climinf - Mathbox for Glauco Siliprandi

	Mathbox for Glauco Siliprandi		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > Mathboxes > climinf		Structured version Visualization version GIF version

Theorem climinf 41429

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 6389	. . . . . . . . . . 11 ⊢ (𝜑 → ran 𝐹 ⊆ ℝ)
3	1	ffnd 6383	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐹 Fn 𝑍)
4		climinf.4	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝑀 ∈ ℤ)
5		uzid 12108	. . . . . . . . . . . . . . 15 ⊢ (𝑀 ∈ ℤ → 𝑀 ∈ (ℤ_≥‘𝑀))
6	4, 5	syl 17	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝑀 ∈ (ℤ_≥‘𝑀))
7		climinf.3	. . . . . . . . . . . . . 14 ⊢ 𝑍 = (ℤ_≥‘𝑀)
8	6, 7	syl6eleqr 2894	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝑀 ∈ 𝑍)
9		fnfvelrn 6713	. . . . . . . . . . . . 13 ⊢ ((𝐹 Fn 𝑍 ∧ 𝑀 ∈ 𝑍) → (𝐹‘𝑀) ∈ ran 𝐹)
10	3, 8, 9	syl2anc 584	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝐹‘𝑀) ∈ ran 𝐹)
11	10	ne0d 4221	. . . . . . . . . . 11 ⊢ (𝜑 → ran 𝐹 ≠ ∅)
12		climinf.7	. . . . . . . . . . . 12 ⊢ (𝜑 → ∃𝑥 ∈ ℝ ∀𝑘 ∈ 𝑍 𝑥 ≤ (𝐹‘𝑘))
13		breq2 4966	. . . . . . . . . . . . . . 15 ⊢ (𝑦 = (𝐹‘𝑘) → (𝑥 ≤ 𝑦 ↔ 𝑥 ≤ (𝐹‘𝑘)))
14	13	ralrn 6719	. . . . . . . . . . . . . 14 ⊢ (𝐹 Fn 𝑍 → (∀𝑦 ∈ ran 𝐹 𝑥 ≤ 𝑦 ↔ ∀𝑘 ∈ 𝑍 𝑥 ≤ (𝐹‘𝑘)))
15	14	rexbidv 3260	. . . . . . . . . . . . 13 ⊢ (𝐹 Fn 𝑍 → (∃𝑥 ∈ ℝ ∀𝑦 ∈ ran 𝐹 𝑥 ≤ 𝑦 ↔ ∃𝑥 ∈ ℝ ∀𝑘 ∈ 𝑍 𝑥 ≤ (𝐹‘𝑘)))
16	3, 15	syl 17	. . . . . . . . . . . 12 ⊢ (𝜑 → (∃𝑥 ∈ ℝ ∀𝑦 ∈ ran 𝐹 𝑥 ≤ 𝑦 ↔ ∃𝑥 ∈ ℝ ∀𝑘 ∈ 𝑍 𝑥 ≤ (𝐹‘𝑘)))
17	12, 16	mpbird 258	. . . . . . . . . . 11 ⊢ (𝜑 → ∃𝑥 ∈ ℝ ∀𝑦 ∈ ran 𝐹 𝑥 ≤ 𝑦)
18	2, 11, 17	3jca 1121	. . . . . . . . . 10 ⊢ (𝜑 → (ran 𝐹 ⊆ ℝ ∧ ran 𝐹 ≠ ∅ ∧ ∃𝑥 ∈ ℝ ∀𝑦 ∈ ran 𝐹 𝑥 ≤ 𝑦))
19	18	adantr 481	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ⁺) → (ran 𝐹 ⊆ ℝ ∧ ran 𝐹 ≠ ∅ ∧ ∃𝑥 ∈ ℝ ∀𝑦 ∈ ran 𝐹 𝑥 ≤ 𝑦))
20		infrecl 11471	. . . . . . . . 9 ⊢ ((ran 𝐹 ⊆ ℝ ∧ ran 𝐹 ≠ ∅ ∧ ∃𝑥 ∈ ℝ ∀𝑦 ∈ ran 𝐹 𝑥 ≤ 𝑦) → inf(ran 𝐹, ℝ, < ) ∈ ℝ)
21	19, 20	syl 17	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ⁺) → inf(ran 𝐹, ℝ, < ) ∈ ℝ)
22		simpr 485	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ⁺) → 𝑦 ∈ ℝ⁺)
23	21, 22	ltaddrpd 12314	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ⁺) → inf(ran 𝐹, ℝ, < ) < (inf(ran 𝐹, ℝ, < ) + 𝑦))
24		rpre 12247	. . . . . . . . . 10 ⊢ (𝑦 ∈ ℝ⁺ → 𝑦 ∈ ℝ)
25	24	adantl 482	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ⁺) → 𝑦 ∈ ℝ)
