Theorem climinfmpt 45959

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

Hypotheses

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

Assertion

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

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

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

Proof of Theorem climinfmpt

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

Step	Hyp	Ref	Expression
1		nfv 1915	. 2 ⊢ Ⅎ𝑖𝜑
2		nfcv 2898	. 2 ⊢ Ⅎ𝑖(𝑘 ∈ 𝑍 ↦ 𝐵)
3		climinfmpt.z	. 2 ⊢ 𝑍 = (ℤ≥‘𝑀)
4		climinfmpt.m	. 2 ⊢ (𝜑 → 𝑀 ∈ ℤ)
5		climinfmpt.p	. . 3 ⊢ Ⅎ𝑘𝜑
6		climinfmpt.b	. . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → 𝐵 ∈ ℝ)
7	5, 6	fmptd2f 45479	. 2 ⊢ (𝜑 → (𝑘 ∈ 𝑍 ↦ 𝐵):𝑍⟶ℝ)
8		nfv 1915	. . . . . . 7 ⊢ Ⅎ𝑘 𝑖 ∈ 𝑍
9	5, 8	nfan 1900	. . . . . 6 ⊢ Ⅎ𝑘(𝜑 ∧ 𝑖 ∈ 𝑍)
10		nfv 1915	. . . . . 6 ⊢ Ⅎ𝑘⦋(𝑖 + 1) / 𝑗⦌𝐶 ≤ ⦋𝑖 / 𝑗⦌𝐶
11	9, 10	nfim 1897	. . . . 5 ⊢ Ⅎ𝑘((𝜑 ∧ 𝑖 ∈ 𝑍) → ⦋(𝑖 + 1) / 𝑗⦌𝐶 ≤ ⦋𝑖 / 𝑗⦌𝐶)
12		eleq1 2824	. . . . . . 7 ⊢ (𝑘 = 𝑖 → (𝑘 ∈ 𝑍 ↔ 𝑖 ∈ 𝑍))
13	12	anbi2d 630	. . . . . 6 ⊢ (𝑘 = 𝑖 → ((𝜑 ∧ 𝑘 ∈ 𝑍) ↔ (𝜑 ∧ 𝑖 ∈ 𝑍)))
14		oveq1 7365	. . . . . . . 8 ⊢ (𝑘 = 𝑖 → (𝑘 + 1) = (𝑖 + 1))
15	14	csbeq1d 3853	. . . . . . 7 ⊢ (𝑘 = 𝑖 → ⦋(𝑘 + 1) / 𝑗⦌𝐶 = ⦋(𝑖 + 1) / 𝑗⦌𝐶)
16		eqidd 2737	. . . . . . . 8 ⊢ (𝑘 = 𝑖 → 𝐵 = 𝐵)
17		csbcow 3864	. . . . . . . . . . 11 ⊢ ⦋𝑘 / 𝑗⦌⦋𝑗 / 𝑘⦌𝐵 = ⦋𝑘 / 𝑘⦌𝐵
18		csbid 3862	. . . . . . . . . . 11 ⊢ ⦋𝑘 / 𝑘⦌𝐵 = 𝐵
19	17, 18	eqtr2i 2760	. . . . . . . . . 10 ⊢ 𝐵 = ⦋𝑘 / 𝑗⦌⦋𝑗 / 𝑘⦌𝐵
20		nfcv 2898	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑗𝐵
21		nfcv 2898	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑘𝐶
22		climinfmpt.c	. . . . . . . . . . . . 13 ⊢ (𝑘 = 𝑗 → 𝐵 = 𝐶)
