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Theorem climsup 14194
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 5952 . . . . . . . . . 10 (𝐹:𝑍⟶ℝ → ran 𝐹 ⊆ ℝ)
31, 2syl 17 . . . . . . . . 9 (𝜑 → ran 𝐹 ⊆ ℝ)
4 ffn 5944 . . . . . . . . . . . 12 (𝐹:𝑍⟶ℝ → 𝐹 Fn 𝑍)
51, 4syl 17 . . . . . . . . . . 11 (𝜑𝐹 Fn 𝑍)
6 climsup.2 . . . . . . . . . . . . 13 (𝜑𝑀 ∈ ℤ)
7 uzid 11534 . . . . . . . . . . . . 13 (𝑀 ∈ ℤ → 𝑀 ∈ (ℤ𝑀))
86, 7syl 17 . . . . . . . . . . . 12 (𝜑𝑀 ∈ (ℤ𝑀))
9 climsup.1 . . . . . . . . . . . 12 𝑍 = (ℤ𝑀)
108, 9syl6eleqr 2698 . . . . . . . . . . 11 (𝜑𝑀𝑍)
11 fnfvelrn 6249 . . . . . . . . . . 11 ((𝐹 Fn 𝑍𝑀𝑍) → (𝐹𝑀) ∈ ran 𝐹)
125, 10, 11syl2anc 690 . . . . . . . . . 10 (𝜑 → (𝐹𝑀) ∈ ran 𝐹)
13 ne0i 3879 . . . . . . . . . 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 4580 . . . . . . . . . . . . 13 (𝑦 = (𝐹𝑘) → (𝑦𝑥 ↔ (𝐹𝑘) ≤ 𝑥))
1716ralrn 6255 . . . . . . . . . . . 12 (𝐹 Fn 𝑍 → (∀𝑦 ∈ ran 𝐹 𝑦𝑥 ↔ ∀𝑘𝑍 (𝐹𝑘) ≤ 𝑥))
1817rexbidv 3033 . . . . . . . . . . 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 245 . . . . . . . . 9 (𝜑 → ∃𝑥 ∈ ℝ ∀𝑦 ∈ ran 𝐹 𝑦𝑥)
213, 14, 203jca 1234 . . . . . . . 8 (𝜑 → (ran 𝐹 ⊆ ℝ ∧ ran 𝐹 ≠ ∅ ∧ ∃𝑥 ∈ ℝ ∀𝑦 ∈ ran 𝐹 𝑦𝑥))
22 suprcl 10832 . . . . . . . 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 11698 . . . . . . 7 ((sup(ran 𝐹, ℝ, < ) ∈ ℝ ∧ 𝑦 ∈ ℝ+) → (sup(ran 𝐹, ℝ, < ) − 𝑦) < sup(ran 𝐹, ℝ, < ))
2523, 24sylan 486 . . . . . 6 ((𝜑𝑦 ∈ ℝ+) → (sup(ran 𝐹, ℝ, < ) − 𝑦) < sup(ran 𝐹, ℝ, < ))
2621adantr 479 . . . . . . 7 ((𝜑𝑦 ∈ ℝ+) → (ran 𝐹 ⊆ ℝ ∧ ran 𝐹 ≠ ∅ ∧ ∃𝑥 ∈ ℝ ∀𝑦 ∈ ran 𝐹 𝑦𝑥))
27 rpre 11671 . . . . . . . 8 (𝑦 ∈ ℝ+𝑦 ∈ ℝ)
28 resubcl 10196 . . . . . . . 8 ((sup(ran 𝐹, ℝ, < ) ∈ ℝ ∧ 𝑦 ∈ ℝ) → (sup(ran 𝐹, ℝ, < ) − 𝑦) ∈ ℝ)
2923, 27, 28syl2an 492 . . . . . . 7 ((𝜑𝑦 ∈ ℝ+) → (sup(ran 𝐹, ℝ, < ) − 𝑦) ∈ ℝ)
