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Theorem climsup 14619
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)
2 frn 6214 . . . . . . . . . 10 (𝐹:𝑍⟶ℝ → ran 𝐹 ⊆ ℝ)
31, 2syl 17 . . . . . . . . 9 (𝜑 → ran 𝐹 ⊆ ℝ)
4 ffn 6206 . . . . . . . . . . . 12 (𝐹:𝑍⟶ℝ → 𝐹 Fn 𝑍)
51, 4syl 17 . . . . . . . . . . 11 (𝜑𝐹 Fn 𝑍)
6 climsup.2 . . . . . . . . . . . . 13 (𝜑𝑀 ∈ ℤ)
7 uzid 11914 . . . . . . . . . . . . 13 (𝑀 ∈ ℤ → 𝑀 ∈ (ℤ𝑀))
86, 7syl 17 . . . . . . . . . . . 12 (𝜑𝑀 ∈ (ℤ𝑀))
9 climsup.1 . . . . . . . . . . . 12 𝑍 = (ℤ𝑀)
108, 9syl6eleqr 2850 . . . . . . . . . . 11 (𝜑𝑀𝑍)
11 fnfvelrn 6520 . . . . . . . . . . 11 ((𝐹 Fn 𝑍𝑀𝑍) → (𝐹𝑀) ∈ ran 𝐹)
125, 10, 11syl2anc 696 . . . . . . . . . 10 (𝜑 → (𝐹𝑀) ∈ ran 𝐹)
13 ne0i 4064 . . . . . . . . . 10 ((𝐹𝑀) ∈ ran 𝐹 → ran 𝐹 ≠ ∅)
1412, 13syl 17 . . . . . . . . 9 (𝜑 → ran 𝐹 ≠ ∅)
15 climsup.5 . . . . . . . . . 10 (𝜑 → ∃𝑥 ∈ ℝ ∀𝑘𝑍 (𝐹𝑘) ≤ 𝑥)
16 breq1 4807 . . . . . . . . . . . . 13 (𝑦 = (𝐹𝑘) → (𝑦𝑥 ↔ (𝐹𝑘) ≤ 𝑥))
1716ralrn 6526 . . . . . . . . . . . 12 (𝐹 Fn 𝑍 → (∀𝑦 ∈ ran 𝐹 𝑦𝑥 ↔ ∀𝑘𝑍 (𝐹𝑘) ≤ 𝑥))
1817rexbidv 3190 . . . . . . . . . . 11 (𝐹 Fn 𝑍 → (∃𝑥 ∈ ℝ ∀𝑦 ∈ ran 𝐹 𝑦𝑥 ↔ ∃𝑥 ∈ ℝ ∀𝑘𝑍 (𝐹𝑘) ≤ 𝑥))
195, 18syl 17 . . . . . . . . . 10 (𝜑 → (∃𝑥 ∈ ℝ ∀𝑦 ∈ ran 𝐹 𝑦𝑥 ↔ ∃𝑥 ∈ ℝ ∀𝑘𝑍 (𝐹𝑘) ≤ 𝑥))
2015, 19mpbird 247 . . . . . . . . 9 (𝜑 → ∃𝑥 ∈ ℝ ∀𝑦 ∈ ran 𝐹 𝑦𝑥)
213, 14, 203jca 1123 . . . . . . . 8 (𝜑 → (ran 𝐹 ⊆ ℝ ∧ ran 𝐹 ≠ ∅ ∧ ∃𝑥 ∈ ℝ ∀𝑦 ∈ ran 𝐹 𝑦𝑥))
22 suprcl 11195 . . . . . . . 8 ((ran 𝐹 ⊆ ℝ ∧ ran 𝐹 ≠ ∅ ∧ ∃𝑥 ∈ ℝ ∀𝑦 ∈ ran 𝐹 𝑦𝑥) → sup(ran 𝐹, ℝ, < ) ∈ ℝ)
2321, 22syl 17 . . . . . . 7 (𝜑 → sup(ran 𝐹, ℝ, < ) ∈ ℝ)
24 ltsubrp 12079 . . . . . . 7 ((sup(ran 𝐹, ℝ, < ) ∈ ℝ ∧ 𝑦 ∈ ℝ+) → (sup(ran 𝐹, ℝ, < ) − 𝑦) < sup(ran 𝐹, ℝ, < ))
