suplesup - Mathbox for Glauco Siliprandi

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Theorem suplesup 46459

Description: If any element of 𝐴 can be approximated from below by members of 𝐵, then the supremum of 𝐴 is less than or equal to the supremum of 𝐵. (Contributed by Glauco Siliprandi, 17-Aug-2020.)

**Hypotheses**
Ref	Expression
suplesup.a	⊢ (𝜑 → 𝐴 ⊆ ℝ)
suplesup.b	⊢ (𝜑 → 𝐵 ⊆ ℝ^*)
suplesup.c	⊢ (𝜑 → ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ ℝ⁺ ∃𝑧 ∈ 𝐵 (𝑥 − 𝑦) < 𝑧)

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

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

Proof of Theorem suplesup Dummy variables 𝑟 𝑤 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		suplesup.a	. . . . . 6 ⊢ (𝜑 → 𝐴 ⊆ ℝ)
2		ressxr 11360	. . . . . 6 ⊢ ℝ ⊆ ℝ^*
3	1, 2	sstrdi 3921	. . . . 5 ⊢ (𝜑 → 𝐴 ⊆ ℝ^*)
4		supxrcl 13452	. . . . 5 ⊢ (𝐴 ⊆ ℝ^* → sup(𝐴, ℝ^, < ) ∈ ℝ^)
5	3, 4	syl 18	. . . 4 ⊢ (𝜑 → sup(𝐴, ℝ^, < ) ∈ ℝ^)
6	5	adantr 486	. . 3 ⊢ ((𝜑 ∧ sup(𝐴, ℝ^, < ) = +∞) → sup(𝐴, ℝ^, < ) ∈ ℝ^*)
7		eqidd 2761	. . . 4 ⊢ ((𝜑 ∧ sup(𝐴, ℝ^*, < ) = +∞) → +∞ = +∞)
8		simpr 490	. . . 4 ⊢ ((𝜑 ∧ sup(𝐴, ℝ^, < ) = +∞) → sup(𝐴, ℝ^, < ) = +∞)
9		peano2re 11490	. . . . . . . . . 10 ⊢ (𝑤 ∈ ℝ → (𝑤 + 1) ∈ ℝ)
10	9	adantl 487	. . . . . . . . 9 ⊢ (((𝜑 ∧ sup(𝐴, ℝ^*, < ) = +∞) ∧ 𝑤 ∈ ℝ) → (𝑤 + 1) ∈ ℝ)
11	3	adantr 486	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ sup(𝐴, ℝ^, < ) = +∞) → 𝐴 ⊆ ℝ^)
12		supxrunb2 13457	. . . . . . . . . . . 12 ⊢ (𝐴 ⊆ ℝ^* → (∀𝑟 ∈ ℝ ∃𝑥 ∈ 𝐴 𝑟 < 𝑥 ↔ sup(𝐴, ℝ^*, < ) = +∞))
13	11, 12	syl 18	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ sup(𝐴, ℝ^, < ) = +∞) → (∀𝑟 ∈ ℝ ∃𝑥 ∈ 𝐴 𝑟 < 𝑥 ↔ sup(𝐴, ℝ^, < ) = +∞))
14	8, 13	mpbird 260	. . . . . . . . . 10 ⊢ ((𝜑 ∧ sup(𝐴, ℝ^*, < ) = +∞) → ∀𝑟 ∈ ℝ ∃𝑥 ∈ 𝐴 𝑟 < 𝑥)
15	14	adantr 486	. . . . . . . . 9 ⊢ (((𝜑 ∧ sup(𝐴, ℝ^*, < ) = +∞) ∧ 𝑤 ∈ ℝ) → ∀𝑟 ∈ ℝ ∃𝑥 ∈ 𝐴 𝑟 < 𝑥)
16		breq1 5105	. . . . . . . . . . 11 ⊢ (𝑟 = (𝑤 + 1) → (𝑟 < 𝑥 ↔ (𝑤 + 1) < 𝑥))
17	16	rexbidv 3186	. . . . . . . . . 10 ⊢ (𝑟 = (𝑤 + 1) → (∃𝑥 ∈ 𝐴 𝑟 < 𝑥 ↔ ∃𝑥 ∈ 𝐴 (𝑤 + 1) < 𝑥))
18	17	rspcva 3574	. . . . . . . . 9 ⊢ (((𝑤 + 1) ∈ ℝ ∧ ∀𝑟 ∈ ℝ ∃𝑥 ∈ 𝐴 𝑟 < 𝑥) → ∃𝑥 ∈ 𝐴 (𝑤 + 1) < 𝑥)
19	10, 15, 18	syl2anc 596	. . . . . . . 8 ⊢ (((𝜑 ∧ sup(𝐴, ℝ^*, < ) = +∞) ∧ 𝑤 ∈ ℝ) → ∃𝑥 ∈ 𝐴 (𝑤 + 1) < 𝑥)
20		1rp 13131	. . . . . . . . . . . . . . . 16 ⊢ 1 ∈ ℝ⁺
21	20	a1i 11	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 1 ∈ ℝ⁺)
