∈ ℂ)
2722, 26syldan 591 . . . . . . . . 9 ((𝜑𝑛 ∈ (ℤ𝑁)) → (seq𝑀( + , 𝐺)‘𝑛) ∈ ℂ)
2827ralrimiva 3187 . . . . . . . 8 (𝜑 → ∀𝑛 ∈ (ℤ𝑁)(seq𝑀( + , 𝐺)‘𝑛) ∈ ℂ)
2928adantr 481 . . . . . . 7 ((𝜑𝑥 ∈ ℝ+) → ∀𝑛 ∈ (ℤ𝑁)(seq𝑀( + , 𝐺)‘𝑛) ∈ ℂ)
30 simpll 763 . . . . . . . . . . . . . . . 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 767 . . . . . . . . . . . . . . . 16 (((𝜑𝑥 ∈ ℝ+) ∧ (𝑚 ∈ (ℤ𝑁) ∧ 𝑛 ∈ (ℤ𝑚))) → 𝑚 ∈ (ℤ𝑁))
341uztrn2 12256 . . . . . . . . . . . . . . . 16 ((𝑁𝑍𝑚 ∈ (ℤ𝑁)) → 𝑚𝑍)
3532, 33, 34syl2anc 584 . . . . . . . . . . . . . . 15 (((𝜑𝑥 ∈ ℝ+) ∧ (𝑚 ∈ (ℤ𝑁) ∧ 𝑛 ∈ (ℤ𝑚))) → 𝑚𝑍)
3631, 35ffvelrnd 6850 . . . . . . . . . . . . . 14 (((𝜑𝑥 ∈ ℝ+) ∧ (𝑚 ∈ (ℤ𝑁) ∧ 𝑛 ∈ (ℤ𝑚))) → (seq𝑀( + , 𝐹)‘𝑚) ∈ ℝ)
37 eqid 2826 . . . . . . . . . . . . . . . . . 18 (ℤ𝑁) = (ℤ𝑁)
3837uztrn2 12256 . . . . . . . . . . . . . . . . 17 ((𝑚 ∈ (ℤ𝑁) ∧ 𝑛 ∈ (ℤ𝑚)) → 𝑛 ∈ (ℤ𝑁))
3938adantl 482 . . . . . . . . . . . . . . . 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, 35ffvelrnd 6850 . . . . . . . . . . . . . . 15 (((𝜑𝑥 ∈ ℝ+) ∧ (𝑚 ∈ (ℤ𝑁) ∧ 𝑛 ∈ (ℤ𝑚))) → (seq𝑀( + , 𝐺)‘𝑚) ∈ ℝ)
4542, 44resubcld 11062 . . . . . . . . . . . . . 14 (((𝜑𝑥 ∈ ℝ+) ∧ (𝑚 ∈ (ℤ𝑁) ∧ 𝑛 ∈ (ℤ𝑚))) → ((seq𝑀( + , 𝐺)‘𝑛) − (seq𝑀( + , 𝐺)‘𝑚)) ∈ ℝ)
4635, 1syl6eleq 2928 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑥 ∈ ℝ+) ∧ (𝑚 ∈ (ℤ𝑁) ∧ 𝑛 ∈ (ℤ𝑚))) → 𝑚 ∈ (ℤ𝑀))
47 simprr 769 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑥 ∈ ℝ+) ∧ (𝑚 ∈ (ℤ𝑁) ∧ 𝑛 ∈ (ℤ𝑚))) → 𝑛 ∈ (ℤ𝑚))
48 elfzuz 12899 . . . . . . . . . . . . . . . . . . . 20 (𝑘 ∈ (𝑀...𝑛) → 𝑘 ∈ (ℤ𝑀))
4948, 1syl6eleqr 2929 . . . . . . . . . . . . . . . . . . 19 (𝑘 ∈ (𝑀...𝑛) → 𝑘𝑍)
50 fveq2 6669 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑚 = 𝑘 → (𝐹𝑚) = (𝐹𝑘))
51 fveq2 6669 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑚 = 𝑘 → (𝐺𝑚) = (𝐺𝑘))
5250, 51oveq12d 7168 . . . . . . . . . . . . . . . . . . . . . 22 (𝑚 = 𝑘 → ((𝐹𝑚) − (𝐺𝑚)) = ((𝐹𝑘) − (𝐺𝑘)))
53 eqid 2826 . . . . . . . . . . . . . . . . . . . . . 22 (𝑚𝑍 ↦ ((𝐹𝑚) − (𝐺𝑚))) = (𝑚𝑍 ↦ ((𝐹𝑚) − (𝐺𝑚)))
54 ovex 7183 . . . . . . . . . . . . . . . . . . . . . 22 ((𝐹𝑘) − (𝐺𝑘)) ∈ V
