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Theorem climsup 15632

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 6676	. . . . . . . . 9 ⊢ (𝜑 → ran 𝐹 ⊆ ℝ)
3	1	ffnd 6669	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐹 Fn 𝑍)
4		climsup.2	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝑀 ∈ ℤ)
5		uzid 12803	. . . . . . . . . . . . 13 ⊢ (𝑀 ∈ ℤ → 𝑀 ∈ (ℤ_≥‘𝑀))
6	4, 5	syl 17	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑀 ∈ (ℤ_≥‘𝑀))
7		climsup.1	. . . . . . . . . . . 12 ⊢ 𝑍 = (ℤ_≥‘𝑀)
8	6, 7	eleqtrrdi 2847	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑀 ∈ 𝑍)
9		fnfvelrn 7032	. . . . . . . . . . 11 ⊢ ((𝐹 Fn 𝑍 ∧ 𝑀 ∈ 𝑍) → (𝐹‘𝑀) ∈ ran 𝐹)
10	3, 8, 9	syl2anc 585	. . . . . . . . . 10 ⊢ (𝜑 → (𝐹‘𝑀) ∈ ran 𝐹)
11	10	ne0d 4282	. . . . . . . . 9 ⊢ (𝜑 → ran 𝐹 ≠ ∅)
12		climsup.5	. . . . . . . . . 10 ⊢ (𝜑 → ∃𝑥 ∈ ℝ ∀𝑘 ∈ 𝑍 (𝐹‘𝑘) ≤ 𝑥)
13		breq1 5088	. . . . . . . . . . . . 13 ⊢ (𝑦 = (𝐹‘𝑘) → (𝑦 ≤ 𝑥 ↔ (𝐹‘𝑘) ≤ 𝑥))
14	13	ralrn 7040	. . . . . . . . . . . 12 ⊢ (𝐹 Fn 𝑍 → (∀𝑦 ∈ ran 𝐹 𝑦 ≤ 𝑥 ↔ ∀𝑘 ∈ 𝑍 (𝐹‘𝑘) ≤ 𝑥))
15	14	rexbidv 3161	. . . . . . . . . . 11 ⊢ (𝐹 Fn 𝑍 → (∃𝑥 ∈ ℝ ∀𝑦 ∈ ran 𝐹 𝑦 ≤ 𝑥 ↔ ∃𝑥 ∈ ℝ ∀𝑘 ∈ 𝑍 (𝐹‘𝑘) ≤ 𝑥))
16	3, 15	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → (∃𝑥 ∈ ℝ ∀𝑦 ∈ ran 𝐹 𝑦 ≤ 𝑥 ↔ ∃𝑥 ∈ ℝ ∀𝑘 ∈ 𝑍 (𝐹‘𝑘) ≤ 𝑥))
17	12, 16	mpbird 257	. . . . . . . . 9 ⊢ (𝜑 → ∃𝑥 ∈ ℝ ∀𝑦 ∈ ran 𝐹 𝑦 ≤ 𝑥)
18	2, 11, 17	3jca 1129	. . . . . . . 8 ⊢ (𝜑 → (ran 𝐹 ⊆ ℝ ∧ ran 𝐹 ≠ ∅ ∧ ∃𝑥 ∈ ℝ ∀𝑦 ∈ ran 𝐹 𝑦 ≤ 𝑥))
19		suprcl 12116	. . . . . . . 8 ⊢ ((ran 𝐹 ⊆ ℝ ∧ ran 𝐹 ≠ ∅ ∧ ∃𝑥 ∈ ℝ ∀𝑦 ∈ ran 𝐹 𝑦 ≤ 𝑥) → sup(ran 𝐹, ℝ, < ) ∈ ℝ)
20	18, 19	syl 17	. . . . . . 7 ⊢ (𝜑 → sup(ran 𝐹, ℝ, < ) ∈ ℝ)
21		ltsubrp 12980	. . . . . . 7 ⊢ ((sup(ran 𝐹, ℝ, < ) ∈ ℝ ∧ 𝑦 ∈ ℝ⁺) → (sup(ran 𝐹, ℝ, < ) − 𝑦) < sup(ran 𝐹, ℝ, < ))
22	20, 21	sylan 581	. . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ⁺) → (sup(ran 𝐹, ℝ, < ) − 𝑦) < sup(ran 𝐹, ℝ, < ))
23	18	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ⁺) → (ran 𝐹 ⊆ ℝ ∧ ran 𝐹 ≠ ∅ ∧ ∃𝑥 ∈ ℝ ∀𝑦 ∈ ran 𝐹 𝑦 ≤ 𝑥))
24		rpre 12951	. . . . . . . 8 ⊢ (𝑦 ∈ ℝ⁺ → 𝑦 ∈ ℝ)
25		resubcl 11458	. . . . . . . 8 ⊢ ((sup(ran 𝐹, ℝ, < ) ∈ ℝ ∧ 𝑦 ∈ ℝ) → (sup(ran 𝐹, ℝ, < ) − 𝑦) ∈ ℝ)
26	20, 24, 25	syl2an 597	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ⁺) → (sup(ran 𝐹, ℝ, < ) − 𝑦) ∈ ℝ)
27		suprlub 12120	. . . . . . 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 585	. . . . . 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 232	. . . . 5 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ⁺) → ∃𝑘 ∈ ran 𝐹(sup(ran 𝐹, ℝ, < ) − 𝑦) < 𝑘)
