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Theorem climinf 42822
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 6553 . . . . . . . . . . 11 (𝜑 → ran 𝐹 ⊆ ℝ)
31ffnd 6546 . . . . . . . . . . . . 13 (𝜑𝐹 Fn 𝑍)
4 climinf.4 . . . . . . . . . . . . . . 15 (𝜑𝑀 ∈ ℤ)
5 uzid 12453 . . . . . . . . . . . . . . 15 (𝑀 ∈ ℤ → 𝑀 ∈ (ℤ𝑀))
64, 5syl 17 . . . . . . . . . . . . . 14 (𝜑𝑀 ∈ (ℤ𝑀))
7 climinf.3 . . . . . . . . . . . . . 14 𝑍 = (ℤ𝑀)
86, 7eleqtrrdi 2849 . . . . . . . . . . . . 13 (𝜑𝑀𝑍)
9 fnfvelrn 6901 . . . . . . . . . . . . 13 ((𝐹 Fn 𝑍𝑀𝑍) → (𝐹𝑀) ∈ ran 𝐹)
103, 8, 9syl2anc 587 . . . . . . . . . . . 12 (𝜑 → (𝐹𝑀) ∈ ran 𝐹)
1110ne0d 4250 . . . . . . . . . . 11 (𝜑 → ran 𝐹 ≠ ∅)
12 climinf.7 . . . . . . . . . . . 12 (𝜑 → ∃𝑥 ∈ ℝ ∀𝑘𝑍 𝑥 ≤ (𝐹𝑘))
13 breq2 5057 . . . . . . . . . . . . . . 15 (𝑦 = (𝐹𝑘) → (𝑥𝑦𝑥 ≤ (𝐹𝑘)))
1413ralrn 6907 . . . . . . . . . . . . . 14 (𝐹 Fn 𝑍 → (∀𝑦 ∈ ran 𝐹 𝑥𝑦 ↔ ∀𝑘𝑍 𝑥 ≤ (𝐹𝑘)))
1514rexbidv 3216 . . . . . . . . . . . . 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 260 . . . . . . . . . . 11 (𝜑 → ∃𝑥 ∈ ℝ ∀𝑦 ∈ ran 𝐹 𝑥𝑦)
182, 11, 173jca 1130 . . . . . . . . . 10 (𝜑 → (ran 𝐹 ⊆ ℝ ∧ ran 𝐹 ≠ ∅ ∧ ∃𝑥 ∈ ℝ ∀𝑦 ∈ ran 𝐹 𝑥𝑦))
1918adantr 484 . . . . . . . . 9 ((𝜑𝑦 ∈ ℝ+) → (ran 𝐹 ⊆ ℝ ∧ ran 𝐹 ≠ ∅ ∧ ∃𝑥 ∈ ℝ ∀𝑦 ∈ ran 𝐹 𝑥𝑦))
20 infrecl 11814 . . . . . . . . 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 488 . . . . . . . 8 ((𝜑𝑦 ∈ ℝ+) → 𝑦 ∈ ℝ+)
2321, 22ltaddrpd 12661 . . . . . . 7 ((𝜑𝑦 ∈ ℝ+) → inf(ran 𝐹, ℝ, < ) < (inf(ran 𝐹, ℝ, < ) + 𝑦))
24 rpre 12594 . . . . . . . . . 10 (𝑦 ∈ ℝ+𝑦 ∈ ℝ)
2524adantl 485 . . . . . . . . 9 ((𝜑𝑦 ∈ ℝ+) → 𝑦 ∈ ℝ)
2621, 25readdcld 10862 . . . . . . . 8 ((𝜑𝑦 ∈ ℝ+) → (inf(ran 𝐹, ℝ, < ) + 𝑦) ∈ ℝ)