2523, 24sylan 489 . . . . . 6 ((𝜑𝑦 ∈ ℝ+) → (sup(ran 𝐹, ℝ, < ) − 𝑦) < sup(ran 𝐹, ℝ, < ))
2621adantr 472 . . . . . . 7 ((𝜑𝑦 ∈ ℝ+) → (ran 𝐹 ⊆ ℝ ∧ ran 𝐹 ≠ ∅ ∧ ∃𝑥 ∈ ℝ ∀𝑦 ∈ ran 𝐹 𝑦𝑥))
27 rpre 12052 . . . . . . . 8 (𝑦 ∈ ℝ+𝑦 ∈ ℝ)
28 resubcl 10557 . . . . . . . 8 ((sup(ran 𝐹, ℝ, < ) ∈ ℝ ∧ 𝑦 ∈ ℝ) → (sup(ran 𝐹, ℝ, < ) − 𝑦) ∈ ℝ)
2923, 27, 28syl2an 495 . . . . . . 7 ((𝜑𝑦 ∈ ℝ+) → (sup(ran 𝐹, ℝ, < ) − 𝑦) ∈ ℝ)
30 suprlub 11199 . . . . . . 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))
3126, 29, 30syl2anc 696 . . . . . 6 ((𝜑𝑦 ∈ ℝ+) → ((sup(ran 𝐹, ℝ, < ) − 𝑦) < sup(ran 𝐹, ℝ, < ) ↔ ∃𝑘 ∈ ran 𝐹(sup(ran 𝐹, ℝ, < ) − 𝑦) < 𝑘))
3225, 31mpbid 222 . . . . 5 ((𝜑𝑦 ∈ ℝ+) → ∃𝑘 ∈ ran 𝐹(sup(ran 𝐹, ℝ, < ) − 𝑦) < 𝑘)
33 breq2 4808 . . . . . . . 8 (𝑘 = (𝐹𝑗) → ((sup(ran 𝐹, ℝ, < ) − 𝑦) < 𝑘 ↔ (sup(ran 𝐹, ℝ, < ) − 𝑦) < (𝐹𝑗)))
3433rexrn 6525 . . . . . . 7 (𝐹 Fn 𝑍 → (∃𝑘 ∈ ran 𝐹(sup(ran 𝐹, ℝ, < ) − 𝑦) < 𝑘 ↔ ∃𝑗𝑍 (sup(ran 𝐹, ℝ, < ) − 𝑦) < (𝐹𝑗)))
355, 34syl 17 . . . . . 6 (𝜑 → (∃𝑘 ∈ ran 𝐹(sup(ran 𝐹, ℝ, < ) − 𝑦) < 𝑘 ↔ ∃𝑗𝑍 (sup(ran 𝐹, ℝ, < ) − 𝑦) < (𝐹𝑗)))
3635biimpa 502 . . . . 5 ((𝜑 ∧ ∃𝑘 ∈ ran 𝐹(sup(ran 𝐹, ℝ, < ) − 𝑦) < 𝑘) → ∃𝑗𝑍 (sup(ran 𝐹, ℝ, < ) − 𝑦) < (𝐹𝑗))
3732, 36syldan 488 . . . 4 ((𝜑𝑦 ∈ ℝ+) → ∃𝑗𝑍 (sup(ran 𝐹, ℝ, < ) − 𝑦) < (𝐹𝑗))
38 ffvelrn 6521 . . . . . . . . . . . 12 ((𝐹:𝑍⟶ℝ ∧ 𝑗𝑍) → (𝐹𝑗) ∈ ℝ)
391, 38sylan 489 . . . . . . . . . . 11 ((𝜑𝑗𝑍) → (𝐹𝑗) ∈ ℝ)
4039ad2ant2r 800 . . . . . . . . . 10 (((𝜑𝑦 ∈ ℝ+) ∧ (𝑗𝑍𝑘 ∈ (ℤ𝑗))) → (𝐹𝑗) ∈ ℝ)
411adantr 472 . . . . . . . . . . 11 ((𝜑𝑦 ∈ ℝ+) → 𝐹:𝑍⟶ℝ)
429uztrn2 11917 . . . . . . . . . . 11 ((𝑗𝑍𝑘 ∈ (ℤ𝑗)) → 𝑘𝑍)
43 ffvelrn 6521 . . . . . . . . . . 11 ((𝐹:𝑍⟶ℝ ∧ 𝑘𝑍) → (𝐹𝑘) ∈ ℝ)
