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Theorem climinf 40156
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 6091 . . . . . . . . . . . 12 (𝐹:𝑍⟶ℝ → ran 𝐹 ⊆ ℝ)
31, 2syl 17 . . . . . . . . . . 11 (𝜑 → ran 𝐹 ⊆ ℝ)
4 ffn 6083 . . . . . . . . . . . . . 14 (𝐹:𝑍⟶ℝ → 𝐹 Fn 𝑍)
51, 4syl 17 . . . . . . . . . . . . 13 (𝜑𝐹 Fn 𝑍)
6 climinf.4 . . . . . . . . . . . . . . 15 (𝜑𝑀 ∈ ℤ)
7 uzid 11740 . . . . . . . . . . . . . . 15 (𝑀 ∈ ℤ → 𝑀 ∈ (ℤ𝑀))
86, 7syl 17 . . . . . . . . . . . . . 14 (𝜑𝑀 ∈ (ℤ𝑀))
9 climinf.3 . . . . . . . . . . . . . 14 𝑍 = (ℤ𝑀)
108, 9syl6eleqr 2741 . . . . . . . . . . . . 13 (𝜑𝑀𝑍)
11 fnfvelrn 6396 . . . . . . . . . . . . 13 ((𝐹 Fn 𝑍𝑀𝑍) → (𝐹𝑀) ∈ ran 𝐹)
125, 10, 11syl2anc 694 . . . . . . . . . . . 12 (𝜑 → (𝐹𝑀) ∈ ran 𝐹)
13 ne0i 3954 . . . . . . . . . . . 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 4689 . . . . . . . . . . . . . . 15 (𝑦 = (𝐹𝑘) → (𝑥𝑦𝑥 ≤ (𝐹𝑘)))
1716ralrn 6402 . . . . . . . . . . . . . 14 (𝐹 Fn 𝑍 → (∀𝑦 ∈ ran 𝐹 𝑥𝑦 ↔ ∀𝑘𝑍 𝑥 ≤ (𝐹𝑘)))
1817rexbidv 3081 . . . . . . . . . . . . 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 1261 . . . . . . . . . 10 (𝜑 → (ran 𝐹 ⊆ ℝ ∧ ran 𝐹 ≠ ∅ ∧ ∃𝑥 ∈ ℝ ∀𝑦 ∈ ran 𝐹 𝑥𝑦))
2221adantr 480 . . . . . . . . 9 ((𝜑𝑦 ∈ ℝ+) → (ran 𝐹 ⊆ ℝ ∧ ran 𝐹 ≠ ∅ ∧ ∃𝑥 ∈ ℝ ∀𝑦 ∈ ran 𝐹 𝑥𝑦))
23 infrecl 11043 . . . . . . . . 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 476 . . . . . . . 8 ((𝜑𝑦 ∈ ℝ+) → 𝑦 ∈ ℝ+)
2624, 25ltaddrpd 11943 . . . . . . 7 ((𝜑𝑦 ∈ ℝ+) → inf(ran 𝐹, ℝ, < ) < (inf(ran 𝐹, ℝ, < ) + 𝑦))
27 rpre 11877 . . . . . . . . . 10 (𝑦 ∈ ℝ+𝑦 ∈ ℝ)
2827adantl 481 . . . . . . . . 9 ((𝜑𝑦 ∈ ℝ+) → 𝑦 ∈ ℝ)
2924, 28readdcld 10107 . . . . . . . 8 ((𝜑𝑦 ∈ ℝ+) → (inf(ran 𝐹, ℝ, < ) + 𝑦) ∈ ℝ)
30 infrglb 40140 . . . . . . . 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 694 . . . . . . 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 3636 . . . . . . . . . . 11 ((𝜑𝑘 ∈ ran 𝐹) → 𝑘 ∈ ℝ)
