climinf - Mathbox for Glauco Siliprandi

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Theorem climinf 46438

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 = (ℤ_≥‘𝑀)
climinf.4	⊢ (𝜑 → 𝑀 ∈ ℤ)
climinf.5	⊢ (𝜑 → 𝐹:𝑍⟶ℝ)
climinf.6	⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘(𝑘 + 1)) ≤ (𝐹‘𝑘))
climinf.7	⊢ (𝜑 → ∃𝑥 ∈ ℝ ∀𝑘 ∈ 𝑍 𝑥 ≤ (𝐹‘𝑘))

**Assertion**
Ref	Expression
climinf	⊢ (𝜑 → 𝐹 ⇝ 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.

Step	Hyp	Ref	Expression
1		climinf.5	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐹:𝑍⟶ℝ)
2	1	frnd 6715	. . . . . . . . . . 11 ⊢ (𝜑 → ran 𝐹 ⊆ ℝ)
3	1	ffnd 6707	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐹 Fn 𝑍)
4		climinf.4	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝑀 ∈ ℤ)
5		uzid 12905	. . . . . . . . . . . . . . 15 ⊢ (𝑀 ∈ ℤ → 𝑀 ∈ (ℤ_≥‘𝑀))
6	4, 5	syl 18	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝑀 ∈ (ℤ_≥‘𝑀))
7		climinf.3	. . . . . . . . . . . . . 14 ⊢ 𝑍 = (ℤ_≥‘𝑀)
8	6, 7	eleqtrrdi 2873	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝑀 ∈ 𝑍)
9		fnfvelrn 7076	. . . . . . . . . . . . 13 ⊢ ((𝐹 Fn 𝑍 ∧ 𝑀 ∈ 𝑍) → (𝐹‘𝑀) ∈ ran 𝐹)
10	3, 8, 9	syl2anc 596	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝐹‘𝑀) ∈ ran 𝐹)
11	10	ne0d 4291	. . . . . . . . . . 11 ⊢ (𝜑 → ran 𝐹 ≠ ∅)
12		climinf.7	. . . . . . . . . . . 12 ⊢ (𝜑 → ∃𝑥 ∈ ℝ ∀𝑘 ∈ 𝑍 𝑥 ≤ (𝐹‘𝑘))
13		breq2 5111	. . . . . . . . . . . . . . 15 ⊢ (𝑦 = (𝐹‘𝑘) → (𝑥 ≤ 𝑦 ↔ 𝑥 ≤ (𝐹‘𝑘)))
14	13	ralrn 7084	. . . . . . . . . . . . . 14 ⊢ (𝐹 Fn 𝑍 → (∀𝑦 ∈ ran 𝐹 𝑥 ≤ 𝑦 ↔ ∀𝑘 ∈ 𝑍 𝑥 ≤ (𝐹‘𝑘)))
15	14	rexbidv 3188	. . . . . . . . . . . . 13 ⊢ (𝐹 Fn 𝑍 → (∃𝑥 ∈ ℝ ∀𝑦 ∈ ran 𝐹 𝑥 ≤ 𝑦 ↔ ∃𝑥 ∈ ℝ ∀𝑘 ∈ 𝑍 𝑥 ≤ (𝐹‘𝑘)))
16	3, 15	syl 18	. . . . . . . . . . . 12 ⊢ (𝜑 → (∃𝑥 ∈ ℝ ∀𝑦 ∈ ran 𝐹 𝑥 ≤ 𝑦 ↔ ∃𝑥 ∈ ℝ ∀𝑘 ∈ 𝑍 𝑥 ≤ (𝐹‘𝑘)))
17	12, 16	mpbird 260	. . . . . . . . . . 11 ⊢ (𝜑 → ∃𝑥 ∈ ℝ ∀𝑦 ∈ ran 𝐹 𝑥 ≤ 𝑦)
18	2, 11, 17	3jca 1146	. . . . . . . . . 10 ⊢ (𝜑 → (ran 𝐹 ⊆ ℝ ∧ ran 𝐹 ≠ ∅ ∧ ∃𝑥 ∈ ℝ ∀𝑦 ∈ ran 𝐹 𝑥 ≤ 𝑦))
19	18	adantr 486	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ⁺) → (ran 𝐹 ⊆ ℝ ∧ ran 𝐹 ≠ ∅ ∧ ∃𝑥 ∈ ℝ ∀𝑦 ∈ ran 𝐹 𝑥 ≤ 𝑦))
