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Theorem climinf 38673
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 5966 . . . . . . . . . . . 12 (𝐹:𝑍⟶ℝ → ran 𝐹 ⊆ ℝ)
31, 2syl 17 . . . . . . . . . . 11 (𝜑 → ran 𝐹 ⊆ ℝ)
4 ffn 5958 . . . . . . . . . . . . . 14 (𝐹:𝑍⟶ℝ → 𝐹 Fn 𝑍)
51, 4syl 17 . . . . . . . . . . . . 13 (𝜑𝐹 Fn 𝑍)
6 climinf.4 . . . . . . . . . . . . . . 15 (𝜑𝑀 ∈ ℤ)
7 uzid 11578 . . . . . . . . . . . . . . 15 (𝑀 ∈ ℤ → 𝑀 ∈ (ℤ𝑀))
86, 7syl 17 . . . . . . . . . . . . . 14 (𝜑𝑀 ∈ (ℤ𝑀))
9 climinf.3 . . . . . . . . . . . . . 14 𝑍 = (ℤ𝑀)
108, 9syl6eleqr 2699 . . . . . . . . . . . . 13 (𝜑𝑀𝑍)
11 fnfvelrn 6264 . . . . . . . . . . . . 13 ((𝐹 Fn 𝑍𝑀𝑍) → (𝐹𝑀) ∈ ran 𝐹)
125, 10, 11syl2anc 691 . . . . . . . . . . . 12 (𝜑 → (𝐹𝑀) ∈ ran 𝐹)
13 ne0i 3880 . . . . . . . . . . . 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 4587 . . . . . . . . . . . . . . 15 (𝑦 = (𝐹𝑘) → (𝑥𝑦𝑥 ≤ (𝐹𝑘)))
1716ralrn 6270 . . . . . . . . . . . . . 14 (𝐹 Fn 𝑍 → (∀𝑦 ∈ ran 𝐹 𝑥𝑦 ↔ ∀𝑘𝑍 𝑥 ≤ (𝐹𝑘)))
1817rexbidv 3034 . . . . . . . . . . . . 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 246 . . . . . . . . . . 11 (𝜑 → ∃𝑥 ∈ ℝ ∀𝑦 ∈ ran 𝐹 𝑥𝑦)
213, 14, 203jca 1235 . . . . . . . . . 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 10882 . . . . . . . . 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 11781 . . . . . . 7 ((𝜑𝑦 ∈ ℝ+) → inf(ran 𝐹, ℝ, < ) < (inf(ran 𝐹, ℝ, < ) + 𝑦))
27 rpre 11715 . . . . . . . . . 10 (𝑦 ∈ ℝ+𝑦 ∈ ℝ)
2827adantl 481 . . . . . . . . 9 ((𝜑𝑦 ∈ ℝ+) → 𝑦 ∈ ℝ)
2924, 28readdcld 9948 . . . . . . . 8 ((𝜑𝑦 ∈ ℝ+) → (inf(ran 𝐹, ℝ, < ) + 𝑦) ∈ ℝ)
30 infrglb 38657 . . . . . . . 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 691 . . . . . . 7 ((𝜑𝑦 ∈ ℝ+) → (inf(ran 𝐹, ℝ, < ) < (inf(ran 𝐹, ℝ, < ) + 𝑦) ↔ ∃𝑘 ∈ ran 𝐹 𝑘 < (inf(ran 𝐹, ℝ, < ) + 𝑦)))
