infrpge - Mathbox for Glauco Siliprandi

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Theorem infrpge 41639

Description: The infimum of a nonempty, 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 4310	. . . . . . 7 ⊢ (𝐴 ≠ ∅ ↔ ∃𝑧 𝑧 ∈ 𝐴)
3	2	biimpi 218	. . . . . 6 ⊢ (𝐴 ≠ ∅ → ∃𝑧 𝑧 ∈ 𝐴)
4	1, 3	syl 17	. . . . 5 ⊢ (𝜑 → ∃𝑧 𝑧 ∈ 𝐴)
5	4	adantr 483	. . . 4 ⊢ ((𝜑 ∧ inf(𝐴, ℝ^*, < ) = +∞) → ∃𝑧 𝑧 ∈ 𝐴)
6		nfv 1915	. . . . 5 ⊢ Ⅎ𝑧(𝜑 ∧ inf(𝐴, ℝ^*, < ) = +∞)
7		simpr 487	. . . . . . 7 ⊢ (((𝜑 ∧ inf(𝐴, ℝ^*, < ) = +∞) ∧ 𝑧 ∈ 𝐴) → 𝑧 ∈ 𝐴)
8		infrpge.a	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐴 ⊆ ℝ^*)
9	8	adantr 483	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐴) → 𝐴 ⊆ ℝ^*)
10		simpr 487	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐴) → 𝑧 ∈ 𝐴)
11	9, 10	sseldd 3968	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐴) → 𝑧 ∈ ℝ^*)
12		pnfge 12526	. . . . . . . . . 10 ⊢ (𝑧 ∈ ℝ^* → 𝑧 ≤ +∞)
13	11, 12	syl 17	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐴) → 𝑧 ≤ +∞)
14	13	adantlr 713	. . . . . . . 8 ⊢ (((𝜑 ∧ inf(𝐴, ℝ^*, < ) = +∞) ∧ 𝑧 ∈ 𝐴) → 𝑧 ≤ +∞)
15		oveq1 7163	. . . . . . . . . . 11 ⊢ (inf(𝐴, ℝ^, < ) = +∞ → (inf(𝐴, ℝ^, < ) +_𝑒 𝐵) = (+∞ +_𝑒 𝐵))
16	15	adantl 484	. . . . . . . . . 10 ⊢ ((𝜑 ∧ inf(𝐴, ℝ^, < ) = +∞) → (inf(𝐴, ℝ^, < ) +_𝑒 𝐵) = (+∞ +_𝑒 𝐵))
17		infrpge.b	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐵 ∈ ℝ⁺)
18	17	rpxrd 12433	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐵 ∈ ℝ^*)
19	17	rpred 12432	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐵 ∈ ℝ)
20		renemnf 10690	. . . . . . . . . . . . 13 ⊢ (𝐵 ∈ ℝ → 𝐵 ≠ -∞)
21	19, 20	syl 17	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐵 ≠ -∞)
22		xaddpnf2 12621	. . . . . . . . . . . 12 ⊢ ((𝐵 ∈ ℝ^* ∧ 𝐵 ≠ -∞) → (+∞ +_𝑒 𝐵) = +∞)
23	18, 21, 22	syl2anc 586	. . . . . . . . . . 11 ⊢ (𝜑 → (+∞ +_𝑒 𝐵) = +∞)
24	23	adantr 483	. . . . . . . . . 10 ⊢ ((𝜑 ∧ inf(𝐴, ℝ^*, < ) = +∞) → (+∞ +_𝑒 𝐵) = +∞)
25	16, 24	eqtr2d 2857	. . . . . . . . 9 ⊢ ((𝜑 ∧ inf(𝐴, ℝ^, < ) = +∞) → +∞ = (inf(𝐴, ℝ^, < ) +_𝑒 𝐵))
26	25	adantr 483	. . . . . . . 8 ⊢ (((𝜑 ∧ inf(𝐴, ℝ^, < ) = +∞) ∧ 𝑧 ∈ 𝐴) → +∞ = (inf(𝐴, ℝ^, < ) +_𝑒 𝐵))