20		infrecl 12224	. . . . . . . . 9 ⊢ ((ran 𝐹 ⊆ ℝ ∧ ran 𝐹 ≠ ∅ ∧ ∃𝑥 ∈ ℝ ∀𝑦 ∈ ran 𝐹 𝑥 ≤ 𝑦) → inf(ran 𝐹, ℝ, < ) ∈ ℝ)
21	19, 20	syl 18	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ⁺) → inf(ran 𝐹, ℝ, < ) ∈ ℝ)
22		simpr 490	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ⁺) → 𝑦 ∈ ℝ⁺)
23	21, 22	ltaddrpd 13121	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ⁺) → inf(ran 𝐹, ℝ, < ) < (inf(ran 𝐹, ℝ, < ) + 𝑦))
24		rpre 13053	. . . . . . . . . 10 ⊢ (𝑦 ∈ ℝ⁺ → 𝑦 ∈ ℝ)
25	24	adantl 487	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ⁺) → 𝑦 ∈ ℝ)
26	21, 25	readdcld 11265	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ⁺) → (inf(ran 𝐹, ℝ, < ) + 𝑦) ∈ ℝ)
27		infrglb 46422	. . . . . . . 8 ⊢ (((ran 𝐹 ⊆ ℝ ∧ ran 𝐹 ≠ ∅ ∧ ∃𝑥 ∈ ℝ ∀𝑦 ∈ ran 𝐹 𝑥 ≤ 𝑦) ∧ (inf(ran 𝐹, ℝ, < ) + 𝑦) ∈ ℝ) → (inf(ran 𝐹, ℝ, < ) < (inf(ran 𝐹, ℝ, < ) + 𝑦) ↔ ∃𝑘 ∈ ran 𝐹 𝑘 < (inf(ran 𝐹, ℝ, < ) + 𝑦)))
28	19, 26, 27	syl2anc 596	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ⁺) → (inf(ran 𝐹, ℝ, < ) < (inf(ran 𝐹, ℝ, < ) + 𝑦) ↔ ∃𝑘 ∈ ran 𝐹 𝑘 < (inf(ran 𝐹, ℝ, < ) + 𝑦)))
29	23, 28	mpbid 235	. . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ⁺) → ∃𝑘 ∈ ran 𝐹 𝑘 < (inf(ran 𝐹, ℝ, < ) + 𝑦))
30	2	sselda 3934	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑘 ∈ ran 𝐹) → 𝑘 ∈ ℝ)
31	30	adantlr 728	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ⁺) ∧ 𝑘 ∈ ran 𝐹) → 𝑘 ∈ ℝ)
32	21	adantr 486	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ⁺) ∧ 𝑘 ∈ ran 𝐹) → inf(ran 𝐹, ℝ, < ) ∈ ℝ)
33	24	ad2antlr 740	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ⁺) ∧ 𝑘 ∈ ran 𝐹) → 𝑦 ∈ ℝ)
34	32, 33	readdcld 11265	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ⁺) ∧ 𝑘 ∈ ran 𝐹) → (inf(ran 𝐹, ℝ, < ) + 𝑦) ∈ ℝ)
35	31, 34, 33	ltsub1d 11850	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ⁺) ∧ 𝑘 ∈ ran 𝐹) → (𝑘 < (inf(ran 𝐹, ℝ, < ) + 𝑦) ↔ (𝑘 − 𝑦) < ((inf(ran 𝐹, ℝ, < ) + 𝑦) − 𝑦)))
36	2, 11, 17, 20	syl3anc 1398	. . . . . . . . . . . . 13 ⊢ (𝜑 → inf(ran 𝐹, ℝ, < ) ∈ ℝ)
37	36	recnd 11264	. . . . . . . . . . . 12 ⊢ (𝜑 → inf(ran 𝐹, ℝ, < ) ∈ ℂ)