5552, 53, 54fvmpt 6767 . . . . . . . . . . . . . . . . . . . . 21 (𝑘𝑍 → ((𝑚𝑍 ↦ ((𝐹𝑚) − (𝐺𝑚)))‘𝑘) = ((𝐹𝑘) − (𝐺𝑘)))
5655adantl 482 . . . . . . . . . . . . . . . . . . . 20 ((𝜑𝑘𝑍) → ((𝑚𝑍 ↦ ((𝐹𝑚) − (𝐺𝑚)))‘𝑘) = ((𝐹𝑘) − (𝐺𝑘)))
5711, 23resubcld 11062 . . . . . . . . . . . . . . . . . . . 20 ((𝜑𝑘𝑍) → ((𝐹𝑘) − (𝐺𝑘)) ∈ ℝ)
5856, 57eqeltrd 2918 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑘𝑍) → ((𝑚𝑍 ↦ ((𝐹𝑚) − (𝐺𝑚)))‘𝑘) ∈ ℝ)
5930, 49, 58syl2an 595 . . . . . . . . . . . . . . . . . 18 ((((𝜑𝑥 ∈ ℝ+) ∧ (𝑚 ∈ (ℤ𝑁) ∧ 𝑛 ∈ (ℤ𝑚))) ∧ 𝑘 ∈ (𝑀...𝑛)) → ((𝑚𝑍 ↦ ((𝐹𝑚) − (𝐺𝑚)))‘𝑘) ∈ ℝ)
60 elfzuz 12899 . . . . . . . . . . . . . . . . . . 19 (𝑘 ∈ ((𝑚 + 1)...𝑛) → 𝑘 ∈ (ℤ‘(𝑚 + 1)))
61 peano2uz 12295 . . . . . . . . . . . . . . . . . . . . . 22 (𝑚 ∈ (ℤ𝑁) → (𝑚 + 1) ∈ (ℤ𝑁))
6233, 61syl 17 . . . . . . . . . . . . . . . . . . . . 21 (((𝜑𝑥 ∈ ℝ+) ∧ (𝑚 ∈ (ℤ𝑁) ∧ 𝑛 ∈ (ℤ𝑚))) → (𝑚 + 1) ∈ (ℤ𝑁))
6337uztrn2 12256 . . . . . . . . . . . . . . . . . . . . 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 12256 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝑁𝑍𝑘 ∈ (ℤ𝑁)) → 𝑘𝑍)
674, 66sylan 580 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝜑𝑘 ∈ (ℤ𝑁)) → 𝑘𝑍)
6811, 23subge0d 11224 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝜑𝑘𝑍) → (0 ≤ ((𝐹𝑘) − (𝐺𝑘)) ↔ (𝐺𝑘) ≤ (𝐹𝑘)))
6967, 68syldan 591 . . . . . . . . . . . . . . . . . . . . . 22 ((𝜑𝑘 ∈ (ℤ𝑁)) → (0 ≤ ((𝐹𝑘) − (𝐺𝑘)) ↔ (𝐺𝑘) ≤ (𝐹𝑘)))
7065, 69mpbird 258 . . . . . . . . . . . . . . . . . . . . 21 ((𝜑𝑘 ∈ (ℤ𝑁)) → 0 ≤ ((𝐹𝑘) − (𝐺𝑘)))
7167, 55syl 17 . . . . . . . . . . . . . . . . . . . . 21 ((𝜑𝑘 ∈ (ℤ𝑁)) → ((𝑚𝑍 ↦ ((𝐹𝑚) − (𝐺𝑚)))‘𝑘) = ((𝐹𝑘) − (𝐺𝑘)))
7270, 71breqtrrd 5091 . . . . . . . . . . . . . . . . . . . 20 ((𝜑𝑘 ∈ (ℤ𝑁)) → 0 ≤ ((𝑚𝑍 ↦ ((𝐹𝑚) − (𝐺𝑚)))‘𝑘))
7330, 64, 72syl2an2r 681 . . . . . . . . . . . . . . . . . . 19 ((((𝜑𝑥 ∈ ℝ+) ∧ (𝑚 ∈ (ℤ𝑁) ∧ 𝑛 ∈ (ℤ𝑚))) ∧ 𝑘 ∈ (ℤ‘(𝑚 + 1))) → 0 ≤ ((𝑚𝑍 ↦ ((𝐹𝑚) − (𝐺𝑚)))‘𝑘))
7460, 73sylan2 592 . . . . . . . . . . . . . . . . . 18 ((((𝜑𝑥 ∈ ℝ+) ∧ (𝑚 ∈ (ℤ𝑁) ∧ 𝑛 ∈ (ℤ𝑚))) ∧ 𝑘 ∈ ((𝑚 + 1)...𝑛)) → 0 ≤ ((𝑚𝑍 ↦ ((𝐹𝑚) − (𝐺𝑚)))‘𝑘))
