MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  climsup Structured version   Visualization version   GIF version

Theorem climsup 15017
Description: A bounded monotonic sequence converges to the supremum of its range. Theorem 12-5.1 of [Gleason] p. 180. (Contributed by NM, 13-Mar-2005.) (Revised by Mario Carneiro, 10-Feb-2014.)
Hypotheses
Ref Expression
climsup.1 𝑍 = (ℤ𝑀)
climsup.2 (𝜑𝑀 ∈ ℤ)
climsup.3 (𝜑𝐹:𝑍⟶ℝ)
climsup.4 ((𝜑𝑘𝑍) → (𝐹𝑘) ≤ (𝐹‘(𝑘 + 1)))
climsup.5 (𝜑 → ∃𝑥 ∈ ℝ ∀𝑘𝑍 (𝐹𝑘) ≤ 𝑥)
Assertion
Ref Expression
climsup (𝜑𝐹 ⇝ sup(ran 𝐹, ℝ, < ))
Distinct variable groups:   𝑥,𝑘,𝐹   𝜑,𝑘   𝑘,𝑍,𝑥
Allowed substitution hints:   𝜑(𝑥)   𝑀(𝑥,𝑘)

Proof of Theorem climsup
Dummy variables 𝑗 𝑛 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 climsup.3 . . . . . . . . . 10 (𝜑𝐹:𝑍⟶ℝ)
21frnd 6501 . . . . . . . . 9 (𝜑 → ran 𝐹 ⊆ ℝ)
31ffnd 6495 . . . . . . . . . . 11 (𝜑𝐹 Fn 𝑍)
4 climsup.2 . . . . . . . . . . . . 13 (𝜑𝑀 ∈ ℤ)
5 uzid 12246 . . . . . . . . . . . . 13 (𝑀 ∈ ℤ → 𝑀 ∈ (ℤ𝑀))
64, 5syl 17 . . . . . . . . . . . 12 (𝜑𝑀 ∈ (ℤ𝑀))
7 climsup.1 . . . . . . . . . . . 12 𝑍 = (ℤ𝑀)
86, 7eleqtrrdi 2925 . . . . . . . . . . 11 (𝜑𝑀𝑍)
9 fnfvelrn 6830 . . . . . . . . . . 11 ((𝐹 Fn 𝑍𝑀𝑍) → (𝐹𝑀) ∈ ran 𝐹)
103, 8, 9syl2anc 587 . . . . . . . . . 10 (𝜑 → (𝐹𝑀) ∈ ran 𝐹)
1110ne0d 4273 . . . . . . . . 9 (𝜑 → ran 𝐹 ≠ ∅)
12 climsup.5 . . . . . . . . . 10 (𝜑 → ∃𝑥 ∈ ℝ ∀𝑘𝑍 (𝐹𝑘) ≤ 𝑥)
13 breq1 5045 . . . . . . . . . . . . 13 (𝑦 = (𝐹𝑘) → (𝑦𝑥 ↔ (𝐹𝑘) ≤ 𝑥))
1413ralrn 6836 . . . . . . . . . . . 12 (𝐹 Fn 𝑍 → (∀𝑦 ∈ ran 𝐹 𝑦𝑥 ↔ ∀𝑘𝑍 (𝐹𝑘) ≤ 𝑥))
1514rexbidv 3283 . . . . . . . . . . 11 (𝐹 Fn 𝑍 → (∃𝑥 ∈ ℝ ∀𝑦 ∈ ran 𝐹 𝑦𝑥 ↔ ∃𝑥 ∈ ℝ ∀𝑘𝑍 (𝐹𝑘) ≤ 𝑥))
163, 15syl 17 . . . . . . . . . 10 (𝜑 → (∃𝑥 ∈ ℝ ∀𝑦 ∈ ran 𝐹 𝑦𝑥 ↔ ∃𝑥 ∈ ℝ ∀𝑘𝑍 (𝐹𝑘) ≤ 𝑥))
1712, 16mpbird 260 . . . . . . . . 9 (𝜑 → ∃𝑥 ∈ ℝ ∀𝑦 ∈ ran 𝐹 𝑦𝑥)
