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Theorem climinf 44095
Description: A bounded monotonic nonincreasing sequence converges to the infimum of its range. (Contributed by Glauco Siliprandi, 29-Jun-2017.) (Revised by AV, 15-Sep-2020.)
Hypotheses
Ref Expression
climinf.3 𝑍 = (ℤ𝑀)
climinf.4 (𝜑𝑀 ∈ ℤ)
climinf.5 (𝜑𝐹:𝑍⟶ℝ)
climinf.6 ((𝜑𝑘𝑍) → (𝐹‘(𝑘 + 1)) ≤ (𝐹𝑘))
climinf.7 (𝜑 → ∃𝑥 ∈ ℝ ∀𝑘𝑍 𝑥 ≤ (𝐹𝑘))
Assertion
Ref Expression
climinf (𝜑𝐹 ⇝ inf(ran 𝐹, ℝ, < ))
Distinct variable groups:   𝜑,𝑘   𝑥,𝑘,𝐹   𝑘,𝑍,𝑥
Allowed substitution hints:   𝜑(𝑥)   𝑀(𝑥,𝑘)

Proof of Theorem climinf
Dummy variables 𝑗 𝑛 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 climinf.5 . . . . . . . . . . . 12 (𝜑𝐹:𝑍⟶ℝ)
21frnd 6712 . . . . . . . . . . 11 (𝜑 → ran 𝐹 ⊆ ℝ)
31ffnd 6705 . . . . . . . . . . . . 13 (𝜑𝐹 Fn 𝑍)
4 climinf.4 . . . . . . . . . . . . . . 15 (𝜑𝑀 ∈ ℤ)
5 uzid 12819 . . . . . . . . . . . . . . 15 (𝑀 ∈ ℤ → 𝑀 ∈ (ℤ𝑀))
64, 5syl 17 . . . . . . . . . . . . . 14 (𝜑𝑀 ∈ (ℤ𝑀))
7 climinf.3 . . . . . . . . . . . . . 14 𝑍 = (ℤ𝑀)
86, 7eleqtrrdi 2843 . . . . . . . . . . . . 13 (𝜑𝑀𝑍)
9 fnfvelrn 7067 . . . . . . . . . . . . 13 ((𝐹 Fn 𝑍𝑀𝑍) → (𝐹𝑀) ∈ ran 𝐹)
103, 8, 9syl2anc 584 . . . . . . . . . . . 12 (𝜑 → (𝐹𝑀) ∈ ran 𝐹)
1110ne0d 4331 . . . . . . . . . . 11 (𝜑 → ran 𝐹 ≠ ∅)
12 climinf.7 . . . . . . . . . . . 12 (𝜑 → ∃𝑥 ∈ ℝ ∀𝑘𝑍 𝑥 ≤ (𝐹𝑘))
13 breq2 5145 . . . . . . . . . . . . . . 15 (𝑦 = (𝐹𝑘) → (𝑥𝑦𝑥 ≤ (𝐹𝑘)))
1413ralrn 7074 . . . . . . . . . . . . . 14 (𝐹 Fn 𝑍 → (∀𝑦 ∈ ran 𝐹 𝑥𝑦 ↔ ∀𝑘𝑍 𝑥 ≤ (𝐹𝑘)))
1514rexbidv 3177 . . . . . . . . . . . . 13 (𝐹 Fn 𝑍 → (∃𝑥 ∈ ℝ ∀𝑦 ∈ ran 𝐹 𝑥𝑦 ↔ ∃𝑥 ∈ ℝ ∀𝑘𝑍 𝑥 ≤ (𝐹𝑘)))
163, 15syl 17 . . . . . . . . . . . 12 (𝜑 → (∃𝑥 ∈ ℝ ∀𝑦 ∈ ran 𝐹 𝑥𝑦 ↔ ∃𝑥 ∈ ℝ ∀𝑘𝑍 𝑥 ≤ (𝐹𝑘)))
