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Theorem climsup 15723

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 6716	. . . . . . . . 9 ⊢ (𝜑 → ran 𝐹 ⊆ ℝ)
3	1	ffnd 6708	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐹 Fn 𝑍)
4		climsup.2	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝑀 ∈ ℤ)
5		uzid 12878	. . . . . . . . . . . . 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 2874	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑀 ∈ 𝑍)
9		fnfvelrn 7077	. . . . . . . . . . 11 ⊢ ((𝐹 Fn 𝑍 ∧ 𝑀 ∈ 𝑍) → (𝐹‘𝑀) ∈ ran 𝐹)
10	3, 8, 9	syl2anc 595	. . . . . . . . . 10 ⊢ (𝜑 → (𝐹‘𝑀) ∈ ran 𝐹)
11	10	ne0d 4296	. . . . . . . . 9 ⊢ (𝜑 → ran 𝐹 ≠ ∅)
12		climsup.5	. . . . . . . . . 10 ⊢ (𝜑 → ∃𝑥 ∈ ℝ ∀𝑘 ∈ 𝑍 (𝐹‘𝑘) ≤ 𝑥)
13		breq1 5113	. . . . . . . . . . . . 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 3189	. . . . . . . . . . 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 12176	. . . . . . . 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 13055	. . . . . . 7 ⊢ ((sup(ran 𝐹, ℝ, < ) ∈ ℝ ∧ 𝑦 ∈ ℝ⁺) → (sup(ran 𝐹, ℝ, < ) − 𝑦) < sup(ran 𝐹, ℝ, < ))
22	20, 21	sylan 591	. . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ⁺) → (sup(ran 𝐹, ℝ, < ) − 𝑦) < sup(ran 𝐹, ℝ, < ))
23	18	adantr 485	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ⁺) → (ran 𝐹 ⊆ ℝ ∧ ran 𝐹 ≠ ∅ ∧ ∃𝑥 ∈ ℝ ∀𝑦 ∈ ran 𝐹 𝑦 ≤ 𝑥))
24		rpre 13026	. . . . . . . 8 ⊢ (𝑦 ∈ ℝ⁺ → 𝑦 ∈ ℝ)
25		resubcl 11523	. . . . . . . 8 ⊢ ((sup(ran 𝐹, ℝ, < ) ∈ ℝ ∧ 𝑦 ∈ ℝ) → (sup(ran 𝐹, ℝ, < ) − 𝑦) ∈ ℝ)
26	20, 24, 25	syl2an 607	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ⁺) → (sup(ran 𝐹, ℝ, < ) − 𝑦) ∈ ℝ)
27		suprlub 12180	. . . . . . 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 595	. . . . . 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 5114	. . . . . . . 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 481	. . . . 5 ⊢ ((𝜑 ∧ ∃𝑘 ∈ ran 𝐹(sup(ran 𝐹, ℝ, < ) − 𝑦) < 𝑘) → ∃𝑗 ∈ 𝑍 (sup(ran 𝐹, ℝ, < ) − 𝑦) < (𝐹‘𝑗))
34	29, 33	syldan 602	. . . 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 591	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → (𝐹‘𝑗) ∈ ℝ)
37	36	ad2ant2r 759	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ⁺) ∧ (𝑗 ∈ 𝑍 ∧ 𝑘 ∈ (ℤ_≥‘𝑗))) → (𝐹‘𝑗) ∈ ℝ)
38	1	adantr 485	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ⁺) → 𝐹:𝑍⟶ℝ)
39	7	uztrn2 12882	. . . . . . . . . . 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 607	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ⁺) ∧ (𝑗 ∈ 𝑍 ∧ 𝑘 ∈ (ℤ_≥‘𝑗))) → (𝐹‘𝑘) ∈ ℝ)