38	37	ad2antrr 739	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ⁺) ∧ 𝑘 ∈ ran 𝐹) → inf(ran 𝐹, ℝ, < ) ∈ ℂ)
39	33	recnd 11264	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ⁺) ∧ 𝑘 ∈ ran 𝐹) → 𝑦 ∈ ℂ)
40	38, 39	pncand 11597	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ⁺) ∧ 𝑘 ∈ ran 𝐹) → ((inf(ran 𝐹, ℝ, < ) + 𝑦) − 𝑦) = inf(ran 𝐹, ℝ, < ))
41	40	breq2d 5119	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ⁺) ∧ 𝑘 ∈ ran 𝐹) → ((𝑘 − 𝑦) < ((inf(ran 𝐹, ℝ, < ) + 𝑦) − 𝑦) ↔ (𝑘 − 𝑦) < inf(ran 𝐹, ℝ, < )))
42	35, 41	bitrd 282	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ⁺) ∧ 𝑘 ∈ ran 𝐹) → (𝑘 < (inf(ran 𝐹, ℝ, < ) + 𝑦) ↔ (𝑘 − 𝑦) < inf(ran 𝐹, ℝ, < )))
43	42	biimpd 232	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ⁺) ∧ 𝑘 ∈ ran 𝐹) → (𝑘 < (inf(ran 𝐹, ℝ, < ) + 𝑦) → (𝑘 − 𝑦) < inf(ran 𝐹, ℝ, < )))
44	43	reximdva 3177	. . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ⁺) → (∃𝑘 ∈ ran 𝐹 𝑘 < (inf(ran 𝐹, ℝ, < ) + 𝑦) → ∃𝑘 ∈ ran 𝐹(𝑘 − 𝑦) < inf(ran 𝐹, ℝ, < )))
45	29, 44	mpd 16	. . . . 5 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ⁺) → ∃𝑘 ∈ ran 𝐹(𝑘 − 𝑦) < inf(ran 𝐹, ℝ, < ))
46		oveq1 7423	. . . . . . . . 9 ⊢ (𝑘 = (𝐹‘𝑗) → (𝑘 − 𝑦) = ((𝐹‘𝑗) − 𝑦))
47	46	breq1d 5117	. . . . . . . 8 ⊢ (𝑘 = (𝐹‘𝑗) → ((𝑘 − 𝑦) < inf(ran 𝐹, ℝ, < ) ↔ ((𝐹‘𝑗) − 𝑦) < inf(ran 𝐹, ℝ, < )))
48	47	rexrn 7083	. . . . . . 7 ⊢ (𝐹 Fn 𝑍 → (∃𝑘 ∈ ran 𝐹(𝑘 − 𝑦) < inf(ran 𝐹, ℝ, < ) ↔ ∃𝑗 ∈ 𝑍 ((𝐹‘𝑗) − 𝑦) < inf(ran 𝐹, ℝ, < )))
49	3, 48	syl 18	. . . . . 6 ⊢ (𝜑 → (∃𝑘 ∈ ran 𝐹(𝑘 − 𝑦) < inf(ran 𝐹, ℝ, < ) ↔ ∃𝑗 ∈ 𝑍 ((𝐹‘𝑗) − 𝑦) < inf(ran 𝐹, ℝ, < )))
50	49	biimpa 482	. . . . 5 ⊢ ((𝜑 ∧ ∃𝑘 ∈ ran 𝐹(𝑘 − 𝑦) < inf(ran 𝐹, ℝ, < )) → ∃𝑗 ∈ 𝑍 ((𝐹‘𝑗) − 𝑦) < inf(ran 𝐹, ℝ, < ))
51	45, 50	syldan 603	. . . 4 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ⁺) → ∃𝑗 ∈ 𝑍 ((𝐹‘𝑗) − 𝑦) < inf(ran 𝐹, ℝ, < ))
52	1	adantr 486	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ⁺) → 𝐹:𝑍⟶ℝ)
53	7	uztrn2 12909	. . . . . . . . . . 11 ⊢ ((𝑗 ∈ 𝑍 ∧ 𝑘 ∈ (ℤ_≥‘𝑗)) → 𝑘 ∈ 𝑍)
54		ffvelcdm 7077	. . . . . . . . . . 11 ⊢ ((𝐹:𝑍⟶ℝ ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) ∈ ℝ)
55	52, 53, 54	syl2an 608	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ⁺) ∧ (𝑗 ∈ 𝑍 ∧ 𝑘 ∈ (ℤ_≥‘𝑗))) → (𝐹‘𝑘) ∈ ℝ)