27 infrglb 42806 . . . . . . . 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 587 . . . . . . 7 ((𝜑𝑦 ∈ ℝ+) → (inf(ran 𝐹, ℝ, < ) < (inf(ran 𝐹, ℝ, < ) + 𝑦) ↔ ∃𝑘 ∈ ran 𝐹 𝑘 < (inf(ran 𝐹, ℝ, < ) + 𝑦)))
2923, 28mpbid 235 . . . . . 6 ((𝜑𝑦 ∈ ℝ+) → ∃𝑘 ∈ ran 𝐹 𝑘 < (inf(ran 𝐹, ℝ, < ) + 𝑦))
302sselda 3901 . . . . . . . . . . 11 ((𝜑𝑘 ∈ ran 𝐹) → 𝑘 ∈ ℝ)
3130adantlr 715 . . . . . . . . . 10 (((𝜑𝑦 ∈ ℝ+) ∧ 𝑘 ∈ ran 𝐹) → 𝑘 ∈ ℝ)
3221adantr 484 . . . . . . . . . . 11 (((𝜑𝑦 ∈ ℝ+) ∧ 𝑘 ∈ ran 𝐹) → inf(ran 𝐹, ℝ, < ) ∈ ℝ)
3324ad2antlr 727 . . . . . . . . . . 11 (((𝜑𝑦 ∈ ℝ+) ∧ 𝑘 ∈ ran 𝐹) → 𝑦 ∈ ℝ)
3432, 33readdcld 10862 . . . . . . . . . 10 (((𝜑𝑦 ∈ ℝ+) ∧ 𝑘 ∈ ran 𝐹) → (inf(ran 𝐹, ℝ, < ) + 𝑦) ∈ ℝ)
3531, 34, 33ltsub1d 11441 . . . . . . . . 9 (((𝜑𝑦 ∈ ℝ+) ∧ 𝑘 ∈ ran 𝐹) → (𝑘 < (inf(ran 𝐹, ℝ, < ) + 𝑦) ↔ (𝑘𝑦) < ((inf(ran 𝐹, ℝ, < ) + 𝑦) − 𝑦)))
362, 11, 17, 20syl3anc 1373 . . . . . . . . . . . . 13 (𝜑 → inf(ran 𝐹, ℝ, < ) ∈ ℝ)
3736recnd 10861 . . . . . . . . . . . 12 (𝜑 → inf(ran 𝐹, ℝ, < ) ∈ ℂ)
3837ad2antrr 726 . . . . . . . . . . 11 (((𝜑𝑦 ∈ ℝ+) ∧ 𝑘 ∈ ran 𝐹) → inf(ran 𝐹, ℝ, < ) ∈ ℂ)
3933recnd 10861 . . . . . . . . . . 11 (((𝜑𝑦 ∈ ℝ+) ∧ 𝑘 ∈ ran 𝐹) → 𝑦 ∈ ℂ)
4038, 39pncand 11190 . . . . . . . . . 10 (((𝜑𝑦 ∈ ℝ+) ∧ 𝑘 ∈ ran 𝐹) → ((inf(ran 𝐹, ℝ, < ) + 𝑦) − 𝑦) = inf(ran 𝐹, ℝ, < ))
4140breq2d 5065 . . . . . . . . 9 (((𝜑𝑦 ∈ ℝ+) ∧ 𝑘 ∈ ran 𝐹) → ((𝑘𝑦) < ((inf(ran 𝐹, ℝ, < ) + 𝑦) − 𝑦) ↔ (𝑘𝑦) < inf(ran 𝐹, ℝ, < )))
4235, 41bitrd 282 . . . . . . . 8 (((𝜑𝑦 ∈ ℝ+) ∧ 𝑘 ∈ ran 𝐹) → (𝑘 < (inf(ran 𝐹, ℝ, < ) + 𝑦) ↔ (𝑘𝑦) < inf(ran 𝐹, ℝ, < )))
4342biimpd 232 . . . . . . 7 (((𝜑𝑦 ∈ ℝ+) ∧ 𝑘 ∈ ran 𝐹) → (𝑘 < (inf(ran 𝐹, ℝ, < ) + 𝑦) → (𝑘𝑦) < inf(ran 𝐹, ℝ, < )))
4443reximdva 3193 . . . . . 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 7220 . . . . . . . . 9 (𝑘 = (𝐹𝑗) → (𝑘𝑦) = ((𝐹𝑗) − 𝑦))