30 suprlub 10834 . . . . . . 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 690 . . . . . 6 ((𝜑𝑦 ∈ ℝ+) → ((sup(ran 𝐹, ℝ, < ) − 𝑦) < sup(ran 𝐹, ℝ, < ) ↔ ∃𝑘 ∈ ran 𝐹(sup(ran 𝐹, ℝ, < ) − 𝑦) < 𝑘))
3225, 31mpbid 220 . . . . 5 ((𝜑𝑦 ∈ ℝ+) → ∃𝑘 ∈ ran 𝐹(sup(ran 𝐹, ℝ, < ) − 𝑦) < 𝑘)
33 breq2 4581 . . . . . . . 8 (𝑘 = (𝐹𝑗) → ((sup(ran 𝐹, ℝ, < ) − 𝑦) < 𝑘 ↔ (sup(ran 𝐹, ℝ, < ) − 𝑦) < (𝐹𝑗)))
3433rexrn 6254 . . . . . . 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 499 . . . . 5 ((𝜑 ∧ ∃𝑘 ∈ ran 𝐹(sup(ran 𝐹, ℝ, < ) − 𝑦) < 𝑘) → ∃𝑗𝑍 (sup(ran 𝐹, ℝ, < ) − 𝑦) < (𝐹𝑗))
3732, 36syldan 485 . . . 4 ((𝜑𝑦 ∈ ℝ+) → ∃𝑗𝑍 (sup(ran 𝐹, ℝ, < ) − 𝑦) < (𝐹𝑗))
38 ffvelrn 6250 . . . . . . . . . . . 12 ((𝐹:𝑍⟶ℝ ∧ 𝑗𝑍) → (𝐹𝑗) ∈ ℝ)
391, 38sylan 486 . . . . . . . . . . 11 ((𝜑𝑗𝑍) → (𝐹𝑗) ∈ ℝ)
4039ad2ant2r 778 . . . . . . . . . 10 (((𝜑𝑦 ∈ ℝ+) ∧ (𝑗𝑍𝑘 ∈ (ℤ𝑗))) → (𝐹𝑗) ∈ ℝ)
411adantr 479 . . . . . . . . . . 11 ((𝜑𝑦 ∈ ℝ+) → 𝐹:𝑍⟶ℝ)
429uztrn2 11537 . . . . . . . . . . 11 ((𝑗𝑍𝑘 ∈ (ℤ𝑗)) → 𝑘𝑍)
43 ffvelrn 6250 . . . . . . . . . . 11 ((𝐹:𝑍⟶ℝ ∧ 𝑘𝑍) → (𝐹𝑘) ∈ ℝ)
4441, 42, 43syl2an 492 . . . . . . . . . 10 (((𝜑𝑦 ∈ ℝ+) ∧ (𝑗𝑍𝑘 ∈ (ℤ𝑗))) → (𝐹𝑘) ∈ ℝ)
4523ad2antrr 757 . . . . . . . . . 10 (((𝜑𝑦 ∈ ℝ+) ∧ (𝑗𝑍𝑘 ∈ (ℤ𝑗))) → sup(ran 𝐹, ℝ, < ) ∈ ℝ)
46 simprr 791 . . . . . . . . . . 11 (((𝜑𝑦 ∈ ℝ+) ∧ (𝑗𝑍𝑘 ∈ (ℤ𝑗))) → 𝑘 ∈ (ℤ𝑗))
47 fzssuz 12208 . . . . . . . . . . . . . 14 (𝑗...𝑘) ⊆ (ℤ𝑗)
48 uzss 11540 . . . . . . . . . . . . . . . . 17 (𝑗 ∈ (ℤ𝑀) → (ℤ𝑗) ⊆ (ℤ𝑀))
4948, 9syl6sseqr 3614 . . . . . . . . . . . . . . . 16 (𝑗 ∈ (ℤ𝑀) → (ℤ𝑗) ⊆ 𝑍)
5049, 9eleq2s 2705 . . . . . . . . . . . . . . 15 (𝑗𝑍 → (ℤ𝑗) ⊆ 𝑍)
5150ad2antrl 759 . . . . . . . . . . . . . 14 (((𝜑𝑦 ∈ ℝ+) ∧ (𝑗𝑍𝑘 ∈ (ℤ𝑗))) → (ℤ𝑗) ⊆ 𝑍)
5247, 51syl5ss 3578 . . . . . . . . . . . . 13 (((𝜑𝑦 ∈ ℝ+) ∧ (𝑗𝑍𝑘 ∈ (ℤ𝑗))) → (𝑗...𝑘) ⊆ 𝑍)
53 ffvelrn 6250 . . . . . . . . . . . . . . . 16 ((𝐹:𝑍⟶ℝ ∧ 𝑛𝑍) → (𝐹𝑛) ∈ ℝ)
5453ralrimiva 2948 . . . . . . . . . . . . . . 15 (𝐹:𝑍⟶ℝ → ∀𝑛𝑍 (𝐹𝑛) ∈ ℝ)
