supxrgere - Mathbox for Glauco Siliprandi

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Theorem supxrgere 46107

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 11270	. . . . . 6 ⊢ (𝐵 ∈ ℝ → 𝐵 ∈ ℝ^*)
3		pnfxr 11278	. . . . . . 7 ⊢ +∞ ∈ ℝ^*
4	3	a1i 11	. . . . . 6 ⊢ (𝐵 ∈ ℝ → +∞ ∈ ℝ^*)
5		ltpnf 13161	. . . . . 6 ⊢ (𝐵 ∈ ℝ → 𝐵 < +∞)
6	2, 4, 5	xrltled 13191	. . . . 5 ⊢ (𝐵 ∈ ℝ → 𝐵 ≤ +∞)
7	1, 6	syl 18	. . . 4 ⊢ (𝜑 → 𝐵 ≤ +∞)
8	7	adantr 486	. . 3 ⊢ ((𝜑 ∧ sup(𝐴, ℝ^*, < ) = +∞) → 𝐵 ≤ +∞)
9		id 23	. . . . 5 ⊢ (sup(𝐴, ℝ^, < ) = +∞ → sup(𝐴, ℝ^, < ) = +∞)
10	9	eqcomd 2771	. . . 4 ⊢ (sup(𝐴, ℝ^, < ) = +∞ → +∞ = sup(𝐴, ℝ^, < ))
11	10	adantl 487	. . 3 ⊢ ((𝜑 ∧ sup(𝐴, ℝ^, < ) = +∞) → +∞ = sup(𝐴, ℝ^, < ))
12	8, 11	breqtrd 5139	. 2 ⊢ ((𝜑 ∧ sup(𝐴, ℝ^, < ) = +∞) → 𝐵 ≤ sup(𝐴, ℝ^, < ))
13		simpl 488	. . 3 ⊢ ((𝜑 ∧ ¬ sup(𝐴, ℝ^*, < ) = +∞) → 𝜑)
14		1rp 13036	. . . . . . . 8 ⊢ 1 ∈ ℝ⁺
15		nfcv 2927	. . . . . . . . . 10 ⊢ Ⅎ𝑥1
16		supxrgere.xph	. . . . . . . . . . . 12 ⊢ Ⅎ𝑥𝜑
17		nfv 1947	. . . . . . . . . . . 12 ⊢ Ⅎ𝑥1 ∈ ℝ⁺
18	16, 17	nfan 1932	. . . . . . . . . . 11 ⊢ Ⅎ𝑥(𝜑 ∧ 1 ∈ ℝ⁺)
19		nfv 1947	. . . . . . . . . . 11 ⊢ Ⅎ𝑥∃𝑦 ∈ 𝐴 (𝐵 − 1) < 𝑦
20	18, 19	nfim 1929	. . . . . . . . . 10 ⊢ Ⅎ𝑥((𝜑 ∧ 1 ∈ ℝ⁺) → ∃𝑦 ∈ 𝐴 (𝐵 − 1) < 𝑦)
21		eleq1 2853	. . . . . . . . . . . 12 ⊢ (𝑥 = 1 → (𝑥 ∈ ℝ⁺ ↔ 1 ∈ ℝ⁺))
22	21	anbi2d 642	. . . . . . . . . . 11 ⊢ (𝑥 = 1 → ((𝜑 ∧ 𝑥 ∈ ℝ⁺) ↔ (𝜑 ∧ 1 ∈ ℝ⁺)))
23		oveq2 7427	. . . . . . . . . . . . 13 ⊢ (𝑥 = 1 → (𝐵 − 𝑥) = (𝐵 − 1))
24	23	breq1d 5121	. . . . . . . . . . . 12 ⊢ (𝑥 = 1 → ((𝐵 − 𝑥) < 𝑦 ↔ (𝐵 − 1) < 𝑦))
25	24	rexbidv 3191	. . . . . . . . . . 11 ⊢ (𝑥 = 1 → (∃𝑦 ∈ 𝐴 (𝐵 − 𝑥) < 𝑦 ↔ ∃𝑦 ∈ 𝐴 (𝐵 − 1) < 𝑦))
26	22, 25	imbi12d 347	. . . . . . . . . 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 3536	. . . . . . . . 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 704	. . . . . . 7 ⊢ (𝜑 → ∃𝑦 ∈ 𝐴 (𝐵 − 1) < 𝑦)
31	30	adantr 486	. . . . . 6 ⊢ ((𝜑 ∧ ¬ sup(𝐴, ℝ^*, < ) = +∞) → ∃𝑦 ∈ 𝐴 (𝐵 − 1) < 𝑦)
32		mnfxr 11281	. . . . . . . . . . 11 ⊢ -∞ ∈ ℝ^*
