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 15761

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	1	frnd 6715	. . . . . . . . 9 ⊢ (𝜑 → ran 𝐹 ⊆ ℝ)
3	1	ffnd 6707	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐹 Fn 𝑍)
4		climsup.2	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝑀 ∈ ℤ)
5		uzid 12906	. . . . . . . . . . . . 13 ⊢ (𝑀 ∈ ℤ → 𝑀 ∈ (ℤ_≥‘𝑀))
6	4, 5	syl 18	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑀 ∈ (ℤ_≥‘𝑀))
7		climsup.1	. . . . . . . . . . . 12 ⊢ 𝑍 = (ℤ_≥‘𝑀)
8	6, 7	eleqtrrdi 2873	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑀 ∈ 𝑍)
9		fnfvelrn 7077	. . . . . . . . . . 11 ⊢ ((𝐹 Fn 𝑍 ∧ 𝑀 ∈ 𝑍) → (𝐹‘𝑀) ∈ ran 𝐹)
10	3, 8, 9	syl2anc 596	. . . . . . . . . 10 ⊢ (𝜑 → (𝐹‘𝑀) ∈ ran 𝐹)
11	10	ne0d 4291	. . . . . . . . 9 ⊢ (𝜑 → ran 𝐹 ≠ ∅)
12		climsup.5	. . . . . . . . . 10 ⊢ (𝜑 → ∃𝑥 ∈ ℝ ∀𝑘 ∈ 𝑍 (𝐹‘𝑘) ≤ 𝑥)
13		breq1 5110	. . . . . . . . . . . . 13 ⊢ (𝑦 = (𝐹‘𝑘) → (𝑦 ≤ 𝑥 ↔ (𝐹‘𝑘) ≤ 𝑥))
14	13	ralrn 7085	. . . . . . . . . . . 12 ⊢ (𝐹 Fn 𝑍 → (∀𝑦 ∈ ran 𝐹 𝑦 ≤ 𝑥 ↔ ∀𝑘 ∈ 𝑍 (𝐹‘𝑘) ≤ 𝑥))
15	14	rexbidv 3188	. . . . . . . . . . 11 ⊢ (𝐹 Fn 𝑍 → (∃𝑥 ∈ ℝ ∀𝑦 ∈ ran 𝐹 𝑦 ≤ 𝑥 ↔ ∃𝑥 ∈ ℝ ∀𝑘 ∈ 𝑍 (𝐹‘𝑘) ≤ 𝑥))
16	3, 15	syl 18	. . . . . . . . . 10 ⊢ (𝜑 → (∃𝑥 ∈ ℝ ∀𝑦 ∈ ran 𝐹 𝑦 ≤ 𝑥 ↔ ∃𝑥 ∈ ℝ ∀𝑘 ∈ 𝑍 (𝐹‘𝑘) ≤ 𝑥))
17	12, 16	mpbird 260	. . . . . . . . 9 ⊢ (𝜑 → ∃𝑥 ∈ ℝ ∀𝑦 ∈ ran 𝐹 𝑦 ≤ 𝑥)
18	2, 11, 17	3jca 1146	. . . . . . . 8 ⊢ (𝜑 → (ran 𝐹 ⊆ ℝ ∧ ran 𝐹 ≠ ∅ ∧ ∃𝑥 ∈ ℝ ∀𝑦 ∈ ran 𝐹 𝑦 ≤ 𝑥))
19		suprcl 12203	. . . . . . . 8 ⊢ ((ran 𝐹 ⊆ ℝ ∧ ran 𝐹 ≠ ∅ ∧ ∃𝑥 ∈ ℝ ∀𝑦 ∈ ran 𝐹 𝑦 ≤ 𝑥) → sup(ran 𝐹, ℝ, < ) ∈ ℝ)
20	18, 19	syl 18	. . . . . . 7 ⊢ (𝜑 → sup(ran 𝐹, ℝ, < ) ∈ ℝ)
21		ltsubrp 13084	. . . . . . 7 ⊢ ((sup(ran 𝐹, ℝ, < ) ∈ ℝ ∧ 𝑦 ∈ ℝ⁺) → (sup(ran 𝐹, ℝ, < ) − 𝑦) < sup(ran 𝐹, ℝ, < ))
22	20, 21	sylan 592	. . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ⁺) → (sup(ran 𝐹, ℝ, < ) − 𝑦) < sup(ran 𝐹, ℝ, < ))
23	18	adantr 486	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ⁺) → (ran 𝐹 ⊆ ℝ ∧ ran 𝐹 ≠ ∅ ∧ ∃𝑥 ∈ ℝ ∀𝑦 ∈ ran 𝐹 𝑦 ≤ 𝑥))
24		rpre 13055	. . . . . . . 8 ⊢ (𝑦 ∈ ℝ⁺ → 𝑦 ∈ ℝ)
