supxrgere - Mathbox for Glauco Siliprandi

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Theorem supxrgere 45336

Description: If a real number can be approximated from below by members of a set, then it is less than 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 11227	. . . . . 6 ⊢ (𝐵 ∈ ℝ → 𝐵 ∈ ℝ^*)
3		pnfxr 11235	. . . . . . 7 ⊢ +∞ ∈ ℝ^*
4	3	a1i 11	. . . . . 6 ⊢ (𝐵 ∈ ℝ → +∞ ∈ ℝ^*)
5		ltpnf 13087	. . . . . 6 ⊢ (𝐵 ∈ ℝ → 𝐵 < +∞)
6	2, 4, 5	xrltled 13117	. . . . 5 ⊢ (𝐵 ∈ ℝ → 𝐵 ≤ +∞)
7	1, 6	syl 17	. . . 4 ⊢ (𝜑 → 𝐵 ≤ +∞)
8	7	adantr 480	. . 3 ⊢ ((𝜑 ∧ sup(𝐴, ℝ^*, < ) = +∞) → 𝐵 ≤ +∞)
9		id 22	. . . . 5 ⊢ (sup(𝐴, ℝ^, < ) = +∞ → sup(𝐴, ℝ^, < ) = +∞)
10	9	eqcomd 2736	. . . 4 ⊢ (sup(𝐴, ℝ^, < ) = +∞ → +∞ = sup(𝐴, ℝ^, < ))
11	10	adantl 481	. . 3 ⊢ ((𝜑 ∧ sup(𝐴, ℝ^, < ) = +∞) → +∞ = sup(𝐴, ℝ^, < ))
12	8, 11	breqtrd 5136	. 2 ⊢ ((𝜑 ∧ sup(𝐴, ℝ^, < ) = +∞) → 𝐵 ≤ sup(𝐴, ℝ^, < ))
13		simpl 482	. . 3 ⊢ ((𝜑 ∧ ¬ sup(𝐴, ℝ^*, < ) = +∞) → 𝜑)
14		1rp 12962	. . . . . . . 8 ⊢ 1 ∈ ℝ⁺
15		nfcv 2892	. . . . . . . . . 10 ⊢ Ⅎ𝑥1
16		supxrgere.xph	. . . . . . . . . . . 12 ⊢ Ⅎ𝑥𝜑
17		nfv 1914	. . . . . . . . . . . 12 ⊢ Ⅎ𝑥1 ∈ ℝ⁺
18	16, 17	nfan 1899	. . . . . . . . . . 11 ⊢ Ⅎ𝑥(𝜑 ∧ 1 ∈ ℝ⁺)
19		nfv 1914	. . . . . . . . . . 11 ⊢ Ⅎ𝑥∃𝑦 ∈ 𝐴 (𝐵 − 1) < 𝑦
20	18, 19	nfim 1896	. . . . . . . . . 10 ⊢ Ⅎ𝑥((𝜑 ∧ 1 ∈ ℝ⁺) → ∃𝑦 ∈ 𝐴 (𝐵 − 1) < 𝑦)
21		eleq1 2817	. . . . . . . . . . . 12 ⊢ (𝑥 = 1 → (𝑥 ∈ ℝ⁺ ↔ 1 ∈ ℝ⁺))
22	21	anbi2d 630	. . . . . . . . . . 11 ⊢ (𝑥 = 1 → ((𝜑 ∧ 𝑥 ∈ ℝ⁺) ↔ (𝜑 ∧ 1 ∈ ℝ⁺)))
23		oveq2 7398	. . . . . . . . . . . . 13 ⊢ (𝑥 = 1 → (𝐵 − 𝑥) = (𝐵 − 1))
24	23	breq1d 5120	. . . . . . . . . . . 12 ⊢ (𝑥 = 1 → ((𝐵 − 𝑥) < 𝑦 ↔ (𝐵 − 1) < 𝑦))
25	24	rexbidv 3158	. . . . . . . . . . 11 ⊢ (𝑥 = 1 → (∃𝑦 ∈ 𝐴 (𝐵 − 𝑥) < 𝑦 ↔ ∃𝑦 ∈ 𝐴 (𝐵 − 1) < 𝑦))
26	22, 25	imbi12d 344	. . . . . . . . . 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 3538	. . . . . . . . 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 691	. . . . . . 7 ⊢ (𝜑 → ∃𝑦 ∈ 𝐴 (𝐵 − 1) < 𝑦)
31	30	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ ¬ sup(𝐴, ℝ^*, < ) = +∞) → ∃𝑦 ∈ 𝐴 (𝐵 − 1) < 𝑦)
32		mnfxr 11238	. . . . . . . . . . 11 ⊢ -∞ ∈ ℝ^*
33	32	a1i 11	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐴 ∧ (𝐵 − 1) < 𝑦) → -∞ ∈ ℝ^*)
