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Theorem cvgcmp 15233
 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 13434 . . 3 seq𝑀( + , 𝐺) ∈ V
32a1i 11 . 2 (𝜑 → seq𝑀( + , 𝐺) ∈ V)
4 cvgcmp.2 . . . . . . . 8 (𝜑𝑁𝑍)
54, 1eleqtrdi 2863 . . . . . . 7 (𝜑𝑁 ∈ (ℤ𝑀))
6 eluzel2 12301 . . . . . . 7 (𝑁 ∈ (ℤ𝑀) → 𝑀 ∈ ℤ)
75, 6syl 17 . . . . . 6 (𝜑𝑀 ∈ ℤ)
8 cvgcmp.5 . . . . . 6 (𝜑 → seq𝑀( + , 𝐹) ∈ dom ⇝ )
91climcau 15089 . . . . . 6 ((𝑀 ∈ ℤ ∧ seq𝑀( + , 𝐹) ∈ dom ⇝ ) → ∀𝑥 ∈ ℝ+𝑚𝑍𝑛 ∈ (ℤ𝑚)(abs‘((seq𝑀( + , 𝐹)‘𝑛) − (seq𝑀( + , 𝐹)‘𝑚))) < 𝑥)
107, 8, 9syl2anc 587 . . . . 5 (𝜑 → ∀𝑥 ∈ ℝ+𝑚𝑍𝑛 ∈ (ℤ𝑚)(abs‘((seq𝑀( + , 𝐹)‘𝑛) − (seq𝑀( + , 𝐹)‘𝑚))) < 𝑥)
11 cvgcmp.3 . . . . . . . . . . 11 ((𝜑𝑘𝑍) → (𝐹𝑘) ∈ ℝ)
121, 7, 11serfre 13463 . . . . . . . . . 10 (𝜑 → seq𝑀( + , 𝐹):𝑍⟶ℝ)
1312ffvelrnda 6849 . . . . . . . . 9 ((𝜑𝑛𝑍) → (seq𝑀( + , 𝐹)‘𝑛) ∈ ℝ)
1413recnd 10721 . . . . . . . 8 ((𝜑𝑛𝑍) → (seq𝑀( + , 𝐹)‘𝑛) ∈ ℂ)
1514ralrimiva 3114 . . . . . . 7 (𝜑 → ∀𝑛𝑍 (seq𝑀( + , 𝐹)‘𝑛) ∈ ℂ)
161r19.29uz 14772 . . . . . . . 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 416 . . . . . . 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 3110 . . . . 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 12315 . . . . . . . . . . 11 ((𝑁𝑍𝑛 ∈ (ℤ𝑁)) → 𝑛𝑍)
224, 21sylan 583 . . . . . . . . . 10 ((𝜑𝑛 ∈ (ℤ𝑁)) → 𝑛𝑍)
23 cvgcmp.4 . . . . . . . . . . . . 13 ((𝜑𝑘𝑍) → (𝐺𝑘) ∈ ℝ)
241, 7, 23serfre 13463 . . . . . . . . . . . 12 (𝜑 → seq𝑀( + , 𝐺):𝑍⟶ℝ)
2524ffvelrnda 6849 . . . . . . . . . . 11 ((𝜑𝑛𝑍) → (seq𝑀( + , 𝐺)‘𝑛) ∈ ℝ)
2625recnd 10721 . . . . . . . . . 10 ((𝜑𝑛𝑍) → (seq𝑀( + , 𝐺)‘𝑛) ∈ ℂ)
2722, 26syldan 594 . . . . . . . . 9 ((𝜑𝑛 ∈ (ℤ𝑁)) → (seq𝑀( + , 𝐺)‘𝑛) ∈ ℂ)
2827ralrimiva 3114 . . . . . . . 8 (𝜑 → ∀𝑛 ∈ (ℤ𝑁)(seq𝑀( + , 𝐺)‘𝑛) ∈ ℂ)
2928adantr 484 . . . . . . 7 ((𝜑𝑥 ∈ ℝ+) → ∀𝑛 ∈ (ℤ𝑁)(seq𝑀( + , 𝐺)‘𝑛) ∈ ℂ)
30 simpll 766 . . . . . . . . . . . . . . . 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 770 . . . . . . . . . . . . . . . 16 (((𝜑𝑥 ∈ ℝ+) ∧ (𝑚 ∈ (ℤ𝑁) ∧ 𝑛 ∈ (ℤ𝑚))) → 𝑚 ∈ (ℤ𝑁))
341uztrn2 12315 . . . . . . . . . . . . . . . 16 ((𝑁𝑍𝑚 ∈ (ℤ𝑁)) → 𝑚𝑍)