25		resubcl 11550	. . . . . . . 8 ⊢ ((sup(ran 𝐹, ℝ, < ) ∈ ℝ ∧ 𝑦 ∈ ℝ) → (sup(ran 𝐹, ℝ, < ) − 𝑦) ∈ ℝ)
26	20, 24, 25	syl2an 608	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ⁺) → (sup(ran 𝐹, ℝ, < ) − 𝑦) ∈ ℝ)
27		suprlub 12207	. . . . . . 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))
28	23, 26, 27	syl2anc 596	. . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ⁺) → ((sup(ran 𝐹, ℝ, < ) − 𝑦) < sup(ran 𝐹, ℝ, < ) ↔ ∃𝑘 ∈ ran 𝐹(sup(ran 𝐹, ℝ, < ) − 𝑦) < 𝑘))
29	22, 28	mpbid 235	. . . . 5 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ⁺) → ∃𝑘 ∈ ran 𝐹(sup(ran 𝐹, ℝ, < ) − 𝑦) < 𝑘)
30		breq2 5111	. . . . . . . 8 ⊢ (𝑘 = (𝐹‘𝑗) → ((sup(ran 𝐹, ℝ, < ) − 𝑦) < 𝑘 ↔ (sup(ran 𝐹, ℝ, < ) − 𝑦) < (𝐹‘𝑗)))
31	30	rexrn 7084	. . . . . . 7 ⊢ (𝐹 Fn 𝑍 → (∃𝑘 ∈ ran 𝐹(sup(ran 𝐹, ℝ, < ) − 𝑦) < 𝑘 ↔ ∃𝑗 ∈ 𝑍 (sup(ran 𝐹, ℝ, < ) − 𝑦) < (𝐹‘𝑗)))
32	3, 31	syl 18	. . . . . 6 ⊢ (𝜑 → (∃𝑘 ∈ ran 𝐹(sup(ran 𝐹, ℝ, < ) − 𝑦) < 𝑘 ↔ ∃𝑗 ∈ 𝑍 (sup(ran 𝐹, ℝ, < ) − 𝑦) < (𝐹‘𝑗)))
33	32	biimpa 482	. . . . 5 ⊢ ((𝜑 ∧ ∃𝑘 ∈ ran 𝐹(sup(ran 𝐹, ℝ, < ) − 𝑦) < 𝑘) → ∃𝑗 ∈ 𝑍 (sup(ran 𝐹, ℝ, < ) − 𝑦) < (𝐹‘𝑗))
34	29, 33	syldan 603	. . . 4 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ⁺) → ∃𝑗 ∈ 𝑍 (sup(ran 𝐹, ℝ, < ) − 𝑦) < (𝐹‘𝑗))
35		ffvelcdm 7078	. . . . . . . . . . . 12 ⊢ ((𝐹:𝑍⟶ℝ ∧ 𝑗 ∈ 𝑍) → (𝐹‘𝑗) ∈ ℝ)
36	1, 35	sylan 592	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → (𝐹‘𝑗) ∈ ℝ)
37	36	ad2ant2r 760	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ⁺) ∧ (𝑗 ∈ 𝑍 ∧ 𝑘 ∈ (ℤ_≥‘𝑗))) → (𝐹‘𝑗) ∈ ℝ)
38	1	adantr 486	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ⁺) → 𝐹:𝑍⟶ℝ)
39	7	uztrn2 12910	. . . . . . . . . . 11 ⊢ ((𝑗 ∈ 𝑍 ∧ 𝑘 ∈ (ℤ_≥‘𝑗)) → 𝑘 ∈ 𝑍)
40		ffvelcdm 7078	. . . . . . . . . . 11 ⊢ ((𝐹:𝑍⟶ℝ ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) ∈ ℝ)
41	38, 39, 40	syl2an 608	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ⁺) ∧ (𝑗 ∈ 𝑍 ∧ 𝑘 ∈ (ℤ_≥‘𝑗))) → (𝐹‘𝑘) ∈ ℝ)
42	20	ad2antrr 739	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ⁺) ∧ (𝑗 ∈ 𝑍 ∧ 𝑘 ∈ (ℤ_≥‘𝑗))) → sup(ran 𝐹, ℝ, < ) ∈ ℝ)