4746breq1d 5063 . . . . . . . 8 (𝑘 = (𝐹𝑗) → ((𝑘𝑦) < inf(ran 𝐹, ℝ, < ) ↔ ((𝐹𝑗) − 𝑦) < inf(ran 𝐹, ℝ, < )))
4847rexrn 6906 . . . . . . 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 480 . . . . 5 ((𝜑 ∧ ∃𝑘 ∈ ran 𝐹(𝑘𝑦) < inf(ran 𝐹, ℝ, < )) → ∃𝑗𝑍 ((𝐹𝑗) − 𝑦) < inf(ran 𝐹, ℝ, < ))
5145, 50syldan 594 . . . 4 ((𝜑𝑦 ∈ ℝ+) → ∃𝑗𝑍 ((𝐹𝑗) − 𝑦) < inf(ran 𝐹, ℝ, < ))
521adantr 484 . . . . . . . . . . 11 ((𝜑𝑦 ∈ ℝ+) → 𝐹:𝑍⟶ℝ)
537uztrn2 12457 . . . . . . . . . . 11 ((𝑗𝑍𝑘 ∈ (ℤ𝑗)) → 𝑘𝑍)
54 ffvelrn 6902 . . . . . . . . . . 11 ((𝐹:𝑍⟶ℝ ∧ 𝑘𝑍) → (𝐹𝑘) ∈ ℝ)
5552, 53, 54syl2an 599 . . . . . . . . . 10 (((𝜑𝑦 ∈ ℝ+) ∧ (𝑗𝑍𝑘 ∈ (ℤ𝑗))) → (𝐹𝑘) ∈ ℝ)
56 simpl 486 . . . . . . . . . . 11 ((𝑗𝑍𝑘 ∈ (ℤ𝑗)) → 𝑗𝑍)
57 ffvelrn 6902 . . . . . . . . . . 11 ((𝐹:𝑍⟶ℝ ∧ 𝑗𝑍) → (𝐹𝑗) ∈ ℝ)
5852, 56, 57syl2an 599 . . . . . . . . . 10 (((𝜑𝑦 ∈ ℝ+) ∧ (𝑗𝑍𝑘 ∈ (ℤ𝑗))) → (𝐹𝑗) ∈ ℝ)
5936ad2antrr 726 . . . . . . . . . 10 (((𝜑𝑦 ∈ ℝ+) ∧ (𝑗𝑍𝑘 ∈ (ℤ𝑗))) → inf(ran 𝐹, ℝ, < ) ∈ ℝ)
60 simprr 773 . . . . . . . . . . 11 (((𝜑𝑦 ∈ ℝ+) ∧ (𝑗𝑍𝑘 ∈ (ℤ𝑗))) → 𝑘 ∈ (ℤ𝑗))
61 fzssuz 13153 . . . . . . . . . . . . . 14 (𝑗...𝑘) ⊆ (ℤ𝑗)
62 uzss 12461 . . . . . . . . . . . . . . . . 17 (𝑗 ∈ (ℤ𝑀) → (ℤ𝑗) ⊆ (ℤ𝑀))
6362, 7sseqtrrdi 3952 . . . . . . . . . . . . . . . 16 (𝑗 ∈ (ℤ𝑀) → (ℤ𝑗) ⊆ 𝑍)
6463, 7eleq2s 2856 . . . . . . . . . . . . . . 15 (𝑗𝑍 → (ℤ𝑗) ⊆ 𝑍)
6564ad2antrl 728 . . . . . . . . . . . . . 14 (((𝜑𝑦 ∈ ℝ+) ∧ (𝑗𝑍𝑘 ∈ (ℤ𝑗))) → (ℤ𝑗) ⊆ 𝑍)
6661, 65sstrid 3912 . . . . . . . . . . . . 13 (((𝜑𝑦 ∈ ℝ+) ∧ (𝑗𝑍𝑘 ∈ (ℤ𝑗))) → (𝑗...𝑘) ⊆ 𝑍)
67 ffvelrn 6902 . . . . . . . . . . . . . . . 16 ((𝐹:𝑍⟶ℝ ∧ 𝑛𝑍) → (𝐹𝑛) ∈ ℝ)
6867ralrimiva 3105 . . . . . . . . . . . . . . 15 (𝐹:𝑍⟶ℝ → ∀𝑛𝑍 (𝐹𝑛) ∈ ℝ)
691, 68syl 17 . . . . . . . . . . . . . 14 (𝜑 → ∀𝑛𝑍 (𝐹𝑛) ∈ ℝ)
7069ad2antrr 726 . . . . . . . . . . . . 13 (((𝜑𝑦 ∈ ℝ+) ∧ (𝑗𝑍𝑘 ∈ (ℤ𝑗))) → ∀𝑛𝑍 (𝐹𝑛) ∈ ℝ)