26	21, 25	readdcld 10516	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ⁺) → (inf(ran 𝐹, ℝ, < ) + 𝑦) ∈ ℝ)
27		infrglb 41413	. . . . . . . 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 584	. . . . . . 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 233	. . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ⁺) → ∃𝑘 ∈ ran 𝐹 𝑘 < (inf(ran 𝐹, ℝ, < ) + 𝑦))
30	2	sselda 3889	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑘 ∈ ran 𝐹) → 𝑘 ∈ ℝ)
31	30	adantlr 711	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ⁺) ∧ 𝑘 ∈ ran 𝐹) → 𝑘 ∈ ℝ)
32	21	adantr 481	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ⁺) ∧ 𝑘 ∈ ran 𝐹) → inf(ran 𝐹, ℝ, < ) ∈ ℝ)
33	24	ad2antlr 723	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ⁺) ∧ 𝑘 ∈ ran 𝐹) → 𝑦 ∈ ℝ)
34	32, 33	readdcld 10516	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ⁺) ∧ 𝑘 ∈ ran 𝐹) → (inf(ran 𝐹, ℝ, < ) + 𝑦) ∈ ℝ)
35	31, 34, 33	ltsub1d 11097	. . . . . . . . 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 1364	. . . . . . . . . . . . 13 ⊢ (𝜑 → inf(ran 𝐹, ℝ, < ) ∈ ℝ)
37	36	recnd 10515	. . . . . . . . . . . 12 ⊢ (𝜑 → inf(ran 𝐹, ℝ, < ) ∈ ℂ)
38	37	ad2antrr 722	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ⁺) ∧ 𝑘 ∈ ran 𝐹) → inf(ran 𝐹, ℝ, < ) ∈ ℂ)
39	33	recnd 10515	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ⁺) ∧ 𝑘 ∈ ran 𝐹) → 𝑦 ∈ ℂ)
40	38, 39	pncand 10846	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ⁺) ∧ 𝑘 ∈ ran 𝐹) → ((inf(ran 𝐹, ℝ, < ) + 𝑦) − 𝑦) = inf(ran 𝐹, ℝ, < ))
41	40	breq2d 4974	. . . . . . . . 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 280	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ⁺) ∧ 𝑘 ∈ ran 𝐹) → (𝑘 < (inf(ran 𝐹, ℝ, < ) + 𝑦) ↔ (𝑘 − 𝑦) < inf(ran 𝐹, ℝ, < )))
43	42	biimpd 230	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ⁺) ∧ 𝑘 ∈ ran 𝐹) → (𝑘 < (inf(ran 𝐹, ℝ, < ) + 𝑦) → (𝑘 − 𝑦) < inf(ran 𝐹, ℝ, < )))
44	43	reximdva 3237	. . . . . 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 15	. . . . 5 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ⁺) → ∃𝑘 ∈ ran 𝐹(𝑘 − 𝑦) < inf(ran 𝐹, ℝ, < ))
46		oveq1 7023	. . . . . . . . 9 ⊢ (𝑘 = (𝐹‘𝑗) → (𝑘 − 𝑦) = ((𝐹‘𝑗) − 𝑦))
47	46	breq1d 4972	. . . . . . . 8 ⊢ (𝑘 = (𝐹‘𝑗) → ((𝑘 − 𝑦) < inf(ran 𝐹, ℝ, < ) ↔ ((𝐹‘𝑗) − 𝑦) < inf(ran 𝐹, ℝ, < )))
48	47	rexrn 6718	. . . . . . 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 17	. . . . . 6 ⊢ (𝜑 → (∃𝑘 ∈ ran 𝐹(𝑘 − 𝑦) < inf(ran 𝐹, ℝ, < ) ↔ ∃𝑗 ∈ 𝑍 ((𝐹‘𝑗) − 𝑦) < inf(ran 𝐹, ℝ, < )))