3433adantlr 751 . . . . . . . . . 10 (((𝜑𝑦 ∈ ℝ+) ∧ 𝑘 ∈ ran 𝐹) → 𝑘 ∈ ℝ)
3524adantr 480 . . . . . . . . . . 11 (((𝜑𝑦 ∈ ℝ+) ∧ 𝑘 ∈ ran 𝐹) → inf(ran 𝐹, ℝ, < ) ∈ ℝ)
3627ad2antlr 763 . . . . . . . . . . 11 (((𝜑𝑦 ∈ ℝ+) ∧ 𝑘 ∈ ran 𝐹) → 𝑦 ∈ ℝ)
3735, 36readdcld 10107 . . . . . . . . . 10 (((𝜑𝑦 ∈ ℝ+) ∧ 𝑘 ∈ ran 𝐹) → (inf(ran 𝐹, ℝ, < ) + 𝑦) ∈ ℝ)
3834, 37, 36ltsub1d 10674 . . . . . . . . 9 (((𝜑𝑦 ∈ ℝ+) ∧ 𝑘 ∈ ran 𝐹) → (𝑘 < (inf(ran 𝐹, ℝ, < ) + 𝑦) ↔ (𝑘𝑦) < ((inf(ran 𝐹, ℝ, < ) + 𝑦) − 𝑦)))
393, 14, 20, 23syl3anc 1366 . . . . . . . . . . . . 13 (𝜑 → inf(ran 𝐹, ℝ, < ) ∈ ℝ)
4039recnd 10106 . . . . . . . . . . . 12 (𝜑 → inf(ran 𝐹, ℝ, < ) ∈ ℂ)
4140ad2antrr 762 . . . . . . . . . . 11 (((𝜑𝑦 ∈ ℝ+) ∧ 𝑘 ∈ ran 𝐹) → inf(ran 𝐹, ℝ, < ) ∈ ℂ)
4236recnd 10106 . . . . . . . . . . 11 (((𝜑𝑦 ∈ ℝ+) ∧ 𝑘 ∈ ran 𝐹) → 𝑦 ∈ ℂ)
4341, 42pncand 10431 . . . . . . . . . 10 (((𝜑𝑦 ∈ ℝ+) ∧ 𝑘 ∈ ran 𝐹) → ((inf(ran 𝐹, ℝ, < ) + 𝑦) − 𝑦) = inf(ran 𝐹, ℝ, < ))
4443breq2d 4697 . . . . . . . . 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 3046 . . . . . 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 6697 . . . . . . . . 9 (𝑘 = (𝐹𝑗) → (𝑘𝑦) = ((𝐹𝑗) − 𝑦))
5049breq1d 4695 . . . . . . . 8 (𝑘 = (𝐹𝑗) → ((𝑘𝑦) < inf(ran 𝐹, ℝ, < ) ↔ ((𝐹𝑗) − 𝑦) < inf(ran 𝐹, ℝ, < )))
5150rexrn 6401 . . . . . . 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 500 . . . . 5 ((𝜑 ∧ ∃𝑘 ∈ ran 𝐹(𝑘𝑦) < inf(ran 𝐹, ℝ, < )) → ∃𝑗𝑍 ((𝐹𝑗) − 𝑦) < inf(ran 𝐹, ℝ, < ))
5448, 53syldan 486 . . . 4 ((𝜑𝑦 ∈ ℝ+) → ∃𝑗𝑍 ((𝐹𝑗) − 𝑦) < inf(ran 𝐹, ℝ, < ))
551adantr 480 . . . . . . . . . . 11 ((𝜑𝑦 ∈ ℝ+) → 𝐹:𝑍⟶ℝ)
569uztrn2 11743 . . . . . . . . . . 11 ((𝑗𝑍𝑘 ∈ (ℤ𝑗)) → 𝑘𝑍)
57 ffvelrn 6397 . . . . . . . . . . 11 ((𝐹:𝑍⟶ℝ ∧ 𝑘𝑍) → (𝐹𝑘) ∈ ℝ)
5855, 56, 57syl2an 493 . . . . . . . . . 10 (((𝜑𝑦 ∈ ℝ+) ∧ (𝑗𝑍𝑘 ∈ (ℤ𝑗))) → (𝐹𝑘) ∈ ℝ)
59 simpl 472 . . . . . . . . . . 11 ((𝑗𝑍𝑘 ∈ (ℤ𝑗)) → 𝑗𝑍)