7546, 47, 59, 74sermono 13397 . . . . . . . . . . . . . . . . 17 (((𝜑𝑥 ∈ ℝ+) ∧ (𝑚 ∈ (ℤ𝑁) ∧ 𝑛 ∈ (ℤ𝑚))) → (seq𝑀( + , (𝑚𝑍 ↦ ((𝐹𝑚) − (𝐺𝑚))))‘𝑚) ≤ (seq𝑀( + , (𝑚𝑍 ↦ ((𝐹𝑚) − (𝐺𝑚))))‘𝑛))
76 elfzuz 12899 . . . . . . . . . . . . . . . . . . . 20 (𝑘 ∈ (𝑀...𝑚) → 𝑘 ∈ (ℤ𝑀))
7776, 1syl6eleqr 2929 . . . . . . . . . . . . . . . . . . 19 (𝑘 ∈ (𝑀...𝑚) → 𝑘𝑍)
7811recnd 10663 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑘𝑍) → (𝐹𝑘) ∈ ℂ)
7930, 77, 78syl2an 595 . . . . . . . . . . . . . . . . . 18 ((((𝜑𝑥 ∈ ℝ+) ∧ (𝑚 ∈ (ℤ𝑁) ∧ 𝑛 ∈ (ℤ𝑚))) ∧ 𝑘 ∈ (𝑀...𝑚)) → (𝐹𝑘) ∈ ℂ)
8023recnd 10663 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑘𝑍) → (𝐺𝑘) ∈ ℂ)
8130, 77, 80syl2an 595 . . . . . . . . . . . . . . . . . 18 ((((𝜑𝑥 ∈ ℝ+) ∧ (𝑚 ∈ (ℤ𝑁) ∧ 𝑛 ∈ (ℤ𝑚))) ∧ 𝑘 ∈ (𝑀...𝑚)) → (𝐺𝑘) ∈ ℂ)
8230, 77, 56syl2an 595 . . . . . . . . . . . . . . . . . 18 ((((𝜑𝑥 ∈ ℝ+) ∧ (𝑚 ∈ (ℤ𝑁) ∧ 𝑛 ∈ (ℤ𝑚))) ∧ 𝑘 ∈ (𝑀...𝑚)) → ((𝑚𝑍 ↦ ((𝐹𝑚) − (𝐺𝑚)))‘𝑘) = ((𝐹𝑘) − (𝐺𝑘)))
8346, 79, 81, 82sersub 13408 . . . . . . . . . . . . . . . . 17 (((𝜑𝑥 ∈ ℝ+) ∧ (𝑚 ∈ (ℤ𝑁) ∧ 𝑛 ∈ (ℤ𝑚))) → (seq𝑀( + , (𝑚𝑍 ↦ ((𝐹𝑚) − (𝐺𝑚))))‘𝑚) = ((seq𝑀( + , 𝐹)‘𝑚) − (seq𝑀( + , 𝐺)‘𝑚)))
8440, 1syl6eleq 2928 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑥 ∈ ℝ+) ∧ (𝑚 ∈ (ℤ𝑁) ∧ 𝑛 ∈ (ℤ𝑚))) → 𝑛 ∈ (ℤ𝑀))
8530, 49, 78syl2an 595 . . . . . . . . . . . . . . . . . 18 ((((𝜑𝑥 ∈ ℝ+) ∧ (𝑚 ∈ (ℤ𝑁) ∧ 𝑛 ∈ (ℤ𝑚))) ∧ 𝑘 ∈ (𝑀...𝑛)) → (𝐹𝑘) ∈ ℂ)
8630, 49, 80syl2an 595 . . . . . . . . . . . . . . . . . 18 ((((𝜑𝑥 ∈ ℝ+) ∧ (𝑚 ∈ (ℤ𝑁) ∧ 𝑛 ∈ (ℤ𝑚))) ∧ 𝑘 ∈ (𝑀...𝑛)) → (𝐺𝑘) ∈ ℂ)
8730, 49, 56syl2an 595 . . . . . . . . . . . . . . . . . 18 ((((𝜑𝑥 ∈ ℝ+) ∧ (𝑚 ∈ (ℤ𝑁) ∧ 𝑛 ∈ (ℤ𝑚))) ∧ 𝑘 ∈ (𝑀...𝑛)) → ((𝑚𝑍 ↦ ((𝐹𝑚) − (𝐺𝑚)))‘𝑘) = ((𝐹𝑘) − (𝐺𝑘)))
8884, 85, 86, 87sersub 13408 . . . . . . . . . . . . . . . . 17 (((𝜑𝑥 ∈ ℝ+) ∧ (𝑚 ∈ (ℤ𝑁) ∧ 𝑛 ∈ (ℤ𝑚))) → (seq𝑀( + , (𝑚𝑍 ↦ ((𝐹𝑚) − (𝐺𝑚))))‘𝑛) = ((seq𝑀( + , 𝐹)‘𝑛) − (seq𝑀( + , 𝐺)‘𝑛)))
8975, 83, 883brtr3d 5094 . . . . . . . . . . . . . . . 16 (((𝜑𝑥 ∈ ℝ+) ∧ (𝑚 ∈ (ℤ𝑁) ∧ 𝑛 ∈ (ℤ𝑚))) → ((seq𝑀( + , 𝐹)‘𝑚) − (seq𝑀( + , 𝐺)‘𝑚)) ≤ ((seq𝑀( + , 𝐹)‘𝑛) − (seq𝑀( + , 𝐺)‘𝑛)))
