infleinf - Mathbox for Glauco Siliprandi

	Mathbox for Glauco Siliprandi		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > Mathboxes > infleinf		Structured version Visualization version GIF version

Theorem infleinf 45352

Description: If any element of 𝐵 can be approximated from above by members of 𝐴, then the infimum of 𝐴 is less than or equal to the infimum of 𝐵. (Contributed by Glauco Siliprandi, 3-Mar-2021.)

**Hypotheses**
Ref	Expression
infleinf.a	⊢ (𝜑 → 𝐴 ⊆ ℝ^*)
infleinf.b	⊢ (𝜑 → 𝐵 ⊆ ℝ^*)
infleinf.c	⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ ℝ⁺) → ∃𝑧 ∈ 𝐴 𝑧 ≤ (𝑥 +_𝑒 𝑦))

**Assertion**
Ref	Expression
infleinf	⊢ (𝜑 → inf(𝐴, ℝ^, < ) ≤ inf(𝐵, ℝ^, < ))

Distinct variable groups: 𝑥,𝐴,𝑦,𝑧 𝑥,𝐵,𝑦,𝑧 𝜑,𝑥,𝑦,𝑧

Proof of Theorem infleinf Dummy variables 𝑟 𝑤 𝑏 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		infleinf.a	. . . . . 6 ⊢ (𝜑 → 𝐴 ⊆ ℝ^*)
2		infxrcl 13254	. . . . . 6 ⊢ (𝐴 ⊆ ℝ^* → inf(𝐴, ℝ^, < ) ∈ ℝ^)
3	1, 2	syl 17	. . . . 5 ⊢ (𝜑 → inf(𝐴, ℝ^, < ) ∈ ℝ^)
4		pnfge 13050	. . . . 5 ⊢ (inf(𝐴, ℝ^, < ) ∈ ℝ^ → inf(𝐴, ℝ^*, < ) ≤ +∞)
5	3, 4	syl 17	. . . 4 ⊢ (𝜑 → inf(𝐴, ℝ^*, < ) ≤ +∞)
6	5	adantr 480	. . 3 ⊢ ((𝜑 ∧ 𝐵 = ∅) → inf(𝐴, ℝ^*, < ) ≤ +∞)
7		infeq1 9386	. . . . . 6 ⊢ (𝐵 = ∅ → inf(𝐵, ℝ^, < ) = inf(∅, ℝ^, < ))
8		xrinf0 13259	. . . . . . 7 ⊢ inf(∅, ℝ^*, < ) = +∞
9	8	a1i 11	. . . . . 6 ⊢ (𝐵 = ∅ → inf(∅, ℝ^*, < ) = +∞)
10	7, 9	eqtrd 2764	. . . . 5 ⊢ (𝐵 = ∅ → inf(𝐵, ℝ^*, < ) = +∞)
11	10	eqcomd 2735	. . . 4 ⊢ (𝐵 = ∅ → +∞ = inf(𝐵, ℝ^*, < ))
12	11	adantl 481	. . 3 ⊢ ((𝜑 ∧ 𝐵 = ∅) → +∞ = inf(𝐵, ℝ^*, < ))
13	6, 12	breqtrd 5121	. 2 ⊢ ((𝜑 ∧ 𝐵 = ∅) → inf(𝐴, ℝ^, < ) ≤ inf(𝐵, ℝ^, < ))
14		neqne 2933	. . . 4 ⊢ (¬ 𝐵 = ∅ → 𝐵 ≠ ∅)
15	14	adantl 481	. . 3 ⊢ ((𝜑 ∧ ¬ 𝐵 = ∅) → 𝐵 ≠ ∅)
16	3	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ inf(𝐵, ℝ^, < ) = -∞) → inf(𝐴, ℝ^, < ) ∈ ℝ^*)
17		id 22	. . . . . . . . . . . . 13 ⊢ (𝑟 ∈ ℝ → 𝑟 ∈ ℝ)
18		2re 12220	. . . . . . . . . . . . . 14 ⊢ 2 ∈ ℝ
19	18	a1i 11	. . . . . . . . . . . . 13 ⊢ (𝑟 ∈ ℝ → 2 ∈ ℝ)
20	17, 19	resubcld 11566	. . . . . . . . . . . 12 ⊢ (𝑟 ∈ ℝ → (𝑟 − 2) ∈ ℝ)
21	20	adantl 481	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ inf(𝐵, ℝ^*, < ) = -∞) ∧ 𝑟 ∈ ℝ) → (𝑟 − 2) ∈ ℝ)
22		simpr 484	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ inf(𝐵, ℝ^, < ) = -∞) → inf(𝐵, ℝ^, < ) = -∞)