4441, 42, 43syl2an 495 . . . . . . . . . 10 (((𝜑𝑦 ∈ ℝ+) ∧ (𝑗𝑍𝑘 ∈ (ℤ𝑗))) → (𝐹𝑘) ∈ ℝ)
4523ad2antrr 764 . . . . . . . . . 10 (((𝜑𝑦 ∈ ℝ+) ∧ (𝑗𝑍𝑘 ∈ (ℤ𝑗))) → sup(ran 𝐹, ℝ, < ) ∈ ℝ)
46 simprr 813 . . . . . . . . . . 11 (((𝜑𝑦 ∈ ℝ+) ∧ (𝑗𝑍𝑘 ∈ (ℤ𝑗))) → 𝑘 ∈ (ℤ𝑗))
47 fzssuz 12595 . . . . . . . . . . . . . 14 (𝑗...𝑘) ⊆ (ℤ𝑗)
48 uzss 11920 . . . . . . . . . . . . . . . . 17 (𝑗 ∈ (ℤ𝑀) → (ℤ𝑗) ⊆ (ℤ𝑀))
4948, 9syl6sseqr 3793 . . . . . . . . . . . . . . . 16 (𝑗 ∈ (ℤ𝑀) → (ℤ𝑗) ⊆ 𝑍)
5049, 9eleq2s 2857 . . . . . . . . . . . . . . 15 (𝑗𝑍 → (ℤ𝑗) ⊆ 𝑍)
5150ad2antrl 766 . . . . . . . . . . . . . 14 (((𝜑𝑦 ∈ ℝ+) ∧ (𝑗𝑍𝑘 ∈ (ℤ𝑗))) → (ℤ𝑗) ⊆ 𝑍)
5247, 51syl5ss 3755 . . . . . . . . . . . . 13 (((𝜑𝑦 ∈ ℝ+) ∧ (𝑗𝑍𝑘 ∈ (ℤ𝑗))) → (𝑗...𝑘) ⊆ 𝑍)
53 ffvelrn 6521 . . . . . . . . . . . . . . . 16 ((𝐹:𝑍⟶ℝ ∧ 𝑛𝑍) → (𝐹𝑛) ∈ ℝ)
5453ralrimiva 3104 . . . . . . . . . . . . . . 15 (𝐹:𝑍⟶ℝ → ∀𝑛𝑍 (𝐹𝑛) ∈ ℝ)
551, 54syl 17 . . . . . . . . . . . . . 14 (𝜑 → ∀𝑛𝑍 (𝐹𝑛) ∈ ℝ)
5655ad2antrr 764 . . . . . . . . . . . . 13 (((𝜑𝑦 ∈ ℝ+) ∧ (𝑗𝑍𝑘 ∈ (ℤ𝑗))) → ∀𝑛𝑍 (𝐹𝑛) ∈ ℝ)
57 ssralv 3807 . . . . . . . . . . . . 13 ((𝑗...𝑘) ⊆ 𝑍 → (∀𝑛𝑍 (𝐹𝑛) ∈ ℝ → ∀𝑛 ∈ (𝑗...𝑘)(𝐹𝑛) ∈ ℝ))
5852, 56, 57sylc 65 . . . . . . . . . . . 12 (((𝜑𝑦 ∈ ℝ+) ∧ (𝑗𝑍𝑘 ∈ (ℤ𝑗))) → ∀𝑛 ∈ (𝑗...𝑘)(𝐹𝑛) ∈ ℝ)
5958r19.21bi 3070 . . . . . . . . . . 11 ((((𝜑𝑦 ∈ ℝ+) ∧ (𝑗𝑍𝑘 ∈ (ℤ𝑗))) ∧ 𝑛 ∈ (𝑗...𝑘)) → (𝐹𝑛) ∈ ℝ)
60 fzssuz 12595 . . . . . . . . . . . . . 14 (𝑗...(𝑘 − 1)) ⊆ (ℤ𝑗)
6160, 51syl5ss 3755 . . . . . . . . . . . . 13 (((𝜑𝑦 ∈ ℝ+) ∧ (𝑗𝑍𝑘 ∈ (ℤ𝑗))) → (𝑗...(𝑘 − 1)) ⊆ 𝑍)
6261sselda 3744 . . . . . . . . . . . 12 ((((𝜑𝑦 ∈ ℝ+) ∧ (𝑗𝑍𝑘 ∈ (ℤ𝑗))) ∧ 𝑛 ∈ (𝑗...(𝑘 − 1))) → 𝑛𝑍)
63 climsup.4 . . . . . . . . . . . . . . 15 ((𝜑𝑘𝑍) → (𝐹𝑘) ≤ (𝐹‘(𝑘 + 1)))