30		breq2 5089	. . . . . . . 8 ⊢ (𝑘 = (𝐹‘𝑗) → ((sup(ran 𝐹, ℝ, < ) − 𝑦) < 𝑘 ↔ (sup(ran 𝐹, ℝ, < ) − 𝑦) < (𝐹‘𝑗)))
31	30	rexrn 7039	. . . . . . 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 17	. . . . . 6 ⊢ (𝜑 → (∃𝑘 ∈ ran 𝐹(sup(ran 𝐹, ℝ, < ) − 𝑦) < 𝑘 ↔ ∃𝑗 ∈ 𝑍 (sup(ran 𝐹, ℝ, < ) − 𝑦) < (𝐹‘𝑗)))
33	32	biimpa 476	. . . . 5 ⊢ ((𝜑 ∧ ∃𝑘 ∈ ran 𝐹(sup(ran 𝐹, ℝ, < ) − 𝑦) < 𝑘) → ∃𝑗 ∈ 𝑍 (sup(ran 𝐹, ℝ, < ) − 𝑦) < (𝐹‘𝑗))
34	29, 33	syldan 592	. . . 4 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ⁺) → ∃𝑗 ∈ 𝑍 (sup(ran 𝐹, ℝ, < ) − 𝑦) < (𝐹‘𝑗))
35		ffvelcdm 7033	. . . . . . . . . . . 12 ⊢ ((𝐹:𝑍⟶ℝ ∧ 𝑗 ∈ 𝑍) → (𝐹‘𝑗) ∈ ℝ)
36	1, 35	sylan 581	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → (𝐹‘𝑗) ∈ ℝ)
37	36	ad2ant2r 748	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ⁺) ∧ (𝑗 ∈ 𝑍 ∧ 𝑘 ∈ (ℤ_≥‘𝑗))) → (𝐹‘𝑗) ∈ ℝ)
38	1	adantr 480	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ⁺) → 𝐹:𝑍⟶ℝ)
39	7	uztrn2 12807	. . . . . . . . . . 11 ⊢ ((𝑗 ∈ 𝑍 ∧ 𝑘 ∈ (ℤ_≥‘𝑗)) → 𝑘 ∈ 𝑍)
40		ffvelcdm 7033	. . . . . . . . . . 11 ⊢ ((𝐹:𝑍⟶ℝ ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) ∈ ℝ)
41	38, 39, 40	syl2an 597	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ⁺) ∧ (𝑗 ∈ 𝑍 ∧ 𝑘 ∈ (ℤ_≥‘𝑗))) → (𝐹‘𝑘) ∈ ℝ)
42	20	ad2antrr 727	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ⁺) ∧ (𝑗 ∈ 𝑍 ∧ 𝑘 ∈ (ℤ_≥‘𝑗))) → sup(ran 𝐹, ℝ, < ) ∈ ℝ)
43		simprr 773	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ⁺) ∧ (𝑗 ∈ 𝑍 ∧ 𝑘 ∈ (ℤ_≥‘𝑗))) → 𝑘 ∈ (ℤ_≥‘𝑗))
44		fzssuz 13519	. . . . . . . . . . . . . 14 ⊢ (𝑗...𝑘) ⊆ (ℤ_≥‘𝑗)
45		uzss 12811	. . . . . . . . . . . . . . . . 17 ⊢ (𝑗 ∈ (ℤ_≥‘𝑀) → (ℤ_≥‘𝑗) ⊆ (ℤ_≥‘𝑀))
46	45, 7	sseqtrrdi 3963	. . . . . . . . . . . . . . . 16 ⊢ (𝑗 ∈ (ℤ_≥‘𝑀) → (ℤ_≥‘𝑗) ⊆ 𝑍)
47	46, 7	eleq2s 2854	. . . . . . . . . . . . . . 15 ⊢ (𝑗 ∈ 𝑍 → (ℤ_≥‘𝑗) ⊆ 𝑍)
48	47	ad2antrl 729	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ⁺) ∧ (𝑗 ∈ 𝑍 ∧ 𝑘 ∈ (ℤ_≥‘𝑗))) → (ℤ_≥‘𝑗) ⊆ 𝑍)
49	44, 48	sstrid 3933	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ⁺) ∧ (𝑗 ∈ 𝑍 ∧ 𝑘 ∈ (ℤ_≥‘𝑗))) → (𝑗...𝑘) ⊆ 𝑍)
50		ffvelcdm 7033	. . . . . . . . . . . . . . . 16 ⊢ ((𝐹:𝑍⟶ℝ ∧ 𝑛 ∈ 𝑍) → (𝐹‘𝑛) ∈ ℝ)
51	50	ralrimiva 3129	. . . . . . . . . . . . . . 15 ⊢ (𝐹:𝑍⟶ℝ → ∀𝑛 ∈ 𝑍 (𝐹‘𝑛) ∈ ℝ)
52	1, 51	syl 17	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ∀𝑛 ∈ 𝑍 (𝐹‘𝑛) ∈ ℝ)