22		suplesup.c	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ ℝ⁺ ∃𝑧 ∈ 𝐵 (𝑥 − 𝑦) < 𝑧)
23	22	r19.21bi 3254	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ∀𝑦 ∈ ℝ⁺ ∃𝑧 ∈ 𝐵 (𝑥 − 𝑦) < 𝑧)
24		oveq2 7432	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑦 = 1 → (𝑥 − 𝑦) = (𝑥 − 1))
25	24	breq1d 5112	. . . . . . . . . . . . . . . . 17 ⊢ (𝑦 = 1 → ((𝑥 − 𝑦) < 𝑧 ↔ (𝑥 − 1) < 𝑧))
26	25	rexbidv 3186	. . . . . . . . . . . . . . . 16 ⊢ (𝑦 = 1 → (∃𝑧 ∈ 𝐵 (𝑥 − 𝑦) < 𝑧 ↔ ∃𝑧 ∈ 𝐵 (𝑥 − 1) < 𝑧))
27	26	rspcva 3574	. . . . . . . . . . . . . . 15 ⊢ ((1 ∈ ℝ⁺ ∧ ∀𝑦 ∈ ℝ⁺ ∃𝑧 ∈ 𝐵 (𝑥 − 𝑦) < 𝑧) → ∃𝑧 ∈ 𝐵 (𝑥 − 1) < 𝑧)
28	21, 23, 27	syl2anc 596	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ∃𝑧 ∈ 𝐵 (𝑥 − 1) < 𝑧)
29	28	adantlr 728	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑤 ∈ ℝ) ∧ 𝑥 ∈ 𝐴) → ∃𝑧 ∈ 𝐵 (𝑥 − 1) < 𝑧)
30	29	3adant3 1150	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑤 ∈ ℝ) ∧ 𝑥 ∈ 𝐴 ∧ (𝑤 + 1) < 𝑥) → ∃𝑧 ∈ 𝐵 (𝑥 − 1) < 𝑧)
31		nfv 1946	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑧((𝜑 ∧ 𝑤 ∈ ℝ) ∧ 𝑥 ∈ 𝐴 ∧ (𝑤 + 1) < 𝑥)
32		simp11r 1304	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ 𝑤 ∈ ℝ) ∧ 𝑥 ∈ 𝐴 ∧ (𝑤 + 1) < 𝑥) ∧ 𝑧 ∈ 𝐵 ∧ (𝑥 − 1) < 𝑧) → 𝑤 ∈ ℝ)
33	2, 32	sselid 3907	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ 𝑤 ∈ ℝ) ∧ 𝑥 ∈ 𝐴 ∧ (𝑤 + 1) < 𝑥) ∧ 𝑧 ∈ 𝐵 ∧ (𝑥 − 1) < 𝑧) → 𝑤 ∈ ℝ^*)
34	1	sselda 3909	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝑥 ∈ ℝ)
35		1red 11316	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 1 ∈ ℝ)
36	34, 35	resubcld 11751	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝑥 − 1) ∈ ℝ)
37	36	adantlr 728	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑤 ∈ ℝ) ∧ 𝑥 ∈ 𝐴) → (𝑥 − 1) ∈ ℝ)
38	37	3adant3 1150	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑤 ∈ ℝ) ∧ 𝑥 ∈ 𝐴 ∧ (𝑤 + 1) < 𝑥) → (𝑥 − 1) ∈ ℝ)
39	38	3ad2ant1 1151	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ 𝑤 ∈ ℝ) ∧ 𝑥 ∈ 𝐴 ∧ (𝑤 + 1) < 𝑥) ∧ 𝑧 ∈ 𝐵 ∧ (𝑥 − 1) < 𝑧) → (𝑥 − 1) ∈ ℝ)
40	2, 39	sselid 3907	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ 𝑤 ∈ ℝ) ∧ 𝑥 ∈ 𝐴 ∧ (𝑤 + 1) < 𝑥) ∧ 𝑧 ∈ 𝐵 ∧ (𝑥 − 1) < 𝑧) → (𝑥 − 1) ∈ ℝ^*)
41		suplesup.b	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → 𝐵 ⊆ ℝ^*)
42	41	sselda 3909	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐵) → 𝑧 ∈ ℝ^*)
43	42	adantlr 728	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑤 ∈ ℝ) ∧ 𝑧 ∈ 𝐵) → 𝑧 ∈ ℝ^*)