71 ssralv 3967 . . . . . . . . . . . . 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 3130 . . . . . . . . . . 11 ((((𝜑𝑦 ∈ ℝ+) ∧ (𝑗𝑍𝑘 ∈ (ℤ𝑗))) ∧ 𝑛 ∈ (𝑗...𝑘)) → (𝐹𝑛) ∈ ℝ)
74 fzssuz 13153 . . . . . . . . . . . . . 14 (𝑗...(𝑘 − 1)) ⊆ (ℤ𝑗)
7574, 65sstrid 3912 . . . . . . . . . . . . 13 (((𝜑𝑦 ∈ ℝ+) ∧ (𝑗𝑍𝑘 ∈ (ℤ𝑗))) → (𝑗...(𝑘 − 1)) ⊆ 𝑍)
7675sselda 3901 . . . . . . . . . . . 12 ((((𝜑𝑦 ∈ ℝ+) ∧ (𝑗𝑍𝑘 ∈ (ℤ𝑗))) ∧ 𝑛 ∈ (𝑗...(𝑘 − 1))) → 𝑛𝑍)
77 climinf.6 . . . . . . . . . . . . . . 15 ((𝜑𝑘𝑍) → (𝐹‘(𝑘 + 1)) ≤ (𝐹𝑘))
7877ralrimiva 3105 . . . . . . . . . . . . . 14 (𝜑 → ∀𝑘𝑍 (𝐹‘(𝑘 + 1)) ≤ (𝐹𝑘))
7978ad2antrr 726 . . . . . . . . . . . . 13 (((𝜑𝑦 ∈ ℝ+) ∧ (𝑗𝑍𝑘 ∈ (ℤ𝑗))) → ∀𝑘𝑍 (𝐹‘(𝑘 + 1)) ≤ (𝐹𝑘))
80 fvoveq1 7236 . . . . . . . . . . . . . . 15 (𝑘 = 𝑛 → (𝐹‘(𝑘 + 1)) = (𝐹‘(𝑛 + 1)))
81 fveq2 6717 . . . . . . . . . . . . . . 15 (𝑘 = 𝑛 → (𝐹𝑘) = (𝐹𝑛))
8280, 81breq12d 5066 . . . . . . . . . . . . . 14 (𝑘 = 𝑛 → ((𝐹‘(𝑘 + 1)) ≤ (𝐹𝑘) ↔ (𝐹‘(𝑛 + 1)) ≤ (𝐹𝑛)))
8382rspccva 3536 . . . . . . . . . . . . 13 ((∀𝑘𝑍 (𝐹‘(𝑘 + 1)) ≤ (𝐹𝑘) ∧ 𝑛𝑍) → (𝐹‘(𝑛 + 1)) ≤ (𝐹𝑛))
8479, 83sylan 583 . . . . . . . . . . . 12 ((((𝜑𝑦 ∈ ℝ+) ∧ (𝑗𝑍𝑘 ∈ (ℤ𝑗))) ∧ 𝑛𝑍) → (𝐹‘(𝑛 + 1)) ≤ (𝐹𝑛))
8576, 84syldan 594 . . . . . . . . . . 11 ((((𝜑𝑦 ∈ ℝ+) ∧ (𝑗𝑍𝑘 ∈ (ℤ𝑗))) ∧ 𝑛 ∈ (𝑗...(𝑘 − 1))) → (𝐹‘(𝑛 + 1)) ≤ (𝐹𝑛))
8660, 73, 85monoord2 13607 . . . . . . . . . 10 (((𝜑𝑦 ∈ ℝ+) ∧ (𝑗𝑍𝑘 ∈ (ℤ𝑗))) → (𝐹𝑘) ≤ (𝐹𝑗))
8755, 58, 59, 86lesub1dd 11448 . . . . . . . . 9 (((𝜑𝑦 ∈ ℝ+) ∧ (𝑗𝑍𝑘 ∈ (ℤ𝑗))) → ((𝐹𝑘) − inf(ran 𝐹, ℝ, < )) ≤ ((𝐹𝑗) − inf(ran 𝐹, ℝ, < )))
8855, 59resubcld 11260 . . . . . . . . . 10 (((𝜑𝑦 ∈ ℝ+) ∧ (𝑗𝑍𝑘 ∈ (ℤ𝑗))) → ((𝐹𝑘) − inf(ran 𝐹, ℝ, < )) ∈ ℝ)
8958, 59resubcld 11260 . . . . . . . . . 10 (((𝜑𝑦 ∈ ℝ+) ∧ (𝑗𝑍𝑘 ∈ (ℤ𝑗))) → ((𝐹𝑗) − inf(ran 𝐹, ℝ, < )) ∈ ℝ)