182, 11, 173jca 1125 . . . . . . . 8 (𝜑 → (ran 𝐹 ⊆ ℝ ∧ ran 𝐹 ≠ ∅ ∧ ∃𝑥 ∈ ℝ ∀𝑦 ∈ ran 𝐹 𝑦𝑥))
19 suprcl 11588 . . . . . . . 8 ((ran 𝐹 ⊆ ℝ ∧ ran 𝐹 ≠ ∅ ∧ ∃𝑥 ∈ ℝ ∀𝑦 ∈ ran 𝐹 𝑦𝑥) → sup(ran 𝐹, ℝ, < ) ∈ ℝ)
2018, 19syl 17 . . . . . . 7 (𝜑 → sup(ran 𝐹, ℝ, < ) ∈ ℝ)
21 ltsubrp 12413 . . . . . . 7 ((sup(ran 𝐹, ℝ, < ) ∈ ℝ ∧ 𝑦 ∈ ℝ+) → (sup(ran 𝐹, ℝ, < ) − 𝑦) < sup(ran 𝐹, ℝ, < ))
2220, 21sylan 583 . . . . . 6 ((𝜑𝑦 ∈ ℝ+) → (sup(ran 𝐹, ℝ, < ) − 𝑦) < sup(ran 𝐹, ℝ, < ))
2318adantr 484 . . . . . . 7 ((𝜑𝑦 ∈ ℝ+) → (ran 𝐹 ⊆ ℝ ∧ ran 𝐹 ≠ ∅ ∧ ∃𝑥 ∈ ℝ ∀𝑦 ∈ ran 𝐹 𝑦𝑥))
24 rpre 12385 . . . . . . . 8 (𝑦 ∈ ℝ+𝑦 ∈ ℝ)
25 resubcl 10939 . . . . . . . 8 ((sup(ran 𝐹, ℝ, < ) ∈ ℝ ∧ 𝑦 ∈ ℝ) → (sup(ran 𝐹, ℝ, < ) − 𝑦) ∈ ℝ)
2620, 24, 25syl2an 598 . . . . . . 7 ((𝜑𝑦 ∈ ℝ+) → (sup(ran 𝐹, ℝ, < ) − 𝑦) ∈ ℝ)
27 suprlub 11592 . . . . . . 7 (((ran 𝐹 ⊆ ℝ ∧ ran 𝐹 ≠ ∅ ∧ ∃𝑥 ∈ ℝ ∀𝑦 ∈ ran 𝐹 𝑦𝑥) ∧ (sup(ran 𝐹, ℝ, < ) − 𝑦) ∈ ℝ) → ((sup(ran 𝐹, ℝ, < ) − 𝑦) < sup(ran 𝐹, ℝ, < ) ↔ ∃𝑘 ∈ ran 𝐹(sup(ran 𝐹, ℝ, < ) − 𝑦) < 𝑘))
2823, 26, 27syl2anc 587 . . . . . 6 ((𝜑𝑦 ∈ ℝ+) → ((sup(ran 𝐹, ℝ, < ) − 𝑦) < sup(ran 𝐹, ℝ, < ) ↔ ∃𝑘 ∈ ran 𝐹(sup(ran 𝐹, ℝ, < ) − 𝑦) < 𝑘))
2922, 28mpbid 235 . . . . 5 ((𝜑𝑦 ∈ ℝ+) → ∃𝑘 ∈ ran 𝐹(sup(ran 𝐹, ℝ, < ) − 𝑦) < 𝑘)
30 breq2 5046 . . . . . . . 8 (𝑘 = (𝐹𝑗) → ((sup(ran 𝐹, ℝ, < ) − 𝑦) < 𝑘 ↔ (sup(ran 𝐹, ℝ, < ) − 𝑦) < (𝐹𝑗)))
3130rexrn 6835 . . . . . . 7 (𝐹 Fn 𝑍 → (∃𝑘 ∈ ran 𝐹(sup(ran 𝐹, ℝ, < ) − 𝑦) < 𝑘 ↔ ∃𝑗𝑍 (sup(ran 𝐹, ℝ, < ) − 𝑦) < (𝐹𝑗)))
323, 31syl 17 . . . . . 6 (𝜑 → (∃𝑘 ∈ ran 𝐹(sup(ran 𝐹, ℝ, < ) − 𝑦) < 𝑘 ↔ ∃𝑗𝑍 (sup(ran 𝐹, ℝ, < ) − 𝑦) < (𝐹𝑗)))
3332biimpa 480 . . . . 5 ((𝜑 ∧ ∃𝑘 ∈ ran 𝐹(sup(ran 𝐹, ℝ, < ) − 𝑦) < 𝑘) → ∃𝑗𝑍 (sup(ran 𝐹, ℝ, < ) − 𝑦) < (𝐹𝑗))
3429, 33syldan 594 . . . 4 ((𝜑𝑦 ∈ ℝ+) → ∃𝑗𝑍 (sup(ran 𝐹, ℝ, < ) − 𝑦) < (𝐹𝑗))