34		supxrgere.a	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐴 ⊆ ℝ^*)
35	34	sselda 3949	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐴) → 𝑦 ∈ ℝ^*)
36	35	3adant3 1132	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐴 ∧ (𝐵 − 1) < 𝑦) → 𝑦 ∈ ℝ^*)
37		supxrcl 13282	. . . . . . . . . . . 12 ⊢ (𝐴 ⊆ ℝ^* → sup(𝐴, ℝ^, < ) ∈ ℝ^)
38	34, 37	syl 17	. . . . . . . . . . 11 ⊢ (𝜑 → sup(𝐴, ℝ^, < ) ∈ ℝ^)
39	38	3ad2ant1 1133	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐴 ∧ (𝐵 − 1) < 𝑦) → sup(𝐴, ℝ^, < ) ∈ ℝ^)
40		peano2rem 11496	. . . . . . . . . . . . . . . 16 ⊢ (𝐵 ∈ ℝ → (𝐵 − 1) ∈ ℝ)
41	1, 40	syl 17	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (𝐵 − 1) ∈ ℝ)
42	41	rexrd 11231	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (𝐵 − 1) ∈ ℝ^*)
43	42	adantr 480	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ ¬ -∞ < 𝑦) → (𝐵 − 1) ∈ ℝ^*)
44	43	3ad2antl1 1186	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝐴 ∧ (𝐵 − 1) < 𝑦) ∧ ¬ -∞ < 𝑦) → (𝐵 − 1) ∈ ℝ^*)
45	36	adantr 480	. . . . . . . . . . . 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 1194	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝐴 ∧ (𝐵 − 1) < 𝑦) ∧ ¬ -∞ < 𝑦) → (𝐵 − 1) < 𝑦)
48		simpr 484	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝐴) ∧ ¬ -∞ < 𝑦) → ¬ -∞ < 𝑦)
49	35	adantr 480	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝐴) ∧ ¬ -∞ < 𝑦) → 𝑦 ∈ ℝ^*)
50	32	a1i 11	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝐴) ∧ ¬ -∞ < 𝑦) → -∞ ∈ ℝ^*)
51		xrlenlt 11246	. . . . . . . . . . . . . . 15 ⊢ ((𝑦 ∈ ℝ^* ∧ -∞ ∈ ℝ^*) → (𝑦 ≤ -∞ ↔ ¬ -∞ < 𝑦))
52	49, 50, 51	syl2anc 584	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝐴) ∧ ¬ -∞ < 𝑦) → (𝑦 ≤ -∞ ↔ ¬ -∞ < 𝑦))
53	48, 52	mpbird 257	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝐴) ∧ ¬ -∞ < 𝑦) → 𝑦 ≤ -∞)
54	53	3adantl3 1169	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝐴 ∧ (𝐵 − 1) < 𝑦) ∧ ¬ -∞ < 𝑦) → 𝑦 ≤ -∞)
55	44, 45, 46, 47, 54	xrltletrd 13128	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝐴 ∧ (𝐵 − 1) < 𝑦) ∧ ¬ -∞ < 𝑦) → (𝐵 − 1) < -∞)
56		nltmnf 13096	. . . . . . . . . . . . . 14 ⊢ ((𝐵 − 1) ∈ ℝ^* → ¬ (𝐵 − 1) < -∞)
57	42, 56	syl 17	. . . . . . . . . . . . 13 ⊢ (𝜑 → ¬ (𝐵 − 1) < -∞)