44	43	3ad2antl1 1204	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ 𝑤 ∈ ℝ) ∧ 𝑥 ∈ 𝐴 ∧ (𝑤 + 1) < 𝑥) ∧ 𝑧 ∈ 𝐵) → 𝑧 ∈ ℝ^*)
45	44	3adant3 1150	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ 𝑤 ∈ ℝ) ∧ 𝑥 ∈ 𝐴 ∧ (𝑤 + 1) < 𝑥) ∧ 𝑧 ∈ 𝐵 ∧ (𝑥 − 1) < 𝑧) → 𝑧 ∈ ℝ^*)
46		simp3 1156	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑤 ∈ ℝ) ∧ 𝑥 ∈ 𝐴 ∧ (𝑤 + 1) < 𝑥) → (𝑤 + 1) < 𝑥)
47		simp1r 1217	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑤 ∈ ℝ) ∧ 𝑥 ∈ 𝐴 ∧ (𝑤 + 1) < 𝑥) → 𝑤 ∈ ℝ)
48		1red 11316	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑤 ∈ ℝ) ∧ 𝑥 ∈ 𝐴 ∧ (𝑤 + 1) < 𝑥) → 1 ∈ ℝ)
49	34	adantlr 728	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ 𝑤 ∈ ℝ) ∧ 𝑥 ∈ 𝐴) → 𝑥 ∈ ℝ)
50	49	3adant3 1150	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑤 ∈ ℝ) ∧ 𝑥 ∈ 𝐴 ∧ (𝑤 + 1) < 𝑥) → 𝑥 ∈ ℝ)
51	47, 48, 50	ltaddsubd 11923	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑤 ∈ ℝ) ∧ 𝑥 ∈ 𝐴 ∧ (𝑤 + 1) < 𝑥) → ((𝑤 + 1) < 𝑥 ↔ 𝑤 < (𝑥 − 1)))
52	46, 51	mpbid 235	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑤 ∈ ℝ) ∧ 𝑥 ∈ 𝐴 ∧ (𝑤 + 1) < 𝑥) → 𝑤 < (𝑥 − 1))
53	52	3ad2ant1 1151	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ 𝑤 ∈ ℝ) ∧ 𝑥 ∈ 𝐴 ∧ (𝑤 + 1) < 𝑥) ∧ 𝑧 ∈ 𝐵 ∧ (𝑥 − 1) < 𝑧) → 𝑤 < (𝑥 − 1))
54		simp3 1156	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ 𝑤 ∈ ℝ) ∧ 𝑥 ∈ 𝐴 ∧ (𝑤 + 1) < 𝑥) ∧ 𝑧 ∈ 𝐵 ∧ (𝑥 − 1) < 𝑧) → (𝑥 − 1) < 𝑧)
55	33, 40, 45, 53, 54	xrlttrd 13295	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑤 ∈ ℝ) ∧ 𝑥 ∈ 𝐴 ∧ (𝑤 + 1) < 𝑥) ∧ 𝑧 ∈ 𝐵 ∧ (𝑥 − 1) < 𝑧) → 𝑤 < 𝑧)
56	55	3exp 1137	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑤 ∈ ℝ) ∧ 𝑥 ∈ 𝐴 ∧ (𝑤 + 1) < 𝑥) → (𝑧 ∈ 𝐵 → ((𝑥 − 1) < 𝑧 → 𝑤 < 𝑧)))
57	31, 56	reximdai 3264	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑤 ∈ ℝ) ∧ 𝑥 ∈ 𝐴 ∧ (𝑤 + 1) < 𝑥) → (∃𝑧 ∈ 𝐵 (𝑥 − 1) < 𝑧 → ∃𝑧 ∈ 𝐵 𝑤 < 𝑧))
58	30, 57	mpd 16	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑤 ∈ ℝ) ∧ 𝑥 ∈ 𝐴 ∧ (𝑤 + 1) < 𝑥) → ∃𝑧 ∈ 𝐵 𝑤 < 𝑧)
59	58	3exp 1137	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑤 ∈ ℝ) → (𝑥 ∈ 𝐴 → ((𝑤 + 1) < 𝑥 → ∃𝑧 ∈ 𝐵 𝑤 < 𝑧)))
60	59	adantlr 728	. . . . . . . . 9 ⊢ (((𝜑 ∧ sup(𝐴, ℝ^*, < ) = +∞) ∧ 𝑤 ∈ ℝ) → (𝑥 ∈ 𝐴 → ((𝑤 + 1) < 𝑥 → ∃𝑧 ∈ 𝐵 𝑤 < 𝑧)))
61	60	rexlimdv 3161	. . . . . . . 8 ⊢ (((𝜑 ∧ sup(𝐴, ℝ^*, < ) = +∞) ∧ 𝑤 ∈ ℝ) → (∃𝑥 ∈ 𝐴 (𝑤 + 1) < 𝑥 → ∃𝑧 ∈ 𝐵 𝑤 < 𝑧))
62	19, 61	mpd 16	. . . . . . 7 ⊢ (((𝜑 ∧ sup(𝐴, ℝ^*, < ) = +∞) ∧ 𝑤 ∈ ℝ) → ∃𝑧 ∈ 𝐵 𝑤 < 𝑧)
