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Theorem climinf 40310
Description: A bounded monotonic non increasing 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)
2 frn 6202 . . . . . . . . . . . 12 (𝐹:𝑍⟶ℝ → ran 𝐹 ⊆ ℝ)
31, 2syl 17 . . . . . . . . . . 11 (𝜑 → ran 𝐹 ⊆ ℝ)
4 ffn 6194 . . . . . . . . . . . . . 14 (𝐹:𝑍⟶ℝ → 𝐹 Fn 𝑍)
51, 4syl 17 . . . . . . . . . . . . 13 (𝜑𝐹 Fn 𝑍)
6 climinf.4 . . . . . . . . . . . . . . 15 (𝜑𝑀 ∈ ℤ)
7 uzid 11865 . . . . . . . . . . . . . . 15 (𝑀 ∈ ℤ → 𝑀 ∈ (ℤ𝑀))
86, 7syl 17 . . . . . . . . . . . . . 14 (𝜑𝑀 ∈ (ℤ𝑀))
9 climinf.3 . . . . . . . . . . . . . 14 𝑍 = (ℤ𝑀)
108, 9syl6eleqr 2838 . . . . . . . . . . . . 13 (𝜑𝑀𝑍)
11 fnfvelrn 6507 . . . . . . . . . . . . 13 ((𝐹 Fn 𝑍𝑀𝑍) → (𝐹𝑀) ∈ ran 𝐹)
125, 10, 11syl2anc 696 . . . . . . . . . . . 12 (𝜑 → (𝐹𝑀) ∈ ran 𝐹)
13 ne0i 4052 . . . . . . . . . . . 12 ((𝐹𝑀) ∈ ran 𝐹 → ran 𝐹 ≠ ∅)
1412, 13syl 17 . . . . . . . . . . 11 (𝜑 → ran 𝐹 ≠ ∅)
15 climinf.7 . . . . . . . . . . . 12 (𝜑 → ∃𝑥 ∈ ℝ ∀𝑘𝑍 𝑥 ≤ (𝐹𝑘))
16 breq2 4796 . . . . . . . . . . . . . . 15 (𝑦 = (𝐹𝑘) → (𝑥𝑦𝑥 ≤ (𝐹𝑘)))
1716ralrn 6513 . . . . . . . . . . . . . 14 (𝐹 Fn 𝑍 → (∀𝑦 ∈ ran 𝐹 𝑥𝑦 ↔ ∀𝑘𝑍 𝑥 ≤ (𝐹𝑘)))
1817rexbidv 3178 . . . . . . . . . . . . 13 (𝐹 Fn 𝑍 → (∃𝑥 ∈ ℝ ∀𝑦 ∈ ran 𝐹 𝑥𝑦 ↔ ∃𝑥 ∈ ℝ ∀𝑘𝑍 𝑥 ≤ (𝐹𝑘)))
195, 18syl 17 . . . . . . . . . . . 12 (𝜑 → (∃𝑥 ∈ ℝ ∀𝑦 ∈ ran 𝐹 𝑥𝑦 ↔ ∃𝑥 ∈ ℝ ∀𝑘𝑍 𝑥 ≤ (𝐹𝑘)))
2015, 19mpbird 247 . . . . . . . . . . 11 (𝜑 → ∃𝑥 ∈ ℝ ∀𝑦 ∈ ran 𝐹 𝑥𝑦)
213, 14, 203jca 1403 . . . . . . . . . 10 (𝜑 → (ran 𝐹 ⊆ ℝ ∧ ran 𝐹 ≠ ∅ ∧ ∃𝑥 ∈ ℝ ∀𝑦 ∈ ran 𝐹 𝑥𝑦))
2221adantr 472 . . . . . . . . 9 ((𝜑𝑦 ∈ ℝ+) → (ran 𝐹 ⊆ ℝ ∧ ran 𝐹 ≠ ∅ ∧ ∃𝑥 ∈ ℝ ∀𝑦 ∈ ran 𝐹 𝑥𝑦))
23 infrecl 11168 . . . . . . . . 9 ((ran 𝐹 ⊆ ℝ ∧ ran 𝐹 ≠ ∅ ∧ ∃𝑥 ∈ ℝ ∀𝑦 ∈ ran 𝐹 𝑥𝑦) → inf(ran 𝐹, ℝ, < ) ∈ ℝ)