3532, 33, 34syl2anc 587 . . . . . . . . . . . . . . 15 (((𝜑𝑥 ∈ ℝ+) ∧ (𝑚 ∈ (ℤ𝑁) ∧ 𝑛 ∈ (ℤ𝑚))) → 𝑚𝑍)
3631, 35ffvelrnd 6850 . . . . . . . . . . . . . 14 (((𝜑𝑥 ∈ ℝ+) ∧ (𝑚 ∈ (ℤ𝑁) ∧ 𝑛 ∈ (ℤ𝑚))) → (seq𝑀( + , 𝐹)‘𝑚) ∈ ℝ)
37 eqid 2759 . . . . . . . . . . . . . . . . . 18 (ℤ𝑁) = (ℤ𝑁)
3837uztrn2 12315 . . . . . . . . . . . . . . . . 17 ((𝑚 ∈ (ℤ𝑁) ∧ 𝑛 ∈ (ℤ𝑚)) → 𝑛 ∈ (ℤ𝑁))
3938adantl 485 . . . . . . . . . . . . . . . 16 (((𝜑𝑥 ∈ ℝ+) ∧ (𝑚 ∈ (ℤ𝑁) ∧ 𝑛 ∈ (ℤ𝑚))) → 𝑛 ∈ (ℤ𝑁))
4032, 39, 21syl2anc 587 . . . . . . . . . . . . . . 15 (((𝜑𝑥 ∈ ℝ+) ∧ (𝑚 ∈ (ℤ𝑁) ∧ 𝑛 ∈ (ℤ𝑚))) → 𝑛𝑍)
4130, 40, 13syl2anc 587 . . . . . . . . . . . . . 14 (((𝜑𝑥 ∈ ℝ+) ∧ (𝑚 ∈ (ℤ𝑁) ∧ 𝑛 ∈ (ℤ𝑚))) → (seq𝑀( + , 𝐹)‘𝑛) ∈ ℝ)
4230, 40, 25syl2anc 587 . . . . . . . . . . . . . . 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 11120 . . . . . . . . . . . . . 14 (((𝜑𝑥 ∈ ℝ+) ∧ (𝑚 ∈ (ℤ𝑁) ∧ 𝑛 ∈ (ℤ𝑚))) → ((seq𝑀( + , 𝐺)‘𝑛) − (seq𝑀( + , 𝐺)‘𝑚)) ∈ ℝ)
4635, 1eleqtrdi 2863 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑥 ∈ ℝ+) ∧ (𝑚 ∈ (ℤ𝑁) ∧ 𝑛 ∈ (ℤ𝑚))) → 𝑚 ∈ (ℤ𝑀))
47 simprr 772 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑥 ∈ ℝ+) ∧ (𝑚 ∈ (ℤ𝑁) ∧ 𝑛 ∈ (ℤ𝑚))) → 𝑛 ∈ (ℤ𝑚))
48 elfzuz 12966 . . . . . . . . . . . . . . . . . . . 20 (𝑘 ∈ (𝑀...𝑛) → 𝑘 ∈ (ℤ𝑀))
4948, 1eleqtrrdi 2864 . . . . . . . . . . . . . . . . . . 19 (𝑘 ∈ (𝑀...𝑛) → 𝑘𝑍)
50 fveq2 6664 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑚 = 𝑘 → (𝐹𝑚) = (𝐹𝑘))
51 fveq2 6664 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑚 = 𝑘 → (𝐺𝑚) = (𝐺𝑘))
5250, 51oveq12d 7175 . . . . . . . . . . . . . . . . . . . . . 22 (𝑚 = 𝑘 → ((𝐹𝑚) − (𝐺𝑚)) = ((𝐹𝑘) − (𝐺𝑘)))
53 eqid 2759 . . . . . . . . . . . . . . . . . . . . . 22 (𝑚𝑍 ↦ ((𝐹𝑚) − (𝐺𝑚))) = (𝑚𝑍 ↦ ((𝐹𝑚) − (𝐺𝑚)))
54 ovex 7190 . . . . . . . . . . . . . . . . . . . . . 22 ((𝐹𝑘) − (𝐺𝑘)) ∈ V
5552, 53, 54fvmpt 6765 . . . . . . . . . . . . . . . . . . . . 21 (𝑘𝑍 → ((𝑚𝑍 ↦ ((𝐹𝑚) − (𝐺𝑚)))‘𝑘) = ((𝐹𝑘) − (𝐺𝑘)))
5655adantl 485 . . . . . . . . . . . . . . . . . . . 20 ((𝜑𝑘𝑍) → ((𝑚𝑍 ↦ ((𝐹𝑚) − (𝐺𝑚)))‘𝑘) = ((𝐹𝑘) − (𝐺𝑘)))
5711, 23resubcld 11120 . . . . . . . . . . . . . . . . . . . 20 ((𝜑𝑘𝑍) → ((𝐹𝑘) − (𝐺𝑘)) ∈ ℝ)