6463ralrimiva 3104 . . . . . . . . . . . . . 14 (𝜑 → ∀𝑘𝑍 (𝐹𝑘) ≤ (𝐹‘(𝑘 + 1)))
6564ad2antrr 764 . . . . . . . . . . . . 13 (((𝜑𝑦 ∈ ℝ+) ∧ (𝑗𝑍𝑘 ∈ (ℤ𝑗))) → ∀𝑘𝑍 (𝐹𝑘) ≤ (𝐹‘(𝑘 + 1)))
66 fveq2 6353 . . . . . . . . . . . . . . 15 (𝑘 = 𝑛 → (𝐹𝑘) = (𝐹𝑛))
67 oveq1 6821 . . . . . . . . . . . . . . . 16 (𝑘 = 𝑛 → (𝑘 + 1) = (𝑛 + 1))
6867fveq2d 6357 . . . . . . . . . . . . . . 15 (𝑘 = 𝑛 → (𝐹‘(𝑘 + 1)) = (𝐹‘(𝑛 + 1)))
6966, 68breq12d 4817 . . . . . . . . . . . . . 14 (𝑘 = 𝑛 → ((𝐹𝑘) ≤ (𝐹‘(𝑘 + 1)) ↔ (𝐹𝑛) ≤ (𝐹‘(𝑛 + 1))))
7069rspccva 3448 . . . . . . . . . . . . 13 ((∀𝑘𝑍 (𝐹𝑘) ≤ (𝐹‘(𝑘 + 1)) ∧ 𝑛𝑍) → (𝐹𝑛) ≤ (𝐹‘(𝑛 + 1)))
7165, 70sylan 489 . . . . . . . . . . . 12 ((((𝜑𝑦 ∈ ℝ+) ∧ (𝑗𝑍𝑘 ∈ (ℤ𝑗))) ∧ 𝑛𝑍) → (𝐹𝑛) ≤ (𝐹‘(𝑛 + 1)))
7262, 71syldan 488 . . . . . . . . . . 11 ((((𝜑𝑦 ∈ ℝ+) ∧ (𝑗𝑍𝑘 ∈ (ℤ𝑗))) ∧ 𝑛 ∈ (𝑗...(𝑘 − 1))) → (𝐹𝑛) ≤ (𝐹‘(𝑛 + 1)))
7346, 59, 72monoord 13045 . . . . . . . . . 10 (((𝜑𝑦 ∈ ℝ+) ∧ (𝑗𝑍𝑘 ∈ (ℤ𝑗))) → (𝐹𝑗) ≤ (𝐹𝑘))
7440, 44, 45, 73lesub2dd 10856 . . . . . . . . 9 (((𝜑𝑦 ∈ ℝ+) ∧ (𝑗𝑍𝑘 ∈ (ℤ𝑗))) → (sup(ran 𝐹, ℝ, < ) − (𝐹𝑘)) ≤ (sup(ran 𝐹, ℝ, < ) − (𝐹𝑗)))
7545, 44resubcld 10670 . . . . . . . . . 10 (((𝜑𝑦 ∈ ℝ+) ∧ (𝑗𝑍𝑘 ∈ (ℤ𝑗))) → (sup(ran 𝐹, ℝ, < ) − (𝐹𝑘)) ∈ ℝ)
7645, 40resubcld 10670 . . . . . . . . . 10 (((𝜑𝑦 ∈ ℝ+) ∧ (𝑗𝑍𝑘 ∈ (ℤ𝑗))) → (sup(ran 𝐹, ℝ, < ) − (𝐹𝑗)) ∈ ℝ)
7727ad2antlr 765 . . . . . . . . . 10 (((𝜑𝑦 ∈ ℝ+) ∧ (𝑗𝑍𝑘 ∈ (ℤ𝑗))) → 𝑦 ∈ ℝ)
78 lelttr 10340 . . . . . . . . . 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))
7975, 76, 77, 78syl3anc 1477 . . . . . . . . 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))
8074, 79mpand 713 . . . . . . . 8 (((𝜑𝑦 ∈ ℝ+) ∧ (𝑗𝑍𝑘 ∈ (ℤ𝑗))) → ((sup(ran 𝐹, ℝ, < ) − (𝐹𝑗)) < 𝑦 → (sup(ran 𝐹, ℝ, < ) − (𝐹𝑘)) < 𝑦))