3226, 31mpbid 221 . . . . . 6 ((𝜑𝑦 ∈ ℝ+) → ∃𝑘 ∈ ran 𝐹 𝑘 < (inf(ran 𝐹, ℝ, < ) + 𝑦))
333sselda 3568 . . . . . . . . . . 11 ((𝜑𝑘 ∈ ran 𝐹) → 𝑘 ∈ ℝ)
3433adantlr 747 . . . . . . . . . 10 (((𝜑𝑦 ∈ ℝ+) ∧ 𝑘 ∈ ran 𝐹) → 𝑘 ∈ ℝ)
3524adantr 480 . . . . . . . . . . 11 (((𝜑𝑦 ∈ ℝ+) ∧ 𝑘 ∈ ran 𝐹) → inf(ran 𝐹, ℝ, < ) ∈ ℝ)
3627ad2antlr 759 . . . . . . . . . . 11 (((𝜑𝑦 ∈ ℝ+) ∧ 𝑘 ∈ ran 𝐹) → 𝑦 ∈ ℝ)
3735, 36readdcld 9948 . . . . . . . . . 10 (((𝜑𝑦 ∈ ℝ+) ∧ 𝑘 ∈ ran 𝐹) → (inf(ran 𝐹, ℝ, < ) + 𝑦) ∈ ℝ)
3834, 37, 36ltsub1d 10515 . . . . . . . . 9 (((𝜑𝑦 ∈ ℝ+) ∧ 𝑘 ∈ ran 𝐹) → (𝑘 < (inf(ran 𝐹, ℝ, < ) + 𝑦) ↔ (𝑘𝑦) < ((inf(ran 𝐹, ℝ, < ) + 𝑦) − 𝑦)))
393, 14, 20, 23syl3anc 1318 . . . . . . . . . . . . 13 (𝜑 → inf(ran 𝐹, ℝ, < ) ∈ ℝ)
4039recnd 9947 . . . . . . . . . . . 12 (𝜑 → inf(ran 𝐹, ℝ, < ) ∈ ℂ)
4140ad2antrr 758 . . . . . . . . . . 11 (((𝜑𝑦 ∈ ℝ+) ∧ 𝑘 ∈ ran 𝐹) → inf(ran 𝐹, ℝ, < ) ∈ ℂ)
4236recnd 9947 . . . . . . . . . . 11 (((𝜑𝑦 ∈ ℝ+) ∧ 𝑘 ∈ ran 𝐹) → 𝑦 ∈ ℂ)
4341, 42pncand 10272 . . . . . . . . . 10 (((𝜑𝑦 ∈ ℝ+) ∧ 𝑘 ∈ ran 𝐹) → ((inf(ran 𝐹, ℝ, < ) + 𝑦) − 𝑦) = inf(ran 𝐹, ℝ, < ))
4443breq2d 4595 . . . . . . . . 9 (((𝜑𝑦 ∈ ℝ+) ∧ 𝑘 ∈ ran 𝐹) → ((𝑘𝑦) < ((inf(ran 𝐹, ℝ, < ) + 𝑦) − 𝑦) ↔ (𝑘𝑦) < inf(ran 𝐹, ℝ, < )))
4538, 44bitrd 267 . . . . . . . 8 (((𝜑𝑦 ∈ ℝ+) ∧ 𝑘 ∈ ran 𝐹) → (𝑘 < (inf(ran 𝐹, ℝ, < ) + 𝑦) ↔ (𝑘𝑦) < inf(ran 𝐹, ℝ, < )))
4645biimpd 218 . . . . . . 7 (((𝜑𝑦 ∈ ℝ+) ∧ 𝑘 ∈ ran 𝐹) → (𝑘 < (inf(ran 𝐹, ℝ, < ) + 𝑦) → (𝑘𝑦) < inf(ran 𝐹, ℝ, < )))
4746reximdva 3000 . . . . . 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 6556 . . . . . . . . 9 (𝑘 = (𝐹𝑗) → (𝑘𝑦) = ((𝐹𝑗) − 𝑦))
5049breq1d 4593 . . . . . . . 8 (𝑘 = (𝐹𝑗) → ((𝑘𝑦) < inf(ran 𝐹, ℝ, < ) ↔ ((𝐹𝑗) − 𝑦) < inf(ran 𝐹, ℝ, < )))
5150rexrn 6269 . . . . . . 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 11581 . . . . . . . . . . 11 ((𝑗𝑍𝑘 ∈ (ℤ𝑗)) → 𝑘𝑍)