5856, 57eqeltrd 2853 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑘𝑍) → ((𝑚𝑍 ↦ ((𝐹𝑚) − (𝐺𝑚)))‘𝑘) ∈ ℝ)
5930, 49, 58syl2an 598 . . . . . . . . . . . . . . . . . 18 ((((𝜑𝑥 ∈ ℝ+) ∧ (𝑚 ∈ (ℤ𝑁) ∧ 𝑛 ∈ (ℤ𝑚))) ∧ 𝑘 ∈ (𝑀...𝑛)) → ((𝑚𝑍 ↦ ((𝐹𝑚) − (𝐺𝑚)))‘𝑘) ∈ ℝ)
60 elfzuz 12966 . . . . . . . . . . . . . . . . . . 19 (𝑘 ∈ ((𝑚 + 1)...𝑛) → 𝑘 ∈ (ℤ‘(𝑚 + 1)))
61 peano2uz 12355 . . . . . . . . . . . . . . . . . . . . . 22 (𝑚 ∈ (ℤ𝑁) → (𝑚 + 1) ∈ (ℤ𝑁))
6233, 61syl 17 . . . . . . . . . . . . . . . . . . . . 21 (((𝜑𝑥 ∈ ℝ+) ∧ (𝑚 ∈ (ℤ𝑁) ∧ 𝑛 ∈ (ℤ𝑚))) → (𝑚 + 1) ∈ (ℤ𝑁))
6337uztrn2 12315 . . . . . . . . . . . . . . . . . . . . 21 (((𝑚 + 1) ∈ (ℤ𝑁) ∧ 𝑘 ∈ (ℤ‘(𝑚 + 1))) → 𝑘 ∈ (ℤ𝑁))
6462, 63sylan 583 . . . . . . . . . . . . . . . . . . . 20 ((((𝜑𝑥 ∈ ℝ+) ∧ (𝑚 ∈ (ℤ𝑁) ∧ 𝑛 ∈ (ℤ𝑚))) ∧ 𝑘 ∈ (ℤ‘(𝑚 + 1))) → 𝑘 ∈ (ℤ𝑁))
65 cvgcmp.7 . . . . . . . . . . . . . . . . . . . . . 22 ((𝜑𝑘 ∈ (ℤ𝑁)) → (𝐺𝑘) ≤ (𝐹𝑘))
661uztrn2 12315 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝑁𝑍𝑘 ∈ (ℤ𝑁)) → 𝑘𝑍)
674, 66sylan 583 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝜑𝑘 ∈ (ℤ𝑁)) → 𝑘𝑍)
6811, 23subge0d 11282 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝜑𝑘𝑍) → (0 ≤ ((𝐹𝑘) − (𝐺𝑘)) ↔ (𝐺𝑘) ≤ (𝐹𝑘)))
6967, 68syldan 594 . . . . . . . . . . . . . . . . . . . . . 22 ((𝜑𝑘 ∈ (ℤ𝑁)) → (0 ≤ ((𝐹𝑘) − (𝐺𝑘)) ↔ (𝐺𝑘) ≤ (𝐹𝑘)))
7065, 69mpbird 260 . . . . . . . . . . . . . . . . . . . . 21 ((𝜑𝑘 ∈ (ℤ𝑁)) → 0 ≤ ((𝐹𝑘) − (𝐺𝑘)))
7167, 55syl 17 . . . . . . . . . . . . . . . . . . . . 21 ((𝜑𝑘 ∈ (ℤ𝑁)) → ((𝑚𝑍 ↦ ((𝐹𝑚) − (𝐺𝑚)))‘𝑘) = ((𝐹𝑘) − (𝐺𝑘)))
7270, 71breqtrrd 5065 . . . . . . . . . . . . . . . . . . . 20 ((𝜑𝑘 ∈ (ℤ𝑁)) → 0 ≤ ((𝑚𝑍 ↦ ((𝐹𝑚) − (𝐺𝑚)))‘𝑘))
7330, 64, 72syl2an2r 684 . . . . . . . . . . . . . . . . . . 19 ((((𝜑𝑥 ∈ ℝ+) ∧ (𝑚 ∈ (ℤ𝑁) ∧ 𝑛 ∈ (ℤ𝑚))) ∧ 𝑘 ∈ (ℤ‘(𝑚 + 1))) → 0 ≤ ((𝑚𝑍 ↦ ((𝐹𝑚) − (𝐺𝑚)))‘𝑘))
7460, 73sylan2 595 . . . . . . . . . . . . . . . . . 18 ((((𝜑𝑥 ∈ ℝ+) ∧ (𝑚 ∈ (ℤ𝑁) ∧ 𝑛 ∈ (ℤ𝑚))) ∧ 𝑘 ∈ ((𝑚 + 1)...𝑛)) → 0 ≤ ((𝑚𝑍 ↦ ((𝐹𝑚) − (𝐺𝑚)))‘𝑘))
7546, 47, 59, 74sermono 13466 . . . . . . . . . . . . . . . . 17 (((𝜑𝑥 ∈ ℝ+) ∧ (𝑚 ∈ (ℤ𝑁) ∧ 𝑛 ∈ (ℤ𝑚))) → (seq𝑀( + , (𝑚𝑍 ↦ ((𝐹𝑚) − (𝐺𝑚))))‘𝑚) ≤ (seq𝑀( + , (𝑚𝑍 ↦ ((𝐹𝑚) − (𝐺𝑚))))‘𝑛))