35 ffvelrn 6831 . . . . . . . . . . . 12 ((𝐹:𝑍⟶ℝ ∧ 𝑗𝑍) → (𝐹𝑗) ∈ ℝ)
361, 35sylan 583 . . . . . . . . . . 11 ((𝜑𝑗𝑍) → (𝐹𝑗) ∈ ℝ)
3736ad2ant2r 746 . . . . . . . . . 10 (((𝜑𝑦 ∈ ℝ+) ∧ (𝑗𝑍𝑘 ∈ (ℤ𝑗))) → (𝐹𝑗) ∈ ℝ)
381adantr 484 . . . . . . . . . . 11 ((𝜑𝑦 ∈ ℝ+) → 𝐹:𝑍⟶ℝ)
397uztrn2 12250 . . . . . . . . . . 11 ((𝑗𝑍𝑘 ∈ (ℤ𝑗)) → 𝑘𝑍)
40 ffvelrn 6831 . . . . . . . . . . 11 ((𝐹:𝑍⟶ℝ ∧ 𝑘𝑍) → (𝐹𝑘) ∈ ℝ)
4138, 39, 40syl2an 598 . . . . . . . . . 10 (((𝜑𝑦 ∈ ℝ+) ∧ (𝑗𝑍𝑘 ∈ (ℤ𝑗))) → (𝐹𝑘) ∈ ℝ)
4220ad2antrr 725 . . . . . . . . . 10 (((𝜑𝑦 ∈ ℝ+) ∧ (𝑗𝑍𝑘 ∈ (ℤ𝑗))) → sup(ran 𝐹, ℝ, < ) ∈ ℝ)
43 simprr 772 . . . . . . . . . . 11 (((𝜑𝑦 ∈ ℝ+) ∧ (𝑗𝑍𝑘 ∈ (ℤ𝑗))) → 𝑘 ∈ (ℤ𝑗))
44 fzssuz 12943 . . . . . . . . . . . . . 14 (𝑗...𝑘) ⊆ (ℤ𝑗)
45 uzss 12253 . . . . . . . . . . . . . . . . 17 (𝑗 ∈ (ℤ𝑀) → (ℤ𝑗) ⊆ (ℤ𝑀))
4645, 7sseqtrrdi 3993 . . . . . . . . . . . . . . . 16 (𝑗 ∈ (ℤ𝑀) → (ℤ𝑗) ⊆ 𝑍)
4746, 7eleq2s 2932 . . . . . . . . . . . . . . 15 (𝑗𝑍 → (ℤ𝑗) ⊆ 𝑍)
4847ad2antrl 727 . . . . . . . . . . . . . 14 (((𝜑𝑦 ∈ ℝ+) ∧ (𝑗𝑍𝑘 ∈ (ℤ𝑗))) → (ℤ𝑗) ⊆ 𝑍)
4944, 48sstrid 3953 . . . . . . . . . . . . 13 (((𝜑𝑦 ∈ ℝ+) ∧ (𝑗𝑍𝑘 ∈ (ℤ𝑗))) → (𝑗...𝑘) ⊆ 𝑍)
50 ffvelrn 6831 . . . . . . . . . . . . . . . 16 ((𝐹:𝑍⟶ℝ ∧ 𝑛𝑍) → (𝐹𝑛) ∈ ℝ)
5150ralrimiva 3174 . . . . . . . . . . . . . . 15 (𝐹:𝑍⟶ℝ → ∀𝑛𝑍 (𝐹𝑛) ∈ ℝ)
521, 51syl 17 . . . . . . . . . . . . . 14 (𝜑 → ∀𝑛𝑍 (𝐹𝑛) ∈ ℝ)
5352ad2antrr 725 . . . . . . . . . . . . 13 (((𝜑𝑦 ∈ ℝ+) ∧ (𝑗𝑍𝑘 ∈ (ℤ𝑗))) → ∀𝑛𝑍 (𝐹𝑛) ∈ ℝ)
54 ssralv 4008 . . . . . . . . . . . . 13 ((𝑗...𝑘) ⊆ 𝑍 → (∀𝑛𝑍 (𝐹𝑛) ∈ ℝ → ∀𝑛 ∈ (𝑗...𝑘)(𝐹𝑛) ∈ ℝ))
5549, 53, 54sylc 65 . . . . . . . . . . . 12 (((𝜑𝑦 ∈ ℝ+) ∧ (𝑗𝑍𝑘 ∈ (ℤ𝑗))) → ∀𝑛 ∈ (𝑗...𝑘)(𝐹𝑛) ∈ ℝ)
5655r19.21bi 3198 . . . . . . . . . . 11 ((((𝜑𝑦 ∈ ℝ+) ∧ (𝑗𝑍𝑘 ∈ (ℤ𝑗))) ∧ 𝑛 ∈ (𝑗...𝑘)) → (𝐹𝑛) ∈ ℝ)
57 fzssuz 12943 . . . . . . . . . . . . . 14 (𝑗...(𝑘 − 1)) ⊆ (ℤ𝑗)