56		simpl 488	. . . . . . . . . . 11 ⊢ ((𝑗 ∈ 𝑍 ∧ 𝑘 ∈ (ℤ_≥‘𝑗)) → 𝑗 ∈ 𝑍)
57		ffvelcdm 7077	. . . . . . . . . . 11 ⊢ ((𝐹:𝑍⟶ℝ ∧ 𝑗 ∈ 𝑍) → (𝐹‘𝑗) ∈ ℝ)
58	52, 56, 57	syl2an 608	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ⁺) ∧ (𝑗 ∈ 𝑍 ∧ 𝑘 ∈ (ℤ_≥‘𝑗))) → (𝐹‘𝑗) ∈ ℝ)
59	36	ad2antrr 739	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ⁺) ∧ (𝑗 ∈ 𝑍 ∧ 𝑘 ∈ (ℤ_≥‘𝑗))) → inf(ran 𝐹, ℝ, < ) ∈ ℝ)
60		simprr 785	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ⁺) ∧ (𝑗 ∈ 𝑍 ∧ 𝑘 ∈ (ℤ_≥‘𝑗))) → 𝑘 ∈ (ℤ_≥‘𝑗))
61		fzssuz 13622	. . . . . . . . . . . . . 14 ⊢ (𝑗...𝑘) ⊆ (ℤ_≥‘𝑗)
62		uzss 12913	. . . . . . . . . . . . . . . . 17 ⊢ (𝑗 ∈ (ℤ_≥‘𝑀) → (ℤ_≥‘𝑗) ⊆ (ℤ_≥‘𝑀))
63	62, 7	sseqtrrdi 3975	. . . . . . . . . . . . . . . 16 ⊢ (𝑗 ∈ (ℤ_≥‘𝑀) → (ℤ_≥‘𝑗) ⊆ 𝑍)
64	63, 7	eleq2s 2880	. . . . . . . . . . . . . . 15 ⊢ (𝑗 ∈ 𝑍 → (ℤ_≥‘𝑗) ⊆ 𝑍)
65	64	ad2antrl 741	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ⁺) ∧ (𝑗 ∈ 𝑍 ∧ 𝑘 ∈ (ℤ_≥‘𝑗))) → (ℤ_≥‘𝑗) ⊆ 𝑍)
66	61, 65	sstrid 3945	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ⁺) ∧ (𝑗 ∈ 𝑍 ∧ 𝑘 ∈ (ℤ_≥‘𝑗))) → (𝑗...𝑘) ⊆ 𝑍)
67		ffvelcdm 7077	. . . . . . . . . . . . . . . 16 ⊢ ((𝐹:𝑍⟶ℝ ∧ 𝑛 ∈ 𝑍) → (𝐹‘𝑛) ∈ ℝ)
68	67	ralrimiva 3156	. . . . . . . . . . . . . . 15 ⊢ (𝐹:𝑍⟶ℝ → ∀𝑛 ∈ 𝑍 (𝐹‘𝑛) ∈ ℝ)
69	1, 68	syl 18	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ∀𝑛 ∈ 𝑍 (𝐹‘𝑛) ∈ ℝ)
70	69	ad2antrr 739	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ⁺) ∧ (𝑗 ∈ 𝑍 ∧ 𝑘 ∈ (ℤ_≥‘𝑗))) → ∀𝑛 ∈ 𝑍 (𝐹‘𝑛) ∈ ℝ)
71		ssralv 4003	. . . . . . . . . . . . 13 ⊢ ((𝑗...𝑘) ⊆ 𝑍 → (∀𝑛 ∈ 𝑍 (𝐹‘𝑛) ∈ ℝ → ∀𝑛 ∈ (𝑗...𝑘)(𝐹‘𝑛) ∈ ℝ))
72	66, 70, 71	sylc 66	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ⁺) ∧ (𝑗 ∈ 𝑍 ∧ 𝑘 ∈ (ℤ_≥‘𝑗))) → ∀𝑛 ∈ (𝑗...𝑘)(𝐹‘𝑛) ∈ ℝ)
73	72	r19.21bi 3256	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑦 ∈ ℝ⁺) ∧ (𝑗 ∈ 𝑍 ∧ 𝑘 ∈ (ℤ_≥‘𝑗))) ∧ 𝑛 ∈ (𝑗...𝑘)) → (𝐹‘𝑛) ∈ ℝ)
74		fzssuz 13622	. . . . . . . . . . . . . 14 ⊢ (𝑗...(𝑘 − 1)) ⊆ (ℤ_≥‘𝑗)
75	74, 65	sstrid 3945	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ⁺) ∧ (𝑗 ∈ 𝑍 ∧ 𝑘 ∈ (ℤ_≥‘𝑗))) → (𝑗...(𝑘 − 1)) ⊆ 𝑍)