33	32	a1i 11	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐴 ∧ (𝐵 − 1) < 𝑦) → -∞ ∈ ℝ^*)
34		supxrgere.a	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐴 ⊆ ℝ^*)
35	34	sselda 3938	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐴) → 𝑦 ∈ ℝ^*)
36	35	3adant3 1150	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐴 ∧ (𝐵 − 1) < 𝑦) → 𝑦 ∈ ℝ^*)
37		supxrcl 13357	. . . . . . . . . . . 12 ⊢ (𝐴 ⊆ ℝ^* → sup(𝐴, ℝ^, < ) ∈ ℝ^)
38	34, 37	syl 18	. . . . . . . . . . 11 ⊢ (𝜑 → sup(𝐴, ℝ^, < ) ∈ ℝ^)
39	38	3ad2ant1 1151	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐴 ∧ (𝐵 − 1) < 𝑦) → sup(𝐴, ℝ^, < ) ∈ ℝ^)
40		peano2rem 11540	. . . . . . . . . . . . . . . 16 ⊢ (𝐵 ∈ ℝ → (𝐵 − 1) ∈ ℝ)
41	1, 40	syl 18	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (𝐵 − 1) ∈ ℝ)
42	41	rexrd 11274	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (𝐵 − 1) ∈ ℝ^*)
43	42	adantr 486	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ ¬ -∞ < 𝑦) → (𝐵 − 1) ∈ ℝ^*)
44	43	3ad2antl1 1204	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝐴 ∧ (𝐵 − 1) < 𝑦) ∧ ¬ -∞ < 𝑦) → (𝐵 − 1) ∈ ℝ^*)
45	36	adantr 486	. . . . . . . . . . . 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 1212	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝐴 ∧ (𝐵 − 1) < 𝑦) ∧ ¬ -∞ < 𝑦) → (𝐵 − 1) < 𝑦)
48		simpr 490	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝐴) ∧ ¬ -∞ < 𝑦) → ¬ -∞ < 𝑦)
49	35	adantr 486	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝐴) ∧ ¬ -∞ < 𝑦) → 𝑦 ∈ ℝ^*)
50	32	a1i 11	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝐴) ∧ ¬ -∞ < 𝑦) → -∞ ∈ ℝ^*)
51		xrlenlt 11289	. . . . . . . . . . . . . . 15 ⊢ ((𝑦 ∈ ℝ^* ∧ -∞ ∈ ℝ^*) → (𝑦 ≤ -∞ ↔ ¬ -∞ < 𝑦))
52	49, 50, 51	syl2anc 596	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝐴) ∧ ¬ -∞ < 𝑦) → (𝑦 ≤ -∞ ↔ ¬ -∞ < 𝑦))
53	48, 52	mpbird 260	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝐴) ∧ ¬ -∞ < 𝑦) → 𝑦 ≤ -∞)
54	53	3adantl3 1187	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝐴 ∧ (𝐵 − 1) < 𝑦) ∧ ¬ -∞ < 𝑦) → 𝑦 ≤ -∞)
