infrpge - Mathbox for Glauco Siliprandi

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

Theorem infrpge 39567

Description: The infimum of a non empty, bounded subset of extended reals can be approximated from above by an element of the set. (Contributed by Glauco Siliprandi, 11-Oct-2020.)

**Hypotheses**
Ref	Expression
infrpge.xph	⊢ Ⅎ𝑥𝜑
infrpge.a	⊢ (𝜑 → 𝐴 ⊆ ℝ^*)
infrpge.an0	⊢ (𝜑 → 𝐴 ≠ ∅)
infrpge.bnd	⊢ (𝜑 → ∃𝑥 ∈ ℝ ∀𝑦 ∈ 𝐴 𝑥 ≤ 𝑦)
infrpge.b	⊢ (𝜑 → 𝐵 ∈ ℝ⁺)

**Assertion**
Ref	Expression
infrpge	⊢ (𝜑 → ∃𝑧 ∈ 𝐴 𝑧 ≤ (inf(𝐴, ℝ^*, < ) +_𝑒 𝐵))

Distinct variable groups: 𝑥,𝐴,𝑦 𝑧,𝐴 𝑧,𝐵 𝜑,𝑧 Allowed substitution hints: 𝜑(𝑥,𝑦) 𝐵(𝑥,𝑦)

**Proof of Theorem infrpge**
Step	Hyp	Ref	Expression
1		infrpge.an0	. . . . . 6 ⊢ (𝜑 → 𝐴 ≠ ∅)
2		n0 3931	. . . . . . 7 ⊢ (𝐴 ≠ ∅ ↔ ∃𝑧 𝑧 ∈ 𝐴)
3	2	biimpi 206	. . . . . 6 ⊢ (𝐴 ≠ ∅ → ∃𝑧 𝑧 ∈ 𝐴)
4	1, 3	syl 17	. . . . 5 ⊢ (𝜑 → ∃𝑧 𝑧 ∈ 𝐴)
5	4	adantr 481	. . . 4 ⊢ ((𝜑 ∧ inf(𝐴, ℝ^*, < ) = +∞) → ∃𝑧 𝑧 ∈ 𝐴)
6		nfv 1843	. . . . 5 ⊢ Ⅎ𝑧(𝜑 ∧ inf(𝐴, ℝ^*, < ) = +∞)
7		simpr 477	. . . . . . 7 ⊢ (((𝜑 ∧ inf(𝐴, ℝ^*, < ) = +∞) ∧ 𝑧 ∈ 𝐴) → 𝑧 ∈ 𝐴)
8		infrpge.a	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐴 ⊆ ℝ^*)
9	8	adantr 481	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐴) → 𝐴 ⊆ ℝ^*)
10		simpr 477	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐴) → 𝑧 ∈ 𝐴)
11	9, 10	sseldd 3604	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐴) → 𝑧 ∈ ℝ^*)
12		pnfge 11964	. . . . . . . . . 10 ⊢ (𝑧 ∈ ℝ^* → 𝑧 ≤ +∞)
13	11, 12	syl 17	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐴) → 𝑧 ≤ +∞)
14	13	adantlr 751	. . . . . . . 8 ⊢ (((𝜑 ∧ inf(𝐴, ℝ^*, < ) = +∞) ∧ 𝑧 ∈ 𝐴) → 𝑧 ≤ +∞)
15		oveq1 6657	. . . . . . . . . . 11 ⊢ (inf(𝐴, ℝ^, < ) = +∞ → (inf(𝐴, ℝ^, < ) +_𝑒 𝐵) = (+∞ +_𝑒 𝐵))