551, 54syl 17 . . . . . . . . . . . . . 14 (𝜑 → ∀𝑛𝑍 (𝐹𝑛) ∈ ℝ)
5655ad2antrr 757 . . . . . . . . . . . . 13 (((𝜑𝑦 ∈ ℝ+) ∧ (𝑗𝑍𝑘 ∈ (ℤ𝑗))) → ∀𝑛𝑍 (𝐹𝑛) ∈ ℝ)
57 ssralv 3628 . . . . . . . . . . . . 13 ((𝑗...𝑘) ⊆ 𝑍 → (∀𝑛𝑍 (𝐹𝑛) ∈ ℝ → ∀𝑛 ∈ (𝑗...𝑘)(𝐹𝑛) ∈ ℝ))
5852, 56, 57sylc 62 . . . . . . . . . . . 12 (((𝜑𝑦 ∈ ℝ+) ∧ (𝑗𝑍𝑘 ∈ (ℤ𝑗))) → ∀𝑛 ∈ (𝑗...𝑘)(𝐹𝑛) ∈ ℝ)
5958r19.21bi 2915 . . . . . . . . . . 11 ((((𝜑𝑦 ∈ ℝ+) ∧ (𝑗𝑍𝑘 ∈ (ℤ𝑗))) ∧ 𝑛 ∈ (𝑗...𝑘)) → (𝐹𝑛) ∈ ℝ)
60 fzssuz 12208 . . . . . . . . . . . . . 14 (𝑗...(𝑘 − 1)) ⊆ (ℤ𝑗)
6160, 51syl5ss 3578 . . . . . . . . . . . . 13 (((𝜑𝑦 ∈ ℝ+) ∧ (𝑗𝑍𝑘 ∈ (ℤ𝑗))) → (𝑗...(𝑘 − 1)) ⊆ 𝑍)
6261sselda 3567 . . . . . . . . . . . 12 ((((𝜑𝑦 ∈ ℝ+) ∧ (𝑗𝑍𝑘 ∈ (ℤ𝑗))) ∧ 𝑛 ∈ (𝑗...(𝑘 − 1))) → 𝑛𝑍)
63 climsup.4 . . . . . . . . . . . . . . 15 ((𝜑𝑘𝑍) → (𝐹𝑘) ≤ (𝐹‘(𝑘 + 1)))
6463ralrimiva 2948 . . . . . . . . . . . . . 14 (𝜑 → ∀𝑘𝑍 (𝐹𝑘) ≤ (𝐹‘(𝑘 + 1)))
6564ad2antrr 757 . . . . . . . . . . . . 13 (((𝜑𝑦 ∈ ℝ+) ∧ (𝑗𝑍𝑘 ∈ (ℤ𝑗))) → ∀𝑘𝑍 (𝐹𝑘) ≤ (𝐹‘(𝑘 + 1)))
66 fveq2 6088 . . . . . . . . . . . . . . 15 (𝑘 = 𝑛 → (𝐹𝑘) = (𝐹𝑛))
67 oveq1 6534 . . . . . . . . . . . . . . . 16 (𝑘 = 𝑛 → (𝑘 + 1) = (𝑛 + 1))
6867fveq2d 6092 . . . . . . . . . . . . . . 15 (𝑘 = 𝑛 → (𝐹‘(𝑘 + 1)) = (𝐹‘(𝑛 + 1)))
6966, 68breq12d 4590 . . . . . . . . . . . . . 14 (𝑘 = 𝑛 → ((𝐹𝑘) ≤ (𝐹‘(𝑘 + 1)) ↔ (𝐹𝑛) ≤ (𝐹‘(𝑛 + 1))))
7069rspccva 3280 . . . . . . . . . . . . 13 ((∀𝑘𝑍 (𝐹𝑘) ≤ (𝐹‘(𝑘 + 1)) ∧ 𝑛𝑍) → (𝐹𝑛) ≤ (𝐹‘(𝑛 + 1)))
7165, 70sylan 486 . . . . . . . . . . . 12 ((((𝜑𝑦 ∈ ℝ+) ∧ (𝑗𝑍𝑘 ∈ (ℤ𝑗))) ∧ 𝑛𝑍) → (𝐹𝑛) ≤ (𝐹‘(𝑛 + 1)))
7262, 71syldan 485 . . . . . . . . . . 11 ((((𝜑𝑦 ∈ ℝ+) ∧ (𝑗𝑍𝑘 ∈ (ℤ𝑗))) ∧ 𝑛 ∈ (𝑗...(𝑘 − 1))) → (𝐹𝑛) ≤ (𝐹‘(𝑛 + 1)))
7346, 59, 72monoord 12648 . . . . . . . . . 10 (((𝜑𝑦 ∈ ℝ+) ∧ (𝑗𝑍𝑘 ∈ (ℤ𝑗))) → (𝐹𝑗) ≤ (𝐹𝑘))
7440, 44, 45, 73lesub2dd 10493 . . . . . . . . 9 (((𝜑𝑦 ∈ ℝ+) ∧ (𝑗𝑍𝑘 ∈ (ℤ𝑗))) → (sup(ran 𝐹, ℝ, < ) − (𝐹𝑘)) ≤ (sup(ran 𝐹, ℝ, < ) − (𝐹𝑗)))