63	2	a1i 11	. . . . . . . . . . . . 13 ⊢ (𝜑 → ℝ ⊆ ℝ^*)
64	63	sselda 3909	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑤 ∈ ℝ) → 𝑤 ∈ ℝ^*)
65	64	ad2antrr 739	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑤 ∈ ℝ) ∧ 𝑧 ∈ 𝐵) ∧ 𝑤 < 𝑧) → 𝑤 ∈ ℝ^*)
66	43	adantr 486	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑤 ∈ ℝ) ∧ 𝑧 ∈ 𝐵) ∧ 𝑤 < 𝑧) → 𝑧 ∈ ℝ^*)
67		simpr 490	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑤 ∈ ℝ) ∧ 𝑧 ∈ 𝐵) ∧ 𝑤 < 𝑧) → 𝑤 < 𝑧)
68	65, 66, 67	xrltled 13286	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑤 ∈ ℝ) ∧ 𝑧 ∈ 𝐵) ∧ 𝑤 < 𝑧) → 𝑤 ≤ 𝑧)
69	68	ex 418	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑤 ∈ ℝ) ∧ 𝑧 ∈ 𝐵) → (𝑤 < 𝑧 → 𝑤 ≤ 𝑧))
70	69	adantllr 732	. . . . . . . 8 ⊢ ((((𝜑 ∧ sup(𝐴, ℝ^*, < ) = +∞) ∧ 𝑤 ∈ ℝ) ∧ 𝑧 ∈ 𝐵) → (𝑤 < 𝑧 → 𝑤 ≤ 𝑧))
71	70	reximdva 3175	. . . . . . 7 ⊢ (((𝜑 ∧ sup(𝐴, ℝ^*, < ) = +∞) ∧ 𝑤 ∈ ℝ) → (∃𝑧 ∈ 𝐵 𝑤 < 𝑧 → ∃𝑧 ∈ 𝐵 𝑤 ≤ 𝑧))
72	62, 71	mpd 16	. . . . . 6 ⊢ (((𝜑 ∧ sup(𝐴, ℝ^*, < ) = +∞) ∧ 𝑤 ∈ ℝ) → ∃𝑧 ∈ 𝐵 𝑤 ≤ 𝑧)
73	72	ralrimiva 3154	. . . . 5 ⊢ ((𝜑 ∧ sup(𝐴, ℝ^*, < ) = +∞) → ∀𝑤 ∈ ℝ ∃𝑧 ∈ 𝐵 𝑤 ≤ 𝑧)
74		supxrunb1 13456	. . . . . . 7 ⊢ (𝐵 ⊆ ℝ^* → (∀𝑤 ∈ ℝ ∃𝑧 ∈ 𝐵 𝑤 ≤ 𝑧 ↔ sup(𝐵, ℝ^*, < ) = +∞))
75	41, 74	syl 18	. . . . . 6 ⊢ (𝜑 → (∀𝑤 ∈ ℝ ∃𝑧 ∈ 𝐵 𝑤 ≤ 𝑧 ↔ sup(𝐵, ℝ^*, < ) = +∞))
76	75	adantr 486	. . . . 5 ⊢ ((𝜑 ∧ sup(𝐴, ℝ^, < ) = +∞) → (∀𝑤 ∈ ℝ ∃𝑧 ∈ 𝐵 𝑤 ≤ 𝑧 ↔ sup(𝐵, ℝ^, < ) = +∞))
77	73, 76	mpbid 235	. . . 4 ⊢ ((𝜑 ∧ sup(𝐴, ℝ^, < ) = +∞) → sup(𝐵, ℝ^, < ) = +∞)
78	7, 8, 77	3eqtr4d 2805	. . 3 ⊢ ((𝜑 ∧ sup(𝐴, ℝ^, < ) = +∞) → sup(𝐴, ℝ^, < ) = sup(𝐵, ℝ^*, < ))
79	6, 78	xreqled 46450	. 2 ⊢ ((𝜑 ∧ sup(𝐴, ℝ^, < ) = +∞) → sup(𝐴, ℝ^, < ) ≤ sup(𝐵, ℝ^*, < ))
80		supeq1 9444	. . . . . . 7 ⊢ (𝐴 = ∅ → sup(𝐴, ℝ^, < ) = sup(∅, ℝ^, < ))
81		xrsup0 13460	. . . . . . . 8 ⊢ sup(∅, ℝ^*, < ) = -∞
82	81	a1i 11	. . . . . . 7 ⊢ (𝐴 = ∅ → sup(∅, ℝ^*, < ) = -∞)
83	80, 82	eqtrd 2795	. . . . . 6 ⊢ (𝐴 = ∅ → sup(𝐴, ℝ^*, < ) = -∞)
84	83	adantl 487	. . . . 5 ⊢ ((𝜑 ∧ 𝐴 = ∅) → sup(𝐴, ℝ^*, < ) = -∞)
85		supxrcl 13452	. . . . . . . 8 ⊢ (𝐵 ⊆ ℝ^* → sup(𝐵, ℝ^, < ) ∈ ℝ^)
86	41, 85	syl 18	. . . . . . 7 ⊢ (𝜑 → sup(𝐵, ℝ^, < ) ∈ ℝ^)
87		mnfle 13271	. . . . . . 7 ⊢ (sup(𝐵, ℝ^, < ) ∈ ℝ^ → -∞ ≤ sup(𝐵, ℝ^*, < ))
88	86, 87	syl 18	. . . . . 6 ⊢ (𝜑 → -∞ ≤ sup(𝐵, ℝ^*, < ))
89	88	adantr 486	. . . . 5 ⊢ ((𝜑 ∧ 𝐴 = ∅) → -∞ ≤ sup(𝐵, ℝ^*, < ))