5857, 48sstrid 3953 . . . . . . . . . . . . 13 (((𝜑𝑦 ∈ ℝ+) ∧ (𝑗𝑍𝑘 ∈ (ℤ𝑗))) → (𝑗...(𝑘 − 1)) ⊆ 𝑍)
5958sselda 3942 . . . . . . . . . . . 12 ((((𝜑𝑦 ∈ ℝ+) ∧ (𝑗𝑍𝑘 ∈ (ℤ𝑗))) ∧ 𝑛 ∈ (𝑗...(𝑘 − 1))) → 𝑛𝑍)
60 climsup.4 . . . . . . . . . . . . . . 15 ((𝜑𝑘𝑍) → (𝐹𝑘) ≤ (𝐹‘(𝑘 + 1)))
6160ralrimiva 3174 . . . . . . . . . . . . . 14 (𝜑 → ∀𝑘𝑍 (𝐹𝑘) ≤ (𝐹‘(𝑘 + 1)))
6261ad2antrr 725 . . . . . . . . . . . . 13 (((𝜑𝑦 ∈ ℝ+) ∧ (𝑗𝑍𝑘 ∈ (ℤ𝑗))) → ∀𝑘𝑍 (𝐹𝑘) ≤ (𝐹‘(𝑘 + 1)))
63 fveq2 6652 . . . . . . . . . . . . . . 15 (𝑘 = 𝑛 → (𝐹𝑘) = (𝐹𝑛))
64 fvoveq1 7163 . . . . . . . . . . . . . . 15 (𝑘 = 𝑛 → (𝐹‘(𝑘 + 1)) = (𝐹‘(𝑛 + 1)))
6563, 64breq12d 5055 . . . . . . . . . . . . . 14 (𝑘 = 𝑛 → ((𝐹𝑘) ≤ (𝐹‘(𝑘 + 1)) ↔ (𝐹𝑛) ≤ (𝐹‘(𝑛 + 1))))
6665rspccva 3597 . . . . . . . . . . . . 13 ((∀𝑘𝑍 (𝐹𝑘) ≤ (𝐹‘(𝑘 + 1)) ∧ 𝑛𝑍) → (𝐹𝑛) ≤ (𝐹‘(𝑛 + 1)))
6762, 66sylan 583 . . . . . . . . . . . 12 ((((𝜑𝑦 ∈ ℝ+) ∧ (𝑗𝑍𝑘 ∈ (ℤ𝑗))) ∧ 𝑛𝑍) → (𝐹𝑛) ≤ (𝐹‘(𝑛 + 1)))
6859, 67syldan 594 . . . . . . . . . . 11 ((((𝜑𝑦 ∈ ℝ+) ∧ (𝑗𝑍𝑘 ∈ (ℤ𝑗))) ∧ 𝑛 ∈ (𝑗...(𝑘 − 1))) → (𝐹𝑛) ≤ (𝐹‘(𝑛 + 1)))
6943, 56, 68monoord 13396 . . . . . . . . . 10 (((𝜑𝑦 ∈ ℝ+) ∧ (𝑗𝑍𝑘 ∈ (ℤ𝑗))) → (𝐹𝑗) ≤ (𝐹𝑘))
7037, 41, 42, 69lesub2dd 11246 . . . . . . . . 9 (((𝜑𝑦 ∈ ℝ+) ∧ (𝑗𝑍𝑘 ∈ (ℤ𝑗))) → (sup(ran 𝐹, ℝ, < ) − (𝐹𝑘)) ≤ (sup(ran 𝐹, ℝ, < ) − (𝐹𝑗)))
7142, 41resubcld 11057 . . . . . . . . . 10 (((𝜑𝑦 ∈ ℝ+) ∧ (𝑗𝑍𝑘 ∈ (ℤ𝑗))) → (sup(ran 𝐹, ℝ, < ) − (𝐹𝑘)) ∈ ℝ)
7242, 37resubcld 11057 . . . . . . . . . 10 (((𝜑𝑦 ∈ ℝ+) ∧ (𝑗𝑍𝑘 ∈ (ℤ𝑗))) → (sup(ran 𝐹, ℝ, < ) − (𝐹𝑗)) ∈ ℝ)
7324ad2antlr 726 . . . . . . . . . 10 (((𝜑𝑦 ∈ ℝ+) ∧ (𝑗𝑍𝑘 ∈ (ℤ𝑗))) → 𝑦 ∈ ℝ)
74 lelttr 10720 . . . . . . . . . 10 (((sup(ran 𝐹, ℝ, < ) − (𝐹𝑘)) ∈ ℝ ∧ (sup(ran 𝐹, ℝ, < ) − (𝐹𝑗)) ∈ ℝ ∧ 𝑦 ∈ ℝ) → (((sup(ran 𝐹, ℝ, < ) − (𝐹𝑘)) ≤ (sup(ran 𝐹, ℝ, < ) − (𝐹𝑗)) ∧ (sup(ran 𝐹, ℝ, < ) − (𝐹𝑗)) < 𝑦) → (sup(ran 𝐹, ℝ, < ) − (𝐹𝑘)) < 𝑦))