1712, 16mpbird 256 . . . . . . . . . . 11 (𝜑 → ∃𝑥 ∈ ℝ ∀𝑦 ∈ ran 𝐹 𝑥𝑦)
182, 11, 173jca 1128 . . . . . . . . . 10 (𝜑 → (ran 𝐹 ⊆ ℝ ∧ ran 𝐹 ≠ ∅ ∧ ∃𝑥 ∈ ℝ ∀𝑦 ∈ ran 𝐹 𝑥𝑦))
1918adantr 481 . . . . . . . . 9 ((𝜑𝑦 ∈ ℝ+) → (ran 𝐹 ⊆ ℝ ∧ ran 𝐹 ≠ ∅ ∧ ∃𝑥 ∈ ℝ ∀𝑦 ∈ ran 𝐹 𝑥𝑦))
20 infrecl 12178 . . . . . . . . 9 ((ran 𝐹 ⊆ ℝ ∧ ran 𝐹 ≠ ∅ ∧ ∃𝑥 ∈ ℝ ∀𝑦 ∈ ran 𝐹 𝑥𝑦) → inf(ran 𝐹, ℝ, < ) ∈ ℝ)
2119, 20syl 17 . . . . . . . 8 ((𝜑𝑦 ∈ ℝ+) → inf(ran 𝐹, ℝ, < ) ∈ ℝ)
22 simpr 485 . . . . . . . 8 ((𝜑𝑦 ∈ ℝ+) → 𝑦 ∈ ℝ+)
2321, 22ltaddrpd 13031 . . . . . . 7 ((𝜑𝑦 ∈ ℝ+) → inf(ran 𝐹, ℝ, < ) < (inf(ran 𝐹, ℝ, < ) + 𝑦))
24 rpre 12964 . . . . . . . . . 10 (𝑦 ∈ ℝ+𝑦 ∈ ℝ)
2524adantl 482 . . . . . . . . 9 ((𝜑𝑦 ∈ ℝ+) → 𝑦 ∈ ℝ)
2621, 25readdcld 11225 . . . . . . . 8 ((𝜑𝑦 ∈ ℝ+) → (inf(ran 𝐹, ℝ, < ) + 𝑦) ∈ ℝ)
27 infrglb 44079 . . . . . . . 8 (((ran 𝐹 ⊆ ℝ ∧ ran 𝐹 ≠ ∅ ∧ ∃𝑥 ∈ ℝ ∀𝑦 ∈ ran 𝐹 𝑥𝑦) ∧ (inf(ran 𝐹, ℝ, < ) + 𝑦) ∈ ℝ) → (inf(ran 𝐹, ℝ, < ) < (inf(ran 𝐹, ℝ, < ) + 𝑦) ↔ ∃𝑘 ∈ ran 𝐹 𝑘 < (inf(ran 𝐹, ℝ, < ) + 𝑦)))
2819, 26, 27syl2anc 584 . . . . . . 7 ((𝜑𝑦 ∈ ℝ+) → (inf(ran 𝐹, ℝ, < ) < (inf(ran 𝐹, ℝ, < ) + 𝑦) ↔ ∃𝑘 ∈ ran 𝐹 𝑘 < (inf(ran 𝐹, ℝ, < ) + 𝑦)))
2923, 28mpbid 231 . . . . . 6 ((𝜑𝑦 ∈ ℝ+) → ∃𝑘 ∈ ran 𝐹 𝑘 < (inf(ran 𝐹, ℝ, < ) + 𝑦))
302sselda 3978 . . . . . . . . . . 11 ((𝜑𝑘 ∈ ran 𝐹) → 𝑘 ∈ ℝ)
3130adantlr 713 . . . . . . . . . 10 (((𝜑𝑦 ∈ ℝ+) ∧ 𝑘 ∈ ran 𝐹) → 𝑘 ∈ ℝ)
3221adantr 481 . . . . . . . . . . 11 (((𝜑𝑦 ∈ ℝ+) ∧ 𝑘 ∈ ran 𝐹) → inf(ran 𝐹, ℝ, < ) ∈ ℝ)
3324ad2antlr 725 . . . . . . . . . . 11 (((𝜑𝑦 ∈ ℝ+) ∧ 𝑘 ∈ ran 𝐹) → 𝑦 ∈ ℝ)
3432, 33readdcld 11225 . . . . . . . . . 10 (((𝜑𝑦 ∈ ℝ+) ∧ 𝑘 ∈ ran 𝐹) → (inf(ran 𝐹, ℝ, < ) + 𝑦) ∈ ℝ)
3531, 34, 33ltsub1d 11805 . . . . . . . . 9 (((𝜑𝑦 ∈ ℝ+) ∧ 𝑘 ∈ ran 𝐹) → (𝑘 < (inf(ran 𝐹, ℝ, < ) + 𝑦) ↔ (𝑘𝑦) < ((inf(ran 𝐹, ℝ, < ) + 𝑦) − 𝑦)))