60 ffvelrn 6397 . . . . . . . . . . 11 ((𝐹:𝑍⟶ℝ ∧ 𝑗𝑍) → (𝐹𝑗) ∈ ℝ)
6155, 59, 60syl2an 493 . . . . . . . . . 10 (((𝜑𝑦 ∈ ℝ+) ∧ (𝑗𝑍𝑘 ∈ (ℤ𝑗))) → (𝐹𝑗) ∈ ℝ)
6239ad2antrr 762 . . . . . . . . . 10 (((𝜑𝑦 ∈ ℝ+) ∧ (𝑗𝑍𝑘 ∈ (ℤ𝑗))) → inf(ran 𝐹, ℝ, < ) ∈ ℝ)
63 simprr 811 . . . . . . . . . . 11 (((𝜑𝑦 ∈ ℝ+) ∧ (𝑗𝑍𝑘 ∈ (ℤ𝑗))) → 𝑘 ∈ (ℤ𝑗))
64 fzssuz 12420 . . . . . . . . . . . . . 14 (𝑗...𝑘) ⊆ (ℤ𝑗)
65 uzss 11746 . . . . . . . . . . . . . . . . 17 (𝑗 ∈ (ℤ𝑀) → (ℤ𝑗) ⊆ (ℤ𝑀))
6665, 9syl6sseqr 3685 . . . . . . . . . . . . . . . 16 (𝑗 ∈ (ℤ𝑀) → (ℤ𝑗) ⊆ 𝑍)
6766, 9eleq2s 2748 . . . . . . . . . . . . . . 15 (𝑗𝑍 → (ℤ𝑗) ⊆ 𝑍)
6867ad2antrl 764 . . . . . . . . . . . . . 14 (((𝜑𝑦 ∈ ℝ+) ∧ (𝑗𝑍𝑘 ∈ (ℤ𝑗))) → (ℤ𝑗) ⊆ 𝑍)
6964, 68syl5ss 3647 . . . . . . . . . . . . 13 (((𝜑𝑦 ∈ ℝ+) ∧ (𝑗𝑍𝑘 ∈ (ℤ𝑗))) → (𝑗...𝑘) ⊆ 𝑍)
70 ffvelrn 6397 . . . . . . . . . . . . . . . 16 ((𝐹:𝑍⟶ℝ ∧ 𝑛𝑍) → (𝐹𝑛) ∈ ℝ)
7170ralrimiva 2995 . . . . . . . . . . . . . . 15 (𝐹:𝑍⟶ℝ → ∀𝑛𝑍 (𝐹𝑛) ∈ ℝ)
721, 71syl 17 . . . . . . . . . . . . . 14 (𝜑 → ∀𝑛𝑍 (𝐹𝑛) ∈ ℝ)
7372ad2antrr 762 . . . . . . . . . . . . 13 (((𝜑𝑦 ∈ ℝ+) ∧ (𝑗𝑍𝑘 ∈ (ℤ𝑗))) → ∀𝑛𝑍 (𝐹𝑛) ∈ ℝ)
74 ssralv 3699 . . . . . . . . . . . . 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 2961 . . . . . . . . . . 11 ((((𝜑𝑦 ∈ ℝ+) ∧ (𝑗𝑍𝑘 ∈ (ℤ𝑗))) ∧ 𝑛 ∈ (𝑗...𝑘)) → (𝐹𝑛) ∈ ℝ)
77 fzssuz 12420 . . . . . . . . . . . . . 14 (𝑗...(𝑘 − 1)) ⊆ (ℤ𝑗)
7877, 68syl5ss 3647 . . . . . . . . . . . . 13 (((𝜑𝑦 ∈ ℝ+) ∧ (𝑗𝑍𝑘 ∈ (ℤ𝑗))) → (𝑗...(𝑘 − 1)) ⊆ 𝑍)
7978sselda 3636 . . . . . . . . . . . 12 ((((𝜑𝑦 ∈ ℝ+) ∧ (𝑗𝑍𝑘 ∈ (ℤ𝑗))) ∧ 𝑛 ∈ (𝑗...(𝑘 − 1))) → 𝑛𝑍)
80 climinf.6 . . . . . . . . . . . . . . 15 ((𝜑𝑘𝑍) → (𝐹‘(𝑘 + 1)) ≤ (𝐹𝑘))
8180ralrimiva 2995 . . . . . . . . . . . . . 14 (𝜑 → ∀𝑘𝑍 (𝐹‘(𝑘 + 1)) ≤ (𝐹𝑘))
8281ad2antrr 762 . . . . . . . . . . . . 13 (((𝜑𝑦 ∈ ℝ+) ∧ (𝑗𝑍𝑘 ∈ (ℤ𝑗))) → ∀𝑘𝑍 (𝐹‘(𝑘 + 1)) ≤ (𝐹𝑘))
83 oveq1 6697 . . . . . . . . . . . . . . . 16 (𝑘 = 𝑛 → (𝑘 + 1) = (𝑛 + 1))
