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Theorem climsup 14741
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 6263 . . . . . . . . 9 (𝜑 → ran 𝐹 ⊆ ℝ)
31ffnd 6257 . . . . . . . . . . 11 (𝜑𝐹 Fn 𝑍)
4 climsup.2 . . . . . . . . . . . . 13 (𝜑𝑀 ∈ ℤ)
5 uzid 11945 . . . . . . . . . . . . 13 (𝑀 ∈ ℤ → 𝑀 ∈ (ℤ𝑀))
64, 5syl 17 . . . . . . . . . . . 12 (𝜑𝑀 ∈ (ℤ𝑀))
7 climsup.1 . . . . . . . . . . . 12 𝑍 = (ℤ𝑀)
86, 7syl6eleqr 2889 . . . . . . . . . . 11 (𝜑𝑀𝑍)
9 fnfvelrn 6582 . . . . . . . . . . 11 ((𝐹 Fn 𝑍𝑀𝑍) → (𝐹𝑀) ∈ ran 𝐹)
103, 8, 9syl2anc 580 . . . . . . . . . 10 (𝜑 → (𝐹𝑀) ∈ ran 𝐹)
1110ne0d 4122 . . . . . . . . 9 (𝜑 → ran 𝐹 ≠ ∅)
12 climsup.5 . . . . . . . . . 10 (𝜑 → ∃𝑥 ∈ ℝ ∀𝑘𝑍 (𝐹𝑘) ≤ 𝑥)
13 breq1 4846 . . . . . . . . . . . . 13 (𝑦 = (𝐹𝑘) → (𝑦𝑥 ↔ (𝐹𝑘) ≤ 𝑥))
1413ralrn 6588 . . . . . . . . . . . 12 (𝐹 Fn 𝑍 → (∀𝑦 ∈ ran 𝐹 𝑦𝑥 ↔ ∀𝑘𝑍 (𝐹𝑘) ≤ 𝑥))
1514rexbidv 3233 . . . . . . . . . . 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 249 . . . . . . . . 9 (𝜑 → ∃𝑥 ∈ ℝ ∀𝑦 ∈ ran 𝐹 𝑦𝑥)
182, 11, 173jca 1159 . . . . . . . 8 (𝜑 → (ran 𝐹 ⊆ ℝ ∧ ran 𝐹 ≠ ∅ ∧ ∃𝑥 ∈ ℝ ∀𝑦 ∈ ran 𝐹 𝑦𝑥))
19 suprcl 11275 . . . . . . . 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 12111 . . . . . . 7 ((sup(ran 𝐹, ℝ, < ) ∈ ℝ ∧ 𝑦 ∈ ℝ+) → (sup(ran 𝐹, ℝ, < ) − 𝑦) < sup(ran 𝐹, ℝ, < ))
2220, 21sylan 576 . . . . . 6 ((𝜑𝑦 ∈ ℝ+) → (sup(ran 𝐹, ℝ, < ) − 𝑦) < sup(ran 𝐹, ℝ, < ))
2318adantr 473 . . . . . . 7 ((𝜑𝑦 ∈ ℝ+) → (ran 𝐹 ⊆ ℝ ∧ ran 𝐹 ≠ ∅ ∧ ∃𝑥 ∈ ℝ ∀𝑦 ∈ ran 𝐹 𝑦𝑥))
24 rpre 12082 . . . . . . . 8 (𝑦 ∈ ℝ+𝑦 ∈ ℝ)
25 resubcl 10637 . . . . . . . 8 ((sup(ran 𝐹, ℝ, < ) ∈ ℝ ∧ 𝑦 ∈ ℝ) → (sup(ran 𝐹, ℝ, < ) − 𝑦) ∈ ℝ)
2620, 24, 25syl2an 590 . . . . . . 7 ((𝜑𝑦 ∈ ℝ+) → (sup(ran 𝐹, ℝ, < ) − 𝑦) ∈ ℝ)