362, 11, 17, 20syl3anc 1371 . . . . . . . . . . . . 13 (𝜑 → inf(ran 𝐹, ℝ, < ) ∈ ℝ)
3736recnd 11224 . . . . . . . . . . . 12 (𝜑 → inf(ran 𝐹, ℝ, < ) ∈ ℂ)
3837ad2antrr 724 . . . . . . . . . . 11 (((𝜑𝑦 ∈ ℝ+) ∧ 𝑘 ∈ ran 𝐹) → inf(ran 𝐹, ℝ, < ) ∈ ℂ)
3933recnd 11224 . . . . . . . . . . 11 (((𝜑𝑦 ∈ ℝ+) ∧ 𝑘 ∈ ran 𝐹) → 𝑦 ∈ ℂ)
4038, 39pncand 11554 . . . . . . . . . 10 (((𝜑𝑦 ∈ ℝ+) ∧ 𝑘 ∈ ran 𝐹) → ((inf(ran 𝐹, ℝ, < ) + 𝑦) − 𝑦) = inf(ran 𝐹, ℝ, < ))
4140breq2d 5153 . . . . . . . . 9 (((𝜑𝑦 ∈ ℝ+) ∧ 𝑘 ∈ ran 𝐹) → ((𝑘𝑦) < ((inf(ran 𝐹, ℝ, < ) + 𝑦) − 𝑦) ↔ (𝑘𝑦) < inf(ran 𝐹, ℝ, < )))
4235, 41bitrd 278 . . . . . . . 8 (((𝜑𝑦 ∈ ℝ+) ∧ 𝑘 ∈ ran 𝐹) → (𝑘 < (inf(ran 𝐹, ℝ, < ) + 𝑦) ↔ (𝑘𝑦) < inf(ran 𝐹, ℝ, < )))
4342biimpd 228 . . . . . . 7 (((𝜑𝑦 ∈ ℝ+) ∧ 𝑘 ∈ ran 𝐹) → (𝑘 < (inf(ran 𝐹, ℝ, < ) + 𝑦) → (𝑘𝑦) < inf(ran 𝐹, ℝ, < )))
4443reximdva 3167 . . . . . 6 ((𝜑𝑦 ∈ ℝ+) → (∃𝑘 ∈ ran 𝐹 𝑘 < (inf(ran 𝐹, ℝ, < ) + 𝑦) → ∃𝑘 ∈ ran 𝐹(𝑘𝑦) < inf(ran 𝐹, ℝ, < )))
4529, 44mpd 15 . . . . 5 ((𝜑𝑦 ∈ ℝ+) → ∃𝑘 ∈ ran 𝐹(𝑘𝑦) < inf(ran 𝐹, ℝ, < ))
46 oveq1 7400 . . . . . . . . 9 (𝑘 = (𝐹𝑗) → (𝑘𝑦) = ((𝐹𝑗) − 𝑦))
4746breq1d 5151 . . . . . . . 8 (𝑘 = (𝐹𝑗) → ((𝑘𝑦) < inf(ran 𝐹, ℝ, < ) ↔ ((𝐹𝑗) − 𝑦) < inf(ran 𝐹, ℝ, < )))
4847rexrn 7073 . . . . . . 7 (𝐹 Fn 𝑍 → (∃𝑘 ∈ ran 𝐹(𝑘𝑦) < inf(ran 𝐹, ℝ, < ) ↔ ∃𝑗𝑍 ((𝐹𝑗) − 𝑦) < inf(ran 𝐹, ℝ, < )))
493, 48syl 17 . . . . . 6 (𝜑 → (∃𝑘 ∈ ran 𝐹(𝑘𝑦) < inf(ran 𝐹, ℝ, < ) ↔ ∃𝑗𝑍 ((𝐹𝑗) − 𝑦) < inf(ran 𝐹, ℝ, < )))
5049biimpa 477 . . . . 5 ((𝜑 ∧ ∃𝑘 ∈ ran 𝐹(𝑘𝑦) < inf(ran 𝐹, ℝ, < )) → ∃𝑗𝑍 ((𝐹𝑗) − 𝑦) < inf(ran 𝐹, ℝ, < ))
5145, 50syldan 591 . . . 4 ((𝜑𝑦 ∈ ℝ+) → ∃𝑗𝑍 ((𝐹𝑗) − 𝑦) < inf(ran 𝐹, ℝ, < ))
521adantr 481 . . . . . . . . . . 11 ((𝜑𝑦 ∈ ℝ+) → 𝐹:𝑍⟶ℝ)
537uztrn2 12823 . . . . . . . . . . 11 ((𝑗𝑍𝑘 ∈ (ℤ𝑗)) → 𝑘𝑍)