76 elfzuz 12966 . . . . . . . . . . . . . . . . . . . 20 (𝑘 ∈ (𝑀...𝑚) → 𝑘 ∈ (ℤ𝑀))
7776, 1eleqtrrdi 2864 . . . . . . . . . . . . . . . . . . 19 (𝑘 ∈ (𝑀...𝑚) → 𝑘𝑍)
7811recnd 10721 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑘𝑍) → (𝐹𝑘) ∈ ℂ)
7930, 77, 78syl2an 598 . . . . . . . . . . . . . . . . . 18 ((((𝜑𝑥 ∈ ℝ+) ∧ (𝑚 ∈ (ℤ𝑁) ∧ 𝑛 ∈ (ℤ𝑚))) ∧ 𝑘 ∈ (𝑀...𝑚)) → (𝐹𝑘) ∈ ℂ)
8023recnd 10721 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑘𝑍) → (𝐺𝑘) ∈ ℂ)
8130, 77, 80syl2an 598 . . . . . . . . . . . . . . . . . 18 ((((𝜑𝑥 ∈ ℝ+) ∧ (𝑚 ∈ (ℤ𝑁) ∧ 𝑛 ∈ (ℤ𝑚))) ∧ 𝑘 ∈ (𝑀...𝑚)) → (𝐺𝑘) ∈ ℂ)
8230, 77, 56syl2an 598 . . . . . . . . . . . . . . . . . 18 ((((𝜑𝑥 ∈ ℝ+) ∧ (𝑚 ∈ (ℤ𝑁) ∧ 𝑛 ∈ (ℤ𝑚))) ∧ 𝑘 ∈ (𝑀...𝑚)) → ((𝑚𝑍 ↦ ((𝐹𝑚) − (𝐺𝑚)))‘𝑘) = ((𝐹𝑘) − (𝐺𝑘)))
8346, 79, 81, 82sersub 13477 . . . . . . . . . . . . . . . . 17 (((𝜑𝑥 ∈ ℝ+) ∧ (𝑚 ∈ (ℤ𝑁) ∧ 𝑛 ∈ (ℤ𝑚))) → (seq𝑀( + , (𝑚𝑍 ↦ ((𝐹𝑚) − (𝐺𝑚))))‘𝑚) = ((seq𝑀( + , 𝐹)‘𝑚) − (seq𝑀( + , 𝐺)‘𝑚)))
8440, 1eleqtrdi 2863 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑥 ∈ ℝ+) ∧ (𝑚 ∈ (ℤ𝑁) ∧ 𝑛 ∈ (ℤ𝑚))) → 𝑛 ∈ (ℤ𝑀))
8530, 49, 78syl2an 598 . . . . . . . . . . . . . . . . . 18 ((((𝜑𝑥 ∈ ℝ+) ∧ (𝑚 ∈ (ℤ𝑁) ∧ 𝑛 ∈ (ℤ𝑚))) ∧ 𝑘 ∈ (𝑀...𝑛)) → (𝐹𝑘) ∈ ℂ)
8630, 49, 80syl2an 598 . . . . . . . . . . . . . . . . . 18 ((((𝜑𝑥 ∈ ℝ+) ∧ (𝑚 ∈ (ℤ𝑁) ∧ 𝑛 ∈ (ℤ𝑚))) ∧ 𝑘 ∈ (𝑀...𝑛)) → (𝐺𝑘) ∈ ℂ)
8730, 49, 56syl2an 598 . . . . . . . . . . . . . . . . . 18 ((((𝜑𝑥 ∈ ℝ+) ∧ (𝑚 ∈ (ℤ𝑁) ∧ 𝑛 ∈ (ℤ𝑚))) ∧ 𝑘 ∈ (𝑀...𝑛)) → ((𝑚𝑍 ↦ ((𝐹𝑚) − (𝐺𝑚)))‘𝑘) = ((𝐹𝑘) − (𝐺𝑘)))
8884, 85, 86, 87sersub 13477 . . . . . . . . . . . . . . . . 17 (((𝜑𝑥 ∈ ℝ+) ∧ (𝑚 ∈ (ℤ𝑁) ∧ 𝑛 ∈ (ℤ𝑚))) → (seq𝑀( + , (𝑚𝑍 ↦ ((𝐹𝑚) − (𝐺𝑚))))‘𝑛) = ((seq𝑀( + , 𝐹)‘𝑛) − (seq𝑀( + , 𝐺)‘𝑛)))
8975, 83, 883brtr3d 5068 . . . . . . . . . . . . . . . 16 (((𝜑𝑥 ∈ ℝ+) ∧ (𝑚 ∈ (ℤ𝑁) ∧ 𝑛 ∈ (ℤ𝑚))) → ((seq𝑀( + , 𝐹)‘𝑚) − (seq𝑀( + , 𝐺)‘𝑚)) ≤ ((seq𝑀( + , 𝐹)‘𝑛) − (seq𝑀( + , 𝐺)‘𝑛)))
9041, 42resubcld 11120 . . . . . . . . . . . . . . . . 17 (((𝜑𝑥 ∈ ℝ+) ∧ (𝑚 ∈ (ℤ𝑁) ∧ 𝑛 ∈ (ℤ𝑚))) → ((seq𝑀( + , 𝐹)‘𝑛) − (seq𝑀( + , 𝐺)‘𝑛)) ∈ ℝ)