23	20, 21, 22	cbvcsbw 3859	. . . . . . . . . . . 12 ⊢ ⦋𝑗 / 𝑘⦌𝐵 = ⦋𝑗 / 𝑗⦌𝐶
24		csbid 3862	. . . . . . . . . . . 12 ⊢ ⦋𝑗 / 𝑗⦌𝐶 = 𝐶
25	23, 24	eqtri 2759	. . . . . . . . . . 11 ⊢ ⦋𝑗 / 𝑘⦌𝐵 = 𝐶
26	25	csbeq2i 3857	. . . . . . . . . 10 ⊢ ⦋𝑘 / 𝑗⦌⦋𝑗 / 𝑘⦌𝐵 = ⦋𝑘 / 𝑗⦌𝐶
27	19, 26	eqtri 2759	. . . . . . . . 9 ⊢ 𝐵 = ⦋𝑘 / 𝑗⦌𝐶
28	27	a1i 11	. . . . . . . 8 ⊢ (𝑘 = 𝑖 → 𝐵 = ⦋𝑘 / 𝑗⦌𝐶)
29		csbeq1 3852	. . . . . . . 8 ⊢ (𝑘 = 𝑖 → ⦋𝑘 / 𝑗⦌𝐶 = ⦋𝑖 / 𝑗⦌𝐶)
30	16, 28, 29	3eqtrd 2775	. . . . . . 7 ⊢ (𝑘 = 𝑖 → 𝐵 = ⦋𝑖 / 𝑗⦌𝐶)
31	15, 30	breq12d 5111	. . . . . 6 ⊢ (𝑘 = 𝑖 → (⦋(𝑘 + 1) / 𝑗⦌𝐶 ≤ 𝐵 ↔ ⦋(𝑖 + 1) / 𝑗⦌𝐶 ≤ ⦋𝑖 / 𝑗⦌𝐶))
32	13, 31	imbi12d 344	. . . . 5 ⊢ (𝑘 = 𝑖 → (((𝜑 ∧ 𝑘 ∈ 𝑍) → ⦋(𝑘 + 1) / 𝑗⦌𝐶 ≤ 𝐵) ↔ ((𝜑 ∧ 𝑖 ∈ 𝑍) → ⦋(𝑖 + 1) / 𝑗⦌𝐶 ≤ ⦋𝑖 / 𝑗⦌𝐶)))
33		simpl 482	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → 𝜑)
34		simpr 484	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → 𝑘 ∈ 𝑍)
35		eqidd 2737	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝑘 + 1) = (𝑘 + 1))
36		climinfmpt.j	. . . . . . . . 9 ⊢ Ⅎ𝑗𝜑
37		nfv 1915	. . . . . . . . 9 ⊢ Ⅎ𝑗 𝑘 ∈ 𝑍
38		nfv 1915	. . . . . . . . 9 ⊢ Ⅎ𝑗(𝑘 + 1) = (𝑘 + 1)
39	36, 37, 38	nf3an 1902	. . . . . . . 8 ⊢ Ⅎ𝑗(𝜑 ∧ 𝑘 ∈ 𝑍 ∧ (𝑘 + 1) = (𝑘 + 1))
40		nfcsb1v 3873	. . . . . . . . 9 ⊢ Ⅎ𝑗⦋(𝑘 + 1) / 𝑗⦌𝐶
41		nfcv 2898	. . . . . . . . 9 ⊢ Ⅎ𝑗 ≤
42	40, 41, 20	nfbr 5145	. . . . . . . 8 ⊢ Ⅎ𝑗⦋(𝑘 + 1) / 𝑗⦌𝐶 ≤ 𝐵
43	39, 42	nfim 1897	. . . . . . 7 ⊢ Ⅎ𝑗((𝜑 ∧ 𝑘 ∈ 𝑍 ∧ (𝑘 + 1) = (𝑘 + 1)) → ⦋(𝑘 + 1) / 𝑗⦌𝐶 ≤ 𝐵)
44		ovex 7391	. . . . . . 7 ⊢ (𝑘 + 1) ∈ V
45		eqeq1 2740	. . . . . . . . 9 ⊢ (𝑗 = (𝑘 + 1) → (𝑗 = (𝑘 + 1) ↔ (𝑘 + 1) = (𝑘 + 1)))
46	45	3anbi3d 1444	. . . . . . . 8 ⊢ (𝑗 = (𝑘 + 1) → ((𝜑 ∧ 𝑘 ∈ 𝑍 ∧ 𝑗 = (𝑘 + 1)) ↔ (𝜑 ∧ 𝑘 ∈ 𝑍 ∧ (𝑘 + 1) = (𝑘 + 1))))