76	75	sselda 3934	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑦 ∈ ℝ⁺) ∧ (𝑗 ∈ 𝑍 ∧ 𝑘 ∈ (ℤ_≥‘𝑗))) ∧ 𝑛 ∈ (𝑗...(𝑘 − 1))) → 𝑛 ∈ 𝑍)
77		climinf.6	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘(𝑘 + 1)) ≤ (𝐹‘𝑘))
78	77	ralrimiva 3156	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ∀𝑘 ∈ 𝑍 (𝐹‘(𝑘 + 1)) ≤ (𝐹‘𝑘))
79	78	ad2antrr 739	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ⁺) ∧ (𝑗 ∈ 𝑍 ∧ 𝑘 ∈ (ℤ_≥‘𝑗))) → ∀𝑘 ∈ 𝑍 (𝐹‘(𝑘 + 1)) ≤ (𝐹‘𝑘))
80		fvoveq1 7439	. . . . . . . . . . . . . . 15 ⊢ (𝑘 = 𝑛 → (𝐹‘(𝑘 + 1)) = (𝐹‘(𝑛 + 1)))
81		fveq2 6882	. . . . . . . . . . . . . . 15 ⊢ (𝑘 = 𝑛 → (𝐹‘𝑘) = (𝐹‘𝑛))
82	80, 81	breq12d 5120	. . . . . . . . . . . . . 14 ⊢ (𝑘 = 𝑛 → ((𝐹‘(𝑘 + 1)) ≤ (𝐹‘𝑘) ↔ (𝐹‘(𝑛 + 1)) ≤ (𝐹‘𝑛)))
83	82	rspccva 3578	. . . . . . . . . . . . 13 ⊢ ((∀𝑘 ∈ 𝑍 (𝐹‘(𝑘 + 1)) ≤ (𝐹‘𝑘) ∧ 𝑛 ∈ 𝑍) → (𝐹‘(𝑛 + 1)) ≤ (𝐹‘𝑛))
84	79, 83	sylan 592	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑦 ∈ ℝ⁺) ∧ (𝑗 ∈ 𝑍 ∧ 𝑘 ∈ (ℤ_≥‘𝑗))) ∧ 𝑛 ∈ 𝑍) → (𝐹‘(𝑛 + 1)) ≤ (𝐹‘𝑛))
85	76, 84	syldan 603	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑦 ∈ ℝ⁺) ∧ (𝑗 ∈ 𝑍 ∧ 𝑘 ∈ (ℤ_≥‘𝑗))) ∧ 𝑛 ∈ (𝑗...(𝑘 − 1))) → (𝐹‘(𝑛 + 1)) ≤ (𝐹‘𝑛))
86	60, 73, 85	monoord2 14099	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ⁺) ∧ (𝑗 ∈ 𝑍 ∧ 𝑘 ∈ (ℤ_≥‘𝑗))) → (𝐹‘𝑘) ≤ (𝐹‘𝑗))
87	55, 58, 59, 86	lesub1dd 11857	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ⁺) ∧ (𝑗 ∈ 𝑍 ∧ 𝑘 ∈ (ℤ_≥‘𝑗))) → ((𝐹‘𝑘) − inf(ran 𝐹, ℝ, < )) ≤ ((𝐹‘𝑗) − inf(ran 𝐹, ℝ, < )))
88	55, 59	resubcld 11669	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ⁺) ∧ (𝑗 ∈ 𝑍 ∧ 𝑘 ∈ (ℤ_≥‘𝑗))) → ((𝐹‘𝑘) − inf(ran 𝐹, ℝ, < )) ∈ ℝ)
89	58, 59	resubcld 11669	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ⁺) ∧ (𝑗 ∈ 𝑍 ∧ 𝑘 ∈ (ℤ_≥‘𝑗))) → ((𝐹‘𝑗) − inf(ran 𝐹, ℝ, < )) ∈ ℝ)
90	24	ad2antlr 740	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ⁺) ∧ (𝑗 ∈ 𝑍 ∧ 𝑘 ∈ (ℤ_≥‘𝑗))) → 𝑦 ∈ ℝ)
91		lelttr 11327	. . . . . . . . . 10 ⊢ ((((𝐹‘𝑘) − inf(ran 𝐹, ℝ, < )) ∈ ℝ ∧ ((𝐹‘𝑗) − inf(ran 𝐹, ℝ, < )) ∈ ℝ ∧ 𝑦 ∈ ℝ) → ((((𝐹‘𝑘) − inf(ran 𝐹, ℝ, < )) ≤ ((𝐹‘𝑗) − inf(ran 𝐹, ℝ, < )) ∧ ((𝐹‘𝑗) − inf(ran 𝐹, ℝ, < )) < 𝑦) → ((𝐹‘𝑘) − inf(ran 𝐹, ℝ, < )) < 𝑦))