7571, 72, 73, 74syl3anc 1368 . . . . . . . . 9 (((𝜑𝑦 ∈ ℝ+) ∧ (𝑗𝑍𝑘 ∈ (ℤ𝑗))) → (((sup(ran 𝐹, ℝ, < ) − (𝐹𝑘)) ≤ (sup(ran 𝐹, ℝ, < ) − (𝐹𝑗)) ∧ (sup(ran 𝐹, ℝ, < ) − (𝐹𝑗)) < 𝑦) → (sup(ran 𝐹, ℝ, < ) − (𝐹𝑘)) < 𝑦))
7670, 75mpand 694 . . . . . . . 8 (((𝜑𝑦 ∈ ℝ+) ∧ (𝑗𝑍𝑘 ∈ (ℤ𝑗))) → ((sup(ran 𝐹, ℝ, < ) − (𝐹𝑗)) < 𝑦 → (sup(ran 𝐹, ℝ, < ) − (𝐹𝑘)) < 𝑦))
77 ltsub23 11109 . . . . . . . . 9 ((sup(ran 𝐹, ℝ, < ) ∈ ℝ ∧ 𝑦 ∈ ℝ ∧ (𝐹𝑗) ∈ ℝ) → ((sup(ran 𝐹, ℝ, < ) − 𝑦) < (𝐹𝑗) ↔ (sup(ran 𝐹, ℝ, < ) − (𝐹𝑗)) < 𝑦))
7842, 73, 37, 77syl3anc 1368 . . . . . . . 8 (((𝜑𝑦 ∈ ℝ+) ∧ (𝑗𝑍𝑘 ∈ (ℤ𝑗))) → ((sup(ran 𝐹, ℝ, < ) − 𝑦) < (𝐹𝑗) ↔ (sup(ran 𝐹, ℝ, < ) − (𝐹𝑗)) < 𝑦))
7918ad2antrr 725 . . . . . . . . . . 11 (((𝜑𝑦 ∈ ℝ+) ∧ (𝑗𝑍𝑘 ∈ (ℤ𝑗))) → (ran 𝐹 ⊆ ℝ ∧ ran 𝐹 ≠ ∅ ∧ ∃𝑥 ∈ ℝ ∀𝑦 ∈ ran 𝐹 𝑦𝑥))
803adantr 484 . . . . . . . . . . . 12 ((𝜑𝑦 ∈ ℝ+) → 𝐹 Fn 𝑍)
81 fnfvelrn 6830 . . . . . . . . . . . 12 ((𝐹 Fn 𝑍𝑘𝑍) → (𝐹𝑘) ∈ ran 𝐹)
8280, 39, 81syl2an 598 . . . . . . . . . . 11 (((𝜑𝑦 ∈ ℝ+) ∧ (𝑗𝑍𝑘 ∈ (ℤ𝑗))) → (𝐹𝑘) ∈ ran 𝐹)
83 suprub 11589 . . . . . . . . . . 11 (((ran 𝐹 ⊆ ℝ ∧ ran 𝐹 ≠ ∅ ∧ ∃𝑥 ∈ ℝ ∀𝑦 ∈ ran 𝐹 𝑦𝑥) ∧ (𝐹𝑘) ∈ ran 𝐹) → (𝐹𝑘) ≤ sup(ran 𝐹, ℝ, < ))
8479, 82, 83syl2anc 587 . . . . . . . . . 10 (((𝜑𝑦 ∈ ℝ+) ∧ (𝑗𝑍𝑘 ∈ (ℤ𝑗))) → (𝐹𝑘) ≤ sup(ran 𝐹, ℝ, < ))
8541, 42, 84abssuble0d 14783 . . . . . . . . 9 (((𝜑𝑦 ∈ ℝ+) ∧ (𝑗𝑍𝑘 ∈ (ℤ𝑗))) → (abs‘((𝐹𝑘) − sup(ran 𝐹, ℝ, < ))) = (sup(ran 𝐹, ℝ, < ) − (𝐹𝑘)))
8685breq1d 5052 . . . . . . . 8 (((𝜑𝑦 ∈ ℝ+) ∧ (𝑗𝑍𝑘 ∈ (ℤ𝑗))) → ((abs‘((𝐹𝑘) − sup(ran 𝐹, ℝ, < ))) < 𝑦 ↔ (sup(ran 𝐹, ℝ, < ) − (𝐹𝑘)) < 𝑦))
8776, 78, 863imtr4d 297 . . . . . . 7 (((𝜑𝑦 ∈ ℝ+) ∧ (𝑗𝑍𝑘 ∈ (ℤ𝑗))) → ((sup(ran 𝐹, ℝ, < ) − 𝑦) < (𝐹𝑗) → (abs‘((𝐹𝑘) − sup(ran 𝐹, ℝ, < ))) < 𝑦))
8887anassrs 471 . . . . . 6 ((((𝜑𝑦 ∈ ℝ+) ∧ 𝑗𝑍) ∧ 𝑘 ∈ (ℤ𝑗)) → ((sup(ran 𝐹, ℝ, < ) − 𝑦) < (𝐹𝑗) → (abs‘((𝐹𝑘) − sup(ran 𝐹, ℝ, < ))) < 𝑦))