50	49	biimpa 477	. . . . 5 ⊢ ((𝜑 ∧ ∃𝑘 ∈ ran 𝐹(𝑘 − 𝑦) < inf(ran 𝐹, ℝ, < )) → ∃𝑗 ∈ 𝑍 ((𝐹‘𝑗) − 𝑦) < inf(ran 𝐹, ℝ, < ))
51	45, 50	syldan 591	. . . 4 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ⁺) → ∃𝑗 ∈ 𝑍 ((𝐹‘𝑗) − 𝑦) < inf(ran 𝐹, ℝ, < ))
52	1	adantr 481	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ⁺) → 𝐹:𝑍⟶ℝ)
53	7	uztrn2 12111	. . . . . . . . . . 11 ⊢ ((𝑗 ∈ 𝑍 ∧ 𝑘 ∈ (ℤ_≥‘𝑗)) → 𝑘 ∈ 𝑍)
54		ffvelrn 6714	. . . . . . . . . . 11 ⊢ ((𝐹:𝑍⟶ℝ ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) ∈ ℝ)
55	52, 53, 54	syl2an 595	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ⁺) ∧ (𝑗 ∈ 𝑍 ∧ 𝑘 ∈ (ℤ_≥‘𝑗))) → (𝐹‘𝑘) ∈ ℝ)
56		simpl 483	. . . . . . . . . . 11 ⊢ ((𝑗 ∈ 𝑍 ∧ 𝑘 ∈ (ℤ_≥‘𝑗)) → 𝑗 ∈ 𝑍)
57		ffvelrn 6714	. . . . . . . . . . 11 ⊢ ((𝐹:𝑍⟶ℝ ∧ 𝑗 ∈ 𝑍) → (𝐹‘𝑗) ∈ ℝ)
58	52, 56, 57	syl2an 595	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ⁺) ∧ (𝑗 ∈ 𝑍 ∧ 𝑘 ∈ (ℤ_≥‘𝑗))) → (𝐹‘𝑗) ∈ ℝ)
59	36	ad2antrr 722	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ⁺) ∧ (𝑗 ∈ 𝑍 ∧ 𝑘 ∈ (ℤ_≥‘𝑗))) → inf(ran 𝐹, ℝ, < ) ∈ ℝ)
60		simprr 769	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ⁺) ∧ (𝑗 ∈ 𝑍 ∧ 𝑘 ∈ (ℤ_≥‘𝑗))) → 𝑘 ∈ (ℤ_≥‘𝑗))
61		fzssuz 12798	. . . . . . . . . . . . . 14 ⊢ (𝑗...𝑘) ⊆ (ℤ_≥‘𝑗)
62		uzss 12114	. . . . . . . . . . . . . . . . 17 ⊢ (𝑗 ∈ (ℤ_≥‘𝑀) → (ℤ_≥‘𝑗) ⊆ (ℤ_≥‘𝑀))
63	62, 7	syl6sseqr 3939	. . . . . . . . . . . . . . . 16 ⊢ (𝑗 ∈ (ℤ_≥‘𝑀) → (ℤ_≥‘𝑗) ⊆ 𝑍)
64	63, 7	eleq2s 2901	. . . . . . . . . . . . . . 15 ⊢ (𝑗 ∈ 𝑍 → (ℤ_≥‘𝑗) ⊆ 𝑍)
65	64	ad2antrl 724	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ⁺) ∧ (𝑗 ∈ 𝑍 ∧ 𝑘 ∈ (ℤ_≥‘𝑗))) → (ℤ_≥‘𝑗) ⊆ 𝑍)
66	61, 65	syl5ss 3900	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ⁺) ∧ (𝑗 ∈ 𝑍 ∧ 𝑘 ∈ (ℤ_≥‘𝑗))) → (𝑗...𝑘) ⊆ 𝑍)
67		ffvelrn 6714	. . . . . . . . . . . . . . . 16 ⊢ ((𝐹:𝑍⟶ℝ ∧ 𝑛 ∈ 𝑍) → (𝐹‘𝑛) ∈ ℝ)
68	67	ralrimiva 3149	. . . . . . . . . . . . . . 15 ⊢ (𝐹:𝑍⟶ℝ → ∀𝑛 ∈ 𝑍 (𝐹‘𝑛) ∈ ℝ)
69	1, 68	syl 17	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ∀𝑛 ∈ 𝑍 (𝐹‘𝑛) ∈ ℝ)
70	69	ad2antrr 722	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ⁺) ∧ (𝑗 ∈ 𝑍 ∧ 𝑘 ∈ (ℤ_≥‘𝑗))) → ∀𝑛 ∈ 𝑍 (𝐹‘𝑛) ∈ ℝ)
71		ssralv 3954	. . . . . . . . . . . . 13 ⊢ ((𝑗...𝑘) ⊆ 𝑍 → (∀𝑛 ∈ 𝑍 (𝐹‘𝑛) ∈ ℝ → ∀𝑛 ∈ (𝑗...𝑘)(𝐹‘𝑛) ∈ ℝ))
72	66, 70, 71	sylc 65	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ⁺) ∧ (𝑗 ∈ 𝑍 ∧ 𝑘 ∈ (ℤ_≥‘𝑗))) → ∀𝑛 ∈ (𝑗...𝑘)(𝐹‘𝑛) ∈ ℝ)