9041, 42resubcld 11062 . . . . . . . . . . . . . . . . 17 (((𝜑𝑥 ∈ ℝ+) ∧ (𝑚 ∈ (ℤ𝑁) ∧ 𝑛 ∈ (ℤ𝑚))) → ((seq𝑀( + , 𝐹)‘𝑛) − (seq𝑀( + , 𝐺)‘𝑛)) ∈ ℝ)
9136, 44, 90lesubaddd 11231 . . . . . . . . . . . . . . . 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 233 . . . . . . . . . . . . . . 15 (((𝜑𝑥 ∈ ℝ+) ∧ (𝑚 ∈ (ℤ𝑁) ∧ 𝑛 ∈ (ℤ𝑚))) → (seq𝑀( + , 𝐹)‘𝑚) ≤ (((seq𝑀( + , 𝐹)‘𝑛) − (seq𝑀( + , 𝐺)‘𝑛)) + (seq𝑀( + , 𝐺)‘𝑚)))
9341recnd 10663 . . . . . . . . . . . . . . . 16 (((𝜑𝑥 ∈ ℝ+) ∧ (𝑚 ∈ (ℤ𝑁) ∧ 𝑛 ∈ (ℤ𝑚))) → (seq𝑀( + , 𝐹)‘𝑛) ∈ ℂ)
9442recnd 10663 . . . . . . . . . . . . . . . 16 (((𝜑𝑥 ∈ ℝ+) ∧ (𝑚 ∈ (ℤ𝑁) ∧ 𝑛 ∈ (ℤ𝑚))) → (seq𝑀( + , 𝐺)‘𝑛) ∈ ℂ)
9544recnd 10663 . . . . . . . . . . . . . . . 16 (((𝜑𝑥 ∈ ℝ+) ∧ (𝑚 ∈ (ℤ𝑁) ∧ 𝑛 ∈ (ℤ𝑚))) → (seq𝑀( + , 𝐺)‘𝑚) ∈ ℂ)
9693, 94, 95subsubd 11019 . . . . . . . . . . . . . . 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 5091 . . . . . . . . . . . . . 14 (((𝜑𝑥 ∈ ℝ+) ∧ (𝑚 ∈ (ℤ𝑁) ∧ 𝑛 ∈ (ℤ𝑚))) → (seq𝑀( + , 𝐹)‘𝑚) ≤ ((seq𝑀( + , 𝐹)‘𝑛) − ((seq𝑀( + , 𝐺)‘𝑛) − (seq𝑀( + , 𝐺)‘𝑚))))
9836, 41, 45, 97lesubd 11238 . . . . . . . . . . . . 13 (((𝜑𝑥 ∈ ℝ+) ∧ (𝑚 ∈ (ℤ𝑁) ∧ 𝑛 ∈ (ℤ𝑚))) → ((seq𝑀( + , 𝐺)‘𝑛) − (seq𝑀( + , 𝐺)‘𝑚)) ≤ ((seq𝑀( + , 𝐹)‘𝑛) − (seq𝑀( + , 𝐹)‘𝑚)))
9941, 36resubcld 11062 . . . . . . . . . . . . . 14 (((𝜑𝑥 ∈ ℝ+) ∧ (𝑚 ∈ (ℤ𝑁) ∧ 𝑛 ∈ (ℤ𝑚))) → ((seq𝑀( + , 𝐹)‘𝑛) − (seq𝑀( + , 𝐹)‘𝑚)) ∈ ℝ)
100 rpre 12392 . . . . . . . . . . . . . . 15 (𝑥 ∈ ℝ+𝑥 ∈ ℝ)
101100ad2antlr 723 . . . . . . . . . . . . . 14 (((𝜑𝑥 ∈ ℝ+) ∧ (𝑚 ∈ (ℤ𝑁) ∧ 𝑛 ∈ (ℤ𝑚))) → 𝑥 ∈ ℝ)
102 lelttr 10725 . . . . . . . . . . . . . 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 1365 . . . . . . . . . . . . 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 691 . . . . . . . . . . . 12 (((𝜑𝑥 ∈ ℝ+) ∧ (𝑚 ∈ (ℤ𝑁) ∧ 𝑛 ∈ (ℤ𝑚))) → (((seq𝑀( + , 𝐹)‘𝑛) − (seq𝑀( + , 𝐹)‘𝑚)) < 𝑥 → ((seq𝑀( + , 𝐺)‘𝑛) − (seq𝑀( + , 𝐺)‘𝑚)) < 𝑥))
10530, 49, 11syl2an 595 . . . . . . . . . . . . . . 15 ((((𝜑𝑥 ∈ ℝ+) ∧ (𝑚 ∈ (ℤ𝑁) ∧ 𝑛 ∈ (ℤ𝑚))) ∧ 𝑘 ∈ (𝑀...𝑛)) → (𝐹𝑘) ∈ ℝ)
10660, 64sylan2 592 . . . . . . . . . . . . . . . 16 ((((𝜑𝑥 ∈ ℝ+) ∧ (𝑚 ∈ (ℤ𝑁) ∧ 𝑛 ∈ (ℤ𝑚))) ∧ 𝑘 ∈ ((𝑚 + 1)...𝑛)) → 𝑘 ∈ (ℤ𝑁))