43		simprr 785	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ⁺) ∧ (𝑗 ∈ 𝑍 ∧ 𝑘 ∈ (ℤ_≥‘𝑗))) → 𝑘 ∈ (ℤ_≥‘𝑗))
44		fzssuz 13624	. . . . . . . . . . . . . 14 ⊢ (𝑗...𝑘) ⊆ (ℤ_≥‘𝑗)
45		uzss 12914	. . . . . . . . . . . . . . . . 17 ⊢ (𝑗 ∈ (ℤ_≥‘𝑀) → (ℤ_≥‘𝑗) ⊆ (ℤ_≥‘𝑀))
46	45, 7	sseqtrrdi 3975	. . . . . . . . . . . . . . . 16 ⊢ (𝑗 ∈ (ℤ_≥‘𝑀) → (ℤ_≥‘𝑗) ⊆ 𝑍)
47	46, 7	eleq2s 2880	. . . . . . . . . . . . . . 15 ⊢ (𝑗 ∈ 𝑍 → (ℤ_≥‘𝑗) ⊆ 𝑍)
48	47	ad2antrl 741	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ⁺) ∧ (𝑗 ∈ 𝑍 ∧ 𝑘 ∈ (ℤ_≥‘𝑗))) → (ℤ_≥‘𝑗) ⊆ 𝑍)
49	44, 48	sstrid 3945	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ⁺) ∧ (𝑗 ∈ 𝑍 ∧ 𝑘 ∈ (ℤ_≥‘𝑗))) → (𝑗...𝑘) ⊆ 𝑍)
50		ffvelcdm 7078	. . . . . . . . . . . . . . . 16 ⊢ ((𝐹:𝑍⟶ℝ ∧ 𝑛 ∈ 𝑍) → (𝐹‘𝑛) ∈ ℝ)
51	50	ralrimiva 3156	. . . . . . . . . . . . . . 15 ⊢ (𝐹:𝑍⟶ℝ → ∀𝑛 ∈ 𝑍 (𝐹‘𝑛) ∈ ℝ)
52	1, 51	syl 18	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ∀𝑛 ∈ 𝑍 (𝐹‘𝑛) ∈ ℝ)
53	52	ad2antrr 739	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ⁺) ∧ (𝑗 ∈ 𝑍 ∧ 𝑘 ∈ (ℤ_≥‘𝑗))) → ∀𝑛 ∈ 𝑍 (𝐹‘𝑛) ∈ ℝ)
54		ssralv 4003	. . . . . . . . . . . . 13 ⊢ ((𝑗...𝑘) ⊆ 𝑍 → (∀𝑛 ∈ 𝑍 (𝐹‘𝑛) ∈ ℝ → ∀𝑛 ∈ (𝑗...𝑘)(𝐹‘𝑛) ∈ ℝ))
55	49, 53, 54	sylc 66	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ⁺) ∧ (𝑗 ∈ 𝑍 ∧ 𝑘 ∈ (ℤ_≥‘𝑗))) → ∀𝑛 ∈ (𝑗...𝑘)(𝐹‘𝑛) ∈ ℝ)
56	55	r19.21bi 3256	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑦 ∈ ℝ⁺) ∧ (𝑗 ∈ 𝑍 ∧ 𝑘 ∈ (ℤ_≥‘𝑗))) ∧ 𝑛 ∈ (𝑗...𝑘)) → (𝐹‘𝑛) ∈ ℝ)
57		fzssuz 13624	. . . . . . . . . . . . . 14 ⊢ (𝑗...(𝑘 − 1)) ⊆ (ℤ_≥‘𝑗)
58	57, 48	sstrid 3945	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ⁺) ∧ (𝑗 ∈ 𝑍 ∧ 𝑘 ∈ (ℤ_≥‘𝑗))) → (𝑗...(𝑘 − 1)) ⊆ 𝑍)
59	58	sselda 3934	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑦 ∈ ℝ⁺) ∧ (𝑗 ∈ 𝑍 ∧ 𝑘 ∈ (ℤ_≥‘𝑗))) ∧ 𝑛 ∈ (𝑗...(𝑘 − 1))) → 𝑛 ∈ 𝑍)
60		climsup.4	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) ≤ (𝐹‘(𝑘 + 1)))
61	60	ralrimiva 3156	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ∀𝑘 ∈ 𝑍 (𝐹‘𝑘) ≤ (𝐹‘(𝑘 + 1)))
62	61	ad2antrr 739	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ⁺) ∧ (𝑗 ∈ 𝑍 ∧ 𝑘 ∈ (ℤ_≥‘𝑗))) → ∀𝑘 ∈ 𝑍 (𝐹‘𝑘) ≤ (𝐹‘(𝑘 + 1)))