55	44, 45, 46, 47, 54	xrltletrd 13202	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝐴 ∧ (𝐵 − 1) < 𝑦) ∧ ¬ -∞ < 𝑦) → (𝐵 − 1) < -∞)
56		nltmnf 13170	. . . . . . . . . . . . . 14 ⊢ ((𝐵 − 1) ∈ ℝ^* → ¬ (𝐵 − 1) < -∞)
57	42, 56	syl 18	. . . . . . . . . . . . 13 ⊢ (𝜑 → ¬ (𝐵 − 1) < -∞)
58	57	adantr 486	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ ¬ -∞ < 𝑦) → ¬ (𝐵 − 1) < -∞)
59	58	3ad2antl1 1204	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝐴 ∧ (𝐵 − 1) < 𝑦) ∧ ¬ -∞ < 𝑦) → ¬ (𝐵 − 1) < -∞)
60	55, 59	condan 830	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐴 ∧ (𝐵 − 1) < 𝑦) → -∞ < 𝑦)
61	34	adantr 486	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐴) → 𝐴 ⊆ ℝ^*)
62		simpr 490	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐴) → 𝑦 ∈ 𝐴)
63		supxrub 13366	. . . . . . . . . . . 12 ⊢ ((𝐴 ⊆ ℝ^* ∧ 𝑦 ∈ 𝐴) → 𝑦 ≤ sup(𝐴, ℝ^*, < ))
64	61, 62, 63	syl2anc 596	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐴) → 𝑦 ≤ sup(𝐴, ℝ^*, < ))
65	64	3adant3 1150	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐴 ∧ (𝐵 − 1) < 𝑦) → 𝑦 ≤ sup(𝐴, ℝ^*, < ))
66	33, 36, 39, 60, 65	xrltletrd 13202	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐴 ∧ (𝐵 − 1) < 𝑦) → -∞ < sup(𝐴, ℝ^*, < ))
67	66	3exp 1137	. . . . . . . 8 ⊢ (𝜑 → (𝑦 ∈ 𝐴 → ((𝐵 − 1) < 𝑦 → -∞ < sup(𝐴, ℝ^*, < ))))
68	67	adantr 486	. . . . . . 7 ⊢ ((𝜑 ∧ ¬ sup(𝐴, ℝ^, < ) = +∞) → (𝑦 ∈ 𝐴 → ((𝐵 − 1) < 𝑦 → -∞ < sup(𝐴, ℝ^, < ))))
69	68	rexlimdv 3166	. . . . . 6 ⊢ ((𝜑 ∧ ¬ sup(𝐴, ℝ^, < ) = +∞) → (∃𝑦 ∈ 𝐴 (𝐵 − 1) < 𝑦 → -∞ < sup(𝐴, ℝ^, < )))
70	31, 69	mpd 16	. . . . 5 ⊢ ((𝜑 ∧ ¬ sup(𝐴, ℝ^, < ) = +∞) → -∞ < sup(𝐴, ℝ^, < ))
71		simpr 490	. . . . . . 7 ⊢ ((𝜑 ∧ ¬ sup(𝐴, ℝ^, < ) = +∞) → ¬ sup(𝐴, ℝ^, < ) = +∞)
72		nltpnft 13206	. . . . . . . . 9 ⊢ (sup(𝐴, ℝ^, < ) ∈ ℝ^ → (sup(𝐴, ℝ^, < ) = +∞ ↔ ¬ sup(𝐴, ℝ^, < ) < +∞))
73	38, 72	syl 18	. . . . . . . 8 ⊢ (𝜑 → (sup(𝐴, ℝ^, < ) = +∞ ↔ ¬ sup(𝐴, ℝ^, < ) < +∞))
74	73	adantr 486	. . . . . . 7 ⊢ ((𝜑 ∧ ¬ sup(𝐴, ℝ^, < ) = +∞) → (sup(𝐴, ℝ^, < ) = +∞ ↔ ¬ sup(𝐴, ℝ^*, < ) < +∞))
75	71, 74	mtbid 327	. . . . . 6 ⊢ ((𝜑 ∧ ¬ sup(𝐴, ℝ^, < ) = +∞) → ¬ ¬ sup(𝐴, ℝ^, < ) < +∞)