58	57	adantr 480	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ ¬ -∞ < 𝑦) → ¬ (𝐵 − 1) < -∞)
59	58	3ad2antl1 1186	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝐴 ∧ (𝐵 − 1) < 𝑦) ∧ ¬ -∞ < 𝑦) → ¬ (𝐵 − 1) < -∞)
60	55, 59	condan 817	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐴 ∧ (𝐵 − 1) < 𝑦) → -∞ < 𝑦)
61	34	adantr 480	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐴) → 𝐴 ⊆ ℝ^*)
62		simpr 484	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐴) → 𝑦 ∈ 𝐴)
63		supxrub 13291	. . . . . . . . . . . 12 ⊢ ((𝐴 ⊆ ℝ^* ∧ 𝑦 ∈ 𝐴) → 𝑦 ≤ sup(𝐴, ℝ^*, < ))
64	61, 62, 63	syl2anc 584	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐴) → 𝑦 ≤ sup(𝐴, ℝ^*, < ))
65	64	3adant3 1132	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐴 ∧ (𝐵 − 1) < 𝑦) → 𝑦 ≤ sup(𝐴, ℝ^*, < ))
66	33, 36, 39, 60, 65	xrltletrd 13128	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐴 ∧ (𝐵 − 1) < 𝑦) → -∞ < sup(𝐴, ℝ^*, < ))
67	66	3exp 1119	. . . . . . . 8 ⊢ (𝜑 → (𝑦 ∈ 𝐴 → ((𝐵 − 1) < 𝑦 → -∞ < sup(𝐴, ℝ^*, < ))))
68	67	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ ¬ sup(𝐴, ℝ^, < ) = +∞) → (𝑦 ∈ 𝐴 → ((𝐵 − 1) < 𝑦 → -∞ < sup(𝐴, ℝ^, < ))))
69	68	rexlimdv 3133	. . . . . 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 484	. . . . . . 7 ⊢ ((𝜑 ∧ ¬ sup(𝐴, ℝ^, < ) = +∞) → ¬ sup(𝐴, ℝ^, < ) = +∞)
72		nltpnft 13131	. . . . . . . . 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 480	. . . . . . 7 ⊢ ((𝜑 ∧ ¬ sup(𝐴, ℝ^, < ) = +∞) → (sup(𝐴, ℝ^, < ) = +∞ ↔ ¬ sup(𝐴, ℝ^*, < ) < +∞))
75	71, 74	mtbid 324	. . . . . 6 ⊢ ((𝜑 ∧ ¬ sup(𝐴, ℝ^, < ) = +∞) → ¬ ¬ sup(𝐴, ℝ^, < ) < +∞)
76	75	notnotrd 133	. . . . 5 ⊢ ((𝜑 ∧ ¬ sup(𝐴, ℝ^, < ) = +∞) → sup(𝐴, ℝ^, < ) < +∞)
77	70, 76	jca 511	. . . 4 ⊢ ((𝜑 ∧ ¬ sup(𝐴, ℝ^, < ) = +∞) → (-∞ < sup(𝐴, ℝ^, < ) ∧ sup(𝐴, ℝ^*, < ) < +∞))
78	38	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ ¬ sup(𝐴, ℝ^, < ) = +∞) → sup(𝐴, ℝ^, < ) ∈ ℝ^*)
79		xrrebnd 13135	. . . . 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 257	. . 3 ⊢ ((𝜑 ∧ ¬ sup(𝐴, ℝ^, < ) = +∞) → sup(𝐴, ℝ^, < ) ∈ ℝ)
82		simpl 482	. . . . 5 ⊢ (((𝜑 ∧ sup(𝐴, ℝ^, < ) ∈ ℝ) ∧ ¬ 𝐵 ≤ sup(𝐴, ℝ^, < )) → (𝜑 ∧ sup(𝐴, ℝ^*, < ) ∈ ℝ))
83		simpr 484	. . . . . 6 ⊢ (((𝜑 ∧ sup(𝐴, ℝ^, < ) ∈ ℝ) ∧ ¬ 𝐵 ≤ sup(𝐴, ℝ^, < )) → ¬ 𝐵 ≤ sup(𝐴, ℝ^*, < ))