54 ffvelcdm 7068 . . . . . . . . . . 11 ((𝐹:𝑍⟶ℝ ∧ 𝑘𝑍) → (𝐹𝑘) ∈ ℝ)
5552, 53, 54syl2an 596 . . . . . . . . . 10 (((𝜑𝑦 ∈ ℝ+) ∧ (𝑗𝑍𝑘 ∈ (ℤ𝑗))) → (𝐹𝑘) ∈ ℝ)
56 simpl 483 . . . . . . . . . . 11 ((𝑗𝑍𝑘 ∈ (ℤ𝑗)) → 𝑗𝑍)
57 ffvelcdm 7068 . . . . . . . . . . 11 ((𝐹:𝑍⟶ℝ ∧ 𝑗𝑍) → (𝐹𝑗) ∈ ℝ)
5852, 56, 57syl2an 596 . . . . . . . . . 10 (((𝜑𝑦 ∈ ℝ+) ∧ (𝑗𝑍𝑘 ∈ (ℤ𝑗))) → (𝐹𝑗) ∈ ℝ)
5936ad2antrr 724 . . . . . . . . . 10 (((𝜑𝑦 ∈ ℝ+) ∧ (𝑗𝑍𝑘 ∈ (ℤ𝑗))) → inf(ran 𝐹, ℝ, < ) ∈ ℝ)
60 simprr 771 . . . . . . . . . . 11 (((𝜑𝑦 ∈ ℝ+) ∧ (𝑗𝑍𝑘 ∈ (ℤ𝑗))) → 𝑘 ∈ (ℤ𝑗))
61 fzssuz 13524 . . . . . . . . . . . . . 14 (𝑗...𝑘) ⊆ (ℤ𝑗)
62 uzss 12827 . . . . . . . . . . . . . . . . 17 (𝑗 ∈ (ℤ𝑀) → (ℤ𝑗) ⊆ (ℤ𝑀))
6362, 7sseqtrrdi 4029 . . . . . . . . . . . . . . . 16 (𝑗 ∈ (ℤ𝑀) → (ℤ𝑗) ⊆ 𝑍)
6463, 7eleq2s 2850 . . . . . . . . . . . . . . 15 (𝑗𝑍 → (ℤ𝑗) ⊆ 𝑍)
6564ad2antrl 726 . . . . . . . . . . . . . 14 (((𝜑𝑦 ∈ ℝ+) ∧ (𝑗𝑍𝑘 ∈ (ℤ𝑗))) → (ℤ𝑗) ⊆ 𝑍)
6661, 65sstrid 3989 . . . . . . . . . . . . 13 (((𝜑𝑦 ∈ ℝ+) ∧ (𝑗𝑍𝑘 ∈ (ℤ𝑗))) → (𝑗...𝑘) ⊆ 𝑍)
67 ffvelcdm 7068 . . . . . . . . . . . . . . . 16 ((𝐹:𝑍⟶ℝ ∧ 𝑛𝑍) → (𝐹𝑛) ∈ ℝ)
6867ralrimiva 3145 . . . . . . . . . . . . . . 15 (𝐹:𝑍⟶ℝ → ∀𝑛𝑍 (𝐹𝑛) ∈ ℝ)
691, 68syl 17 . . . . . . . . . . . . . 14 (𝜑 → ∀𝑛𝑍 (𝐹𝑛) ∈ ℝ)
7069ad2antrr 724 . . . . . . . . . . . . 13 (((𝜑𝑦 ∈ ℝ+) ∧ (𝑗𝑍𝑘 ∈ (ℤ𝑗))) → ∀𝑛𝑍 (𝐹𝑛) ∈ ℝ)
71 ssralv 4046 . . . . . . . . . . . . 13 ((𝑗...𝑘) ⊆ 𝑍 → (∀𝑛𝑍 (𝐹𝑛) ∈ ℝ → ∀𝑛 ∈ (𝑗...𝑘)(𝐹𝑛) ∈ ℝ))
7266, 70, 71sylc 65 . . . . . . . . . . . 12 (((𝜑𝑦 ∈ ℝ+) ∧ (𝑗𝑍𝑘 ∈ (ℤ𝑗))) → ∀𝑛 ∈ (𝑗...𝑘)(𝐹𝑛) ∈ ℝ)
7372r19.21bi 3247 . . . . . . . . . . 11 ((((𝜑𝑦 ∈ ℝ+) ∧ (𝑗𝑍𝑘 ∈ (ℤ𝑗))) ∧ 𝑛 ∈ (𝑗...𝑘)) → (𝐹𝑛) ∈ ℝ)
74 fzssuz 13524 . . . . . . . . . . . . . 14 (𝑗...(𝑘 − 1)) ⊆ (ℤ𝑗)
7574, 65sstrid 3989 . . . . . . . . . . . . 13 (((𝜑𝑦 ∈ ℝ+) ∧ (𝑗𝑍𝑘 ∈ (ℤ𝑗))) → (𝑗...(𝑘 − 1)) ⊆ 𝑍)
