Theorem climinf2mpt 41993

Description: A bounded below, monotonic nonincreasing sequence converges to the infimum of its range. (Contributed by Glauco Siliprandi, 23-Oct-2021.)

Hypotheses

Ref	Expression
climinf2mpt.p	⊢ Ⅎ𝑘𝜑
climinf2mpt.j	⊢ Ⅎ𝑗𝜑
climinf2mpt.m	⊢ (𝜑 → 𝑀 ∈ ℤ)
climinf2mpt.z	⊢ 𝑍 = (ℤ≥‘𝑀)
climinf2mpt.b	⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → 𝐵 ∈ ℝ)
climinf2mpt.c	⊢ (𝑘 = 𝑗 → 𝐵 = 𝐶)
climinf2mpt.l	⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍 ∧ 𝑗 = (𝑘 + 1)) → 𝐶 ≤ 𝐵)
climinf2mpt.e	⊢ (𝜑 → (𝑘 ∈ 𝑍 ↦ 𝐵) ∈ dom ⇝ )

Assertion

Ref	Expression
climinf2mpt	⊢ (𝜑 → (𝑘 ∈ 𝑍 ↦ 𝐵) ⇝ inf(ran (𝑘 ∈ 𝑍 ↦ 𝐵), ℝ*, < ))

Distinct variable groups:   𝐵,𝑗   𝐶,𝑘   𝑗,𝑍,𝑘

Allowed substitution hints:   𝜑(𝑗,𝑘)   𝐵(𝑘)   𝐶(𝑗)   𝑀(𝑗,𝑘)