8483fveq2d 6233 . . . . . . . . . . . . . . 15 (𝑘 = 𝑛 → (𝐹‘(𝑘 + 1)) = (𝐹‘(𝑛 + 1)))
85 fveq2 6229 . . . . . . . . . . . . . . 15 (𝑘 = 𝑛 → (𝐹𝑘) = (𝐹𝑛))
8684, 85breq12d 4698 . . . . . . . . . . . . . 14 (𝑘 = 𝑛 → ((𝐹‘(𝑘 + 1)) ≤ (𝐹𝑘) ↔ (𝐹‘(𝑛 + 1)) ≤ (𝐹𝑛)))
8786rspccva 3339 . . . . . . . . . . . . 13 ((∀𝑘𝑍 (𝐹‘(𝑘 + 1)) ≤ (𝐹𝑘) ∧ 𝑛𝑍) → (𝐹‘(𝑛 + 1)) ≤ (𝐹𝑛))
8882, 87sylan 487 . . . . . . . . . . . 12 ((((𝜑𝑦 ∈ ℝ+) ∧ (𝑗𝑍𝑘 ∈ (ℤ𝑗))) ∧ 𝑛𝑍) → (𝐹‘(𝑛 + 1)) ≤ (𝐹𝑛))
8979, 88syldan 486 . . . . . . . . . . 11 ((((𝜑𝑦 ∈ ℝ+) ∧ (𝑗𝑍𝑘 ∈ (ℤ𝑗))) ∧ 𝑛 ∈ (𝑗...(𝑘 − 1))) → (𝐹‘(𝑛 + 1)) ≤ (𝐹𝑛))
9063, 76, 89monoord2 12872 . . . . . . . . . 10 (((𝜑𝑦 ∈ ℝ+) ∧ (𝑗𝑍𝑘 ∈ (ℤ𝑗))) → (𝐹𝑘) ≤ (𝐹𝑗))
9158, 61, 62, 90lesub1dd 10681 . . . . . . . . 9 (((𝜑𝑦 ∈ ℝ+) ∧ (𝑗𝑍𝑘 ∈ (ℤ𝑗))) → ((𝐹𝑘) − inf(ran 𝐹, ℝ, < )) ≤ ((𝐹𝑗) − inf(ran 𝐹, ℝ, < )))
9258, 62resubcld 10496 . . . . . . . . . 10 (((𝜑𝑦 ∈ ℝ+) ∧ (𝑗𝑍𝑘 ∈ (ℤ𝑗))) → ((𝐹𝑘) − inf(ran 𝐹, ℝ, < )) ∈ ℝ)
9361, 62resubcld 10496 . . . . . . . . . 10 (((𝜑𝑦 ∈ ℝ+) ∧ (𝑗𝑍𝑘 ∈ (ℤ𝑗))) → ((𝐹𝑗) − inf(ran 𝐹, ℝ, < )) ∈ ℝ)
9427ad2antlr 763 . . . . . . . . . 10 (((𝜑𝑦 ∈ ℝ+) ∧ (𝑗𝑍𝑘 ∈ (ℤ𝑗))) → 𝑦 ∈ ℝ)
95 lelttr 10166 . . . . . . . . . 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 1366 . . . . . . . . 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 711 . . . . . . . 8 (((𝜑𝑦 ∈ ℝ+) ∧ (𝑗𝑍𝑘 ∈ (ℤ𝑗))) → (((𝐹𝑗) − inf(ran 𝐹, ℝ, < )) < 𝑦 → ((𝐹𝑘) − inf(ran 𝐹, ℝ, < )) < 𝑦))
98 ltsub23 10546 . . . . . . . . 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 1366 . . . . . . . 8 (((𝜑𝑦 ∈ ℝ+) ∧ (𝑗𝑍𝑘 ∈ (ℤ𝑗))) → (((𝐹𝑗) − 𝑦) < inf(ran 𝐹, ℝ, < ) ↔ ((𝐹𝑗) − inf(ran 𝐹, ℝ, < )) < 𝑦))
1003ad2antrr 762 . . . . . . . . . . 11 (((𝜑𝑦 ∈ ℝ+) ∧ (𝑗𝑍𝑘 ∈ (ℤ𝑗))) → ran 𝐹 ⊆ ℝ)
1015adantr 480 . . . . . . . . . . . 12 ((𝜑𝑦 ∈ ℝ+) → 𝐹 Fn 𝑍)
102 fnfvelrn 6396 . . . . . . . . . . . 12 ((𝐹 Fn 𝑍𝑘𝑍) → (𝐹𝑘) ∈ ran 𝐹)
