supxrgere - Mathbox for Glauco Siliprandi

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Theorem supxrgere 39549

Description: If a real number can be approximated from below by members of a set, then it is smaller or equal to the supremum of the set. (Contributed by Glauco Siliprandi, 17-Aug-2020.)

**Hypotheses**
Ref	Expression
supxrgere.xph	⊢ Ⅎ𝑥𝜑
supxrgere.a	⊢ (𝜑 → 𝐴 ⊆ ℝ^*)
supxrgere.b	⊢ (𝜑 → 𝐵 ∈ ℝ)
supxrgere.y	⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → ∃𝑦 ∈ 𝐴 (𝐵 − 𝑥) < 𝑦)

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

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

**Proof of Theorem supxrgere**
Step	Hyp	Ref	Expression
1		supxrgere.b	. . . . 5 ⊢ (𝜑 → 𝐵 ∈ ℝ)
2		rexr 10085	. . . . . 6 ⊢ (𝐵 ∈ ℝ → 𝐵 ∈ ℝ^*)
3		pnfxr 10092	. . . . . . 7 ⊢ +∞ ∈ ℝ^*
4	3	a1i 11	. . . . . 6 ⊢ (𝐵 ∈ ℝ → +∞ ∈ ℝ^*)
5		ltpnf 11954	. . . . . 6 ⊢ (𝐵 ∈ ℝ → 𝐵 < +∞)
6	2, 4, 5	xrltled 39486	. . . . 5 ⊢ (𝐵 ∈ ℝ → 𝐵 ≤ +∞)
7	1, 6	syl 17	. . . 4 ⊢ (𝜑 → 𝐵 ≤ +∞)
8	7	adantr 481	. . 3 ⊢ ((𝜑 ∧ sup(𝐴, ℝ^*, < ) = +∞) → 𝐵 ≤ +∞)
9		id 22	. . . . 5 ⊢ (sup(𝐴, ℝ^, < ) = +∞ → sup(𝐴, ℝ^, < ) = +∞)
10	9	eqcomd 2628	. . . 4 ⊢ (sup(𝐴, ℝ^, < ) = +∞ → +∞ = sup(𝐴, ℝ^, < ))
11	10	adantl 482	. . 3 ⊢ ((𝜑 ∧ sup(𝐴, ℝ^, < ) = +∞) → +∞ = sup(𝐴, ℝ^, < ))
12	8, 11	breqtrd 4679	. 2 ⊢ ((𝜑 ∧ sup(𝐴, ℝ^, < ) = +∞) → 𝐵 ≤ sup(𝐴, ℝ^, < ))
13		simpl 473	. . 3 ⊢ ((𝜑 ∧ ¬ sup(𝐴, ℝ^*, < ) = +∞) → 𝜑)
14		1rp 11836	. . . . . . . 8 ⊢ 1 ∈ ℝ⁺
15		nfcv 2764	. . . . . . . . . 10 ⊢ Ⅎ𝑥1
16		supxrgere.xph	. . . . . . . . . . . 12 ⊢ Ⅎ𝑥𝜑
17		nfv 1843	. . . . . . . . . . . 12 ⊢ Ⅎ𝑥1 ∈ ℝ⁺
18	16, 17	nfan 1828	. . . . . . . . . . 11 ⊢ Ⅎ𝑥(𝜑 ∧ 1 ∈ ℝ⁺)
19		nfv 1843	. . . . . . . . . . 11 ⊢ Ⅎ𝑥∃𝑦 ∈ 𝐴 (𝐵 − 1) < 𝑦
20	18, 19	nfim 1825	. . . . . . . . . 10 ⊢ Ⅎ𝑥((𝜑 ∧ 1 ∈ ℝ⁺) → ∃𝑦 ∈ 𝐴 (𝐵 − 1) < 𝑦)
21		eleq1 2689	. . . . . . . . . . . 12 ⊢ (𝑥 = 1 → (𝑥 ∈ ℝ⁺ ↔ 1 ∈ ℝ⁺))