53	52	ad2antrr 727	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ⁺) ∧ (𝑗 ∈ 𝑍 ∧ 𝑘 ∈ (ℤ_≥‘𝑗))) → ∀𝑛 ∈ 𝑍 (𝐹‘𝑛) ∈ ℝ)
54		ssralv 3990	. . . . . . . . . . . . 13 ⊢ ((𝑗...𝑘) ⊆ 𝑍 → (∀𝑛 ∈ 𝑍 (𝐹‘𝑛) ∈ ℝ → ∀𝑛 ∈ (𝑗...𝑘)(𝐹‘𝑛) ∈ ℝ))
55	49, 53, 54	sylc 65	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ⁺) ∧ (𝑗 ∈ 𝑍 ∧ 𝑘 ∈ (ℤ_≥‘𝑗))) → ∀𝑛 ∈ (𝑗...𝑘)(𝐹‘𝑛) ∈ ℝ)
56	55	r19.21bi 3229	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑦 ∈ ℝ⁺) ∧ (𝑗 ∈ 𝑍 ∧ 𝑘 ∈ (ℤ_≥‘𝑗))) ∧ 𝑛 ∈ (𝑗...𝑘)) → (𝐹‘𝑛) ∈ ℝ)
57		fzssuz 13519	. . . . . . . . . . . . . 14 ⊢ (𝑗...(𝑘 − 1)) ⊆ (ℤ_≥‘𝑗)
58	57, 48	sstrid 3933	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ⁺) ∧ (𝑗 ∈ 𝑍 ∧ 𝑘 ∈ (ℤ_≥‘𝑗))) → (𝑗...(𝑘 − 1)) ⊆ 𝑍)
59	58	sselda 3921	. . . . . . . . . . . 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 3129	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ∀𝑘 ∈ 𝑍 (𝐹‘𝑘) ≤ (𝐹‘(𝑘 + 1)))
62	61	ad2antrr 727	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ⁺) ∧ (𝑗 ∈ 𝑍 ∧ 𝑘 ∈ (ℤ_≥‘𝑗))) → ∀𝑘 ∈ 𝑍 (𝐹‘𝑘) ≤ (𝐹‘(𝑘 + 1)))
63		fveq2 6840	. . . . . . . . . . . . . . 15 ⊢ (𝑘 = 𝑛 → (𝐹‘𝑘) = (𝐹‘𝑛))
64		fvoveq1 7390	. . . . . . . . . . . . . . 15 ⊢ (𝑘 = 𝑛 → (𝐹‘(𝑘 + 1)) = (𝐹‘(𝑛 + 1)))
65	63, 64	breq12d 5098	. . . . . . . . . . . . . 14 ⊢ (𝑘 = 𝑛 → ((𝐹‘𝑘) ≤ (𝐹‘(𝑘 + 1)) ↔ (𝐹‘𝑛) ≤ (𝐹‘(𝑛 + 1))))
66	65	rspccva 3563	. . . . . . . . . . . . 13 ⊢ ((∀𝑘 ∈ 𝑍 (𝐹‘𝑘) ≤ (𝐹‘(𝑘 + 1)) ∧ 𝑛 ∈ 𝑍) → (𝐹‘𝑛) ≤ (𝐹‘(𝑛 + 1)))
67	62, 66	sylan 581	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑦 ∈ ℝ⁺) ∧ (𝑗 ∈ 𝑍 ∧ 𝑘 ∈ (ℤ_≥‘𝑗))) ∧ 𝑛 ∈ 𝑍) → (𝐹‘𝑛) ≤ (𝐹‘(𝑛 + 1)))
68	59, 67	syldan 592	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑦 ∈ ℝ⁺) ∧ (𝑗 ∈ 𝑍 ∧ 𝑘 ∈ (ℤ_≥‘𝑗))) ∧ 𝑛 ∈ (𝑗...(𝑘 − 1))) → (𝐹‘𝑛) ≤ (𝐹‘(𝑛 + 1)))
69	43, 56, 68	monoord 13994	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ⁺) ∧ (𝑗 ∈ 𝑍 ∧ 𝑘 ∈ (ℤ_≥‘𝑗))) → (𝐹‘𝑗) ≤ (𝐹‘𝑘))
70	37, 41, 42, 69	lesub2dd 11767	. . . . . . . . 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 11578	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ⁺) ∧ (𝑗 ∈ 𝑍 ∧ 𝑘 ∈ (ℤ_≥‘𝑗))) → (sup(ran 𝐹, ℝ, < ) − (𝐹‘𝑘)) ∈ ℝ)
72	42, 37	resubcld 11578	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ⁺) ∧ (𝑗 ∈ 𝑍 ∧ 𝑘 ∈ (ℤ_≥‘𝑗))) → (sup(ran 𝐹, ℝ, < ) − (𝐹‘𝑗)) ∈ ℝ)
73	24	ad2antlr 728	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ⁺) ∧ (𝑗 ∈ 𝑍 ∧ 𝑘 ∈ (ℤ_≥‘𝑗))) → 𝑦 ∈ ℝ)