2422, 23syl 17 . . . . . . . 8 ((𝜑𝑦 ∈ ℝ+) → inf(ran 𝐹, ℝ, < ) ∈ ℝ)
25 simpr 479 . . . . . . . 8 ((𝜑𝑦 ∈ ℝ+) → 𝑦 ∈ ℝ+)
2624, 25ltaddrpd 12069 . . . . . . 7 ((𝜑𝑦 ∈ ℝ+) → inf(ran 𝐹, ℝ, < ) < (inf(ran 𝐹, ℝ, < ) + 𝑦))
27 rpre 12003 . . . . . . . . . 10 (𝑦 ∈ ℝ+𝑦 ∈ ℝ)
2827adantl 473 . . . . . . . . 9 ((𝜑𝑦 ∈ ℝ+) → 𝑦 ∈ ℝ)
2924, 28readdcld 10232 . . . . . . . 8 ((𝜑𝑦 ∈ ℝ+) → (inf(ran 𝐹, ℝ, < ) + 𝑦) ∈ ℝ)
30 infrglb 40294 . . . . . . . 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)))
3122, 29, 30syl2anc 696 . . . . . . 7 ((𝜑𝑦 ∈ ℝ+) → (inf(ran 𝐹, ℝ, < ) < (inf(ran 𝐹, ℝ, < ) + 𝑦) ↔ ∃𝑘 ∈ ran 𝐹 𝑘 < (inf(ran 𝐹, ℝ, < ) + 𝑦)))
3226, 31mpbid 222 . . . . . 6 ((𝜑𝑦 ∈ ℝ+) → ∃𝑘 ∈ ran 𝐹 𝑘 < (inf(ran 𝐹, ℝ, < ) + 𝑦))
333sselda 3732 . . . . . . . . . . 11 ((𝜑𝑘 ∈ ran 𝐹) → 𝑘 ∈ ℝ)
3433adantlr 753 . . . . . . . . . 10 (((𝜑𝑦 ∈ ℝ+) ∧ 𝑘 ∈ ran 𝐹) → 𝑘 ∈ ℝ)
3524adantr 472 . . . . . . . . . . 11 (((𝜑𝑦 ∈ ℝ+) ∧ 𝑘 ∈ ran 𝐹) → inf(ran 𝐹, ℝ, < ) ∈ ℝ)
3627ad2antlr 765 . . . . . . . . . . 11 (((𝜑𝑦 ∈ ℝ+) ∧ 𝑘 ∈ ran 𝐹) → 𝑦 ∈ ℝ)
3735, 36readdcld 10232 . . . . . . . . . 10 (((𝜑𝑦 ∈ ℝ+) ∧ 𝑘 ∈ ran 𝐹) → (inf(ran 𝐹, ℝ, < ) + 𝑦) ∈ ℝ)
3834, 37, 36ltsub1d 10799 . . . . . . . . 9 (((𝜑𝑦 ∈ ℝ+) ∧ 𝑘 ∈ ran 𝐹) → (𝑘 < (inf(ran 𝐹, ℝ, < ) + 𝑦) ↔ (𝑘𝑦) < ((inf(ran 𝐹, ℝ, < ) + 𝑦) − 𝑦)))
393, 14, 20, 23syl3anc 1463 . . . . . . . . . . . . 13 (𝜑 → inf(ran 𝐹, ℝ, < ) ∈ ℝ)
4039recnd 10231 . . . . . . . . . . . 12 (𝜑 → inf(ran 𝐹, ℝ, < ) ∈ ℂ)
4140ad2antrr 764 . . . . . . . . . . 11 (((𝜑𝑦 ∈ ℝ+) ∧ 𝑘 ∈ ran 𝐹) → inf(ran 𝐹, ℝ, < ) ∈ ℂ)
4236recnd 10231 . . . . . . . . . . 11 (((𝜑𝑦 ∈ ℝ+) ∧ 𝑘 ∈ ran 𝐹) → 𝑦 ∈ ℂ)
4341, 42pncand 10556 . . . . . . . . . 10 (((𝜑𝑦 ∈ ℝ+) ∧ 𝑘 ∈ ran 𝐹) → ((inf(ran 𝐹, ℝ, < ) + 𝑦) − 𝑦) = inf(ran 𝐹, ℝ, < ))
