climsup - Metamath Proof Explorer

	Metamath Proof Explorer		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > climsup		Structured version Visualization version GIF version

Theorem climsup 14248

Description: A bounded monotonic sequence converges to the supremum of its range. Theorem 12-5.1 of [Gleason] p. 180. (Contributed by NM, 13-Mar-2005.) (Revised by Mario Carneiro, 10-Feb-2014.)

**Hypotheses**
Ref	Expression
climsup.1	⊢ 𝑍 = (ℤ_≥‘𝑀)
climsup.2	⊢ (𝜑 → 𝑀 ∈ ℤ)
climsup.3	⊢ (𝜑 → 𝐹:𝑍⟶ℝ)
climsup.4	⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) ≤ (𝐹‘(𝑘 + 1)))
climsup.5	⊢ (𝜑 → ∃𝑥 ∈ ℝ ∀𝑘 ∈ 𝑍 (𝐹‘𝑘) ≤ 𝑥)

**Assertion**
Ref	Expression
climsup	⊢ (𝜑 → 𝐹 ⇝ sup(ran 𝐹, ℝ, < ))

Distinct variable groups: 𝑥,𝑘,𝐹 𝜑,𝑘 𝑘,𝑍,𝑥 Allowed substitution hints: 𝜑(𝑥) 𝑀(𝑥,𝑘)

Proof of Theorem climsup Dummy variables 𝑗 𝑛 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		climsup.3	. . . . . . . . . 10 ⊢ (𝜑 → 𝐹:𝑍⟶ℝ)
2		frn 5966	. . . . . . . . . 10 ⊢ (𝐹:𝑍⟶ℝ → ran 𝐹 ⊆ ℝ)
3	1, 2	syl 17	. . . . . . . . 9 ⊢ (𝜑 → ran 𝐹 ⊆ ℝ)
4		ffn 5958	. . . . . . . . . . . 12 ⊢ (𝐹:𝑍⟶ℝ → 𝐹 Fn 𝑍)
5	1, 4	syl 17	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐹 Fn 𝑍)
6		climsup.2	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝑀 ∈ ℤ)
7		uzid 11578	. . . . . . . . . . . . 13 ⊢ (𝑀 ∈ ℤ → 𝑀 ∈ (ℤ_≥‘𝑀))
8	6, 7	syl 17	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑀 ∈ (ℤ_≥‘𝑀))
9		climsup.1	. . . . . . . . . . . 12 ⊢ 𝑍 = (ℤ_≥‘𝑀)
10	8, 9	syl6eleqr 2699	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑀 ∈ 𝑍)
11		fnfvelrn 6264	. . . . . . . . . . 11 ⊢ ((𝐹 Fn 𝑍 ∧ 𝑀 ∈ 𝑍) → (𝐹‘𝑀) ∈ ran 𝐹)
12	5, 10, 11	syl2anc 691	. . . . . . . . . 10 ⊢ (𝜑 → (𝐹‘𝑀) ∈ ran 𝐹)
13		ne0i 3880	. . . . . . . . . 10 ⊢ ((𝐹‘𝑀) ∈ ran 𝐹 → ran 𝐹 ≠ ∅)
14	12, 13	syl 17	. . . . . . . . 9 ⊢ (𝜑 → ran 𝐹 ≠ ∅)
15		climsup.5	. . . . . . . . . 10 ⊢ (𝜑 → ∃𝑥 ∈ ℝ ∀𝑘 ∈ 𝑍 (𝐹‘𝑘) ≤ 𝑥)