76	75	notnotrd 134	. . . . 5 ⊢ ((𝜑 ∧ ¬ sup(𝐴, ℝ^, < ) = +∞) → sup(𝐴, ℝ^, < ) < +∞)
77	70, 76	jca 521	. . . 4 ⊢ ((𝜑 ∧ ¬ sup(𝐴, ℝ^, < ) = +∞) → (-∞ < sup(𝐴, ℝ^, < ) ∧ sup(𝐴, ℝ^*, < ) < +∞))
78	38	adantr 486	. . . . 5 ⊢ ((𝜑 ∧ ¬ sup(𝐴, ℝ^, < ) = +∞) → sup(𝐴, ℝ^, < ) ∈ ℝ^*)
79		xrrebnd 13210	. . . . 5 ⊢ (sup(𝐴, ℝ^, < ) ∈ ℝ^ → (sup(𝐴, ℝ^, < ) ∈ ℝ ↔ (-∞ < sup(𝐴, ℝ^, < ) ∧ sup(𝐴, ℝ^*, < ) < +∞)))
80	78, 79	syl 18	. . . 4 ⊢ ((𝜑 ∧ ¬ sup(𝐴, ℝ^, < ) = +∞) → (sup(𝐴, ℝ^, < ) ∈ ℝ ↔ (-∞ < sup(𝐴, ℝ^, < ) ∧ sup(𝐴, ℝ^, < ) < +∞)))
81	77, 80	mpbird 260	. . 3 ⊢ ((𝜑 ∧ ¬ sup(𝐴, ℝ^, < ) = +∞) → sup(𝐴, ℝ^, < ) ∈ ℝ)
82		simpl 488	. . . . 5 ⊢ (((𝜑 ∧ sup(𝐴, ℝ^, < ) ∈ ℝ) ∧ ¬ 𝐵 ≤ sup(𝐴, ℝ^, < )) → (𝜑 ∧ sup(𝐴, ℝ^*, < ) ∈ ℝ))
83		simpr 490	. . . . . 6 ⊢ (((𝜑 ∧ sup(𝐴, ℝ^, < ) ∈ ℝ) ∧ ¬ 𝐵 ≤ sup(𝐴, ℝ^, < )) → ¬ 𝐵 ≤ sup(𝐴, ℝ^*, < ))
84	82	simprd 501	. . . . . . 7 ⊢ (((𝜑 ∧ sup(𝐴, ℝ^, < ) ∈ ℝ) ∧ ¬ 𝐵 ≤ sup(𝐴, ℝ^, < )) → sup(𝐴, ℝ^*, < ) ∈ ℝ)
85	1	ad2antrr 739	. . . . . . 7 ⊢ (((𝜑 ∧ sup(𝐴, ℝ^, < ) ∈ ℝ) ∧ ¬ 𝐵 ≤ sup(𝐴, ℝ^, < )) → 𝐵 ∈ ℝ)
86	84, 85	ltnled 11372	. . . . . 6 ⊢ (((𝜑 ∧ sup(𝐴, ℝ^, < ) ∈ ℝ) ∧ ¬ 𝐵 ≤ sup(𝐴, ℝ^, < )) → (sup(𝐴, ℝ^, < ) < 𝐵 ↔ ¬ 𝐵 ≤ sup(𝐴, ℝ^, < )))
87	83, 86	mpbird 260	. . . . 5 ⊢ (((𝜑 ∧ sup(𝐴, ℝ^, < ) ∈ ℝ) ∧ ¬ 𝐵 ≤ sup(𝐴, ℝ^, < )) → sup(𝐴, ℝ^*, < ) < 𝐵)
88		simpll 779	. . . . . . 7 ⊢ (((𝜑 ∧ sup(𝐴, ℝ^, < ) ∈ ℝ) ∧ sup(𝐴, ℝ^, < ) < 𝐵) → 𝜑)
89	1	adantr 486	. . . . . . . . . 10 ⊢ ((𝜑 ∧ sup(𝐴, ℝ^*, < ) ∈ ℝ) → 𝐵 ∈ ℝ)
90		simpr 490	. . . . . . . . . 10 ⊢ ((𝜑 ∧ sup(𝐴, ℝ^, < ) ∈ ℝ) → sup(𝐴, ℝ^, < ) ∈ ℝ)
91	89, 90	resubcld 11657	. . . . . . . . 9 ⊢ ((𝜑 ∧ sup(𝐴, ℝ^, < ) ∈ ℝ) → (𝐵 − sup(𝐴, ℝ^, < )) ∈ ℝ)
92	91	adantr 486	. . . . . . . 8 ⊢ (((𝜑 ∧ sup(𝐴, ℝ^, < ) ∈ ℝ) ∧ sup(𝐴, ℝ^, < ) < 𝐵) → (𝐵 − sup(𝐴, ℝ^*, < )) ∈ ℝ)
93		simpr 490	. . . . . . . . 9 ⊢ (((𝜑 ∧ sup(𝐴, ℝ^, < ) ∈ ℝ) ∧ sup(𝐴, ℝ^, < ) < 𝐵) → sup(𝐴, ℝ^*, < ) < 𝐵)
94	90	adantr 486	. . . . . . . . . 10 ⊢ (((𝜑 ∧ sup(𝐴, ℝ^, < ) ∈ ℝ) ∧ sup(𝐴, ℝ^, < ) < 𝐵) → sup(𝐴, ℝ^*, < ) ∈ ℝ)