81 ltsub23 10720 . . . . . . . . 9 ((sup(ran 𝐹, ℝ, < ) ∈ ℝ ∧ 𝑦 ∈ ℝ ∧ (𝐹𝑗) ∈ ℝ) → ((sup(ran 𝐹, ℝ, < ) − 𝑦) < (𝐹𝑗) ↔ (sup(ran 𝐹, ℝ, < ) − (𝐹𝑗)) < 𝑦))
8245, 77, 40, 81syl3anc 1477 . . . . . . . 8 (((𝜑𝑦 ∈ ℝ+) ∧ (𝑗𝑍𝑘 ∈ (ℤ𝑗))) → ((sup(ran 𝐹, ℝ, < ) − 𝑦) < (𝐹𝑗) ↔ (sup(ran 𝐹, ℝ, < ) − (𝐹𝑗)) < 𝑦))
8321ad2antrr 764 . . . . . . . . . . 11 (((𝜑𝑦 ∈ ℝ+) ∧ (𝑗𝑍𝑘 ∈ (ℤ𝑗))) → (ran 𝐹 ⊆ ℝ ∧ ran 𝐹 ≠ ∅ ∧ ∃𝑥 ∈ ℝ ∀𝑦 ∈ ran 𝐹 𝑦𝑥))
845adantr 472 . . . . . . . . . . . 12 ((𝜑𝑦 ∈ ℝ+) → 𝐹 Fn 𝑍)
85 fnfvelrn 6520 . . . . . . . . . . . 12 ((𝐹 Fn 𝑍𝑘𝑍) → (𝐹𝑘) ∈ ran 𝐹)
8684, 42, 85syl2an 495 . . . . . . . . . . 11 (((𝜑𝑦 ∈ ℝ+) ∧ (𝑗𝑍𝑘 ∈ (ℤ𝑗))) → (𝐹𝑘) ∈ ran 𝐹)
87 suprub 11196 . . . . . . . . . . 11 (((ran 𝐹 ⊆ ℝ ∧ ran 𝐹 ≠ ∅ ∧ ∃𝑥 ∈ ℝ ∀𝑦 ∈ ran 𝐹 𝑦𝑥) ∧ (𝐹𝑘) ∈ ran 𝐹) → (𝐹𝑘) ≤ sup(ran 𝐹, ℝ, < ))
8883, 86, 87syl2anc 696 . . . . . . . . . 10 (((𝜑𝑦 ∈ ℝ+) ∧ (𝑗𝑍𝑘 ∈ (ℤ𝑗))) → (𝐹𝑘) ≤ sup(ran 𝐹, ℝ, < ))
8944, 45, 88abssuble0d 14390 . . . . . . . . 9 (((𝜑𝑦 ∈ ℝ+) ∧ (𝑗𝑍𝑘 ∈ (ℤ𝑗))) → (abs‘((𝐹𝑘) − sup(ran 𝐹, ℝ, < ))) = (sup(ran 𝐹, ℝ, < ) − (𝐹𝑘)))
9089breq1d 4814 . . . . . . . 8 (((𝜑𝑦 ∈ ℝ+) ∧ (𝑗𝑍𝑘 ∈ (ℤ𝑗))) → ((abs‘((𝐹𝑘) − sup(ran 𝐹, ℝ, < ))) < 𝑦 ↔ (sup(ran 𝐹, ℝ, < ) − (𝐹𝑘)) < 𝑦))
9180, 82, 903imtr4d 283 . . . . . . 7 (((𝜑𝑦 ∈ ℝ+) ∧ (𝑗𝑍𝑘 ∈ (ℤ𝑗))) → ((sup(ran 𝐹, ℝ, < ) − 𝑦) < (𝐹𝑗) → (abs‘((𝐹𝑘) − sup(ran 𝐹, ℝ, < ))) < 𝑦))
9291anassrs 683 . . . . . 6 ((((𝜑𝑦 ∈ ℝ+) ∧ 𝑗𝑍) ∧ 𝑘 ∈ (ℤ𝑗)) → ((sup(ran 𝐹, ℝ, < ) − 𝑦) < (𝐹𝑗) → (abs‘((𝐹𝑘) − sup(ran 𝐹, ℝ, < ))) < 𝑦))
9392ralrimdva 3107 . . . . 5 (((𝜑𝑦 ∈ ℝ+) ∧ 𝑗𝑍) → ((sup(ran 𝐹, ℝ, < ) − 𝑦) < (𝐹𝑗) → ∀𝑘 ∈ (ℤ𝑗)(abs‘((𝐹𝑘) − sup(ran 𝐹, ℝ, < ))) < 𝑦))