16		breq1 4586	. . . . . . . . . . . . 13 ⊢ (𝑦 = (𝐹‘𝑘) → (𝑦 ≤ 𝑥 ↔ (𝐹‘𝑘) ≤ 𝑥))
17	16	ralrn 6270	. . . . . . . . . . . 12 ⊢ (𝐹 Fn 𝑍 → (∀𝑦 ∈ ran 𝐹 𝑦 ≤ 𝑥 ↔ ∀𝑘 ∈ 𝑍 (𝐹‘𝑘) ≤ 𝑥))
18	17	rexbidv 3034	. . . . . . . . . . 11 ⊢ (𝐹 Fn 𝑍 → (∃𝑥 ∈ ℝ ∀𝑦 ∈ ran 𝐹 𝑦 ≤ 𝑥 ↔ ∃𝑥 ∈ ℝ ∀𝑘 ∈ 𝑍 (𝐹‘𝑘) ≤ 𝑥))
19	5, 18	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → (∃𝑥 ∈ ℝ ∀𝑦 ∈ ran 𝐹 𝑦 ≤ 𝑥 ↔ ∃𝑥 ∈ ℝ ∀𝑘 ∈ 𝑍 (𝐹‘𝑘) ≤ 𝑥))
20	15, 19	mpbird 246	. . . . . . . . 9 ⊢ (𝜑 → ∃𝑥 ∈ ℝ ∀𝑦 ∈ ran 𝐹 𝑦 ≤ 𝑥)
21	3, 14, 20	3jca 1235	. . . . . . . 8 ⊢ (𝜑 → (ran 𝐹 ⊆ ℝ ∧ ran 𝐹 ≠ ∅ ∧ ∃𝑥 ∈ ℝ ∀𝑦 ∈ ran 𝐹 𝑦 ≤ 𝑥))
22		suprcl 10862	. . . . . . . 8 ⊢ ((ran 𝐹 ⊆ ℝ ∧ ran 𝐹 ≠ ∅ ∧ ∃𝑥 ∈ ℝ ∀𝑦 ∈ ran 𝐹 𝑦 ≤ 𝑥) → sup(ran 𝐹, ℝ, < ) ∈ ℝ)
23	21, 22	syl 17	. . . . . . 7 ⊢ (𝜑 → sup(ran 𝐹, ℝ, < ) ∈ ℝ)
24		ltsubrp 11742	. . . . . . 7 ⊢ ((sup(ran 𝐹, ℝ, < ) ∈ ℝ ∧ 𝑦 ∈ ℝ⁺) → (sup(ran 𝐹, ℝ, < ) − 𝑦) < sup(ran 𝐹, ℝ, < ))
25	23, 24	sylan 487	. . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ⁺) → (sup(ran 𝐹, ℝ, < ) − 𝑦) < sup(ran 𝐹, ℝ, < ))
26	21	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ⁺) → (ran 𝐹 ⊆ ℝ ∧ ran 𝐹 ≠ ∅ ∧ ∃𝑥 ∈ ℝ ∀𝑦 ∈ ran 𝐹 𝑦 ≤ 𝑥))
27		rpre 11715	. . . . . . . 8 ⊢ (𝑦 ∈ ℝ⁺ → 𝑦 ∈ ℝ)
28		resubcl 10224	. . . . . . . 8 ⊢ ((sup(ran 𝐹, ℝ, < ) ∈ ℝ ∧ 𝑦 ∈ ℝ) → (sup(ran 𝐹, ℝ, < ) − 𝑦) ∈ ℝ)
29	23, 27, 28	syl2an 493	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ⁺) → (sup(ran 𝐹, ℝ, < ) − 𝑦) ∈ ℝ)
30		suprlub 10864	. . . . . . 7 ⊢ (((ran 𝐹 ⊆ ℝ ∧ ran 𝐹 ≠ ∅ ∧ ∃𝑥 ∈ ℝ ∀𝑦 ∈ ran 𝐹 𝑦 ≤ 𝑥) ∧ (sup(ran 𝐹, ℝ, < ) − 𝑦) ∈ ℝ) → ((sup(ran 𝐹, ℝ, < ) − 𝑦) < sup(ran 𝐹, ℝ, < ) ↔ ∃𝑘 ∈ ran 𝐹(sup(ran 𝐹, ℝ, < ) − 𝑦) < 𝑘))
31	26, 29, 30	syl2anc 691	. . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ⁺) → ((sup(ran 𝐹, ℝ, < ) − 𝑦) < sup(ran 𝐹, ℝ, < ) ↔ ∃𝑘 ∈ ran 𝐹(sup(ran 𝐹, ℝ, < ) − 𝑦) < 𝑘))
32	25, 31	mpbid 221	. . . . 5 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ⁺) → ∃𝑘 ∈ ran 𝐹(sup(ran 𝐹, ℝ, < ) − 𝑦) < 𝑘)