107 0red 10638 . . . . . . . . . . . . . . . . 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 10791 . . . . . . . . . . . . . . . 16 ((𝜑𝑘 ∈ (ℤ𝑁)) → 0 ≤ (𝐹𝑘))
11230, 106, 111syl2an2r 681 . . . . . . . . . . . . . . 15 ((((𝜑𝑥 ∈ ℝ+) ∧ (𝑚 ∈ (ℤ𝑁) ∧ 𝑛 ∈ (ℤ𝑚))) ∧ 𝑘 ∈ ((𝑚 + 1)...𝑛)) → 0 ≤ (𝐹𝑘))
11346, 47, 105, 112sermono 13397 . . . . . . . . . . . . . 14 (((𝜑𝑥 ∈ ℝ+) ∧ (𝑚 ∈ (ℤ𝑁) ∧ 𝑛 ∈ (ℤ𝑚))) → (seq𝑀( + , 𝐹)‘𝑚) ≤ (seq𝑀( + , 𝐹)‘𝑛))
11436, 41, 113abssubge0d 14786 . . . . . . . . . . . . 13 (((𝜑𝑥 ∈ ℝ+) ∧ (𝑚 ∈ (ℤ𝑁) ∧ 𝑛 ∈ (ℤ𝑚))) → (abs‘((seq𝑀( + , 𝐹)‘𝑛) − (seq𝑀( + , 𝐹)‘𝑚))) = ((seq𝑀( + , 𝐹)‘𝑛) − (seq𝑀( + , 𝐹)‘𝑚)))
115114breq1d 5073 . . . . . . . . . . . 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 595 . . . . . . . . . . . . . . 15 ((((𝜑𝑥 ∈ ℝ+) ∧ (𝑚 ∈ (ℤ𝑁) ∧ 𝑛 ∈ (ℤ𝑚))) ∧ 𝑘 ∈ (𝑀...𝑛)) → (𝐺𝑘) ∈ ℝ)
11730, 64, 110syl2an2r 681 . . . . . . . . . . . . . . . 16 ((((𝜑𝑥 ∈ ℝ+) ∧ (𝑚 ∈ (ℤ𝑁) ∧ 𝑛 ∈ (ℤ𝑚))) ∧ 𝑘 ∈ (ℤ‘(𝑚 + 1))) → 0 ≤ (𝐺𝑘))
11860, 117sylan2 592 . . . . . . . . . . . . . . 15 ((((𝜑𝑥 ∈ ℝ+) ∧ (𝑚 ∈ (ℤ𝑁) ∧ 𝑛 ∈ (ℤ𝑚))) ∧ 𝑘 ∈ ((𝑚 + 1)...𝑛)) → 0 ≤ (𝐺𝑘))
11946, 47, 116, 118sermono 13397 . . . . . . . . . . . . . 14 (((𝜑𝑥 ∈ ℝ+) ∧ (𝑚 ∈ (ℤ𝑁) ∧ 𝑛 ∈ (ℤ𝑚))) → (seq𝑀( + , 𝐺)‘𝑚) ≤ (seq𝑀( + , 𝐺)‘𝑛))
12044, 42, 119abssubge0d 14786 . . . . . . . . . . . . 13 (((𝜑𝑥 ∈ ℝ+) ∧ (𝑚 ∈ (ℤ𝑁) ∧ 𝑛 ∈ (ℤ𝑚))) → (abs‘((seq𝑀( + , 𝐺)‘𝑛) − (seq𝑀( + , 𝐺)‘𝑚))) = ((seq𝑀( + , 𝐺)‘𝑛) − (seq𝑀( + , 𝐺)‘𝑚)))
121120breq1d 5073 . . . . . . . . . . . 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 295 . . . . . . . . . . 11 (((𝜑𝑥 ∈ ℝ+) ∧ (𝑚 ∈ (ℤ𝑁) ∧ 𝑛 ∈ (ℤ𝑚))) → ((abs‘((seq𝑀( + , 𝐹)‘𝑛) − (seq𝑀( + , 𝐹)‘𝑚))) < 𝑥 → (abs‘((seq𝑀( + , 𝐺)‘𝑛) − (seq𝑀( + , 𝐺)‘𝑚))) < 𝑥))
123122anassrs 468 . . . . . . . . . 10 ((((𝜑𝑥 ∈ ℝ+) ∧ 𝑚 ∈ (ℤ𝑁)) ∧ 𝑛 ∈ (ℤ𝑚)) → ((abs‘((seq𝑀( + , 𝐹)‘𝑛) − (seq𝑀( + , 𝐹)‘𝑚))) < 𝑥 → (abs‘((seq𝑀( + , 𝐺)‘𝑛) − (seq𝑀( + , 𝐺)‘𝑚))) < 𝑥))