9024ad2antlr 727 . . . . . . . . . 10 (((𝜑𝑦 ∈ ℝ+) ∧ (𝑗𝑍𝑘 ∈ (ℤ𝑗))) → 𝑦 ∈ ℝ)
91 lelttr 10923 . . . . . . . . . 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 1373 . . . . . . . . 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 695 . . . . . . . 8 (((𝜑𝑦 ∈ ℝ+) ∧ (𝑗𝑍𝑘 ∈ (ℤ𝑗))) → (((𝐹𝑗) − inf(ran 𝐹, ℝ, < )) < 𝑦 → ((𝐹𝑘) − inf(ran 𝐹, ℝ, < )) < 𝑦))
94 ltsub23 11312 . . . . . . . . 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 1373 . . . . . . . 8 (((𝜑𝑦 ∈ ℝ+) ∧ (𝑗𝑍𝑘 ∈ (ℤ𝑗))) → (((𝐹𝑗) − 𝑦) < inf(ran 𝐹, ℝ, < ) ↔ ((𝐹𝑗) − inf(ran 𝐹, ℝ, < )) < 𝑦))
962ad2antrr 726 . . . . . . . . . . 11 (((𝜑𝑦 ∈ ℝ+) ∧ (𝑗𝑍𝑘 ∈ (ℤ𝑗))) → ran 𝐹 ⊆ ℝ)
973adantr 484 . . . . . . . . . . . 12 ((𝜑𝑦 ∈ ℝ+) → 𝐹 Fn 𝑍)
98 fnfvelrn 6901 . . . . . . . . . . . 12 ((𝐹 Fn 𝑍𝑘𝑍) → (𝐹𝑘) ∈ ran 𝐹)
9997, 53, 98syl2an 599 . . . . . . . . . . 11 (((𝜑𝑦 ∈ ℝ+) ∧ (𝑗𝑍𝑘 ∈ (ℤ𝑗))) → (𝐹𝑘) ∈ ran 𝐹)
10096, 99sseldd 3902 . . . . . . . . . 10 (((𝜑𝑦 ∈ ℝ+) ∧ (𝑗𝑍𝑘 ∈ (ℤ𝑗))) → (𝐹𝑘) ∈ ℝ)
10117ad2antrr 726 . . . . . . . . . . 11 (((𝜑𝑦 ∈ ℝ+) ∧ (𝑗𝑍𝑘 ∈ (ℤ𝑗))) → ∃𝑥 ∈ ℝ ∀𝑦 ∈ ran 𝐹 𝑥𝑦)
102 infrelb 11817 . . . . . . . . . . 11 ((ran 𝐹 ⊆ ℝ ∧ ∃𝑥 ∈ ℝ ∀𝑦 ∈ ran 𝐹 𝑥𝑦 ∧ (𝐹𝑘) ∈ ran 𝐹) → inf(ran 𝐹, ℝ, < ) ≤ (𝐹𝑘))
10396, 101, 99, 102syl3anc 1373 . . . . . . . . . 10 (((𝜑𝑦 ∈ ℝ+) ∧ (𝑗𝑍𝑘 ∈ (ℤ𝑗))) → inf(ran 𝐹, ℝ, < ) ≤ (𝐹𝑘))
10459, 100, 103abssubge0d 14995 . . . . . . . . 9 (((𝜑𝑦 ∈ ℝ+) ∧ (𝑗𝑍𝑘 ∈ (ℤ𝑗))) → (abs‘((𝐹𝑘) − inf(ran 𝐹, ℝ, < ))) = ((𝐹𝑘) − inf(ran 𝐹, ℝ, < )))
105104breq1d 5063 . . . . . . . 8 (((𝜑𝑦 ∈ ℝ+) ∧ (𝑗𝑍𝑘 ∈ (ℤ𝑗))) → ((abs‘((𝐹𝑘) − inf(ran 𝐹, ℝ, < ))) < 𝑦 ↔ ((𝐹𝑘) − inf(ran 𝐹, ℝ, < )) < 𝑦))
10693, 95, 1053imtr4d 297 . . . . . . 7 (((𝜑𝑦 ∈ ℝ+) ∧ (𝑗𝑍𝑘 ∈ (ℤ𝑗))) → (((𝐹𝑗) − 𝑦) < inf(ran 𝐹, ℝ, < ) → (abs‘((𝐹𝑘) − inf(ran 𝐹, ℝ, < ))) < 𝑦))