Proof of Theorem climinf2mpt

Dummy variables 𝑖 𝑥 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		nfv 1911	. 2 ⊢ Ⅎ𝑖𝜑
2		nfcv 2977	. 2 ⊢ Ⅎ𝑖(𝑘 ∈ 𝑍 ↦ 𝐵)
3		climinf2mpt.z	. 2 ⊢ 𝑍 = (ℤ≥‘𝑀)
4		climinf2mpt.m	. 2 ⊢ (𝜑 → 𝑀 ∈ ℤ)
5		climinf2mpt.p	. . 3 ⊢ Ⅎ𝑘𝜑
6		climinf2mpt.b	. . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → 𝐵 ∈ ℝ)
7	5, 6	fmptd2f 41503	. 2 ⊢ (𝜑 → (𝑘 ∈ 𝑍 ↦ 𝐵):𝑍⟶ℝ)
8		nfv 1911	. . . . . . 7 ⊢ Ⅎ𝑘 𝑖 ∈ 𝑍
9	5, 8	nfan 1896	. . . . . 6 ⊢ Ⅎ𝑘(𝜑 ∧ 𝑖 ∈ 𝑍)
10		nfv 1911	. . . . . 6 ⊢ Ⅎ𝑘⦋(𝑖 + 1) / 𝑗⦌𝐶 ≤ ⦋𝑖 / 𝑗⦌𝐶
11	9, 10	nfim 1893	. . . . 5 ⊢ Ⅎ𝑘((𝜑 ∧ 𝑖 ∈ 𝑍) → ⦋(𝑖 + 1) / 𝑗⦌𝐶 ≤ ⦋𝑖 / 𝑗⦌𝐶)
12		eleq1 2900	. . . . . . 7 ⊢ (𝑘 = 𝑖 → (𝑘 ∈ 𝑍 ↔ 𝑖 ∈ 𝑍))
13	12	anbi2d 630	. . . . . 6 ⊢ (𝑘 = 𝑖 → ((𝜑 ∧ 𝑘 ∈ 𝑍) ↔ (𝜑 ∧ 𝑖 ∈ 𝑍)))
14		oveq1 7162	. . . . . . . 8 ⊢ (𝑘 = 𝑖 → (𝑘 + 1) = (𝑖 + 1))
15	14	csbeq1d 3886	. . . . . . 7 ⊢ (𝑘 = 𝑖 → ⦋(𝑘 + 1) / 𝑗⦌𝐶 = ⦋(𝑖 + 1) / 𝑗⦌𝐶)
16		eqidd 2822	. . . . . . . 8 ⊢ (𝑘 = 𝑖 → 𝐵 = 𝐵)
17		csbcow 3897	. . . . . . . . . . 11 ⊢ ⦋𝑘 / 𝑗⦌⦋𝑗 / 𝑘⦌𝐵 = ⦋𝑘 / 𝑘⦌𝐵
18		csbid 3895	. . . . . . . . . . 11 ⊢ ⦋𝑘 / 𝑘⦌𝐵 = 𝐵
19	17, 18	eqtr2i 2845	. . . . . . . . . 10 ⊢ 𝐵 = ⦋𝑘 / 𝑗⦌⦋𝑗 / 𝑘⦌𝐵
20		nfcv 2977	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑗𝐵
21		nfcv 2977	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑘𝐶
22		climinf2mpt.c	. . . . . . . . . . . . 13 ⊢ (𝑘 = 𝑗 → 𝐵 = 𝐶)
23	20, 21, 22	cbvcsbw 3892	. . . . . . . . . . . 12 ⊢ ⦋𝑗 / 𝑘⦌𝐵 = ⦋𝑗 / 𝑗⦌𝐶
24		csbid 3895	. . . . . . . . . . . 12 ⊢ ⦋𝑗 / 𝑗⦌𝐶 = 𝐶
25	23, 24	eqtri 2844	. . . . . . . . . . 11 ⊢ ⦋𝑗 / 𝑘⦌𝐵 = 𝐶
26	25	csbeq2i 3890	. . . . . . . . . 10 ⊢ ⦋𝑘 / 𝑗⦌⦋𝑗 / 𝑘⦌𝐵 = ⦋𝑘 / 𝑗⦌𝐶
27	19, 26	eqtri 2844	. . . . . . . . 9 ⊢ 𝐵 = ⦋𝑘 / 𝑗⦌𝐶
28	27	a1i 11	. . . . . . . 8 ⊢ (𝑘 = 𝑖 → 𝐵 = ⦋𝑘 / 𝑗⦌𝐶)
29		csbeq1 3885	. . . . . . . 8 ⊢ (𝑘 = 𝑖 → ⦋𝑘 / 𝑗⦌𝐶 = ⦋𝑖 / 𝑗⦌𝐶)
30	16, 28, 29	3eqtrd 2860	. . . . . . 7 ⊢ (𝑘 = 𝑖 → 𝐵 = ⦋𝑖 / 𝑗⦌𝐶)
31	15, 30	breq12d 5078	. . . . . 6 ⊢ (𝑘 = 𝑖 → (⦋(𝑘 + 1) / 𝑗⦌𝐶 ≤ 𝐵 ↔ ⦋(𝑖 + 1) / 𝑗⦌𝐶 ≤ ⦋𝑖 / 𝑗⦌𝐶))
32	13, 31	imbi12d 347	. . . . 5 ⊢ (𝑘 = 𝑖 → (((𝜑 ∧ 𝑘 ∈ 𝑍) → ⦋(𝑘 + 1) / 𝑗⦌𝐶 ≤ 𝐵) ↔ ((𝜑 ∧ 𝑖 ∈ 𝑍) → ⦋(𝑖 + 1) / 𝑗⦌𝐶 ≤ ⦋𝑖 / 𝑗⦌𝐶)))
33		simpl 485	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → 𝜑)
34		simpr 487	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → 𝑘 ∈ 𝑍)