57 ffvelrn 6265 . . . . . . . . . . 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 6265 . . . . . . . . . . 11 ((𝐹:𝑍⟶ℝ ∧ 𝑗𝑍) → (𝐹𝑗) ∈ ℝ)
6155, 59, 60syl2an 493 . . . . . . . . . 10 (((𝜑𝑦 ∈ ℝ+) ∧ (𝑗𝑍𝑘 ∈ (ℤ𝑗))) → (𝐹𝑗) ∈ ℝ)
6239ad2antrr 758 . . . . . . . . . 10 (((𝜑𝑦 ∈ ℝ+) ∧ (𝑗𝑍𝑘 ∈ (ℤ𝑗))) → inf(ran 𝐹, ℝ, < ) ∈ ℝ)
63 simprr 792 . . . . . . . . . . 11 (((𝜑𝑦 ∈ ℝ+) ∧ (𝑗𝑍𝑘 ∈ (ℤ𝑗))) → 𝑘 ∈ (ℤ𝑗))
64 fzssuz 12253 . . . . . . . . . . . . . 14 (𝑗...𝑘) ⊆ (ℤ𝑗)
65 uzss 11584 . . . . . . . . . . . . . . . . 17 (𝑗 ∈ (ℤ𝑀) → (ℤ𝑗) ⊆ (ℤ𝑀))
6665, 9syl6sseqr 3615 . . . . . . . . . . . . . . . 16 (𝑗 ∈ (ℤ𝑀) → (ℤ𝑗) ⊆ 𝑍)
6766, 9eleq2s 2706 . . . . . . . . . . . . . . 15 (𝑗𝑍 → (ℤ𝑗) ⊆ 𝑍)
6867ad2antrl 760 . . . . . . . . . . . . . 14 (((𝜑𝑦 ∈ ℝ+) ∧ (𝑗𝑍𝑘 ∈ (ℤ𝑗))) → (ℤ𝑗) ⊆ 𝑍)
6964, 68syl5ss 3579 . . . . . . . . . . . . 13 (((𝜑𝑦 ∈ ℝ+) ∧ (𝑗𝑍𝑘 ∈ (ℤ𝑗))) → (𝑗...𝑘) ⊆ 𝑍)
70 ffvelrn 6265 . . . . . . . . . . . . . . . 16 ((𝐹:𝑍⟶ℝ ∧ 𝑛𝑍) → (𝐹𝑛) ∈ ℝ)
7170ralrimiva 2949 . . . . . . . . . . . . . . 15 (𝐹:𝑍⟶ℝ → ∀𝑛𝑍 (𝐹𝑛) ∈ ℝ)
721, 71syl 17 . . . . . . . . . . . . . 14 (𝜑 → ∀𝑛𝑍 (𝐹𝑛) ∈ ℝ)
7372ad2antrr 758 . . . . . . . . . . . . 13 (((𝜑𝑦 ∈ ℝ+) ∧ (𝑗𝑍𝑘 ∈ (ℤ𝑗))) → ∀𝑛𝑍 (𝐹𝑛) ∈ ℝ)
74 ssralv 3629 . . . . . . . . . . . . 13 ((𝑗...𝑘) ⊆ 𝑍 → (∀𝑛𝑍 (𝐹𝑛) ∈ ℝ → ∀𝑛 ∈ (𝑗...𝑘)(𝐹𝑛) ∈ ℝ))
7569, 73, 74sylc 63 . . . . . . . . . . . 12 (((𝜑𝑦 ∈ ℝ+) ∧ (𝑗𝑍𝑘 ∈ (ℤ𝑗))) → ∀𝑛 ∈ (𝑗...𝑘)(𝐹𝑛) ∈ ℝ)
7675r19.21bi 2916 . . . . . . . . . . 11 ((((𝜑𝑦 ∈ ℝ+) ∧ (𝑗𝑍𝑘 ∈ (ℤ𝑗))) ∧ 𝑛 ∈ (𝑗...𝑘)) → (𝐹𝑛) ∈ ℝ)
77 fzssuz 12253 . . . . . . . . . . . . . 14 (𝑗...(𝑘 − 1)) ⊆ (ℤ𝑗)
7877, 68syl5ss 3579 . . . . . . . . . . . . 13 (((𝜑𝑦 ∈ ℝ+) ∧ (𝑗𝑍𝑘 ∈ (ℤ𝑗))) → (𝑗...(𝑘 − 1)) ⊆ 𝑍)
7978sselda 3568 . . . . . . . . . . . 12 ((((𝜑𝑦 ∈ ℝ+) ∧ (𝑗𝑍𝑘 ∈ (ℤ𝑗))) ∧ 𝑛 ∈ (𝑗...(𝑘 − 1))) → 𝑛𝑍)