124123adantld 491 . . . . . . . . 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 3182 . . . . . . . 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 3279 . . . . . . 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 14705 . . . . . . 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 680 . . . . . 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 3182 . . . . 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 14711 . . . . . 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 14711 . . . . . 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 295 . . . 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 12256 . . . . . . . 8 ((𝑚𝑍𝑛 ∈ (ℤ𝑚)) → 𝑛𝑍)
137 simpr 485 . . . . . . . . 9 (((seq𝑀( + , 𝐺)‘𝑛) ∈ ℂ ∧ (abs‘((seq𝑀( + , 𝐺)‘𝑛) − (seq𝑀( + , 𝐺)‘𝑚))) < 𝑥) → (abs‘((seq𝑀( + , 𝐺)‘𝑛) − (seq𝑀( + , 𝐺)‘𝑚))) < 𝑥)
13825biantrurd 533 . . . . . . . . 9 ((𝜑𝑛𝑍) → ((abs‘((seq𝑀( + , 𝐺)‘𝑛) − (seq𝑀( + , 𝐺)‘𝑚))) < 𝑥 ↔ ((seq𝑀( + , 𝐺)‘𝑛) ∈ ℝ ∧ (abs‘((seq𝑀( + , 𝐺)‘𝑛) − (seq𝑀( + , 𝐺)‘𝑚))) < 𝑥)))
139137, 138syl5ib 245 . . . . . . . 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 592 . . . . . . 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 468 . . . . . 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 3182 . . . . 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 3279 . . . 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 3183 . . 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 15029 1 (𝜑 → seq𝑀( + , 𝐺) ∈ dom ⇝ )
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 207  wa 396   = wceq 1530  wcel 2107  wral 3143  wrex 3144  Vcvv 3500   class class class wbr 5063  cmpt 5143  dom cdm 5554  wf 6350  cfv 6354  (class class class)co 7150  cc 10529  cr 10530  0cc0 10531  1c1 10532   + caddc 10534   < clt 10669  cle 10670  cmin 10864  cz 11975  cuz 12237  +crp 12384  ...cfz 12887  seqcseq 13364  abscabs 14588  cli 14836