27 suprlub 11279 . . . . . . 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 580 . . . . . 6 ((𝜑𝑦 ∈ ℝ+) → ((sup(ran 𝐹, ℝ, < ) − 𝑦) < sup(ran 𝐹, ℝ, < ) ↔ ∃𝑘 ∈ ran 𝐹(sup(ran 𝐹, ℝ, < ) − 𝑦) < 𝑘))
2922, 28mpbid 224 . . . . 5 ((𝜑𝑦 ∈ ℝ+) → ∃𝑘 ∈ ran 𝐹(sup(ran 𝐹, ℝ, < ) − 𝑦) < 𝑘)
30 breq2 4847 . . . . . . . 8 (𝑘 = (𝐹𝑗) → ((sup(ran 𝐹, ℝ, < ) − 𝑦) < 𝑘 ↔ (sup(ran 𝐹, ℝ, < ) − 𝑦) < (𝐹𝑗)))
3130rexrn 6587 . . . . . . 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 469 . . . . 5 ((𝜑 ∧ ∃𝑘 ∈ ran 𝐹(sup(ran 𝐹, ℝ, < ) − 𝑦) < 𝑘) → ∃𝑗𝑍 (sup(ran 𝐹, ℝ, < ) − 𝑦) < (𝐹𝑗))
3429, 33syldan 586 . . . 4 ((𝜑𝑦 ∈ ℝ+) → ∃𝑗𝑍 (sup(ran 𝐹, ℝ, < ) − 𝑦) < (𝐹𝑗))
35 ffvelrn 6583 . . . . . . . . . . . 12 ((𝐹:𝑍⟶ℝ ∧ 𝑗𝑍) → (𝐹𝑗) ∈ ℝ)
361, 35sylan 576 . . . . . . . . . . 11 ((𝜑𝑗𝑍) → (𝐹𝑗) ∈ ℝ)
3736ad2ant2r 754 . . . . . . . . . 10 (((𝜑𝑦 ∈ ℝ+) ∧ (𝑗𝑍𝑘 ∈ (ℤ𝑗))) → (𝐹𝑗) ∈ ℝ)
381adantr 473 . . . . . . . . . . 11 ((𝜑𝑦 ∈ ℝ+) → 𝐹:𝑍⟶ℝ)
397uztrn2 11948 . . . . . . . . . . 11 ((𝑗𝑍𝑘 ∈ (ℤ𝑗)) → 𝑘𝑍)
40 ffvelrn 6583 . . . . . . . . . . 11 ((𝐹:𝑍⟶ℝ ∧ 𝑘𝑍) → (𝐹𝑘) ∈ ℝ)
4138, 39, 40syl2an 590 . . . . . . . . . 10 (((𝜑𝑦 ∈ ℝ+) ∧ (𝑗𝑍𝑘 ∈ (ℤ𝑗))) → (𝐹𝑘) ∈ ℝ)
4220ad2antrr 718 . . . . . . . . . 10 (((𝜑𝑦 ∈ ℝ+) ∧ (𝑗𝑍𝑘 ∈ (ℤ𝑗))) → sup(ran 𝐹, ℝ, < ) ∈ ℝ)
43 simprr 790 . . . . . . . . . . 11 (((𝜑𝑦 ∈ ℝ+) ∧ (𝑗𝑍𝑘 ∈ (ℤ𝑗))) → 𝑘 ∈ (ℤ𝑗))
44 fzssuz 12636 . . . . . . . . . . . . . 14 (𝑗...𝑘) ⊆ (ℤ𝑗)
45 uzss 11951 . . . . . . . . . . . . . . . . 17 (𝑗 ∈ (ℤ𝑀) → (ℤ𝑗) ⊆ (ℤ𝑀))
4645, 7syl6sseqr 3848 . . . . . . . . . . . . . . . 16 (𝑗 ∈ (ℤ𝑀) → (ℤ𝑗) ⊆ 𝑍)
4746, 7eleq2s 2896 . . . . . . . . . . . . . . 15 (𝑗𝑍 → (ℤ𝑗) ⊆ 𝑍)
4847ad2antrl 720 . . . . . . . . . . . . . 14 (((𝜑𝑦 ∈ ℝ+) ∧ (𝑗𝑍𝑘 ∈ (ℤ𝑗))) → (ℤ𝑗) ⊆ 𝑍)
4944, 48syl5ss 3809 . . . . . . . . . . . . 13 (((𝜑𝑦 ∈ ℝ+) ∧ (𝑗𝑍𝑘 ∈ (ℤ𝑗))) → (𝑗...𝑘) ⊆ 𝑍)
50 ffvelrn 6583 . . . . . . . . . . . . . . . 16 ((𝐹:𝑍⟶ℝ ∧ 𝑛𝑍) → (𝐹𝑛) ∈ ℝ)