22	21	anbi2d 740	. . . . . . . . . . 11 ⊢ (𝑥 = 1 → ((𝜑 ∧ 𝑥 ∈ ℝ⁺) ↔ (𝜑 ∧ 1 ∈ ℝ⁺)))
23		oveq2 6658	. . . . . . . . . . . . 13 ⊢ (𝑥 = 1 → (𝐵 − 𝑥) = (𝐵 − 1))
24	23	breq1d 4663	. . . . . . . . . . . 12 ⊢ (𝑥 = 1 → ((𝐵 − 𝑥) < 𝑦 ↔ (𝐵 − 1) < 𝑦))
25	24	rexbidv 3052	. . . . . . . . . . 11 ⊢ (𝑥 = 1 → (∃𝑦 ∈ 𝐴 (𝐵 − 𝑥) < 𝑦 ↔ ∃𝑦 ∈ 𝐴 (𝐵 − 1) < 𝑦))
26	22, 25	imbi12d 334	. . . . . . . . . 10 ⊢ (𝑥 = 1 → (((𝜑 ∧ 𝑥 ∈ ℝ⁺) → ∃𝑦 ∈ 𝐴 (𝐵 − 𝑥) < 𝑦) ↔ ((𝜑 ∧ 1 ∈ ℝ⁺) → ∃𝑦 ∈ 𝐴 (𝐵 − 1) < 𝑦)))
27		supxrgere.y	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → ∃𝑦 ∈ 𝐴 (𝐵 − 𝑥) < 𝑦)
28	15, 20, 26, 27	vtoclgf 3264	. . . . . . . . 9 ⊢ (1 ∈ ℝ⁺ → ((𝜑 ∧ 1 ∈ ℝ⁺) → ∃𝑦 ∈ 𝐴 (𝐵 − 1) < 𝑦))
29	14, 28	ax-mp 5	. . . . . . . 8 ⊢ ((𝜑 ∧ 1 ∈ ℝ⁺) → ∃𝑦 ∈ 𝐴 (𝐵 − 1) < 𝑦)
30	14, 29	mpan2 707	. . . . . . 7 ⊢ (𝜑 → ∃𝑦 ∈ 𝐴 (𝐵 − 1) < 𝑦)
31	30	adantr 481	. . . . . 6 ⊢ ((𝜑 ∧ ¬ sup(𝐴, ℝ^*, < ) = +∞) → ∃𝑦 ∈ 𝐴 (𝐵 − 1) < 𝑦)
32		mnfxr 10096	. . . . . . . . . . 11 ⊢ -∞ ∈ ℝ^*
33	32	a1i 11	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐴 ∧ (𝐵 − 1) < 𝑦) → -∞ ∈ ℝ^*)
34		supxrgere.a	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐴 ⊆ ℝ^*)
35	34	sselda 3603	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐴) → 𝑦 ∈ ℝ^*)
36	35	3adant3 1081	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐴 ∧ (𝐵 − 1) < 𝑦) → 𝑦 ∈ ℝ^*)
37		supxrcl 12145	. . . . . . . . . . . 12 ⊢ (𝐴 ⊆ ℝ^* → sup(𝐴, ℝ^, < ) ∈ ℝ^)
38	34, 37	syl 17	. . . . . . . . . . 11 ⊢ (𝜑 → sup(𝐴, ℝ^, < ) ∈ ℝ^)
39	38	3ad2ant1 1082	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐴 ∧ (𝐵 − 1) < 𝑦) → sup(𝐴, ℝ^, < ) ∈ ℝ^)
40		peano2rem 10348	. . . . . . . . . . . . . . . 16 ⊢ (𝐵 ∈ ℝ → (𝐵 − 1) ∈ ℝ)
41	1, 40	syl 17	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (𝐵 − 1) ∈ ℝ)
42	41	rexrd 10089	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (𝐵 − 1) ∈ ℝ^*)