84	82	simprd 495	. . . . . . 7 ⊢ (((𝜑 ∧ sup(𝐴, ℝ^, < ) ∈ ℝ) ∧ ¬ 𝐵 ≤ sup(𝐴, ℝ^, < )) → sup(𝐴, ℝ^*, < ) ∈ ℝ)
85	1	ad2antrr 726	. . . . . . 7 ⊢ (((𝜑 ∧ sup(𝐴, ℝ^, < ) ∈ ℝ) ∧ ¬ 𝐵 ≤ sup(𝐴, ℝ^, < )) → 𝐵 ∈ ℝ)
86	84, 85	ltnled 11328	. . . . . 6 ⊢ (((𝜑 ∧ sup(𝐴, ℝ^, < ) ∈ ℝ) ∧ ¬ 𝐵 ≤ sup(𝐴, ℝ^, < )) → (sup(𝐴, ℝ^, < ) < 𝐵 ↔ ¬ 𝐵 ≤ sup(𝐴, ℝ^, < )))
87	83, 86	mpbird 257	. . . . 5 ⊢ (((𝜑 ∧ sup(𝐴, ℝ^, < ) ∈ ℝ) ∧ ¬ 𝐵 ≤ sup(𝐴, ℝ^, < )) → sup(𝐴, ℝ^*, < ) < 𝐵)
88		simpll 766	. . . . . . 7 ⊢ (((𝜑 ∧ sup(𝐴, ℝ^, < ) ∈ ℝ) ∧ sup(𝐴, ℝ^, < ) < 𝐵) → 𝜑)
89	1	adantr 480	. . . . . . . . . 10 ⊢ ((𝜑 ∧ sup(𝐴, ℝ^*, < ) ∈ ℝ) → 𝐵 ∈ ℝ)
90		simpr 484	. . . . . . . . . 10 ⊢ ((𝜑 ∧ sup(𝐴, ℝ^, < ) ∈ ℝ) → sup(𝐴, ℝ^, < ) ∈ ℝ)
91	89, 90	resubcld 11613	. . . . . . . . 9 ⊢ ((𝜑 ∧ sup(𝐴, ℝ^, < ) ∈ ℝ) → (𝐵 − sup(𝐴, ℝ^, < )) ∈ ℝ)
92	91	adantr 480	. . . . . . . 8 ⊢ (((𝜑 ∧ sup(𝐴, ℝ^, < ) ∈ ℝ) ∧ sup(𝐴, ℝ^, < ) < 𝐵) → (𝐵 − sup(𝐴, ℝ^*, < )) ∈ ℝ)
93		simpr 484	. . . . . . . . 9 ⊢ (((𝜑 ∧ sup(𝐴, ℝ^, < ) ∈ ℝ) ∧ sup(𝐴, ℝ^, < ) < 𝐵) → sup(𝐴, ℝ^*, < ) < 𝐵)
94	90	adantr 480	. . . . . . . . . 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 11772	. . . . . . . . 9 ⊢ (((𝜑 ∧ sup(𝐴, ℝ^, < ) ∈ ℝ) ∧ sup(𝐴, ℝ^, < ) < 𝐵) → (sup(𝐴, ℝ^, < ) < 𝐵 ↔ 0 < (𝐵 − sup(𝐴, ℝ^, < ))))
97	93, 96	mpbid 232	. . . . . . . 8 ⊢ (((𝜑 ∧ sup(𝐴, ℝ^, < ) ∈ ℝ) ∧ sup(𝐴, ℝ^, < ) < 𝐵) → 0 < (𝐵 − sup(𝐴, ℝ^*, < )))
98	92, 97	elrpd 12999	. . . . . . 7 ⊢ (((𝜑 ∧ sup(𝐴, ℝ^, < ) ∈ ℝ) ∧ sup(𝐴, ℝ^, < ) < 𝐵) → (𝐵 − sup(𝐴, ℝ^*, < )) ∈ ℝ⁺)
99		ovex 7423	. . . . . . . 8 ⊢ (𝐵 − sup(𝐴, ℝ^*, < )) ∈ V
100		nfcv 2892	. . . . . . . . 9 ⊢ Ⅎ𝑥(𝐵 − sup(𝐴, ℝ^*, < ))
101		nfv 1914	. . . . . . . . . . 11 ⊢ Ⅎ𝑥(𝐵 − sup(𝐴, ℝ^*, < )) ∈ ℝ⁺
102	16, 101	nfan 1899	. . . . . . . . . 10 ⊢ Ⅎ𝑥(𝜑 ∧ (𝐵 − sup(𝐴, ℝ^*, < )) ∈ ℝ⁺)
103		nfv 1914	. . . . . . . . . 10 ⊢ Ⅎ𝑥∃𝑦 ∈ 𝐴 (𝐵 − (𝐵 − sup(𝐴, ℝ^*, < ))) < 𝑦
104	102, 103	nfim 1896	. . . . . . . . 9 ⊢ Ⅎ𝑥((𝜑 ∧ (𝐵 − sup(𝐴, ℝ^, < )) ∈ ℝ⁺) → ∃𝑦 ∈ 𝐴 (𝐵 − (𝐵 − sup(𝐴, ℝ^, < ))) < 𝑦)
105		eleq1 2817	. . . . . . . . . . 11 ⊢ (𝑥 = (𝐵 − sup(𝐴, ℝ^, < )) → (𝑥 ∈ ℝ⁺ ↔ (𝐵 − sup(𝐴, ℝ^, < )) ∈ ℝ⁺))
106	105	anbi2d 630	. . . . . . . . . 10 ⊢ (𝑥 = (𝐵 − sup(𝐴, ℝ^, < )) → ((𝜑 ∧ 𝑥 ∈ ℝ⁺) ↔ (𝜑 ∧ (𝐵 − sup(𝐴, ℝ^, < )) ∈ ℝ⁺)))
