infrpge - Mathbox for Glauco Siliprandi

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Theorem infrpge 45347

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 4316	. . . . . . 7 ⊢ (𝐴 ≠ ∅ ↔ ∃𝑧 𝑧 ∈ 𝐴)
3	2	biimpi 216	. . . . . 6 ⊢ (𝐴 ≠ ∅ → ∃𝑧 𝑧 ∈ 𝐴)
4	1, 3	syl 17	. . . . 5 ⊢ (𝜑 → ∃𝑧 𝑧 ∈ 𝐴)
5	4	adantr 480	. . . 4 ⊢ ((𝜑 ∧ inf(𝐴, ℝ^*, < ) = +∞) → ∃𝑧 𝑧 ∈ 𝐴)
6		nfv 1914	. . . . 5 ⊢ Ⅎ𝑧(𝜑 ∧ inf(𝐴, ℝ^*, < ) = +∞)
7		simpr 484	. . . . . . 7 ⊢ (((𝜑 ∧ inf(𝐴, ℝ^*, < ) = +∞) ∧ 𝑧 ∈ 𝐴) → 𝑧 ∈ 𝐴)
8		infrpge.a	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐴 ⊆ ℝ^*)
9	8	adantr 480	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐴) → 𝐴 ⊆ ℝ^*)
10		simpr 484	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐴) → 𝑧 ∈ 𝐴)
11	9, 10	sseldd 3947	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐴) → 𝑧 ∈ ℝ^*)
12		pnfge 13090	. . . . . . . . . 10 ⊢ (𝑧 ∈ ℝ^* → 𝑧 ≤ +∞)
13	11, 12	syl 17	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐴) → 𝑧 ≤ +∞)
14	13	adantlr 715	. . . . . . . 8 ⊢ (((𝜑 ∧ inf(𝐴, ℝ^*, < ) = +∞) ∧ 𝑧 ∈ 𝐴) → 𝑧 ≤ +∞)
15		oveq1 7394	. . . . . . . . . . 11 ⊢ (inf(𝐴, ℝ^, < ) = +∞ → (inf(𝐴, ℝ^, < ) +_𝑒 𝐵) = (+∞ +_𝑒 𝐵))
16	15	adantl 481	. . . . . . . . . 10 ⊢ ((𝜑 ∧ inf(𝐴, ℝ^, < ) = +∞) → (inf(𝐴, ℝ^, < ) +_𝑒 𝐵) = (+∞ +_𝑒 𝐵))
17		infrpge.b	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐵 ∈ ℝ⁺)
18	17	rpxrd 12996	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐵 ∈ ℝ^*)
19	17	rpred 12995	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐵 ∈ ℝ)
20		renemnf 11223	. . . . . . . . . . . . 13 ⊢ (𝐵 ∈ ℝ → 𝐵 ≠ -∞)
21	19, 20	syl 17	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐵 ≠ -∞)
22		xaddpnf2 13187	. . . . . . . . . . . 12 ⊢ ((𝐵 ∈ ℝ^* ∧ 𝐵 ≠ -∞) → (+∞ +_𝑒 𝐵) = +∞)
23	18, 21, 22	syl2anc 584	. . . . . . . . . . 11 ⊢ (𝜑 → (+∞ +_𝑒 𝐵) = +∞)
24	23	adantr 480	. . . . . . . . . 10 ⊢ ((𝜑 ∧ inf(𝐴, ℝ^*, < ) = +∞) → (+∞ +_𝑒 𝐵) = +∞)
25	16, 24	eqtr2d 2765	. . . . . . . . 9 ⊢ ((𝜑 ∧ inf(𝐴, ℝ^, < ) = +∞) → +∞ = (inf(𝐴, ℝ^, < ) +_𝑒 𝐵))