16	15	adantl 482	. . . . . . . . . 10 ⊢ ((𝜑 ∧ inf(𝐴, ℝ^, < ) = +∞) → (inf(𝐴, ℝ^, < ) +_𝑒 𝐵) = (+∞ +_𝑒 𝐵))
17		infrpge.b	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐵 ∈ ℝ⁺)
18	17	rpxrd 11873	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐵 ∈ ℝ^*)
19	17	rpred 11872	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐵 ∈ ℝ)
20		renemnf 10088	. . . . . . . . . . . . 13 ⊢ (𝐵 ∈ ℝ → 𝐵 ≠ -∞)
21	19, 20	syl 17	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐵 ≠ -∞)
22		xaddpnf2 12058	. . . . . . . . . . . 12 ⊢ ((𝐵 ∈ ℝ^* ∧ 𝐵 ≠ -∞) → (+∞ +_𝑒 𝐵) = +∞)
23	18, 21, 22	syl2anc 693	. . . . . . . . . . 11 ⊢ (𝜑 → (+∞ +_𝑒 𝐵) = +∞)
24	23	adantr 481	. . . . . . . . . 10 ⊢ ((𝜑 ∧ inf(𝐴, ℝ^*, < ) = +∞) → (+∞ +_𝑒 𝐵) = +∞)
25	16, 24	eqtr2d 2657	. . . . . . . . 9 ⊢ ((𝜑 ∧ inf(𝐴, ℝ^, < ) = +∞) → +∞ = (inf(𝐴, ℝ^, < ) +_𝑒 𝐵))
26	25	adantr 481	. . . . . . . 8 ⊢ (((𝜑 ∧ inf(𝐴, ℝ^, < ) = +∞) ∧ 𝑧 ∈ 𝐴) → +∞ = (inf(𝐴, ℝ^, < ) +_𝑒 𝐵))
27	14, 26	breqtrd 4679	. . . . . . 7 ⊢ (((𝜑 ∧ inf(𝐴, ℝ^, < ) = +∞) ∧ 𝑧 ∈ 𝐴) → 𝑧 ≤ (inf(𝐴, ℝ^, < ) +_𝑒 𝐵))
28	7, 27	jca 554	. . . . . 6 ⊢ (((𝜑 ∧ inf(𝐴, ℝ^, < ) = +∞) ∧ 𝑧 ∈ 𝐴) → (𝑧 ∈ 𝐴 ∧ 𝑧 ≤ (inf(𝐴, ℝ^, < ) +_𝑒 𝐵)))
29	28	ex 450	. . . . 5 ⊢ ((𝜑 ∧ inf(𝐴, ℝ^, < ) = +∞) → (𝑧 ∈ 𝐴 → (𝑧 ∈ 𝐴 ∧ 𝑧 ≤ (inf(𝐴, ℝ^, < ) +_𝑒 𝐵))))
30	6, 29	eximd 2085	. . . 4 ⊢ ((𝜑 ∧ inf(𝐴, ℝ^, < ) = +∞) → (∃𝑧 𝑧 ∈ 𝐴 → ∃𝑧(𝑧 ∈ 𝐴 ∧ 𝑧 ≤ (inf(𝐴, ℝ^, < ) +_𝑒 𝐵))))
31	5, 30	mpd 15	. . 3 ⊢ ((𝜑 ∧ inf(𝐴, ℝ^, < ) = +∞) → ∃𝑧(𝑧 ∈ 𝐴 ∧ 𝑧 ≤ (inf(𝐴, ℝ^, < ) +_𝑒 𝐵)))
32		df-rex 2918	. . 3 ⊢ (∃𝑧 ∈ 𝐴 𝑧 ≤ (inf(𝐴, ℝ^, < ) +_𝑒 𝐵) ↔ ∃𝑧(𝑧 ∈ 𝐴 ∧ 𝑧 ≤ (inf(𝐴, ℝ^, < ) +_𝑒 𝐵)))
33	31, 32	sylibr 224	. 2 ⊢ ((𝜑 ∧ inf(𝐴, ℝ^, < ) = +∞) → ∃𝑧 ∈ 𝐴 𝑧 ≤ (inf(𝐴, ℝ^, < ) +_𝑒 𝐵))
34		simpl 473	. . . 4 ⊢ ((𝜑 ∧ ¬ inf(𝐴, ℝ^*, < ) = +∞) → 𝜑)
35		infrpge.bnd	. . . . . . . . 9 ⊢ (𝜑 → ∃𝑥 ∈ ℝ ∀𝑦 ∈ 𝐴 𝑥 ≤ 𝑦)