5150ralrimiva 3147 . . . . . . . . . . . . . . 15 (𝐹:𝑍⟶ℝ → ∀𝑛𝑍 (𝐹𝑛) ∈ ℝ)
521, 51syl 17 . . . . . . . . . . . . . 14 (𝜑 → ∀𝑛𝑍 (𝐹𝑛) ∈ ℝ)
5352ad2antrr 718 . . . . . . . . . . . . 13 (((𝜑𝑦 ∈ ℝ+) ∧ (𝑗𝑍𝑘 ∈ (ℤ𝑗))) → ∀𝑛𝑍 (𝐹𝑛) ∈ ℝ)
54 ssralv 3862 . . . . . . . . . . . . 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 3113 . . . . . . . . . . 11 ((((𝜑𝑦 ∈ ℝ+) ∧ (𝑗𝑍𝑘 ∈ (ℤ𝑗))) ∧ 𝑛 ∈ (𝑗...𝑘)) → (𝐹𝑛) ∈ ℝ)
57 fzssuz 12636 . . . . . . . . . . . . . 14 (𝑗...(𝑘 − 1)) ⊆ (ℤ𝑗)
5857, 48syl5ss 3809 . . . . . . . . . . . . 13 (((𝜑𝑦 ∈ ℝ+) ∧ (𝑗𝑍𝑘 ∈ (ℤ𝑗))) → (𝑗...(𝑘 − 1)) ⊆ 𝑍)
5958sselda 3798 . . . . . . . . . . . 12 ((((𝜑𝑦 ∈ ℝ+) ∧ (𝑗𝑍𝑘 ∈ (ℤ𝑗))) ∧ 𝑛 ∈ (𝑗...(𝑘 − 1))) → 𝑛𝑍)
60 climsup.4 . . . . . . . . . . . . . . 15 ((𝜑𝑘𝑍) → (𝐹𝑘) ≤ (𝐹‘(𝑘 + 1)))
6160ralrimiva 3147 . . . . . . . . . . . . . 14 (𝜑 → ∀𝑘𝑍 (𝐹𝑘) ≤ (𝐹‘(𝑘 + 1)))
6261ad2antrr 718 . . . . . . . . . . . . 13 (((𝜑𝑦 ∈ ℝ+) ∧ (𝑗𝑍𝑘 ∈ (ℤ𝑗))) → ∀𝑘𝑍 (𝐹𝑘) ≤ (𝐹‘(𝑘 + 1)))
63 fveq2 6411 . . . . . . . . . . . . . . 15 (𝑘 = 𝑛 → (𝐹𝑘) = (𝐹𝑛))
64 fvoveq1 6901 . . . . . . . . . . . . . . 15 (𝑘 = 𝑛 → (𝐹‘(𝑘 + 1)) = (𝐹‘(𝑛 + 1)))
6563, 64breq12d 4856 . . . . . . . . . . . . . 14 (𝑘 = 𝑛 → ((𝐹𝑘) ≤ (𝐹‘(𝑘 + 1)) ↔ (𝐹𝑛) ≤ (𝐹‘(𝑛 + 1))))
6665rspccva 3496 . . . . . . . . . . . . 13 ((∀𝑘𝑍 (𝐹𝑘) ≤ (𝐹‘(𝑘 + 1)) ∧ 𝑛𝑍) → (𝐹𝑛) ≤ (𝐹‘(𝑛 + 1)))
6762, 66sylan 576 . . . . . . . . . . . 12 ((((𝜑𝑦 ∈ ℝ+) ∧ (𝑗𝑍𝑘 ∈ (ℤ𝑗))) ∧ 𝑛𝑍) → (𝐹𝑛) ≤ (𝐹‘(𝑛 + 1)))
6859, 67syldan 586 . . . . . . . . . . 11 ((((𝜑𝑦 ∈ ℝ+) ∧ (𝑗𝑍𝑘 ∈ (ℤ𝑗))) ∧ 𝑛 ∈ (𝑗...(𝑘 − 1))) → (𝐹𝑛) ≤ (𝐹‘(𝑛 + 1)))
6943, 56, 68monoord 13085 . . . . . . . . . 10 (((𝜑𝑦 ∈ ℝ+) ∧ (𝑗𝑍𝑘 ∈ (ℤ𝑗))) → (𝐹𝑗) ≤ (𝐹𝑘))
7037, 41, 42, 69lesub2dd 10936 . . . . . . . . 9 (((𝜑𝑦 ∈ ℝ+) ∧ (𝑗𝑍𝑘 ∈ (ℤ𝑗))) → (sup(ran 𝐹, ℝ, < ) − (𝐹𝑘)) ≤ (sup(ran 𝐹, ℝ, < ) − (𝐹𝑗)))
7142, 41resubcld 10750 . . . . . . . . . 10 (((𝜑𝑦 ∈ ℝ+) ∧ (𝑗𝑍𝑘 ∈ (ℤ𝑗))) → (sup(ran 𝐹, ℝ, < ) − (𝐹𝑘)) ∈ ℝ)
