infleinf - Mathbox for Glauco Siliprandi

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Theorem infleinf 45368

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 13294	. . . . . 6 ⊢ (𝐴 ⊆ ℝ^* → inf(𝐴, ℝ^, < ) ∈ ℝ^)
3	1, 2	syl 17	. . . . 5 ⊢ (𝜑 → inf(𝐴, ℝ^, < ) ∈ ℝ^)
4		pnfge 13090	. . . . 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 9428	. . . . . 6 ⊢ (𝐵 = ∅ → inf(𝐵, ℝ^, < ) = inf(∅, ℝ^, < ))
8		xrinf0 13299	. . . . . . 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 5133	. 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 12260	. . . . . . . . . . . . . 14 ⊢ 2 ∈ ℝ
19	18	a1i 11	. . . . . . . . . . . . 13 ⊢ (𝑟 ∈ ℝ → 2 ∈ ℝ)
20	17, 19	resubcld 11606	. . . . . . . . . . . 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 45364	. . . . . . . . . . . . . . 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 5111	. . . . . . . . . . . . 13 ⊢ (𝑦 = (𝑟 − 2) → (𝑥 < 𝑦 ↔ 𝑥 < (𝑟 − 2)))
30	29	rexbidv 3157	. . . . . . . . . . . 12 ⊢ (𝑦 = (𝑟 − 2) → (∃𝑥 ∈ 𝐵 𝑥 < 𝑦 ↔ ∃𝑥 ∈ 𝐵 𝑥 < (𝑟 − 2)))
31	30	rspcva 3586	. . . . . . . . . . 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 12955	. . . . . . . . . . . . . . . . . 18 ⊢ 1 ∈ ℝ⁺
36	35	a1i 11	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → 1 ∈ ℝ⁺)
37		1ex 11170	. . . . . . . . . . . . . . . . . 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 7395	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑦 = 1 → (𝑥 +_𝑒 𝑦) = (𝑥 +_𝑒 1))
41	40	breq2d 5119	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑦 = 1 → (𝑧 ≤ (𝑥 +_𝑒 𝑦) ↔ 𝑧 ≤ (𝑥 +_𝑒 1)))
42	41	rexbidv 3157	. . . . . . . . . . . . . . . . . . 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 3524	. . . . . . . . . . . . . . . . 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 45367	. . . . . . . . . . . . . . . 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 3146	. . . . . . . . . . . . . 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 3132	. . . . . . . . . 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 3125	. . . . . . . 8 ⊢ ((𝜑 ∧ inf(𝐵, ℝ^*, < ) = -∞) → ∀𝑟 ∈ ℝ ∃𝑧 ∈ 𝐴 𝑧 < 𝑟)
69		infxrunb2 45364	. . . . . . . . . 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 45326	. . . . 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 11231	. . . . . . . 8 ⊢ -∞ ∈ ℝ^*
77	76	a1i 11	. . . . . . 7 ⊢ (𝜑 → -∞ ∈ ℝ^*)
78	77	ad2antrr 726	. . . . . 6 ⊢ (((𝜑 ∧ 𝐵 ≠ ∅) ∧ ¬ inf(𝐵, ℝ^, < ) = -∞) → -∞ ∈ ℝ^)
79		infxrcl 13294	. . . . . . . 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 13095	. . . . . . 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 45319	. . . . 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 45365	. . . . . . . . . . . 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 13007	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝐵 ≠ ∅) ∧ -∞ < inf(𝐵, ℝ^*, < )) ∧ 𝑤 ∈ ℝ⁺) → (𝑤 / 2) ∈ ℝ⁺)
101	90, 91, 92, 98, 100	infrpge 45347	. . . . . . 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 12980	. . . . . . . . . . . . . 14 ⊢ (𝑤 ∈ ℝ⁺ → (𝑤 / 2) ∈ ℝ⁺)
105	104	ad2antlr 727	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑤 ∈ ℝ⁺) ∧ 𝑥 ∈ 𝐵) → (𝑤 / 2) ∈ ℝ⁺)
106		ovex 7420	. . . . . . . . . . . . . 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 7395	. . . . . . . . . . . . . . . . 17 ⊢ (𝑦 = (𝑤 / 2) → (𝑥 +_𝑒 𝑦) = (𝑥 +_𝑒 (𝑤 / 2)))
110	109	breq2d 5119	. . . . . . . . . . . . . . . 16 ⊢ (𝑦 = (𝑤 / 2) → (𝑧 ≤ (𝑥 +_𝑒 𝑦) ↔ 𝑧 ≤ (𝑥 +_𝑒 (𝑤 / 2))))
111	110	rexbidv 3157	. . . . . . . . . . . . . . 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 3524	. . . . . . . . . . . . 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 45366	. . . . . . . . . . . . 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 3132	. . . . . . . . . . 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 3132	. . . . . . . 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 45348	. . . . 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 3914 ∅c0 4296 class class class wbr 5107 (class class class)co 7387 infcinf 9392 ℝcr 11067 1c1 11069 +∞cpnf 11205 -∞cmnf 11206 ℝ^*cxr 11207 < clt 11208 ≤ cle 11209 − cmin 11405 / cdiv 11835 2c2 12241 ℝ⁺crp 12951 +_𝑒 cxad 13070