43	42	adantr 481	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ ¬ -∞ < 𝑦) → (𝐵 − 1) ∈ ℝ^*)
44	43	3ad2antl1 1223	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝐴 ∧ (𝐵 − 1) < 𝑦) ∧ ¬ -∞ < 𝑦) → (𝐵 − 1) ∈ ℝ^*)
45	36	adantr 481	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝐴 ∧ (𝐵 − 1) < 𝑦) ∧ ¬ -∞ < 𝑦) → 𝑦 ∈ ℝ^*)
46	32	a1i 11	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝐴 ∧ (𝐵 − 1) < 𝑦) ∧ ¬ -∞ < 𝑦) → -∞ ∈ ℝ^*)
47		simpl3 1066	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝐴 ∧ (𝐵 − 1) < 𝑦) ∧ ¬ -∞ < 𝑦) → (𝐵 − 1) < 𝑦)
48		simpr 477	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝐴) ∧ ¬ -∞ < 𝑦) → ¬ -∞ < 𝑦)
49	35	adantr 481	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝐴) ∧ ¬ -∞ < 𝑦) → 𝑦 ∈ ℝ^*)
50	32	a1i 11	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝐴) ∧ ¬ -∞ < 𝑦) → -∞ ∈ ℝ^*)
51		xrlenlt 10103	. . . . . . . . . . . . . . 15 ⊢ ((𝑦 ∈ ℝ^* ∧ -∞ ∈ ℝ^*) → (𝑦 ≤ -∞ ↔ ¬ -∞ < 𝑦))
52	49, 50, 51	syl2anc 693	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝐴) ∧ ¬ -∞ < 𝑦) → (𝑦 ≤ -∞ ↔ ¬ -∞ < 𝑦))
53	48, 52	mpbird 247	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝐴) ∧ ¬ -∞ < 𝑦) → 𝑦 ≤ -∞)
54	53	3adantl3 1219	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝐴 ∧ (𝐵 − 1) < 𝑦) ∧ ¬ -∞ < 𝑦) → 𝑦 ≤ -∞)
55	44, 45, 46, 47, 54	xrltletrd 11992	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝐴 ∧ (𝐵 − 1) < 𝑦) ∧ ¬ -∞ < 𝑦) → (𝐵 − 1) < -∞)
56		nltmnf 11963	. . . . . . . . . . . . . 14 ⊢ ((𝐵 − 1) ∈ ℝ^* → ¬ (𝐵 − 1) < -∞)
57	42, 56	syl 17	. . . . . . . . . . . . 13 ⊢ (𝜑 → ¬ (𝐵 − 1) < -∞)
58	57	adantr 481	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ ¬ -∞ < 𝑦) → ¬ (𝐵 − 1) < -∞)
59	58	3ad2antl1 1223	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝐴 ∧ (𝐵 − 1) < 𝑦) ∧ ¬ -∞ < 𝑦) → ¬ (𝐵 − 1) < -∞)
60	55, 59	condan 835	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐴 ∧ (𝐵 − 1) < 𝑦) → -∞ < 𝑦)