23		infleinf.b	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝐵 ⊆ ℝ^*)
24		infxrunb2 45348	. . . . . . . . . . . . . . 15 ⊢ (𝐵 ⊆ ℝ^* → (∀𝑦 ∈ ℝ ∃𝑥 ∈ 𝐵 𝑥 < 𝑦 ↔ inf(𝐵, ℝ^*, < ) = -∞))
25	23, 24	syl 17	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (∀𝑦 ∈ ℝ ∃𝑥 ∈ 𝐵 𝑥 < 𝑦 ↔ inf(𝐵, ℝ^*, < ) = -∞))
26	25	adantr 480	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ inf(𝐵, ℝ^, < ) = -∞) → (∀𝑦 ∈ ℝ ∃𝑥 ∈ 𝐵 𝑥 < 𝑦 ↔ inf(𝐵, ℝ^, < ) = -∞))
27	22, 26	mpbird 257	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ inf(𝐵, ℝ^*, < ) = -∞) → ∀𝑦 ∈ ℝ ∃𝑥 ∈ 𝐵 𝑥 < 𝑦)
28	27	adantr 480	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ inf(𝐵, ℝ^*, < ) = -∞) ∧ 𝑟 ∈ ℝ) → ∀𝑦 ∈ ℝ ∃𝑥 ∈ 𝐵 𝑥 < 𝑦)
29		breq2 5099	. . . . . . . . . . . . 13 ⊢ (𝑦 = (𝑟 − 2) → (𝑥 < 𝑦 ↔ 𝑥 < (𝑟 − 2)))
30	29	rexbidv 3153	. . . . . . . . . . . 12 ⊢ (𝑦 = (𝑟 − 2) → (∃𝑥 ∈ 𝐵 𝑥 < 𝑦 ↔ ∃𝑥 ∈ 𝐵 𝑥 < (𝑟 − 2)))
31	30	rspcva 3577	. . . . . . . . . . 11 ⊢ (((𝑟 − 2) ∈ ℝ ∧ ∀𝑦 ∈ ℝ ∃𝑥 ∈ 𝐵 𝑥 < 𝑦) → ∃𝑥 ∈ 𝐵 𝑥 < (𝑟 − 2))
32	21, 28, 31	syl2anc 584	. . . . . . . . . 10 ⊢ (((𝜑 ∧ inf(𝐵, ℝ^*, < ) = -∞) ∧ 𝑟 ∈ ℝ) → ∃𝑥 ∈ 𝐵 𝑥 < (𝑟 − 2))
33		simpl 482	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → 𝜑)
34		simpr 484	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → 𝑥 ∈ 𝐵)
35		1rp 12915	. . . . . . . . . . . . . . . . . 18 ⊢ 1 ∈ ℝ⁺
36	35	a1i 11	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → 1 ∈ ℝ⁺)
37		1ex 11130	. . . . . . . . . . . . . . . . . 18 ⊢ 1 ∈ V
38		eleq1 2816	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑦 = 1 → (𝑦 ∈ ℝ⁺ ↔ 1 ∈ ℝ⁺))
39	38	3anbi3d 1444	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑦 = 1 → ((𝜑 ∧ 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ ℝ⁺) ↔ (𝜑 ∧ 𝑥 ∈ 𝐵 ∧ 1 ∈ ℝ⁺)))
40		oveq2 7361	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑦 = 1 → (𝑥 +_𝑒 𝑦) = (𝑥 +_𝑒 1))
41	40	breq2d 5107	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑦 = 1 → (𝑧 ≤ (𝑥 +_𝑒 𝑦) ↔ 𝑧 ≤ (𝑥 +_𝑒 1)))
42	41	rexbidv 3153	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑦 = 1 → (∃𝑧 ∈ 𝐴 𝑧 ≤ (𝑥 +_𝑒 𝑦) ↔ ∃𝑧 ∈ 𝐴 𝑧 ≤ (𝑥 +_𝑒 1)))
43	39, 42	imbi12d 344	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑦 = 1 → (((𝜑 ∧ 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ ℝ⁺) → ∃𝑧 ∈ 𝐴 𝑧 ≤ (𝑥 +_𝑒 𝑦)) ↔ ((𝜑 ∧ 𝑥 ∈ 𝐵 ∧ 1 ∈ ℝ⁺) → ∃𝑧 ∈ 𝐴 𝑧 ≤ (𝑥 +_𝑒 1))))