9136, 44, 90lesubaddd 11289 . . . . . . . . . . . . . . . 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 235 . . . . . . . . . . . . . . 15 (((𝜑𝑥 ∈ ℝ+) ∧ (𝑚 ∈ (ℤ𝑁) ∧ 𝑛 ∈ (ℤ𝑚))) → (seq𝑀( + , 𝐹)‘𝑚) ≤ (((seq𝑀( + , 𝐹)‘𝑛) − (seq𝑀( + , 𝐺)‘𝑛)) + (seq𝑀( + , 𝐺)‘𝑚)))
9341recnd 10721 . . . . . . . . . . . . . . . 16 (((𝜑𝑥 ∈ ℝ+) ∧ (𝑚 ∈ (ℤ𝑁) ∧ 𝑛 ∈ (ℤ𝑚))) → (seq𝑀( + , 𝐹)‘𝑛) ∈ ℂ)
9442recnd 10721 . . . . . . . . . . . . . . . 16 (((𝜑𝑥 ∈ ℝ+) ∧ (𝑚 ∈ (ℤ𝑁) ∧ 𝑛 ∈ (ℤ𝑚))) → (seq𝑀( + , 𝐺)‘𝑛) ∈ ℂ)
9544recnd 10721 . . . . . . . . . . . . . . . 16 (((𝜑𝑥 ∈ ℝ+) ∧ (𝑚 ∈ (ℤ𝑁) ∧ 𝑛 ∈ (ℤ𝑚))) → (seq𝑀( + , 𝐺)‘𝑚) ∈ ℂ)
9693, 94, 95subsubd 11077 . . . . . . . . . . . . . . 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 5065 . . . . . . . . . . . . . 14 (((𝜑𝑥 ∈ ℝ+) ∧ (𝑚 ∈ (ℤ𝑁) ∧ 𝑛 ∈ (ℤ𝑚))) → (seq𝑀( + , 𝐹)‘𝑚) ≤ ((seq𝑀( + , 𝐹)‘𝑛) − ((seq𝑀( + , 𝐺)‘𝑛) − (seq𝑀( + , 𝐺)‘𝑚))))
9836, 41, 45, 97lesubd 11296 . . . . . . . . . . . . 13 (((𝜑𝑥 ∈ ℝ+) ∧ (𝑚 ∈ (ℤ𝑁) ∧ 𝑛 ∈ (ℤ𝑚))) → ((seq𝑀( + , 𝐺)‘𝑛) − (seq𝑀( + , 𝐺)‘𝑚)) ≤ ((seq𝑀( + , 𝐹)‘𝑛) − (seq𝑀( + , 𝐹)‘𝑚)))
9941, 36resubcld 11120 . . . . . . . . . . . . . 14 (((𝜑𝑥 ∈ ℝ+) ∧ (𝑚 ∈ (ℤ𝑁) ∧ 𝑛 ∈ (ℤ𝑚))) → ((seq𝑀( + , 𝐹)‘𝑛) − (seq𝑀( + , 𝐹)‘𝑚)) ∈ ℝ)
100 rpre 12452 . . . . . . . . . . . . . . 15 (𝑥 ∈ ℝ+𝑥 ∈ ℝ)
101100ad2antlr 726 . . . . . . . . . . . . . 14 (((𝜑𝑥 ∈ ℝ+) ∧ (𝑚 ∈ (ℤ𝑁) ∧ 𝑛 ∈ (ℤ𝑚))) → 𝑥 ∈ ℝ)
102 lelttr 10783 . . . . . . . . . . . . . 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 1369 . . . . . . . . . . . . 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 694 . . . . . . . . . . . 12 (((𝜑𝑥 ∈ ℝ+) ∧ (𝑚 ∈ (ℤ𝑁) ∧ 𝑛 ∈ (ℤ𝑚))) → (((seq𝑀( + , 𝐹)‘𝑛) − (seq𝑀( + , 𝐹)‘𝑚)) < 𝑥 → ((seq𝑀( + , 𝐺)‘𝑛) − (seq𝑀( + , 𝐺)‘𝑚)) < 𝑥))
10530, 49, 11syl2an 598 . . . . . . . . . . . . . . 15 ((((𝜑𝑥 ∈ ℝ+) ∧ (𝑚 ∈ (ℤ𝑁) ∧ 𝑛 ∈ (ℤ𝑚))) ∧ 𝑘 ∈ (𝑀...𝑛)) → (𝐹𝑘) ∈ ℝ)
10660, 64sylan2 595 . . . . . . . . . . . . . . . 16 ((((𝜑𝑥 ∈ ℝ+) ∧ (𝑚 ∈ (ℤ𝑁) ∧ 𝑛 ∈ (ℤ𝑚))) ∧ 𝑘 ∈ ((𝑚 + 1)...𝑛)) → 𝑘 ∈ (ℤ𝑁))
107 0red 10696 . . . . . . . . . . . . . . . . 17 ((𝜑𝑘 ∈ (ℤ𝑁)) → 0 ∈ ℝ)
10867, 23syldan 594 . . . . . . . . . . . . . . . . 17 ((𝜑𝑘 ∈ (ℤ𝑁)) → (𝐺𝑘) ∈ ℝ)