73	72	r19.21bi 3175	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑦 ∈ ℝ⁺) ∧ (𝑗 ∈ 𝑍 ∧ 𝑘 ∈ (ℤ_≥‘𝑗))) ∧ 𝑛 ∈ (𝑗...𝑘)) → (𝐹‘𝑛) ∈ ℝ)
74		fzssuz 12798	. . . . . . . . . . . . . 14 ⊢ (𝑗...(𝑘 − 1)) ⊆ (ℤ_≥‘𝑗)
75	74, 65	syl5ss 3900	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ⁺) ∧ (𝑗 ∈ 𝑍 ∧ 𝑘 ∈ (ℤ_≥‘𝑗))) → (𝑗...(𝑘 − 1)) ⊆ 𝑍)
76	75	sselda 3889	. . . . . . . . . . . 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 3149	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ∀𝑘 ∈ 𝑍 (𝐹‘(𝑘 + 1)) ≤ (𝐹‘𝑘))
79	78	ad2antrr 722	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ⁺) ∧ (𝑗 ∈ 𝑍 ∧ 𝑘 ∈ (ℤ_≥‘𝑗))) → ∀𝑘 ∈ 𝑍 (𝐹‘(𝑘 + 1)) ≤ (𝐹‘𝑘))
80		fvoveq1 7039	. . . . . . . . . . . . . . 15 ⊢ (𝑘 = 𝑛 → (𝐹‘(𝑘 + 1)) = (𝐹‘(𝑛 + 1)))
81		fveq2 6538	. . . . . . . . . . . . . . 15 ⊢ (𝑘 = 𝑛 → (𝐹‘𝑘) = (𝐹‘𝑛))
82	80, 81	breq12d 4975	. . . . . . . . . . . . . 14 ⊢ (𝑘 = 𝑛 → ((𝐹‘(𝑘 + 1)) ≤ (𝐹‘𝑘) ↔ (𝐹‘(𝑛 + 1)) ≤ (𝐹‘𝑛)))
83	82	rspccva 3558	. . . . . . . . . . . . 13 ⊢ ((∀𝑘 ∈ 𝑍 (𝐹‘(𝑘 + 1)) ≤ (𝐹‘𝑘) ∧ 𝑛 ∈ 𝑍) → (𝐹‘(𝑛 + 1)) ≤ (𝐹‘𝑛))
84	79, 83	sylan 580	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑦 ∈ ℝ⁺) ∧ (𝑗 ∈ 𝑍 ∧ 𝑘 ∈ (ℤ_≥‘𝑗))) ∧ 𝑛 ∈ 𝑍) → (𝐹‘(𝑛 + 1)) ≤ (𝐹‘𝑛))
85	76, 84	syldan 591	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑦 ∈ ℝ⁺) ∧ (𝑗 ∈ 𝑍 ∧ 𝑘 ∈ (ℤ_≥‘𝑗))) ∧ 𝑛 ∈ (𝑗...(𝑘 − 1))) → (𝐹‘(𝑛 + 1)) ≤ (𝐹‘𝑛))
86	60, 73, 85	monoord2 13251	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ⁺) ∧ (𝑗 ∈ 𝑍 ∧ 𝑘 ∈ (ℤ_≥‘𝑗))) → (𝐹‘𝑘) ≤ (𝐹‘𝑗))
87	55, 58, 59, 86	lesub1dd 11104	. . . . . . . . 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 10916	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ⁺) ∧ (𝑗 ∈ 𝑍 ∧ 𝑘 ∈ (ℤ_≥‘𝑗))) → ((𝐹‘𝑘) − inf(ran 𝐹, ℝ, < )) ∈ ℝ)
89	58, 59	resubcld 10916	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ⁺) ∧ (𝑗 ∈ 𝑍 ∧ 𝑘 ∈ (ℤ_≥‘𝑗))) → ((𝐹‘𝑗) − inf(ran 𝐹, ℝ, < )) ∈ ℝ)
90	24	ad2antlr 723	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ⁺) ∧ (𝑗 ∈ 𝑍 ∧ 𝑘 ∈ (ℤ_≥‘𝑗))) → 𝑦 ∈ ℝ)
91		lelttr 10578	. . . . . . . . . 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 1364	. . . . . . . . 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 691	. . . . . . . 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 10968	. . . . . . . . 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 1364	. . . . . . . 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 722	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ⁺) ∧ (𝑗 ∈ 𝑍 ∧ 𝑘 ∈ (ℤ_≥‘𝑗))) → ran 𝐹 ⊆ ℝ)