44		infleinf.c	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ ℝ⁺) → ∃𝑧 ∈ 𝐴 𝑧 ≤ (𝑥 +_𝑒 𝑦))
45	37, 43, 44	vtocl 3515	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵 ∧ 1 ∈ ℝ⁺) → ∃𝑧 ∈ 𝐴 𝑧 ≤ (𝑥 +_𝑒 1))
46	33, 34, 36, 45	syl3anc 1373	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → ∃𝑧 ∈ 𝐴 𝑧 ≤ (𝑥 +_𝑒 1))
47	46	adantlr 715	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑟 ∈ ℝ) ∧ 𝑥 ∈ 𝐵) → ∃𝑧 ∈ 𝐴 𝑧 ≤ (𝑥 +_𝑒 1))
48	47	3adant3 1132	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑟 ∈ ℝ) ∧ 𝑥 ∈ 𝐵 ∧ 𝑥 < (𝑟 − 2)) → ∃𝑧 ∈ 𝐴 𝑧 ≤ (𝑥 +_𝑒 1))
49		simp1l 1198	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ 𝑟 ∈ ℝ) ∧ 𝑥 ∈ 𝐵 ∧ 𝑥 < (𝑟 − 2)) → 𝜑)
50	49	ad2antrr 726	. . . . . . . . . . . . . . . . . 18 ⊢ (((((𝜑 ∧ 𝑟 ∈ ℝ) ∧ 𝑥 ∈ 𝐵 ∧ 𝑥 < (𝑟 − 2)) ∧ 𝑧 ∈ 𝐴) ∧ 𝑧 ≤ (𝑥 +_𝑒 1)) → 𝜑)
51	50, 1	syl 17	. . . . . . . . . . . . . . . . 17 ⊢ (((((𝜑 ∧ 𝑟 ∈ ℝ) ∧ 𝑥 ∈ 𝐵 ∧ 𝑥 < (𝑟 − 2)) ∧ 𝑧 ∈ 𝐴) ∧ 𝑧 ≤ (𝑥 +_𝑒 1)) → 𝐴 ⊆ ℝ^*)
52	50, 23	syl 17	. . . . . . . . . . . . . . . . 17 ⊢ (((((𝜑 ∧ 𝑟 ∈ ℝ) ∧ 𝑥 ∈ 𝐵 ∧ 𝑥 < (𝑟 − 2)) ∧ 𝑧 ∈ 𝐴) ∧ 𝑧 ≤ (𝑥 +_𝑒 1)) → 𝐵 ⊆ ℝ^*)
53		simp1r 1199	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑟 ∈ ℝ) ∧ 𝑥 ∈ 𝐵 ∧ 𝑥 < (𝑟 − 2)) → 𝑟 ∈ ℝ)
54	53	ad2antrr 726	. . . . . . . . . . . . . . . . 17 ⊢ (((((𝜑 ∧ 𝑟 ∈ ℝ) ∧ 𝑥 ∈ 𝐵 ∧ 𝑥 < (𝑟 − 2)) ∧ 𝑧 ∈ 𝐴) ∧ 𝑧 ≤ (𝑥 +_𝑒 1)) → 𝑟 ∈ ℝ)
55		simp2 1137	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑟 ∈ ℝ) ∧ 𝑥 ∈ 𝐵 ∧ 𝑥 < (𝑟 − 2)) → 𝑥 ∈ 𝐵)
56	55	ad2antrr 726	. . . . . . . . . . . . . . . . 17 ⊢ (((((𝜑 ∧ 𝑟 ∈ ℝ) ∧ 𝑥 ∈ 𝐵 ∧ 𝑥 < (𝑟 − 2)) ∧ 𝑧 ∈ 𝐴) ∧ 𝑧 ≤ (𝑥 +_𝑒 1)) → 𝑥 ∈ 𝐵)
57		simpll3 1215	. . . . . . . . . . . . . . . . 17 ⊢ (((((𝜑 ∧ 𝑟 ∈ ℝ) ∧ 𝑥 ∈ 𝐵 ∧ 𝑥 < (𝑟 − 2)) ∧ 𝑧 ∈ 𝐴) ∧ 𝑧 ≤ (𝑥 +_𝑒 1)) → 𝑥 < (𝑟 − 2))
58		simplr 768	. . . . . . . . . . . . . . . . 17 ⊢ (((((𝜑 ∧ 𝑟 ∈ ℝ) ∧ 𝑥 ∈ 𝐵 ∧ 𝑥 < (𝑟 − 2)) ∧ 𝑧 ∈ 𝐴) ∧ 𝑧 ≤ (𝑥 +_𝑒 1)) → 𝑧 ∈ 𝐴)
59		simpr 484	. . . . . . . . . . . . . . . . 17 ⊢ (((((𝜑 ∧ 𝑟 ∈ ℝ) ∧ 𝑥 ∈ 𝐵 ∧ 𝑥 < (𝑟 − 2)) ∧ 𝑧 ∈ 𝐴) ∧ 𝑧 ≤ (𝑥 +_𝑒 1)) → 𝑧 ≤ (𝑥 +_𝑒 1))
60	51, 52, 54, 56, 57, 58, 59	infleinflem2 45351	. . . . . . . . . . . . . . . 16 ⊢ (((((𝜑 ∧ 𝑟 ∈ ℝ) ∧ 𝑥 ∈ 𝐵 ∧ 𝑥 < (𝑟 − 2)) ∧ 𝑧 ∈ 𝐴) ∧ 𝑧 ≤ (𝑥 +_𝑒 1)) → 𝑧 < 𝑟)
61	60	ex 412	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ 𝑟 ∈ ℝ) ∧ 𝑥 ∈ 𝐵 ∧ 𝑥 < (𝑟 − 2)) ∧ 𝑧 ∈ 𝐴) → (𝑧 ≤ (𝑥 +_𝑒 1) → 𝑧 < 𝑟))
62	61	reximdva 3142	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑟 ∈ ℝ) ∧ 𝑥 ∈ 𝐵 ∧ 𝑥 < (𝑟 − 2)) → (∃𝑧 ∈ 𝐴 𝑧 ≤ (𝑥 +_𝑒 1) → ∃𝑧 ∈ 𝐴 𝑧 < 𝑟))