35		eqidd 2822	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝑘 + 1) = (𝑘 + 1))
36		climinf2mpt.j	. . . . . . . . 9 ⊢ Ⅎ𝑗𝜑
37		nfv 1911	. . . . . . . . 9 ⊢ Ⅎ𝑗 𝑘 ∈ 𝑍
38		nfv 1911	. . . . . . . . 9 ⊢ Ⅎ𝑗(𝑘 + 1) = (𝑘 + 1)
39	36, 37, 38	nf3an 1898	. . . . . . . 8 ⊢ Ⅎ𝑗(𝜑 ∧ 𝑘 ∈ 𝑍 ∧ (𝑘 + 1) = (𝑘 + 1))
40		nfcsb1v 3906	. . . . . . . . 9 ⊢ Ⅎ𝑗⦋(𝑘 + 1) / 𝑗⦌𝐶
41		nfcv 2977	. . . . . . . . 9 ⊢ Ⅎ𝑗 ≤
42	40, 41, 20	nfbr 5112	. . . . . . . 8 ⊢ Ⅎ𝑗⦋(𝑘 + 1) / 𝑗⦌𝐶 ≤ 𝐵
43	39, 42	nfim 1893	. . . . . . 7 ⊢ Ⅎ𝑗((𝜑 ∧ 𝑘 ∈ 𝑍 ∧ (𝑘 + 1) = (𝑘 + 1)) → ⦋(𝑘 + 1) / 𝑗⦌𝐶 ≤ 𝐵)
44		ovex 7188	. . . . . . 7 ⊢ (𝑘 + 1) ∈ V
45		eqeq1 2825	. . . . . . . . 9 ⊢ (𝑗 = (𝑘 + 1) → (𝑗 = (𝑘 + 1) ↔ (𝑘 + 1) = (𝑘 + 1)))
46	45	3anbi3d 1438	. . . . . . . 8 ⊢ (𝑗 = (𝑘 + 1) → ((𝜑 ∧ 𝑘 ∈ 𝑍 ∧ 𝑗 = (𝑘 + 1)) ↔ (𝜑 ∧ 𝑘 ∈ 𝑍 ∧ (𝑘 + 1) = (𝑘 + 1))))
47		csbeq1a 3896	. . . . . . . . 9 ⊢ (𝑗 = (𝑘 + 1) → 𝐶 = ⦋(𝑘 + 1) / 𝑗⦌𝐶)
48	47	breq1d 5075	. . . . . . . 8 ⊢ (𝑗 = (𝑘 + 1) → (𝐶 ≤ 𝐵 ↔ ⦋(𝑘 + 1) / 𝑗⦌𝐶 ≤ 𝐵))
49	46, 48	imbi12d 347	. . . . . . 7 ⊢ (𝑗 = (𝑘 + 1) → (((𝜑 ∧ 𝑘 ∈ 𝑍 ∧ 𝑗 = (𝑘 + 1)) → 𝐶 ≤ 𝐵) ↔ ((𝜑 ∧ 𝑘 ∈ 𝑍 ∧ (𝑘 + 1) = (𝑘 + 1)) → ⦋(𝑘 + 1) / 𝑗⦌𝐶 ≤ 𝐵)))
50		climinf2mpt.l	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍 ∧ 𝑗 = (𝑘 + 1)) → 𝐶 ≤ 𝐵)
51	43, 44, 49, 50	vtoclf 3558	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍 ∧ (𝑘 + 1) = (𝑘 + 1)) → ⦋(𝑘 + 1) / 𝑗⦌𝐶 ≤ 𝐵)
52	33, 34, 35, 51	syl3anc 1367	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → ⦋(𝑘 + 1) / 𝑗⦌𝐶 ≤ 𝐵)
53	11, 32, 52	chvarfv 2238	. . . 4 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑍) → ⦋(𝑖 + 1) / 𝑗⦌𝐶 ≤ ⦋𝑖 / 𝑗⦌𝐶)
54	20, 21, 22	cbvcsbw 3892	. . . . . 6 ⊢ ⦋(𝑖 + 1) / 𝑘⦌𝐵 = ⦋(𝑖 + 1) / 𝑗⦌𝐶
55	54	a1i 11	. . . . 5 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑍) → ⦋(𝑖 + 1) / 𝑘⦌𝐵 = ⦋(𝑖 + 1) / 𝑗⦌𝐶)
56		eqidd 2822	. . . . 5 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑍) → ⦋𝑖 / 𝑗⦌𝐶 = ⦋𝑖 / 𝑗⦌𝐶)
57	55, 56	breq12d 5078	. . . 4 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑍) → (⦋(𝑖 + 1) / 𝑘⦌𝐵 ≤ ⦋𝑖 / 𝑗⦌𝐶 ↔ ⦋(𝑖 + 1) / 𝑗⦌𝐶 ≤ ⦋𝑖 / 𝑗⦌𝐶))
58	53, 57	mpbird 259	. . 3 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑍) → ⦋(𝑖 + 1) / 𝑘⦌𝐵 ≤ ⦋𝑖 / 𝑗⦌𝐶)
59	3	peano2uzs 12301	. . . . . 6 ⊢ (𝑖 ∈ 𝑍 → (𝑖 + 1) ∈ 𝑍)
60	59	adantl 484	. . . . 5 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑍) → (𝑖 + 1) ∈ 𝑍)