97	3	adantr 481	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ⁺) → 𝐹 Fn 𝑍)
98		fnfvelrn 6713	. . . . . . . . . . . 12 ⊢ ((𝐹 Fn 𝑍 ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) ∈ ran 𝐹)
99	97, 53, 98	syl2an 595	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ⁺) ∧ (𝑗 ∈ 𝑍 ∧ 𝑘 ∈ (ℤ_≥‘𝑗))) → (𝐹‘𝑘) ∈ ran 𝐹)
100	96, 99	sseldd 3890	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ⁺) ∧ (𝑗 ∈ 𝑍 ∧ 𝑘 ∈ (ℤ_≥‘𝑗))) → (𝐹‘𝑘) ∈ ℝ)
101	17	ad2antrr 722	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ⁺) ∧ (𝑗 ∈ 𝑍 ∧ 𝑘 ∈ (ℤ_≥‘𝑗))) → ∃𝑥 ∈ ℝ ∀𝑦 ∈ ran 𝐹 𝑥 ≤ 𝑦)
102		infrelb 11474	. . . . . . . . . . 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 1364	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ⁺) ∧ (𝑗 ∈ 𝑍 ∧ 𝑘 ∈ (ℤ_≥‘𝑗))) → inf(ran 𝐹, ℝ, < ) ≤ (𝐹‘𝑘))
104	59, 100, 103	abssubge0d 14625	. . . . . . . . 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 4972	. . . . . . . 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 295	. . . . . . 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 468	. . . . . 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 3156	. . . . 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 3237	. . . 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 15	. . 3 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ⁺) → ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ_≥‘𝑗)(abs‘((𝐹‘𝑘) − inf(ran 𝐹, ℝ, < ))) < 𝑦)
111	110	ralrimiva 3149	. 2 ⊢ (𝜑 → ∀𝑦 ∈ ℝ⁺ ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ_≥‘𝑗)(abs‘((𝐹‘𝑘) − inf(ran 𝐹, ℝ, < ))) < 𝑦)
112	7	fvexi 6552	. . . 4 ⊢ 𝑍 ∈ V
113		fex 6855	. . . 4 ⊢ ((𝐹:𝑍⟶ℝ ∧ 𝑍 ∈ V) → 𝐹 ∈ V)
114	1, 112, 113	sylancl 586	. . 3 ⊢ (𝜑 → 𝐹 ∈ V)
115		eqidd 2796	. . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) = (𝐹‘𝑘))
116	1	ffvelrnda 6716	. . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) ∈ ℝ)
117	116	recnd 10515	. . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) ∈ ℂ)
118	7, 4, 114, 115, 37, 117	clim2c 14696	. 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 258	1 ⊢ (𝜑 → 𝐹 ⇝ inf(ran 𝐹, ℝ, < ))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 207 ∧ wa 396 ∧ w3a 1080 = wceq 1522 ∈ wcel 2081 ≠ wne 2984 ∀wral 3105 ∃wrex 3106 Vcvv 3437 ⊆ wss 3859 ∅c0 4211 class class class wbr 4962 ran crn 5444 Fn wfn 6220 ⟶wf 6221 ‘cfv 6225 (class class class)co 7016 infcinf 8751 ℂcc 10381 ℝcr 10382 1c1 10384 + caddc 10386 < clt 10521 ≤ cle 10522 − cmin 10717 ℤcz 11829 ℤ_≥cuz 12093 ℝ⁺crp 12239 ...cfz 12742 abscabs 14427 ⇝ cli 14675