4443breq2d 4804 . . . . . . . . 9 (((𝜑𝑦 ∈ ℝ+) ∧ 𝑘 ∈ ran 𝐹) → ((𝑘𝑦) < ((inf(ran 𝐹, ℝ, < ) + 𝑦) − 𝑦) ↔ (𝑘𝑦) < inf(ran 𝐹, ℝ, < )))
4538, 44bitrd 268 . . . . . . . 8 (((𝜑𝑦 ∈ ℝ+) ∧ 𝑘 ∈ ran 𝐹) → (𝑘 < (inf(ran 𝐹, ℝ, < ) + 𝑦) ↔ (𝑘𝑦) < inf(ran 𝐹, ℝ, < )))
4645biimpd 219 . . . . . . 7 (((𝜑𝑦 ∈ ℝ+) ∧ 𝑘 ∈ ran 𝐹) → (𝑘 < (inf(ran 𝐹, ℝ, < ) + 𝑦) → (𝑘𝑦) < inf(ran 𝐹, ℝ, < )))
4746reximdva 3143 . . . . . 6 ((𝜑𝑦 ∈ ℝ+) → (∃𝑘 ∈ ran 𝐹 𝑘 < (inf(ran 𝐹, ℝ, < ) + 𝑦) → ∃𝑘 ∈ ran 𝐹(𝑘𝑦) < inf(ran 𝐹, ℝ, < )))
4832, 47mpd 15 . . . . 5 ((𝜑𝑦 ∈ ℝ+) → ∃𝑘 ∈ ran 𝐹(𝑘𝑦) < inf(ran 𝐹, ℝ, < ))
49 oveq1 6808 . . . . . . . . 9 (𝑘 = (𝐹𝑗) → (𝑘𝑦) = ((𝐹𝑗) − 𝑦))
5049breq1d 4802 . . . . . . . 8 (𝑘 = (𝐹𝑗) → ((𝑘𝑦) < inf(ran 𝐹, ℝ, < ) ↔ ((𝐹𝑗) − 𝑦) < inf(ran 𝐹, ℝ, < )))
5150rexrn 6512 . . . . . . 7 (𝐹 Fn 𝑍 → (∃𝑘 ∈ ran 𝐹(𝑘𝑦) < inf(ran 𝐹, ℝ, < ) ↔ ∃𝑗𝑍 ((𝐹𝑗) − 𝑦) < inf(ran 𝐹, ℝ, < )))
525, 51syl 17 . . . . . 6 (𝜑 → (∃𝑘 ∈ ran 𝐹(𝑘𝑦) < inf(ran 𝐹, ℝ, < ) ↔ ∃𝑗𝑍 ((𝐹𝑗) − 𝑦) < inf(ran 𝐹, ℝ, < )))
5352biimpa 502 . . . . 5 ((𝜑 ∧ ∃𝑘 ∈ ran 𝐹(𝑘𝑦) < inf(ran 𝐹, ℝ, < )) → ∃𝑗𝑍 ((𝐹𝑗) − 𝑦) < inf(ran 𝐹, ℝ, < ))
5448, 53syldan 488 . . . 4 ((𝜑𝑦 ∈ ℝ+) → ∃𝑗𝑍 ((𝐹𝑗) − 𝑦) < inf(ran 𝐹, ℝ, < ))
551adantr 472 . . . . . . . . . . 11 ((𝜑𝑦 ∈ ℝ+) → 𝐹:𝑍⟶ℝ)
569uztrn2 11868 . . . . . . . . . . 11 ((𝑗𝑍𝑘 ∈ (ℤ𝑗)) → 𝑘𝑍)
57 ffvelrn 6508 . . . . . . . . . . 11 ((𝐹:𝑍⟶ℝ ∧ 𝑘𝑍) → (𝐹𝑘) ∈ ℝ)
5855, 56, 57syl2an 495 . . . . . . . . . 10 (((𝜑𝑦 ∈ ℝ+) ∧ (𝑗𝑍𝑘 ∈ (ℤ𝑗))) → (𝐹𝑘) ∈ ℝ)
59 simpl 474 . . . . . . . . . . 11 ((𝑗𝑍𝑘 ∈ (ℤ𝑗)) → 𝑗𝑍)
60 ffvelrn 6508 . . . . . . . . . . 11 ((𝐹:𝑍⟶ℝ ∧ 𝑗𝑍) → (𝐹𝑗) ∈ ℝ)
6155, 59, 60syl2an 495 . . . . . . . . . 10 (((𝜑𝑦 ∈ ℝ+) ∧ (𝑗𝑍𝑘 ∈ (ℤ𝑗))) → (𝐹𝑗) ∈ ℝ)