26	25	adantr 480	. . . . . . . 8 ⊢ (((𝜑 ∧ inf(𝐴, ℝ^, < ) = +∞) ∧ 𝑧 ∈ 𝐴) → +∞ = (inf(𝐴, ℝ^, < ) +_𝑒 𝐵))
27	14, 26	breqtrd 5133	. . . . . . 7 ⊢ (((𝜑 ∧ inf(𝐴, ℝ^, < ) = +∞) ∧ 𝑧 ∈ 𝐴) → 𝑧 ≤ (inf(𝐴, ℝ^, < ) +_𝑒 𝐵))
28	7, 27	jca 511	. . . . . 6 ⊢ (((𝜑 ∧ inf(𝐴, ℝ^, < ) = +∞) ∧ 𝑧 ∈ 𝐴) → (𝑧 ∈ 𝐴 ∧ 𝑧 ≤ (inf(𝐴, ℝ^, < ) +_𝑒 𝐵)))
29	28	ex 412	. . . . 5 ⊢ ((𝜑 ∧ inf(𝐴, ℝ^, < ) = +∞) → (𝑧 ∈ 𝐴 → (𝑧 ∈ 𝐴 ∧ 𝑧 ≤ (inf(𝐴, ℝ^, < ) +_𝑒 𝐵))))
30	6, 29	eximd 2217	. . . 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 3054	. . 3 ⊢ (∃𝑧 ∈ 𝐴 𝑧 ≤ (inf(𝐴, ℝ^, < ) +_𝑒 𝐵) ↔ ∃𝑧(𝑧 ∈ 𝐴 ∧ 𝑧 ≤ (inf(𝐴, ℝ^, < ) +_𝑒 𝐵)))
33	31, 32	sylibr 234	. 2 ⊢ ((𝜑 ∧ inf(𝐴, ℝ^, < ) = +∞) → ∃𝑧 ∈ 𝐴 𝑧 ≤ (inf(𝐴, ℝ^, < ) +_𝑒 𝐵))
34		simpl 482	. . . 4 ⊢ ((𝜑 ∧ ¬ inf(𝐴, ℝ^*, < ) = +∞) → 𝜑)
35		infrpge.bnd	. . . . . . . . 9 ⊢ (𝜑 → ∃𝑥 ∈ ℝ ∀𝑦 ∈ 𝐴 𝑥 ≤ 𝑦)
36		infrpge.xph	. . . . . . . . . 10 ⊢ Ⅎ𝑥𝜑
37		nfv 1914	. . . . . . . . . 10 ⊢ Ⅎ𝑥-∞ < inf(𝐴, ℝ^*, < )
38		mnfxr 11231	. . . . . . . . . . . . 13 ⊢ -∞ ∈ ℝ^*
39	38	a1i 11	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ ∧ ∀𝑦 ∈ 𝐴 𝑥 ≤ 𝑦) → -∞ ∈ ℝ^*)
40		rexr 11220	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ ℝ → 𝑥 ∈ ℝ^*)
41	40	3ad2ant2 1134	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ ∧ ∀𝑦 ∈ 𝐴 𝑥 ≤ 𝑦) → 𝑥 ∈ ℝ^*)
42		infxrcl 13294	. . . . . . . . . . . . . 14 ⊢ (𝐴 ⊆ ℝ^* → inf(𝐴, ℝ^, < ) ∈ ℝ^)
43	8, 42	syl 17	. . . . . . . . . . . . 13 ⊢ (𝜑 → inf(𝐴, ℝ^, < ) ∈ ℝ^)
44	43	3ad2ant1 1133	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ ∧ ∀𝑦 ∈ 𝐴 𝑥 ≤ 𝑦) → inf(𝐴, ℝ^, < ) ∈ ℝ^)
45		mnflt 13083	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ ℝ → -∞ < 𝑥)
46	45	3ad2ant2 1134	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ ∧ ∀𝑦 ∈ 𝐴 𝑥 ≤ 𝑦) → -∞ < 𝑥)
47		simp3 1138	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ ∧ ∀𝑦 ∈ 𝐴 𝑥 ≤ 𝑦) → ∀𝑦 ∈ 𝐴 𝑥 ≤ 𝑦)
48	8	adantr 480	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → 𝐴 ⊆ ℝ^*)
49	40	adantl 481	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → 𝑥 ∈ ℝ^*)