61	34	adantr 481	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐴) → 𝐴 ⊆ ℝ^*)
62		simpr 477	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐴) → 𝑦 ∈ 𝐴)
63		supxrub 12154	. . . . . . . . . . . 12 ⊢ ((𝐴 ⊆ ℝ^* ∧ 𝑦 ∈ 𝐴) → 𝑦 ≤ sup(𝐴, ℝ^*, < ))
64	61, 62, 63	syl2anc 693	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐴) → 𝑦 ≤ sup(𝐴, ℝ^*, < ))
65	64	3adant3 1081	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐴 ∧ (𝐵 − 1) < 𝑦) → 𝑦 ≤ sup(𝐴, ℝ^*, < ))
66	33, 36, 39, 60, 65	xrltletrd 11992	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐴 ∧ (𝐵 − 1) < 𝑦) → -∞ < sup(𝐴, ℝ^*, < ))
67	66	3exp 1264	. . . . . . . 8 ⊢ (𝜑 → (𝑦 ∈ 𝐴 → ((𝐵 − 1) < 𝑦 → -∞ < sup(𝐴, ℝ^*, < ))))
68	67	adantr 481	. . . . . . 7 ⊢ ((𝜑 ∧ ¬ sup(𝐴, ℝ^, < ) = +∞) → (𝑦 ∈ 𝐴 → ((𝐵 − 1) < 𝑦 → -∞ < sup(𝐴, ℝ^, < ))))
69	68	rexlimdv 3030	. . . . . 6 ⊢ ((𝜑 ∧ ¬ sup(𝐴, ℝ^, < ) = +∞) → (∃𝑦 ∈ 𝐴 (𝐵 − 1) < 𝑦 → -∞ < sup(𝐴, ℝ^, < )))
70	31, 69	mpd 15	. . . . 5 ⊢ ((𝜑 ∧ ¬ sup(𝐴, ℝ^, < ) = +∞) → -∞ < sup(𝐴, ℝ^, < ))
71		simpr 477	. . . . . . 7 ⊢ ((𝜑 ∧ ¬ sup(𝐴, ℝ^, < ) = +∞) → ¬ sup(𝐴, ℝ^, < ) = +∞)
72		nltpnft 11995	. . . . . . . . 9 ⊢ (sup(𝐴, ℝ^, < ) ∈ ℝ^ → (sup(𝐴, ℝ^, < ) = +∞ ↔ ¬ sup(𝐴, ℝ^, < ) < +∞))
73	38, 72	syl 17	. . . . . . . 8 ⊢ (𝜑 → (sup(𝐴, ℝ^, < ) = +∞ ↔ ¬ sup(𝐴, ℝ^, < ) < +∞))
74	73	adantr 481	. . . . . . 7 ⊢ ((𝜑 ∧ ¬ sup(𝐴, ℝ^, < ) = +∞) → (sup(𝐴, ℝ^, < ) = +∞ ↔ ¬ sup(𝐴, ℝ^*, < ) < +∞))
75	71, 74	mtbid 314	. . . . . 6 ⊢ ((𝜑 ∧ ¬ sup(𝐴, ℝ^, < ) = +∞) → ¬ ¬ sup(𝐴, ℝ^, < ) < +∞)
76	75	notnotrd 128	. . . . 5 ⊢ ((𝜑 ∧ ¬ sup(𝐴, ℝ^, < ) = +∞) → sup(𝐴, ℝ^, < ) < +∞)
77	70, 76	jca 554	. . . 4 ⊢ ((𝜑 ∧ ¬ sup(𝐴, ℝ^, < ) = +∞) → (-∞ < sup(𝐴, ℝ^, < ) ∧ sup(𝐴, ℝ^*, < ) < +∞))
78	38	adantr 481	. . . . 5 ⊢ ((𝜑 ∧ ¬ sup(𝐴, ℝ^, < ) = +∞) → sup(𝐴, ℝ^, < ) ∈ ℝ^*)