95	88, 1	syl 18	. . . . . . . . . 10 ⊢ (((𝜑 ∧ sup(𝐴, ℝ^, < ) ∈ ℝ) ∧ sup(𝐴, ℝ^, < ) < 𝐵) → 𝐵 ∈ ℝ)
96	94, 95	posdifd 11816	. . . . . . . . 9 ⊢ (((𝜑 ∧ sup(𝐴, ℝ^, < ) ∈ ℝ) ∧ sup(𝐴, ℝ^, < ) < 𝐵) → (sup(𝐴, ℝ^, < ) < 𝐵 ↔ 0 < (𝐵 − sup(𝐴, ℝ^, < ))))
97	93, 96	mpbid 235	. . . . . . . 8 ⊢ (((𝜑 ∧ sup(𝐴, ℝ^, < ) ∈ ℝ) ∧ sup(𝐴, ℝ^, < ) < 𝐵) → 0 < (𝐵 − sup(𝐴, ℝ^*, < )))
98	92, 97	elrpd 13073	. . . . . . 7 ⊢ (((𝜑 ∧ sup(𝐴, ℝ^, < ) ∈ ℝ) ∧ sup(𝐴, ℝ^, < ) < 𝐵) → (𝐵 − sup(𝐴, ℝ^*, < )) ∈ ℝ⁺)
99		ovex 7452	. . . . . . . 8 ⊢ (𝐵 − sup(𝐴, ℝ^*, < )) ∈ V
100		nfcv 2927	. . . . . . . . 9 ⊢ Ⅎ𝑥(𝐵 − sup(𝐴, ℝ^*, < ))
101		nfv 1947	. . . . . . . . . . 11 ⊢ Ⅎ𝑥(𝐵 − sup(𝐴, ℝ^*, < )) ∈ ℝ⁺
102	16, 101	nfan 1932	. . . . . . . . . 10 ⊢ Ⅎ𝑥(𝜑 ∧ (𝐵 − sup(𝐴, ℝ^*, < )) ∈ ℝ⁺)
103		nfv 1947	. . . . . . . . . 10 ⊢ Ⅎ𝑥∃𝑦 ∈ 𝐴 (𝐵 − (𝐵 − sup(𝐴, ℝ^*, < ))) < 𝑦
104	102, 103	nfim 1929	. . . . . . . . 9 ⊢ Ⅎ𝑥((𝜑 ∧ (𝐵 − sup(𝐴, ℝ^, < )) ∈ ℝ⁺) → ∃𝑦 ∈ 𝐴 (𝐵 − (𝐵 − sup(𝐴, ℝ^, < ))) < 𝑦)
105		eleq1 2853	. . . . . . . . . . 11 ⊢ (𝑥 = (𝐵 − sup(𝐴, ℝ^, < )) → (𝑥 ∈ ℝ⁺ ↔ (𝐵 − sup(𝐴, ℝ^, < )) ∈ ℝ⁺))
106	105	anbi2d 642	. . . . . . . . . 10 ⊢ (𝑥 = (𝐵 − sup(𝐴, ℝ^, < )) → ((𝜑 ∧ 𝑥 ∈ ℝ⁺) ↔ (𝜑 ∧ (𝐵 − sup(𝐴, ℝ^, < )) ∈ ℝ⁺)))
107		oveq2 7427	. . . . . . . . . . . 12 ⊢ (𝑥 = (𝐵 − sup(𝐴, ℝ^, < )) → (𝐵 − 𝑥) = (𝐵 − (𝐵 − sup(𝐴, ℝ^, < ))))
108	107	breq1d 5121	. . . . . . . . . . 11 ⊢ (𝑥 = (𝐵 − sup(𝐴, ℝ^, < )) → ((𝐵 − 𝑥) < 𝑦 ↔ (𝐵 − (𝐵 − sup(𝐴, ℝ^, < ))) < 𝑦))
109	108	rexbidv 3191	. . . . . . . . . 10 ⊢ (𝑥 = (𝐵 − sup(𝐴, ℝ^, < )) → (∃𝑦 ∈ 𝐴 (𝐵 − 𝑥) < 𝑦 ↔ ∃𝑦 ∈ 𝐴 (𝐵 − (𝐵 − sup(𝐴, ℝ^, < ))) < 𝑦))
110	106, 109	imbi12d 347	. . . . . . . . 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 3536	. . . . . . . 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 596	. . . . . 6 ⊢ (((𝜑 ∧ sup(𝐴, ℝ^, < ) ∈ ℝ) ∧ sup(𝐴, ℝ^, < ) < 𝐵) → ∃𝑦 ∈ 𝐴 (𝐵 − (𝐵 − sup(𝐴, ℝ^*, < ))) < 𝑦)
114	1	recnd 11252	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐵 ∈ ℂ)
115	114	ad3antrrr 743	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ sup(𝐴, ℝ^, < ) ∈ ℝ) ∧ sup(𝐴, ℝ^, < ) < 𝐵) ∧ (𝐵 − (𝐵 − sup(𝐴, ℝ^*, < ))) < 𝑦) → 𝐵 ∈ ℂ)
