supxrgelem - Mathbox for Glauco Siliprandi

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Theorem supxrgelem 45375

Description: If an extended 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
supxrgelem.xph	⊢ Ⅎ𝑥𝜑
supxrgelem.a	⊢ (𝜑 → 𝐴 ⊆ ℝ^*)
supxrgelem.b	⊢ (𝜑 → 𝐵 ∈ ℝ^*)
supxrgelem.y	⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → ∃𝑦 ∈ 𝐴 𝐵 < (𝑦 +_𝑒 𝑥))

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

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

**Proof of Theorem supxrgelem**
Step	Hyp	Ref	Expression
1		supxrgelem.b	. . . . 5 ⊢ (𝜑 → 𝐵 ∈ ℝ^*)
2		pnfge 13026	. . . . 5 ⊢ (𝐵 ∈ ℝ^* → 𝐵 ≤ +∞)
3	1, 2	syl 17	. . . 4 ⊢ (𝜑 → 𝐵 ≤ +∞)
4	3	adantr 480	. . 3 ⊢ ((𝜑 ∧ sup(𝐴, ℝ^*, < ) = +∞) → 𝐵 ≤ +∞)
5		id 22	. . . . 5 ⊢ (sup(𝐴, ℝ^, < ) = +∞ → sup(𝐴, ℝ^, < ) = +∞)
6	5	eqcomd 2737	. . . 4 ⊢ (sup(𝐴, ℝ^, < ) = +∞ → +∞ = sup(𝐴, ℝ^, < ))
7	6	adantl 481	. . 3 ⊢ ((𝜑 ∧ sup(𝐴, ℝ^, < ) = +∞) → +∞ = sup(𝐴, ℝ^, < ))
8	4, 7	breqtrd 5117	. 2 ⊢ ((𝜑 ∧ sup(𝐴, ℝ^, < ) = +∞) → 𝐵 ≤ sup(𝐴, ℝ^, < ))
9		simpl 482	. . 3 ⊢ ((𝜑 ∧ ¬ sup(𝐴, ℝ^*, < ) = +∞) → 𝜑)
10		1rp 12891	. . . . . . . 8 ⊢ 1 ∈ ℝ⁺
11		nfcv 2894	. . . . . . . . . 10 ⊢ Ⅎ𝑥1
12		supxrgelem.xph	. . . . . . . . . . . 12 ⊢ Ⅎ𝑥𝜑
13		nfv 1915	. . . . . . . . . . . 12 ⊢ Ⅎ𝑥1 ∈ ℝ⁺
14	12, 13	nfan 1900	. . . . . . . . . . 11 ⊢ Ⅎ𝑥(𝜑 ∧ 1 ∈ ℝ⁺)
15		nfv 1915	. . . . . . . . . . 11 ⊢ Ⅎ𝑥∃𝑦 ∈ 𝐴 𝐵 < (𝑦 +_𝑒 1)
16	14, 15	nfim 1897	. . . . . . . . . 10 ⊢ Ⅎ𝑥((𝜑 ∧ 1 ∈ ℝ⁺) → ∃𝑦 ∈ 𝐴 𝐵 < (𝑦 +_𝑒 1))
17		eleq1 2819	. . . . . . . . . . . 12 ⊢ (𝑥 = 1 → (𝑥 ∈ ℝ⁺ ↔ 1 ∈ ℝ⁺))
18	17	anbi2d 630	. . . . . . . . . . 11 ⊢ (𝑥 = 1 → ((𝜑 ∧ 𝑥 ∈ ℝ⁺) ↔ (𝜑 ∧ 1 ∈ ℝ⁺)))
19		oveq2 7354	. . . . . . . . . . . . 13 ⊢ (𝑥 = 1 → (𝑦 +_𝑒 𝑥) = (𝑦 +_𝑒 1))
20	19	breq2d 5103	. . . . . . . . . . . 12 ⊢ (𝑥 = 1 → (𝐵 < (𝑦 +_𝑒 𝑥) ↔ 𝐵 < (𝑦 +_𝑒 1)))
21	20	rexbidv 3156	. . . . . . . . . . 11 ⊢ (𝑥 = 1 → (∃𝑦 ∈ 𝐴 𝐵 < (𝑦 +_𝑒 𝑥) ↔ ∃𝑦 ∈ 𝐴 𝐵 < (𝑦 +_𝑒 1)))
22	18, 21	imbi12d 344	. . . . . . . . . 10 ⊢ (𝑥 = 1 → (((𝜑 ∧ 𝑥 ∈ ℝ⁺) → ∃𝑦 ∈ 𝐴 𝐵 < (𝑦 +_𝑒 𝑥)) ↔ ((𝜑 ∧ 1 ∈ ℝ⁺) → ∃𝑦 ∈ 𝐴 𝐵 < (𝑦 +_𝑒 1))))
23		supxrgelem.y	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → ∃𝑦 ∈ 𝐴 𝐵 < (𝑦 +_𝑒 𝑥))
24	11, 16, 22, 23	vtoclgf 3524	. . . . . . . . 9 ⊢ (1 ∈ ℝ⁺ → ((𝜑 ∧ 1 ∈ ℝ⁺) → ∃𝑦 ∈ 𝐴 𝐵 < (𝑦 +_𝑒 1)))
25	10, 24	ax-mp 5	. . . . . . . 8 ⊢ ((𝜑 ∧ 1 ∈ ℝ⁺) → ∃𝑦 ∈ 𝐴 𝐵 < (𝑦 +_𝑒 1))
26	10, 25	mpan2 691	. . . . . . 7 ⊢ (𝜑 → ∃𝑦 ∈ 𝐴 𝐵 < (𝑦 +_𝑒 1))
27	26	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ ¬ sup(𝐴, ℝ^*, < ) = +∞) → ∃𝑦 ∈ 𝐴 𝐵 < (𝑦 +_𝑒 1))
28		mnfxr 11166	. . . . . . . . . . 11 ⊢ -∞ ∈ ℝ^*
29	28	a1i 11	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐴 ∧ 𝐵 < (𝑦 +_𝑒 1)) → -∞ ∈ ℝ^*)
30		supxrgelem.a	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐴 ⊆ ℝ^*)
31	30	sselda 3934	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐴) → 𝑦 ∈ ℝ^*)
32	31	3adant3 1132	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐴 ∧ 𝐵 < (𝑦 +_𝑒 1)) → 𝑦 ∈ ℝ^*)
33		supxrcl 13211	. . . . . . . . . . . 12 ⊢ (𝐴 ⊆ ℝ^* → sup(𝐴, ℝ^, < ) ∈ ℝ^)
34	30, 33	syl 17	. . . . . . . . . . 11 ⊢ (𝜑 → sup(𝐴, ℝ^, < ) ∈ ℝ^)
35	34	3ad2ant1 1133	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐴 ∧ 𝐵 < (𝑦 +_𝑒 1)) → sup(𝐴, ℝ^, < ) ∈ ℝ^)
36		simpl3 1194	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝐴 ∧ 𝐵 < (𝑦 +_𝑒 1)) ∧ ¬ -∞ < 𝑦) → 𝐵 < (𝑦 +_𝑒 1))
37		simpr 484	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝐴) ∧ ¬ -∞ < 𝑦) → ¬ -∞ < 𝑦)
38	31	adantr 480	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝐴) ∧ ¬ -∞ < 𝑦) → 𝑦 ∈ ℝ^*)
39		ngtmnft 13062	. . . . . . . . . . . . . . . . 17 ⊢ (𝑦 ∈ ℝ^* → (𝑦 = -∞ ↔ ¬ -∞ < 𝑦))
40	38, 39	syl 17	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝐴) ∧ ¬ -∞ < 𝑦) → (𝑦 = -∞ ↔ ¬ -∞ < 𝑦))
41	37, 40	mpbird 257	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝐴) ∧ ¬ -∞ < 𝑦) → 𝑦 = -∞)