7242, 37resubcld 10750 . . . . . . . . . 10 (((𝜑𝑦 ∈ ℝ+) ∧ (𝑗𝑍𝑘 ∈ (ℤ𝑗))) → (sup(ran 𝐹, ℝ, < ) − (𝐹𝑗)) ∈ ℝ)
7324ad2antlr 719 . . . . . . . . . 10 (((𝜑𝑦 ∈ ℝ+) ∧ (𝑗𝑍𝑘 ∈ (ℤ𝑗))) → 𝑦 ∈ ℝ)
74 lelttr 10418 . . . . . . . . . 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 1491 . . . . . . . . 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 687 . . . . . . . 8 (((𝜑𝑦 ∈ ℝ+) ∧ (𝑗𝑍𝑘 ∈ (ℤ𝑗))) → ((sup(ran 𝐹, ℝ, < ) − (𝐹𝑗)) < 𝑦 → (sup(ran 𝐹, ℝ, < ) − (𝐹𝑘)) < 𝑦))
77 ltsub23 10800 . . . . . . . . 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 1491 . . . . . . . 8 (((𝜑𝑦 ∈ ℝ+) ∧ (𝑗𝑍𝑘 ∈ (ℤ𝑗))) → ((sup(ran 𝐹, ℝ, < ) − 𝑦) < (𝐹𝑗) ↔ (sup(ran 𝐹, ℝ, < ) − (𝐹𝑗)) < 𝑦))
7918ad2antrr 718 . . . . . . . . . . 11 (((𝜑𝑦 ∈ ℝ+) ∧ (𝑗𝑍𝑘 ∈ (ℤ𝑗))) → (ran 𝐹 ⊆ ℝ ∧ ran 𝐹 ≠ ∅ ∧ ∃𝑥 ∈ ℝ ∀𝑦 ∈ ran 𝐹 𝑦𝑥))
803adantr 473 . . . . . . . . . . . 12 ((𝜑𝑦 ∈ ℝ+) → 𝐹 Fn 𝑍)
81 fnfvelrn 6582 . . . . . . . . . . . 12 ((𝐹 Fn 𝑍𝑘𝑍) → (𝐹𝑘) ∈ ran 𝐹)
8280, 39, 81syl2an 590 . . . . . . . . . . 11 (((𝜑𝑦 ∈ ℝ+) ∧ (𝑗𝑍𝑘 ∈ (ℤ𝑗))) → (𝐹𝑘) ∈ ran 𝐹)
83 suprub 11276 . . . . . . . . . . 11 (((ran 𝐹 ⊆ ℝ ∧ ran 𝐹 ≠ ∅ ∧ ∃𝑥 ∈ ℝ ∀𝑦 ∈ ran 𝐹 𝑦𝑥) ∧ (𝐹𝑘) ∈ ran 𝐹) → (𝐹𝑘) ≤ sup(ran 𝐹, ℝ, < ))
8479, 82, 83syl2anc 580 . . . . . . . . . 10 (((𝜑𝑦 ∈ ℝ+) ∧ (𝑗𝑍𝑘 ∈ (ℤ𝑗))) → (𝐹𝑘) ≤ sup(ran 𝐹, ℝ, < ))
8541, 42, 84abssuble0d 14512 . . . . . . . . 9 (((𝜑𝑦 ∈ ℝ+) ∧ (𝑗𝑍𝑘 ∈ (ℤ𝑗))) → (abs‘((𝐹𝑘) − sup(ran 𝐹, ℝ, < ))) = (sup(ran 𝐹, ℝ, < ) − (𝐹𝑘)))
8685breq1d 4853 . . . . . . . 8 (((𝜑𝑦 ∈ ℝ+) ∧ (𝑗𝑍𝑘 ∈ (ℤ𝑗))) → ((abs‘((𝐹𝑘) − sup(ran 𝐹, ℝ, < ))) < 𝑦 ↔ (sup(ran 𝐹, ℝ, < ) − (𝐹𝑘)) < 𝑦))
8776, 78, 863imtr4d 286 . . . . . . 7 (((𝜑𝑦 ∈ ℝ+) ∧ (𝑗𝑍𝑘 ∈ (ℤ𝑗))) → ((sup(ran 𝐹, ℝ, < ) − 𝑦) < (𝐹𝑗) → (abs‘((𝐹𝑘) − sup(ran 𝐹, ℝ, < ))) < 𝑦))
8887anassrs 460 . . . . . 6 ((((𝜑𝑦 ∈ ℝ+) ∧ 𝑗𝑍) ∧ 𝑘 ∈ (ℤ𝑗)) → ((sup(ran 𝐹, ℝ, < ) − 𝑦) < (𝐹𝑗) → (abs‘((𝐹𝑘) − sup(ran 𝐹, ℝ, < ))) < 𝑦))