90	84, 89	eqbrtrd 5126	. . . 4 ⊢ ((𝜑 ∧ 𝐴 = ∅) → sup(𝐴, ℝ^, < ) ≤ sup(𝐵, ℝ^, < ))
91	90	adantlr 728	. . 3 ⊢ (((𝜑 ∧ ¬ sup(𝐴, ℝ^, < ) = +∞) ∧ 𝐴 = ∅) → sup(𝐴, ℝ^, < ) ≤ sup(𝐵, ℝ^*, < ))
92		simpll 779	. . . 4 ⊢ (((𝜑 ∧ ¬ sup(𝐴, ℝ^*, < ) = +∞) ∧ ¬ 𝐴 = ∅) → 𝜑)
93	1	adantr 486	. . . . . . . 8 ⊢ ((𝜑 ∧ ¬ 𝐴 = ∅) → 𝐴 ⊆ ℝ)
94		neqne 2963	. . . . . . . . 9 ⊢ (¬ 𝐴 = ∅ → 𝐴 ≠ ∅)
95	94	adantl 487	. . . . . . . 8 ⊢ ((𝜑 ∧ ¬ 𝐴 = ∅) → 𝐴 ≠ ∅)
96		supxrgtmnf 13466	. . . . . . . 8 ⊢ ((𝐴 ⊆ ℝ ∧ 𝐴 ≠ ∅) → -∞ < sup(𝐴, ℝ^*, < ))
97	93, 95, 96	syl2anc 596	. . . . . . 7 ⊢ ((𝜑 ∧ ¬ 𝐴 = ∅) → -∞ < sup(𝐴, ℝ^*, < ))
98	97	adantlr 728	. . . . . 6 ⊢ (((𝜑 ∧ ¬ sup(𝐴, ℝ^, < ) = +∞) ∧ ¬ 𝐴 = ∅) → -∞ < sup(𝐴, ℝ^, < ))
99		simpr 490	. . . . . . . . 9 ⊢ ((𝜑 ∧ ¬ sup(𝐴, ℝ^, < ) = +∞) → ¬ sup(𝐴, ℝ^, < ) = +∞)
100		simpl 488	. . . . . . . . . 10 ⊢ ((𝜑 ∧ ¬ sup(𝐴, ℝ^*, < ) = +∞) → 𝜑)
101		nltpnft 13301	. . . . . . . . . 10 ⊢ (sup(𝐴, ℝ^, < ) ∈ ℝ^ → (sup(𝐴, ℝ^, < ) = +∞ ↔ ¬ sup(𝐴, ℝ^, < ) < +∞))
102	100, 5, 101	3syl 19	. . . . . . . . 9 ⊢ ((𝜑 ∧ ¬ sup(𝐴, ℝ^, < ) = +∞) → (sup(𝐴, ℝ^, < ) = +∞ ↔ ¬ sup(𝐴, ℝ^*, < ) < +∞))
103	99, 102	mtbid 327	. . . . . . . 8 ⊢ ((𝜑 ∧ ¬ sup(𝐴, ℝ^, < ) = +∞) → ¬ ¬ sup(𝐴, ℝ^, < ) < +∞)
104		notnotr 131	. . . . . . . 8 ⊢ (¬ ¬ sup(𝐴, ℝ^, < ) < +∞ → sup(𝐴, ℝ^, < ) < +∞)
105	103, 104	syl 18	. . . . . . 7 ⊢ ((𝜑 ∧ ¬ sup(𝐴, ℝ^, < ) = +∞) → sup(𝐴, ℝ^, < ) < +∞)
106	105	adantr 486	. . . . . 6 ⊢ (((𝜑 ∧ ¬ sup(𝐴, ℝ^, < ) = +∞) ∧ ¬ 𝐴 = ∅) → sup(𝐴, ℝ^, < ) < +∞)
107	98, 106	jca 521	. . . . 5 ⊢ (((𝜑 ∧ ¬ sup(𝐴, ℝ^, < ) = +∞) ∧ ¬ 𝐴 = ∅) → (-∞ < sup(𝐴, ℝ^, < ) ∧ sup(𝐴, ℝ^*, < ) < +∞))
108	92, 5	syl 18	. . . . . 6 ⊢ (((𝜑 ∧ ¬ sup(𝐴, ℝ^, < ) = +∞) ∧ ¬ 𝐴 = ∅) → sup(𝐴, ℝ^, < ) ∈ ℝ^*)
109		xrrebnd 13305	. . . . . 6 ⊢ (sup(𝐴, ℝ^, < ) ∈ ℝ^ → (sup(𝐴, ℝ^, < ) ∈ ℝ ↔ (-∞ < sup(𝐴, ℝ^, < ) ∧ sup(𝐴, ℝ^*, < ) < +∞)))
110	108, 109	syl 18	. . . . 5 ⊢ (((𝜑 ∧ ¬ sup(𝐴, ℝ^, < ) = +∞) ∧ ¬ 𝐴 = ∅) → (sup(𝐴, ℝ^, < ) ∈ ℝ ↔ (-∞ < sup(𝐴, ℝ^, < ) ∧ sup(𝐴, ℝ^, < ) < +∞)))
111	107, 110	mpbird 260	. . . 4 ⊢ (((𝜑 ∧ ¬ sup(𝐴, ℝ^, < ) = +∞) ∧ ¬ 𝐴 = ∅) → sup(𝐴, ℝ^, < ) ∈ ℝ)
112		nfv 1946	. . . . 5 ⊢ Ⅎ𝑤(𝜑 ∧ sup(𝐴, ℝ^*, < ) ∈ ℝ)
113	41	adantr 486	. . . . 5 ⊢ ((𝜑 ∧ sup(𝐴, ℝ^, < ) ∈ ℝ) → 𝐵 ⊆ ℝ^)
114		simpr 490	. . . . 5 ⊢ ((𝜑 ∧ sup(𝐴, ℝ^, < ) ∈ ℝ) → sup(𝐴, ℝ^, < ) ∈ ℝ)
115	114	adantr 486	. . . . . . . 8 ⊢ (((𝜑 ∧ sup(𝐴, ℝ^, < ) ∈ ℝ) ∧ 𝑤 ∈ ℝ⁺) → sup(𝐴, ℝ^, < ) ∈ ℝ)
116		simpr 490	. . . . . . . . 9 ⊢ (((𝜑 ∧ sup(𝐴, ℝ^*, < ) ∈ ℝ) ∧ 𝑤 ∈ ℝ⁺) → 𝑤 ∈ ℝ⁺)