9493reximdva 3155 . . . 4 ((𝜑𝑦 ∈ ℝ+) → (∃𝑗𝑍 (sup(ran 𝐹, ℝ, < ) − 𝑦) < (𝐹𝑗) → ∃𝑗𝑍𝑘 ∈ (ℤ𝑗)(abs‘((𝐹𝑘) − sup(ran 𝐹, ℝ, < ))) < 𝑦))
9537, 94mpd 15 . . 3 ((𝜑𝑦 ∈ ℝ+) → ∃𝑗𝑍𝑘 ∈ (ℤ𝑗)(abs‘((𝐹𝑘) − sup(ran 𝐹, ℝ, < ))) < 𝑦)
9695ralrimiva 3104 . 2 (𝜑 → ∀𝑦 ∈ ℝ+𝑗𝑍𝑘 ∈ (ℤ𝑗)(abs‘((𝐹𝑘) − sup(ran 𝐹, ℝ, < ))) < 𝑦)
97 fvex 6363 . . . . 5 (ℤ𝑀) ∈ V
989, 97eqeltri 2835 . . . 4 𝑍 ∈ V
99 fex 6654 . . . 4 ((𝐹:𝑍⟶ℝ ∧ 𝑍 ∈ V) → 𝐹 ∈ V)
1001, 98, 99sylancl 697 . . 3 (𝜑𝐹 ∈ V)
101 eqidd 2761 . . 3 ((𝜑𝑘𝑍) → (𝐹𝑘) = (𝐹𝑘))
10223recnd 10280 . . 3 (𝜑 → sup(ran 𝐹, ℝ, < ) ∈ ℂ)
1031, 43sylan 489 . . . 4 ((𝜑𝑘𝑍) → (𝐹𝑘) ∈ ℝ)
104103recnd 10280 . . 3 ((𝜑𝑘𝑍) → (𝐹𝑘) ∈ ℂ)
1059, 6, 100, 101, 102, 104clim2c 14455 . 2 (𝜑 → (𝐹 ⇝ sup(ran 𝐹, ℝ, < ) ↔ ∀𝑦 ∈ ℝ+𝑗𝑍𝑘 ∈ (ℤ𝑗)(abs‘((𝐹𝑘) − sup(ran 𝐹, ℝ, < ))) < 𝑦))
10696, 105mpbird 247 1 (𝜑𝐹 ⇝ sup(ran 𝐹, ℝ, < ))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383  w3a 1072   = wceq 1632  wcel 2139  wne 2932  wral 3050  wrex 3051  Vcvv 3340  wss 3715  c0 4058   class class class wbr 4804  ran crn 5267   Fn wfn 6044  wf 6045  cfv 6049  (class class class)co 6814  supcsup 8513  cr 10147  1c1 10149   + caddc 10151   < clt 10286  cle 10287  cmin 10478  cz 11589  cuz 11899  +crp 12045  ...cfz 12539  abscabs 14193  cli 14434
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-rep 4923  ax-sep 4933  ax-nul 4941  ax-pow 4992  ax-pr 5055  ax-un 7115  ax-cnex 10204  ax-resscn 10205  ax-1cn 10206  ax-icn 10207  ax-addcl 10208  ax-addrcl 10209  ax-mulcl 10210  ax-mulrcl 10211  ax-mulcom 10212  ax-addass 10213  ax-mulass 10214  ax-distr 10215  ax-i2m1 10216  ax-1ne0 10217  ax-1rid 10218  ax-rnegex 10219  ax-rrecex 10220  ax-cnre 10221  ax-pre-lttri 10222  ax-pre-lttrn 10223  ax-pre-ltadd 10224  ax-pre-mulgt0 10225  ax-pre-sup 10226