63		fveq2 6882	. . . . . . . . . . . . . . 15 ⊢ (𝑘 = 𝑛 → (𝐹‘𝑘) = (𝐹‘𝑛))
64		fvoveq1 7440	. . . . . . . . . . . . . . 15 ⊢ (𝑘 = 𝑛 → (𝐹‘(𝑘 + 1)) = (𝐹‘(𝑛 + 1)))
65	63, 64	breq12d 5120	. . . . . . . . . . . . . 14 ⊢ (𝑘 = 𝑛 → ((𝐹‘𝑘) ≤ (𝐹‘(𝑘 + 1)) ↔ (𝐹‘𝑛) ≤ (𝐹‘(𝑛 + 1))))
66	65	rspccva 3578	. . . . . . . . . . . . 13 ⊢ ((∀𝑘 ∈ 𝑍 (𝐹‘𝑘) ≤ (𝐹‘(𝑘 + 1)) ∧ 𝑛 ∈ 𝑍) → (𝐹‘𝑛) ≤ (𝐹‘(𝑛 + 1)))
67	62, 66	sylan 592	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑦 ∈ ℝ⁺) ∧ (𝑗 ∈ 𝑍 ∧ 𝑘 ∈ (ℤ_≥‘𝑗))) ∧ 𝑛 ∈ 𝑍) → (𝐹‘𝑛) ≤ (𝐹‘(𝑛 + 1)))
68	59, 67	syldan 603	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑦 ∈ ℝ⁺) ∧ (𝑗 ∈ 𝑍 ∧ 𝑘 ∈ (ℤ_≥‘𝑗))) ∧ 𝑛 ∈ (𝑗...(𝑘 − 1))) → (𝐹‘𝑛) ≤ (𝐹‘(𝑛 + 1)))
69	43, 56, 68	monoord 14100	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ⁺) ∧ (𝑗 ∈ 𝑍 ∧ 𝑘 ∈ (ℤ_≥‘𝑗))) → (𝐹‘𝑗) ≤ (𝐹‘𝑘))
70	37, 41, 42, 69	lesub2dd 11859	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ⁺) ∧ (𝑗 ∈ 𝑍 ∧ 𝑘 ∈ (ℤ_≥‘𝑗))) → (sup(ran 𝐹, ℝ, < ) − (𝐹‘𝑘)) ≤ (sup(ran 𝐹, ℝ, < ) − (𝐹‘𝑗)))
71	42, 41	resubcld 11670	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ⁺) ∧ (𝑗 ∈ 𝑍 ∧ 𝑘 ∈ (ℤ_≥‘𝑗))) → (sup(ran 𝐹, ℝ, < ) − (𝐹‘𝑘)) ∈ ℝ)
72	42, 37	resubcld 11670	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ⁺) ∧ (𝑗 ∈ 𝑍 ∧ 𝑘 ∈ (ℤ_≥‘𝑗))) → (sup(ran 𝐹, ℝ, < ) − (𝐹‘𝑗)) ∈ ℝ)
73	24	ad2antlr 740	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ⁺) ∧ (𝑗 ∈ 𝑍 ∧ 𝑘 ∈ (ℤ_≥‘𝑗))) → 𝑦 ∈ ℝ)
74		lelttr 11328	. . . . . . . . . 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))
75	71, 72, 73, 74	syl3anc 1398	. . . . . . . . 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))
76	70, 75	mpand 708	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ⁺) ∧ (𝑗 ∈ 𝑍 ∧ 𝑘 ∈ (ℤ_≥‘𝑗))) → ((sup(ran 𝐹, ℝ, < ) − (𝐹‘𝑗)) < 𝑦 → (sup(ran 𝐹, ℝ, < ) − (𝐹‘𝑘)) < 𝑦))
77		ltsub23 11722	. . . . . . . . 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))
78	42, 73, 37, 77	syl3anc 1398	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ⁺) ∧ (𝑗 ∈ 𝑍 ∧ 𝑘 ∈ (ℤ_≥‘𝑗))) → ((sup(ran 𝐹, ℝ, < ) − 𝑦) < (𝐹‘𝑗) ↔ (sup(ran 𝐹, ℝ, < ) − (𝐹‘𝑗)) < 𝑦))