117	116	rphalfcld 13183	. . . . . . . 8 ⊢ (((𝜑 ∧ sup(𝐴, ℝ^*, < ) ∈ ℝ) ∧ 𝑤 ∈ ℝ⁺) → (𝑤 / 2) ∈ ℝ⁺)
118	115, 117	ltsubrpd 13203	. . . . . . 7 ⊢ (((𝜑 ∧ sup(𝐴, ℝ^, < ) ∈ ℝ) ∧ 𝑤 ∈ ℝ⁺) → (sup(𝐴, ℝ^, < ) − (𝑤 / 2)) < sup(𝐴, ℝ^*, < ))
119	3	ad2antrr 739	. . . . . . . 8 ⊢ (((𝜑 ∧ sup(𝐴, ℝ^, < ) ∈ ℝ) ∧ 𝑤 ∈ ℝ⁺) → 𝐴 ⊆ ℝ^)
120		rpre 13136	. . . . . . . . . . . 12 ⊢ (𝑤 ∈ ℝ⁺ → 𝑤 ∈ ℝ)
121		2re 12424	. . . . . . . . . . . . 13 ⊢ 2 ∈ ℝ
122	121	a1i 11	. . . . . . . . . . . 12 ⊢ (𝑤 ∈ ℝ⁺ → 2 ∈ ℝ)
123		2ne0 12456	. . . . . . . . . . . . 13 ⊢ 2 ≠ 0
124	123	a1i 11	. . . . . . . . . . . 12 ⊢ (𝑤 ∈ ℝ⁺ → 2 ≠ 0)
125	120, 122, 124	redivcld 12152	. . . . . . . . . . 11 ⊢ (𝑤 ∈ ℝ⁺ → (𝑤 / 2) ∈ ℝ)
126	125	adantl 487	. . . . . . . . . 10 ⊢ (((𝜑 ∧ sup(𝐴, ℝ^*, < ) ∈ ℝ) ∧ 𝑤 ∈ ℝ⁺) → (𝑤 / 2) ∈ ℝ)
127	115, 126	resubcld 11751	. . . . . . . . 9 ⊢ (((𝜑 ∧ sup(𝐴, ℝ^, < ) ∈ ℝ) ∧ 𝑤 ∈ ℝ⁺) → (sup(𝐴, ℝ^, < ) − (𝑤 / 2)) ∈ ℝ)
128	2, 127	sselid 3907	. . . . . . . 8 ⊢ (((𝜑 ∧ sup(𝐴, ℝ^, < ) ∈ ℝ) ∧ 𝑤 ∈ ℝ⁺) → (sup(𝐴, ℝ^, < ) − (𝑤 / 2)) ∈ ℝ^*)
129		supxrlub 13462	. . . . . . . 8 ⊢ ((𝐴 ⊆ ℝ^* ∧ (sup(𝐴, ℝ^, < ) − (𝑤 / 2)) ∈ ℝ^) → ((sup(𝐴, ℝ^, < ) − (𝑤 / 2)) < sup(𝐴, ℝ^, < ) ↔ ∃𝑥 ∈ 𝐴 (sup(𝐴, ℝ^*, < ) − (𝑤 / 2)) < 𝑥))
130	119, 128, 129	syl2anc 596	. . . . . . 7 ⊢ (((𝜑 ∧ sup(𝐴, ℝ^, < ) ∈ ℝ) ∧ 𝑤 ∈ ℝ⁺) → ((sup(𝐴, ℝ^, < ) − (𝑤 / 2)) < sup(𝐴, ℝ^, < ) ↔ ∃𝑥 ∈ 𝐴 (sup(𝐴, ℝ^, < ) − (𝑤 / 2)) < 𝑥))
131	118, 130	mpbid 235	. . . . . 6 ⊢ (((𝜑 ∧ sup(𝐴, ℝ^, < ) ∈ ℝ) ∧ 𝑤 ∈ ℝ⁺) → ∃𝑥 ∈ 𝐴 (sup(𝐴, ℝ^, < ) − (𝑤 / 2)) < 𝑥)
132		rphalfcl 13156	. . . . . . . . . . 11 ⊢ (𝑤 ∈ ℝ⁺ → (𝑤 / 2) ∈ ℝ⁺)
133	132	3ad2ant2 1152	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑤 ∈ ℝ⁺ ∧ 𝑥 ∈ 𝐴) → (𝑤 / 2) ∈ ℝ⁺)
134	23	3adant2 1149	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑤 ∈ ℝ⁺ ∧ 𝑥 ∈ 𝐴) → ∀𝑦 ∈ ℝ⁺ ∃𝑧 ∈ 𝐵 (𝑥 − 𝑦) < 𝑧)
135		oveq2 7432	. . . . . . . . . . . . 13 ⊢ (𝑦 = (𝑤 / 2) → (𝑥 − 𝑦) = (𝑥 − (𝑤 / 2)))
136	135	breq1d 5112	. . . . . . . . . . . 12 ⊢ (𝑦 = (𝑤 / 2) → ((𝑥 − 𝑦) < 𝑧 ↔ (𝑥 − (𝑤 / 2)) < 𝑧))
137	136	rexbidv 3186	. . . . . . . . . . 11 ⊢ (𝑦 = (𝑤 / 2) → (∃𝑧 ∈ 𝐵 (𝑥 − 𝑦) < 𝑧 ↔ ∃𝑧 ∈ 𝐵 (𝑥 − (𝑤 / 2)) < 𝑧))
138	137	rspcva 3574	. . . . . . . . . 10 ⊢ (((𝑤 / 2) ∈ ℝ⁺ ∧ ∀𝑦 ∈ ℝ⁺ ∃𝑧 ∈ 𝐵 (𝑥 − 𝑦) < 𝑧) → ∃𝑧 ∈ 𝐵 (𝑥 − (𝑤 / 2)) < 𝑧)
139	133, 134, 138	syl2anc 596	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑤 ∈ ℝ⁺ ∧ 𝑥 ∈ 𝐴) → ∃𝑧 ∈ 𝐵 (𝑥 − (𝑤 / 2)) < 𝑧)