33		breq2 4587	. . . . . . . 8 ⊢ (𝑘 = (𝐹‘𝑗) → ((sup(ran 𝐹, ℝ, < ) − 𝑦) < 𝑘 ↔ (sup(ran 𝐹, ℝ, < ) − 𝑦) < (𝐹‘𝑗)))
34	33	rexrn 6269	. . . . . . 7 ⊢ (𝐹 Fn 𝑍 → (∃𝑘 ∈ ran 𝐹(sup(ran 𝐹, ℝ, < ) − 𝑦) < 𝑘 ↔ ∃𝑗 ∈ 𝑍 (sup(ran 𝐹, ℝ, < ) − 𝑦) < (𝐹‘𝑗)))
35	5, 34	syl 17	. . . . . 6 ⊢ (𝜑 → (∃𝑘 ∈ ran 𝐹(sup(ran 𝐹, ℝ, < ) − 𝑦) < 𝑘 ↔ ∃𝑗 ∈ 𝑍 (sup(ran 𝐹, ℝ, < ) − 𝑦) < (𝐹‘𝑗)))
36	35	biimpa 500	. . . . 5 ⊢ ((𝜑 ∧ ∃𝑘 ∈ ran 𝐹(sup(ran 𝐹, ℝ, < ) − 𝑦) < 𝑘) → ∃𝑗 ∈ 𝑍 (sup(ran 𝐹, ℝ, < ) − 𝑦) < (𝐹‘𝑗))
37	32, 36	syldan 486	. . . 4 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ⁺) → ∃𝑗 ∈ 𝑍 (sup(ran 𝐹, ℝ, < ) − 𝑦) < (𝐹‘𝑗))
38		ffvelrn 6265	. . . . . . . . . . . 12 ⊢ ((𝐹:𝑍⟶ℝ ∧ 𝑗 ∈ 𝑍) → (𝐹‘𝑗) ∈ ℝ)
39	1, 38	sylan 487	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → (𝐹‘𝑗) ∈ ℝ)
40	39	ad2ant2r 779	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ⁺) ∧ (𝑗 ∈ 𝑍 ∧ 𝑘 ∈ (ℤ_≥‘𝑗))) → (𝐹‘𝑗) ∈ ℝ)
41	1	adantr 480	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ⁺) → 𝐹:𝑍⟶ℝ)
42	9	uztrn2 11581	. . . . . . . . . . 11 ⊢ ((𝑗 ∈ 𝑍 ∧ 𝑘 ∈ (ℤ_≥‘𝑗)) → 𝑘 ∈ 𝑍)
43		ffvelrn 6265	. . . . . . . . . . 11 ⊢ ((𝐹:𝑍⟶ℝ ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) ∈ ℝ)
44	41, 42, 43	syl2an 493	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ⁺) ∧ (𝑗 ∈ 𝑍 ∧ 𝑘 ∈ (ℤ_≥‘𝑗))) → (𝐹‘𝑘) ∈ ℝ)
45	23	ad2antrr 758	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ⁺) ∧ (𝑗 ∈ 𝑍 ∧ 𝑘 ∈ (ℤ_≥‘𝑗))) → sup(ran 𝐹, ℝ, < ) ∈ ℝ)
46		simprr 792	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ⁺) ∧ (𝑗 ∈ 𝑍 ∧ 𝑘 ∈ (ℤ_≥‘𝑗))) → 𝑘 ∈ (ℤ_≥‘𝑗))
47		fzssuz 12253	. . . . . . . . . . . . . 14 ⊢ (𝑗...𝑘) ⊆ (ℤ_≥‘𝑗)
48		uzss 11584	. . . . . . . . . . . . . . . . 17 ⊢ (𝑗 ∈ (ℤ_≥‘𝑀) → (ℤ_≥‘𝑗) ⊆ (ℤ_≥‘𝑀))
49	48, 9	syl6sseqr 3615	. . . . . . . . . . . . . . . 16 ⊢ (𝑗 ∈ (ℤ_≥‘𝑀) → (ℤ_≥‘𝑗) ⊆ 𝑍)
50	49, 9	eleq2s 2706	. . . . . . . . . . . . . . 15 ⊢ (𝑗 ∈ 𝑍 → (ℤ_≥‘𝑗) ⊆ 𝑍)