79		xrrebnd 11999	. . . . 5 ⊢ (sup(𝐴, ℝ^, < ) ∈ ℝ^ → (sup(𝐴, ℝ^, < ) ∈ ℝ ↔ (-∞ < sup(𝐴, ℝ^, < ) ∧ sup(𝐴, ℝ^*, < ) < +∞)))
80	78, 79	syl 17	. . . 4 ⊢ ((𝜑 ∧ ¬ sup(𝐴, ℝ^, < ) = +∞) → (sup(𝐴, ℝ^, < ) ∈ ℝ ↔ (-∞ < sup(𝐴, ℝ^, < ) ∧ sup(𝐴, ℝ^, < ) < +∞)))
81	77, 80	mpbird 247	. . 3 ⊢ ((𝜑 ∧ ¬ sup(𝐴, ℝ^, < ) = +∞) → sup(𝐴, ℝ^, < ) ∈ ℝ)
82		simpl 473	. . . . 5 ⊢ (((𝜑 ∧ sup(𝐴, ℝ^, < ) ∈ ℝ) ∧ ¬ 𝐵 ≤ sup(𝐴, ℝ^, < )) → (𝜑 ∧ sup(𝐴, ℝ^*, < ) ∈ ℝ))
83		simpr 477	. . . . . 6 ⊢ (((𝜑 ∧ sup(𝐴, ℝ^, < ) ∈ ℝ) ∧ ¬ 𝐵 ≤ sup(𝐴, ℝ^, < )) → ¬ 𝐵 ≤ sup(𝐴, ℝ^*, < ))
84	82	simprd 479	. . . . . . 7 ⊢ (((𝜑 ∧ sup(𝐴, ℝ^, < ) ∈ ℝ) ∧ ¬ 𝐵 ≤ sup(𝐴, ℝ^, < )) → sup(𝐴, ℝ^*, < ) ∈ ℝ)
85	1	ad2antrr 762	. . . . . . 7 ⊢ (((𝜑 ∧ sup(𝐴, ℝ^, < ) ∈ ℝ) ∧ ¬ 𝐵 ≤ sup(𝐴, ℝ^, < )) → 𝐵 ∈ ℝ)
86	84, 85	ltnled 10184	. . . . . 6 ⊢ (((𝜑 ∧ sup(𝐴, ℝ^, < ) ∈ ℝ) ∧ ¬ 𝐵 ≤ sup(𝐴, ℝ^, < )) → (sup(𝐴, ℝ^, < ) < 𝐵 ↔ ¬ 𝐵 ≤ sup(𝐴, ℝ^, < )))
87	83, 86	mpbird 247	. . . . 5 ⊢ (((𝜑 ∧ sup(𝐴, ℝ^, < ) ∈ ℝ) ∧ ¬ 𝐵 ≤ sup(𝐴, ℝ^, < )) → sup(𝐴, ℝ^*, < ) < 𝐵)
88		simpll 790	. . . . . . 7 ⊢ (((𝜑 ∧ sup(𝐴, ℝ^, < ) ∈ ℝ) ∧ sup(𝐴, ℝ^, < ) < 𝐵) → 𝜑)
89	1	adantr 481	. . . . . . . . . 10 ⊢ ((𝜑 ∧ sup(𝐴, ℝ^*, < ) ∈ ℝ) → 𝐵 ∈ ℝ)
90		simpr 477	. . . . . . . . . 10 ⊢ ((𝜑 ∧ sup(𝐴, ℝ^, < ) ∈ ℝ) → sup(𝐴, ℝ^, < ) ∈ ℝ)
91	89, 90	resubcld 10458	. . . . . . . . 9 ⊢ ((𝜑 ∧ sup(𝐴, ℝ^, < ) ∈ ℝ) → (𝐵 − sup(𝐴, ℝ^, < )) ∈ ℝ)
92	91	adantr 481	. . . . . . . 8 ⊢ (((𝜑 ∧ sup(𝐴, ℝ^, < ) ∈ ℝ) ∧ sup(𝐴, ℝ^, < ) < 𝐵) → (𝐵 − sup(𝐴, ℝ^*, < )) ∈ ℝ)
93		simpr 477	. . . . . . . . 9 ⊢ (((𝜑 ∧ sup(𝐴, ℝ^, < ) ∈ ℝ) ∧ sup(𝐴, ℝ^, < ) < 𝐵) → sup(𝐴, ℝ^*, < ) < 𝐵)
94	90	adantr 481	. . . . . . . . . 10 ⊢ (((𝜑 ∧ sup(𝐴, ℝ^, < ) ∈ ℝ) ∧ sup(𝐴, ℝ^, < ) < 𝐵) → sup(𝐴, ℝ^*, < ) ∈ ℝ)
95	88, 1	syl 17	. . . . . . . . . 10 ⊢ (((𝜑 ∧ sup(𝐴, ℝ^, < ) ∈ ℝ) ∧ sup(𝐴, ℝ^, < ) < 𝐵) → 𝐵 ∈ ℝ)
96	94, 95	posdifd 10614	. . . . . . . . 9 ⊢ (((𝜑 ∧ sup(𝐴, ℝ^, < ) ∈ ℝ) ∧ sup(𝐴, ℝ^, < ) < 𝐵) → (sup(𝐴, ℝ^, < ) < 𝐵 ↔ 0 < (𝐵 − sup(𝐴, ℝ^, < ))))