80 climinf.6 . . . . . . . . . . . . . . 15 ((𝜑𝑘𝑍) → (𝐹‘(𝑘 + 1)) ≤ (𝐹𝑘))
8180ralrimiva 2949 . . . . . . . . . . . . . 14 (𝜑 → ∀𝑘𝑍 (𝐹‘(𝑘 + 1)) ≤ (𝐹𝑘))
8281ad2antrr 758 . . . . . . . . . . . . 13 (((𝜑𝑦 ∈ ℝ+) ∧ (𝑗𝑍𝑘 ∈ (ℤ𝑗))) → ∀𝑘𝑍 (𝐹‘(𝑘 + 1)) ≤ (𝐹𝑘))
83 oveq1 6556 . . . . . . . . . . . . . . . 16 (𝑘 = 𝑛 → (𝑘 + 1) = (𝑛 + 1))
8483fveq2d 6107 . . . . . . . . . . . . . . 15 (𝑘 = 𝑛 → (𝐹‘(𝑘 + 1)) = (𝐹‘(𝑛 + 1)))
85 fveq2 6103 . . . . . . . . . . . . . . 15 (𝑘 = 𝑛 → (𝐹𝑘) = (𝐹𝑛))
8684, 85breq12d 4596 . . . . . . . . . . . . . 14 (𝑘 = 𝑛 → ((𝐹‘(𝑘 + 1)) ≤ (𝐹𝑘) ↔ (𝐹‘(𝑛 + 1)) ≤ (𝐹𝑛)))
8786rspccva 3281 . . . . . . . . . . . . 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 12694 . . . . . . . . . 10 (((𝜑𝑦 ∈ ℝ+) ∧ (𝑗𝑍𝑘 ∈ (ℤ𝑗))) → (𝐹𝑘) ≤ (𝐹𝑗))
9158, 61, 62, 90lesub1dd 10522 . . . . . . . . 9 (((𝜑𝑦 ∈ ℝ+) ∧ (𝑗𝑍𝑘 ∈ (ℤ𝑗))) → ((𝐹𝑘) − inf(ran 𝐹, ℝ, < )) ≤ ((𝐹𝑗) − inf(ran 𝐹, ℝ, < )))
9258, 62resubcld 10337 . . . . . . . . . 10 (((𝜑𝑦 ∈ ℝ+) ∧ (𝑗𝑍𝑘 ∈ (ℤ𝑗))) → ((𝐹𝑘) − inf(ran 𝐹, ℝ, < )) ∈ ℝ)
9361, 62resubcld 10337 . . . . . . . . . 10 (((𝜑𝑦 ∈ ℝ+) ∧ (𝑗𝑍𝑘 ∈ (ℤ𝑗))) → ((𝐹𝑗) − inf(ran 𝐹, ℝ, < )) ∈ ℝ)
9427ad2antlr 759 . . . . . . . . . 10 (((𝜑𝑦 ∈ ℝ+) ∧ (𝑗𝑍𝑘 ∈ (ℤ𝑗))) → 𝑦 ∈ ℝ)
95 lelttr 10007 . . . . . . . . . 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 1318 . . . . . . . . 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 707 . . . . . . . 8 (((𝜑𝑦 ∈ ℝ+) ∧ (𝑗𝑍𝑘 ∈ (ℤ𝑗))) → (((𝐹𝑗) − inf(ran 𝐹, ℝ, < )) < 𝑦 → ((𝐹𝑘) − inf(ran 𝐹, ℝ, < )) < 𝑦))
98 ltsub23 10387 . . . . . . . . 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 1318 . . . . . . . 8 (((𝜑𝑦 ∈ ℝ+) ∧ (𝑗𝑍𝑘 ∈ (ℤ𝑗))) → (((𝐹𝑗) − 𝑦) < inf(ran 𝐹, ℝ, < ) ↔ ((𝐹𝑗) − inf(ran 𝐹, ℝ, < )) < 𝑦))
1003ad2antrr 758 . . . . . . . . . . 11 (((𝜑𝑦 ∈ ℝ+) ∧ (𝑗𝑍𝑘 ∈ (ℤ𝑗))) → ran 𝐹 ⊆ ℝ)
1015adantr 480 . . . . . . . . . . . 12 ((𝜑𝑦 ∈ ℝ+) → 𝐹 Fn 𝑍)