36		infrpge.xph	. . . . . . . . . 10 ⊢ Ⅎ𝑥𝜑
37		nfv 1843	. . . . . . . . . 10 ⊢ Ⅎ𝑥-∞ < inf(𝐴, ℝ^*, < )
38		mnfxr 10096	. . . . . . . . . . . . 13 ⊢ -∞ ∈ ℝ^*
39	38	a1i 11	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ ∧ ∀𝑦 ∈ 𝐴 𝑥 ≤ 𝑦) → -∞ ∈ ℝ^*)
40		rexr 10085	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ ℝ → 𝑥 ∈ ℝ^*)
41	40	3ad2ant2 1083	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ ∧ ∀𝑦 ∈ 𝐴 𝑥 ≤ 𝑦) → 𝑥 ∈ ℝ^*)
42		infxrcl 12163	. . . . . . . . . . . . . 14 ⊢ (𝐴 ⊆ ℝ^* → inf(𝐴, ℝ^, < ) ∈ ℝ^)
43	8, 42	syl 17	. . . . . . . . . . . . 13 ⊢ (𝜑 → inf(𝐴, ℝ^, < ) ∈ ℝ^)
44	43	3ad2ant1 1082	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ ∧ ∀𝑦 ∈ 𝐴 𝑥 ≤ 𝑦) → inf(𝐴, ℝ^, < ) ∈ ℝ^)
45		mnflt 11957	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ ℝ → -∞ < 𝑥)
46	45	3ad2ant2 1083	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ ∧ ∀𝑦 ∈ 𝐴 𝑥 ≤ 𝑦) → -∞ < 𝑥)
47		simp3 1063	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ ∧ ∀𝑦 ∈ 𝐴 𝑥 ≤ 𝑦) → ∀𝑦 ∈ 𝐴 𝑥 ≤ 𝑦)
48	8	adantr 481	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → 𝐴 ⊆ ℝ^*)
49	40	adantl 482	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → 𝑥 ∈ ℝ^*)
50		infxrgelb 12165	. . . . . . . . . . . . . . 15 ⊢ ((𝐴 ⊆ ℝ^* ∧ 𝑥 ∈ ℝ^) → (𝑥 ≤ inf(𝐴, ℝ^, < ) ↔ ∀𝑦 ∈ 𝐴 𝑥 ≤ 𝑦))
51	48, 49, 50	syl2anc 693	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → (𝑥 ≤ inf(𝐴, ℝ^*, < ) ↔ ∀𝑦 ∈ 𝐴 𝑥 ≤ 𝑦))
52	51	3adant3 1081	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ ∧ ∀𝑦 ∈ 𝐴 𝑥 ≤ 𝑦) → (𝑥 ≤ inf(𝐴, ℝ^*, < ) ↔ ∀𝑦 ∈ 𝐴 𝑥 ≤ 𝑦))
53	47, 52	mpbird 247	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ ∧ ∀𝑦 ∈ 𝐴 𝑥 ≤ 𝑦) → 𝑥 ≤ inf(𝐴, ℝ^*, < ))
54	39, 41, 44, 46, 53	xrltletrd 11992	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ ∧ ∀𝑦 ∈ 𝐴 𝑥 ≤ 𝑦) → -∞ < inf(𝐴, ℝ^*, < ))