103101, 56, 102syl2an 493 . . . . . . . . . . 11 (((𝜑𝑦 ∈ ℝ+) ∧ (𝑗𝑍𝑘 ∈ (ℤ𝑗))) → (𝐹𝑘) ∈ ran 𝐹)
104100, 103sseldd 3637 . . . . . . . . . 10 (((𝜑𝑦 ∈ ℝ+) ∧ (𝑗𝑍𝑘 ∈ (ℤ𝑗))) → (𝐹𝑘) ∈ ℝ)
10520ad2antrr 762 . . . . . . . . . . 11 (((𝜑𝑦 ∈ ℝ+) ∧ (𝑗𝑍𝑘 ∈ (ℤ𝑗))) → ∃𝑥 ∈ ℝ ∀𝑦 ∈ ran 𝐹 𝑥𝑦)
106 infrelb 11046 . . . . . . . . . . 11 ((ran 𝐹 ⊆ ℝ ∧ ∃𝑥 ∈ ℝ ∀𝑦 ∈ ran 𝐹 𝑥𝑦 ∧ (𝐹𝑘) ∈ ran 𝐹) → inf(ran 𝐹, ℝ, < ) ≤ (𝐹𝑘))
107100, 105, 103, 106syl3anc 1366 . . . . . . . . . 10 (((𝜑𝑦 ∈ ℝ+) ∧ (𝑗𝑍𝑘 ∈ (ℤ𝑗))) → inf(ran 𝐹, ℝ, < ) ≤ (𝐹𝑘))
10862, 104, 107abssubge0d 14214 . . . . . . . . 9 (((𝜑𝑦 ∈ ℝ+) ∧ (𝑗𝑍𝑘 ∈ (ℤ𝑗))) → (abs‘((𝐹𝑘) − inf(ran 𝐹, ℝ, < ))) = ((𝐹𝑘) − inf(ran 𝐹, ℝ, < )))
109108breq1d 4695 . . . . . . . 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 681 . . . . . 6 ((((𝜑𝑦 ∈ ℝ+) ∧ 𝑗𝑍) ∧ 𝑘 ∈ (ℤ𝑗)) → (((𝐹𝑗) − 𝑦) < inf(ran 𝐹, ℝ, < ) → (abs‘((𝐹𝑘) − inf(ran 𝐹, ℝ, < ))) < 𝑦))
112111ralrimdva 2998 . . . . 5 (((𝜑𝑦 ∈ ℝ+) ∧ 𝑗𝑍) → (((𝐹𝑗) − 𝑦) < inf(ran 𝐹, ℝ, < ) → ∀𝑘 ∈ (ℤ𝑗)(abs‘((𝐹𝑘) − inf(ran 𝐹, ℝ, < ))) < 𝑦))
113112reximdva 3046 . . . 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 2995 . 2 (𝜑 → ∀𝑦 ∈ ℝ+𝑗𝑍𝑘 ∈ (ℤ𝑗)(abs‘((𝐹𝑘) − inf(ran 𝐹, ℝ, < ))) < 𝑦)
116 fvex 6239 . . . . 5 (ℤ𝑀) ∈ V
1179, 116eqeltri 2726 . . . 4 𝑍 ∈ V
118 fex 6530 . . . 4 ((𝐹:𝑍⟶ℝ ∧ 𝑍 ∈ V) → 𝐹 ∈ V)
1191, 117, 118sylancl 695 . . 3 (𝜑𝐹 ∈ V)
120 eqidd 2652 . . 3 ((𝜑𝑘𝑍) → (𝐹𝑘) = (𝐹𝑘))
1211ffvelrnda 6399 . . . 4 ((𝜑𝑘𝑍) → (𝐹𝑘) ∈ ℝ)
122121recnd 10106 . . 3 ((𝜑𝑘𝑍) → (𝐹𝑘) ∈ ℂ)
1239, 6, 119, 120, 40, 122clim2c 14280 . 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 1054   = wceq 1523  wcel 2030  wne 2823  wral 2941  wrex 2942  Vcvv 3231  wss 3607  c0 3948   class class class wbr 4685  ran crn 5144   Fn wfn 5921  wf 5922  cfv 5926  (class class class)co 6690  infcinf 8388  cc 9972  cr 9973  1c1 9975   + caddc 9977   < clt 10112  cle 10113  cmin 10304  cz 11415  cuz 11725  +crp 11870  ...cfz 12364  abscabs 14018  cli 14259