8988ralrimdva 3150 . . . . 5 (((𝜑𝑦 ∈ ℝ+) ∧ 𝑗𝑍) → ((sup(ran 𝐹, ℝ, < ) − 𝑦) < (𝐹𝑗) → ∀𝑘 ∈ (ℤ𝑗)(abs‘((𝐹𝑘) − sup(ran 𝐹, ℝ, < ))) < 𝑦))
9089reximdva 3197 . . . 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 3147 . 2 (𝜑 → ∀𝑦 ∈ ℝ+𝑗𝑍𝑘 ∈ (ℤ𝑗)(abs‘((𝐹𝑘) − sup(ran 𝐹, ℝ, < ))) < 𝑦)
937fvexi 6425 . . . 4 𝑍 ∈ V
94 fex 6718 . . . 4 ((𝐹:𝑍⟶ℝ ∧ 𝑍 ∈ V) → 𝐹 ∈ V)
951, 93, 94sylancl 581 . . 3 (𝜑𝐹 ∈ V)
96 eqidd 2800 . . 3 ((𝜑𝑘𝑍) → (𝐹𝑘) = (𝐹𝑘))
9720recnd 10357 . . 3 (𝜑 → sup(ran 𝐹, ℝ, < ) ∈ ℂ)
981, 40sylan 576 . . . 4 ((𝜑𝑘𝑍) → (𝐹𝑘) ∈ ℝ)
9998recnd 10357 . . 3 ((𝜑𝑘𝑍) → (𝐹𝑘) ∈ ℂ)
1007, 4, 95, 96, 97, 99clim2c 14577 . 2 (𝜑 → (𝐹 ⇝ sup(ran 𝐹, ℝ, < ) ↔ ∀𝑦 ∈ ℝ+𝑗𝑍𝑘 ∈ (ℤ𝑗)(abs‘((𝐹𝑘) − sup(ran 𝐹, ℝ, < ))) < 𝑦))
10192, 100mpbird 249 1 (𝜑𝐹 ⇝ sup(ran 𝐹, ℝ, < ))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 198  wa 385  w3a 1108   = wceq 1653  wcel 2157  wne 2971  wral 3089  wrex 3090  Vcvv 3385  wss 3769  c0 4115   class class class wbr 4843  ran crn 5313   Fn wfn 6096  wf 6097  cfv 6101  (class class class)co 6878  supcsup 8588  cr 10223  1c1 10225   + caddc 10227   < clt 10363  cle 10364  cmin 10556  cz 11666  cuz 11930  +crp 12074  ...cfz 12580  abscabs 14315  cli 14556
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-8 2159  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2377  ax-ext 2777  ax-rep 4964  ax-sep 4975  ax-nul 4983  ax-pow 5035  ax-pr 5097  ax-un 7183  ax-cnex 10280  ax-resscn 10281  ax-1cn 10282  ax-icn 10283  ax-addcl 10284  ax-addrcl 10285  ax-mulcl 10286  ax-mulrcl 10287  ax-mulcom 10288  ax-addass 10289  ax-mulass 10290  ax-distr 10291  ax-i2m1 10292  ax-1ne0 10293  ax-1rid 10294  ax-rnegex 10295  ax-rrecex 10296  ax-cnre 10297  ax-pre-lttri 10298  ax-pre-lttrn 10299  ax-pre-ltadd 10300  ax-pre-mulgt0 10301  ax-pre-sup 10302