140	139	ad5ant134 1392	. . . . . . . 8 ⊢ (((((𝜑 ∧ sup(𝐴, ℝ^, < ) ∈ ℝ) ∧ 𝑤 ∈ ℝ⁺) ∧ 𝑥 ∈ 𝐴) ∧ (sup(𝐴, ℝ^, < ) − (𝑤 / 2)) < 𝑥) → ∃𝑧 ∈ 𝐵 (𝑥 − (𝑤 / 2)) < 𝑧)
141		recn 11297	. . . . . . . . . . . . . . . . 17 ⊢ (sup(𝐴, ℝ^, < ) ∈ ℝ → sup(𝐴, ℝ^, < ) ∈ ℂ)
142	141	adantr 486	. . . . . . . . . . . . . . . 16 ⊢ ((sup(𝐴, ℝ^, < ) ∈ ℝ ∧ 𝑤 ∈ ℝ⁺) → sup(𝐴, ℝ^, < ) ∈ ℂ)
143	120	recnd 11344	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑤 ∈ ℝ⁺ → 𝑤 ∈ ℂ)
144	143	adantl 487	. . . . . . . . . . . . . . . . 17 ⊢ ((sup(𝐴, ℝ^*, < ) ∈ ℝ ∧ 𝑤 ∈ ℝ⁺) → 𝑤 ∈ ℂ)
145	144	halfcld 12598	. . . . . . . . . . . . . . . 16 ⊢ ((sup(𝐴, ℝ^*, < ) ∈ ℝ ∧ 𝑤 ∈ ℝ⁺) → (𝑤 / 2) ∈ ℂ)
146	142, 145, 145	subsub4d 11707	. . . . . . . . . . . . . . 15 ⊢ ((sup(𝐴, ℝ^, < ) ∈ ℝ ∧ 𝑤 ∈ ℝ⁺) → ((sup(𝐴, ℝ^, < ) − (𝑤 / 2)) − (𝑤 / 2)) = (sup(𝐴, ℝ^*, < ) − ((𝑤 / 2) + (𝑤 / 2))))
147	143	2halvesd 12599	. . . . . . . . . . . . . . . . 17 ⊢ (𝑤 ∈ ℝ⁺ → ((𝑤 / 2) + (𝑤 / 2)) = 𝑤)
148	147	oveq2d 7440	. . . . . . . . . . . . . . . 16 ⊢ (𝑤 ∈ ℝ⁺ → (sup(𝐴, ℝ^, < ) − ((𝑤 / 2) + (𝑤 / 2))) = (sup(𝐴, ℝ^, < ) − 𝑤))
149	148	adantl 487	. . . . . . . . . . . . . . 15 ⊢ ((sup(𝐴, ℝ^, < ) ∈ ℝ ∧ 𝑤 ∈ ℝ⁺) → (sup(𝐴, ℝ^, < ) − ((𝑤 / 2) + (𝑤 / 2))) = (sup(𝐴, ℝ^*, < ) − 𝑤))
150	146, 149	eqtr2d 2796	. . . . . . . . . . . . . 14 ⊢ ((sup(𝐴, ℝ^, < ) ∈ ℝ ∧ 𝑤 ∈ ℝ⁺) → (sup(𝐴, ℝ^, < ) − 𝑤) = ((sup(𝐴, ℝ^*, < ) − (𝑤 / 2)) − (𝑤 / 2)))
151	150	adantll 727	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ sup(𝐴, ℝ^, < ) ∈ ℝ) ∧ 𝑤 ∈ ℝ⁺) → (sup(𝐴, ℝ^, < ) − 𝑤) = ((sup(𝐴, ℝ^*, < ) − (𝑤 / 2)) − (𝑤 / 2)))
152	151	adantr 486	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ sup(𝐴, ℝ^, < ) ∈ ℝ) ∧ 𝑤 ∈ ℝ⁺) ∧ 𝑥 ∈ 𝐴) → (sup(𝐴, ℝ^, < ) − 𝑤) = ((sup(𝐴, ℝ^*, < ) − (𝑤 / 2)) − (𝑤 / 2)))
153	152	ad3antrrr 743	. . . . . . . . . . 11 ⊢ (((((((𝜑 ∧ sup(𝐴, ℝ^, < ) ∈ ℝ) ∧ 𝑤 ∈ ℝ⁺) ∧ 𝑥 ∈ 𝐴) ∧ (sup(𝐴, ℝ^, < ) − (𝑤 / 2)) < 𝑥) ∧ 𝑧 ∈ 𝐵) ∧ (𝑥 − (𝑤 / 2)) < 𝑧) → (sup(𝐴, ℝ^, < ) − 𝑤) = ((sup(𝐴, ℝ^, < ) − (𝑤 / 2)) − (𝑤 / 2)))
154	127, 126	resubcld 11751	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ sup(𝐴, ℝ^, < ) ∈ ℝ) ∧ 𝑤 ∈ ℝ⁺) → ((sup(𝐴, ℝ^, < ) − (𝑤 / 2)) − (𝑤 / 2)) ∈ ℝ)
155	154	adantr 486	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ sup(𝐴, ℝ^, < ) ∈ ℝ) ∧ 𝑤 ∈ ℝ⁺) ∧ 𝑥 ∈ 𝐴) → ((sup(𝐴, ℝ^, < ) − (𝑤 / 2)) − (𝑤 / 2)) ∈ ℝ)