6239ad2antrr 764 . . . . . . . . . 10 (((𝜑𝑦 ∈ ℝ+) ∧ (𝑗𝑍𝑘 ∈ (ℤ𝑗))) → inf(ran 𝐹, ℝ, < ) ∈ ℝ)
63 simprr 813 . . . . . . . . . . 11 (((𝜑𝑦 ∈ ℝ+) ∧ (𝑗𝑍𝑘 ∈ (ℤ𝑗))) → 𝑘 ∈ (ℤ𝑗))
64 fzssuz 12546 . . . . . . . . . . . . . 14 (𝑗...𝑘) ⊆ (ℤ𝑗)
65 uzss 11871 . . . . . . . . . . . . . . . . 17 (𝑗 ∈ (ℤ𝑀) → (ℤ𝑗) ⊆ (ℤ𝑀))
6665, 9syl6sseqr 3781 . . . . . . . . . . . . . . . 16 (𝑗 ∈ (ℤ𝑀) → (ℤ𝑗) ⊆ 𝑍)
6766, 9eleq2s 2845 . . . . . . . . . . . . . . 15 (𝑗𝑍 → (ℤ𝑗) ⊆ 𝑍)
6867ad2antrl 766 . . . . . . . . . . . . . 14 (((𝜑𝑦 ∈ ℝ+) ∧ (𝑗𝑍𝑘 ∈ (ℤ𝑗))) → (ℤ𝑗) ⊆ 𝑍)
6964, 68syl5ss 3743 . . . . . . . . . . . . 13 (((𝜑𝑦 ∈ ℝ+) ∧ (𝑗𝑍𝑘 ∈ (ℤ𝑗))) → (𝑗...𝑘) ⊆ 𝑍)
70 ffvelrn 6508 . . . . . . . . . . . . . . . 16 ((𝐹:𝑍⟶ℝ ∧ 𝑛𝑍) → (𝐹𝑛) ∈ ℝ)
7170ralrimiva 3092 . . . . . . . . . . . . . . 15 (𝐹:𝑍⟶ℝ → ∀𝑛𝑍 (𝐹𝑛) ∈ ℝ)
721, 71syl 17 . . . . . . . . . . . . . 14 (𝜑 → ∀𝑛𝑍 (𝐹𝑛) ∈ ℝ)
7372ad2antrr 764 . . . . . . . . . . . . 13 (((𝜑𝑦 ∈ ℝ+) ∧ (𝑗𝑍𝑘 ∈ (ℤ𝑗))) → ∀𝑛𝑍 (𝐹𝑛) ∈ ℝ)
74 ssralv 3795 . . . . . . . . . . . . 13 ((𝑗...𝑘) ⊆ 𝑍 → (∀𝑛𝑍 (𝐹𝑛) ∈ ℝ → ∀𝑛 ∈ (𝑗...𝑘)(𝐹𝑛) ∈ ℝ))
7569, 73, 74sylc 65 . . . . . . . . . . . 12 (((𝜑𝑦 ∈ ℝ+) ∧ (𝑗𝑍𝑘 ∈ (ℤ𝑗))) → ∀𝑛 ∈ (𝑗...𝑘)(𝐹𝑛) ∈ ℝ)
7675r19.21bi 3058 . . . . . . . . . . 11 ((((𝜑𝑦 ∈ ℝ+) ∧ (𝑗𝑍𝑘 ∈ (ℤ𝑗))) ∧ 𝑛 ∈ (𝑗...𝑘)) → (𝐹𝑛) ∈ ℝ)
77 fzssuz 12546 . . . . . . . . . . . . . 14 (𝑗...(𝑘 − 1)) ⊆ (ℤ𝑗)
7877, 68syl5ss 3743 . . . . . . . . . . . . 13 (((𝜑𝑦 ∈ ℝ+) ∧ (𝑗𝑍𝑘 ∈ (ℤ𝑗))) → (𝑗...(𝑘 − 1)) ⊆ 𝑍)
7978sselda 3732 . . . . . . . . . . . 12 ((((𝜑𝑦 ∈ ℝ+) ∧ (𝑗𝑍𝑘 ∈ (ℤ𝑗))) ∧ 𝑛 ∈ (𝑗...(𝑘 − 1))) → 𝑛𝑍)
80 climinf.6 . . . . . . . . . . . . . . 15 ((𝜑𝑘𝑍) → (𝐹‘(𝑘 + 1)) ≤ (𝐹𝑘))
8180ralrimiva 3092 . . . . . . . . . . . . . 14 (𝜑 → ∀𝑘𝑍 (𝐹‘(𝑘 + 1)) ≤ (𝐹𝑘))
8281ad2antrr 764 . . . . . . . . . . . . 13 (((𝜑𝑦 ∈ ℝ+) ∧ (𝑗𝑍𝑘 ∈ (ℤ𝑗))) → ∀𝑘𝑍 (𝐹‘(𝑘 + 1)) ≤ (𝐹𝑘))