116	90	recnd 11252	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ sup(𝐴, ℝ^, < ) ∈ ℝ) → sup(𝐴, ℝ^, < ) ∈ ℂ)
117	116	ad2antrr 739	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ sup(𝐴, ℝ^, < ) ∈ ℝ) ∧ sup(𝐴, ℝ^, < ) < 𝐵) ∧ (𝐵 − (𝐵 − sup(𝐴, ℝ^, < ))) < 𝑦) → sup(𝐴, ℝ^, < ) ∈ ℂ)
118	115, 117	nncand 11589	. . . . . . . . . . 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 2771	. . . . . . . . . 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 490	. . . . . . . . . 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 5135	. . . . . . . . 9 ⊢ ((((𝜑 ∧ sup(𝐴, ℝ^, < ) ∈ ℝ) ∧ sup(𝐴, ℝ^, < ) < 𝐵) ∧ (𝐵 − (𝐵 − sup(𝐴, ℝ^, < ))) < 𝑦) → sup(𝐴, ℝ^, < ) < 𝑦)
122	121	ex 418	. . . . . . . 8 ⊢ (((𝜑 ∧ sup(𝐴, ℝ^, < ) ∈ ℝ) ∧ sup(𝐴, ℝ^, < ) < 𝐵) → ((𝐵 − (𝐵 − sup(𝐴, ℝ^, < ))) < 𝑦 → sup(𝐴, ℝ^, < ) < 𝑦))
123	122	adantr 486	. . . . . . 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 3180	. . . . . 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 16	. . . . 5 ⊢ (((𝜑 ∧ sup(𝐴, ℝ^, < ) ∈ ℝ) ∧ sup(𝐴, ℝ^, < ) < 𝐵) → ∃𝑦 ∈ 𝐴 sup(𝐴, ℝ^*, < ) < 𝑦)
126	82, 87, 125	syl2anc 596	. . . 4 ⊢ (((𝜑 ∧ sup(𝐴, ℝ^, < ) ∈ ℝ) ∧ ¬ 𝐵 ≤ sup(𝐴, ℝ^, < )) → ∃𝑦 ∈ 𝐴 sup(𝐴, ℝ^*, < ) < 𝑦)
127	61, 37	syl 18	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐴) → sup(𝐴, ℝ^, < ) ∈ ℝ^)
128	35, 127	xrlenltd 11290	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐴) → (𝑦 ≤ sup(𝐴, ℝ^, < ) ↔ ¬ sup(𝐴, ℝ^, < ) < 𝑦))
129	64, 128	mpbid 235	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐴) → ¬ sup(𝐴, ℝ^*, < ) < 𝑦)
130	129	ralrimiva 3159	. . . . . 6 ⊢ (𝜑 → ∀𝑦 ∈ 𝐴 ¬ sup(𝐴, ℝ^*, < ) < 𝑦)
131		ralnex 3093	. . . . . 6 ⊢ (∀𝑦 ∈ 𝐴 ¬ sup(𝐴, ℝ^, < ) < 𝑦 ↔ ¬ ∃𝑦 ∈ 𝐴 sup(𝐴, ℝ^, < ) < 𝑦)
132	130, 131	sylib 221	. . . . 5 ⊢ (𝜑 → ¬ ∃𝑦 ∈ 𝐴 sup(𝐴, ℝ^*, < ) < 𝑦)
133	132	ad2antrr 739	. . . 4 ⊢ (((𝜑 ∧ sup(𝐴, ℝ^, < ) ∈ ℝ) ∧ ¬ 𝐵 ≤ sup(𝐴, ℝ^, < )) → ¬ ∃𝑦 ∈ 𝐴 sup(𝐴, ℝ^*, < ) < 𝑦)
134	126, 133	condan 830	. . 3 ⊢ ((𝜑 ∧ sup(𝐴, ℝ^, < ) ∈ ℝ) → 𝐵 ≤ sup(𝐴, ℝ^, < ))
135	13, 81, 134	syl2anc 596	. 2 ⊢ ((𝜑 ∧ ¬ sup(𝐴, ℝ^, < ) = +∞) → 𝐵 ≤ sup(𝐴, ℝ^, < ))