42	41	oveq1d 7361	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝐴) ∧ ¬ -∞ < 𝑦) → (𝑦 +_𝑒 1) = (-∞ +_𝑒 1))
43		1xr 11168	. . . . . . . . . . . . . . . 16 ⊢ 1 ∈ ℝ^*
44	43	a1i 11	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝐴) ∧ ¬ -∞ < 𝑦) → 1 ∈ ℝ^*)
45		1re 11109	. . . . . . . . . . . . . . . . 17 ⊢ 1 ∈ ℝ
46		renepnf 11157	. . . . . . . . . . . . . . . . 17 ⊢ (1 ∈ ℝ → 1 ≠ +∞)
47	45, 46	ax-mp 5	. . . . . . . . . . . . . . . 16 ⊢ 1 ≠ +∞
48	47	a1i 11	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝐴) ∧ ¬ -∞ < 𝑦) → 1 ≠ +∞)
49		xaddmnf2 13125	. . . . . . . . . . . . . . 15 ⊢ ((1 ∈ ℝ^* ∧ 1 ≠ +∞) → (-∞ +_𝑒 1) = -∞)
50	44, 48, 49	syl2anc 584	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝐴) ∧ ¬ -∞ < 𝑦) → (-∞ +_𝑒 1) = -∞)
51	42, 50	eqtrd 2766	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝐴) ∧ ¬ -∞ < 𝑦) → (𝑦 +_𝑒 1) = -∞)
52	51	3adantl3 1169	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝐴 ∧ 𝐵 < (𝑦 +_𝑒 1)) ∧ ¬ -∞ < 𝑦) → (𝑦 +_𝑒 1) = -∞)
53	36, 52	breqtrd 5117	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝐴 ∧ 𝐵 < (𝑦 +_𝑒 1)) ∧ ¬ -∞ < 𝑦) → 𝐵 < -∞)
54		nltmnf 13025	. . . . . . . . . . . . . 14 ⊢ (𝐵 ∈ ℝ^* → ¬ 𝐵 < -∞)
55	1, 54	syl 17	. . . . . . . . . . . . 13 ⊢ (𝜑 → ¬ 𝐵 < -∞)
56	55	adantr 480	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ ¬ -∞ < 𝑦) → ¬ 𝐵 < -∞)
57	56	3ad2antl1 1186	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝐴 ∧ 𝐵 < (𝑦 +_𝑒 1)) ∧ ¬ -∞ < 𝑦) → ¬ 𝐵 < -∞)
58	53, 57	condan 817	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐴 ∧ 𝐵 < (𝑦 +_𝑒 1)) → -∞ < 𝑦)
59	30	adantr 480	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐴) → 𝐴 ⊆ ℝ^*)
60		simpr 484	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐴) → 𝑦 ∈ 𝐴)
61		supxrub 13220	. . . . . . . . . . . 12 ⊢ ((𝐴 ⊆ ℝ^* ∧ 𝑦 ∈ 𝐴) → 𝑦 ≤ sup(𝐴, ℝ^*, < ))
62	59, 60, 61	syl2anc 584	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐴) → 𝑦 ≤ sup(𝐴, ℝ^*, < ))
63	62	3adant3 1132	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐴 ∧ 𝐵 < (𝑦 +_𝑒 1)) → 𝑦 ≤ sup(𝐴, ℝ^*, < ))
64	29, 32, 35, 58, 63	xrltletrd 13057	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐴 ∧ 𝐵 < (𝑦 +_𝑒 1)) → -∞ < sup(𝐴, ℝ^*, < ))
65	64	3exp 1119	. . . . . . . 8 ⊢ (𝜑 → (𝑦 ∈ 𝐴 → (𝐵 < (𝑦 +_𝑒 1) → -∞ < sup(𝐴, ℝ^*, < ))))
66	65	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ ¬ sup(𝐴, ℝ^, < ) = +∞) → (𝑦 ∈ 𝐴 → (𝐵 < (𝑦 +_𝑒 1) → -∞ < sup(𝐴, ℝ^, < ))))
67	66	rexlimdv 3131	. . . . . 6 ⊢ ((𝜑 ∧ ¬ sup(𝐴, ℝ^, < ) = +∞) → (∃𝑦 ∈ 𝐴 𝐵 < (𝑦 +_𝑒 1) → -∞ < sup(𝐴, ℝ^, < )))
68	27, 67	mpd 15	. . . . 5 ⊢ ((𝜑 ∧ ¬ sup(𝐴, ℝ^, < ) = +∞) → -∞ < sup(𝐴, ℝ^, < ))
69		simpr 484	. . . . . . 7 ⊢ ((𝜑 ∧ ¬ sup(𝐴, ℝ^, < ) = +∞) → ¬ sup(𝐴, ℝ^, < ) = +∞)
70		nltpnft 13060	. . . . . . . . 9 ⊢ (sup(𝐴, ℝ^, < ) ∈ ℝ^ → (sup(𝐴, ℝ^, < ) = +∞ ↔ ¬ sup(𝐴, ℝ^, < ) < +∞))
71	34, 70	syl 17	. . . . . . . 8 ⊢ (𝜑 → (sup(𝐴, ℝ^, < ) = +∞ ↔ ¬ sup(𝐴, ℝ^, < ) < +∞))
72	71	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ ¬ sup(𝐴, ℝ^, < ) = +∞) → (sup(𝐴, ℝ^, < ) = +∞ ↔ ¬ sup(𝐴, ℝ^*, < ) < +∞))
73	69, 72	mtbid 324	. . . . . 6 ⊢ ((𝜑 ∧ ¬ sup(𝐴, ℝ^, < ) = +∞) → ¬ ¬ sup(𝐴, ℝ^, < ) < +∞)
74	73	notnotrd 133	. . . . 5 ⊢ ((𝜑 ∧ ¬ sup(𝐴, ℝ^, < ) = +∞) → sup(𝐴, ℝ^, < ) < +∞)
75	68, 74	jca 511	. . . 4 ⊢ ((𝜑 ∧ ¬ sup(𝐴, ℝ^, < ) = +∞) → (-∞ < sup(𝐴, ℝ^, < ) ∧ sup(𝐴, ℝ^*, < ) < +∞))
76	34	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ ¬ sup(𝐴, ℝ^, < ) = +∞) → sup(𝐴, ℝ^, < ) ∈ ℝ^*)
77		xrrebnd 13064	. . . . 5 ⊢ (sup(𝐴, ℝ^, < ) ∈ ℝ^ → (sup(𝐴, ℝ^, < ) ∈ ℝ ↔ (-∞ < sup(𝐴, ℝ^, < ) ∧ sup(𝐴, ℝ^*, < ) < +∞)))
78	76, 77	syl 17	. . . 4 ⊢ ((𝜑 ∧ ¬ sup(𝐴, ℝ^, < ) = +∞) → (sup(𝐴, ℝ^, < ) ∈ ℝ ↔ (-∞ < sup(𝐴, ℝ^, < ) ∧ sup(𝐴, ℝ^, < ) < +∞)))