97	93, 96	mpbid 222	. . . . . . . 8 ⊢ (((𝜑 ∧ sup(𝐴, ℝ^, < ) ∈ ℝ) ∧ sup(𝐴, ℝ^, < ) < 𝐵) → 0 < (𝐵 − sup(𝐴, ℝ^*, < )))
98	92, 97	elrpd 11869	. . . . . . 7 ⊢ (((𝜑 ∧ sup(𝐴, ℝ^, < ) ∈ ℝ) ∧ sup(𝐴, ℝ^, < ) < 𝐵) → (𝐵 − sup(𝐴, ℝ^*, < )) ∈ ℝ⁺)
99		ovex 6678	. . . . . . . 8 ⊢ (𝐵 − sup(𝐴, ℝ^*, < )) ∈ V
100		nfcv 2764	. . . . . . . . 9 ⊢ Ⅎ𝑥(𝐵 − sup(𝐴, ℝ^*, < ))
101		nfv 1843	. . . . . . . . . . 11 ⊢ Ⅎ𝑥(𝐵 − sup(𝐴, ℝ^*, < )) ∈ ℝ⁺
102	16, 101	nfan 1828	. . . . . . . . . 10 ⊢ Ⅎ𝑥(𝜑 ∧ (𝐵 − sup(𝐴, ℝ^*, < )) ∈ ℝ⁺)
103		nfv 1843	. . . . . . . . . 10 ⊢ Ⅎ𝑥∃𝑦 ∈ 𝐴 (𝐵 − (𝐵 − sup(𝐴, ℝ^*, < ))) < 𝑦
104	102, 103	nfim 1825	. . . . . . . . 9 ⊢ Ⅎ𝑥((𝜑 ∧ (𝐵 − sup(𝐴, ℝ^, < )) ∈ ℝ⁺) → ∃𝑦 ∈ 𝐴 (𝐵 − (𝐵 − sup(𝐴, ℝ^, < ))) < 𝑦)
105		eleq1 2689	. . . . . . . . . . 11 ⊢ (𝑥 = (𝐵 − sup(𝐴, ℝ^, < )) → (𝑥 ∈ ℝ⁺ ↔ (𝐵 − sup(𝐴, ℝ^, < )) ∈ ℝ⁺))
106	105	anbi2d 740	. . . . . . . . . 10 ⊢ (𝑥 = (𝐵 − sup(𝐴, ℝ^, < )) → ((𝜑 ∧ 𝑥 ∈ ℝ⁺) ↔ (𝜑 ∧ (𝐵 − sup(𝐴, ℝ^, < )) ∈ ℝ⁺)))
107		oveq2 6658	. . . . . . . . . . . 12 ⊢ (𝑥 = (𝐵 − sup(𝐴, ℝ^, < )) → (𝐵 − 𝑥) = (𝐵 − (𝐵 − sup(𝐴, ℝ^, < ))))
108	107	breq1d 4663	. . . . . . . . . . 11 ⊢ (𝑥 = (𝐵 − sup(𝐴, ℝ^, < )) → ((𝐵 − 𝑥) < 𝑦 ↔ (𝐵 − (𝐵 − sup(𝐴, ℝ^, < ))) < 𝑦))
109	108	rexbidv 3052	. . . . . . . . . 10 ⊢ (𝑥 = (𝐵 − sup(𝐴, ℝ^, < )) → (∃𝑦 ∈ 𝐴 (𝐵 − 𝑥) < 𝑦 ↔ ∃𝑦 ∈ 𝐴 (𝐵 − (𝐵 − sup(𝐴, ℝ^, < ))) < 𝑦))
110	106, 109	imbi12d 334	. . . . . . . . 9 ⊢ (𝑥 = (𝐵 − sup(𝐴, ℝ^, < )) → (((𝜑 ∧ 𝑥 ∈ ℝ⁺) → ∃𝑦 ∈ 𝐴 (𝐵 − 𝑥) < 𝑦) ↔ ((𝜑 ∧ (𝐵 − sup(𝐴, ℝ^, < )) ∈ ℝ⁺) → ∃𝑦 ∈ 𝐴 (𝐵 − (𝐵 − sup(𝐴, ℝ^*, < ))) < 𝑦)))
111	100, 104, 110, 27	vtoclgf 3264	. . . . . . . 8 ⊢ ((𝐵 − sup(𝐴, ℝ^, < )) ∈ V → ((𝜑 ∧ (𝐵 − sup(𝐴, ℝ^, < )) ∈ ℝ⁺) → ∃𝑦 ∈ 𝐴 (𝐵 − (𝐵 − sup(𝐴, ℝ^*, < ))) < 𝑦))
112	99, 111	ax-mp 5	. . . . . . 7 ⊢ ((𝜑 ∧ (𝐵 − sup(𝐴, ℝ^, < )) ∈ ℝ⁺) → ∃𝑦 ∈ 𝐴 (𝐵 − (𝐵 − sup(𝐴, ℝ^, < ))) < 𝑦)