47		csbeq1a 3863	. . . . . . . . 9 ⊢ (𝑗 = (𝑘 + 1) → 𝐶 = ⦋(𝑘 + 1) / 𝑗⦌𝐶)
48	47	breq1d 5108	. . . . . . . 8 ⊢ (𝑗 = (𝑘 + 1) → (𝐶 ≤ 𝐵 ↔ ⦋(𝑘 + 1) / 𝑗⦌𝐶 ≤ 𝐵))
49	46, 48	imbi12d 344	. . . . . . 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		climinfmpt.l	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍 ∧ 𝑗 = (𝑘 + 1)) → 𝐶 ≤ 𝐵)
51	43, 44, 49, 50	vtoclf 3521	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍 ∧ (𝑘 + 1) = (𝑘 + 1)) → ⦋(𝑘 + 1) / 𝑗⦌𝐶 ≤ 𝐵)
52	33, 34, 35, 51	syl3anc 1373	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → ⦋(𝑘 + 1) / 𝑗⦌𝐶 ≤ 𝐵)
53	11, 32, 52	chvarfv 2247	. . . 4 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑍) → ⦋(𝑖 + 1) / 𝑗⦌𝐶 ≤ ⦋𝑖 / 𝑗⦌𝐶)
54	20, 21, 22	cbvcsbw 3859	. . . . . 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 2737	. . . . 5 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑍) → ⦋𝑖 / 𝑗⦌𝐶 = ⦋𝑖 / 𝑗⦌𝐶)
57	55, 56	breq12d 5111	. . . 4 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑍) → (⦋(𝑖 + 1) / 𝑘⦌𝐵 ≤ ⦋𝑖 / 𝑗⦌𝐶 ↔ ⦋(𝑖 + 1) / 𝑗⦌𝐶 ≤ ⦋𝑖 / 𝑗⦌𝐶))
58	53, 57	mpbird 257	. . 3 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑍) → ⦋(𝑖 + 1) / 𝑘⦌𝐵 ≤ ⦋𝑖 / 𝑗⦌𝐶)
59	3	peano2uzs 12815	. . . . . 6 ⊢ (𝑖 ∈ 𝑍 → (𝑖 + 1) ∈ 𝑍)
60	59	adantl 481	. . . . 5 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑍) → (𝑖 + 1) ∈ 𝑍)
61		nfv 1915	. . . . . . . . 9 ⊢ Ⅎ𝑘(𝑖 + 1) ∈ 𝑍
62	5, 61	nfan 1900	. . . . . . . 8 ⊢ Ⅎ𝑘(𝜑 ∧ (𝑖 + 1) ∈ 𝑍)
63		nfcv 2898	. . . . . . . . . 10 ⊢ Ⅎ𝑘(𝑖 + 1)
64	63	nfcsb1 3872	. . . . . . . . 9 ⊢ Ⅎ𝑘⦋(𝑖 + 1) / 𝑘⦌𝐵
65	64	nfel1 2915	. . . . . . . 8 ⊢ Ⅎ𝑘⦋(𝑖 + 1) / 𝑘⦌𝐵 ∈ ℝ
66	62, 65	nfim 1897	. . . . . . 7 ⊢ Ⅎ𝑘((𝜑 ∧ (𝑖 + 1) ∈ 𝑍) → ⦋(𝑖 + 1) / 𝑘⦌𝐵 ∈ ℝ)