136	12, 135	pm2.61dan 825	1 ⊢ (𝜑 → 𝐵 ≤ sup(𝐴, ℝ^*, < ))

Colors of variables: wff setvar class

This proof depends on syntax axioms: ¬ wn 3 → wi 4 ↔ wb 209 ∧ wa 401 ∧ w3a 1103 = wceq 1570 Ⅎwnf 1816 ∈ wcel 2146 ∀wral 3081 ∃wrex 3091 Vcvv 3457 ⊆ wss 3906 class class class wbr 5111 (class class class)co 7419 supcsup 9407 ℂcc 11113 ℝcr 11114 0cc0 11115 1c1 11116 +∞cpnf 11255 -∞cmnf 11256 ℝ^*cxr 11257 < clt 11258 ≤ cle 11259 − cmin 11456 ℝ⁺crp 13032

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1828 ax-4 1842 ax-5 1943 ax-6 2000 ax-7 2041 ax-8 2148 ax-9 2156 ax-10 2179 ax-11 2195 ax-12 2216 ax-ext 2737 ax-sep 5259 ax-nul 5271 ax-pow 5338 ax-pr 5406 ax-un 7742 ax-cnex 11171 ax-resscn 11172 ax-1cn 11173 ax-icn 11174 ax-addcl 11175 ax-addrcl 11176 ax-mulcl 11177 ax-mulrcl 11178 ax-mulcom 11179 ax-addass 11180 ax-mulass 11181 ax-distr 11182 ax-i2m1 11183 ax-1ne0 11184 ax-1rid 11185 ax-rnegex 11186 ax-rrecex 11187 ax-cnre 11188 ax-pre-lttri 11189 ax-pre-lttrn 11190 ax-pre-ltadd 11191 ax-pre-mulgt0 11192 ax-pre-sup 11193

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1813 df-nf 1817 df-sb 2100 df-mo 2569 df-eu 2599 df-clab 2744 df-cleq 2757 df-clel 2840 df-nfc 2914 df-ne 2961 df-nel 3067 df-ral 3082 df-rex 3092 df-rmo 3371 df-reu 3372 df-rab 3419 df-v 3459 df-sbc 3747 df-csb 3855 df-dif 3909 df-un 3911 df-in 3913 df-ss 3923 df-nul 4287 df-if 4490 df-pw 4566 df-sn 4592 df-pr 4594 df-op 4598 df-uni 4875 df-br 5112 df-opab 5176 df-mpt 5195 df-id 5558 df-po 5571 df-so 5572 df-xp 5669 df-rel 5670 df-cnv 5671 df-co 5672 df-dm 5673 df-rn 5674 df-res 5675 df-ima 5676 df-iota 6496 df-fun 6542 df-fn 6543 df-f 6544 df-f1 6545 df-fo 6546 df-f1o 6547 df-fv 6548 df-riota 7376 df-ov 7422 df-oprab 7423 df-mpo 7424 df-er 8700 df-en 8950 df-dom 8951 df-sdom 8952 df-sup 9409 df-pnf 11260 df-mnf 11261 df-xr 11262 df-ltxr 11263 df-le 11264 df-sub 11458 df-neg 11459 df-rp 13033

This theorem is used by: suplesup 46113

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