107		oveq2 7398	. . . . . . . . . . . 12 ⊢ (𝑥 = (𝐵 − sup(𝐴, ℝ^, < )) → (𝐵 − 𝑥) = (𝐵 − (𝐵 − sup(𝐴, ℝ^, < ))))
108	107	breq1d 5120	. . . . . . . . . . 11 ⊢ (𝑥 = (𝐵 − sup(𝐴, ℝ^, < )) → ((𝐵 − 𝑥) < 𝑦 ↔ (𝐵 − (𝐵 − sup(𝐴, ℝ^, < ))) < 𝑦))
109	108	rexbidv 3158	. . . . . . . . . 10 ⊢ (𝑥 = (𝐵 − sup(𝐴, ℝ^, < )) → (∃𝑦 ∈ 𝐴 (𝐵 − 𝑥) < 𝑦 ↔ ∃𝑦 ∈ 𝐴 (𝐵 − (𝐵 − sup(𝐴, ℝ^, < ))) < 𝑦))
110	106, 109	imbi12d 344	. . . . . . . . 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 3538	. . . . . . . 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 584	. . . . . 6 ⊢ (((𝜑 ∧ sup(𝐴, ℝ^, < ) ∈ ℝ) ∧ sup(𝐴, ℝ^, < ) < 𝐵) → ∃𝑦 ∈ 𝐴 (𝐵 − (𝐵 − sup(𝐴, ℝ^*, < ))) < 𝑦)
114	1	recnd 11209	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐵 ∈ ℂ)
115	114	ad3antrrr 730	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ sup(𝐴, ℝ^, < ) ∈ ℝ) ∧ sup(𝐴, ℝ^, < ) < 𝐵) ∧ (𝐵 − (𝐵 − sup(𝐴, ℝ^*, < ))) < 𝑦) → 𝐵 ∈ ℂ)
116	90	recnd 11209	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ sup(𝐴, ℝ^, < ) ∈ ℝ) → sup(𝐴, ℝ^, < ) ∈ ℂ)
117	116	ad2antrr 726	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ sup(𝐴, ℝ^, < ) ∈ ℝ) ∧ sup(𝐴, ℝ^, < ) < 𝐵) ∧ (𝐵 − (𝐵 − sup(𝐴, ℝ^, < ))) < 𝑦) → sup(𝐴, ℝ^, < ) ∈ ℂ)
118	115, 117	nncand 11545	. . . . . . . . . . 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 2736	. . . . . . . . . 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 484	. . . . . . . . . 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 5132	. . . . . . . . 9 ⊢ ((((𝜑 ∧ sup(𝐴, ℝ^, < ) ∈ ℝ) ∧ sup(𝐴, ℝ^, < ) < 𝐵) ∧ (𝐵 − (𝐵 − sup(𝐴, ℝ^, < ))) < 𝑦) → sup(𝐴, ℝ^, < ) < 𝑦)
122	121	ex 412	. . . . . . . 8 ⊢ (((𝜑 ∧ sup(𝐴, ℝ^, < ) ∈ ℝ) ∧ sup(𝐴, ℝ^, < ) < 𝐵) → ((𝐵 − (𝐵 − sup(𝐴, ℝ^, < ))) < 𝑦 → sup(𝐴, ℝ^, < ) < 𝑦))
123	122	adantr 480	. . . . . . 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 3147	. . . . . 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 584	. . . 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 11247	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐴) → (𝑦 ≤ sup(𝐴, ℝ^, < ) ↔ ¬ sup(𝐴, ℝ^, < ) < 𝑦))
129	64, 128	mpbid 232	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐴) → ¬ sup(𝐴, ℝ^*, < ) < 𝑦)
130	129	ralrimiva 3126	. . . . . 6 ⊢ (𝜑 → ∀𝑦 ∈ 𝐴 ¬ sup(𝐴, ℝ^*, < ) < 𝑦)
131		ralnex 3056	. . . . . 6 ⊢ (∀𝑦 ∈ 𝐴 ¬ sup(𝐴, ℝ^, < ) < 𝑦 ↔ ¬ ∃𝑦 ∈ 𝐴 sup(𝐴, ℝ^, < ) < 𝑦)