42	20	ad2antrr 738	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ⁺) ∧ (𝑗 ∈ 𝑍 ∧ 𝑘 ∈ (ℤ_≥‘𝑗))) → sup(ran 𝐹, ℝ, < ) ∈ ℝ)
43		simprr 784	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ⁺) ∧ (𝑗 ∈ 𝑍 ∧ 𝑘 ∈ (ℤ_≥‘𝑗))) → 𝑘 ∈ (ℤ_≥‘𝑗))
44		fzssuz 13595	. . . . . . . . . . . . . 14 ⊢ (𝑗...𝑘) ⊆ (ℤ_≥‘𝑗)
45		uzss 12886	. . . . . . . . . . . . . . . . 17 ⊢ (𝑗 ∈ (ℤ_≥‘𝑀) → (ℤ_≥‘𝑗) ⊆ (ℤ_≥‘𝑀))
46	45, 7	sseqtrrdi 3979	. . . . . . . . . . . . . . . 16 ⊢ (𝑗 ∈ (ℤ_≥‘𝑀) → (ℤ_≥‘𝑗) ⊆ 𝑍)
47	46, 7	eleq2s 2881	. . . . . . . . . . . . . . 15 ⊢ (𝑗 ∈ 𝑍 → (ℤ_≥‘𝑗) ⊆ 𝑍)
48	47	ad2antrl 740	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ⁺) ∧ (𝑗 ∈ 𝑍 ∧ 𝑘 ∈ (ℤ_≥‘𝑗))) → (ℤ_≥‘𝑗) ⊆ 𝑍)
49	44, 48	sstrid 3949	. . . . . . . . . . . . 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 3157	. . . . . . . . . . . . . . 15 ⊢ (𝐹:𝑍⟶ℝ → ∀𝑛 ∈ 𝑍 (𝐹‘𝑛) ∈ ℝ)
52	1, 51	syl 18	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ∀𝑛 ∈ 𝑍 (𝐹‘𝑛) ∈ ℝ)
53	52	ad2antrr 738	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ⁺) ∧ (𝑗 ∈ 𝑍 ∧ 𝑘 ∈ (ℤ_≥‘𝑗))) → ∀𝑛 ∈ 𝑍 (𝐹‘𝑛) ∈ ℝ)
54		ssralv 4007	. . . . . . . . . . . . 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 3257	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑦 ∈ ℝ⁺) ∧ (𝑗 ∈ 𝑍 ∧ 𝑘 ∈ (ℤ_≥‘𝑗))) ∧ 𝑛 ∈ (𝑗...𝑘)) → (𝐹‘𝑛) ∈ ℝ)
57		fzssuz 13595	. . . . . . . . . . . . . 14 ⊢ (𝑗...(𝑘 − 1)) ⊆ (ℤ_≥‘𝑗)
58	57, 48	sstrid 3949	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ⁺) ∧ (𝑗 ∈ 𝑍 ∧ 𝑘 ∈ (ℤ_≥‘𝑗))) → (𝑗...(𝑘 − 1)) ⊆ 𝑍)
59	58	sselda 3938	. . . . . . . . . . . 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 3157	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ∀𝑘 ∈ 𝑍 (𝐹‘𝑘) ≤ (𝐹‘(𝑘 + 1)))
62	61	ad2antrr 738	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ⁺) ∧ (𝑗 ∈ 𝑍 ∧ 𝑘 ∈ (ℤ_≥‘𝑗))) → ∀𝑘 ∈ 𝑍 (𝐹‘𝑘) ≤ (𝐹‘(𝑘 + 1)))
63		fveq2 6883	. . . . . . . . . . . . . . 15 ⊢ (𝑘 = 𝑛 → (𝐹‘𝑘) = (𝐹‘𝑛))
64		fvoveq1 7435	. . . . . . . . . . . . . . 15 ⊢ (𝑘 = 𝑛 → (𝐹‘(𝑘 + 1)) = (𝐹‘(𝑛 + 1)))
65	63, 64	breq12d 5123	. . . . . . . . . . . . . 14 ⊢ (𝑘 = 𝑛 → ((𝐹‘𝑘) ≤ (𝐹‘(𝑘 + 1)) ↔ (𝐹‘𝑛) ≤ (𝐹‘(𝑛 + 1))))
66	65	rspccva 3581	. . . . . . . . . . . . 13 ⊢ ((∀𝑘 ∈ 𝑍 (𝐹‘𝑘) ≤ (𝐹‘(𝑘 + 1)) ∧ 𝑛 ∈ 𝑍) → (𝐹‘𝑛) ≤ (𝐹‘(𝑛 + 1)))
67	62, 66	sylan 591	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑦 ∈ ℝ⁺) ∧ (𝑗 ∈ 𝑍 ∧ 𝑘 ∈ (ℤ_≥‘𝑗))) ∧ 𝑛 ∈ 𝑍) → (𝐹‘𝑛) ≤ (𝐹‘(𝑛 + 1)))
68	59, 67	syldan 602	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑦 ∈ ℝ⁺) ∧ (𝑗 ∈ 𝑍 ∧ 𝑘 ∈ (ℤ_≥‘𝑗))) ∧ 𝑛 ∈ (𝑗...(𝑘 − 1))) → (𝐹‘𝑛) ≤ (𝐹‘(𝑛 + 1)))