27	14, 26	breqtrd 5092	. . . . . . 7 ⊢ (((𝜑 ∧ inf(𝐴, ℝ^, < ) = +∞) ∧ 𝑧 ∈ 𝐴) → 𝑧 ≤ (inf(𝐴, ℝ^, < ) +_𝑒 𝐵))
28	7, 27	jca 514	. . . . . 6 ⊢ (((𝜑 ∧ inf(𝐴, ℝ^, < ) = +∞) ∧ 𝑧 ∈ 𝐴) → (𝑧 ∈ 𝐴 ∧ 𝑧 ≤ (inf(𝐴, ℝ^, < ) +_𝑒 𝐵)))
29	28	ex 415	. . . . 5 ⊢ ((𝜑 ∧ inf(𝐴, ℝ^, < ) = +∞) → (𝑧 ∈ 𝐴 → (𝑧 ∈ 𝐴 ∧ 𝑧 ≤ (inf(𝐴, ℝ^, < ) +_𝑒 𝐵))))
30	6, 29	eximd 2216	. . . 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 3144	. . 3 ⊢ (∃𝑧 ∈ 𝐴 𝑧 ≤ (inf(𝐴, ℝ^, < ) +_𝑒 𝐵) ↔ ∃𝑧(𝑧 ∈ 𝐴 ∧ 𝑧 ≤ (inf(𝐴, ℝ^, < ) +_𝑒 𝐵)))
33	31, 32	sylibr 236	. 2 ⊢ ((𝜑 ∧ inf(𝐴, ℝ^, < ) = +∞) → ∃𝑧 ∈ 𝐴 𝑧 ≤ (inf(𝐴, ℝ^, < ) +_𝑒 𝐵))
34		simpl 485	. . . 4 ⊢ ((𝜑 ∧ ¬ inf(𝐴, ℝ^*, < ) = +∞) → 𝜑)
35		infrpge.bnd	. . . . . . . . 9 ⊢ (𝜑 → ∃𝑥 ∈ ℝ ∀𝑦 ∈ 𝐴 𝑥 ≤ 𝑦)
36		infrpge.xph	. . . . . . . . . 10 ⊢ Ⅎ𝑥𝜑
37		nfv 1915	. . . . . . . . . 10 ⊢ Ⅎ𝑥-∞ < inf(𝐴, ℝ^*, < )
38		mnfxr 10698	. . . . . . . . . . . . 13 ⊢ -∞ ∈ ℝ^*
39	38	a1i 11	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ ∧ ∀𝑦 ∈ 𝐴 𝑥 ≤ 𝑦) → -∞ ∈ ℝ^*)
40		rexr 10687	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ ℝ → 𝑥 ∈ ℝ^*)
41	40	3ad2ant2 1130	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ ∧ ∀𝑦 ∈ 𝐴 𝑥 ≤ 𝑦) → 𝑥 ∈ ℝ^*)
42		infxrcl 12727	. . . . . . . . . . . . . 14 ⊢ (𝐴 ⊆ ℝ^* → inf(𝐴, ℝ^, < ) ∈ ℝ^)
43	8, 42	syl 17	. . . . . . . . . . . . 13 ⊢ (𝜑 → inf(𝐴, ℝ^, < ) ∈ ℝ^)
44	43	3ad2ant1 1129	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ ∧ ∀𝑦 ∈ 𝐴 𝑥 ≤ 𝑦) → inf(𝐴, ℝ^, < ) ∈ ℝ^)
45		mnflt 12519	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ ℝ → -∞ < 𝑥)
46	45	3ad2ant2 1130	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ ∧ ∀𝑦 ∈ 𝐴 𝑥 ≤ 𝑦) → -∞ < 𝑥)
47		simp3 1134	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ ∧ ∀𝑦 ∈ 𝐴 𝑥 ≤ 𝑦) → ∀𝑦 ∈ 𝐴 𝑥 ≤ 𝑦)
48	8	adantr 483	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → 𝐴 ⊆ ℝ^*)
49	40	adantl 484	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → 𝑥 ∈ ℝ^*)
50		infxrgelb 12729	. . . . . . . . . . . . . . 15 ⊢ ((𝐴 ⊆ ℝ^* ∧ 𝑥 ∈ ℝ^) → (𝑥 ≤ inf(𝐴, ℝ^, < ) ↔ ∀𝑦 ∈ 𝐴 𝑥 ≤ 𝑦))
51	48, 49, 50	syl2anc 586	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → (𝑥 ≤ inf(𝐴, ℝ^*, < ) ↔ ∀𝑦 ∈ 𝐴 𝑥 ≤ 𝑦))
52	51	3adant3 1128	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ ∧ ∀𝑦 ∈ 𝐴 𝑥 ≤ 𝑦) → (𝑥 ≤ inf(𝐴, ℝ^*, < ) ↔ ∀𝑦 ∈ 𝐴 𝑥 ≤ 𝑦))