51	50	ad2antrl 760	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ⁺) ∧ (𝑗 ∈ 𝑍 ∧ 𝑘 ∈ (ℤ_≥‘𝑗))) → (ℤ_≥‘𝑗) ⊆ 𝑍)
52	47, 51	syl5ss 3579	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ⁺) ∧ (𝑗 ∈ 𝑍 ∧ 𝑘 ∈ (ℤ_≥‘𝑗))) → (𝑗...𝑘) ⊆ 𝑍)
53		ffvelrn 6265	. . . . . . . . . . . . . . . 16 ⊢ ((𝐹:𝑍⟶ℝ ∧ 𝑛 ∈ 𝑍) → (𝐹‘𝑛) ∈ ℝ)
54	53	ralrimiva 2949	. . . . . . . . . . . . . . 15 ⊢ (𝐹:𝑍⟶ℝ → ∀𝑛 ∈ 𝑍 (𝐹‘𝑛) ∈ ℝ)
55	1, 54	syl 17	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ∀𝑛 ∈ 𝑍 (𝐹‘𝑛) ∈ ℝ)
56	55	ad2antrr 758	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ⁺) ∧ (𝑗 ∈ 𝑍 ∧ 𝑘 ∈ (ℤ_≥‘𝑗))) → ∀𝑛 ∈ 𝑍 (𝐹‘𝑛) ∈ ℝ)
57		ssralv 3629	. . . . . . . . . . . . 13 ⊢ ((𝑗...𝑘) ⊆ 𝑍 → (∀𝑛 ∈ 𝑍 (𝐹‘𝑛) ∈ ℝ → ∀𝑛 ∈ (𝑗...𝑘)(𝐹‘𝑛) ∈ ℝ))
58	52, 56, 57	sylc 63	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ⁺) ∧ (𝑗 ∈ 𝑍 ∧ 𝑘 ∈ (ℤ_≥‘𝑗))) → ∀𝑛 ∈ (𝑗...𝑘)(𝐹‘𝑛) ∈ ℝ)
59	58	r19.21bi 2916	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑦 ∈ ℝ⁺) ∧ (𝑗 ∈ 𝑍 ∧ 𝑘 ∈ (ℤ_≥‘𝑗))) ∧ 𝑛 ∈ (𝑗...𝑘)) → (𝐹‘𝑛) ∈ ℝ)
60		fzssuz 12253	. . . . . . . . . . . . . 14 ⊢ (𝑗...(𝑘 − 1)) ⊆ (ℤ_≥‘𝑗)
61	60, 51	syl5ss 3579	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ⁺) ∧ (𝑗 ∈ 𝑍 ∧ 𝑘 ∈ (ℤ_≥‘𝑗))) → (𝑗...(𝑘 − 1)) ⊆ 𝑍)
62	61	sselda 3568	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑦 ∈ ℝ⁺) ∧ (𝑗 ∈ 𝑍 ∧ 𝑘 ∈ (ℤ_≥‘𝑗))) ∧ 𝑛 ∈ (𝑗...(𝑘 − 1))) → 𝑛 ∈ 𝑍)
63		climsup.4	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) ≤ (𝐹‘(𝑘 + 1)))
64	63	ralrimiva 2949	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ∀𝑘 ∈ 𝑍 (𝐹‘𝑘) ≤ (𝐹‘(𝑘 + 1)))
65	64	ad2antrr 758	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ⁺) ∧ (𝑗 ∈ 𝑍 ∧ 𝑘 ∈ (ℤ_≥‘𝑗))) → ∀𝑘 ∈ 𝑍 (𝐹‘𝑘) ≤ (𝐹‘(𝑘 + 1)))
66		fveq2 6103	. . . . . . . . . . . . . . 15 ⊢ (𝑘 = 𝑛 → (𝐹‘𝑘) = (𝐹‘𝑛))
67		oveq1 6556	. . . . . . . . . . . . . . . 16 ⊢ (𝑘 = 𝑛 → (𝑘 + 1) = (𝑛 + 1))
68	67	fveq2d 6107	. . . . . . . . . . . . . . 15 ⊢ (𝑘 = 𝑛 → (𝐹‘(𝑘 + 1)) = (𝐹‘(𝑛 + 1)))