79	75, 78	mpbird 257	. . 3 ⊢ ((𝜑 ∧ ¬ sup(𝐴, ℝ^, < ) = +∞) → sup(𝐴, ℝ^, < ) ∈ ℝ)
80		simpl 482	. . . . 5 ⊢ (((𝜑 ∧ sup(𝐴, ℝ^, < ) ∈ ℝ) ∧ ¬ 𝐵 ≤ sup(𝐴, ℝ^, < )) → (𝜑 ∧ sup(𝐴, ℝ^*, < ) ∈ ℝ))
81		simpr 484	. . . . . 6 ⊢ (((𝜑 ∧ sup(𝐴, ℝ^, < ) ∈ ℝ) ∧ ¬ 𝐵 ≤ sup(𝐴, ℝ^, < )) → ¬ 𝐵 ≤ sup(𝐴, ℝ^*, < ))
82	34	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ ¬ 𝐵 ≤ sup(𝐴, ℝ^, < )) → sup(𝐴, ℝ^, < ) ∈ ℝ^*)
83	1	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ ¬ 𝐵 ≤ sup(𝐴, ℝ^, < )) → 𝐵 ∈ ℝ^)
84		xrltnle 11176	. . . . . . . 8 ⊢ ((sup(𝐴, ℝ^, < ) ∈ ℝ^ ∧ 𝐵 ∈ ℝ^) → (sup(𝐴, ℝ^, < ) < 𝐵 ↔ ¬ 𝐵 ≤ sup(𝐴, ℝ^*, < )))
85	82, 83, 84	syl2anc 584	. . . . . . 7 ⊢ ((𝜑 ∧ ¬ 𝐵 ≤ sup(𝐴, ℝ^, < )) → (sup(𝐴, ℝ^, < ) < 𝐵 ↔ ¬ 𝐵 ≤ sup(𝐴, ℝ^*, < )))
86	85	adantlr 715	. . . . . 6 ⊢ (((𝜑 ∧ sup(𝐴, ℝ^, < ) ∈ ℝ) ∧ ¬ 𝐵 ≤ sup(𝐴, ℝ^, < )) → (sup(𝐴, ℝ^, < ) < 𝐵 ↔ ¬ 𝐵 ≤ sup(𝐴, ℝ^, < )))
87	81, 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	28	a1i 11	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ sup(𝐴, ℝ^, < ) ∈ ℝ) ∧ sup(𝐴, ℝ^, < ) < 𝐵) → -∞ ∈ ℝ^*)
90	88, 34	syl 17	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ sup(𝐴, ℝ^, < ) ∈ ℝ) ∧ sup(𝐴, ℝ^, < ) < 𝐵) → sup(𝐴, ℝ^, < ) ∈ ℝ^)
91	88, 1	syl 17	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ sup(𝐴, ℝ^, < ) ∈ ℝ) ∧ sup(𝐴, ℝ^, < ) < 𝐵) → 𝐵 ∈ ℝ^*)
92		mnfle 13031	. . . . . . . . . . . . . 14 ⊢ (sup(𝐴, ℝ^, < ) ∈ ℝ^ → -∞ ≤ sup(𝐴, ℝ^*, < ))
93	34, 92	syl 17	. . . . . . . . . . . . 13 ⊢ (𝜑 → -∞ ≤ sup(𝐴, ℝ^*, < ))
94	93	ad2antrr 726	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ sup(𝐴, ℝ^, < ) ∈ ℝ) ∧ sup(𝐴, ℝ^, < ) < 𝐵) → -∞ ≤ sup(𝐴, ℝ^*, < ))
95		simpr 484	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ sup(𝐴, ℝ^, < ) ∈ ℝ) ∧ sup(𝐴, ℝ^, < ) < 𝐵) → sup(𝐴, ℝ^*, < ) < 𝐵)
96	89, 90, 91, 94, 95	xrlelttrd 13056	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ sup(𝐴, ℝ^, < ) ∈ ℝ) ∧ sup(𝐴, ℝ^, < ) < 𝐵) → -∞ < 𝐵)
97		id 22	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝜑)
98	10	a1i 11	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 1 ∈ ℝ⁺)
99	97, 98, 25	syl2anc 584	. . . . . . . . . . . . 13 ⊢ (𝜑 → ∃𝑦 ∈ 𝐴 𝐵 < (𝑦 +_𝑒 1))
100	99	ad2antrr 726	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ sup(𝐴, ℝ^, < ) ∈ ℝ) ∧ sup(𝐴, ℝ^, < ) < 𝐵) → ∃𝑦 ∈ 𝐴 𝐵 < (𝑦 +_𝑒 1))
101	1	3ad2ant1 1133	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐴 ∧ 𝐵 < (𝑦 +_𝑒 1)) → 𝐵 ∈ ℝ^*)
102	43	a1i 11	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐴 ∧ 𝐵 < (𝑦 +_𝑒 1)) → 1 ∈ ℝ^*)
103	32, 102	jca 511	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐴 ∧ 𝐵 < (𝑦 +_𝑒 1)) → (𝑦 ∈ ℝ^* ∧ 1 ∈ ℝ^*))
104		xaddcl 13135	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑦 ∈ ℝ^* ∧ 1 ∈ ℝ^) → (𝑦 +_𝑒 1) ∈ ℝ^)
105	103, 104	syl 17	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐴 ∧ 𝐵 < (𝑦 +_𝑒 1)) → (𝑦 +_𝑒 1) ∈ ℝ^*)
106		pnfxr 11163	. . . . . . . . . . . . . . . . 17 ⊢ +∞ ∈ ℝ^*
107	106	a1i 11	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐴 ∧ 𝐵 < (𝑦 +_𝑒 1)) → +∞ ∈ ℝ^*)
108		simp3 1138	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐴 ∧ 𝐵 < (𝑦 +_𝑒 1)) → 𝐵 < (𝑦 +_𝑒 1))
109	31, 43, 104	sylancl 586	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐴) → (𝑦 +_𝑒 1) ∈ ℝ^*)
110		pnfge 13026	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑦 +_𝑒 1) ∈ ℝ^* → (𝑦 +_𝑒 1) ≤ +∞)
111	109, 110	syl 17	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐴) → (𝑦 +_𝑒 1) ≤ +∞)
112	111	3adant3 1132	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐴 ∧ 𝐵 < (𝑦 +_𝑒 1)) → (𝑦 +_𝑒 1) ≤ +∞)
113	101, 105, 107, 108, 112	xrltletrd 13057	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐴 ∧ 𝐵 < (𝑦 +_𝑒 1)) → 𝐵 < +∞)
114	113	3exp 1119	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (𝑦 ∈ 𝐴 → (𝐵 < (𝑦 +_𝑒 1) → 𝐵 < +∞)))