7675sselda 3978 . . . . . . . . . . . 12 ((((𝜑𝑦 ∈ ℝ+) ∧ (𝑗𝑍𝑘 ∈ (ℤ𝑗))) ∧ 𝑛 ∈ (𝑗...(𝑘 − 1))) → 𝑛𝑍)
77 climinf.6 . . . . . . . . . . . . . . 15 ((𝜑𝑘𝑍) → (𝐹‘(𝑘 + 1)) ≤ (𝐹𝑘))
7877ralrimiva 3145 . . . . . . . . . . . . . 14 (𝜑 → ∀𝑘𝑍 (𝐹‘(𝑘 + 1)) ≤ (𝐹𝑘))
7978ad2antrr 724 . . . . . . . . . . . . 13 (((𝜑𝑦 ∈ ℝ+) ∧ (𝑗𝑍𝑘 ∈ (ℤ𝑗))) → ∀𝑘𝑍 (𝐹‘(𝑘 + 1)) ≤ (𝐹𝑘))
80 fvoveq1 7416 . . . . . . . . . . . . . . 15 (𝑘 = 𝑛 → (𝐹‘(𝑘 + 1)) = (𝐹‘(𝑛 + 1)))
81 fveq2 6878 . . . . . . . . . . . . . . 15 (𝑘 = 𝑛 → (𝐹𝑘) = (𝐹𝑛))
8280, 81breq12d 5154 . . . . . . . . . . . . . 14 (𝑘 = 𝑛 → ((𝐹‘(𝑘 + 1)) ≤ (𝐹𝑘) ↔ (𝐹‘(𝑛 + 1)) ≤ (𝐹𝑛)))
8382rspccva 3608 . . . . . . . . . . . . 13 ((∀𝑘𝑍 (𝐹‘(𝑘 + 1)) ≤ (𝐹𝑘) ∧ 𝑛𝑍) → (𝐹‘(𝑛 + 1)) ≤ (𝐹𝑛))
8479, 83sylan 580 . . . . . . . . . . . 12 ((((𝜑𝑦 ∈ ℝ+) ∧ (𝑗𝑍𝑘 ∈ (ℤ𝑗))) ∧ 𝑛𝑍) → (𝐹‘(𝑛 + 1)) ≤ (𝐹𝑛))
8576, 84syldan 591 . . . . . . . . . . 11 ((((𝜑𝑦 ∈ ℝ+) ∧ (𝑗𝑍𝑘 ∈ (ℤ𝑗))) ∧ 𝑛 ∈ (𝑗...(𝑘 − 1))) → (𝐹‘(𝑛 + 1)) ≤ (𝐹𝑛))
8660, 73, 85monoord2 13981 . . . . . . . . . 10 (((𝜑𝑦 ∈ ℝ+) ∧ (𝑗𝑍𝑘 ∈ (ℤ𝑗))) → (𝐹𝑘) ≤ (𝐹𝑗))
8755, 58, 59, 86lesub1dd 11812 . . . . . . . . 9 (((𝜑𝑦 ∈ ℝ+) ∧ (𝑗𝑍𝑘 ∈ (ℤ𝑗))) → ((𝐹𝑘) − inf(ran 𝐹, ℝ, < )) ≤ ((𝐹𝑗) − inf(ran 𝐹, ℝ, < )))
8855, 59resubcld 11624 . . . . . . . . . 10 (((𝜑𝑦 ∈ ℝ+) ∧ (𝑗𝑍𝑘 ∈ (ℤ𝑗))) → ((𝐹𝑘) − inf(ran 𝐹, ℝ, < )) ∈ ℝ)
8958, 59resubcld 11624 . . . . . . . . . 10 (((𝜑𝑦 ∈ ℝ+) ∧ (𝑗𝑍𝑘 ∈ (ℤ𝑗))) → ((𝐹𝑗) − inf(ran 𝐹, ℝ, < )) ∈ ℝ)
9024ad2antlr 725 . . . . . . . . . 10 (((𝜑𝑦 ∈ ℝ+) ∧ (𝑗𝑍𝑘 ∈ (ℤ𝑗))) → 𝑦 ∈ ℝ)
91 lelttr 11286 . . . . . . . . . 10 ((((𝐹𝑘) − inf(ran 𝐹, ℝ, < )) ∈ ℝ ∧ ((𝐹𝑗) − inf(ran 𝐹, ℝ, < )) ∈ ℝ ∧ 𝑦 ∈ ℝ) → ((((𝐹𝑘) − inf(ran 𝐹, ℝ, < )) ≤ ((𝐹𝑗) − inf(ran 𝐹, ℝ, < )) ∧ ((𝐹𝑗) − inf(ran 𝐹, ℝ, < )) < 𝑦) → ((𝐹𝑘) − inf(ran 𝐹, ℝ, < )) < 𝑦))