74		lelttr 11236	. . . . . . . . . 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 1374	. . . . . . . . 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 696	. . . . . . . 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 11630	. . . . . . . . 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 1374	. . . . . . . 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 727	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ⁺) ∧ (𝑗 ∈ 𝑍 ∧ 𝑘 ∈ (ℤ_≥‘𝑗))) → (ran 𝐹 ⊆ ℝ ∧ ran 𝐹 ≠ ∅ ∧ ∃𝑥 ∈ ℝ ∀𝑦 ∈ ran 𝐹 𝑦 ≤ 𝑥))
80	3	adantr 480	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ⁺) → 𝐹 Fn 𝑍)
81		fnfvelrn 7032	. . . . . . . . . . . 12 ⊢ ((𝐹 Fn 𝑍 ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) ∈ ran 𝐹)
82	80, 39, 81	syl2an 597	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ⁺) ∧ (𝑗 ∈ 𝑍 ∧ 𝑘 ∈ (ℤ_≥‘𝑗))) → (𝐹‘𝑘) ∈ ran 𝐹)
83		suprub 12117	. . . . . . . . . . 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 585	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ⁺) ∧ (𝑗 ∈ 𝑍 ∧ 𝑘 ∈ (ℤ_≥‘𝑗))) → (𝐹‘𝑘) ≤ sup(ran 𝐹, ℝ, < ))
85	41, 42, 84	abssuble0d 15397	. . . . . . . . 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 5095	. . . . . . . 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 294	. . . . . . 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 467	. . . . . 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 3137	. . . . 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 3150	. . . 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 15	. . 3 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ⁺) → ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ_≥‘𝑗)(abs‘((𝐹‘𝑘) − sup(ran 𝐹, ℝ, < ))) < 𝑦)
92	91	ralrimiva 3129	. 2 ⊢ (𝜑 → ∀𝑦 ∈ ℝ⁺ ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ_≥‘𝑗)(abs‘((𝐹‘𝑘) − sup(ran 𝐹, ℝ, < ))) < 𝑦)
93	7	fvexi 6854	. . . 4 ⊢ 𝑍 ∈ V
94		fex 7181	. . . 4 ⊢ ((𝐹:𝑍⟶ℝ ∧ 𝑍 ∈ V) → 𝐹 ∈ V)
95	1, 93, 94	sylancl 587	. . 3 ⊢ (𝜑 → 𝐹 ∈ V)
96		eqidd 2737	. . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) = (𝐹‘𝑘))
97	20	recnd 11173	. . 3 ⊢ (𝜑 → sup(ran 𝐹, ℝ, < ) ∈ ℂ)
98	1, 40	sylan 581	. . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) ∈ ℝ)
99	98	recnd 11173	. . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) ∈ ℂ)
100	7, 4, 95, 96, 97, 99	clim2c 15467	. 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 257	1 ⊢ (𝜑 → 𝐹 ⇝ sup(ran 𝐹, ℝ, < ))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 ∧ w3a 1087 = wceq 1542 ∈ wcel 2114 ≠ wne 2932 ∀wral 3051 ∃wrex 3061 Vcvv 3429 ⊆ wss 3889 ∅c0 4273 class class class wbr 5085 ran crn 5632 Fn wfn 6493 ⟶wf 6494 ‘cfv 6498 (class class class)co 7367 supcsup 9353 ℝcr 11037 1c1 11039 + caddc 11041 < clt 11179 ≤ cle 11180 − cmin 11377 ℤcz 12524 ℤ_≥cuz 12788 ℝ⁺crp 12942 ...cfz 13461 abscabs 15196 ⇝ cli 15446