67		ovex 7391	. . . . . . 7 ⊢ (𝑖 + 1) ∈ V
68		eleq1 2824	. . . . . . . . 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 3863	. . . . . . . . 9 ⊢ (𝑘 = (𝑖 + 1) → 𝐵 = ⦋(𝑖 + 1) / 𝑘⦌𝐵)
71	70	eleq1d 2821	. . . . . . . 8 ⊢ (𝑘 = (𝑖 + 1) → (𝐵 ∈ ℝ ↔ ⦋(𝑖 + 1) / 𝑘⦌𝐵 ∈ ℝ))
72	69, 71	imbi12d 344	. . . . . . 7 ⊢ (𝑘 = (𝑖 + 1) → (((𝜑 ∧ 𝑘 ∈ 𝑍) → 𝐵 ∈ ℝ) ↔ ((𝜑 ∧ (𝑖 + 1) ∈ 𝑍) → ⦋(𝑖 + 1) / 𝑘⦌𝐵 ∈ ℝ)))
73	66, 67, 72, 6	vtoclf 3521	. . . . . 6 ⊢ ((𝜑 ∧ (𝑖 + 1) ∈ 𝑍) → ⦋(𝑖 + 1) / 𝑘⦌𝐵 ∈ ℝ)
74	59, 73	sylan2 593	. . . . 5 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑍) → ⦋(𝑖 + 1) / 𝑘⦌𝐵 ∈ ℝ)
75		eqid 2736	. . . . . 6 ⊢ (𝑘 ∈ 𝑍 ↦ 𝐵) = (𝑘 ∈ 𝑍 ↦ 𝐵)
76	63, 64, 70, 75	fvmptf 6962	. . . . 5 ⊢ (((𝑖 + 1) ∈ 𝑍 ∧ ⦋(𝑖 + 1) / 𝑘⦌𝐵 ∈ ℝ) → ((𝑘 ∈ 𝑍 ↦ 𝐵)‘(𝑖 + 1)) = ⦋(𝑖 + 1) / 𝑘⦌𝐵)
77	60, 74, 76	syl2anc 584	. . . 4 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑍) → ((𝑘 ∈ 𝑍 ↦ 𝐵)‘(𝑖 + 1)) = ⦋(𝑖 + 1) / 𝑘⦌𝐵)
78		simpr 484	. . . . 5 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑍) → 𝑖 ∈ 𝑍)
79		nfv 1915	. . . . . . . 8 ⊢ Ⅎ𝑗 𝑖 ∈ 𝑍
80	36, 79	nfan 1900	. . . . . . 7 ⊢ Ⅎ𝑗(𝜑 ∧ 𝑖 ∈ 𝑍)
81		nfcsb1v 3873	. . . . . . . 8 ⊢ Ⅎ𝑗⦋𝑖 / 𝑗⦌𝐶
82		nfcv 2898	. . . . . . . 8 ⊢ Ⅎ𝑗ℝ
83	81, 82	nfel 2913	. . . . . . 7 ⊢ Ⅎ𝑗⦋𝑖 / 𝑗⦌𝐶 ∈ ℝ
84	80, 83	nfim 1897	. . . . . 6 ⊢ Ⅎ𝑗((𝜑 ∧ 𝑖 ∈ 𝑍) → ⦋𝑖 / 𝑗⦌𝐶 ∈ ℝ)
85		eleq1 2824	. . . . . . . 8 ⊢ (𝑗 = 𝑖 → (𝑗 ∈ 𝑍 ↔ 𝑖 ∈ 𝑍))
86	85	anbi2d 630	. . . . . . 7 ⊢ (𝑗 = 𝑖 → ((𝜑 ∧ 𝑗 ∈ 𝑍) ↔ (𝜑 ∧ 𝑖 ∈ 𝑍)))
87		csbeq1a 3863	. . . . . . . 8 ⊢ (𝑗 = 𝑖 → 𝐶 = ⦋𝑖 / 𝑗⦌𝐶)
88	87	eleq1d 2821	. . . . . . 7 ⊢ (𝑗 = 𝑖 → (𝐶 ∈ ℝ ↔ ⦋𝑖 / 𝑗⦌𝐶 ∈ ℝ))
89	86, 88	imbi12d 344	. . . . . 6 ⊢ (𝑗 = 𝑖 → (((𝜑 ∧ 𝑗 ∈ 𝑍) → 𝐶 ∈ ℝ) ↔ ((𝜑 ∧ 𝑖 ∈ 𝑍) → ⦋𝑖 / 𝑗⦌𝐶 ∈ ℝ)))
90		nfv 1915	. . . . . . . . 9 ⊢ Ⅎ𝑘 𝑗 ∈ 𝑍