63	48, 62	mpd 15	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑟 ∈ ℝ) ∧ 𝑥 ∈ 𝐵 ∧ 𝑥 < (𝑟 − 2)) → ∃𝑧 ∈ 𝐴 𝑧 < 𝑟)
64	63	3exp 1119	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑟 ∈ ℝ) → (𝑥 ∈ 𝐵 → (𝑥 < (𝑟 − 2) → ∃𝑧 ∈ 𝐴 𝑧 < 𝑟)))
65	64	adantlr 715	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ inf(𝐵, ℝ^*, < ) = -∞) ∧ 𝑟 ∈ ℝ) → (𝑥 ∈ 𝐵 → (𝑥 < (𝑟 − 2) → ∃𝑧 ∈ 𝐴 𝑧 < 𝑟)))
66	65	rexlimdv 3128	. . . . . . . . . 10 ⊢ (((𝜑 ∧ inf(𝐵, ℝ^*, < ) = -∞) ∧ 𝑟 ∈ ℝ) → (∃𝑥 ∈ 𝐵 𝑥 < (𝑟 − 2) → ∃𝑧 ∈ 𝐴 𝑧 < 𝑟))
67	32, 66	mpd 15	. . . . . . . . 9 ⊢ (((𝜑 ∧ inf(𝐵, ℝ^*, < ) = -∞) ∧ 𝑟 ∈ ℝ) → ∃𝑧 ∈ 𝐴 𝑧 < 𝑟)
68	67	ralrimiva 3121	. . . . . . . 8 ⊢ ((𝜑 ∧ inf(𝐵, ℝ^*, < ) = -∞) → ∀𝑟 ∈ ℝ ∃𝑧 ∈ 𝐴 𝑧 < 𝑟)
69		infxrunb2 45348	. . . . . . . . . 10 ⊢ (𝐴 ⊆ ℝ^* → (∀𝑟 ∈ ℝ ∃𝑧 ∈ 𝐴 𝑧 < 𝑟 ↔ inf(𝐴, ℝ^*, < ) = -∞))
70	1, 69	syl 17	. . . . . . . . 9 ⊢ (𝜑 → (∀𝑟 ∈ ℝ ∃𝑧 ∈ 𝐴 𝑧 < 𝑟 ↔ inf(𝐴, ℝ^*, < ) = -∞))
71	70	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ inf(𝐵, ℝ^, < ) = -∞) → (∀𝑟 ∈ ℝ ∃𝑧 ∈ 𝐴 𝑧 < 𝑟 ↔ inf(𝐴, ℝ^, < ) = -∞))
72	68, 71	mpbid 232	. . . . . . 7 ⊢ ((𝜑 ∧ inf(𝐵, ℝ^, < ) = -∞) → inf(𝐴, ℝ^, < ) = -∞)
73	72, 22	eqtr4d 2767	. . . . . 6 ⊢ ((𝜑 ∧ inf(𝐵, ℝ^, < ) = -∞) → inf(𝐴, ℝ^, < ) = inf(𝐵, ℝ^*, < ))
74	16, 73	xreqled 45310	. . . . 5 ⊢ ((𝜑 ∧ inf(𝐵, ℝ^, < ) = -∞) → inf(𝐴, ℝ^, < ) ≤ inf(𝐵, ℝ^*, < ))
75	74	adantlr 715	. . . 4 ⊢ (((𝜑 ∧ 𝐵 ≠ ∅) ∧ inf(𝐵, ℝ^, < ) = -∞) → inf(𝐴, ℝ^, < ) ≤ inf(𝐵, ℝ^*, < ))
76		mnfxr 11191	. . . . . . . 8 ⊢ -∞ ∈ ℝ^*
77	76	a1i 11	. . . . . . 7 ⊢ (𝜑 → -∞ ∈ ℝ^*)
78	77	ad2antrr 726	. . . . . 6 ⊢ (((𝜑 ∧ 𝐵 ≠ ∅) ∧ ¬ inf(𝐵, ℝ^, < ) = -∞) → -∞ ∈ ℝ^)
79		infxrcl 13254	. . . . . . . 8 ⊢ (𝐵 ⊆ ℝ^* → inf(𝐵, ℝ^, < ) ∈ ℝ^)
80	23, 79	syl 17	. . . . . . 7 ⊢ (𝜑 → inf(𝐵, ℝ^, < ) ∈ ℝ^)
81	80	ad2antrr 726	. . . . . 6 ⊢ (((𝜑 ∧ 𝐵 ≠ ∅) ∧ ¬ inf(𝐵, ℝ^, < ) = -∞) → inf(𝐵, ℝ^, < ) ∈ ℝ^*)
82		mnfle 13055	. . . . . . 7 ⊢ (inf(𝐵, ℝ^, < ) ∈ ℝ^ → -∞ ≤ inf(𝐵, ℝ^*, < ))
83	81, 82	syl 17	. . . . . 6 ⊢ (((𝜑 ∧ 𝐵 ≠ ∅) ∧ ¬ inf(𝐵, ℝ^, < ) = -∞) → -∞ ≤ inf(𝐵, ℝ^, < ))
84		neqne 2933	. . . . . . . 8 ⊢ (¬ inf(𝐵, ℝ^, < ) = -∞ → inf(𝐵, ℝ^, < ) ≠ -∞)
85	84	necomd 2980	. . . . . . 7 ⊢ (¬ inf(𝐵, ℝ^, < ) = -∞ → -∞ ≠ inf(𝐵, ℝ^, < ))
86	85	adantl 481	. . . . . 6 ⊢ (((𝜑 ∧ 𝐵 ≠ ∅) ∧ ¬ inf(𝐵, ℝ^, < ) = -∞) → -∞ ≠ inf(𝐵, ℝ^, < ))