This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-3or 1109  df-3an 1110  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-mo 2591  df-eu 2609  df-clab 2786  df-cleq 2792  df-clel 2795  df-nfc 2930  df-ne 2972  df-nel 3075  df-ral 3094  df-rex 3095  df-reu 3096  df-rmo 3097  df-rab 3098  df-v 3387  df-sbc 3634  df-csb 3729  df-dif 3772  df-un 3774  df-in 3776  df-ss 3783  df-pss 3785  df-nul 4116  df-if 4278  df-pw 4351  df-sn 4369  df-pr 4371  df-tp 4373  df-op 4375  df-uni 4629  df-iun 4712  df-br 4844  df-opab 4906  df-mpt 4923  df-tr 4946  df-id 5220  df-eprel 5225  df-po 5233  df-so 5234  df-fr 5271  df-we 5273  df-xp 5318  df-rel 5319  df-cnv 5320  df-co 5321  df-dm 5322  df-rn 5323  df-res 5324  df-ima 5325  df-pred 5898  df-ord 5944  df-on 5945  df-lim 5946  df-suc 5947  df-iota 6064  df-fun 6103  df-fn 6104  df-f 6105  df-f1 6106  df-fo 6107  df-f1o 6108  df-fv 6109  df-riota 6839  df-ov 6881  df-oprab 6882  df-mpt2 6883  df-om 7300  df-1st 7401  df-2nd 7402  df-wrecs 7645  df-recs 7707  df-rdg 7745  df-er 7982  df-en 8196  df-dom 8197  df-sdom 8198  df-sup 8590  df-pnf 10365  df-mnf 10366  df-xr 10367  df-ltxr 10368  df-le 10369  df-sub 10558  df-neg 10559  df-div 10977  df-nn 11313  df-2 11376  df-3 11377  df-n0 11581  df-z 11667  df-uz 11931  df-rp 12075  df-fz 12581  df-seq 13056  df-exp 13115  df-cj 14180  df-re 14181  df-im 14182  df-sqrt 14316  df-abs 14317  df-clim 14560
This theorem is referenced by:  isumsup2  14916  climcnds  14921  itg1climres  23822  itg2monolem1  23858  itg2i1fseq  23863  itg2i1fseq2  23864  emcllem6  25079  lmdvg  30515  esumpcvgval  30656  meaiuninclem  41440
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