91	5, 90	nfan 1900	. . . . . . . 8 ⊢ Ⅎ𝑘(𝜑 ∧ 𝑗 ∈ 𝑍)
92		nfv 1915	. . . . . . . 8 ⊢ Ⅎ𝑘 𝐶 ∈ ℝ
93	91, 92	nfim 1897	. . . . . . 7 ⊢ Ⅎ𝑘((𝜑 ∧ 𝑗 ∈ 𝑍) → 𝐶 ∈ ℝ)
94		eleq1 2824	. . . . . . . . 9 ⊢ (𝑘 = 𝑗 → (𝑘 ∈ 𝑍 ↔ 𝑗 ∈ 𝑍))
95	94	anbi2d 630	. . . . . . . 8 ⊢ (𝑘 = 𝑗 → ((𝜑 ∧ 𝑘 ∈ 𝑍) ↔ (𝜑 ∧ 𝑗 ∈ 𝑍)))
96	22	eleq1d 2821	. . . . . . . 8 ⊢ (𝑘 = 𝑗 → (𝐵 ∈ ℝ ↔ 𝐶 ∈ ℝ))
97	95, 96	imbi12d 344	. . . . . . 7 ⊢ (𝑘 = 𝑗 → (((𝜑 ∧ 𝑘 ∈ 𝑍) → 𝐵 ∈ ℝ) ↔ ((𝜑 ∧ 𝑗 ∈ 𝑍) → 𝐶 ∈ ℝ)))
98	93, 97, 6	chvarfv 2247	. . . . . 6 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → 𝐶 ∈ ℝ)
99	84, 89, 98	chvarfv 2247	. . . . 5 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑍) → ⦋𝑖 / 𝑗⦌𝐶 ∈ ℝ)
100		nfcv 2898	. . . . . 6 ⊢ Ⅎ𝑘𝑖
101		nfcv 2898	. . . . . 6 ⊢ Ⅎ𝑘⦋𝑖 / 𝑗⦌𝐶
102	100, 101, 30, 75	fvmptf 6962	. . . . 5 ⊢ ((𝑖 ∈ 𝑍 ∧ ⦋𝑖 / 𝑗⦌𝐶 ∈ ℝ) → ((𝑘 ∈ 𝑍 ↦ 𝐵)‘𝑖) = ⦋𝑖 / 𝑗⦌𝐶)
103	78, 99, 102	syl2anc 584	. . . 4 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑍) → ((𝑘 ∈ 𝑍 ↦ 𝐵)‘𝑖) = ⦋𝑖 / 𝑗⦌𝐶)
104	77, 103	breq12d 5111	. . 3 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑍) → (((𝑘 ∈ 𝑍 ↦ 𝐵)‘(𝑖 + 1)) ≤ ((𝑘 ∈ 𝑍 ↦ 𝐵)‘𝑖) ↔ ⦋(𝑖 + 1) / 𝑘⦌𝐵 ≤ ⦋𝑖 / 𝑗⦌𝐶))
105	58, 104	mpbird 257	. 2 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑍) → ((𝑘 ∈ 𝑍 ↦ 𝐵)‘(𝑖 + 1)) ≤ ((𝑘 ∈ 𝑍 ↦ 𝐵)‘𝑖))
106		climinfmpt.e	. . . 4 ⊢ (𝜑 → ∃𝑥 ∈ ℝ ∀𝑘 ∈ 𝑍 𝑥 ≤ 𝐵)
107		breq1 5101	. . . . . 6 ⊢ (𝑥 = 𝑦 → (𝑥 ≤ 𝐵 ↔ 𝑦 ≤ 𝐵))
108	107	ralbidv 3159	. . . . 5 ⊢ (𝑥 = 𝑦 → (∀𝑘 ∈ 𝑍 𝑥 ≤ 𝐵 ↔ ∀𝑘 ∈ 𝑍 𝑦 ≤ 𝐵))
109	108	cbvrexvw 3215	. . . 4 ⊢ (∃𝑥 ∈ ℝ ∀𝑘 ∈ 𝑍 𝑥 ≤ 𝐵 ↔ ∃𝑦 ∈ ℝ ∀𝑘 ∈ 𝑍 𝑦 ≤ 𝐵)
110	106, 109	sylib 218	. . 3 ⊢ (𝜑 → ∃𝑦 ∈ ℝ ∀𝑘 ∈ 𝑍 𝑦 ≤ 𝐵)
111		nfcv 2898	. . . . . . . 8 ⊢ Ⅎ𝑘𝑦
112		nfcv 2898	. . . . . . . 8 ⊢ Ⅎ𝑘 ≤