102 fnfvelrn 6264 . . . . . . . . . . . 12 ((𝐹 Fn 𝑍𝑘𝑍) → (𝐹𝑘) ∈ ran 𝐹)
103101, 56, 102syl2an 493 . . . . . . . . . . 11 (((𝜑𝑦 ∈ ℝ+) ∧ (𝑗𝑍𝑘 ∈ (ℤ𝑗))) → (𝐹𝑘) ∈ ran 𝐹)
104100, 103sseldd 3569 . . . . . . . . . 10 (((𝜑𝑦 ∈ ℝ+) ∧ (𝑗𝑍𝑘 ∈ (ℤ𝑗))) → (𝐹𝑘) ∈ ℝ)
10520ad2antrr 758 . . . . . . . . . . 11 (((𝜑𝑦 ∈ ℝ+) ∧ (𝑗𝑍𝑘 ∈ (ℤ𝑗))) → ∃𝑥 ∈ ℝ ∀𝑦 ∈ ran 𝐹 𝑥𝑦)
106 infrelb 10885 . . . . . . . . . . 11 ((ran 𝐹 ⊆ ℝ ∧ ∃𝑥 ∈ ℝ ∀𝑦 ∈ ran 𝐹 𝑥𝑦 ∧ (𝐹𝑘) ∈ ran 𝐹) → inf(ran 𝐹, ℝ, < ) ≤ (𝐹𝑘))
107100, 105, 103, 106syl3anc 1318 . . . . . . . . . 10 (((𝜑𝑦 ∈ ℝ+) ∧ (𝑗𝑍𝑘 ∈ (ℤ𝑗))) → inf(ran 𝐹, ℝ, < ) ≤ (𝐹𝑘))
10862, 104, 107abssubge0d 14018 . . . . . . . . 9 (((𝜑𝑦 ∈ ℝ+) ∧ (𝑗𝑍𝑘 ∈ (ℤ𝑗))) → (abs‘((𝐹𝑘) − inf(ran 𝐹, ℝ, < ))) = ((𝐹𝑘) − inf(ran 𝐹, ℝ, < )))
109108breq1d 4593 . . . . . . . 8 (((𝜑𝑦 ∈ ℝ+) ∧ (𝑗𝑍𝑘 ∈ (ℤ𝑗))) → ((abs‘((𝐹𝑘) − inf(ran 𝐹, ℝ, < ))) < 𝑦 ↔ ((𝐹𝑘) − inf(ran 𝐹, ℝ, < )) < 𝑦))
11097, 99, 1093imtr4d 282 . . . . . . 7 (((𝜑𝑦 ∈ ℝ+) ∧ (𝑗𝑍𝑘 ∈ (ℤ𝑗))) → (((𝐹𝑗) − 𝑦) < inf(ran 𝐹, ℝ, < ) → (abs‘((𝐹𝑘) − inf(ran 𝐹, ℝ, < ))) < 𝑦))
111110anassrs 678 . . . . . 6 ((((𝜑𝑦 ∈ ℝ+) ∧ 𝑗𝑍) ∧ 𝑘 ∈ (ℤ𝑗)) → (((𝐹𝑗) − 𝑦) < inf(ran 𝐹, ℝ, < ) → (abs‘((𝐹𝑘) − inf(ran 𝐹, ℝ, < ))) < 𝑦))
112111ralrimdva 2952 . . . . 5 (((𝜑𝑦 ∈ ℝ+) ∧ 𝑗𝑍) → (((𝐹𝑗) − 𝑦) < inf(ran 𝐹, ℝ, < ) → ∀𝑘 ∈ (ℤ𝑗)(abs‘((𝐹𝑘) − inf(ran 𝐹, ℝ, < ))) < 𝑦))
113112reximdva 3000 . . . 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 2949 . 2 (𝜑 → ∀𝑦 ∈ ℝ+𝑗𝑍𝑘 ∈ (ℤ𝑗)(abs‘((𝐹𝑘) − inf(ran 𝐹, ℝ, < ))) < 𝑦)
116 fvex 6113 . . . . 5 (ℤ𝑀) ∈ V
1179, 116eqeltri 2684 . . . 4 𝑍 ∈ V
118 fex 6394 . . . 4 ((𝐹:𝑍⟶ℝ ∧ 𝑍 ∈ V) → 𝐹 ∈ V)
1191, 117, 118sylancl 693 . . 3 (𝜑𝐹 ∈ V)
120 eqidd 2611 . . 3 ((𝜑𝑘𝑍) → (𝐹𝑘) = (𝐹𝑘))
1211ffvelrnda 6267 . . . 4 ((𝜑𝑘𝑍) → (𝐹𝑘) ∈ ℝ)
122121recnd 9947 . . 3 ((𝜑𝑘𝑍) → (𝐹𝑘) ∈ ℂ)