53	47, 52	mpbird 259	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ ∧ ∀𝑦 ∈ 𝐴 𝑥 ≤ 𝑦) → 𝑥 ≤ inf(𝐴, ℝ^*, < ))
54	39, 41, 44, 46, 53	xrltletrd 12555	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ ∧ ∀𝑦 ∈ 𝐴 𝑥 ≤ 𝑦) → -∞ < inf(𝐴, ℝ^*, < ))
55	54	3exp 1115	. . . . . . . . . 10 ⊢ (𝜑 → (𝑥 ∈ ℝ → (∀𝑦 ∈ 𝐴 𝑥 ≤ 𝑦 → -∞ < inf(𝐴, ℝ^*, < ))))
56	36, 37, 55	rexlimd 3317	. . . . . . . . 9 ⊢ (𝜑 → (∃𝑥 ∈ ℝ ∀𝑦 ∈ 𝐴 𝑥 ≤ 𝑦 → -∞ < inf(𝐴, ℝ^*, < )))
57	35, 56	mpd 15	. . . . . . . 8 ⊢ (𝜑 → -∞ < inf(𝐴, ℝ^*, < ))
58	57	adantr 483	. . . . . . 7 ⊢ ((𝜑 ∧ ¬ inf(𝐴, ℝ^, < ) = +∞) → -∞ < inf(𝐴, ℝ^, < ))
59		neqne 3024	. . . . . . . . 9 ⊢ (¬ inf(𝐴, ℝ^, < ) = +∞ → inf(𝐴, ℝ^, < ) ≠ +∞)
60	59	adantl 484	. . . . . . . 8 ⊢ ((𝜑 ∧ ¬ inf(𝐴, ℝ^, < ) = +∞) → inf(𝐴, ℝ^, < ) ≠ +∞)
61	43	adantr 483	. . . . . . . 8 ⊢ ((𝜑 ∧ ¬ inf(𝐴, ℝ^, < ) = +∞) → inf(𝐴, ℝ^, < ) ∈ ℝ^*)
62	60, 61	nepnfltpnf 41630	. . . . . . 7 ⊢ ((𝜑 ∧ ¬ inf(𝐴, ℝ^, < ) = +∞) → inf(𝐴, ℝ^, < ) < +∞)
63	58, 62	jca 514	. . . . . 6 ⊢ ((𝜑 ∧ ¬ inf(𝐴, ℝ^, < ) = +∞) → (-∞ < inf(𝐴, ℝ^, < ) ∧ inf(𝐴, ℝ^*, < ) < +∞))
64		xrrebnd 12562	. . . . . . . 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 483	. . . . . 6 ⊢ ((𝜑 ∧ ¬ inf(𝐴, ℝ^, < ) = +∞) → (inf(𝐴, ℝ^, < ) ∈ ℝ ↔ (-∞ < inf(𝐴, ℝ^, < ) ∧ inf(𝐴, ℝ^, < ) < +∞)))
67	63, 66	mpbird 259	. . . . 5 ⊢ ((𝜑 ∧ ¬ inf(𝐴, ℝ^, < ) = +∞) → inf(𝐴, ℝ^, < ) ∈ ℝ)
68		simpr 487	. . . . . . . 8 ⊢ ((𝜑 ∧ inf(𝐴, ℝ^, < ) ∈ ℝ) → inf(𝐴, ℝ^, < ) ∈ ℝ)
69	17	adantr 483	. . . . . . . 8 ⊢ ((𝜑 ∧ inf(𝐴, ℝ^*, < ) ∈ ℝ) → 𝐵 ∈ ℝ⁺)
70	68, 69	ltaddrpd 12465	. . . . . . 7 ⊢ ((𝜑 ∧ inf(𝐴, ℝ^, < ) ∈ ℝ) → inf(𝐴, ℝ^, < ) < (inf(𝐴, ℝ^*, < ) + 𝐵))
71	19	adantr 483	. . . . . . . . 9 ⊢ ((𝜑 ∧ inf(𝐴, ℝ^*, < ) ∈ ℝ) → 𝐵 ∈ ℝ)
72		rexadd 12626	. . . . . . . . 9 ⊢ ((inf(𝐴, ℝ^, < ) ∈ ℝ ∧ 𝐵 ∈ ℝ) → (inf(𝐴, ℝ^, < ) +_𝑒 𝐵) = (inf(𝐴, ℝ^*, < ) + 𝐵))
73	68, 71, 72	syl2anc 586	. . . . . . . 8 ⊢ ((𝜑 ∧ inf(𝐴, ℝ^, < ) ∈ ℝ) → (inf(𝐴, ℝ^, < ) +_𝑒 𝐵) = (inf(𝐴, ℝ^*, < ) + 𝐵))
74	73	eqcomd 2827	. . . . . . 7 ⊢ ((𝜑 ∧ inf(𝐴, ℝ^, < ) ∈ ℝ) → (inf(𝐴, ℝ^, < ) + 𝐵) = (inf(𝐴, ℝ^*, < ) +_𝑒 𝐵))
75	70, 74	breqtrd 5092	. . . . . 6 ⊢ ((𝜑 ∧ inf(𝐴, ℝ^, < ) ∈ ℝ) → inf(𝐴, ℝ^, < ) < (inf(𝐴, ℝ^*, < ) +_𝑒 𝐵))