9288, 89, 90, 91syl3anc 1371 . . . . . . . . 9 (((𝜑𝑦 ∈ ℝ+) ∧ (𝑗𝑍𝑘 ∈ (ℤ𝑗))) → ((((𝐹𝑘) − inf(ran 𝐹, ℝ, < )) ≤ ((𝐹𝑗) − inf(ran 𝐹, ℝ, < )) ∧ ((𝐹𝑗) − inf(ran 𝐹, ℝ, < )) < 𝑦) → ((𝐹𝑘) − inf(ran 𝐹, ℝ, < )) < 𝑦))
9387, 92mpand 693 . . . . . . . 8 (((𝜑𝑦 ∈ ℝ+) ∧ (𝑗𝑍𝑘 ∈ (ℤ𝑗))) → (((𝐹𝑗) − inf(ran 𝐹, ℝ, < )) < 𝑦 → ((𝐹𝑘) − inf(ran 𝐹, ℝ, < )) < 𝑦))
94 ltsub23 11676 . . . . . . . . 9 (((𝐹𝑗) ∈ ℝ ∧ 𝑦 ∈ ℝ ∧ inf(ran 𝐹, ℝ, < ) ∈ ℝ) → (((𝐹𝑗) − 𝑦) < inf(ran 𝐹, ℝ, < ) ↔ ((𝐹𝑗) − inf(ran 𝐹, ℝ, < )) < 𝑦))
9558, 90, 59, 94syl3anc 1371 . . . . . . . 8 (((𝜑𝑦 ∈ ℝ+) ∧ (𝑗𝑍𝑘 ∈ (ℤ𝑗))) → (((𝐹𝑗) − 𝑦) < inf(ran 𝐹, ℝ, < ) ↔ ((𝐹𝑗) − inf(ran 𝐹, ℝ, < )) < 𝑦))
962ad2antrr 724 . . . . . . . . . . 11 (((𝜑𝑦 ∈ ℝ+) ∧ (𝑗𝑍𝑘 ∈ (ℤ𝑗))) → ran 𝐹 ⊆ ℝ)
973adantr 481 . . . . . . . . . . . 12 ((𝜑𝑦 ∈ ℝ+) → 𝐹 Fn 𝑍)
98 fnfvelrn 7067 . . . . . . . . . . . 12 ((𝐹 Fn 𝑍𝑘𝑍) → (𝐹𝑘) ∈ ran 𝐹)
9997, 53, 98syl2an 596 . . . . . . . . . . 11 (((𝜑𝑦 ∈ ℝ+) ∧ (𝑗𝑍𝑘 ∈ (ℤ𝑗))) → (𝐹𝑘) ∈ ran 𝐹)
10096, 99sseldd 3979 . . . . . . . . . 10 (((𝜑𝑦 ∈ ℝ+) ∧ (𝑗𝑍𝑘 ∈ (ℤ𝑗))) → (𝐹𝑘) ∈ ℝ)
10117ad2antrr 724 . . . . . . . . . . 11 (((𝜑𝑦 ∈ ℝ+) ∧ (𝑗𝑍𝑘 ∈ (ℤ𝑗))) → ∃𝑥 ∈ ℝ ∀𝑦 ∈ ran 𝐹 𝑥𝑦)
102 infrelb 12181 . . . . . . . . . . 11 ((ran 𝐹 ⊆ ℝ ∧ ∃𝑥 ∈ ℝ ∀𝑦 ∈ ran 𝐹 𝑥𝑦 ∧ (𝐹𝑘) ∈ ran 𝐹) → inf(ran 𝐹, ℝ, < ) ≤ (𝐹𝑘))
10396, 101, 99, 102syl3anc 1371 . . . . . . . . . 10 (((𝜑𝑦 ∈ ℝ+) ∧ (𝑗𝑍𝑘 ∈ (ℤ𝑗))) → inf(ran 𝐹, ℝ, < ) ≤ (𝐹𝑘))
10459, 100, 103abssubge0d 15360 . . . . . . . . 9 (((𝜑𝑦 ∈ ℝ+) ∧ (𝑗𝑍𝑘 ∈ (ℤ𝑗))) → (abs‘((𝐹𝑘) − inf(ran 𝐹, ℝ, < ))) = ((𝐹𝑘) − inf(ran 𝐹, ℝ, < )))
105104breq1d 5151 . . . . . . . 8 (((𝜑𝑦 ∈ ℝ+) ∧ (𝑗𝑍𝑘 ∈ (ℤ𝑗))) → ((abs‘((𝐹𝑘) − inf(ran 𝐹, ℝ, < ))) < 𝑦 ↔ ((𝐹𝑘) − inf(ran 𝐹, ℝ, < )) < 𝑦))
10693, 95, 1053imtr4d 293 . . . . . . 7 (((𝜑𝑦 ∈ ℝ+) ∧ (𝑗𝑍𝑘 ∈ (ℤ𝑗))) → (((𝐹𝑗) − 𝑦) < inf(ran 𝐹, ℝ, < ) → (abs‘((𝐹𝑘) − inf(ran 𝐹, ℝ, < ))) < 𝑦))