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1777 ax-4 1791 ax-5 1888 ax-6 1947 ax-7 1992 ax-8 2083 ax-9 2091 ax-10 2112 ax-11 2126 ax-12 2141 ax-13 2344 ax-ext 2769 ax-rep 5081 ax-sep 5094 ax-nul 5101 ax-pow 5157 ax-pr 5221 ax-un 7319 ax-cnex 10439 ax-resscn 10440 ax-1cn 10441 ax-icn 10442 ax-addcl 10443 ax-addrcl 10444 ax-mulcl 10445 ax-mulrcl 10446 ax-mulcom 10447 ax-addass 10448 ax-mulass 10449 ax-distr 10450 ax-i2m1 10451 ax-1ne0 10452 ax-1rid 10453 ax-rnegex 10454 ax-rrecex 10455 ax-cnre 10456 ax-pre-lttri 10457 ax-pre-lttrn 10458 ax-pre-ltadd 10459 ax-pre-mulgt0 10460 ax-pre-sup 10461

This theorem depends on definitions: df-bi 208 df-an 397 df-or 843 df-3or 1081 df-3an 1082 df-tru 1525 df-ex 1762 df-nf 1766 df-sb 2043 df-mo 2576 df-eu 2612 df-clab 2776 df-cleq 2788 df-clel 2863 df-nfc 2935 df-ne 2985 df-nel 3091 df-ral 3110 df-rex 3111 df-reu 3112 df-rmo 3113 df-rab 3114 df-v 3439 df-sbc 3707 df-csb 3812 df-dif 3862 df-un 3864 df-in 3866 df-ss 3874 df-pss 3876 df-nul 4212 df-if 4382 df-pw 4455 df-sn 4473 df-pr 4475 df-tp 4477 df-op 4479 df-uni 4746 df-iun 4827 df-br 4963 df-opab 5025 df-mpt 5042 df-tr 5064 df-id 5348 df-eprel 5353 df-po 5362 df-so 5363 df-fr 5402 df-we 5404 df-xp 5449 df-rel 5450 df-cnv 5451 df-co 5452 df-dm 5453 df-rn 5454 df-res 5455 df-ima 5456 df-pred 6023 df-ord 6069 df-on 6070 df-lim 6071 df-suc 6072 df-iota 6189 df-fun 6227 df-fn 6228 df-f 6229 df-f1 6230 df-fo 6231 df-f1o 6232 df-fv 6233 df-riota 6977 df-ov 7019 df-oprab 7020 df-mpo 7021 df-om 7437 df-1st 7545 df-2nd 7546 df-wrecs 7798 df-recs 7860 df-rdg 7898 df-er 8139 df-en 8358 df-dom 8359 df-sdom 8360 df-sup 8752 df-inf 8753 df-pnf 10523 df-mnf 10524 df-xr 10525 df-ltxr 10526 df-le 10527 df-sub 10719 df-neg 10720 df-div 11146 df-nn 11487 df-2 11548 df-3 11549 df-n0 11746 df-z 11830 df-uz 12094 df-rp 12240 df-fz 12743 df-seq 13220 df-exp 13280 df-cj 14292 df-re 14293 df-im 14294 df-sqrt 14428 df-abs 14429 df-clim 14679

This theorem is referenced by: climinff 41434 climinf2lem 41529 supcnvlimsup 41563 stirlinglem13 41913

W3C validator