83 oveq1 6808 . . . . . . . . . . . . . . . 16 (𝑘 = 𝑛 → (𝑘 + 1) = (𝑛 + 1))
8483fveq2d 6344 . . . . . . . . . . . . . . 15 (𝑘 = 𝑛 → (𝐹‘(𝑘 + 1)) = (𝐹‘(𝑛 + 1)))
85 fveq2 6340 . . . . . . . . . . . . . . 15 (𝑘 = 𝑛 → (𝐹𝑘) = (𝐹𝑛))
8684, 85breq12d 4805 . . . . . . . . . . . . . 14 (𝑘 = 𝑛 → ((𝐹‘(𝑘 + 1)) ≤ (𝐹𝑘) ↔ (𝐹‘(𝑛 + 1)) ≤ (𝐹𝑛)))
8786rspccva 3436 . . . . . . . . . . . . 13 ((∀𝑘𝑍 (𝐹‘(𝑘 + 1)) ≤ (𝐹𝑘) ∧ 𝑛𝑍) → (𝐹‘(𝑛 + 1)) ≤ (𝐹𝑛))
8882, 87sylan 489 . . . . . . . . . . . 12 ((((𝜑𝑦 ∈ ℝ+) ∧ (𝑗𝑍𝑘 ∈ (ℤ𝑗))) ∧ 𝑛𝑍) → (𝐹‘(𝑛 + 1)) ≤ (𝐹𝑛))
8979, 88syldan 488 . . . . . . . . . . 11 ((((𝜑𝑦 ∈ ℝ+) ∧ (𝑗𝑍𝑘 ∈ (ℤ𝑗))) ∧ 𝑛 ∈ (𝑗...(𝑘 − 1))) → (𝐹‘(𝑛 + 1)) ≤ (𝐹𝑛))
9063, 76, 89monoord2 12997 . . . . . . . . . 10 (((𝜑𝑦 ∈ ℝ+) ∧ (𝑗𝑍𝑘 ∈ (ℤ𝑗))) → (𝐹𝑘) ≤ (𝐹𝑗))
9158, 61, 62, 90lesub1dd 10806 . . . . . . . . 9 (((𝜑𝑦 ∈ ℝ+) ∧ (𝑗𝑍𝑘 ∈ (ℤ𝑗))) → ((𝐹𝑘) − inf(ran 𝐹, ℝ, < )) ≤ ((𝐹𝑗) − inf(ran 𝐹, ℝ, < )))
9258, 62resubcld 10621 . . . . . . . . . 10 (((𝜑𝑦 ∈ ℝ+) ∧ (𝑗𝑍𝑘 ∈ (ℤ𝑗))) → ((𝐹𝑘) − inf(ran 𝐹, ℝ, < )) ∈ ℝ)
9361, 62resubcld 10621 . . . . . . . . . 10 (((𝜑𝑦 ∈ ℝ+) ∧ (𝑗𝑍𝑘 ∈ (ℤ𝑗))) → ((𝐹𝑗) − inf(ran 𝐹, ℝ, < )) ∈ ℝ)
9427ad2antlr 765 . . . . . . . . . 10 (((𝜑𝑦 ∈ ℝ+) ∧ (𝑗𝑍𝑘 ∈ (ℤ𝑗))) → 𝑦 ∈ ℝ)
95 lelttr 10291 . . . . . . . . . 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))
9692, 93, 94, 95syl3anc 1463 . . . . . . . . 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))
9791, 96mpand 713 . . . . . . . 8 (((𝜑𝑦 ∈ ℝ+) ∧ (𝑗𝑍𝑘 ∈ (ℤ𝑗))) → (((𝐹𝑗) − inf(ran 𝐹, ℝ, < )) < 𝑦 → ((𝐹𝑘) − inf(ran 𝐹, ℝ, < )) < 𝑦))
98 ltsub23 10671 . . . . . . . . 9 (((𝐹𝑗) ∈ ℝ ∧ 𝑦 ∈ ℝ ∧ inf(ran 𝐹, ℝ, < ) ∈ ℝ) → (((𝐹𝑗) − 𝑦) < inf(ran 𝐹, ℝ, < ) ↔ ((𝐹𝑗) − inf(ran 𝐹, ℝ, < )) < 𝑦))