115	114	rexlimdv 3131	. . . . . . . . . . . . 13 ⊢ (𝜑 → (∃𝑦 ∈ 𝐴 𝐵 < (𝑦 +_𝑒 1) → 𝐵 < +∞))
116	88, 115	syl 17	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ sup(𝐴, ℝ^, < ) ∈ ℝ) ∧ sup(𝐴, ℝ^, < ) < 𝐵) → (∃𝑦 ∈ 𝐴 𝐵 < (𝑦 +_𝑒 1) → 𝐵 < +∞))
117	100, 116	mpd 15	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ sup(𝐴, ℝ^, < ) ∈ ℝ) ∧ sup(𝐴, ℝ^, < ) < 𝐵) → 𝐵 < +∞)
118	96, 117	jca 511	. . . . . . . . . 10 ⊢ (((𝜑 ∧ sup(𝐴, ℝ^, < ) ∈ ℝ) ∧ sup(𝐴, ℝ^, < ) < 𝐵) → (-∞ < 𝐵 ∧ 𝐵 < +∞))
119		xrrebnd 13064	. . . . . . . . . . 11 ⊢ (𝐵 ∈ ℝ^* → (𝐵 ∈ ℝ ↔ (-∞ < 𝐵 ∧ 𝐵 < +∞)))
120	91, 119	syl 17	. . . . . . . . . 10 ⊢ (((𝜑 ∧ sup(𝐴, ℝ^, < ) ∈ ℝ) ∧ sup(𝐴, ℝ^, < ) < 𝐵) → (𝐵 ∈ ℝ ↔ (-∞ < 𝐵 ∧ 𝐵 < +∞)))
121	118, 120	mpbird 257	. . . . . . . . 9 ⊢ (((𝜑 ∧ sup(𝐴, ℝ^, < ) ∈ ℝ) ∧ sup(𝐴, ℝ^, < ) < 𝐵) → 𝐵 ∈ ℝ)
122		simpr 484	. . . . . . . . . 10 ⊢ ((𝜑 ∧ sup(𝐴, ℝ^, < ) ∈ ℝ) → sup(𝐴, ℝ^, < ) ∈ ℝ)
123	122	adantr 480	. . . . . . . . 9 ⊢ (((𝜑 ∧ sup(𝐴, ℝ^, < ) ∈ ℝ) ∧ sup(𝐴, ℝ^, < ) < 𝐵) → sup(𝐴, ℝ^*, < ) ∈ ℝ)
124	121, 123	resubcld 11542	. . . . . . . 8 ⊢ (((𝜑 ∧ sup(𝐴, ℝ^, < ) ∈ ℝ) ∧ sup(𝐴, ℝ^, < ) < 𝐵) → (𝐵 − sup(𝐴, ℝ^*, < )) ∈ ℝ)
125	26, 115	mpd 15	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐵 < +∞)
126	125	ad2antrr 726	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ sup(𝐴, ℝ^, < ) ∈ ℝ) ∧ sup(𝐴, ℝ^, < ) < 𝐵) → 𝐵 < +∞)
127	96, 126	jca 511	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ sup(𝐴, ℝ^, < ) ∈ ℝ) ∧ sup(𝐴, ℝ^, < ) < 𝐵) → (-∞ < 𝐵 ∧ 𝐵 < +∞))
128	127, 120	mpbird 257	. . . . . . . . . 10 ⊢ (((𝜑 ∧ sup(𝐴, ℝ^, < ) ∈ ℝ) ∧ sup(𝐴, ℝ^, < ) < 𝐵) → 𝐵 ∈ ℝ)
129	123, 128	posdifd 11701	. . . . . . . . 9 ⊢ (((𝜑 ∧ sup(𝐴, ℝ^, < ) ∈ ℝ) ∧ sup(𝐴, ℝ^, < ) < 𝐵) → (sup(𝐴, ℝ^, < ) < 𝐵 ↔ 0 < (𝐵 − sup(𝐴, ℝ^, < ))))
130	95, 129	mpbid 232	. . . . . . . 8 ⊢ (((𝜑 ∧ sup(𝐴, ℝ^, < ) ∈ ℝ) ∧ sup(𝐴, ℝ^, < ) < 𝐵) → 0 < (𝐵 − sup(𝐴, ℝ^*, < )))
131	124, 130	elrpd 12928	. . . . . . 7 ⊢ (((𝜑 ∧ sup(𝐴, ℝ^, < ) ∈ ℝ) ∧ sup(𝐴, ℝ^, < ) < 𝐵) → (𝐵 − sup(𝐴, ℝ^*, < )) ∈ ℝ⁺)
132		ovex 7379	. . . . . . . 8 ⊢ (𝐵 − sup(𝐴, ℝ^*, < )) ∈ V
133		nfcv 2894	. . . . . . . . 9 ⊢ Ⅎ𝑥(𝐵 − sup(𝐴, ℝ^*, < ))
134		nfv 1915	. . . . . . . . . . 11 ⊢ Ⅎ𝑥(𝐵 − sup(𝐴, ℝ^*, < )) ∈ ℝ⁺
135	12, 134	nfan 1900	. . . . . . . . . 10 ⊢ Ⅎ𝑥(𝜑 ∧ (𝐵 − sup(𝐴, ℝ^*, < )) ∈ ℝ⁺)
136		nfv 1915	. . . . . . . . . 10 ⊢ Ⅎ𝑥∃𝑦 ∈ 𝐴 𝐵 < (𝑦 +_𝑒 (𝐵 − sup(𝐴, ℝ^*, < )))
137	135, 136	nfim 1897	. . . . . . . . 9 ⊢ Ⅎ𝑥((𝜑 ∧ (𝐵 − sup(𝐴, ℝ^, < )) ∈ ℝ⁺) → ∃𝑦 ∈ 𝐴 𝐵 < (𝑦 +_𝑒 (𝐵 − sup(𝐴, ℝ^, < ))))
138		eleq1 2819	. . . . . . . . . . 11 ⊢ (𝑥 = (𝐵 − sup(𝐴, ℝ^, < )) → (𝑥 ∈ ℝ⁺ ↔ (𝐵 − sup(𝐴, ℝ^, < )) ∈ ℝ⁺))
139	138	anbi2d 630	. . . . . . . . . 10 ⊢ (𝑥 = (𝐵 − sup(𝐴, ℝ^, < )) → ((𝜑 ∧ 𝑥 ∈ ℝ⁺) ↔ (𝜑 ∧ (𝐵 − sup(𝐴, ℝ^, < )) ∈ ℝ⁺)))
140		oveq2 7354	. . . . . . . . . . . 12 ⊢ (𝑥 = (𝐵 − sup(𝐴, ℝ^, < )) → (𝑦 +_𝑒 𝑥) = (𝑦 +_𝑒 (𝐵 − sup(𝐴, ℝ^, < ))))
141	140	breq2d 5103	. . . . . . . . . . 11 ⊢ (𝑥 = (𝐵 − sup(𝐴, ℝ^, < )) → (𝐵 < (𝑦 +_𝑒 𝑥) ↔ 𝐵 < (𝑦 +_𝑒 (𝐵 − sup(𝐴, ℝ^, < )))))
142	141	rexbidv 3156	. . . . . . . . . 10 ⊢ (𝑥 = (𝐵 − sup(𝐴, ℝ^, < )) → (∃𝑦 ∈ 𝐴 𝐵 < (𝑦 +_𝑒 𝑥) ↔ ∃𝑦 ∈ 𝐴 𝐵 < (𝑦 +_𝑒 (𝐵 − sup(𝐴, ℝ^, < )))))
143	139, 142	imbi12d 344	. . . . . . . . 9 ⊢ (𝑥 = (𝐵 − sup(𝐴, ℝ^, < )) → (((𝜑 ∧ 𝑥 ∈ ℝ⁺) → ∃𝑦 ∈ 𝐴 𝐵 < (𝑦 +_𝑒 𝑥)) ↔ ((𝜑 ∧ (𝐵 − sup(𝐴, ℝ^, < )) ∈ ℝ⁺) → ∃𝑦 ∈ 𝐴 𝐵 < (𝑦 +_𝑒 (𝐵 − sup(𝐴, ℝ^*, < ))))))
144	133, 137, 143, 23	vtoclgf 3524	. . . . . . . 8 ⊢ ((𝐵 − sup(𝐴, ℝ^, < )) ∈ V → ((𝜑 ∧ (𝐵 − sup(𝐴, ℝ^, < )) ∈ ℝ⁺) → ∃𝑦 ∈ 𝐴 𝐵 < (𝑦 +_𝑒 (𝐵 − sup(𝐴, ℝ^*, < )))))
145	132, 144	ax-mp 5	. . . . . . 7 ⊢ ((𝜑 ∧ (𝐵 − sup(𝐴, ℝ^, < )) ∈ ℝ⁺) → ∃𝑦 ∈ 𝐴 𝐵 < (𝑦 +_𝑒 (𝐵 − sup(𝐴, ℝ^, < ))))
146	88, 131, 145	syl2anc 584	. . . . . 6 ⊢ (((𝜑 ∧ sup(𝐴, ℝ^, < ) ∈ ℝ) ∧ sup(𝐴, ℝ^, < ) < 𝐵) → ∃𝑦 ∈ 𝐴 𝐵 < (𝑦 +_𝑒 (𝐵 − sup(𝐴, ℝ^*, < ))))
147		ltpnf 13016	. . . . . . . . . . . . 13 ⊢ (sup(𝐴, ℝ^, < ) ∈ ℝ → sup(𝐴, ℝ^, < ) < +∞)