50		infxrgelb 13296	. . . . . . . . . . . . . . 15 ⊢ ((𝐴 ⊆ ℝ^* ∧ 𝑥 ∈ ℝ^) → (𝑥 ≤ inf(𝐴, ℝ^, < ) ↔ ∀𝑦 ∈ 𝐴 𝑥 ≤ 𝑦))
51	48, 49, 50	syl2anc 584	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → (𝑥 ≤ inf(𝐴, ℝ^*, < ) ↔ ∀𝑦 ∈ 𝐴 𝑥 ≤ 𝑦))
52	51	3adant3 1132	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ ∧ ∀𝑦 ∈ 𝐴 𝑥 ≤ 𝑦) → (𝑥 ≤ inf(𝐴, ℝ^*, < ) ↔ ∀𝑦 ∈ 𝐴 𝑥 ≤ 𝑦))
53	47, 52	mpbird 257	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ ∧ ∀𝑦 ∈ 𝐴 𝑥 ≤ 𝑦) → 𝑥 ≤ inf(𝐴, ℝ^*, < ))
54	39, 41, 44, 46, 53	xrltletrd 13121	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ ∧ ∀𝑦 ∈ 𝐴 𝑥 ≤ 𝑦) → -∞ < inf(𝐴, ℝ^*, < ))
55	54	3exp 1119	. . . . . . . . . 10 ⊢ (𝜑 → (𝑥 ∈ ℝ → (∀𝑦 ∈ 𝐴 𝑥 ≤ 𝑦 → -∞ < inf(𝐴, ℝ^*, < ))))
56	36, 37, 55	rexlimd 3244	. . . . . . . . 9 ⊢ (𝜑 → (∃𝑥 ∈ ℝ ∀𝑦 ∈ 𝐴 𝑥 ≤ 𝑦 → -∞ < inf(𝐴, ℝ^*, < )))
57	35, 56	mpd 15	. . . . . . . 8 ⊢ (𝜑 → -∞ < inf(𝐴, ℝ^*, < ))
58	57	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ ¬ inf(𝐴, ℝ^, < ) = +∞) → -∞ < inf(𝐴, ℝ^, < ))
59		neqne 2933	. . . . . . . . 9 ⊢ (¬ inf(𝐴, ℝ^, < ) = +∞ → inf(𝐴, ℝ^, < ) ≠ +∞)
60	59	adantl 481	. . . . . . . 8 ⊢ ((𝜑 ∧ ¬ inf(𝐴, ℝ^, < ) = +∞) → inf(𝐴, ℝ^, < ) ≠ +∞)
61	43	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ ¬ inf(𝐴, ℝ^, < ) = +∞) → inf(𝐴, ℝ^, < ) ∈ ℝ^*)
62	60, 61	nepnfltpnf 45338	. . . . . . 7 ⊢ ((𝜑 ∧ ¬ inf(𝐴, ℝ^, < ) = +∞) → inf(𝐴, ℝ^, < ) < +∞)
63	58, 62	jca 511	. . . . . 6 ⊢ ((𝜑 ∧ ¬ inf(𝐴, ℝ^, < ) = +∞) → (-∞ < inf(𝐴, ℝ^, < ) ∧ inf(𝐴, ℝ^*, < ) < +∞))
64		xrrebnd 13128	. . . . . . . 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 480	. . . . . 6 ⊢ ((𝜑 ∧ ¬ inf(𝐴, ℝ^, < ) = +∞) → (inf(𝐴, ℝ^, < ) ∈ ℝ ↔ (-∞ < inf(𝐴, ℝ^, < ) ∧ inf(𝐴, ℝ^, < ) < +∞)))
67	63, 66	mpbird 257	. . . . 5 ⊢ ((𝜑 ∧ ¬ inf(𝐴, ℝ^, < ) = +∞) → inf(𝐴, ℝ^, < ) ∈ ℝ)
68		simpr 484	. . . . . . . 8 ⊢ ((𝜑 ∧ inf(𝐴, ℝ^, < ) ∈ ℝ) → inf(𝐴, ℝ^, < ) ∈ ℝ)
69	17	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ inf(𝐴, ℝ^*, < ) ∈ ℝ) → 𝐵 ∈ ℝ⁺)
70	68, 69	ltaddrpd 13028	. . . . . . 7 ⊢ ((𝜑 ∧ inf(𝐴, ℝ^, < ) ∈ ℝ) → inf(𝐴, ℝ^, < ) < (inf(𝐴, ℝ^*, < ) + 𝐵))