8988ralrimdva 3179 . . . . 5 (((𝜑𝑦 ∈ ℝ+) ∧ 𝑗𝑍) → ((sup(ran 𝐹, ℝ, < ) − 𝑦) < (𝐹𝑗) → ∀𝑘 ∈ (ℤ𝑗)(abs‘((𝐹𝑘) − sup(ran 𝐹, ℝ, < ))) < 𝑦))
9089reximdva 3260 . . . 4 ((𝜑𝑦 ∈ ℝ+) → (∃𝑗𝑍 (sup(ran 𝐹, ℝ, < ) − 𝑦) < (𝐹𝑗) → ∃𝑗𝑍𝑘 ∈ (ℤ𝑗)(abs‘((𝐹𝑘) − sup(ran 𝐹, ℝ, < ))) < 𝑦))
9134, 90mpd 15 . . 3 ((𝜑𝑦 ∈ ℝ+) → ∃𝑗𝑍𝑘 ∈ (ℤ𝑗)(abs‘((𝐹𝑘) − sup(ran 𝐹, ℝ, < ))) < 𝑦)
9291ralrimiva 3174 . 2 (𝜑 → ∀𝑦 ∈ ℝ+𝑗𝑍𝑘 ∈ (ℤ𝑗)(abs‘((𝐹𝑘) − sup(ran 𝐹, ℝ, < ))) < 𝑦)
937fvexi 6666 . . . 4 𝑍 ∈ V
94 fex 6971 . . . 4 ((𝐹:𝑍⟶ℝ ∧ 𝑍 ∈ V) → 𝐹 ∈ V)
951, 93, 94sylancl 589 . . 3 (𝜑𝐹 ∈ V)
96 eqidd 2823 . . 3 ((𝜑𝑘𝑍) → (𝐹𝑘) = (𝐹𝑘))
9720recnd 10658 . . 3 (𝜑 → sup(ran 𝐹, ℝ, < ) ∈ ℂ)
981, 40sylan 583 . . . 4 ((𝜑𝑘𝑍) → (𝐹𝑘) ∈ ℝ)
9998recnd 10658 . . 3 ((𝜑𝑘𝑍) → (𝐹𝑘) ∈ ℂ)
1007, 4, 95, 96, 97, 99clim2c 14853 . 2 (𝜑 → (𝐹 ⇝ sup(ran 𝐹, ℝ, < ) ↔ ∀𝑦 ∈ ℝ+𝑗𝑍𝑘 ∈ (ℤ𝑗)(abs‘((𝐹𝑘) − sup(ran 𝐹, ℝ, < ))) < 𝑦))
10192, 100mpbird 260 1 (𝜑𝐹 ⇝ sup(ran 𝐹, ℝ, < ))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wa 399  w3a 1084   = wceq 1538  wcel 2114  wne 3011  wral 3130  wrex 3131  Vcvv 3469  wss 3908  c0 4265   class class class wbr 5042  ran crn 5533   Fn wfn 6329  wf 6330  cfv 6334  (class class class)co 7140  supcsup 8892  cr 10525  1c1 10527   + caddc 10529   < clt 10664  cle 10665  cmin 10859  cz 11969  cuz 12231  +crp 12377  ...cfz 12885  abscabs 14584  cli 14832
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2178  ax-ext 2794  ax-rep 5166  ax-sep 5179  ax-nul 5186  ax-pow 5243  ax-pr 5307  ax-un 7446  ax-cnex 10582  ax-resscn 10583  ax-1cn 10584  ax-icn 10585  ax-addcl 10586  ax-addrcl 10587  ax-mulcl 10588  ax-mulrcl 10589  ax-mulcom 10590  ax-addass 10591  ax-mulass 10592  ax-distr 10593  ax-i2m1 10594  ax-1ne0 10595  ax-1rid 10596  ax-rnegex 10597  ax-rrecex 10598  ax-cnre 10599  ax-pre-lttri 10600  ax-pre-lttrn 10601  ax-pre-ltadd 10602  ax-pre-mulgt0 10603  ax-pre-sup 10604