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-rep 4804  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991  ax-cnex 10030  ax-resscn 10031  ax-1cn 10032  ax-icn 10033  ax-addcl 10034  ax-addrcl 10035  ax-mulcl 10036  ax-mulrcl 10037  ax-mulcom 10038  ax-addass 10039  ax-mulass 10040  ax-distr 10041  ax-i2m1 10042  ax-1ne0 10043  ax-1rid 10044  ax-rnegex 10045  ax-rrecex 10046  ax-cnre 10047  ax-pre-lttri 10048  ax-pre-lttrn 10049  ax-pre-ltadd 10050  ax-pre-mulgt0 10051  ax-pre-sup 10052
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-nel 2927  df-ral 2946  df-rex 2947  df-reu 2948  df-rmo 2949  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-pss 3623  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-tp 4215  df-op 4217  df-uni 4469  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-tr 4786  df-id 5053  df-eprel 5058  df-po 5064  df-so 5065  df-fr 5102  df-we 5104  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-pred 5718  df-ord 5764  df-on 5765  df-lim 5766  df-suc 5767  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-riota 6651  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-om 7108  df-1st 7210  df-2nd 7211  df-wrecs 7452  df-recs 7513  df-rdg 7551  df-er 7787  df-en 7998  df-dom 7999  df-sdom 8000  df-sup 8389  df-inf 8390  df-pnf 10114  df-mnf 10115  df-xr 10116  df-ltxr 10117  df-le 10118  df-sub 10306  df-neg 10307  df-div 10723  df-nn 11059  df-2 11117  df-3 11118  df-n0 11331  df-z 11416  df-uz 11726  df-rp 11871  df-fz 12365  df-seq 12842  df-exp 12901  df-cj 13883  df-re 13884  df-im 13885  df-sqrt 14019  df-abs 14020  df-clim 14263
This theorem is referenced by:  climinff  40161  climinf2lem  40256  supcnvlimsup  40290  stirlinglem13  40621
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