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1912 ax-6 1969 ax-7 2010 ax-8 2116 ax-9 2124 ax-10 2147 ax-11 2163 ax-12 2185 ax-ext 2708 ax-rep 5212 ax-sep 5231 ax-nul 5241 ax-pow 5307 ax-pr 5375 ax-un 7689 ax-cnex 11094 ax-resscn 11095 ax-1cn 11096 ax-icn 11097 ax-addcl 11098 ax-addrcl 11099 ax-mulcl 11100 ax-mulrcl 11101 ax-mulcom 11102 ax-addass 11103 ax-mulass 11104 ax-distr 11105 ax-i2m1 11106 ax-1ne0 11107 ax-1rid 11108 ax-rnegex 11109 ax-rrecex 11110 ax-cnre 11111 ax-pre-lttri 11112 ax-pre-lttrn 11113 ax-pre-ltadd 11114 ax-pre-mulgt0 11115 ax-pre-sup 11116

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1545 df-fal 1555 df-ex 1782 df-nf 1786 df-sb 2069 df-mo 2539 df-eu 2569 df-clab 2715 df-cleq 2728 df-clel 2811 df-nfc 2885 df-ne 2933 df-nel 3037 df-ral 3052 df-rex 3062 df-rmo 3342 df-reu 3343 df-rab 3390 df-v 3431 df-sbc 3729 df-csb 3838 df-dif 3892 df-un 3894 df-in 3896 df-ss 3906 df-pss 3909 df-nul 4274 df-if 4467 df-pw 4543 df-sn 4568 df-pr 4570 df-op 4574 df-uni 4851 df-iun 4935 df-br 5086 df-opab 5148 df-mpt 5167 df-tr 5193 df-id 5526 df-eprel 5531 df-po 5539 df-so 5540 df-fr 5584 df-we 5586 df-xp 5637 df-rel 5638 df-cnv 5639 df-co 5640 df-dm 5641 df-rn 5642 df-res 5643 df-ima 5644 df-pred 6265 df-ord 6326 df-on 6327 df-lim 6328 df-suc 6329 df-iota 6454 df-fun 6500 df-fn 6501 df-f 6502 df-f1 6503 df-fo 6504 df-f1o 6505 df-fv 6506 df-riota 7324 df-ov 7370 df-oprab 7371 df-mpo 7372 df-om 7818 df-1st 7942 df-2nd 7943 df-frecs 8231 df-wrecs 8262 df-recs 8311 df-rdg 8349 df-er 8643 df-en 8894 df-dom 8895 df-sdom 8896 df-sup 9355 df-pnf 11181 df-mnf 11182 df-xr 11183 df-ltxr 11184 df-le 11185 df-sub 11379 df-neg 11380 df-div 11808 df-nn 12175 df-2 12244 df-3 12245 df-n0 12438 df-z 12525 df-uz 12789 df-rp 12943 df-fz 13462 df-seq 13964 df-exp 14024 df-cj 15061 df-re 15062 df-im 15063 df-sqrt 15197 df-abs 15198 df-clim 15450

This theorem is referenced by: isumsup2 15811 climcnds 15816 itg1climres 25681 itg2monolem1 25717 itg2i1fseq 25722 itg2i1fseq2 25723 emcllem6 26964 lmdvg 34097 esumpcvgval 34222 meaiuninclem 46908

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