87	78, 81, 83, 86	xrleneltd 45303	. . . . 5 ⊢ (((𝜑 ∧ 𝐵 ≠ ∅) ∧ ¬ inf(𝐵, ℝ^, < ) = -∞) → -∞ < inf(𝐵, ℝ^, < ))
88	3	ad2antrr 726	. . . . . 6 ⊢ (((𝜑 ∧ 𝐵 ≠ ∅) ∧ -∞ < inf(𝐵, ℝ^, < )) → inf(𝐴, ℝ^, < ) ∈ ℝ^*)
89	80	ad2antrr 726	. . . . . 6 ⊢ (((𝜑 ∧ 𝐵 ≠ ∅) ∧ -∞ < inf(𝐵, ℝ^, < )) → inf(𝐵, ℝ^, < ) ∈ ℝ^*)
90		nfv 1914	. . . . . . . 8 ⊢ Ⅎ𝑏(((𝜑 ∧ 𝐵 ≠ ∅) ∧ -∞ < inf(𝐵, ℝ^*, < )) ∧ 𝑤 ∈ ℝ⁺)
91	23	ad3antrrr 730	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝐵 ≠ ∅) ∧ -∞ < inf(𝐵, ℝ^, < )) ∧ 𝑤 ∈ ℝ⁺) → 𝐵 ⊆ ℝ^)
92		simpllr 775	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝐵 ≠ ∅) ∧ -∞ < inf(𝐵, ℝ^*, < )) ∧ 𝑤 ∈ ℝ⁺) → 𝐵 ≠ ∅)
93		simpr 484	. . . . . . . . . 10 ⊢ ((𝜑 ∧ -∞ < inf(𝐵, ℝ^, < )) → -∞ < inf(𝐵, ℝ^, < ))
94		infxrbnd2 45349	. . . . . . . . . . . 12 ⊢ (𝐵 ⊆ ℝ^* → (∃𝑏 ∈ ℝ ∀𝑥 ∈ 𝐵 𝑏 ≤ 𝑥 ↔ -∞ < inf(𝐵, ℝ^*, < )))
95	23, 94	syl 17	. . . . . . . . . . 11 ⊢ (𝜑 → (∃𝑏 ∈ ℝ ∀𝑥 ∈ 𝐵 𝑏 ≤ 𝑥 ↔ -∞ < inf(𝐵, ℝ^*, < )))
96	95	adantr 480	. . . . . . . . . 10 ⊢ ((𝜑 ∧ -∞ < inf(𝐵, ℝ^, < )) → (∃𝑏 ∈ ℝ ∀𝑥 ∈ 𝐵 𝑏 ≤ 𝑥 ↔ -∞ < inf(𝐵, ℝ^, < )))
97	93, 96	mpbird 257	. . . . . . . . 9 ⊢ ((𝜑 ∧ -∞ < inf(𝐵, ℝ^*, < )) → ∃𝑏 ∈ ℝ ∀𝑥 ∈ 𝐵 𝑏 ≤ 𝑥)
98	97	ad4ant13 751	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝐵 ≠ ∅) ∧ -∞ < inf(𝐵, ℝ^*, < )) ∧ 𝑤 ∈ ℝ⁺) → ∃𝑏 ∈ ℝ ∀𝑥 ∈ 𝐵 𝑏 ≤ 𝑥)
99		simpr 484	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝐵 ≠ ∅) ∧ -∞ < inf(𝐵, ℝ^*, < )) ∧ 𝑤 ∈ ℝ⁺) → 𝑤 ∈ ℝ⁺)
100	99	rphalfcld 12967	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝐵 ≠ ∅) ∧ -∞ < inf(𝐵, ℝ^*, < )) ∧ 𝑤 ∈ ℝ⁺) → (𝑤 / 2) ∈ ℝ⁺)
101	90, 91, 92, 98, 100	infrpge 45331	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝐵 ≠ ∅) ∧ -∞ < inf(𝐵, ℝ^, < )) ∧ 𝑤 ∈ ℝ⁺) → ∃𝑥 ∈ 𝐵 𝑥 ≤ (inf(𝐵, ℝ^, < ) +_𝑒 (𝑤 / 2)))
102		simpll 766	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑤 ∈ ℝ⁺) ∧ 𝑥 ∈ 𝐵) → 𝜑)
103		simpr 484	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑤 ∈ ℝ⁺) ∧ 𝑥 ∈ 𝐵) → 𝑥 ∈ 𝐵)
104		rphalfcl 12940	. . . . . . . . . . . . . 14 ⊢ (𝑤 ∈ ℝ⁺ → (𝑤 / 2) ∈ ℝ⁺)
105	104	ad2antlr 727	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑤 ∈ ℝ⁺) ∧ 𝑥 ∈ 𝐵) → (𝑤 / 2) ∈ ℝ⁺)
106		ovex 7386	. . . . . . . . . . . . . 14 ⊢ (𝑤 / 2) ∈ V
107		eleq1 2816	. . . . . . . . . . . . . . . 16 ⊢ (𝑦 = (𝑤 / 2) → (𝑦 ∈ ℝ⁺ ↔ (𝑤 / 2) ∈ ℝ⁺))
108	107	3anbi3d 1444	. . . . . . . . . . . . . . 15 ⊢ (𝑦 = (𝑤 / 2) → ((𝜑 ∧ 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ ℝ⁺) ↔ (𝜑 ∧ 𝑥 ∈ 𝐵 ∧ (𝑤 / 2) ∈ ℝ⁺)))