55	54	3exp 1264	. . . . . . . . . 10 ⊢ (𝜑 → (𝑥 ∈ ℝ → (∀𝑦 ∈ 𝐴 𝑥 ≤ 𝑦 → -∞ < inf(𝐴, ℝ^*, < ))))
56	36, 37, 55	rexlimd 3026	. . . . . . . . 9 ⊢ (𝜑 → (∃𝑥 ∈ ℝ ∀𝑦 ∈ 𝐴 𝑥 ≤ 𝑦 → -∞ < inf(𝐴, ℝ^*, < )))
57	35, 56	mpd 15	. . . . . . . 8 ⊢ (𝜑 → -∞ < inf(𝐴, ℝ^*, < ))
58	57	adantr 481	. . . . . . 7 ⊢ ((𝜑 ∧ ¬ inf(𝐴, ℝ^, < ) = +∞) → -∞ < inf(𝐴, ℝ^, < ))
59		neqne 2802	. . . . . . . . 9 ⊢ (¬ inf(𝐴, ℝ^, < ) = +∞ → inf(𝐴, ℝ^, < ) ≠ +∞)
60	59	adantl 482	. . . . . . . 8 ⊢ ((𝜑 ∧ ¬ inf(𝐴, ℝ^, < ) = +∞) → inf(𝐴, ℝ^, < ) ≠ +∞)
61	43	adantr 481	. . . . . . . 8 ⊢ ((𝜑 ∧ ¬ inf(𝐴, ℝ^, < ) = +∞) → inf(𝐴, ℝ^, < ) ∈ ℝ^*)
62	60, 61	nepnfltpnf 39558	. . . . . . 7 ⊢ ((𝜑 ∧ ¬ inf(𝐴, ℝ^, < ) = +∞) → inf(𝐴, ℝ^, < ) < +∞)
63	58, 62	jca 554	. . . . . 6 ⊢ ((𝜑 ∧ ¬ inf(𝐴, ℝ^, < ) = +∞) → (-∞ < inf(𝐴, ℝ^, < ) ∧ inf(𝐴, ℝ^*, < ) < +∞))
64		xrrebnd 11999	. . . . . . . 8 ⊢ (inf(𝐴, ℝ^, < ) ∈ ℝ^ → (inf(𝐴, ℝ^, < ) ∈ ℝ ↔ (-∞ < inf(𝐴, ℝ^, < ) ∧ inf(𝐴, ℝ^*, < ) < +∞)))
65	43, 64	syl 17	. . . . . . 7 ⊢ (𝜑 → (inf(𝐴, ℝ^, < ) ∈ ℝ ↔ (-∞ < inf(𝐴, ℝ^, < ) ∧ inf(𝐴, ℝ^*, < ) < +∞)))
66	65	adantr 481	. . . . . 6 ⊢ ((𝜑 ∧ ¬ inf(𝐴, ℝ^, < ) = +∞) → (inf(𝐴, ℝ^, < ) ∈ ℝ ↔ (-∞ < inf(𝐴, ℝ^, < ) ∧ inf(𝐴, ℝ^, < ) < +∞)))
67	63, 66	mpbird 247	. . . . 5 ⊢ ((𝜑 ∧ ¬ inf(𝐴, ℝ^, < ) = +∞) → inf(𝐴, ℝ^, < ) ∈ ℝ)
68		simpr 477	. . . . . . . 8 ⊢ ((𝜑 ∧ inf(𝐴, ℝ^, < ) ∈ ℝ) → inf(𝐴, ℝ^, < ) ∈ ℝ)
69	17	adantr 481	. . . . . . . 8 ⊢ ((𝜑 ∧ inf(𝐴, ℝ^*, < ) ∈ ℝ) → 𝐵 ∈ ℝ⁺)
70	68, 69	ltaddrpd 11905	. . . . . . 7 ⊢ ((𝜑 ∧ inf(𝐴, ℝ^, < ) ∈ ℝ) → inf(𝐴, ℝ^, < ) < (inf(𝐴, ℝ^*, < ) + 𝐵))
71	19	adantr 481	. . . . . . . . 9 ⊢ ((𝜑 ∧ inf(𝐴, ℝ^*, < ) ∈ ℝ) → 𝐵 ∈ ℝ)
72		rexadd 12063	. . . . . . . . 9 ⊢ ((inf(𝐴, ℝ^, < ) ∈ ℝ ∧ 𝐵 ∈ ℝ) → (inf(𝐴, ℝ^, < ) +_𝑒 𝐵) = (inf(𝐴, ℝ^*, < ) + 𝐵))