79	18	ad2antrr 739	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ⁺) ∧ (𝑗 ∈ 𝑍 ∧ 𝑘 ∈ (ℤ_≥‘𝑗))) → (ran 𝐹 ⊆ ℝ ∧ ran 𝐹 ≠ ∅ ∧ ∃𝑥 ∈ ℝ ∀𝑦 ∈ ran 𝐹 𝑦 ≤ 𝑥))
80	3	adantr 486	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ⁺) → 𝐹 Fn 𝑍)
81		fnfvelrn 7077	. . . . . . . . . . . 12 ⊢ ((𝐹 Fn 𝑍 ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) ∈ ran 𝐹)
82	80, 39, 81	syl2an 608	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ⁺) ∧ (𝑗 ∈ 𝑍 ∧ 𝑘 ∈ (ℤ_≥‘𝑗))) → (𝐹‘𝑘) ∈ ran 𝐹)
83		suprub 12204	. . . . . . . . . . 11 ⊢ (((ran 𝐹 ⊆ ℝ ∧ ran 𝐹 ≠ ∅ ∧ ∃𝑥 ∈ ℝ ∀𝑦 ∈ ran 𝐹 𝑦 ≤ 𝑥) ∧ (𝐹‘𝑘) ∈ ran 𝐹) → (𝐹‘𝑘) ≤ sup(ran 𝐹, ℝ, < ))
84	79, 82, 83	syl2anc 596	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ⁺) ∧ (𝑗 ∈ 𝑍 ∧ 𝑘 ∈ (ℤ_≥‘𝑗))) → (𝐹‘𝑘) ≤ sup(ran 𝐹, ℝ, < ))
85	41, 42, 84	abssuble0d 15526	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ⁺) ∧ (𝑗 ∈ 𝑍 ∧ 𝑘 ∈ (ℤ_≥‘𝑗))) → (abs‘((𝐹‘𝑘) − sup(ran 𝐹, ℝ, < ))) = (sup(ran 𝐹, ℝ, < ) − (𝐹‘𝑘)))
86	85	breq1d 5117	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ⁺) ∧ (𝑗 ∈ 𝑍 ∧ 𝑘 ∈ (ℤ_≥‘𝑗))) → ((abs‘((𝐹‘𝑘) − sup(ran 𝐹, ℝ, < ))) < 𝑦 ↔ (sup(ran 𝐹, ℝ, < ) − (𝐹‘𝑘)) < 𝑦))
87	76, 78, 86	3imtr4d 297	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ⁺) ∧ (𝑗 ∈ 𝑍 ∧ 𝑘 ∈ (ℤ_≥‘𝑗))) → ((sup(ran 𝐹, ℝ, < ) − 𝑦) < (𝐹‘𝑗) → (abs‘((𝐹‘𝑘) − sup(ran 𝐹, ℝ, < ))) < 𝑦))
88	87	anassrs 473	. . . . . 6 ⊢ ((((𝜑 ∧ 𝑦 ∈ ℝ⁺) ∧ 𝑗 ∈ 𝑍) ∧ 𝑘 ∈ (ℤ_≥‘𝑗)) → ((sup(ran 𝐹, ℝ, < ) − 𝑦) < (𝐹‘𝑗) → (abs‘((𝐹‘𝑘) − sup(ran 𝐹, ℝ, < ))) < 𝑦))
89	88	ralrimdva 3164	. . . . 5 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ⁺) ∧ 𝑗 ∈ 𝑍) → ((sup(ran 𝐹, ℝ, < ) − 𝑦) < (𝐹‘𝑗) → ∀𝑘 ∈ (ℤ_≥‘𝑗)(abs‘((𝐹‘𝑘) − sup(ran 𝐹, ℝ, < ))) < 𝑦))
90	89	reximdva 3177	. . . 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))
91	34, 90	mpd 16	. . 3 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ⁺) → ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ_≥‘𝑗)(abs‘((𝐹‘𝑘) − sup(ran 𝐹, ℝ, < ))) < 𝑦)
92	91	ralrimiva 3156	. 2 ⊢ (𝜑 → ∀𝑦 ∈ ℝ⁺ ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ_≥‘𝑗)(abs‘((𝐹‘𝑘) − sup(ran 𝐹, ℝ, < ))) < 𝑦)
93	7	fvexi 6896	. . . 4 ⊢ 𝑍 ∈ V