92	88, 89, 90, 91	syl3anc 1398	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ⁺) ∧ (𝑗 ∈ 𝑍 ∧ 𝑘 ∈ (ℤ_≥‘𝑗))) → ((((𝐹‘𝑘) − inf(ran 𝐹, ℝ, < )) ≤ ((𝐹‘𝑗) − inf(ran 𝐹, ℝ, < )) ∧ ((𝐹‘𝑗) − inf(ran 𝐹, ℝ, < )) < 𝑦) → ((𝐹‘𝑘) − inf(ran 𝐹, ℝ, < )) < 𝑦))
93	87, 92	mpand 708	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ⁺) ∧ (𝑗 ∈ 𝑍 ∧ 𝑘 ∈ (ℤ_≥‘𝑗))) → (((𝐹‘𝑗) − inf(ran 𝐹, ℝ, < )) < 𝑦 → ((𝐹‘𝑘) − inf(ran 𝐹, ℝ, < )) < 𝑦))
94		ltsub23 11721	. . . . . . . . 9 ⊢ (((𝐹‘𝑗) ∈ ℝ ∧ 𝑦 ∈ ℝ ∧ inf(ran 𝐹, ℝ, < ) ∈ ℝ) → (((𝐹‘𝑗) − 𝑦) < inf(ran 𝐹, ℝ, < ) ↔ ((𝐹‘𝑗) − inf(ran 𝐹, ℝ, < )) < 𝑦))
95	58, 90, 59, 94	syl3anc 1398	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ⁺) ∧ (𝑗 ∈ 𝑍 ∧ 𝑘 ∈ (ℤ_≥‘𝑗))) → (((𝐹‘𝑗) − 𝑦) < inf(ran 𝐹, ℝ, < ) ↔ ((𝐹‘𝑗) − inf(ran 𝐹, ℝ, < )) < 𝑦))
96	2	ad2antrr 739	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ⁺) ∧ (𝑗 ∈ 𝑍 ∧ 𝑘 ∈ (ℤ_≥‘𝑗))) → ran 𝐹 ⊆ ℝ)
97	3	adantr 486	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ⁺) → 𝐹 Fn 𝑍)
98		fnfvelrn 7076	. . . . . . . . . . . 12 ⊢ ((𝐹 Fn 𝑍 ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) ∈ ran 𝐹)
99	97, 53, 98	syl2an 608	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ⁺) ∧ (𝑗 ∈ 𝑍 ∧ 𝑘 ∈ (ℤ_≥‘𝑗))) → (𝐹‘𝑘) ∈ ran 𝐹)
100	96, 99	sseldd 3935	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ⁺) ∧ (𝑗 ∈ 𝑍 ∧ 𝑘 ∈ (ℤ_≥‘𝑗))) → (𝐹‘𝑘) ∈ ℝ)
101	17	ad2antrr 739	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ⁺) ∧ (𝑗 ∈ 𝑍 ∧ 𝑘 ∈ (ℤ_≥‘𝑗))) → ∃𝑥 ∈ ℝ ∀𝑦 ∈ ran 𝐹 𝑥 ≤ 𝑦)
102		infrelb 12227	. . . . . . . . . . 11 ⊢ ((ran 𝐹 ⊆ ℝ ∧ ∃𝑥 ∈ ℝ ∀𝑦 ∈ ran 𝐹 𝑥 ≤ 𝑦 ∧ (𝐹‘𝑘) ∈ ran 𝐹) → inf(ran 𝐹, ℝ, < ) ≤ (𝐹‘𝑘))
103	96, 101, 99, 102	syl3anc 1398	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ⁺) ∧ (𝑗 ∈ 𝑍 ∧ 𝑘 ∈ (ℤ_≥‘𝑗))) → inf(ran 𝐹, ℝ, < ) ≤ (𝐹‘𝑘))
104	59, 100, 103	abssubge0d 15523	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ⁺) ∧ (𝑗 ∈ 𝑍 ∧ 𝑘 ∈ (ℤ_≥‘𝑗))) → (abs‘((𝐹‘𝑘) − inf(ran 𝐹, ℝ, < ))) = ((𝐹‘𝑘) − inf(ran 𝐹, ℝ, < )))
105	104	breq1d 5117	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ⁺) ∧ (𝑗 ∈ 𝑍 ∧ 𝑘 ∈ (ℤ_≥‘𝑗))) → ((abs‘((𝐹‘𝑘) − inf(ran 𝐹, ℝ, < ))) < 𝑦 ↔ ((𝐹‘𝑘) − inf(ran 𝐹, ℝ, < )) < 𝑦))