61		nfv 1911	. . . . . . . . 9 ⊢ Ⅎ𝑘(𝑖 + 1) ∈ 𝑍
62	5, 61	nfan 1896	. . . . . . . 8 ⊢ Ⅎ𝑘(𝜑 ∧ (𝑖 + 1) ∈ 𝑍)
63		nfcv 2977	. . . . . . . . . 10 ⊢ Ⅎ𝑘(𝑖 + 1)
64	63	nfcsb1 3905	. . . . . . . . 9 ⊢ Ⅎ𝑘⦋(𝑖 + 1) / 𝑘⦌𝐵
65	64	nfel1 2994	. . . . . . . 8 ⊢ Ⅎ𝑘⦋(𝑖 + 1) / 𝑘⦌𝐵 ∈ ℝ
66	62, 65	nfim 1893	. . . . . . 7 ⊢ Ⅎ𝑘((𝜑 ∧ (𝑖 + 1) ∈ 𝑍) → ⦋(𝑖 + 1) / 𝑘⦌𝐵 ∈ ℝ)
67		ovex 7188	. . . . . . 7 ⊢ (𝑖 + 1) ∈ V
68		eleq1 2900	. . . . . . . . 9 ⊢ (𝑘 = (𝑖 + 1) → (𝑘 ∈ 𝑍 ↔ (𝑖 + 1) ∈ 𝑍))
69	68	anbi2d 630	. . . . . . . 8 ⊢ (𝑘 = (𝑖 + 1) → ((𝜑 ∧ 𝑘 ∈ 𝑍) ↔ (𝜑 ∧ (𝑖 + 1) ∈ 𝑍)))
70		csbeq1a 3896	. . . . . . . . 9 ⊢ (𝑘 = (𝑖 + 1) → 𝐵 = ⦋(𝑖 + 1) / 𝑘⦌𝐵)
71	70	eleq1d 2897	. . . . . . . 8 ⊢ (𝑘 = (𝑖 + 1) → (𝐵 ∈ ℝ ↔ ⦋(𝑖 + 1) / 𝑘⦌𝐵 ∈ ℝ))
72	69, 71	imbi12d 347	. . . . . . 7 ⊢ (𝑘 = (𝑖 + 1) → (((𝜑 ∧ 𝑘 ∈ 𝑍) → 𝐵 ∈ ℝ) ↔ ((𝜑 ∧ (𝑖 + 1) ∈ 𝑍) → ⦋(𝑖 + 1) / 𝑘⦌𝐵 ∈ ℝ)))
73	66, 67, 72, 6	vtoclf 3558	. . . . . 6 ⊢ ((𝜑 ∧ (𝑖 + 1) ∈ 𝑍) → ⦋(𝑖 + 1) / 𝑘⦌𝐵 ∈ ℝ)
74	59, 73	sylan2 594	. . . . 5 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑍) → ⦋(𝑖 + 1) / 𝑘⦌𝐵 ∈ ℝ)
75		eqid 2821	. . . . . 6 ⊢ (𝑘 ∈ 𝑍 ↦ 𝐵) = (𝑘 ∈ 𝑍 ↦ 𝐵)
76	63, 64, 70, 75	fvmptf 6788	. . . . 5 ⊢ (((𝑖 + 1) ∈ 𝑍 ∧ ⦋(𝑖 + 1) / 𝑘⦌𝐵 ∈ ℝ) → ((𝑘 ∈ 𝑍 ↦ 𝐵)‘(𝑖 + 1)) = ⦋(𝑖 + 1) / 𝑘⦌𝐵)
77	60, 74, 76	syl2anc 586	. . . 4 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑍) → ((𝑘 ∈ 𝑍 ↦ 𝐵)‘(𝑖 + 1)) = ⦋(𝑖 + 1) / 𝑘⦌𝐵)
78		simpr 487	. . . . 5 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑍) → 𝑖 ∈ 𝑍)
79		nfv 1911	. . . . . . . 8 ⊢ Ⅎ𝑗 𝑖 ∈ 𝑍
80	36, 79	nfan 1896	. . . . . . 7 ⊢ Ⅎ𝑗(𝜑 ∧ 𝑖 ∈ 𝑍)
81		nfcsb1v 3906	. . . . . . . 8 ⊢ Ⅎ𝑗⦋𝑖 / 𝑗⦌𝐶
82		nfcv 2977	. . . . . . . 8 ⊢ Ⅎ𝑗ℝ
83	81, 82	nfel 2992	. . . . . . 7 ⊢ Ⅎ𝑗⦋𝑖 / 𝑗⦌𝐶 ∈ ℝ
84	80, 83	nfim 1893	. . . . . 6 ⊢ Ⅎ𝑗((𝜑 ∧ 𝑖 ∈ 𝑍) → ⦋𝑖 / 𝑗⦌𝐶 ∈ ℝ)
85		eleq1 2900	. . . . . . . 8 ⊢ (𝑗 = 𝑖 → (𝑗 ∈ 𝑍 ↔ 𝑖 ∈ 𝑍))
86	85	anbi2d 630	. . . . . . 7 ⊢ (𝑗 = 𝑖 → ((𝜑 ∧ 𝑗 ∈ 𝑍) ↔ (𝜑 ∧ 𝑖 ∈ 𝑍)))
87		csbeq1a 3896	. . . . . . . 8 ⊢ (𝑗 = 𝑖 → 𝐶 = ⦋𝑖 / 𝑗⦌𝐶)
88	87	eleq1d 2897	. . . . . . 7 ⊢ (𝑗 = 𝑖 → (𝐶 ∈ ℝ ↔ ⦋𝑖 / 𝑗⦌𝐶 ∈ ℝ))
89	86, 88	imbi12d 347	. . . . . 6 ⊢ (𝑗 = 𝑖 → (((𝜑 ∧ 𝑗 ∈ 𝑍) → 𝐶 ∈ ℝ) ↔ ((𝜑 ∧ 𝑖 ∈ 𝑍) → ⦋𝑖 / 𝑗⦌𝐶 ∈ ℝ)))
90		nfv 1911	. . . . . . . . 9 ⊢ Ⅎ𝑘 𝑗 ∈ 𝑍
91	5, 90	nfan 1896	. . . . . . . 8 ⊢ Ⅎ𝑘(𝜑 ∧ 𝑗 ∈ 𝑍)