113		nfmpt1 5197	. . . . . . . . 9 ⊢ Ⅎ𝑘(𝑘 ∈ 𝑍 ↦ 𝐵)
114	113, 100	nffv 6844	. . . . . . . 8 ⊢ Ⅎ𝑘((𝑘 ∈ 𝑍 ↦ 𝐵)‘𝑖)
115	111, 112, 114	nfbr 5145	. . . . . . 7 ⊢ Ⅎ𝑘 𝑦 ≤ ((𝑘 ∈ 𝑍 ↦ 𝐵)‘𝑖)
116		nfv 1915	. . . . . . 7 ⊢ Ⅎ𝑖 𝑦 ≤ ((𝑘 ∈ 𝑍 ↦ 𝐵)‘𝑘)
117		fveq2 6834	. . . . . . . 8 ⊢ (𝑖 = 𝑘 → ((𝑘 ∈ 𝑍 ↦ 𝐵)‘𝑖) = ((𝑘 ∈ 𝑍 ↦ 𝐵)‘𝑘))
118	117	breq2d 5110	. . . . . . 7 ⊢ (𝑖 = 𝑘 → (𝑦 ≤ ((𝑘 ∈ 𝑍 ↦ 𝐵)‘𝑖) ↔ 𝑦 ≤ ((𝑘 ∈ 𝑍 ↦ 𝐵)‘𝑘)))
119	115, 116, 118	cbvralw 3278	. . . . . 6 ⊢ (∀𝑖 ∈ 𝑍 𝑦 ≤ ((𝑘 ∈ 𝑍 ↦ 𝐵)‘𝑖) ↔ ∀𝑘 ∈ 𝑍 𝑦 ≤ ((𝑘 ∈ 𝑍 ↦ 𝐵)‘𝑘))
120	119	a1i 11	. . . . 5 ⊢ (𝜑 → (∀𝑖 ∈ 𝑍 𝑦 ≤ ((𝑘 ∈ 𝑍 ↦ 𝐵)‘𝑖) ↔ ∀𝑘 ∈ 𝑍 𝑦 ≤ ((𝑘 ∈ 𝑍 ↦ 𝐵)‘𝑘)))
121	75	a1i 11	. . . . . . . 8 ⊢ (𝜑 → (𝑘 ∈ 𝑍 ↦ 𝐵) = (𝑘 ∈ 𝑍 ↦ 𝐵))
122	121, 6	fvmpt2d 6954	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → ((𝑘 ∈ 𝑍 ↦ 𝐵)‘𝑘) = 𝐵)
123	122	breq2d 5110	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝑦 ≤ ((𝑘 ∈ 𝑍 ↦ 𝐵)‘𝑘) ↔ 𝑦 ≤ 𝐵))
124	5, 123	ralbida 3247	. . . . 5 ⊢ (𝜑 → (∀𝑘 ∈ 𝑍 𝑦 ≤ ((𝑘 ∈ 𝑍 ↦ 𝐵)‘𝑘) ↔ ∀𝑘 ∈ 𝑍 𝑦 ≤ 𝐵))
125	120, 124	bitrd 279	. . . 4 ⊢ (𝜑 → (∀𝑖 ∈ 𝑍 𝑦 ≤ ((𝑘 ∈ 𝑍 ↦ 𝐵)‘𝑖) ↔ ∀𝑘 ∈ 𝑍 𝑦 ≤ 𝐵))
126	125	rexbidv 3160	. . 3 ⊢ (𝜑 → (∃𝑦 ∈ ℝ ∀𝑖 ∈ 𝑍 𝑦 ≤ ((𝑘 ∈ 𝑍 ↦ 𝐵)‘𝑖) ↔ ∃𝑦 ∈ ℝ ∀𝑘 ∈ 𝑍 𝑦 ≤ 𝐵))
127	110, 126	mpbird 257	. 2 ⊢ (𝜑 → ∃𝑦 ∈ ℝ ∀𝑖 ∈ 𝑍 𝑦 ≤ ((𝑘 ∈ 𝑍 ↦ 𝐵)‘𝑖))
128	1, 2, 3, 4, 7, 105, 127	climinf2 45951	1 ⊢ (𝜑 → (𝑘 ∈ 𝑍 ↦ 𝐵) ⇝ inf(ran (𝑘 ∈ 𝑍 ↦ 𝐵), ℝ*, < ))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1086   = wceq 1541  Ⅎwnf 1784   ∈ wcel 2113  ∀wral 3051  ∃wrex 3060  ⦋csb 3849   class class class wbr 5098   ↦ cmpt 5179  ran crn 5625  ‘cfv 6492  (class class class)co 7358  infcinf 9344  ℝcr 11025  1c1 11027   + caddc 11029  ℝ^*cxr 11165   < clt 11166   ≤ cle 11167  ℤcz 12488  ℤ_≥cuz 12751   ⇝ cli 15407