69	43, 56, 68	monoord 14070	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ⁺) ∧ (𝑗 ∈ 𝑍 ∧ 𝑘 ∈ (ℤ_≥‘𝑗))) → (𝐹‘𝑗) ≤ (𝐹‘𝑘))
70	37, 41, 42, 69	lesub2dd 11832	. . . . . . . . 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 11643	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ⁺) ∧ (𝑗 ∈ 𝑍 ∧ 𝑘 ∈ (ℤ_≥‘𝑗))) → (sup(ran 𝐹, ℝ, < ) − (𝐹‘𝑘)) ∈ ℝ)
72	42, 37	resubcld 11643	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ⁺) ∧ (𝑗 ∈ 𝑍 ∧ 𝑘 ∈ (ℤ_≥‘𝑗))) → (sup(ran 𝐹, ℝ, < ) − (𝐹‘𝑗)) ∈ ℝ)
73	24	ad2antlr 739	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ⁺) ∧ (𝑗 ∈ 𝑍 ∧ 𝑘 ∈ (ℤ_≥‘𝑗))) → 𝑦 ∈ ℝ)
74		lelttr 11301	. . . . . . . . . 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 707	. . . . . . . 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 11695	. . . . . . . . 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 738	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ⁺) ∧ (𝑗 ∈ 𝑍 ∧ 𝑘 ∈ (ℤ_≥‘𝑗))) → (ran 𝐹 ⊆ ℝ ∧ ran 𝐹 ≠ ∅ ∧ ∃𝑥 ∈ ℝ ∀𝑦 ∈ ran 𝐹 𝑦 ≤ 𝑥))
80	3	adantr 485	. . . . . . . . . . . 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 607	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ⁺) ∧ (𝑗 ∈ 𝑍 ∧ 𝑘 ∈ (ℤ_≥‘𝑗))) → (𝐹‘𝑘) ∈ ran 𝐹)
83		suprub 12177	. . . . . . . . . . 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 595	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ⁺) ∧ (𝑗 ∈ 𝑍 ∧ 𝑘 ∈ (ℤ_≥‘𝑗))) → (𝐹‘𝑘) ≤ sup(ran 𝐹, ℝ, < ))
85	41, 42, 84	abssuble0d 15488	. . . . . . . . 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 5120	. . . . . . . 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 472	. . . . . 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 3165	. . . . 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 3178	. . . 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 3157	. 2 ⊢ (𝜑 → ∀𝑦 ∈ ℝ⁺ ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ_≥‘𝑗)(abs‘((𝐹‘𝑘) − sup(ran 𝐹, ℝ, < ))) < 𝑦)
93	7	fvexi 6897	. . . 4 ⊢ 𝑍 ∈ V
94		fex 7226	. . . 4 ⊢ ((𝐹:𝑍⟶ℝ ∧ 𝑍 ∈ V) → 𝐹 ∈ V)
95	1, 93, 94	sylancl 597	. . 3 ⊢ (𝜑 → 𝐹 ∈ V)
96		eqidd 2764	. . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) = (𝐹‘𝑘))
97	20	recnd 11238	. . 3 ⊢ (𝜑 → sup(ran 𝐹, ℝ, < ) ∈ ℂ)
98	1, 40	sylan 591	. . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) ∈ ℝ)
99	98	recnd 11238	. . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) ∈ ℂ)
100	7, 4, 95, 96, 97, 99	clim2c 15558	. 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