92		nfv 1911	. . . . . . . 8 ⊢ Ⅎ𝑘 𝐶 ∈ ℝ
93	91, 92	nfim 1893	. . . . . . 7 ⊢ Ⅎ𝑘((𝜑 ∧ 𝑗 ∈ 𝑍) → 𝐶 ∈ ℝ)
94		eleq1 2900	. . . . . . . . 9 ⊢ (𝑘 = 𝑗 → (𝑘 ∈ 𝑍 ↔ 𝑗 ∈ 𝑍))
95	94	anbi2d 630	. . . . . . . 8 ⊢ (𝑘 = 𝑗 → ((𝜑 ∧ 𝑘 ∈ 𝑍) ↔ (𝜑 ∧ 𝑗 ∈ 𝑍)))
96	22	eleq1d 2897	. . . . . . . 8 ⊢ (𝑘 = 𝑗 → (𝐵 ∈ ℝ ↔ 𝐶 ∈ ℝ))
97	95, 96	imbi12d 347	. . . . . . 7 ⊢ (𝑘 = 𝑗 → (((𝜑 ∧ 𝑘 ∈ 𝑍) → 𝐵 ∈ ℝ) ↔ ((𝜑 ∧ 𝑗 ∈ 𝑍) → 𝐶 ∈ ℝ)))
98	93, 97, 6	chvarfv 2238	. . . . . 6 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → 𝐶 ∈ ℝ)
99	84, 89, 98	chvarfv 2238	. . . . 5 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑍) → ⦋𝑖 / 𝑗⦌𝐶 ∈ ℝ)
100		nfcv 2977	. . . . . 6 ⊢ Ⅎ𝑘𝑖
101		nfcv 2977	. . . . . 6 ⊢ Ⅎ𝑘⦋𝑖 / 𝑗⦌𝐶
102	100, 101, 30, 75	fvmptf 6788	. . . . 5 ⊢ ((𝑖 ∈ 𝑍 ∧ ⦋𝑖 / 𝑗⦌𝐶 ∈ ℝ) → ((𝑘 ∈ 𝑍 ↦ 𝐵)‘𝑖) = ⦋𝑖 / 𝑗⦌𝐶)
103	78, 99, 102	syl2anc 586	. . . 4 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑍) → ((𝑘 ∈ 𝑍 ↦ 𝐵)‘𝑖) = ⦋𝑖 / 𝑗⦌𝐶)
104	77, 103	breq12d 5078	. . 3 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑍) → (((𝑘 ∈ 𝑍 ↦ 𝐵)‘(𝑖 + 1)) ≤ ((𝑘 ∈ 𝑍 ↦ 𝐵)‘𝑖) ↔ ⦋(𝑖 + 1) / 𝑘⦌𝐵 ≤ ⦋𝑖 / 𝑗⦌𝐶))
105	58, 104	mpbird 259	. 2 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑍) → ((𝑘 ∈ 𝑍 ↦ 𝐵)‘(𝑖 + 1)) ≤ ((𝑘 ∈ 𝑍 ↦ 𝐵)‘𝑖))
106		climinf2mpt.e	. . . . 5 ⊢ (𝜑 → (𝑘 ∈ 𝑍 ↦ 𝐵) ∈ dom ⇝ )
107	103, 99	eqeltrd 2913	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑍) → ((𝑘 ∈ 𝑍 ↦ 𝐵)‘𝑖) ∈ ℝ)
108	107	recnd 10668	. . . . . 6 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑍) → ((𝑘 ∈ 𝑍 ↦ 𝐵)‘𝑖) ∈ ℂ)
109	108	ralrimiva 3182	. . . . 5 ⊢ (𝜑 → ∀𝑖 ∈ 𝑍 ((𝑘 ∈ 𝑍 ↦ 𝐵)‘𝑖) ∈ ℂ)
110	2, 3	climbddf 41966	. . . . 5 ⊢ ((𝑀 ∈ ℤ ∧ (𝑘 ∈ 𝑍 ↦ 𝐵) ∈ dom ⇝ ∧ ∀𝑖 ∈ 𝑍 ((𝑘 ∈ 𝑍 ↦ 𝐵)‘𝑖) ∈ ℂ) → ∃𝑥 ∈ ℝ ∀𝑖 ∈ 𝑍 (abs‘((𝑘 ∈ 𝑍 ↦ 𝐵)‘𝑖)) ≤ 𝑥)
111	4, 106, 109, 110	syl3anc 1367	. . . 4 ⊢ (𝜑 → ∃𝑥 ∈ ℝ ∀𝑖 ∈ 𝑍 (abs‘((𝑘 ∈ 𝑍 ↦ 𝐵)‘𝑖)) ≤ 𝑥)
112	1, 107	rexabsle2 41699	. . . 4 ⊢ (𝜑 → (∃𝑥 ∈ ℝ ∀𝑖 ∈ 𝑍 (abs‘((𝑘 ∈ 𝑍 ↦ 𝐵)‘𝑖)) ≤ 𝑥 ↔ (∃𝑥 ∈ ℝ ∀𝑖 ∈ 𝑍 ((𝑘 ∈ 𝑍 ↦ 𝐵)‘𝑖) ≤ 𝑥 ∧ ∃𝑥 ∈ ℝ ∀𝑖 ∈ 𝑍 𝑥 ≤ ((𝑘 ∈ 𝑍 ↦ 𝐵)‘𝑖))))
113	111, 112	mpbid 234	. . 3 ⊢ (𝜑 → (∃𝑥 ∈ ℝ ∀𝑖 ∈ 𝑍 ((𝑘 ∈ 𝑍 ↦ 𝐵)‘𝑖) ≤ 𝑥 ∧ ∃𝑥 ∈ ℝ ∀𝑖 ∈ 𝑍 𝑥 ≤ ((𝑘 ∈ 𝑍 ↦ 𝐵)‘𝑖)))
114	113	simprd 498	. 2 ⊢ (𝜑 → ∃𝑥 ∈ ℝ ∀𝑖 ∈ 𝑍 𝑥 ≤ ((𝑘 ∈ 𝑍 ↦ 𝐵)‘𝑖))
115	1, 2, 3, 4, 7, 105, 114	climinf2 41986	1 ⊢ (𝜑 → (𝑘 ∈ 𝑍 ↦ 𝐵) ⇝ inf(ran (𝑘 ∈ 𝑍 ↦ 𝐵), ℝ*, < ))