69	66, 68	breq12d 4596	. . . . . . . . . . . . . 14 ⊢ (𝑘 = 𝑛 → ((𝐹‘𝑘) ≤ (𝐹‘(𝑘 + 1)) ↔ (𝐹‘𝑛) ≤ (𝐹‘(𝑛 + 1))))
70	69	rspccva 3281	. . . . . . . . . . . . 13 ⊢ ((∀𝑘 ∈ 𝑍 (𝐹‘𝑘) ≤ (𝐹‘(𝑘 + 1)) ∧ 𝑛 ∈ 𝑍) → (𝐹‘𝑛) ≤ (𝐹‘(𝑛 + 1)))
71	65, 70	sylan 487	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑦 ∈ ℝ⁺) ∧ (𝑗 ∈ 𝑍 ∧ 𝑘 ∈ (ℤ_≥‘𝑗))) ∧ 𝑛 ∈ 𝑍) → (𝐹‘𝑛) ≤ (𝐹‘(𝑛 + 1)))
72	62, 71	syldan 486	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑦 ∈ ℝ⁺) ∧ (𝑗 ∈ 𝑍 ∧ 𝑘 ∈ (ℤ_≥‘𝑗))) ∧ 𝑛 ∈ (𝑗...(𝑘 − 1))) → (𝐹‘𝑛) ≤ (𝐹‘(𝑛 + 1)))
73	46, 59, 72	monoord 12693	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ⁺) ∧ (𝑗 ∈ 𝑍 ∧ 𝑘 ∈ (ℤ_≥‘𝑗))) → (𝐹‘𝑗) ≤ (𝐹‘𝑘))
74	40, 44, 45, 73	lesub2dd 10523	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ⁺) ∧ (𝑗 ∈ 𝑍 ∧ 𝑘 ∈ (ℤ_≥‘𝑗))) → (sup(ran 𝐹, ℝ, < ) − (𝐹‘𝑘)) ≤ (sup(ran 𝐹, ℝ, < ) − (𝐹‘𝑗)))
75	45, 44	resubcld 10337	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ⁺) ∧ (𝑗 ∈ 𝑍 ∧ 𝑘 ∈ (ℤ_≥‘𝑗))) → (sup(ran 𝐹, ℝ, < ) − (𝐹‘𝑘)) ∈ ℝ)
76	45, 40	resubcld 10337	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ⁺) ∧ (𝑗 ∈ 𝑍 ∧ 𝑘 ∈ (ℤ_≥‘𝑗))) → (sup(ran 𝐹, ℝ, < ) − (𝐹‘𝑗)) ∈ ℝ)
77	27	ad2antlr 759	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ⁺) ∧ (𝑗 ∈ 𝑍 ∧ 𝑘 ∈ (ℤ_≥‘𝑗))) → 𝑦 ∈ ℝ)
78		lelttr 10007	. . . . . . . . . 10 ⊢ (((sup(ran 𝐹, ℝ, < ) − (𝐹‘𝑘)) ∈ ℝ ∧ (sup(ran 𝐹, ℝ, < ) − (𝐹‘𝑗)) ∈ ℝ ∧ 𝑦 ∈ ℝ) → (((sup(ran 𝐹, ℝ, < ) − (𝐹‘𝑘)) ≤ (sup(ran 𝐹, ℝ, < ) − (𝐹‘𝑗)) ∧ (sup(ran 𝐹, ℝ, < ) − (𝐹‘𝑗)) < 𝑦) → (sup(ran 𝐹, ℝ, < ) − (𝐹‘𝑘)) < 𝑦))
79	75, 76, 77, 78	syl3anc 1318	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ⁺) ∧ (𝑗 ∈ 𝑍 ∧ 𝑘 ∈ (ℤ_≥‘𝑗))) → (((sup(ran 𝐹, ℝ, < ) − (𝐹‘𝑘)) ≤ (sup(ran 𝐹, ℝ, < ) − (𝐹‘𝑗)) ∧ (sup(ran 𝐹, ℝ, < ) − (𝐹‘𝑗)) < 𝑦) → (sup(ran 𝐹, ℝ, < ) − (𝐹‘𝑘)) < 𝑦))
80	74, 79	mpand 707	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ⁺) ∧ (𝑗 ∈ 𝑍 ∧ 𝑘 ∈ (ℤ_≥‘𝑗))) → ((sup(ran 𝐹, ℝ, < ) − (𝐹‘𝑗)) < 𝑦 → (sup(ran 𝐹, ℝ, < ) − (𝐹‘𝑘)) < 𝑦))