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1073  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-nel 3036  df-ral 3055  df-rex 3056  df-reu 3057  df-rmo 3058  df-rab 3059  df-v 3342  df-sbc 3577  df-csb 3675  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-pss 3731  df-nul 4059  df-if 4231  df-pw 4304  df-sn 4322  df-pr 4324  df-tp 4326  df-op 4328  df-uni 4589  df-iun 4674  df-br 4805  df-opab 4865  df-mpt 4882  df-tr 4905  df-id 5174  df-eprel 5179  df-po 5187  df-so 5188  df-fr 5225  df-we 5227  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-res 5278  df-ima 5279  df-pred 5841  df-ord 5887  df-on 5888  df-lim 5889  df-suc 5890  df-iota 6012  df-fun 6051  df-fn 6052  df-f 6053  df-f1 6054  df-fo 6055  df-f1o 6056  df-fv 6057  df-riota 6775  df-ov 6817  df-oprab 6818  df-mpt2 6819  df-om 7232  df-1st 7334  df-2nd 7335  df-wrecs 7577  df-recs 7638  df-rdg 7676  df-er 7913  df-en 8124  df-dom 8125  df-sdom 8126  df-sup 8515  df-pnf 10288  df-mnf 10289  df-xr 10290  df-ltxr 10291  df-le 10292  df-sub 10480  df-neg 10481  df-div 10897  df-nn 11233  df-2 11291  df-3 11292  df-n0 11505  df-z 11590  df-uz 11900  df-rp 12046  df-fz 12540  df-seq 13016  df-exp 13075  df-cj 14058  df-re 14059  df-im 14060  df-sqrt 14194  df-abs 14195  df-clim 14438
This theorem is referenced by:  isumsup2  14797  climcnds  14802  itg1climres  23700  itg2monolem1  23736  itg2i1fseq  23741  itg2i1fseq2  23742  emcllem6  24947  lmdvg  30329  esumpcvgval  30470  meaiuninclem  41218
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