10967, 11syldan 594 . . . . . . . . . . . . . . . . 17 ((𝜑𝑘 ∈ (ℤ𝑁)) → (𝐹𝑘) ∈ ℝ)
110 cvgcmp.6 . . . . . . . . . . . . . . . . 17 ((𝜑𝑘 ∈ (ℤ𝑁)) → 0 ≤ (𝐺𝑘))
111107, 108, 109, 110, 65letrd 10849 . . . . . . . . . . . . . . . 16 ((𝜑𝑘 ∈ (ℤ𝑁)) → 0 ≤ (𝐹𝑘))
11230, 106, 111syl2an2r 684 . . . . . . . . . . . . . . 15 ((((𝜑𝑥 ∈ ℝ+) ∧ (𝑚 ∈ (ℤ𝑁) ∧ 𝑛 ∈ (ℤ𝑚))) ∧ 𝑘 ∈ ((𝑚 + 1)...𝑛)) → 0 ≤ (𝐹𝑘))
11346, 47, 105, 112sermono 13466 . . . . . . . . . . . . . 14 (((𝜑𝑥 ∈ ℝ+) ∧ (𝑚 ∈ (ℤ𝑁) ∧ 𝑛 ∈ (ℤ𝑚))) → (seq𝑀( + , 𝐹)‘𝑚) ≤ (seq𝑀( + , 𝐹)‘𝑛))
11436, 41, 113abssubge0d 14853 . . . . . . . . . . . . 13 (((𝜑𝑥 ∈ ℝ+) ∧ (𝑚 ∈ (ℤ𝑁) ∧ 𝑛 ∈ (ℤ𝑚))) → (abs‘((seq𝑀( + , 𝐹)‘𝑛) − (seq𝑀( + , 𝐹)‘𝑚))) = ((seq𝑀( + , 𝐹)‘𝑛) − (seq𝑀( + , 𝐹)‘𝑚)))
115114breq1d 5047 . . . . . . . . . . . 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 598 . . . . . . . . . . . . . . 15 ((((𝜑𝑥 ∈ ℝ+) ∧ (𝑚 ∈ (ℤ𝑁) ∧ 𝑛 ∈ (ℤ𝑚))) ∧ 𝑘 ∈ (𝑀...𝑛)) → (𝐺𝑘) ∈ ℝ)
11730, 64, 110syl2an2r 684 . . . . . . . . . . . . . . . 16 ((((𝜑𝑥 ∈ ℝ+) ∧ (𝑚 ∈ (ℤ𝑁) ∧ 𝑛 ∈ (ℤ𝑚))) ∧ 𝑘 ∈ (ℤ‘(𝑚 + 1))) → 0 ≤ (𝐺𝑘))
11860, 117sylan2 595 . . . . . . . . . . . . . . 15 ((((𝜑𝑥 ∈ ℝ+) ∧ (𝑚 ∈ (ℤ𝑁) ∧ 𝑛 ∈ (ℤ𝑚))) ∧ 𝑘 ∈ ((𝑚 + 1)...𝑛)) → 0 ≤ (𝐺𝑘))
11946, 47, 116, 118sermono 13466 . . . . . . . . . . . . . 14 (((𝜑𝑥 ∈ ℝ+) ∧ (𝑚 ∈ (ℤ𝑁) ∧ 𝑛 ∈ (ℤ𝑚))) → (seq𝑀( + , 𝐺)‘𝑚) ≤ (seq𝑀( + , 𝐺)‘𝑛))
12044, 42, 119abssubge0d 14853 . . . . . . . . . . . . 13 (((𝜑𝑥 ∈ ℝ+) ∧ (𝑚 ∈ (ℤ𝑁) ∧ 𝑛 ∈ (ℤ𝑚))) → (abs‘((seq𝑀( + , 𝐺)‘𝑛) − (seq𝑀( + , 𝐺)‘𝑚))) = ((seq𝑀( + , 𝐺)‘𝑛) − (seq𝑀( + , 𝐺)‘𝑚)))
121120breq1d 5047 . . . . . . . . . . . 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 297 . . . . . . . . . . 11 (((𝜑𝑥 ∈ ℝ+) ∧ (𝑚 ∈ (ℤ𝑁) ∧ 𝑛 ∈ (ℤ𝑚))) → ((abs‘((seq𝑀( + , 𝐹)‘𝑛) − (seq𝑀( + , 𝐹)‘𝑚))) < 𝑥 → (abs‘((seq𝑀( + , 𝐺)‘𝑛) − (seq𝑀( + , 𝐺)‘𝑚))) < 𝑥))
123122anassrs 471 . . . . . . . . . 10 ((((𝜑𝑥 ∈ ℝ+) ∧ 𝑚 ∈ (ℤ𝑁)) ∧ 𝑛 ∈ (ℤ𝑚)) → ((abs‘((seq𝑀( + , 𝐹)‘𝑛) − (seq𝑀( + , 𝐹)‘𝑚))) < 𝑥 → (abs‘((seq𝑀( + , 𝐺)‘𝑛) − (seq𝑀( + , 𝐺)‘𝑚))) < 𝑥))