109		oveq2 7361	. . . . . . . . . . . . . . . . 17 ⊢ (𝑦 = (𝑤 / 2) → (𝑥 +_𝑒 𝑦) = (𝑥 +_𝑒 (𝑤 / 2)))
110	109	breq2d 5107	. . . . . . . . . . . . . . . 16 ⊢ (𝑦 = (𝑤 / 2) → (𝑧 ≤ (𝑥 +_𝑒 𝑦) ↔ 𝑧 ≤ (𝑥 +_𝑒 (𝑤 / 2))))
111	110	rexbidv 3153	. . . . . . . . . . . . . . 15 ⊢ (𝑦 = (𝑤 / 2) → (∃𝑧 ∈ 𝐴 𝑧 ≤ (𝑥 +_𝑒 𝑦) ↔ ∃𝑧 ∈ 𝐴 𝑧 ≤ (𝑥 +_𝑒 (𝑤 / 2))))
112	108, 111	imbi12d 344	. . . . . . . . . . . . . 14 ⊢ (𝑦 = (𝑤 / 2) → (((𝜑 ∧ 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ ℝ⁺) → ∃𝑧 ∈ 𝐴 𝑧 ≤ (𝑥 +_𝑒 𝑦)) ↔ ((𝜑 ∧ 𝑥 ∈ 𝐵 ∧ (𝑤 / 2) ∈ ℝ⁺) → ∃𝑧 ∈ 𝐴 𝑧 ≤ (𝑥 +_𝑒 (𝑤 / 2)))))
113	106, 112, 44	vtocl 3515	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵 ∧ (𝑤 / 2) ∈ ℝ⁺) → ∃𝑧 ∈ 𝐴 𝑧 ≤ (𝑥 +_𝑒 (𝑤 / 2)))
114	102, 103, 105, 113	syl3anc 1373	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑤 ∈ ℝ⁺) ∧ 𝑥 ∈ 𝐵) → ∃𝑧 ∈ 𝐴 𝑧 ≤ (𝑥 +_𝑒 (𝑤 / 2)))
115	114	3adant3 1132	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑤 ∈ ℝ⁺) ∧ 𝑥 ∈ 𝐵 ∧ 𝑥 ≤ (inf(𝐵, ℝ^*, < ) +_𝑒 (𝑤 / 2))) → ∃𝑧 ∈ 𝐴 𝑧 ≤ (𝑥 +_𝑒 (𝑤 / 2)))
116		simp11l 1285	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ 𝑤 ∈ ℝ⁺) ∧ 𝑥 ∈ 𝐵 ∧ 𝑥 ≤ (inf(𝐵, ℝ^*, < ) +_𝑒 (𝑤 / 2))) ∧ 𝑧 ∈ 𝐴 ∧ 𝑧 ≤ (𝑥 +_𝑒 (𝑤 / 2))) → 𝜑)
117	116, 1	syl 17	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑤 ∈ ℝ⁺) ∧ 𝑥 ∈ 𝐵 ∧ 𝑥 ≤ (inf(𝐵, ℝ^, < ) +_𝑒 (𝑤 / 2))) ∧ 𝑧 ∈ 𝐴 ∧ 𝑧 ≤ (𝑥 +_𝑒 (𝑤 / 2))) → 𝐴 ⊆ ℝ^)
118	116, 23	syl 17	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑤 ∈ ℝ⁺) ∧ 𝑥 ∈ 𝐵 ∧ 𝑥 ≤ (inf(𝐵, ℝ^, < ) +_𝑒 (𝑤 / 2))) ∧ 𝑧 ∈ 𝐴 ∧ 𝑧 ≤ (𝑥 +_𝑒 (𝑤 / 2))) → 𝐵 ⊆ ℝ^)
119		simp11 1204	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ 𝑤 ∈ ℝ⁺) ∧ 𝑥 ∈ 𝐵 ∧ 𝑥 ≤ (inf(𝐵, ℝ^*, < ) +_𝑒 (𝑤 / 2))) ∧ 𝑧 ∈ 𝐴 ∧ 𝑧 ≤ (𝑥 +_𝑒 (𝑤 / 2))) → (𝜑 ∧ 𝑤 ∈ ℝ⁺))
120	119	simprd 495	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑤 ∈ ℝ⁺) ∧ 𝑥 ∈ 𝐵 ∧ 𝑥 ≤ (inf(𝐵, ℝ^*, < ) +_𝑒 (𝑤 / 2))) ∧ 𝑧 ∈ 𝐴 ∧ 𝑧 ≤ (𝑥 +_𝑒 (𝑤 / 2))) → 𝑤 ∈ ℝ⁺)
121		simp12 1205	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑤 ∈ ℝ⁺) ∧ 𝑥 ∈ 𝐵 ∧ 𝑥 ≤ (inf(𝐵, ℝ^*, < ) +_𝑒 (𝑤 / 2))) ∧ 𝑧 ∈ 𝐴 ∧ 𝑧 ≤ (𝑥 +_𝑒 (𝑤 / 2))) → 𝑥 ∈ 𝐵)
122		simp3 1138	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑤 ∈ ℝ⁺) ∧ 𝑥 ∈ 𝐵 ∧ 𝑥 ≤ (inf(𝐵, ℝ^, < ) +_𝑒 (𝑤 / 2))) → 𝑥 ≤ (inf(𝐵, ℝ^, < ) +_𝑒 (𝑤 / 2)))