107106anassrs 468 . . . . . 6 ((((𝜑𝑦 ∈ ℝ+) ∧ 𝑗𝑍) ∧ 𝑘 ∈ (ℤ𝑗)) → (((𝐹𝑗) − 𝑦) < inf(ran 𝐹, ℝ, < ) → (abs‘((𝐹𝑘) − inf(ran 𝐹, ℝ, < ))) < 𝑦))
108107ralrimdva 3153 . . . . 5 (((𝜑𝑦 ∈ ℝ+) ∧ 𝑗𝑍) → (((𝐹𝑗) − 𝑦) < inf(ran 𝐹, ℝ, < ) → ∀𝑘 ∈ (ℤ𝑗)(abs‘((𝐹𝑘) − inf(ran 𝐹, ℝ, < ))) < 𝑦))
109108reximdva 3167 . . . 4 ((𝜑𝑦 ∈ ℝ+) → (∃𝑗𝑍 ((𝐹𝑗) − 𝑦) < inf(ran 𝐹, ℝ, < ) → ∃𝑗𝑍𝑘 ∈ (ℤ𝑗)(abs‘((𝐹𝑘) − inf(ran 𝐹, ℝ, < ))) < 𝑦))
11051, 109mpd 15 . . 3 ((𝜑𝑦 ∈ ℝ+) → ∃𝑗𝑍𝑘 ∈ (ℤ𝑗)(abs‘((𝐹𝑘) − inf(ran 𝐹, ℝ, < ))) < 𝑦)
111110ralrimiva 3145 . 2 (𝜑 → ∀𝑦 ∈ ℝ+𝑗𝑍𝑘 ∈ (ℤ𝑗)(abs‘((𝐹𝑘) − inf(ran 𝐹, ℝ, < ))) < 𝑦)
1127fvexi 6892 . . . 4 𝑍 ∈ V
113 fex 7212 . . . 4 ((𝐹:𝑍⟶ℝ ∧ 𝑍 ∈ V) → 𝐹 ∈ V)
1141, 112, 113sylancl 586 . . 3 (𝜑𝐹 ∈ V)
115 eqidd 2732 . . 3 ((𝜑𝑘𝑍) → (𝐹𝑘) = (𝐹𝑘))
1161ffvelcdmda 7071 . . . 4 ((𝜑𝑘𝑍) → (𝐹𝑘) ∈ ℝ)
117116recnd 11224 . . 3 ((𝜑𝑘𝑍) → (𝐹𝑘) ∈ ℂ)
1187, 4, 114, 115, 37, 117clim2c 15431 . 2 (𝜑 → (𝐹 ⇝ inf(ran 𝐹, ℝ, < ) ↔ ∀𝑦 ∈ ℝ+𝑗𝑍𝑘 ∈ (ℤ𝑗)(abs‘((𝐹𝑘) − inf(ran 𝐹, ℝ, < ))) < 𝑦))
119111, 118mpbird 256 1 (𝜑𝐹 ⇝ inf(ran 𝐹, ℝ, < ))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 396  w3a 1087   = wceq 1541  wcel 2106  wne 2939  wral 3060  wrex 3069  Vcvv 3473  wss 3944  c0 4318   class class class wbr 5141  ran crn 5670   Fn wfn 6527  wf 6528  cfv 6532  (class class class)co 7393  infcinf 9418  cc 11090  cr 11091  1c1 11093   + caddc 11095   < clt 11230  cle 11231  cmin 11426  cz 12540  cuz 12804  +crp 12956  ...cfz 13466  abscabs 15163  cli 15410