76	43	adantr 483	. . . . . . 7 ⊢ ((𝜑 ∧ inf(𝐴, ℝ^, < ) ∈ ℝ) → inf(𝐴, ℝ^, < ) ∈ ℝ^*)
77	43, 18	xaddcld 12695	. . . . . . . 8 ⊢ (𝜑 → (inf(𝐴, ℝ^, < ) +_𝑒 𝐵) ∈ ℝ^)
78	77	adantr 483	. . . . . . 7 ⊢ ((𝜑 ∧ inf(𝐴, ℝ^, < ) ∈ ℝ) → (inf(𝐴, ℝ^, < ) +_𝑒 𝐵) ∈ ℝ^*)
79		xrltnle 10708	. . . . . . 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 586	. . . . . 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 234	. . . . 5 ⊢ ((𝜑 ∧ inf(𝐴, ℝ^, < ) ∈ ℝ) → ¬ (inf(𝐴, ℝ^, < ) +_𝑒 𝐵) ≤ inf(𝐴, ℝ^*, < ))
82	34, 67, 81	syl2anc 586	. . . 4 ⊢ ((𝜑 ∧ ¬ inf(𝐴, ℝ^, < ) = +∞) → ¬ (inf(𝐴, ℝ^, < ) +_𝑒 𝐵) ≤ inf(𝐴, ℝ^*, < ))
83		simpr 487	. . . . . 6 ⊢ ((𝜑 ∧ ¬ (inf(𝐴, ℝ^, < ) +_𝑒 𝐵) ≤ inf(𝐴, ℝ^, < )) → ¬ (inf(𝐴, ℝ^, < ) +_𝑒 𝐵) ≤ inf(𝐴, ℝ^, < ))
84		simpl 485	. . . . . . 7 ⊢ ((𝜑 ∧ ¬ (inf(𝐴, ℝ^, < ) +_𝑒 𝐵) ≤ inf(𝐴, ℝ^, < )) → 𝜑)
85		infxrgelb 12729	. . . . . . . 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 586	. . . . . . 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 326	. . . . 5 ⊢ ((𝜑 ∧ ¬ (inf(𝐴, ℝ^, < ) +_𝑒 𝐵) ≤ inf(𝐴, ℝ^, < )) → ¬ ∀𝑧 ∈ 𝐴 (inf(𝐴, ℝ^*, < ) +_𝑒 𝐵) ≤ 𝑧)
89		rexnal 3238	. . . . 5 ⊢ (∃𝑧 ∈ 𝐴 ¬ (inf(𝐴, ℝ^, < ) +_𝑒 𝐵) ≤ 𝑧 ↔ ¬ ∀𝑧 ∈ 𝐴 (inf(𝐴, ℝ^, < ) +_𝑒 𝐵) ≤ 𝑧)
90	88, 89	sylibr 236	. . . 4 ⊢ ((𝜑 ∧ ¬ (inf(𝐴, ℝ^, < ) +_𝑒 𝐵) ≤ inf(𝐴, ℝ^, < )) → ∃𝑧 ∈ 𝐴 ¬ (inf(𝐴, ℝ^*, < ) +_𝑒 𝐵) ≤ 𝑧)
91	34, 82, 90	syl2anc 586	. . 3 ⊢ ((𝜑 ∧ ¬ inf(𝐴, ℝ^, < ) = +∞) → ∃𝑧 ∈ 𝐴 ¬ (inf(𝐴, ℝ^, < ) +_𝑒 𝐵) ≤ 𝑧)
92	11	adantr 483	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝐴) ∧ ¬ (inf(𝐴, ℝ^, < ) +_𝑒 𝐵) ≤ 𝑧) → 𝑧 ∈ ℝ^)
93	77	ad2antrr 724	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝐴) ∧ ¬ (inf(𝐴, ℝ^, < ) +_𝑒 𝐵) ≤ 𝑧) → (inf(𝐴, ℝ^, < ) +_𝑒 𝐵) ∈ ℝ^*)
94		simpr 487	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝐴) ∧ ¬ (inf(𝐴, ℝ^, < ) +_𝑒 𝐵) ≤ 𝑧) → ¬ (inf(𝐴, ℝ^, < ) +_𝑒 𝐵) ≤ 𝑧)
95		xrltnle 10708	. . . . . . . . 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 586	. . . . . . . 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 259	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝐴) ∧ ¬ (inf(𝐴, ℝ^, < ) +_𝑒 𝐵) ≤ 𝑧) → 𝑧 < (inf(𝐴, ℝ^, < ) +_𝑒 𝐵))