148	147	adantr 480	. . . . . . . . . . . 12 ⊢ ((sup(𝐴, ℝ^, < ) ∈ ℝ ∧ 𝑦 = +∞) → sup(𝐴, ℝ^, < ) < +∞)
149		id 22	. . . . . . . . . . . . . 14 ⊢ (𝑦 = +∞ → 𝑦 = +∞)
150	149	eqcomd 2737	. . . . . . . . . . . . 13 ⊢ (𝑦 = +∞ → +∞ = 𝑦)
151	150	adantl 481	. . . . . . . . . . . 12 ⊢ ((sup(𝐴, ℝ^*, < ) ∈ ℝ ∧ 𝑦 = +∞) → +∞ = 𝑦)
152	148, 151	breqtrd 5117	. . . . . . . . . . 11 ⊢ ((sup(𝐴, ℝ^, < ) ∈ ℝ ∧ 𝑦 = +∞) → sup(𝐴, ℝ^, < ) < 𝑦)
153	152	adantll 714	. . . . . . . . . 10 ⊢ (((𝜑 ∧ sup(𝐴, ℝ^, < ) ∈ ℝ) ∧ 𝑦 = +∞) → sup(𝐴, ℝ^, < ) < 𝑦)
154	153	ad5ant15 758	. . . . . . . . 9 ⊢ ((((((𝜑 ∧ sup(𝐴, ℝ^, < ) ∈ ℝ) ∧ sup(𝐴, ℝ^, < ) < 𝐵) ∧ 𝑦 ∈ 𝐴) ∧ 𝐵 < (𝑦 +_𝑒 (𝐵 − sup(𝐴, ℝ^, < )))) ∧ 𝑦 = +∞) → sup(𝐴, ℝ^, < ) < 𝑦)
155		simplll 774	. . . . . . . . . 10 ⊢ ((((((𝜑 ∧ sup(𝐴, ℝ^, < ) ∈ ℝ) ∧ sup(𝐴, ℝ^, < ) < 𝐵) ∧ 𝑦 ∈ 𝐴) ∧ 𝐵 < (𝑦 +_𝑒 (𝐵 − sup(𝐴, ℝ^, < )))) ∧ ¬ 𝑦 = +∞) → ((𝜑 ∧ sup(𝐴, ℝ^, < ) ∈ ℝ) ∧ sup(𝐴, ℝ^*, < ) < 𝐵))
156		simpl 482	. . . . . . . . . . . . . . 15 ⊢ ((((((𝜑 ∧ sup(𝐴, ℝ^, < ) ∈ ℝ) ∧ sup(𝐴, ℝ^, < ) < 𝐵) ∧ 𝑦 ∈ 𝐴) ∧ 𝐵 < (𝑦 +_𝑒 (𝐵 − sup(𝐴, ℝ^, < )))) ∧ ¬ -∞ < 𝑦) → ((((𝜑 ∧ sup(𝐴, ℝ^, < ) ∈ ℝ) ∧ sup(𝐴, ℝ^, < ) < 𝐵) ∧ 𝑦 ∈ 𝐴) ∧ 𝐵 < (𝑦 +_𝑒 (𝐵 − sup(𝐴, ℝ^, < )))))
157	88, 41	sylanl1 680	. . . . . . . . . . . . . . . 16 ⊢ (((((𝜑 ∧ sup(𝐴, ℝ^, < ) ∈ ℝ) ∧ sup(𝐴, ℝ^, < ) < 𝐵) ∧ 𝑦 ∈ 𝐴) ∧ ¬ -∞ < 𝑦) → 𝑦 = -∞)
158	157	adantlr 715	. . . . . . . . . . . . . . 15 ⊢ ((((((𝜑 ∧ sup(𝐴, ℝ^, < ) ∈ ℝ) ∧ sup(𝐴, ℝ^, < ) < 𝐵) ∧ 𝑦 ∈ 𝐴) ∧ 𝐵 < (𝑦 +_𝑒 (𝐵 − sup(𝐴, ℝ^*, < )))) ∧ ¬ -∞ < 𝑦) → 𝑦 = -∞)
159		simplr 768	. . . . . . . . . . . . . . . 16 ⊢ ((((((𝜑 ∧ sup(𝐴, ℝ^, < ) ∈ ℝ) ∧ sup(𝐴, ℝ^, < ) < 𝐵) ∧ 𝑦 ∈ 𝐴) ∧ 𝐵 < (𝑦 +_𝑒 (𝐵 − sup(𝐴, ℝ^, < )))) ∧ 𝑦 = -∞) → 𝐵 < (𝑦 +_𝑒 (𝐵 − sup(𝐴, ℝ^, < ))))
160		oveq1 7353	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑦 = -∞ → (𝑦 +_𝑒 (𝐵 − sup(𝐴, ℝ^, < ))) = (-∞ +_𝑒 (𝐵 − sup(𝐴, ℝ^, < ))))
161	160	adantl 481	. . . . . . . . . . . . . . . . 17 ⊢ ((((((𝜑 ∧ sup(𝐴, ℝ^, < ) ∈ ℝ) ∧ sup(𝐴, ℝ^, < ) < 𝐵) ∧ 𝑦 ∈ 𝐴) ∧ 𝐵 < (𝑦 +_𝑒 (𝐵 − sup(𝐴, ℝ^, < )))) ∧ 𝑦 = -∞) → (𝑦 +_𝑒 (𝐵 − sup(𝐴, ℝ^, < ))) = (-∞ +_𝑒 (𝐵 − sup(𝐴, ℝ^*, < ))))
162	128, 123	resubcld 11542	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ sup(𝐴, ℝ^, < ) ∈ ℝ) ∧ sup(𝐴, ℝ^, < ) < 𝐵) → (𝐵 − sup(𝐴, ℝ^*, < )) ∈ ℝ)
163	162	rexrd 11159	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ sup(𝐴, ℝ^, < ) ∈ ℝ) ∧ sup(𝐴, ℝ^, < ) < 𝐵) → (𝐵 − sup(𝐴, ℝ^, < )) ∈ ℝ^)
164	163	ad3antrrr 730	. . . . . . . . . . . . . . . . . 18 ⊢ ((((((𝜑 ∧ sup(𝐴, ℝ^, < ) ∈ ℝ) ∧ sup(𝐴, ℝ^, < ) < 𝐵) ∧ 𝑦 ∈ 𝐴) ∧ 𝐵 < (𝑦 +_𝑒 (𝐵 − sup(𝐴, ℝ^, < )))) ∧ 𝑦 = -∞) → (𝐵 − sup(𝐴, ℝ^, < )) ∈ ℝ^*)
165		renepnf 11157	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝐵 − sup(𝐴, ℝ^, < )) ∈ ℝ → (𝐵 − sup(𝐴, ℝ^, < )) ≠ +∞)
166	124, 165	syl 17	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ sup(𝐴, ℝ^, < ) ∈ ℝ) ∧ sup(𝐴, ℝ^, < ) < 𝐵) → (𝐵 − sup(𝐴, ℝ^*, < )) ≠ +∞)
167	166	ad3antrrr 730	. . . . . . . . . . . . . . . . . 18 ⊢ ((((((𝜑 ∧ sup(𝐴, ℝ^, < ) ∈ ℝ) ∧ sup(𝐴, ℝ^, < ) < 𝐵) ∧ 𝑦 ∈ 𝐴) ∧ 𝐵 < (𝑦 +_𝑒 (𝐵 − sup(𝐴, ℝ^, < )))) ∧ 𝑦 = -∞) → (𝐵 − sup(𝐴, ℝ^, < )) ≠ +∞)
168		xaddmnf2 13125	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝐵 − sup(𝐴, ℝ^, < )) ∈ ℝ^ ∧ (𝐵 − sup(𝐴, ℝ^, < )) ≠ +∞) → (-∞ +_𝑒 (𝐵 − sup(𝐴, ℝ^, < ))) = -∞)
169	164, 167, 168	syl2anc 584	. . . . . . . . . . . . . . . . 17 ⊢ ((((((𝜑 ∧ sup(𝐴, ℝ^, < ) ∈ ℝ) ∧ sup(𝐴, ℝ^, < ) < 𝐵) ∧ 𝑦 ∈ 𝐴) ∧ 𝐵 < (𝑦 +_𝑒 (𝐵 − sup(𝐴, ℝ^, < )))) ∧ 𝑦 = -∞) → (-∞ +_𝑒 (𝐵 − sup(𝐴, ℝ^, < ))) = -∞)
170	161, 169	eqtrd 2766	. . . . . . . . . . . . . . . 16 ⊢ ((((((𝜑 ∧ sup(𝐴, ℝ^, < ) ∈ ℝ) ∧ sup(𝐴, ℝ^, < ) < 𝐵) ∧ 𝑦 ∈ 𝐴) ∧ 𝐵 < (𝑦 +_𝑒 (𝐵 − sup(𝐴, ℝ^, < )))) ∧ 𝑦 = -∞) → (𝑦 +_𝑒 (𝐵 − sup(𝐴, ℝ^, < ))) = -∞)
171	159, 170	breqtrd 5117	. . . . . . . . . . . . . . 15 ⊢ ((((((𝜑 ∧ sup(𝐴, ℝ^, < ) ∈ ℝ) ∧ sup(𝐴, ℝ^, < ) < 𝐵) ∧ 𝑦 ∈ 𝐴) ∧ 𝐵 < (𝑦 +_𝑒 (𝐵 − sup(𝐴, ℝ^*, < )))) ∧ 𝑦 = -∞) → 𝐵 < -∞)