7545, 44resubcld 10309 . . . . . . . . . 10 (((𝜑𝑦 ∈ ℝ+) ∧ (𝑗𝑍𝑘 ∈ (ℤ𝑗))) → (sup(ran 𝐹, ℝ, < ) − (𝐹𝑘)) ∈ ℝ)
7645, 40resubcld 10309 . . . . . . . . . 10 (((𝜑𝑦 ∈ ℝ+) ∧ (𝑗𝑍𝑘 ∈ (ℤ𝑗))) → (sup(ran 𝐹, ℝ, < ) − (𝐹𝑗)) ∈ ℝ)
7727ad2antlr 758 . . . . . . . . . 10 (((𝜑𝑦 ∈ ℝ+) ∧ (𝑗𝑍𝑘 ∈ (ℤ𝑗))) → 𝑦 ∈ ℝ)
78 lelttr 9979 . . . . . . . . . 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 1317 . . . . . . . . 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 706 . . . . . . . 8 (((𝜑𝑦 ∈ ℝ+) ∧ (𝑗𝑍𝑘 ∈ (ℤ𝑗))) → ((sup(ran 𝐹, ℝ, < ) − (𝐹𝑗)) < 𝑦 → (sup(ran 𝐹, ℝ, < ) − (𝐹𝑘)) < 𝑦))
81 ltsub23 10357 . . . . . . . . 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 1317 . . . . . . . 8 (((𝜑𝑦 ∈ ℝ+) ∧ (𝑗𝑍𝑘 ∈ (ℤ𝑗))) → ((sup(ran 𝐹, ℝ, < ) − 𝑦) < (𝐹𝑗) ↔ (sup(ran 𝐹, ℝ, < ) − (𝐹𝑗)) < 𝑦))
8321ad2antrr 757 . . . . . . . . . . 11 (((𝜑𝑦 ∈ ℝ+) ∧ (𝑗𝑍𝑘 ∈ (ℤ𝑗))) → (ran 𝐹 ⊆ ℝ ∧ ran 𝐹 ≠ ∅ ∧ ∃𝑥 ∈ ℝ ∀𝑦 ∈ ran 𝐹 𝑦𝑥))
845adantr 479 . . . . . . . . . . . 12 ((𝜑𝑦 ∈ ℝ+) → 𝐹 Fn 𝑍)
85 fnfvelrn 6249 . . . . . . . . . . . 12 ((𝐹 Fn 𝑍𝑘𝑍) → (𝐹𝑘) ∈ ran 𝐹)
8684, 42, 85syl2an 492 . . . . . . . . . . 11 (((𝜑𝑦 ∈ ℝ+) ∧ (𝑗𝑍𝑘 ∈ (ℤ𝑗))) → (𝐹𝑘) ∈ ran 𝐹)
87 suprub 10833 . . . . . . . . . . 11 (((ran 𝐹 ⊆ ℝ ∧ ran 𝐹 ≠ ∅ ∧ ∃𝑥 ∈ ℝ ∀𝑦 ∈ ran 𝐹 𝑦𝑥) ∧ (𝐹𝑘) ∈ ran 𝐹) → (𝐹𝑘) ≤ sup(ran 𝐹, ℝ, < ))
8883, 86, 87syl2anc 690 . . . . . . . . . 10 (((𝜑𝑦 ∈ ℝ+) ∧ (𝑗𝑍𝑘 ∈ (ℤ𝑗))) → (𝐹𝑘) ≤ sup(ran 𝐹, ℝ, < ))
8944, 45, 88abssuble0d 13965 . . . . . . . . 9 (((𝜑𝑦 ∈ ℝ+) ∧ (𝑗𝑍𝑘 ∈ (ℤ𝑗))) → (abs‘((𝐹𝑘) − sup(ran 𝐹, ℝ, < ))) = (sup(ran 𝐹, ℝ, < ) − (𝐹𝑘)))
9089breq1d 4587 . . . . . . . 8 (((𝜑𝑦 ∈ ℝ+) ∧ (𝑗𝑍𝑘 ∈ (ℤ𝑗))) → ((abs‘((𝐹𝑘) − sup(ran 𝐹, ℝ, < ))) < 𝑦 ↔ (sup(ran 𝐹, ℝ, < ) − (𝐹𝑘)) < 𝑦))
9180, 82, 903imtr4d 281 . . . . . . 7 (((𝜑𝑦 ∈ ℝ+) ∧ (𝑗𝑍𝑘 ∈ (ℤ𝑗))) → ((sup(ran 𝐹, ℝ, < ) − 𝑦) < (𝐹𝑗) → (abs‘((𝐹𝑘) − sup(ran 𝐹, ℝ, < ))) < 𝑦))