94		fex 7229	. . . 4 ⊢ ((𝐹:𝑍⟶ℝ ∧ 𝑍 ∈ V) → 𝐹 ∈ V)
95	1, 93, 94	sylancl 598	. . 3 ⊢ (𝜑 → 𝐹 ∈ V)
96		eqidd 2763	. . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) = (𝐹‘𝑘))
97	20	recnd 11265	. . 3 ⊢ (𝜑 → sup(ran 𝐹, ℝ, < ) ∈ ℂ)
98	1, 40	sylan 592	. . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) ∈ ℝ)
99	98	recnd 11265	. . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) ∈ ℂ)
100	7, 4, 95, 96, 97, 99	clim2c 15596	. 2 ⊢ (𝜑 → (𝐹 ⇝ sup(ran 𝐹, ℝ, < ) ↔ ∀𝑦 ∈ ℝ⁺ ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ_≥‘𝑗)(abs‘((𝐹‘𝑘) − sup(ran 𝐹, ℝ, < ))) < 𝑦))
101	92, 100	mpbird 260	1 ⊢ (𝜑 → 𝐹 ⇝ sup(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 7417 supcsup 9414 ℝcr 11127 1c1 11129 + caddc 11131 < clt 11271 ≤ cle 11272 − cmin 11469 ℤcz 12619 ℤ_≥cuz 12891 ℝ⁺crp 13046 ...cfz 13565 abscabs 15325 ⇝ cli 15575

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 7740 ax-cnex 11184 ax-resscn 11185 ax-1cn 11186 ax-icn 11187 ax-addcl 11188 ax-addrcl 11189 ax-mulcl 11190 ax-mulrcl 11191 ax-mulcom 11192 ax-addass 11193 ax-mulass 11194 ax-distr 11195 ax-i2m1 11196 ax-1ne0 11197 ax-1rid 11198 ax-rnegex 11199 ax-rrecex 11200 ax-cnre 11201 ax-pre-lttri 11202 ax-pre-lttrn 11203 ax-pre-ltadd 11204 ax-pre-mulgt0 11205 ax-pre-sup 11206

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 7374 df-ov 7420 df-oprab 7421 df-mpo 7422 df-om 7867 df-1st 7990 df-2nd 7991 df-frecs 8284 df-wrecs 8315 df-recs 8364 df-rdg 8403 df-er 8700 df-en 8957 df-dom 8958 df-sdom 8959 df-sup 9416 df-pnf 11273 df-mnf 11274 df-xr 11275 df-ltxr 11276 df-le 11277 df-sub 11471 df-neg 11472 df-div 11900 df-nn 12262 df-2 12331 df-3 12332 df-n0 12533 df-z 12620 df-uz 12892 df-rp 13047 df-fz 13566 df-seq 14070 df-exp 14130 df-cj 15190 df-re 15191 df-im 15192 df-sqrt 15326 df-abs 15327 df-clim 15579

This theorem is used by: isumsup2 15939 climcnds 15944 itg1climres 25948 itg2monolem1 25984 itg2i1fseq 25989 itg2i1fseq2 25990 emcllem6 27245 lmdvg 34471 esumpcvgval 34596 meaiuninclem 47316

W3C validator