106	93, 95, 105	3imtr4d 297	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ⁺) ∧ (𝑗 ∈ 𝑍 ∧ 𝑘 ∈ (ℤ_≥‘𝑗))) → (((𝐹‘𝑗) − 𝑦) < inf(ran 𝐹, ℝ, < ) → (abs‘((𝐹‘𝑘) − inf(ran 𝐹, ℝ, < ))) < 𝑦))
107	106	anassrs 473	. . . . . 6 ⊢ ((((𝜑 ∧ 𝑦 ∈ ℝ⁺) ∧ 𝑗 ∈ 𝑍) ∧ 𝑘 ∈ (ℤ_≥‘𝑗)) → (((𝐹‘𝑗) − 𝑦) < inf(ran 𝐹, ℝ, < ) → (abs‘((𝐹‘𝑘) − inf(ran 𝐹, ℝ, < ))) < 𝑦))
108	107	ralrimdva 3164	. . . . 5 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ⁺) ∧ 𝑗 ∈ 𝑍) → (((𝐹‘𝑗) − 𝑦) < inf(ran 𝐹, ℝ, < ) → ∀𝑘 ∈ (ℤ_≥‘𝑗)(abs‘((𝐹‘𝑘) − inf(ran 𝐹, ℝ, < ))) < 𝑦))
109	108	reximdva 3177	. . . 4 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ⁺) → (∃𝑗 ∈ 𝑍 ((𝐹‘𝑗) − 𝑦) < inf(ran 𝐹, ℝ, < ) → ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ_≥‘𝑗)(abs‘((𝐹‘𝑘) − inf(ran 𝐹, ℝ, < ))) < 𝑦))
110	51, 109	mpd 16	. . 3 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ⁺) → ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ_≥‘𝑗)(abs‘((𝐹‘𝑘) − inf(ran 𝐹, ℝ, < ))) < 𝑦)
111	110	ralrimiva 3156	. 2 ⊢ (𝜑 → ∀𝑦 ∈ ℝ⁺ ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ_≥‘𝑗)(abs‘((𝐹‘𝑘) − inf(ran 𝐹, ℝ, < ))) < 𝑦)
112	7	fvexi 6896	. . . 4 ⊢ 𝑍 ∈ V
113		fex 7228	. . . 4 ⊢ ((𝐹:𝑍⟶ℝ ∧ 𝑍 ∈ V) → 𝐹 ∈ V)
114	1, 112, 113	sylancl 598	. . 3 ⊢ (𝜑 → 𝐹 ∈ V)
115		eqidd 2763	. . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) = (𝐹‘𝑘))
116	1	ffvelcdmda 7080	. . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) ∈ ℝ)
117	116	recnd 11264	. . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) ∈ ℂ)
118	7, 4, 114, 115, 37, 117	clim2c 15594	. 2 ⊢ (𝜑 → (𝐹 ⇝ inf(ran 𝐹, ℝ, < ) ↔ ∀𝑦 ∈ ℝ⁺ ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ_≥‘𝑗)(abs‘((𝐹‘𝑘) − inf(ran 𝐹, ℝ, < ))) < 𝑦))
119	111, 118	mpbird 260	1 ⊢ (𝜑 → 𝐹 ⇝ inf(ran 𝐹, ℝ, < ))

Colors of variables: wff setvar class

This proof depends on syntax axioms: → wi 4 ↔ wb 209 ∧ wa 401 ∧ w3a 1103 = wceq 1570 ∈ wcel 2145 ≠ wne 2957 ∀wral 3078 ∃wrex 3088 Vcvv 3453 ⊆ wss 3902 ∅c0 4282 class class class wbr 5107 ran crn 5660 Fn wfn 6532 ⟶wf 6533 ‘cfv 6537 (class class class)co 7416 infcinf 9414 ℂcc 11125 ℝcr 11126 1c1 11128 + caddc 11130 < clt 11270 ≤ cle 11271 − cmin 11468 ℤcz 12618 ℤ_≥cuz 12890 ℝ⁺crp 13044 ...cfz 13563 abscabs 15323 ⇝ cli 15573