9291anassrs 677 . . . . . 6 ((((𝜑𝑦 ∈ ℝ+) ∧ 𝑗𝑍) ∧ 𝑘 ∈ (ℤ𝑗)) → ((sup(ran 𝐹, ℝ, < ) − 𝑦) < (𝐹𝑗) → (abs‘((𝐹𝑘) − sup(ran 𝐹, ℝ, < ))) < 𝑦))
9392ralrimdva 2951 . . . . 5 (((𝜑𝑦 ∈ ℝ+) ∧ 𝑗𝑍) → ((sup(ran 𝐹, ℝ, < ) − 𝑦) < (𝐹𝑗) → ∀𝑘 ∈ (ℤ𝑗)(abs‘((𝐹𝑘) − sup(ran 𝐹, ℝ, < ))) < 𝑦))
9493reximdva 2999 . . . 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 2948 . 2 (𝜑 → ∀𝑦 ∈ ℝ+𝑗𝑍𝑘 ∈ (ℤ𝑗)(abs‘((𝐹𝑘) − sup(ran 𝐹, ℝ, < ))) < 𝑦)
97 fvex 6098 . . . . 5 (ℤ𝑀) ∈ V
989, 97eqeltri 2683 . . . 4 𝑍 ∈ V
99 fex 6372 . . . 4 ((𝐹:𝑍⟶ℝ ∧ 𝑍 ∈ V) → 𝐹 ∈ V)
1001, 98, 99sylancl 692 . . 3 (𝜑𝐹 ∈ V)
101 eqidd 2610 . . 3 ((𝜑𝑘𝑍) → (𝐹𝑘) = (𝐹𝑘))
10223recnd 9924 . . 3 (𝜑 → sup(ran 𝐹, ℝ, < ) ∈ ℂ)
1031, 43sylan 486 . . . 4 ((𝜑𝑘𝑍) → (𝐹𝑘) ∈ ℝ)
104103recnd 9924 . . 3 ((𝜑𝑘𝑍) → (𝐹𝑘) ∈ ℂ)
1059, 6, 100, 101, 102, 104clim2c 14030 . 2 (𝜑 → (𝐹 ⇝ sup(ran 𝐹, ℝ, < ) ↔ ∀𝑦 ∈ ℝ+𝑗𝑍𝑘 ∈ (ℤ𝑗)(abs‘((𝐹𝑘) − sup(ran 𝐹, ℝ, < ))) < 𝑦))
10696, 105mpbird 245 1 (𝜑𝐹 ⇝ sup(ran 𝐹, ℝ, < ))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 194  wa 382  w3a 1030   = wceq 1474  wcel 1976  wne 2779  wral 2895  wrex 2896  Vcvv 3172  wss 3539  c0 3873   class class class wbr 4577  ran crn 5029   Fn wfn 5785  wf 5786  cfv 5790  (class class class)co 6527  supcsup 8206  cr 9791  1c1 9793   + caddc 9795   < clt 9930  cle 9931  cmin 10117  cz 11210  cuz 11519  +crp 11664  ...cfz 12152  abscabs 13768  cli 14009
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-8 1978  ax-9 1985  ax-10 2005  ax-11 2020  ax-12 2033  ax-13 2233  ax-ext 2589  ax-rep 4693  ax-sep 4703  ax-nul 4712  ax-pow 4764  ax-pr 4828  ax-un 6824  ax-cnex 9848  ax-resscn 9849  ax-1cn 9850  ax-icn 9851  ax-addcl 9852  ax-addrcl 9853  ax-mulcl 9854  ax-mulrcl 9855  ax-mulcom 9856  ax-addass 9857  ax-mulass 9858  ax-distr 9859  ax-i2m1 9860  ax-1ne0 9861  ax-1rid 9862  ax-rnegex 9863  ax-rrecex 9864  ax-cnre 9865  ax-pre-lttri 9866  ax-pre-lttrn 9867  ax-pre-ltadd 9868  ax-pre-mulgt0 9869  ax-pre-sup 9870