1239, 6, 119, 120, 40, 122clim2c 14084 . 2 (𝜑 → (𝐹 ⇝ inf(ran 𝐹, ℝ, < ) ↔ ∀𝑦 ∈ ℝ+𝑗𝑍𝑘 ∈ (ℤ𝑗)(abs‘((𝐹𝑘) − inf(ran 𝐹, ℝ, < ))) < 𝑦))
124115, 123mpbird 246 1 (𝜑𝐹 ⇝ inf(ran 𝐹, ℝ, < ))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 195  wa 383  w3a 1031   = wceq 1475  wcel 1977  wne 2780  wral 2896  wrex 2897  Vcvv 3173  wss 3540  c0 3874   class class class wbr 4583  ran crn 5039   Fn wfn 5799  wf 5800  cfv 5804  (class class class)co 6549  infcinf 8230  cc 9813  cr 9814  1c1 9816   + caddc 9818   < clt 9953  cle 9954  cmin 10145  cz 11254  cuz 11563  +crp 11708  ...cfz 12197  abscabs 13822  cli 14063
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-cnex 9871  ax-resscn 9872  ax-1cn 9873  ax-icn 9874  ax-addcl 9875  ax-addrcl 9876  ax-mulcl 9877  ax-mulrcl 9878  ax-mulcom 9879  ax-addass 9880  ax-mulass 9881  ax-distr 9882  ax-i2m1 9883  ax-1ne0 9884  ax-1rid 9885  ax-rnegex 9886  ax-rrecex 9887  ax-cnre 9888  ax-pre-lttri 9889  ax-pre-lttrn 9890  ax-pre-ltadd 9891  ax-pre-mulgt0 9892  ax-pre-sup 9893
This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-nel 2783  df-ral 2901  df-rex 2902  df-reu 2903  df-rmo 2904  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-om 6958  df-1st 7059  df-2nd 7060  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-er 7629  df-en 7842  df-dom 7843  df-sdom 7844  df-sup 8231  df-inf 8232  df-pnf 9955  df-mnf 9956  df-xr 9957  df-ltxr 9958  df-le 9959  df-sub 10147  df-neg 10148  df-div 10564  df-nn 10898  df-2 10956  df-3 10957  df-n0 11170  df-z 11255  df-uz 11564  df-rp 11709  df-fz 12198  df-seq 12664  df-exp 12723  df-cj 13687  df-re 13688  df-im 13689  df-sqrt 13823  df-abs 13824  df-clim 14067
This theorem is referenced by:  climinff  38678  stirlinglem13  38979
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