71	19	adantr 480	. . . . . . . . 9 ⊢ ((𝜑 ∧ inf(𝐴, ℝ^*, < ) ∈ ℝ) → 𝐵 ∈ ℝ)
72		rexadd 13192	. . . . . . . . 9 ⊢ ((inf(𝐴, ℝ^, < ) ∈ ℝ ∧ 𝐵 ∈ ℝ) → (inf(𝐴, ℝ^, < ) +_𝑒 𝐵) = (inf(𝐴, ℝ^*, < ) + 𝐵))
73	68, 71, 72	syl2anc 584	. . . . . . . 8 ⊢ ((𝜑 ∧ inf(𝐴, ℝ^, < ) ∈ ℝ) → (inf(𝐴, ℝ^, < ) +_𝑒 𝐵) = (inf(𝐴, ℝ^*, < ) + 𝐵))
74	73	eqcomd 2735	. . . . . . 7 ⊢ ((𝜑 ∧ inf(𝐴, ℝ^, < ) ∈ ℝ) → (inf(𝐴, ℝ^, < ) + 𝐵) = (inf(𝐴, ℝ^*, < ) +_𝑒 𝐵))
75	70, 74	breqtrd 5133	. . . . . 6 ⊢ ((𝜑 ∧ inf(𝐴, ℝ^, < ) ∈ ℝ) → inf(𝐴, ℝ^, < ) < (inf(𝐴, ℝ^*, < ) +_𝑒 𝐵))
76	43	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ inf(𝐴, ℝ^, < ) ∈ ℝ) → inf(𝐴, ℝ^, < ) ∈ ℝ^*)
77	43, 18	xaddcld 13261	. . . . . . . 8 ⊢ (𝜑 → (inf(𝐴, ℝ^, < ) +_𝑒 𝐵) ∈ ℝ^)
78	77	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ inf(𝐴, ℝ^, < ) ∈ ℝ) → (inf(𝐴, ℝ^, < ) +_𝑒 𝐵) ∈ ℝ^*)
79		xrltnle 11241	. . . . . . 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 584	. . . . . 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 232	. . . . 5 ⊢ ((𝜑 ∧ inf(𝐴, ℝ^, < ) ∈ ℝ) → ¬ (inf(𝐴, ℝ^, < ) +_𝑒 𝐵) ≤ inf(𝐴, ℝ^*, < ))
82	34, 67, 81	syl2anc 584	. . . 4 ⊢ ((𝜑 ∧ ¬ inf(𝐴, ℝ^, < ) = +∞) → ¬ (inf(𝐴, ℝ^, < ) +_𝑒 𝐵) ≤ inf(𝐴, ℝ^*, < ))
83		simpr 484	. . . . . 6 ⊢ ((𝜑 ∧ ¬ (inf(𝐴, ℝ^, < ) +_𝑒 𝐵) ≤ inf(𝐴, ℝ^, < )) → ¬ (inf(𝐴, ℝ^, < ) +_𝑒 𝐵) ≤ inf(𝐴, ℝ^, < ))
84		simpl 482	. . . . . . 7 ⊢ ((𝜑 ∧ ¬ (inf(𝐴, ℝ^, < ) +_𝑒 𝐵) ≤ inf(𝐴, ℝ^, < )) → 𝜑)
85		infxrgelb 13296	. . . . . . . 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 584	. . . . . . 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 324	. . . . 5 ⊢ ((𝜑 ∧ ¬ (inf(𝐴, ℝ^, < ) +_𝑒 𝐵) ≤ inf(𝐴, ℝ^, < )) → ¬ ∀𝑧 ∈ 𝐴 (inf(𝐴, ℝ^*, < ) +_𝑒 𝐵) ≤ 𝑧)
89		rexnal 3082	. . . . 5 ⊢ (∃𝑧 ∈ 𝐴 ¬ (inf(𝐴, ℝ^, < ) +_𝑒 𝐵) ≤ 𝑧 ↔ ¬ ∀𝑧 ∈ 𝐴 (inf(𝐴, ℝ^, < ) +_𝑒 𝐵) ≤ 𝑧)
90	88, 89	sylibr 234	. . . 4 ⊢ ((𝜑 ∧ ¬ (inf(𝐴, ℝ^, < ) +_𝑒 𝐵) ≤ inf(𝐴, ℝ^, < )) → ∃𝑧 ∈ 𝐴 ¬ (inf(𝐴, ℝ^*, < ) +_𝑒 𝐵) ≤ 𝑧)
91	34, 82, 90	syl2anc 584	. . 3 ⊢ ((𝜑 ∧ ¬ inf(𝐴, ℝ^, < ) = +∞) → ∃𝑧 ∈ 𝐴 ¬ (inf(𝐴, ℝ^, < ) +_𝑒 𝐵) ≤ 𝑧)