98	92, 93, 97	xrltled 12544	. . . . . 6 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝐴) ∧ ¬ (inf(𝐴, ℝ^, < ) +_𝑒 𝐵) ≤ 𝑧) → 𝑧 ≤ (inf(𝐴, ℝ^, < ) +_𝑒 𝐵))
99	98	ex 415	. . . . 5 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐴) → (¬ (inf(𝐴, ℝ^, < ) +_𝑒 𝐵) ≤ 𝑧 → 𝑧 ≤ (inf(𝐴, ℝ^, < ) +_𝑒 𝐵)))
100	99	adantlr 713	. . . 4 ⊢ (((𝜑 ∧ ¬ inf(𝐴, ℝ^, < ) = +∞) ∧ 𝑧 ∈ 𝐴) → (¬ (inf(𝐴, ℝ^, < ) +_𝑒 𝐵) ≤ 𝑧 → 𝑧 ≤ (inf(𝐴, ℝ^*, < ) +_𝑒 𝐵)))
101	100	reximdva 3274	. . 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 811	1 ⊢ (𝜑 → ∃𝑧 ∈ 𝐴 𝑧 ≤ (inf(𝐴, ℝ^*, < ) +_𝑒 𝐵))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 208 ∧ wa 398 ∧ w3a 1083 = wceq 1537 ∃wex 1780 Ⅎwnf 1784 ∈ wcel 2114 ≠ wne 3016 ∀wral 3138 ∃wrex 3139 ⊆ wss 3936 ∅c0 4291 class class class wbr 5066 (class class class)co 7156 infcinf 8905 ℝcr 10536 + caddc 10540 +∞cpnf 10672 -∞cmnf 10673 ℝ^*cxr 10674 < clt 10675 ≤ cle 10676 ℝ⁺crp 12390 +_𝑒 cxad 12506

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 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 ax-cnex 10593 ax-resscn 10594 ax-1cn 10595 ax-icn 10596 ax-addcl 10597 ax-addrcl 10598 ax-mulcl 10599 ax-mulrcl 10600 ax-mulcom 10601 ax-addass 10602 ax-mulass 10603 ax-distr 10604 ax-i2m1 10605 ax-1ne0 10606 ax-1rid 10607 ax-rnegex 10608 ax-rrecex 10609 ax-cnre 10610 ax-pre-lttri 10611 ax-pre-lttrn 10612 ax-pre-ltadd 10613 ax-pre-mulgt0 10614 ax-pre-sup 10615

This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 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 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-op 4574 df-uni 4839 df-iun 4921 df-br 5067 df-opab 5129 df-mpt 5147 df-id 5460 df-po 5474 df-so 5475 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-f1 6360 df-fo 6361 df-f1o 6362 df-fv 6363 df-riota 7114 df-ov 7159 df-oprab 7160 df-mpo 7161 df-1st 7689 df-2nd 7690 df-er 8289 df-en 8510 df-dom 8511 df-sdom 8512 df-sup 8906 df-inf 8907 df-pnf 10677 df-mnf 10678 df-xr 10679 df-ltxr 10680 df-le 10681 df-sub 10872 df-neg 10873 df-rp 12391 df-xadd 12509

This theorem is referenced by: infleinf 41660 infrpgernmpt 41761 ovnlerp 42864

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