9961, 94, 62, 98syl3anc 1463 . . . . . . . 8 (((𝜑𝑦 ∈ ℝ+) ∧ (𝑗𝑍𝑘 ∈ (ℤ𝑗))) → (((𝐹𝑗) − 𝑦) < inf(ran 𝐹, ℝ, < ) ↔ ((𝐹𝑗) − inf(ran 𝐹, ℝ, < )) < 𝑦))
1003ad2antrr 764 . . . . . . . . . . 11 (((𝜑𝑦 ∈ ℝ+) ∧ (𝑗𝑍𝑘 ∈ (ℤ𝑗))) → ran 𝐹 ⊆ ℝ)
1015adantr 472 . . . . . . . . . . . 12 ((𝜑𝑦 ∈ ℝ+) → 𝐹 Fn 𝑍)
102 fnfvelrn 6507 . . . . . . . . . . . 12 ((𝐹 Fn 𝑍𝑘𝑍) → (𝐹𝑘) ∈ ran 𝐹)
103101, 56, 102syl2an 495 . . . . . . . . . . 11 (((𝜑𝑦 ∈ ℝ+) ∧ (𝑗𝑍𝑘 ∈ (ℤ𝑗))) → (𝐹𝑘) ∈ ran 𝐹)
104100, 103sseldd 3733 . . . . . . . . . 10 (((𝜑𝑦 ∈ ℝ+) ∧ (𝑗𝑍𝑘 ∈ (ℤ𝑗))) → (𝐹𝑘) ∈ ℝ)
10520ad2antrr 764 . . . . . . . . . . 11 (((𝜑𝑦 ∈ ℝ+) ∧ (𝑗𝑍𝑘 ∈ (ℤ𝑗))) → ∃𝑥 ∈ ℝ ∀𝑦 ∈ ran 𝐹 𝑥𝑦)
106 infrelb 11171 . . . . . . . . . . 11 ((ran 𝐹 ⊆ ℝ ∧ ∃𝑥 ∈ ℝ ∀𝑦 ∈ ran 𝐹 𝑥𝑦 ∧ (𝐹𝑘) ∈ ran 𝐹) → inf(ran 𝐹, ℝ, < ) ≤ (𝐹𝑘))
107100, 105, 103, 106syl3anc 1463 . . . . . . . . . 10 (((𝜑𝑦 ∈ ℝ+) ∧ (𝑗𝑍𝑘 ∈ (ℤ𝑗))) → inf(ran 𝐹, ℝ, < ) ≤ (𝐹𝑘))
10862, 104, 107abssubge0d 14340 . . . . . . . . 9 (((𝜑𝑦 ∈ ℝ+) ∧ (𝑗𝑍𝑘 ∈ (ℤ𝑗))) → (abs‘((𝐹𝑘) − inf(ran 𝐹, ℝ, < ))) = ((𝐹𝑘) − inf(ran 𝐹, ℝ, < )))
109108breq1d 4802 . . . . . . . 8 (((𝜑𝑦 ∈ ℝ+) ∧ (𝑗𝑍𝑘 ∈ (ℤ𝑗))) → ((abs‘((𝐹𝑘) − inf(ran 𝐹, ℝ, < ))) < 𝑦 ↔ ((𝐹𝑘) − inf(ran 𝐹, ℝ, < )) < 𝑦))
11097, 99, 1093imtr4d 283 . . . . . . 7 (((𝜑𝑦 ∈ ℝ+) ∧ (𝑗𝑍𝑘 ∈ (ℤ𝑗))) → (((𝐹𝑗) − 𝑦) < inf(ran 𝐹, ℝ, < ) → (abs‘((𝐹𝑘) − inf(ran 𝐹, ℝ, < ))) < 𝑦))
111110anassrs 683 . . . . . 6 ((((𝜑𝑦 ∈ ℝ+) ∧ 𝑗𝑍) ∧ 𝑘 ∈ (ℤ𝑗)) → (((𝐹𝑗) − 𝑦) < inf(ran 𝐹, ℝ, < ) → (abs‘((𝐹𝑘) − inf(ran 𝐹, ℝ, < ))) < 𝑦))
112111ralrimdva 3095 . . . . 5 (((𝜑𝑦 ∈ ℝ+) ∧ 𝑗𝑍) → (((𝐹𝑗) − 𝑦) < inf(ran 𝐹, ℝ, < ) → ∀𝑘 ∈ (ℤ𝑗)(abs‘((𝐹𝑘) − inf(ran 𝐹, ℝ, < ))) < 𝑦))
113112reximdva 3143 . . . 4 ((𝜑𝑦 ∈ ℝ+) → (∃𝑗𝑍 ((𝐹𝑗) − 𝑦) < inf(ran 𝐹, ℝ, < ) → ∃𝑗𝑍𝑘 ∈ (ℤ𝑗)(abs‘((𝐹𝑘) − inf(ran 𝐹, ℝ, < ))) < 𝑦))