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2622  df-eu 2653  df-clab 2801  df-cleq 2815  df-clel 2894  df-nfc 2962  df-ne 3012  df-nel 3116  df-ral 3135  df-rex 3136  df-reu 3137  df-rmo 3138  df-rab 3139  df-v 3471  df-sbc 3748  df-csb 3856  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-pss 3927  df-nul 4266  df-if 4440  df-pw 4513  df-sn 4540  df-pr 4542  df-tp 4544  df-op 4546  df-uni 4814  df-iun 4896  df-br 5043  df-opab 5105  df-mpt 5123  df-tr 5149  df-id 5437  df-eprel 5442  df-po 5451  df-so 5452  df-fr 5491  df-we 5493  df-xp 5538  df-rel 5539  df-cnv 5540  df-co 5541  df-dm 5542  df-rn 5543  df-res 5544  df-ima 5545  df-pred 6126  df-ord 6172  df-on 6173  df-lim 6174  df-suc 6175  df-iota 6293  df-fun 6336  df-fn 6337  df-f 6338  df-f1 6339  df-fo 6340  df-f1o 6341  df-fv 6342  df-riota 7098  df-ov 7143  df-oprab 7144  df-mpo 7145  df-om 7566  df-1st 7675  df-2nd 7676  df-wrecs 7934  df-recs 7995  df-rdg 8033  df-er 8276  df-en 8497  df-dom 8498  df-sdom 8499  df-sup 8894  df-pnf 10666  df-mnf 10667  df-xr 10668  df-ltxr 10669  df-le 10670  df-sub 10861  df-neg 10862  df-div 11287  df-nn 11626  df-2 11688  df-3 11689  df-n0 11886  df-z 11970  df-uz 12232  df-rp 12378  df-fz 12886  df-seq 13365  df-exp 13426  df-cj 14449  df-re 14450  df-im 14451  df-sqrt 14585  df-abs 14586  df-clim 14836
This theorem is referenced by:  isumsup2  15192  climcnds  15197  itg1climres  24316  itg2monolem1  24352  itg2i1fseq  24357  itg2i1fseq2  24358  emcllem6  25584  lmdvg  31270  esumpcvgval  31411  meaiuninclem  43059
  Copyright terms: Public domain W3C validator