81		ltsub23 10387	. . . . . . . . 9 ⊢ ((sup(ran 𝐹, ℝ, < ) ∈ ℝ ∧ 𝑦 ∈ ℝ ∧ (𝐹‘𝑗) ∈ ℝ) → ((sup(ran 𝐹, ℝ, < ) − 𝑦) < (𝐹‘𝑗) ↔ (sup(ran 𝐹, ℝ, < ) − (𝐹‘𝑗)) < 𝑦))
82	45, 77, 40, 81	syl3anc 1318	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ⁺) ∧ (𝑗 ∈ 𝑍 ∧ 𝑘 ∈ (ℤ_≥‘𝑗))) → ((sup(ran 𝐹, ℝ, < ) − 𝑦) < (𝐹‘𝑗) ↔ (sup(ran 𝐹, ℝ, < ) − (𝐹‘𝑗)) < 𝑦))
83	21	ad2antrr 758	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ⁺) ∧ (𝑗 ∈ 𝑍 ∧ 𝑘 ∈ (ℤ_≥‘𝑗))) → (ran 𝐹 ⊆ ℝ ∧ ran 𝐹 ≠ ∅ ∧ ∃𝑥 ∈ ℝ ∀𝑦 ∈ ran 𝐹 𝑦 ≤ 𝑥))
84	5	adantr 480	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ⁺) → 𝐹 Fn 𝑍)
85		fnfvelrn 6264	. . . . . . . . . . . 12 ⊢ ((𝐹 Fn 𝑍 ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) ∈ ran 𝐹)
86	84, 42, 85	syl2an 493	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ⁺) ∧ (𝑗 ∈ 𝑍 ∧ 𝑘 ∈ (ℤ_≥‘𝑗))) → (𝐹‘𝑘) ∈ ran 𝐹)
87		suprub 10863	. . . . . . . . . . 11 ⊢ (((ran 𝐹 ⊆ ℝ ∧ ran 𝐹 ≠ ∅ ∧ ∃𝑥 ∈ ℝ ∀𝑦 ∈ ran 𝐹 𝑦 ≤ 𝑥) ∧ (𝐹‘𝑘) ∈ ran 𝐹) → (𝐹‘𝑘) ≤ sup(ran 𝐹, ℝ, < ))
88	83, 86, 87	syl2anc 691	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ⁺) ∧ (𝑗 ∈ 𝑍 ∧ 𝑘 ∈ (ℤ_≥‘𝑗))) → (𝐹‘𝑘) ≤ sup(ran 𝐹, ℝ, < ))
89	44, 45, 88	abssuble0d 14019	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ⁺) ∧ (𝑗 ∈ 𝑍 ∧ 𝑘 ∈ (ℤ_≥‘𝑗))) → (abs‘((𝐹‘𝑘) − sup(ran 𝐹, ℝ, < ))) = (sup(ran 𝐹, ℝ, < ) − (𝐹‘𝑘)))
90	89	breq1d 4593	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ⁺) ∧ (𝑗 ∈ 𝑍 ∧ 𝑘 ∈ (ℤ_≥‘𝑗))) → ((abs‘((𝐹‘𝑘) − sup(ran 𝐹, ℝ, < ))) < 𝑦 ↔ (sup(ran 𝐹, ℝ, < ) − (𝐹‘𝑘)) < 𝑦))
91	80, 82, 90	3imtr4d 282	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ⁺) ∧ (𝑗 ∈ 𝑍 ∧ 𝑘 ∈ (ℤ_≥‘𝑗))) → ((sup(ran 𝐹, ℝ, < ) − 𝑦) < (𝐹‘𝑗) → (abs‘((𝐹‘𝑘) − sup(ran 𝐹, ℝ, < ))) < 𝑦))
92	91	anassrs 678	. . . . . 6 ⊢ ((((𝜑 ∧ 𝑦 ∈ ℝ⁺) ∧ 𝑗 ∈ 𝑍) ∧ 𝑘 ∈ (ℤ_≥‘𝑗)) → ((sup(ran 𝐹, ℝ, < ) − 𝑦) < (𝐹‘𝑗) → (abs‘((𝐹‘𝑘) − sup(ran 𝐹, ℝ, < ))) < 𝑦))