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1828 ax-4 1842 ax-5 1943 ax-6 2000 ax-7 2041 ax-8 2147 ax-9 2155 ax-10 2178 ax-11 2194 ax-12 2215 ax-ext 2734 ax-rep 5236 ax-sep 5255 ax-nul 5267 ax-pow 5334 ax-pr 5402 ax-un 7739 ax-cnex 11183 ax-resscn 11184 ax-1cn 11185 ax-icn 11186 ax-addcl 11187 ax-addrcl 11188 ax-mulcl 11189 ax-mulrcl 11190 ax-mulcom 11191 ax-addass 11192 ax-mulass 11193 ax-distr 11194 ax-i2m1 11195 ax-1ne0 11196 ax-1rid 11197 ax-rnegex 11198 ax-rrecex 11199 ax-cnre 11200 ax-pre-lttri 11201 ax-pre-lttrn 11202 ax-pre-ltadd 11203 ax-pre-mulgt0 11204 ax-pre-sup 11205

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1813 df-nf 1817 df-sb 2100 df-mo 2566 df-eu 2596 df-clab 2741 df-cleq 2754 df-clel 2837 df-nfc 2911 df-ne 2958 df-nel 3064 df-ral 3079 df-rex 3089 df-rmo 3367 df-reu 3368 df-rab 3415 df-v 3455 df-sbc 3743 df-csb 3851 df-dif 3905 df-un 3907 df-in 3909 df-ss 3919 df-pss 3922 df-nul 4283 df-if 4486 df-pw 4562 df-sn 4588 df-pr 4590 df-op 4594 df-uni 4871 df-iun 4956 df-br 5108 df-opab 5172 df-mpt 5191 df-tr 5217 df-id 5554 df-eprel 5559 df-po 5567 df-so 5568 df-fr 5612 df-we 5614 df-xp 5665 df-rel 5666 df-cnv 5667 df-co 5668 df-dm 5669 df-rn 5670 df-res 5671 df-ima 5672 df-pred 6303 df-ord 6364 df-on 6365 df-lim 6366 df-suc 6367 df-iota 6493 df-fun 6539 df-fn 6540 df-f 6541 df-f1 6542 df-fo 6543 df-f1o 6544 df-fv 6545 df-riota 7373 df-ov 7419 df-oprab 7420 df-mpo 7421 df-om 7866 df-1st 7989 df-2nd 7990 df-frecs 8283 df-wrecs 8314 df-recs 8363 df-rdg 8402 df-er 8699 df-en 8956 df-dom 8957 df-sdom 8958 df-sup 9415 df-inf 9416 df-pnf 11272 df-mnf 11273 df-xr 11274 df-ltxr 11275 df-le 11276 df-sub 11470 df-neg 11471 df-div 11899 df-nn 12261 df-2 12330 df-3 12331 df-n0 12532 df-z 12619 df-uz 12891 df-rp 13045 df-fz 13564 df-seq 14068 df-exp 14128 df-cj 15188 df-re 15189 df-im 15190 df-sqrt 15324 df-abs 15325 df-clim 15577

This theorem is used by: climinff 46443 climinf2lem 46536 supcnvlimsup 46570 stirlinglem13 46916

W3C validator