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 5251 ax-nul 5261 ax-pow 5320 ax-pr 5387 ax-un 7711 ax-cnex 11124 ax-resscn 11125 ax-1cn 11126 ax-icn 11127 ax-addcl 11128 ax-addrcl 11129 ax-mulcl 11130 ax-mulrcl 11131 ax-mulcom 11132 ax-addass 11133 ax-mulass 11134 ax-distr 11135 ax-i2m1 11136 ax-1ne0 11137 ax-1rid 11138 ax-rnegex 11139 ax-rrecex 11140 ax-cnre 11141 ax-pre-lttri 11142 ax-pre-lttrn 11143 ax-pre-ltadd 11144 ax-pre-mulgt0 11145 ax-pre-sup 11146

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 3354 df-reu 3355 df-rab 3406 df-v 3449 df-sbc 3754 df-csb 3863 df-dif 3917 df-un 3919 df-in 3921 df-ss 3931 df-pss 3934 df-nul 4297 df-if 4489 df-pw 4565 df-sn 4590 df-pr 4592 df-op 4596 df-uni 4872 df-iun 4957 df-br 5108 df-opab 5170 df-mpt 5189 df-tr 5215 df-id 5533 df-eprel 5538 df-po 5546 df-so 5547 df-fr 5591 df-we 5593 df-xp 5644 df-rel 5645 df-cnv 5646 df-co 5647 df-dm 5648 df-rn 5649 df-res 5650 df-ima 5651 df-pred 6274 df-ord 6335 df-on 6336 df-lim 6337 df-suc 6338 df-iota 6464 df-fun 6513 df-fn 6514 df-f 6515 df-f1 6516 df-fo 6517 df-f1o 6518 df-fv 6519 df-riota 7344 df-ov 7390 df-oprab 7391 df-mpo 7392 df-om 7843 df-1st 7968 df-2nd 7969 df-frecs 8260 df-wrecs 8291 df-recs 8340 df-rdg 8378 df-er 8671 df-en 8919 df-dom 8920 df-sdom 8921 df-sup 9393 df-inf 9394 df-pnf 11210 df-mnf 11211 df-xr 11212 df-ltxr 11213 df-le 11214 df-sub 11407 df-neg 11408 df-div 11836 df-nn 12187 df-2 12249 df-n0 12443 df-z 12530 df-uz 12794 df-q 12908 df-rp 12952 df-xneg 13072 df-xadd 13073

This theorem is referenced by: ovolval5lem3 46652

W3C validator