123	122	3ad2ant1 1133	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑤 ∈ ℝ⁺) ∧ 𝑥 ∈ 𝐵 ∧ 𝑥 ≤ (inf(𝐵, ℝ^, < ) +_𝑒 (𝑤 / 2))) ∧ 𝑧 ∈ 𝐴 ∧ 𝑧 ≤ (𝑥 +_𝑒 (𝑤 / 2))) → 𝑥 ≤ (inf(𝐵, ℝ^, < ) +_𝑒 (𝑤 / 2)))
124		simp2 1137	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑤 ∈ ℝ⁺) ∧ 𝑥 ∈ 𝐵 ∧ 𝑥 ≤ (inf(𝐵, ℝ^*, < ) +_𝑒 (𝑤 / 2))) ∧ 𝑧 ∈ 𝐴 ∧ 𝑧 ≤ (𝑥 +_𝑒 (𝑤 / 2))) → 𝑧 ∈ 𝐴)
125		simp3 1138	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑤 ∈ ℝ⁺) ∧ 𝑥 ∈ 𝐵 ∧ 𝑥 ≤ (inf(𝐵, ℝ^*, < ) +_𝑒 (𝑤 / 2))) ∧ 𝑧 ∈ 𝐴 ∧ 𝑧 ≤ (𝑥 +_𝑒 (𝑤 / 2))) → 𝑧 ≤ (𝑥 +_𝑒 (𝑤 / 2)))
126	117, 118, 120, 121, 123, 124, 125	infleinflem1 45350	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑤 ∈ ℝ⁺) ∧ 𝑥 ∈ 𝐵 ∧ 𝑥 ≤ (inf(𝐵, ℝ^, < ) +_𝑒 (𝑤 / 2))) ∧ 𝑧 ∈ 𝐴 ∧ 𝑧 ≤ (𝑥 +_𝑒 (𝑤 / 2))) → inf(𝐴, ℝ^, < ) ≤ (inf(𝐵, ℝ^*, < ) +_𝑒 𝑤))
127	126	3exp 1119	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑤 ∈ ℝ⁺) ∧ 𝑥 ∈ 𝐵 ∧ 𝑥 ≤ (inf(𝐵, ℝ^, < ) +_𝑒 (𝑤 / 2))) → (𝑧 ∈ 𝐴 → (𝑧 ≤ (𝑥 +_𝑒 (𝑤 / 2)) → inf(𝐴, ℝ^, < ) ≤ (inf(𝐵, ℝ^*, < ) +_𝑒 𝑤))))
128	127	rexlimdv 3128	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑤 ∈ ℝ⁺) ∧ 𝑥 ∈ 𝐵 ∧ 𝑥 ≤ (inf(𝐵, ℝ^, < ) +_𝑒 (𝑤 / 2))) → (∃𝑧 ∈ 𝐴 𝑧 ≤ (𝑥 +_𝑒 (𝑤 / 2)) → inf(𝐴, ℝ^, < ) ≤ (inf(𝐵, ℝ^*, < ) +_𝑒 𝑤)))
129	115, 128	mpd 15	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑤 ∈ ℝ⁺) ∧ 𝑥 ∈ 𝐵 ∧ 𝑥 ≤ (inf(𝐵, ℝ^, < ) +_𝑒 (𝑤 / 2))) → inf(𝐴, ℝ^, < ) ≤ (inf(𝐵, ℝ^*, < ) +_𝑒 𝑤))
130	129	3exp 1119	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑤 ∈ ℝ⁺) → (𝑥 ∈ 𝐵 → (𝑥 ≤ (inf(𝐵, ℝ^, < ) +_𝑒 (𝑤 / 2)) → inf(𝐴, ℝ^, < ) ≤ (inf(𝐵, ℝ^*, < ) +_𝑒 𝑤))))
131	130	rexlimdv 3128	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑤 ∈ ℝ⁺) → (∃𝑥 ∈ 𝐵 𝑥 ≤ (inf(𝐵, ℝ^, < ) +_𝑒 (𝑤 / 2)) → inf(𝐴, ℝ^, < ) ≤ (inf(𝐵, ℝ^*, < ) +_𝑒 𝑤)))
132	131	ad4ant14 752	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝐵 ≠ ∅) ∧ -∞ < inf(𝐵, ℝ^, < )) ∧ 𝑤 ∈ ℝ⁺) → (∃𝑥 ∈ 𝐵 𝑥 ≤ (inf(𝐵, ℝ^, < ) +_𝑒 (𝑤 / 2)) → inf(𝐴, ℝ^, < ) ≤ (inf(𝐵, ℝ^, < ) +_𝑒 𝑤)))
133	101, 132	mpd 15	. . . . . 6 ⊢ ((((𝜑 ∧ 𝐵 ≠ ∅) ∧ -∞ < inf(𝐵, ℝ^, < )) ∧ 𝑤 ∈ ℝ⁺) → inf(𝐴, ℝ^, < ) ≤ (inf(𝐵, ℝ^*, < ) +_𝑒 𝑤))
134	88, 89, 133	xrlexaddrp 45332	. . . . 5 ⊢ (((𝜑 ∧ 𝐵 ≠ ∅) ∧ -∞ < inf(𝐵, ℝ^, < )) → inf(𝐴, ℝ^, < ) ≤ inf(𝐵, ℝ^*, < ))
135	87, 134	syldan 591	. . . 4 ⊢ (((𝜑 ∧ 𝐵 ≠ ∅) ∧ ¬ inf(𝐵, ℝ^, < ) = -∞) → inf(𝐴, ℝ^, < ) ≤ inf(𝐵, ℝ^*, < ))
136	75, 135	pm2.61dan 812	. . 3 ⊢ ((𝜑 ∧ 𝐵 ≠ ∅) → inf(𝐴, ℝ^, < ) ≤ inf(𝐵, ℝ^, < ))