124123adantld 494 . . . . . . . . 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 3109 . . . . . . . 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 3199 . . . . . . 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 14772 . . . . . . 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 683 . . . . . 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 3109 . . . . 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 14778 . . . . . 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 14778 . . . . . 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 297 . . . 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 12315 . . . . . . . 8 ((𝑚𝑍𝑛 ∈ (ℤ𝑚)) → 𝑛𝑍)
137 simpr 488 . . . . . . . . 9 (((seq𝑀( + , 𝐺)‘𝑛) ∈ ℂ ∧ (abs‘((seq𝑀( + , 𝐺)‘𝑛) − (seq𝑀( + , 𝐺)‘𝑚))) < 𝑥) → (abs‘((seq𝑀( + , 𝐺)‘𝑛) − (seq𝑀( + , 𝐺)‘𝑚))) < 𝑥)
13825biantrurd 536 . . . . . . . . 9 ((𝜑𝑛𝑍) → ((abs‘((seq𝑀( + , 𝐺)‘𝑛) − (seq𝑀( + , 𝐺)‘𝑚))) < 𝑥 ↔ ((seq𝑀( + , 𝐺)‘𝑛) ∈ ℝ ∧ (abs‘((seq𝑀( + , 𝐺)‘𝑛) − (seq𝑀( + , 𝐺)‘𝑚))) < 𝑥)))
139137, 138syl5ib 247 . . . . . . . 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 595 . . . . . . 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 471 . . . . . 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 3109 . . . . 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 3199 . . . 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 3110 . . 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 15096 1 (𝜑 → seq𝑀( + , 𝐺) ∈ dom ⇝ )
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399   = wceq 1539   ∈ wcel 2112  ∀wral 3071  ∃wrex 3072  Vcvv 3410   class class class wbr 5037   ↦ cmpt 5117  dom cdm 5529  ⟶wf 6337  ‘cfv 6341  (class class class)co 7157  ℂcc 10587  ℝcr 10588  0cc0 10589  1c1 10590   + caddc 10592   < clt 10727   ≤ cle 10728   − cmin 10922  ℤcz 12034  ℤ≥cuz 12296  ℝ+crp 12444  ...cfz 12953  seqcseq 13432  abscabs 14655   ⇝ cli 14903 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 1912  ax-6 1971  ax-7 2016  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2730  ax-rep 5161  ax-sep 5174  ax-nul 5181  ax-pow 5239  ax-pr 5303  ax-un 7466  ax-inf2 9151  