107106anassrs 471 . . . . . 6 ((((𝜑𝑦 ∈ ℝ+) ∧ 𝑗𝑍) ∧ 𝑘 ∈ (ℤ𝑗)) → (((𝐹𝑗) − 𝑦) < inf(ran 𝐹, ℝ, < ) → (abs‘((𝐹𝑘) − inf(ran 𝐹, ℝ, < ))) < 𝑦))
108107ralrimdva 3110 . . . . 5 (((𝜑𝑦 ∈ ℝ+) ∧ 𝑗𝑍) → (((𝐹𝑗) − 𝑦) < inf(ran 𝐹, ℝ, < ) → ∀𝑘 ∈ (ℤ𝑗)(abs‘((𝐹𝑘) − inf(ran 𝐹, ℝ, < ))) < 𝑦))
109108reximdva 3193 . . . 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 3105 . 2 (𝜑 → ∀𝑦 ∈ ℝ+𝑗𝑍𝑘 ∈ (ℤ𝑗)(abs‘((𝐹𝑘) − inf(ran 𝐹, ℝ, < ))) < 𝑦)
1127fvexi 6731 . . . 4 𝑍 ∈ V
113 fex 7042 . . . 4 ((𝐹:𝑍⟶ℝ ∧ 𝑍 ∈ V) → 𝐹 ∈ V)
1141, 112, 113sylancl 589 . . 3 (𝜑𝐹 ∈ V)
115 eqidd 2738 . . 3 ((𝜑𝑘𝑍) → (𝐹𝑘) = (𝐹𝑘))
1161ffvelrnda 6904 . . . 4 ((𝜑𝑘𝑍) → (𝐹𝑘) ∈ ℝ)
117116recnd 10861 . . 3 ((𝜑𝑘𝑍) → (𝐹𝑘) ∈ ℂ)
1187, 4, 114, 115, 37, 117clim2c 15066 . 2 (𝜑 → (𝐹 ⇝ inf(ran 𝐹, ℝ, < ) ↔ ∀𝑦 ∈ ℝ+𝑗𝑍𝑘 ∈ (ℤ𝑗)(abs‘((𝐹𝑘) − inf(ran 𝐹, ℝ, < ))) < 𝑦))
119111, 118mpbird 260 1 (𝜑𝐹 ⇝ inf(ran 𝐹, ℝ, < ))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wa 399  w3a 1089   = wceq 1543  wcel 2110  wne 2940  wral 3061  wrex 3062  Vcvv 3408  wss 3866  c0 4237   class class class wbr 5053  ran crn 5552   Fn wfn 6375  wf 6376  cfv 6380  (class class class)co 7213  infcinf 9057  cc 10727  cr 10728  1c1 10730   + caddc 10732   < clt 10867  cle 10868  cmin 11062  cz 12176  cuz 12438  +crp 12586  ...cfz 13095  abscabs 14797  cli 15045
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2158  ax-12 2175  ax-ext 2708  ax-rep 5179  ax-sep 5192  ax-nul 5199  ax-pow 5258  ax-pr 5322  ax-un 7523  ax-cnex 10785  ax-resscn 10786  ax-1cn 10787  ax-icn 10788  ax-addcl 10789  ax-addrcl 10790  ax-mulcl 10791  ax-mulrcl 10792  ax-mulcom 10793  ax-addass 10794  ax-mulass 10795  ax-distr 10796  ax-i2m1 10797  ax-1ne0 10798  ax-1rid 10799  ax-rnegex 10800  ax-rrecex 10801  ax-cnre 10802  ax-pre-lttri 10803  ax-pre-lttrn 10804  ax-pre-ltadd 10805  ax-pre-mulgt0 10806  ax-pre-sup 10807