93	92	ralrimdva 2952	. . . . 5 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ⁺) ∧ 𝑗 ∈ 𝑍) → ((sup(ran 𝐹, ℝ, < ) − 𝑦) < (𝐹‘𝑗) → ∀𝑘 ∈ (ℤ_≥‘𝑗)(abs‘((𝐹‘𝑘) − sup(ran 𝐹, ℝ, < ))) < 𝑦))
94	93	reximdva 3000	. . . 4 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ⁺) → (∃𝑗 ∈ 𝑍 (sup(ran 𝐹, ℝ, < ) − 𝑦) < (𝐹‘𝑗) → ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ_≥‘𝑗)(abs‘((𝐹‘𝑘) − sup(ran 𝐹, ℝ, < ))) < 𝑦))
95	37, 94	mpd 15	. . 3 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ⁺) → ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ_≥‘𝑗)(abs‘((𝐹‘𝑘) − sup(ran 𝐹, ℝ, < ))) < 𝑦)
96	95	ralrimiva 2949	. 2 ⊢ (𝜑 → ∀𝑦 ∈ ℝ⁺ ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ_≥‘𝑗)(abs‘((𝐹‘𝑘) − sup(ran 𝐹, ℝ, < ))) < 𝑦)
97		fvex 6113	. . . . 5 ⊢ (ℤ_≥‘𝑀) ∈ V
98	9, 97	eqeltri 2684	. . . 4 ⊢ 𝑍 ∈ V
99		fex 6394	. . . 4 ⊢ ((𝐹:𝑍⟶ℝ ∧ 𝑍 ∈ V) → 𝐹 ∈ V)
100	1, 98, 99	sylancl 693	. . 3 ⊢ (𝜑 → 𝐹 ∈ V)
101		eqidd 2611	. . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) = (𝐹‘𝑘))
102	23	recnd 9947	. . 3 ⊢ (𝜑 → sup(ran 𝐹, ℝ, < ) ∈ ℂ)
103	1, 43	sylan 487	. . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) ∈ ℝ)
104	103	recnd 9947	. . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) ∈ ℂ)
105	9, 6, 100, 101, 102, 104	clim2c 14084	. 2 ⊢ (𝜑 → (𝐹 ⇝ sup(ran 𝐹, ℝ, < ) ↔ ∀𝑦 ∈ ℝ⁺ ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ_≥‘𝑗)(abs‘((𝐹‘𝑘) − sup(ran 𝐹, ℝ, < ))) < 𝑦))
106	96, 105	mpbird 246	1 ⊢ (𝜑 → 𝐹 ⇝ sup(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 supcsup 8229 ℝcr 9814 1c1 9816 + caddc 9818 < clt 9953 ≤ cle 9954 − cmin 10145 ℤcz 11254 ℤ_≥cuz 11563 ℝ⁺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-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: isumsup2 14417 climcnds 14422 itg1climres 23287 itg2monolem1 23323 itg2i1fseq 23328 itg2i1fseq2 23329 emcllem6 24527 lmdvg 29327 esumpcvgval 29467 meaiuninclem 39373

W3C validator