Syntax hints:   → wi 4   ∧ wa 398   ∧ w3a 1083   = wceq 1533  Ⅎwnf 1780   ∈ wcel 2110  ∀wral 3138  ∃wrex 3139  ⦋csb 3882   class class class wbr 5065   ↦ cmpt 5145  dom cdm 5554  ran crn 5555  ‘cfv 6354  (class class class)co 7155  infcinf 8904  ℂcc 10534  ℝcr 10535  1c1 10537   + caddc 10539  ℝ^*cxr 10673   < clt 10674   ≤ cle 10675  ℤcz 11980  ℤ_≥cuz 12242  abscabs 14592   ⇝ cli 14840

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2157  ax-12 2173  ax-ext 2793  ax-rep 5189  ax-sep 5202  ax-nul 5209  ax-pow 5265  ax-pr 5329  ax-un 7460  ax-cnex 10592  ax-resscn 10593  ax-1cn 10594  ax-icn 10595  ax-addcl 10596  ax-addrcl 10597  ax-mulcl 10598  ax-mulrcl 10599  ax-mulcom 10600  ax-addass 10601  ax-mulass 10602  ax-distr 10603  ax-i2m1 10604  ax-1ne0 10605  ax-1rid 10606  ax-rnegex 10607  ax-rrecex 10608  ax-cnre 10609  ax-pre-lttri 10610  ax-pre-lttrn 10611  ax-pre-ltadd 10612  ax-pre-mulgt0 10613  ax-pre-sup 10614