92	11	adantr 480	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝐴) ∧ ¬ (inf(𝐴, ℝ^, < ) +_𝑒 𝐵) ≤ 𝑧) → 𝑧 ∈ ℝ^)
93	77	ad2antrr 726	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝐴) ∧ ¬ (inf(𝐴, ℝ^, < ) +_𝑒 𝐵) ≤ 𝑧) → (inf(𝐴, ℝ^, < ) +_𝑒 𝐵) ∈ ℝ^*)
94		simpr 484	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝐴) ∧ ¬ (inf(𝐴, ℝ^, < ) +_𝑒 𝐵) ≤ 𝑧) → ¬ (inf(𝐴, ℝ^, < ) +_𝑒 𝐵) ≤ 𝑧)
95		xrltnle 11241	. . . . . . . . 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 584	. . . . . . . 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 257	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝐴) ∧ ¬ (inf(𝐴, ℝ^, < ) +_𝑒 𝐵) ≤ 𝑧) → 𝑧 < (inf(𝐴, ℝ^, < ) +_𝑒 𝐵))
98	92, 93, 97	xrltled 13110	. . . . . 6 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝐴) ∧ ¬ (inf(𝐴, ℝ^, < ) +_𝑒 𝐵) ≤ 𝑧) → 𝑧 ≤ (inf(𝐴, ℝ^, < ) +_𝑒 𝐵))
99	98	ex 412	. . . . 5 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐴) → (¬ (inf(𝐴, ℝ^, < ) +_𝑒 𝐵) ≤ 𝑧 → 𝑧 ≤ (inf(𝐴, ℝ^, < ) +_𝑒 𝐵)))
100	99	adantlr 715	. . . 4 ⊢ (((𝜑 ∧ ¬ inf(𝐴, ℝ^, < ) = +∞) ∧ 𝑧 ∈ 𝐴) → (¬ (inf(𝐴, ℝ^, < ) +_𝑒 𝐵) ≤ 𝑧 → 𝑧 ≤ (inf(𝐴, ℝ^*, < ) +_𝑒 𝐵)))
101	100	reximdva 3146	. . 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 812	1 ⊢ (𝜑 → ∃𝑧 ∈ 𝐴 𝑧 ≤ (inf(𝐴, ℝ^*, < ) +_𝑒 𝐵))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 206 ∧ wa 395 ∧ w3a 1086 = wceq 1540 ∃wex 1779 Ⅎwnf 1783 ∈ 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 + caddc 11071 +∞cpnf 11205 -∞cmnf 11206 ℝ^*cxr 11207 < clt 11208 ≤ cle 11209 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-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-id 5533 df-po 5546 df-so 5547 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-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-1st 7968 df-2nd 7969 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-rp 12952 df-xadd 13073

This theorem is referenced by: infleinf 45368 infrpgernmpt 45461 ovnlerp 46560

W3C validator