73	68, 71, 72	syl2anc 693	. . . . . . . 8 ⊢ ((𝜑 ∧ inf(𝐴, ℝ^, < ) ∈ ℝ) → (inf(𝐴, ℝ^, < ) +_𝑒 𝐵) = (inf(𝐴, ℝ^*, < ) + 𝐵))
74	73	eqcomd 2628	. . . . . . 7 ⊢ ((𝜑 ∧ inf(𝐴, ℝ^, < ) ∈ ℝ) → (inf(𝐴, ℝ^, < ) + 𝐵) = (inf(𝐴, ℝ^*, < ) +_𝑒 𝐵))
75	70, 74	breqtrd 4679	. . . . . 6 ⊢ ((𝜑 ∧ inf(𝐴, ℝ^, < ) ∈ ℝ) → inf(𝐴, ℝ^, < ) < (inf(𝐴, ℝ^*, < ) +_𝑒 𝐵))
76	43	adantr 481	. . . . . . 7 ⊢ ((𝜑 ∧ inf(𝐴, ℝ^, < ) ∈ ℝ) → inf(𝐴, ℝ^, < ) ∈ ℝ^*)
77	43, 18	xaddcld 12131	. . . . . . . 8 ⊢ (𝜑 → (inf(𝐴, ℝ^, < ) +_𝑒 𝐵) ∈ ℝ^)
78	77	adantr 481	. . . . . . 7 ⊢ ((𝜑 ∧ inf(𝐴, ℝ^, < ) ∈ ℝ) → (inf(𝐴, ℝ^, < ) +_𝑒 𝐵) ∈ ℝ^*)
79		xrltnle 10105	. . . . . . 7 ⊢ ((inf(𝐴, ℝ^, < ) ∈ ℝ^ ∧ (inf(𝐴, ℝ^, < ) +_𝑒 𝐵) ∈ ℝ^) → (inf(𝐴, ℝ^, < ) < (inf(𝐴, ℝ^, < ) +_𝑒 𝐵) ↔ ¬ (inf(𝐴, ℝ^, < ) +_𝑒 𝐵) ≤ inf(𝐴, ℝ^, < )))
80	76, 78, 79	syl2anc 693	. . . . . 6 ⊢ ((𝜑 ∧ inf(𝐴, ℝ^, < ) ∈ ℝ) → (inf(𝐴, ℝ^, < ) < (inf(𝐴, ℝ^, < ) +_𝑒 𝐵) ↔ ¬ (inf(𝐴, ℝ^, < ) +_𝑒 𝐵) ≤ inf(𝐴, ℝ^*, < )))
81	75, 80	mpbid 222	. . . . 5 ⊢ ((𝜑 ∧ inf(𝐴, ℝ^, < ) ∈ ℝ) → ¬ (inf(𝐴, ℝ^, < ) +_𝑒 𝐵) ≤ inf(𝐴, ℝ^*, < ))
82	34, 67, 81	syl2anc 693	. . . 4 ⊢ ((𝜑 ∧ ¬ inf(𝐴, ℝ^, < ) = +∞) → ¬ (inf(𝐴, ℝ^, < ) +_𝑒 𝐵) ≤ inf(𝐴, ℝ^*, < ))
83		simpr 477	. . . . . 6 ⊢ ((𝜑 ∧ ¬ (inf(𝐴, ℝ^, < ) +_𝑒 𝐵) ≤ inf(𝐴, ℝ^, < )) → ¬ (inf(𝐴, ℝ^, < ) +_𝑒 𝐵) ≤ inf(𝐴, ℝ^, < ))
84		simpl 473	. . . . . . 7 ⊢ ((𝜑 ∧ ¬ (inf(𝐴, ℝ^, < ) +_𝑒 𝐵) ≤ inf(𝐴, ℝ^, < )) → 𝜑)
85		infxrgelb 12165	. . . . . . . 8 ⊢ ((𝐴 ⊆ ℝ^* ∧ (inf(𝐴, ℝ^, < ) +_𝑒 𝐵) ∈ ℝ^) → ((inf(𝐴, ℝ^, < ) +_𝑒 𝐵) ≤ inf(𝐴, ℝ^, < ) ↔ ∀𝑧 ∈ 𝐴 (inf(𝐴, ℝ^*, < ) +_𝑒 𝐵) ≤ 𝑧))
86	8, 77, 85	syl2anc 693	. . . . . . 7 ⊢ (𝜑 → ((inf(𝐴, ℝ^, < ) +_𝑒 𝐵) ≤ inf(𝐴, ℝ^, < ) ↔ ∀𝑧 ∈ 𝐴 (inf(𝐴, ℝ^*, < ) +_𝑒 𝐵) ≤ 𝑧))
87	84, 86	syl 17	. . . . . 6 ⊢ ((𝜑 ∧ ¬ (inf(𝐴, ℝ^, < ) +_𝑒 𝐵) ≤ inf(𝐴, ℝ^, < )) → ((inf(𝐴, ℝ^, < ) +_𝑒 𝐵) ≤ inf(𝐴, ℝ^, < ) ↔ ∀𝑧 ∈ 𝐴 (inf(𝐴, ℝ^*, < ) +_𝑒 𝐵) ≤ 𝑧))