172	156, 158, 171	syl2anc 584	. . . . . . . . . . . . . 14 ⊢ ((((((𝜑 ∧ sup(𝐴, ℝ^, < ) ∈ ℝ) ∧ sup(𝐴, ℝ^, < ) < 𝐵) ∧ 𝑦 ∈ 𝐴) ∧ 𝐵 < (𝑦 +_𝑒 (𝐵 − sup(𝐴, ℝ^*, < )))) ∧ ¬ -∞ < 𝑦) → 𝐵 < -∞)
173	55	ad5antr 734	. . . . . . . . . . . . . 14 ⊢ ((((((𝜑 ∧ sup(𝐴, ℝ^, < ) ∈ ℝ) ∧ sup(𝐴, ℝ^, < ) < 𝐵) ∧ 𝑦 ∈ 𝐴) ∧ 𝐵 < (𝑦 +_𝑒 (𝐵 − sup(𝐴, ℝ^*, < )))) ∧ ¬ -∞ < 𝑦) → ¬ 𝐵 < -∞)
174	172, 173	condan 817	. . . . . . . . . . . . 13 ⊢ (((((𝜑 ∧ sup(𝐴, ℝ^, < ) ∈ ℝ) ∧ sup(𝐴, ℝ^, < ) < 𝐵) ∧ 𝑦 ∈ 𝐴) ∧ 𝐵 < (𝑦 +_𝑒 (𝐵 − sup(𝐴, ℝ^*, < )))) → -∞ < 𝑦)
175	174	adantr 480	. . . . . . . . . . . 12 ⊢ ((((((𝜑 ∧ sup(𝐴, ℝ^, < ) ∈ ℝ) ∧ sup(𝐴, ℝ^, < ) < 𝐵) ∧ 𝑦 ∈ 𝐴) ∧ 𝐵 < (𝑦 +_𝑒 (𝐵 − sup(𝐴, ℝ^*, < )))) ∧ ¬ 𝑦 = +∞) → -∞ < 𝑦)
176		simp3 1138	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐴 ∧ ¬ 𝑦 = +∞) → ¬ 𝑦 = +∞)
177	31	3adant3 1132	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐴 ∧ ¬ 𝑦 = +∞) → 𝑦 ∈ ℝ^*)
178		nltpnft 13060	. . . . . . . . . . . . . . . . 17 ⊢ (𝑦 ∈ ℝ^* → (𝑦 = +∞ ↔ ¬ 𝑦 < +∞))
179	177, 178	syl 17	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐴 ∧ ¬ 𝑦 = +∞) → (𝑦 = +∞ ↔ ¬ 𝑦 < +∞))
180	176, 179	mtbid 324	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐴 ∧ ¬ 𝑦 = +∞) → ¬ ¬ 𝑦 < +∞)
181	180	notnotrd 133	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐴 ∧ ¬ 𝑦 = +∞) → 𝑦 < +∞)
182	181	3adant1r 1178	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ sup(𝐴, ℝ^*, < ) ∈ ℝ) ∧ 𝑦 ∈ 𝐴 ∧ ¬ 𝑦 = +∞) → 𝑦 < +∞)
183	182	ad5ant135 1370	. . . . . . . . . . . 12 ⊢ ((((((𝜑 ∧ sup(𝐴, ℝ^, < ) ∈ ℝ) ∧ sup(𝐴, ℝ^, < ) < 𝐵) ∧ 𝑦 ∈ 𝐴) ∧ 𝐵 < (𝑦 +_𝑒 (𝐵 − sup(𝐴, ℝ^*, < )))) ∧ ¬ 𝑦 = +∞) → 𝑦 < +∞)
184	175, 183	jca 511	. . . . . . . . . . 11 ⊢ ((((((𝜑 ∧ sup(𝐴, ℝ^, < ) ∈ ℝ) ∧ sup(𝐴, ℝ^, < ) < 𝐵) ∧ 𝑦 ∈ 𝐴) ∧ 𝐵 < (𝑦 +_𝑒 (𝐵 − sup(𝐴, ℝ^*, < )))) ∧ ¬ 𝑦 = +∞) → (-∞ < 𝑦 ∧ 𝑦 < +∞))
185	31	adantlr 715	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ sup(𝐴, ℝ^, < ) ∈ ℝ) ∧ 𝑦 ∈ 𝐴) → 𝑦 ∈ ℝ^)
186	185	ad5ant13 756	. . . . . . . . . . . 12 ⊢ ((((((𝜑 ∧ sup(𝐴, ℝ^, < ) ∈ ℝ) ∧ sup(𝐴, ℝ^, < ) < 𝐵) ∧ 𝑦 ∈ 𝐴) ∧ 𝐵 < (𝑦 +_𝑒 (𝐵 − sup(𝐴, ℝ^, < )))) ∧ ¬ 𝑦 = +∞) → 𝑦 ∈ ℝ^)
187		xrrebnd 13064	. . . . . . . . . . . 12 ⊢ (𝑦 ∈ ℝ^* → (𝑦 ∈ ℝ ↔ (-∞ < 𝑦 ∧ 𝑦 < +∞)))
188	186, 187	syl 17	. . . . . . . . . . 11 ⊢ ((((((𝜑 ∧ sup(𝐴, ℝ^, < ) ∈ ℝ) ∧ sup(𝐴, ℝ^, < ) < 𝐵) ∧ 𝑦 ∈ 𝐴) ∧ 𝐵 < (𝑦 +_𝑒 (𝐵 − sup(𝐴, ℝ^*, < )))) ∧ ¬ 𝑦 = +∞) → (𝑦 ∈ ℝ ↔ (-∞ < 𝑦 ∧ 𝑦 < +∞)))
189	184, 188	mpbird 257	. . . . . . . . . 10 ⊢ ((((((𝜑 ∧ sup(𝐴, ℝ^, < ) ∈ ℝ) ∧ sup(𝐴, ℝ^, < ) < 𝐵) ∧ 𝑦 ∈ 𝐴) ∧ 𝐵 < (𝑦 +_𝑒 (𝐵 − sup(𝐴, ℝ^*, < )))) ∧ ¬ 𝑦 = +∞) → 𝑦 ∈ ℝ)
190		simplr 768	. . . . . . . . . 10 ⊢ ((((((𝜑 ∧ sup(𝐴, ℝ^, < ) ∈ ℝ) ∧ sup(𝐴, ℝ^, < ) < 𝐵) ∧ 𝑦 ∈ 𝐴) ∧ 𝐵 < (𝑦 +_𝑒 (𝐵 − sup(𝐴, ℝ^, < )))) ∧ ¬ 𝑦 = +∞) → 𝐵 < (𝑦 +_𝑒 (𝐵 − sup(𝐴, ℝ^, < ))))
191	121	ad2antrr 726	. . . . . . . . . . . . 13 ⊢ (((((𝜑 ∧ sup(𝐴, ℝ^, < ) ∈ ℝ) ∧ sup(𝐴, ℝ^, < ) < 𝐵) ∧ 𝑦 ∈ ℝ) ∧ 𝐵 < (𝑦 +_𝑒 (𝐵 − sup(𝐴, ℝ^*, < )))) → 𝐵 ∈ ℝ)
192		simpr 484	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ sup(𝐴, ℝ^, < ) ∈ ℝ) ∧ sup(𝐴, ℝ^, < ) < 𝐵) ∧ 𝑦 ∈ ℝ) → 𝑦 ∈ ℝ)
193	124	adantr 480	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ sup(𝐴, ℝ^, < ) ∈ ℝ) ∧ sup(𝐴, ℝ^, < ) < 𝐵) ∧ 𝑦 ∈ ℝ) → (𝐵 − sup(𝐴, ℝ^*, < )) ∈ ℝ)
194		rexadd 13128	. . . . . . . . . . . . . . . 16 ⊢ ((𝑦 ∈ ℝ ∧ (𝐵 − sup(𝐴, ℝ^, < )) ∈ ℝ) → (𝑦 +_𝑒 (𝐵 − sup(𝐴, ℝ^, < ))) = (𝑦 + (𝐵 − sup(𝐴, ℝ^*, < ))))
195	192, 193, 194	syl2anc 584	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ sup(𝐴, ℝ^, < ) ∈ ℝ) ∧ sup(𝐴, ℝ^, < ) < 𝐵) ∧ 𝑦 ∈ ℝ) → (𝑦 +_𝑒 (𝐵 − sup(𝐴, ℝ^, < ))) = (𝑦 + (𝐵 − sup(𝐴, ℝ^, < ))))
196	192, 193	readdcld 11138	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ sup(𝐴, ℝ^, < ) ∈ ℝ) ∧ sup(𝐴, ℝ^, < ) < 𝐵) ∧ 𝑦 ∈ ℝ) → (𝑦 + (𝐵 − sup(𝐴, ℝ^*, < ))) ∈ ℝ)