137	15, 136	syldan 591	. 2 ⊢ ((𝜑 ∧ ¬ 𝐵 = ∅) → inf(𝐴, ℝ^, < ) ≤ inf(𝐵, ℝ^, < ))
138	13, 137	pm2.61dan 812	1 ⊢ (𝜑 → inf(𝐴, ℝ^, < ) ≤ inf(𝐵, ℝ^, < ))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 206 ∧ wa 395 ∧ w3a 1086 = wceq 1540 ∈ wcel 2109 ≠ wne 2925 ∀wral 3044 ∃wrex 3053 ⊆ wss 3905 ∅c0 4286 class class class wbr 5095 (class class class)co 7353 infcinf 9350 ℝcr 11027 1c1 11029 +∞cpnf 11165 -∞cmnf 11166 ℝ^*cxr 11167 < clt 11168 ≤ cle 11169 − cmin 11365 / cdiv 11795 2c2 12201 ℝ⁺crp 12911 +_𝑒 cxad 13030

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2008 ax-8 2111 ax-9 2119 ax-10 2142 ax-11 2158 ax-12 2178 ax-ext 2701 ax-sep 5238 ax-nul 5248 ax-pow 5307 ax-pr 5374 ax-un 7675 ax-cnex 11084 ax-resscn 11085 ax-1cn 11086 ax-icn 11087 ax-addcl 11088 ax-addrcl 11089 ax-mulcl 11090 ax-mulrcl 11091 ax-mulcom 11092 ax-addass 11093 ax-mulass 11094 ax-distr 11095 ax-i2m1 11096 ax-1ne0 11097 ax-1rid 11098 ax-rnegex 11099 ax-rrecex 11100 ax-cnre 11101 ax-pre-lttri 11102 ax-pre-lttrn 11103 ax-pre-ltadd 11104 ax-pre-mulgt0 11105 ax-pre-sup 11106

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2533 df-eu 2562 df-clab 2708 df-cleq 2721 df-clel 2803 df-nfc 2878 df-ne 2926 df-nel 3030 df-ral 3045 df-rex 3054 df-rmo 3345 df-reu 3346 df-rab 3397 df-v 3440 df-sbc 3745 df-csb 3854 df-dif 3908 df-un 3910 df-in 3912 df-ss 3922 df-pss 3925 df-nul 4287 df-if 4479 df-pw 4555 df-sn 4580 df-pr 4582 df-op 4586 df-uni 4862 df-iun 4946 df-br 5096 df-opab 5158 df-mpt 5177 df-tr 5203 df-id 5518 df-eprel 5523 df-po 5531 df-so 5532 df-fr 5576 df-we 5578 df-xp 5629 df-rel 5630 df-cnv 5631 df-co 5632 df-dm 5633 df-rn 5634 df-res 5635 df-ima 5636 df-pred 6253 df-ord 6314 df-on 6315 df-lim 6316 df-suc 6317 df-iota 6442 df-fun 6488 df-fn 6489 df-f 6490 df-f1 6491 df-fo 6492 df-f1o 6493 df-fv 6494 df-riota 7310 df-ov 7356 df-oprab 7357 df-mpo 7358 df-om 7807 df-1st 7931 df-2nd 7932 df-frecs 8221 df-wrecs 8252 df-recs 8301 df-rdg 8339 df-er 8632 df-en 8880 df-dom 8881 df-sdom 8882 df-sup 9351 df-inf 9352 df-pnf 11170 df-mnf 11171 df-xr 11172 df-ltxr 11173 df-le 11174 df-sub 11367 df-neg 11368 df-div 11796 df-nn 12147 df-2 12209 df-n0 12403 df-z 12490 df-uz 12754 df-q 12868 df-rp 12912 df-xneg 13032 df-xadd 13033

This theorem is referenced by: ovolval5lem3 46636

W3C validator