113	88, 98, 112	syl2anc 693	. . . . . 6 ⊢ (((𝜑 ∧ sup(𝐴, ℝ^, < ) ∈ ℝ) ∧ sup(𝐴, ℝ^, < ) < 𝐵) → ∃𝑦 ∈ 𝐴 (𝐵 − (𝐵 − sup(𝐴, ℝ^*, < ))) < 𝑦)
114	1	recnd 10068	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐵 ∈ ℂ)
115	114	ad3antrrr 766	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ sup(𝐴, ℝ^, < ) ∈ ℝ) ∧ sup(𝐴, ℝ^, < ) < 𝐵) ∧ (𝐵 − (𝐵 − sup(𝐴, ℝ^*, < ))) < 𝑦) → 𝐵 ∈ ℂ)
116	90	recnd 10068	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ sup(𝐴, ℝ^, < ) ∈ ℝ) → sup(𝐴, ℝ^, < ) ∈ ℂ)
117	116	ad2antrr 762	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ sup(𝐴, ℝ^, < ) ∈ ℝ) ∧ sup(𝐴, ℝ^, < ) < 𝐵) ∧ (𝐵 − (𝐵 − sup(𝐴, ℝ^, < ))) < 𝑦) → sup(𝐴, ℝ^, < ) ∈ ℂ)
118	115, 117	nncand 10397	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ sup(𝐴, ℝ^, < ) ∈ ℝ) ∧ sup(𝐴, ℝ^, < ) < 𝐵) ∧ (𝐵 − (𝐵 − sup(𝐴, ℝ^, < ))) < 𝑦) → (𝐵 − (𝐵 − sup(𝐴, ℝ^, < ))) = sup(𝐴, ℝ^*, < ))
119	118	eqcomd 2628	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ sup(𝐴, ℝ^, < ) ∈ ℝ) ∧ sup(𝐴, ℝ^, < ) < 𝐵) ∧ (𝐵 − (𝐵 − sup(𝐴, ℝ^, < ))) < 𝑦) → sup(𝐴, ℝ^, < ) = (𝐵 − (𝐵 − sup(𝐴, ℝ^*, < ))))
120		simpr 477	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ sup(𝐴, ℝ^, < ) ∈ ℝ) ∧ sup(𝐴, ℝ^, < ) < 𝐵) ∧ (𝐵 − (𝐵 − sup(𝐴, ℝ^, < ))) < 𝑦) → (𝐵 − (𝐵 − sup(𝐴, ℝ^, < ))) < 𝑦)
121	119, 120	eqbrtrd 4675	. . . . . . . . 9 ⊢ ((((𝜑 ∧ sup(𝐴, ℝ^, < ) ∈ ℝ) ∧ sup(𝐴, ℝ^, < ) < 𝐵) ∧ (𝐵 − (𝐵 − sup(𝐴, ℝ^, < ))) < 𝑦) → sup(𝐴, ℝ^, < ) < 𝑦)
122	121	ex 450	. . . . . . . 8 ⊢ (((𝜑 ∧ sup(𝐴, ℝ^, < ) ∈ ℝ) ∧ sup(𝐴, ℝ^, < ) < 𝐵) → ((𝐵 − (𝐵 − sup(𝐴, ℝ^, < ))) < 𝑦 → sup(𝐴, ℝ^, < ) < 𝑦))
123	122	adantr 481	. . . . . . 7 ⊢ ((((𝜑 ∧ sup(𝐴, ℝ^, < ) ∈ ℝ) ∧ sup(𝐴, ℝ^, < ) < 𝐵) ∧ 𝑦 ∈ 𝐴) → ((𝐵 − (𝐵 − sup(𝐴, ℝ^, < ))) < 𝑦 → sup(𝐴, ℝ^, < ) < 𝑦))
124	123	reximdva 3017	. . . . . 6 ⊢ (((𝜑 ∧ sup(𝐴, ℝ^, < ) ∈ ℝ) ∧ sup(𝐴, ℝ^, < ) < 𝐵) → (∃𝑦 ∈ 𝐴 (𝐵 − (𝐵 − sup(𝐴, ℝ^, < ))) < 𝑦 → ∃𝑦 ∈ 𝐴 sup(𝐴, ℝ^, < ) < 𝑦))