197	195, 196	eqeltrd 2831	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ sup(𝐴, ℝ^, < ) ∈ ℝ) ∧ sup(𝐴, ℝ^, < ) < 𝐵) ∧ 𝑦 ∈ ℝ) → (𝑦 +_𝑒 (𝐵 − sup(𝐴, ℝ^*, < ))) ∈ ℝ)
198	197	adantr 480	. . . . . . . . . . . . 13 ⊢ (((((𝜑 ∧ sup(𝐴, ℝ^, < ) ∈ ℝ) ∧ sup(𝐴, ℝ^, < ) < 𝐵) ∧ 𝑦 ∈ ℝ) ∧ 𝐵 < (𝑦 +_𝑒 (𝐵 − sup(𝐴, ℝ^, < )))) → (𝑦 +_𝑒 (𝐵 − sup(𝐴, ℝ^, < ))) ∈ ℝ)
199		simpr 484	. . . . . . . . . . . . 13 ⊢ (((((𝜑 ∧ sup(𝐴, ℝ^, < ) ∈ ℝ) ∧ sup(𝐴, ℝ^, < ) < 𝐵) ∧ 𝑦 ∈ ℝ) ∧ 𝐵 < (𝑦 +_𝑒 (𝐵 − sup(𝐴, ℝ^, < )))) → 𝐵 < (𝑦 +_𝑒 (𝐵 − sup(𝐴, ℝ^, < ))))
200	191, 198, 191, 199	ltsub1dd 11726	. . . . . . . . . . . 12 ⊢ (((((𝜑 ∧ sup(𝐴, ℝ^, < ) ∈ ℝ) ∧ sup(𝐴, ℝ^, < ) < 𝐵) ∧ 𝑦 ∈ ℝ) ∧ 𝐵 < (𝑦 +_𝑒 (𝐵 − sup(𝐴, ℝ^, < )))) → (𝐵 − 𝐵) < ((𝑦 +_𝑒 (𝐵 − sup(𝐴, ℝ^, < ))) − 𝐵))
201	121	recnd 11137	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ sup(𝐴, ℝ^, < ) ∈ ℝ) ∧ sup(𝐴, ℝ^, < ) < 𝐵) → 𝐵 ∈ ℂ)
202	201	subidd 11457	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ sup(𝐴, ℝ^, < ) ∈ ℝ) ∧ sup(𝐴, ℝ^, < ) < 𝐵) → (𝐵 − 𝐵) = 0)
203	202	ad2antrr 726	. . . . . . . . . . . . 13 ⊢ (((((𝜑 ∧ sup(𝐴, ℝ^, < ) ∈ ℝ) ∧ sup(𝐴, ℝ^, < ) < 𝐵) ∧ 𝑦 ∈ ℝ) ∧ 𝐵 < (𝑦 +_𝑒 (𝐵 − sup(𝐴, ℝ^*, < )))) → (𝐵 − 𝐵) = 0)
204	201	adantr 480	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ sup(𝐴, ℝ^, < ) ∈ ℝ) ∧ sup(𝐴, ℝ^, < ) < 𝐵) ∧ 𝑦 ∈ ℝ) → 𝐵 ∈ ℂ)
205	192	recnd 11137	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ sup(𝐴, ℝ^, < ) ∈ ℝ) ∧ sup(𝐴, ℝ^, < ) < 𝐵) ∧ 𝑦 ∈ ℝ) → 𝑦 ∈ ℂ)
206	122	recnd 11137	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ sup(𝐴, ℝ^, < ) ∈ ℝ) → sup(𝐴, ℝ^, < ) ∈ ℂ)
207	206	ad2antrr 726	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ sup(𝐴, ℝ^, < ) ∈ ℝ) ∧ sup(𝐴, ℝ^, < ) < 𝐵) ∧ 𝑦 ∈ ℝ) → sup(𝐴, ℝ^*, < ) ∈ ℂ)
208	205, 207	subcld 11469	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ sup(𝐴, ℝ^, < ) ∈ ℝ) ∧ sup(𝐴, ℝ^, < ) < 𝐵) ∧ 𝑦 ∈ ℝ) → (𝑦 − sup(𝐴, ℝ^*, < )) ∈ ℂ)
209	205, 204, 207	addsub12d 11492	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ sup(𝐴, ℝ^, < ) ∈ ℝ) ∧ sup(𝐴, ℝ^, < ) < 𝐵) ∧ 𝑦 ∈ ℝ) → (𝑦 + (𝐵 − sup(𝐴, ℝ^, < ))) = (𝐵 + (𝑦 − sup(𝐴, ℝ^, < ))))
210	195, 209	eqtrd 2766	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ sup(𝐴, ℝ^, < ) ∈ ℝ) ∧ sup(𝐴, ℝ^, < ) < 𝐵) ∧ 𝑦 ∈ ℝ) → (𝑦 +_𝑒 (𝐵 − sup(𝐴, ℝ^, < ))) = (𝐵 + (𝑦 − sup(𝐴, ℝ^, < ))))
211	204, 208, 210	mvrladdd 11527	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ sup(𝐴, ℝ^, < ) ∈ ℝ) ∧ sup(𝐴, ℝ^, < ) < 𝐵) ∧ 𝑦 ∈ ℝ) → ((𝑦 +_𝑒 (𝐵 − sup(𝐴, ℝ^, < ))) − 𝐵) = (𝑦 − sup(𝐴, ℝ^, < )))
212	211	adantr 480	. . . . . . . . . . . . 13 ⊢ (((((𝜑 ∧ sup(𝐴, ℝ^, < ) ∈ ℝ) ∧ sup(𝐴, ℝ^, < ) < 𝐵) ∧ 𝑦 ∈ ℝ) ∧ 𝐵 < (𝑦 +_𝑒 (𝐵 − sup(𝐴, ℝ^, < )))) → ((𝑦 +_𝑒 (𝐵 − sup(𝐴, ℝ^, < ))) − 𝐵) = (𝑦 − sup(𝐴, ℝ^*, < )))
213	203, 212	breq12d 5104	. . . . . . . . . . . 12 ⊢ (((((𝜑 ∧ sup(𝐴, ℝ^, < ) ∈ ℝ) ∧ sup(𝐴, ℝ^, < ) < 𝐵) ∧ 𝑦 ∈ ℝ) ∧ 𝐵 < (𝑦 +_𝑒 (𝐵 − sup(𝐴, ℝ^, < )))) → ((𝐵 − 𝐵) < ((𝑦 +_𝑒 (𝐵 − sup(𝐴, ℝ^, < ))) − 𝐵) ↔ 0 < (𝑦 − sup(𝐴, ℝ^*, < ))))
214	200, 213	mpbid 232	. . . . . . . . . . 11 ⊢ (((((𝜑 ∧ sup(𝐴, ℝ^, < ) ∈ ℝ) ∧ sup(𝐴, ℝ^, < ) < 𝐵) ∧ 𝑦 ∈ ℝ) ∧ 𝐵 < (𝑦 +_𝑒 (𝐵 − sup(𝐴, ℝ^, < )))) → 0 < (𝑦 − sup(𝐴, ℝ^, < )))
215	123	ad2antrr 726	. . . . . . . . . . . 12 ⊢ (((((𝜑 ∧ sup(𝐴, ℝ^, < ) ∈ ℝ) ∧ sup(𝐴, ℝ^, < ) < 𝐵) ∧ 𝑦 ∈ ℝ) ∧ 𝐵 < (𝑦 +_𝑒 (𝐵 − sup(𝐴, ℝ^, < )))) → sup(𝐴, ℝ^, < ) ∈ ℝ)
216		simplr 768	. . . . . . . . . . . 12 ⊢ (((((𝜑 ∧ sup(𝐴, ℝ^, < ) ∈ ℝ) ∧ sup(𝐴, ℝ^, < ) < 𝐵) ∧ 𝑦 ∈ ℝ) ∧ 𝐵 < (𝑦 +_𝑒 (𝐵 − sup(𝐴, ℝ^*, < )))) → 𝑦 ∈ ℝ)
217	215, 216	posdifd 11701	. . . . . . . . . . 11 ⊢ (((((𝜑 ∧ sup(𝐴, ℝ^, < ) ∈ ℝ) ∧ sup(𝐴, ℝ^, < ) < 𝐵) ∧ 𝑦 ∈ ℝ) ∧ 𝐵 < (𝑦 +_𝑒 (𝐵 − sup(𝐴, ℝ^, < )))) → (sup(𝐴, ℝ^, < ) < 𝑦 ↔ 0 < (𝑦 − sup(𝐴, ℝ^*, < ))))
218	214, 217	mpbird 257	. . . . . . . . . 10 ⊢ (((((𝜑 ∧ sup(𝐴, ℝ^, < ) ∈ ℝ) ∧ sup(𝐴, ℝ^, < ) < 𝐵) ∧ 𝑦 ∈ ℝ) ∧ 𝐵 < (𝑦 +_𝑒 (𝐵 − sup(𝐴, ℝ^, < )))) → sup(𝐴, ℝ^, < ) < 𝑦)