Syntax hints: → wi 4 ↔ wb 209 ∧ wa 400 ∧ w3a 1103 = wceq 1570 ∈ wcel 2143 ≠ wne 2958 ∀wral 3079 ∃wrex 3089 Vcvv 3455 ⊆ wss 3906 ∅c0 4287 class class class wbr 5110 ran crn 5664 Fn wfn 6533 ⟶wf 6534 ‘cfv 6538 (class class class)co 7412 supcsup 9401 ℝcr 11100 1c1 11102 + caddc 11104 < clt 11244 ≤ cle 11245 − cmin 11442 ℤcz 12592 ℤ_≥cuz 12863 ℝ⁺crp 13017 ...cfz 13536 abscabs 15287 ⇝ cli 15537

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1825 ax-4 1839 ax-5 1940 ax-6 1997 ax-7 2038 ax-8 2145 ax-9 2153 ax-10 2176 ax-11 2192 ax-12 2213 ax-ext 2735 ax-rep 5239 ax-sep 5258 ax-nul 5270 ax-pow 5338 ax-pr 5406 ax-un 7734 ax-cnex 11157 ax-resscn 11158 ax-1cn 11159 ax-icn 11160 ax-addcl 11161 ax-addrcl 11162 ax-mulcl 11163 ax-mulrcl 11164 ax-mulcom 11165 ax-addass 11166 ax-mulass 11167 ax-distr 11168 ax-i2m1 11169 ax-1ne0 11170 ax-1rid 11171 ax-rnegex 11172 ax-rrecex 11173 ax-cnre 11174 ax-pre-lttri 11175 ax-pre-lttrn 11176 ax-pre-ltadd 11177 ax-pre-mulgt0 11178 ax-pre-sup 11179

This theorem depends on definitions: df-bi 210 df-an 401 df-or 861 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1810 df-nf 1814 df-sb 2097 df-mo 2567 df-eu 2597 df-clab 2742 df-cleq 2755 df-clel 2838 df-nfc 2912 df-ne 2959 df-nel 3065 df-ral 3080 df-rex 3090 df-rmo 3369 df-reu 3370 df-rab 3417 df-v 3457 df-sbc 3746 df-csb 3855 df-dif 3909 df-un 3911 df-in 3913 df-ss 3923 df-pss 3926 df-nul 4288 df-if 4489 df-pw 4565 df-sn 4591 df-pr 4593 df-op 4597 df-uni 4874 df-iun 4959 df-br 5111 df-opab 5175 df-mpt 5194 df-tr 5220 df-id 5558 df-eprel 5563 df-po 5571 df-so 5572 df-fr 5616 df-we 5618 df-xp 5669 df-rel 5670 df-cnv 5671 df-co 5672 df-dm 5673 df-rn 5674 df-res 5675 df-ima 5676 df-pred 6304 df-ord 6365 df-on 6366 df-lim 6367 df-suc 6368 df-iota 6494 df-fun 6540 df-fn 6541 df-f 6542 df-f1 6543 df-fo 6544 df-f1o 6545 df-fv 6546 df-riota 7369 df-ov 7415 df-oprab 7416 df-mpo 7417 df-om 7864 df-1st 7987 df-2nd 7988 df-frecs 8279 df-wrecs 8310 df-recs 8359 df-rdg 8398 df-er 8695 df-en 8945 df-dom 8946 df-sdom 8947 df-sup 9403 df-pnf 11246 df-mnf 11247 df-xr 11248 df-ltxr 11249 df-le 11250 df-sub 11444 df-neg 11445 df-div 11873 df-nn 12235 df-2 12304 df-3 12305 df-n0 12506 df-z 12593 df-uz 12864 df-rp 13018 df-fz 13537 df-seq 14040 df-exp 14100 df-cj 15152 df-re 15153 df-im 15154 df-sqrt 15288 df-abs 15289 df-clim 15541

This theorem is referenced by: isumsup2 15902 climcnds 15907 itg1climres 25854 itg2monolem1 25890 itg2i1fseq 25895 itg2i1fseq2 25896 emcllem6 27143 lmdvg 34321 esumpcvgval 34446 meaiuninclem 47174

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