125	113, 124	mpd 15	. . . . 5 ⊢ (((𝜑 ∧ sup(𝐴, ℝ^, < ) ∈ ℝ) ∧ sup(𝐴, ℝ^, < ) < 𝐵) → ∃𝑦 ∈ 𝐴 sup(𝐴, ℝ^*, < ) < 𝑦)
126	82, 87, 125	syl2anc 693	. . . 4 ⊢ (((𝜑 ∧ sup(𝐴, ℝ^, < ) ∈ ℝ) ∧ ¬ 𝐵 ≤ sup(𝐴, ℝ^, < )) → ∃𝑦 ∈ 𝐴 sup(𝐴, ℝ^*, < ) < 𝑦)
127	61, 37	syl 17	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐴) → sup(𝐴, ℝ^, < ) ∈ ℝ^)
128	35, 127	xrlenltd 10104	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐴) → (𝑦 ≤ sup(𝐴, ℝ^, < ) ↔ ¬ sup(𝐴, ℝ^, < ) < 𝑦))
129	64, 128	mpbid 222	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐴) → ¬ sup(𝐴, ℝ^*, < ) < 𝑦)
130	129	ralrimiva 2966	. . . . . 6 ⊢ (𝜑 → ∀𝑦 ∈ 𝐴 ¬ sup(𝐴, ℝ^*, < ) < 𝑦)
131		ralnex 2992	. . . . . 6 ⊢ (∀𝑦 ∈ 𝐴 ¬ sup(𝐴, ℝ^, < ) < 𝑦 ↔ ¬ ∃𝑦 ∈ 𝐴 sup(𝐴, ℝ^, < ) < 𝑦)
132	130, 131	sylib 208	. . . . 5 ⊢ (𝜑 → ¬ ∃𝑦 ∈ 𝐴 sup(𝐴, ℝ^*, < ) < 𝑦)
133	132	ad2antrr 762	. . . 4 ⊢ (((𝜑 ∧ sup(𝐴, ℝ^, < ) ∈ ℝ) ∧ ¬ 𝐵 ≤ sup(𝐴, ℝ^, < )) → ¬ ∃𝑦 ∈ 𝐴 sup(𝐴, ℝ^*, < ) < 𝑦)
134	126, 133	condan 835	. . 3 ⊢ ((𝜑 ∧ sup(𝐴, ℝ^, < ) ∈ ℝ) → 𝐵 ≤ sup(𝐴, ℝ^, < ))
135	13, 81, 134	syl2anc 693	. 2 ⊢ ((𝜑 ∧ ¬ sup(𝐴, ℝ^, < ) = +∞) → 𝐵 ≤ sup(𝐴, ℝ^, < ))
136	12, 135	pm2.61dan 832	1 ⊢ (𝜑 → 𝐵 ≤ sup(𝐴, ℝ^*, < ))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 196 ∧ wa 384 ∧ w3a 1037 = wceq 1483 Ⅎwnf 1708 ∈ wcel 1990 ∀wral 2912 ∃wrex 2913 Vcvv 3200 ⊆ wss 3574 class class class wbr 4653 (class class class)co 6650 supcsup 8346 ℂcc 9934 ℝcr 9935 0cc0 9936 1c1 9937 +∞cpnf 10071 -∞cmnf 10072 ℝ^*cxr 10073 < clt 10074 ≤ cle 10075 − cmin 10266 ℝ⁺crp 11832

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-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-er 7742 df-en 7956 df-dom 7957 df-sdom 7958 df-sup 8348 df-pnf 10076 df-mnf 10077 df-xr 10078 df-ltxr 10079 df-le 10080 df-sub 10268 df-neg 10269 df-rp 11833

This theorem is referenced by: suplesup 39555

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