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 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2798  ax-rep 5187  ax-sep 5200  ax-nul 5207  ax-pow 5263  ax-pr 5326  ax-un 7455  ax-inf2 9098  ax-cnex 10587  ax-resscn 10588  ax-1cn 10589  ax-icn 10590  ax-addcl 10591  ax-addrcl 10592  ax-mulcl 10593  ax-mulrcl 10594  ax-mulcom 10595  ax-addass 10596  ax-mulass 10597  ax-distr 10598  ax-i2m1 10599  ax-1ne0 10600  ax-1rid 10601  ax-rnegex 10602  ax-rrecex 10603  ax-cnre 10604  ax-pre-lttri 10605  ax-pre-lttrn 10606  ax-pre-ltadd 10607  ax-pre-mulgt0 10608  ax-pre-sup 10609
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-3or 1082  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2620  df-eu 2652  df-clab 2805  df-cleq 2819  df-clel 2898  df-nfc 2968  df-ne 3022  df-nel 3129  df-ral 3148  df-rex 3149  df-reu 3150  df-rmo 3151  df-rab 3152  df-v 3502  df-sbc 3777  df-csb 3888  df-dif 3943  df-un 3945  df-in 3947  df-ss 3956  df-pss 3958  df-nul 4296  df-if 4471  df-pw 4544  df-sn 4565  df-pr 4567  df-tp 4569  df-op 4571  df-uni 4838  df-iun 4919  df-br 5064  df-opab 5126  df-mpt 5144  df-tr 5170  df-id 5459  df-eprel 5464  df-po 5473  df-so 5474  df-fr 5513  df-we 5515  df-xp 5560  df-rel 5561  df-cnv 5562  df-co 5563  df-dm 5564  df-rn 5565  df-res 5566  df-ima 5567  df-pred 6147  df-ord 6193  df-on 6194  df-lim 6195  df-suc 6196  df-iota 6313  df-fun 6356  df-fn 6357  df-f 6358  df-f1 6359  df-fo 6360  df-f1o 6361  df-fv 6362  df-riota 7108  df-ov 7153  df-oprab 7154  df-mpo 7155  df-om 7574  df-1st 7685  df-2nd 7686  df-wrecs 7943  df-recs 8004  df-rdg 8042  df-er 8284  df-pm 8404  df-en 8504  df-dom 8505  df-sdom 8506  df-sup 8900  df-inf 8901  df-pnf 10671  df-mnf 10672  df-xr 10673  df-ltxr 10674  df-le 10675  df-sub 10866  df-neg 10867  df-div 11292  df-nn 11633  df-2 11694  df-3 11695  df-n0 11892  df-z 11976  df-uz 12238  df-rp 12385  df-ico 12739  df-fz 12888  df-fzo 13029  df-fl 13157  df-seq 13365  df-exp 13425  df-cj 14453  df-re 14454  df-im 14455  df-sqrt 14589  df-abs 14590  df-limsup 14823  df-clim 14840  df-rlim 14841
This theorem is referenced by:  cvgcmpce  15168  rpnnen2lem5  15566  aaliou3lem3  24867
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