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2184  ax-ext 2708  ax-rep 5224  ax-sep 5241  ax-nul 5251  ax-pow 5310  ax-pr 5377  ax-un 7680  ax-cnex 11082  ax-resscn 11083  ax-1cn 11084  ax-icn 11085  ax-addcl 11086  ax-addrcl 11087  ax-mulcl 11088  ax-mulrcl 11089  ax-mulcom 11090  ax-addass 11091  ax-mulass 11092  ax-distr 11093  ax-i2m1 11094  ax-1ne0 11095  ax-1rid 11096  ax-rnegex 11097  ax-rrecex 11098  ax-cnre 11099  ax-pre-lttri 11100  ax-pre-lttrn 11101  ax-pre-ltadd 11102  ax-pre-mulgt0 11103  ax-pre-sup 11104

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  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 3061  df-rmo 3350  df-reu 3351  df-rab 3400  df-v 3442  df-sbc 3741  df-csb 3850  df-dif 3904  df-un 3906  df-in 3908  df-ss 3918  df-pss 3921  df-nul 4286  df-if 4480  df-pw 4556  df-sn 4581  df-pr 4583  df-op 4587  df-uni 4864  df-iun 4948  df-br 5099  df-opab 5161  df-mpt 5180  df-tr 5206  df-id 5519  df-eprel 5524  df-po 5532  df-so 5533  df-fr 5577  df-we 5579  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-rn 5635  df-res 5636  df-ima 5637  df-pred 6259  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6494  df-fn 6495  df-f 6496  df-f1 6497  df-fo 6498  df-f1o 6499  df-fv 6500  df-riota 7315  df-ov 7361  df-oprab 7362  df-mpo 7363  df-om 7809  df-1st 7933  df-2nd 7934  df-frecs 8223  df-wrecs 8254  df-recs 8303  df-rdg 8341  df-er 8635  df-en 8884  df-dom 8885  df-sdom 8886  df-sup 9345  df-inf 9346  df-pnf 11168  df-mnf 11169  df-xr 11170  df-ltxr 11171  df-le 11172  df-sub 11366  df-neg 11367  df-div 11795  df-nn 12146  df-2 12208  df-3 12209  df-n0 12402  df-z 12489  df-uz 12752  df-rp 12906  df-fz 13424  df-seq 13925  df-exp 13985  df-cj 15022  df-re 15023  df-im 15024  df-sqrt 15158  df-abs 15159  df-clim 15411

This theorem is referenced by: (None)