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3or 1090  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-nf 1792  df-sb 2071  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2886  df-ne 2941  df-nel 3047  df-ral 3066  df-rex 3067  df-reu 3068  df-rmo 3069  df-rab 3070  df-v 3410  df-sbc 3695  df-csb 3812  df-dif 3869  df-un 3871  df-in 3873  df-ss 3883  df-pss 3885  df-nul 4238  df-if 4440  df-pw 4515  df-sn 4542  df-pr 4544  df-tp 4546  df-op 4548  df-uni 4820  df-iun 4906  df-br 5054  df-opab 5116  df-mpt 5136  df-tr 5162  df-id 5455  df-eprel 5460  df-po 5468  df-so 5469  df-fr 5509  df-we 5511  df-xp 5557  df-rel 5558  df-cnv 5559  df-co 5560  df-dm 5561  df-rn 5562  df-res 5563  df-ima 5564  df-pred 6160  df-ord 6216  df-on 6217  df-lim 6218  df-suc 6219  df-iota 6338  df-fun 6382  df-fn 6383  df-f 6384  df-f1 6385  df-fo 6386  df-f1o 6387  df-fv 6388  df-riota 7170  df-ov 7216  df-oprab 7217  df-mpo 7218  df-om 7645  df-1st 7761  df-2nd 7762  df-wrecs 8047  df-recs 8108  df-rdg 8146  df-er 8391  df-en 8627  df-dom 8628  df-sdom 8629  df-sup 9058  df-inf 9059  df-pnf 10869  df-mnf 10870  df-xr 10871  df-ltxr 10872  df-le 10873  df-sub 11064  df-neg 11065  df-div 11490  df-nn 11831  df-2 11893  df-3 11894  df-n0 12091  df-z 12177  df-uz 12439  df-rp 12587  df-fz 13096  df-seq 13575  df-exp 13636  df-cj 14662  df-re 14663  df-im 14664  df-sqrt 14798  df-abs 14799  df-clim 15049
This theorem is referenced by:  climinff  42827  climinf2lem  42922  supcnvlimsup  42956  stirlinglem13  43302
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