132	130, 131	sylib 218	. . . . 5 ⊢ (𝜑 → ¬ ∃𝑦 ∈ 𝐴 sup(𝐴, ℝ^*, < ) < 𝑦)
133	132	ad2antrr 726	. . . 4 ⊢ (((𝜑 ∧ sup(𝐴, ℝ^, < ) ∈ ℝ) ∧ ¬ 𝐵 ≤ sup(𝐴, ℝ^, < )) → ¬ ∃𝑦 ∈ 𝐴 sup(𝐴, ℝ^*, < ) < 𝑦)
134	126, 133	condan 817	. . 3 ⊢ ((𝜑 ∧ sup(𝐴, ℝ^, < ) ∈ ℝ) → 𝐵 ≤ sup(𝐴, ℝ^, < ))
135	13, 81, 134	syl2anc 584	. 2 ⊢ ((𝜑 ∧ ¬ sup(𝐴, ℝ^, < ) = +∞) → 𝐵 ≤ sup(𝐴, ℝ^, < ))
136	12, 135	pm2.61dan 812	1 ⊢ (𝜑 → 𝐵 ≤ sup(𝐴, ℝ^*, < ))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 206 ∧ wa 395 ∧ w3a 1086 = wceq 1540 Ⅎwnf 1783 ∈ wcel 2109 ∀wral 3045 ∃wrex 3054 Vcvv 3450 ⊆ wss 3917 class class class wbr 5110 (class class class)co 7390 supcsup 9398 ℂcc 11073 ℝcr 11074 0cc0 11075 1c1 11076 +∞cpnf 11212 -∞cmnf 11213 ℝ^*cxr 11214 < clt 11215 ≤ cle 11216 − cmin 11412 ℝ⁺crp 12958

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 2702 ax-sep 5254 ax-nul 5264 ax-pow 5323 ax-pr 5390 ax-un 7714 ax-cnex 11131 ax-resscn 11132 ax-1cn 11133 ax-icn 11134 ax-addcl 11135 ax-addrcl 11136 ax-mulcl 11137 ax-mulrcl 11138 ax-mulcom 11139 ax-addass 11140 ax-mulass 11141 ax-distr 11142 ax-i2m1 11143 ax-1ne0 11144 ax-1rid 11145 ax-rnegex 11146 ax-rrecex 11147 ax-cnre 11148 ax-pre-lttri 11149 ax-pre-lttrn 11150 ax-pre-ltadd 11151 ax-pre-mulgt0 11152 ax-pre-sup 11153

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 2534 df-eu 2563 df-clab 2709 df-cleq 2722 df-clel 2804 df-nfc 2879 df-ne 2927 df-nel 3031 df-ral 3046 df-rex 3055 df-rmo 3356 df-reu 3357 df-rab 3409 df-v 3452 df-sbc 3757 df-csb 3866 df-dif 3920 df-un 3922 df-in 3924 df-ss 3934 df-nul 4300 df-if 4492 df-pw 4568 df-sn 4593 df-pr 4595 df-op 4599 df-uni 4875 df-br 5111 df-opab 5173 df-mpt 5192 df-id 5536 df-po 5549 df-so 5550 df-xp 5647 df-rel 5648 df-cnv 5649 df-co 5650 df-dm 5651 df-rn 5652 df-res 5653 df-ima 5654 df-iota 6467 df-fun 6516 df-fn 6517 df-f 6518 df-f1 6519 df-fo 6520 df-f1o 6521 df-fv 6522 df-riota 7347 df-ov 7393 df-oprab 7394 df-mpo 7395 df-er 8674 df-en 8922 df-dom 8923 df-sdom 8924 df-sup 9400 df-pnf 11217 df-mnf 11218 df-xr 11219 df-ltxr 11220 df-le 11221 df-sub 11414 df-neg 11415 df-rp 12959

This theorem is referenced by: suplesup 45342

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