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3or 1031  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1867  df-eu 2461  df-mo 2462  df-clab 2596  df-cleq 2602  df-clel 2605  df-nfc 2739  df-ne 2781  df-nel 2782  df-ral 2900  df-rex 2901  df-reu 2902  df-rmo 2903  df-rab 2904  df-v 3174  df-sbc 3402  df-csb 3499  df-dif 3542  df-un 3544  df-in 3546  df-ss 3553  df-pss 3555  df-nul 3874  df-if 4036  df-pw 4109  df-sn 4125  df-pr 4127  df-tp 4129  df-op 4131  df-uni 4367  df-iun 4451  df-br 4578  df-opab 4638  df-mpt 4639  df-tr 4675  df-eprel 4939  df-id 4943  df-po 4949  df-so 4950  df-fr 4987  df-we 4989  df-xp 5034  df-rel 5035  df-cnv 5036  df-co 5037  df-dm 5038  df-rn 5039  df-res 5040  df-ima 5041  df-pred 5583  df-ord 5629  df-on 5630  df-lim 5631  df-suc 5632  df-iota 5754  df-fun 5792  df-fn 5793  df-f 5794  df-f1 5795  df-fo 5796  df-f1o 5797  df-fv 5798  df-riota 6489  df-ov 6530  df-oprab 6531  df-mpt2 6532  df-om 6935  df-1st 7036  df-2nd 7037  df-wrecs 7271  df-recs 7332  df-rdg 7370  df-er 7606  df-en 7819  df-dom 7820  df-sdom 7821  df-sup 8208  df-pnf 9932  df-mnf 9933  df-xr 9934  df-ltxr 9935  df-le 9936  df-sub 10119  df-neg 10120  df-div 10534  df-nn 10868  df-2 10926  df-3 10927  df-n0 11140  df-z 11211  df-uz 11520  df-rp 11665  df-fz 12153  df-seq 12619  df-exp 12678  df-cj 13633  df-re 13634  df-im 13635  df-sqrt 13769  df-abs 13770  df-clim 14013
This theorem is referenced by:  isumsup2  14363  climcnds  14368  itg1climres  23204  itg2monolem1  23240  itg2i1fseq  23245  itg2i1fseq2  23246  emcllem6  24444  lmdvg  29133  esumpcvgval  29273  meaiuninclem  39170
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