88	83, 87	mtbid 314	. . . . 5 ⊢ ((𝜑 ∧ ¬ (inf(𝐴, ℝ^, < ) +_𝑒 𝐵) ≤ inf(𝐴, ℝ^, < )) → ¬ ∀𝑧 ∈ 𝐴 (inf(𝐴, ℝ^*, < ) +_𝑒 𝐵) ≤ 𝑧)
89		rexnal 2995	. . . . 5 ⊢ (∃𝑧 ∈ 𝐴 ¬ (inf(𝐴, ℝ^, < ) +_𝑒 𝐵) ≤ 𝑧 ↔ ¬ ∀𝑧 ∈ 𝐴 (inf(𝐴, ℝ^, < ) +_𝑒 𝐵) ≤ 𝑧)
90	88, 89	sylibr 224	. . . 4 ⊢ ((𝜑 ∧ ¬ (inf(𝐴, ℝ^, < ) +_𝑒 𝐵) ≤ inf(𝐴, ℝ^, < )) → ∃𝑧 ∈ 𝐴 ¬ (inf(𝐴, ℝ^*, < ) +_𝑒 𝐵) ≤ 𝑧)
91	34, 82, 90	syl2anc 693	. . 3 ⊢ ((𝜑 ∧ ¬ inf(𝐴, ℝ^, < ) = +∞) → ∃𝑧 ∈ 𝐴 ¬ (inf(𝐴, ℝ^, < ) +_𝑒 𝐵) ≤ 𝑧)
92	11	adantr 481	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝐴) ∧ ¬ (inf(𝐴, ℝ^, < ) +_𝑒 𝐵) ≤ 𝑧) → 𝑧 ∈ ℝ^)
93	77	ad2antrr 762	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝐴) ∧ ¬ (inf(𝐴, ℝ^, < ) +_𝑒 𝐵) ≤ 𝑧) → (inf(𝐴, ℝ^, < ) +_𝑒 𝐵) ∈ ℝ^*)
94		simpr 477	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝐴) ∧ ¬ (inf(𝐴, ℝ^, < ) +_𝑒 𝐵) ≤ 𝑧) → ¬ (inf(𝐴, ℝ^, < ) +_𝑒 𝐵) ≤ 𝑧)
95		xrltnle 10105	. . . . . . . . 9 ⊢ ((𝑧 ∈ ℝ^* ∧ (inf(𝐴, ℝ^, < ) +_𝑒 𝐵) ∈ ℝ^) → (𝑧 < (inf(𝐴, ℝ^, < ) +_𝑒 𝐵) ↔ ¬ (inf(𝐴, ℝ^, < ) +_𝑒 𝐵) ≤ 𝑧))
96	92, 93, 95	syl2anc 693	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝐴) ∧ ¬ (inf(𝐴, ℝ^, < ) +_𝑒 𝐵) ≤ 𝑧) → (𝑧 < (inf(𝐴, ℝ^, < ) +_𝑒 𝐵) ↔ ¬ (inf(𝐴, ℝ^*, < ) +_𝑒 𝐵) ≤ 𝑧))
97	94, 96	mpbird 247	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝐴) ∧ ¬ (inf(𝐴, ℝ^, < ) +_𝑒 𝐵) ≤ 𝑧) → 𝑧 < (inf(𝐴, ℝ^, < ) +_𝑒 𝐵))
98	92, 93, 97	xrltled 39486	. . . . . 6 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝐴) ∧ ¬ (inf(𝐴, ℝ^, < ) +_𝑒 𝐵) ≤ 𝑧) → 𝑧 ≤ (inf(𝐴, ℝ^, < ) +_𝑒 𝐵))
99	98	ex 450	. . . . 5 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐴) → (¬ (inf(𝐴, ℝ^, < ) +_𝑒 𝐵) ≤ 𝑧 → 𝑧 ≤ (inf(𝐴, ℝ^, < ) +_𝑒 𝐵)))
100	99	adantlr 751	. . . 4 ⊢ (((𝜑 ∧ ¬ inf(𝐴, ℝ^, < ) = +∞) ∧ 𝑧 ∈ 𝐴) → (¬ (inf(𝐴, ℝ^, < ) +_𝑒 𝐵) ≤ 𝑧 → 𝑧 ≤ (inf(𝐴, ℝ^*, < ) +_𝑒 𝐵)))