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2702  ax-rep 5278  ax-sep 5292  ax-nul 5299  ax-pow 5356  ax-pr 5420  ax-un 7708  ax-cnex 11148  ax-resscn 11149  ax-1cn 11150  ax-icn 11151  ax-addcl 11152  ax-addrcl 11153  ax-mulcl 11154  ax-mulrcl 11155  ax-mulcom 11156  ax-addass 11157  ax-mulass 11158  ax-distr 11159  ax-i2m1 11160  ax-1ne0 11161  ax-1rid 11162  ax-rnegex 11163  ax-rrecex 11164  ax-cnre 11165  ax-pre-lttri 11166  ax-pre-lttrn 11167  ax-pre-ltadd 11168  ax-pre-mulgt0 11169  ax-pre-sup 11170
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-nel 3046  df-ral 3061  df-rex 3070  df-rmo 3375  df-reu 3376  df-rab 3432  df-v 3475  df-sbc 3774  df-csb 3890  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-pss 3963  df-nul 4319  df-if 4523  df-pw 4598  df-sn 4623  df-pr 4625  df-op 4629  df-uni 4902  df-iun 4992  df-br 5142  df-opab 5204  df-mpt 5225  df-tr 5259  df-id 5567  df-eprel 5573  df-po 5581  df-so 5582  df-fr 5624  df-we 5626  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-res 5681  df-ima 5682  df-pred 6289  df-ord 6356  df-on 6357  df-lim 6358  df-suc 6359  df-iota 6484  df-fun 6534  df-fn 6535  df-f 6536  df-f1 6537  df-fo 6538  df-f1o 6539  df-fv 6540  df-riota 7349  df-ov 7396  df-oprab 7397  df-mpo 7398  df-om 7839  df-1st 7957  df-2nd 7958  df-frecs 8248  df-wrecs 8279  df-recs 8353  df-rdg 8392  df-er 8686  df-en 8923  df-dom 8924  df-sdom 8925  df-sup 9419  df-inf 9420  df-pnf 11232  df-mnf 11233  df-xr 11234  df-ltxr 11235  df-le 11236  df-sub 11428  df-neg 11429  df-div 11854  df-nn 12195  df-2 12257  df-3 12258  df-n0 12455  df-z 12541  df-uz 12805  df-rp 12957  df-fz 13467  df-seq 13949  df-exp 14010  df-cj 15028  df-re 15029  df-im 15030  df-sqrt 15164  df-abs 15165  df-clim 15414
This theorem is referenced by:  climinff  44100  climinf2lem  44195  supcnvlimsup  44229  stirlinglem13  44575
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