219	155, 189, 190, 218	syl21anc 837	. . . . . . . . 9 ⊢ ((((((𝜑 ∧ sup(𝐴, ℝ^, < ) ∈ ℝ) ∧ sup(𝐴, ℝ^, < ) < 𝐵) ∧ 𝑦 ∈ 𝐴) ∧ 𝐵 < (𝑦 +_𝑒 (𝐵 − sup(𝐴, ℝ^, < )))) ∧ ¬ 𝑦 = +∞) → sup(𝐴, ℝ^, < ) < 𝑦)
220	154, 219	pm2.61dan 812	. . . . . . . 8 ⊢ (((((𝜑 ∧ sup(𝐴, ℝ^, < ) ∈ ℝ) ∧ sup(𝐴, ℝ^, < ) < 𝐵) ∧ 𝑦 ∈ 𝐴) ∧ 𝐵 < (𝑦 +_𝑒 (𝐵 − sup(𝐴, ℝ^, < )))) → sup(𝐴, ℝ^, < ) < 𝑦)
221	220	ex 412	. . . . . . 7 ⊢ ((((𝜑 ∧ sup(𝐴, ℝ^, < ) ∈ ℝ) ∧ sup(𝐴, ℝ^, < ) < 𝐵) ∧ 𝑦 ∈ 𝐴) → (𝐵 < (𝑦 +_𝑒 (𝐵 − sup(𝐴, ℝ^, < ))) → sup(𝐴, ℝ^, < ) < 𝑦))
222	221	reximdva 3145	. . . . . 6 ⊢ (((𝜑 ∧ sup(𝐴, ℝ^, < ) ∈ ℝ) ∧ sup(𝐴, ℝ^, < ) < 𝐵) → (∃𝑦 ∈ 𝐴 𝐵 < (𝑦 +_𝑒 (𝐵 − sup(𝐴, ℝ^, < ))) → ∃𝑦 ∈ 𝐴 sup(𝐴, ℝ^, < ) < 𝑦))
223	146, 222	mpd 15	. . . . 5 ⊢ (((𝜑 ∧ sup(𝐴, ℝ^, < ) ∈ ℝ) ∧ sup(𝐴, ℝ^, < ) < 𝐵) → ∃𝑦 ∈ 𝐴 sup(𝐴, ℝ^*, < ) < 𝑦)
224	80, 87, 223	syl2anc 584	. . . 4 ⊢ (((𝜑 ∧ sup(𝐴, ℝ^, < ) ∈ ℝ) ∧ ¬ 𝐵 ≤ sup(𝐴, ℝ^, < )) → ∃𝑦 ∈ 𝐴 sup(𝐴, ℝ^*, < ) < 𝑦)
225	59, 33	syl 17	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐴) → sup(𝐴, ℝ^, < ) ∈ ℝ^)
226	31, 225	xrlenltd 11175	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐴) → (𝑦 ≤ sup(𝐴, ℝ^, < ) ↔ ¬ sup(𝐴, ℝ^, < ) < 𝑦))
227	62, 226	mpbid 232	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐴) → ¬ sup(𝐴, ℝ^*, < ) < 𝑦)
228	227	ralrimiva 3124	. . . . . 6 ⊢ (𝜑 → ∀𝑦 ∈ 𝐴 ¬ sup(𝐴, ℝ^*, < ) < 𝑦)
229		ralnex 3058	. . . . . 6 ⊢ (∀𝑦 ∈ 𝐴 ¬ sup(𝐴, ℝ^, < ) < 𝑦 ↔ ¬ ∃𝑦 ∈ 𝐴 sup(𝐴, ℝ^, < ) < 𝑦)
230	228, 229	sylib 218	. . . . 5 ⊢ (𝜑 → ¬ ∃𝑦 ∈ 𝐴 sup(𝐴, ℝ^*, < ) < 𝑦)
231	230	ad2antrr 726	. . . 4 ⊢ (((𝜑 ∧ sup(𝐴, ℝ^, < ) ∈ ℝ) ∧ ¬ 𝐵 ≤ sup(𝐴, ℝ^, < )) → ¬ ∃𝑦 ∈ 𝐴 sup(𝐴, ℝ^*, < ) < 𝑦)
232	224, 231	condan 817	. . 3 ⊢ ((𝜑 ∧ sup(𝐴, ℝ^, < ) ∈ ℝ) → 𝐵 ≤ sup(𝐴, ℝ^, < ))
233	9, 79, 232	syl2anc 584	. 2 ⊢ ((𝜑 ∧ ¬ sup(𝐴, ℝ^, < ) = +∞) → 𝐵 ≤ sup(𝐴, ℝ^, < ))
234	8, 233	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 1541 Ⅎwnf 1784 ∈ wcel 2111 ≠ wne 2928 ∀wral 3047 ∃wrex 3056 Vcvv 3436 ⊆ wss 3902 class class class wbr 5091 (class class class)co 7346 supcsup 9324 ℂcc 11001 ℝcr 11002 0cc0 11003 1c1 11004 + caddc 11006 +∞cpnf 11140 -∞cmnf 11141 ℝ^*cxr 11142 < clt 11143 ≤ cle 11144 − cmin 11341 ℝ⁺crp 12887 +_𝑒 cxad 13006

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 1968 ax-7 2009 ax-8 2113 ax-9 2121 ax-10 2144 ax-11 2160 ax-12 2180 ax-ext 2703 ax-sep 5234 ax-nul 5244 ax-pow 5303 ax-pr 5370 ax-un 7668 ax-cnex 11059 ax-resscn 11060 ax-1cn 11061 ax-icn 11062 ax-addcl 11063 ax-addrcl 11064 ax-mulcl 11065 ax-mulrcl 11066 ax-mulcom 11067 ax-addass 11068 ax-mulass 11069 ax-distr 11070 ax-i2m1 11071 ax-1ne0 11072 ax-1rid 11073 ax-rnegex 11074 ax-rrecex 11075 ax-cnre 11076 ax-pre-lttri 11077 ax-pre-lttrn 11078 ax-pre-ltadd 11079 ax-pre-mulgt0 11080 ax-pre-sup 11081

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1544 df-fal 1554 df-ex 1781 df-nf 1785 df-sb 2068 df-mo 2535 df-eu 2564 df-clab 2710 df-cleq 2723 df-clel 2806 df-nfc 2881 df-ne 2929 df-nel 3033 df-ral 3048 df-rex 3057 df-rmo 3346 df-reu 3347 df-rab 3396 df-v 3438 df-sbc 3742 df-csb 3851 df-dif 3905 df-un 3907 df-in 3909 df-ss 3919 df-nul 4284 df-if 4476 df-pw 4552 df-sn 4577 df-pr 4579 df-op 4583 df-uni 4860 df-iun 4943 df-br 5092 df-opab 5154 df-mpt 5173 df-id 5511 df-po 5524 df-so 5525 df-xp 5622 df-rel 5623 df-cnv 5624 df-co 5625 df-dm 5626 df-rn 5627 df-res 5628 df-ima 5629 df-iota 6437 df-fun 6483 df-fn 6484 df-f 6485 df-f1 6486 df-fo 6487 df-f1o 6488 df-fv 6489 df-riota 7303 df-ov 7349 df-oprab 7350 df-mpo 7351 df-1st 7921 df-2nd 7922 df-er 8622 df-en 8870 df-dom 8871 df-sdom 8872 df-sup 9326 df-pnf 11145 df-mnf 11146 df-xr 11147 df-ltxr 11148 df-le 11149 df-sub 11343 df-neg 11344 df-rp 12888 df-xadd 13009

This theorem is referenced by: supxrge 45376

W3C validator