11454, 113mpd 15 . . 3 ((𝜑𝑦 ∈ ℝ+) → ∃𝑗𝑍𝑘 ∈ (ℤ𝑗)(abs‘((𝐹𝑘) − inf(ran 𝐹, ℝ, < ))) < 𝑦)
115114ralrimiva 3092 . 2 (𝜑 → ∀𝑦 ∈ ℝ+𝑗𝑍𝑘 ∈ (ℤ𝑗)(abs‘((𝐹𝑘) − inf(ran 𝐹, ℝ, < ))) < 𝑦)
116 fvex 6350 . . . . 5 (ℤ𝑀) ∈ V
1179, 116eqeltri 2823 . . . 4 𝑍 ∈ V
118 fex 6641 . . . 4 ((𝐹:𝑍⟶ℝ ∧ 𝑍 ∈ V) → 𝐹 ∈ V)
1191, 117, 118sylancl 697 . . 3 (𝜑𝐹 ∈ V)
120 eqidd 2749 . . 3 ((𝜑𝑘𝑍) → (𝐹𝑘) = (𝐹𝑘))
1211ffvelrnda 6510 . . . 4 ((𝜑𝑘𝑍) → (𝐹𝑘) ∈ ℝ)
122121recnd 10231 . . 3 ((𝜑𝑘𝑍) → (𝐹𝑘) ∈ ℂ)
1239, 6, 119, 120, 40, 122clim2c 14406 . 2 (𝜑 → (𝐹 ⇝ inf(ran 𝐹, ℝ, < ) ↔ ∀𝑦 ∈ ℝ+𝑗𝑍𝑘 ∈ (ℤ𝑗)(abs‘((𝐹𝑘) − inf(ran 𝐹, ℝ, < ))) < 𝑦))
124115, 123mpbird 247 1 (𝜑𝐹 ⇝ inf(ran 𝐹, ℝ, < ))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383  w3a 1072   = wceq 1620  wcel 2127  wne 2920  wral 3038  wrex 3039  Vcvv 3328  wss 3703  c0 4046   class class class wbr 4792  ran crn 5255   Fn wfn 6032  wf 6033  cfv 6037  (class class class)co 6801  infcinf 8500  cc 10097  cr 10098  1c1 10100   + caddc 10102   < clt 10237  cle 10238  cmin 10429  cz 11540  cuz 11850  +crp 11996  ...cfz 12490  abscabs 14144  cli 14385
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1859  ax-4 1874  ax-5 1976  ax-6 2042  ax-7 2078  ax-8 2129  ax-9 2136  ax-10 2156  ax-11 2171  ax-12 2184  ax-13 2379  ax-ext 2728  ax-rep 4911  ax-sep 4921  ax-nul 4929  ax-pow 4980  ax-pr 5043  ax-un 7102  ax-cnex 10155  ax-resscn 10156  ax-1cn 10157  ax-icn 10158  ax-addcl 10159  ax-addrcl 10160  ax-mulcl 10161  ax-mulrcl 10162  ax-mulcom 10163  ax-addass 10164  ax-mulass 10165  ax-distr 10166  ax-i2m1 10167  ax-1ne0 10168  ax-1rid 10169  ax-rnegex 10170  ax-rrecex 10171  ax-cnre 10172  ax-pre-lttri 10173  ax-pre-lttrn 10174  ax-pre-ltadd 10175  ax-pre-mulgt0 10176  ax-pre-sup 10177