ax-cnex 10645  ax-resscn 10646  ax-1cn 10647  ax-icn 10648  ax-addcl 10649  ax-addrcl 10650  ax-mulcl 10651  ax-mulrcl 10652  ax-mulcom 10653  ax-addass 10654  ax-mulass 10655  ax-distr 10656  ax-i2m1 10657  ax-1ne0 10658  ax-1rid 10659  ax-rnegex 10660  ax-rrecex 10661  ax-cnre 10662  ax-pre-lttri 10663  ax-pre-lttrn 10664  ax-pre-ltadd 10665  ax-pre-mulgt0 10666  ax-pre-sup 10667 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1086  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2071  df-mo 2558  df-eu 2589  df-clab 2737  df-cleq 2751  df-clel 2831  df-nfc 2902  df-ne 2953  df-nel 3057  df-ral 3076  df-rex 3077  df-reu 3078  df-rmo 3079  df-rab 3080  df-v 3412  df-sbc 3700  df-csb 3809  df-dif 3864  df-un 3866  df-in 3868  df-ss 3878  df-pss 3880  df-nul 4229  df-if 4425  df-pw 4500  df-sn 4527  df-pr 4529  df-tp 4531  df-op 4533  df-uni 4803  df-iun 4889  df-br 5038  df-opab 5100  df-mpt 5118  df-tr 5144  df-id 5435  df-eprel 5440  df-po 5448  df-so 5449  df-fr 5488  df-we 5490  df-xp 5535  df-rel 5536  df-cnv 5537  df-co 5538  df-dm 5539  df-rn 5540  df-res 5541  df-ima 5542  df-pred 6132  df-ord 6178  df-on 6179  df-lim 6180  df-suc 6181  df-iota 6300  df-fun 6343  df-fn 6344  df-f 6345  df-f1 6346  df-fo 6347  df-f1o 6348  df-fv 6349  df-riota 7115  df-ov 7160  df-oprab 7161  df-mpo 7162  df-om 7587  df-1st 7700  df-2nd 7701  df-wrecs 7964  df-recs 8025  df-rdg 8063  df-er 8306  df-pm 8426  df-en 8542  df-dom 8543  df-sdom 8544  df-sup 8953  df-inf 8954  df-pnf 10729  df-mnf 10730  df-xr 10731  df-ltxr 10732  df-le 10733  df-sub 10924  df-neg 10925  df-div 11350  df-nn 11689  df-2 11751  df-3 11752  df-n0 11949  df-z 12035  df-uz 12297  df-rp 12445  df-ico 12799  df-fz 12954  df-fzo 13097  df-fl 13225  df-seq 13433  df-exp 13494  df-cj 14520  df-re 14521  df-im 14522  df-sqrt 14656  df-abs 14657  df-limsup 14890  df-clim 14907  df-rlim 14908 This theorem is referenced by:  cvgcmpce  15235  rpnnen2lem5  15633  aaliou3lem3  25054
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