101	100	reximdva 3017	. . 3 ⊢ ((𝜑 ∧ ¬ inf(𝐴, ℝ^, < ) = +∞) → (∃𝑧 ∈ 𝐴 ¬ (inf(𝐴, ℝ^, < ) +_𝑒 𝐵) ≤ 𝑧 → ∃𝑧 ∈ 𝐴 𝑧 ≤ (inf(𝐴, ℝ^*, < ) +_𝑒 𝐵)))
102	91, 101	mpd 15	. 2 ⊢ ((𝜑 ∧ ¬ inf(𝐴, ℝ^, < ) = +∞) → ∃𝑧 ∈ 𝐴 𝑧 ≤ (inf(𝐴, ℝ^, < ) +_𝑒 𝐵))
103	33, 102	pm2.61dan 832	1 ⊢ (𝜑 → ∃𝑧 ∈ 𝐴 𝑧 ≤ (inf(𝐴, ℝ^*, < ) +_𝑒 𝐵))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 196 ∧ wa 384 ∧ w3a 1037 = wceq 1483 ∃wex 1704 Ⅎwnf 1708 ∈ wcel 1990 ≠ wne 2794 ∀wral 2912 ∃wrex 2913 ⊆ wss 3574 ∅c0 3915 class class class wbr 4653 (class class class)co 6650 infcinf 8347 ℝcr 9935 + caddc 9939 +∞cpnf 10071 -∞cmnf 10072 ℝ^*cxr 10073 < clt 10074 ≤ cle 10075 ℝ⁺crp 11832 +_𝑒 cxad 11944

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-cnex 9992 ax-resscn 9993 ax-1cn 9994 ax-icn 9995 ax-addcl 9996 ax-addrcl 9997 ax-mulcl 9998 ax-mulrcl 9999 ax-mulcom 10000 ax-addass 10001 ax-mulass 10002 ax-distr 10003 ax-i2m1 10004 ax-1ne0 10005 ax-1rid 10006 ax-rnegex 10007 ax-rrecex 10008 ax-cnre 10009 ax-pre-lttri 10010 ax-pre-lttrn 10011 ax-pre-ltadd 10012 ax-pre-mulgt0 10013 ax-pre-sup 10014

This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-nel 2898 df-ral 2917 df-rex 2918 df-reu 2919 df-rmo 2920 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-op 4184 df-uni 4437 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-id 5024 df-po 5035 df-so 5036 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-riota 6611 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-1st 7168 df-2nd 7169 df-er 7742 df-en 7956 df-dom 7957 df-sdom 7958 df-sup 8348 df-inf 8349 df-pnf 10076 df-mnf 10077 df-xr 10078 df-ltxr 10079 df-le 10080 df-sub 10268 df-neg 10269 df-rp 11833 df-xadd 11947

This theorem is referenced by: infleinf 39588 infrpgernmpt 39695 ovnlerp 40776

W3C validator