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1084  df-3an 1085  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-mo 2618  df-eu 2650  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-nel 3124  df-ral 3143  df-rex 3144  df-reu 3145  df-rmo 3146  df-rab 3147  df-v 3496  df-sbc 3772  df-csb 3883  df-dif 3938  df-un 3940  df-in 3942  df-ss 3951  df-pss 3953  df-nul 4291  df-if 4467  df-pw 4540  df-sn 4567  df-pr 4569  df-tp 4571  df-op 4573  df-uni 4838  df-int 4876  df-iun 4920  df-br 5066  df-opab 5128  df-mpt 5146  df-tr 5172  df-id 5459  df-eprel 5464  df-po 5473  df-so 5474  df-fr 5513  df-we 5515  df-xp 5560  df-rel 5561  df-cnv 5562  df-co 5563  df-dm 5564  df-rn 5565  df-res 5566  df-ima 5567  df-pred 6147  df-ord 6193  df-on 6194  df-lim 6195  df-suc 6196  df-iota 6313  df-fun 6356  df-fn 6357  df-f 6358  df-f1 6359  df-fo 6360  df-f1o 6361  df-fv 6362  df-riota 7113  df-ov 7158  df-oprab 7159  df-mpo 7160  df-om 7580  df-1st 7688  df-2nd 7689  df-wrecs 7946  df-recs 8007  df-rdg 8045  df-1o 8101  df-oadd 8105  df-er 8288  df-en 8509  df-dom 8510  df-sdom 8511  df-fin 8512  df-sup 8905  df-inf 8906  df-pnf 10676  df-mnf 10677  df-xr 10678  df-ltxr 10679  df-le 10680  df-sub 10871  df-neg 10872  df-div 11297  df-nn 11638  df-2 11699  df-3 11700  df-n0 11897  df-z 11981  df-uz 12243  df-rp 12389  df-fz 12892  df-seq 13369  df-exp 13429  df-cj 14457  df-re 14458  df-im 14459  df-sqrt 14593  df-abs 14594  df-clim 14844

This theorem is referenced by:  smflimsuplem4  43096