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1073  df-3an 1074  df-tru 1623  df-ex 1842  df-nf 1847  df-sb 2035  df-eu 2599  df-mo 2600  df-clab 2735  df-cleq 2741  df-clel 2744  df-nfc 2879  df-ne 2921  df-nel 3024  df-ral 3043  df-rex 3044  df-reu 3045  df-rmo 3046  df-rab 3047  df-v 3330  df-sbc 3565  df-csb 3663  df-dif 3706  df-un 3708  df-in 3710  df-ss 3717  df-pss 3719  df-nul 4047  df-if 4219  df-pw 4292  df-sn 4310  df-pr 4312  df-tp 4314  df-op 4316  df-uni 4577  df-iun 4662  df-br 4793  df-opab 4853  df-mpt 4870  df-tr 4893  df-id 5162  df-eprel 5167  df-po 5175  df-so 5176  df-fr 5213  df-we 5215  df-xp 5260  df-rel 5261  df-cnv 5262  df-co 5263  df-dm 5264  df-rn 5265  df-res 5266  df-ima 5267  df-pred 5829  df-ord 5875  df-on 5876  df-lim 5877  df-suc 5878  df-iota 6000  df-fun 6039  df-fn 6040  df-f 6041  df-f1 6042  df-fo 6043  df-f1o 6044  df-fv 6045  df-riota 6762  df-ov 6804  df-oprab 6805  df-mpt2 6806  df-om 7219  df-1st 7321  df-2nd 7322  df-wrecs 7564  df-recs 7625  df-rdg 7663  df-er 7899  df-en 8110  df-dom 8111  df-sdom 8112  df-sup 8501  df-inf 8502  df-pnf 10239  df-mnf 10240  df-xr 10241  df-ltxr 10242  df-le 10243  df-sub 10431  df-neg 10432  df-div 10848  df-nn 11184  df-2 11242  df-3 11243  df-n0 11456  df-z 11541  df-uz 11851  df-rp 11997  df-fz 12491  df-seq 12967  df-exp 13026  df-cj 14009  df-re 14010  df-im 14011  df-sqrt 14145  df-abs 14146  df-clim 14389
This theorem is referenced by:  climinff  40315  climinf2lem  40410  supcnvlimsup  40444  stirlinglem13  40775
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