156	155	ad3antrrr 743	. . . . . . . . . . . . 13 ⊢ (((((((𝜑 ∧ sup(𝐴, ℝ^, < ) ∈ ℝ) ∧ 𝑤 ∈ ℝ⁺) ∧ 𝑥 ∈ 𝐴) ∧ (sup(𝐴, ℝ^, < ) − (𝑤 / 2)) < 𝑥) ∧ 𝑧 ∈ 𝐵) ∧ (𝑥 − (𝑤 / 2)) < 𝑧) → ((sup(𝐴, ℝ^*, < ) − (𝑤 / 2)) − (𝑤 / 2)) ∈ ℝ)
157	2, 156	sselid 3907	. . . . . . . . . . . 12 ⊢ (((((((𝜑 ∧ sup(𝐴, ℝ^, < ) ∈ ℝ) ∧ 𝑤 ∈ ℝ⁺) ∧ 𝑥 ∈ 𝐴) ∧ (sup(𝐴, ℝ^, < ) − (𝑤 / 2)) < 𝑥) ∧ 𝑧 ∈ 𝐵) ∧ (𝑥 − (𝑤 / 2)) < 𝑧) → ((sup(𝐴, ℝ^, < ) − (𝑤 / 2)) − (𝑤 / 2)) ∈ ℝ^)
158	120, 49	sylanl2 694	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑤 ∈ ℝ⁺) ∧ 𝑥 ∈ 𝐴) → 𝑥 ∈ ℝ)
159	125	ad2antlr 740	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑤 ∈ ℝ⁺) ∧ 𝑥 ∈ 𝐴) → (𝑤 / 2) ∈ ℝ)
160	158, 159	resubcld 11751	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑤 ∈ ℝ⁺) ∧ 𝑥 ∈ 𝐴) → (𝑥 − (𝑤 / 2)) ∈ ℝ)
161	160	adantllr 732	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ sup(𝐴, ℝ^*, < ) ∈ ℝ) ∧ 𝑤 ∈ ℝ⁺) ∧ 𝑥 ∈ 𝐴) → (𝑥 − (𝑤 / 2)) ∈ ℝ)
162	161	ad3antrrr 743	. . . . . . . . . . . . 13 ⊢ (((((((𝜑 ∧ sup(𝐴, ℝ^, < ) ∈ ℝ) ∧ 𝑤 ∈ ℝ⁺) ∧ 𝑥 ∈ 𝐴) ∧ (sup(𝐴, ℝ^, < ) − (𝑤 / 2)) < 𝑥) ∧ 𝑧 ∈ 𝐵) ∧ (𝑥 − (𝑤 / 2)) < 𝑧) → (𝑥 − (𝑤 / 2)) ∈ ℝ)
163	2, 162	sselid 3907	. . . . . . . . . . . 12 ⊢ (((((((𝜑 ∧ sup(𝐴, ℝ^, < ) ∈ ℝ) ∧ 𝑤 ∈ ℝ⁺) ∧ 𝑥 ∈ 𝐴) ∧ (sup(𝐴, ℝ^, < ) − (𝑤 / 2)) < 𝑥) ∧ 𝑧 ∈ 𝐵) ∧ (𝑥 − (𝑤 / 2)) < 𝑧) → (𝑥 − (𝑤 / 2)) ∈ ℝ^*)
164		simp-6l 799	. . . . . . . . . . . . 13 ⊢ (((((((𝜑 ∧ sup(𝐴, ℝ^, < ) ∈ ℝ) ∧ 𝑤 ∈ ℝ⁺) ∧ 𝑥 ∈ 𝐴) ∧ (sup(𝐴, ℝ^, < ) − (𝑤 / 2)) < 𝑥) ∧ 𝑧 ∈ 𝐵) ∧ (𝑥 − (𝑤 / 2)) < 𝑧) → 𝜑)
165		simplr 781	. . . . . . . . . . . . 13 ⊢ (((((((𝜑 ∧ sup(𝐴, ℝ^, < ) ∈ ℝ) ∧ 𝑤 ∈ ℝ⁺) ∧ 𝑥 ∈ 𝐴) ∧ (sup(𝐴, ℝ^, < ) − (𝑤 / 2)) < 𝑥) ∧ 𝑧 ∈ 𝐵) ∧ (𝑥 − (𝑤 / 2)) < 𝑧) → 𝑧 ∈ 𝐵)
166	164, 165, 42	syl2anc 596	. . . . . . . . . . . 12 ⊢ (((((((𝜑 ∧ sup(𝐴, ℝ^, < ) ∈ ℝ) ∧ 𝑤 ∈ ℝ⁺) ∧ 𝑥 ∈ 𝐴) ∧ (sup(𝐴, ℝ^, < ) − (𝑤 / 2)) < 𝑥) ∧ 𝑧 ∈ 𝐵) ∧ (𝑥 − (𝑤 / 2)) < 𝑧) → 𝑧 ∈ ℝ^*)
167		simp-6r 800	. . . . . . . . . . . . . 14 ⊢ (((((((𝜑 ∧ sup(𝐴, ℝ^, < ) ∈ ℝ) ∧ 𝑤 ∈ ℝ⁺) ∧ 𝑥 ∈ 𝐴) ∧ (sup(𝐴, ℝ^, < ) − (𝑤 / 2)) < 𝑥) ∧ 𝑧 ∈ 𝐵) ∧ (𝑥 − (𝑤 / 2)) < 𝑧) → sup(𝐴, ℝ^*, < ) ∈ ℝ)
168	120	ad5antlr 748	. . . . . . . . . . . . . . 15 ⊢ (((((((𝜑 ∧ sup(𝐴, ℝ^, < ) ∈ ℝ) ∧ 𝑤 ∈ ℝ⁺) ∧ 𝑥 ∈ 𝐴) ∧ (sup(𝐴, ℝ^, < ) − (𝑤 / 2)) < 𝑥) ∧ 𝑧 ∈ 𝐵) ∧ (𝑥 − (𝑤 / 2)) < 𝑧) → 𝑤 ∈ ℝ)
169	168	rehalfcld 12600	. . . . . . . . . . . . . 14 ⊢ (((((((𝜑 ∧ sup(𝐴, ℝ^, < ) ∈ ℝ) ∧ 𝑤 ∈ ℝ⁺) ∧ 𝑥 ∈ 𝐴) ∧ (sup(𝐴, ℝ^, < ) − (𝑤 / 2)) < 𝑥) ∧ 𝑧 ∈ 𝐵) ∧ (𝑥 − (𝑤 / 2)) < 𝑧) → (𝑤 / 2) ∈ ℝ)
170	167, 169	resubcld 11751	. . . . . . . . . . . . 13 ⊢ (((((((𝜑 ∧ sup(𝐴, ℝ^, < ) ∈ ℝ) ∧ 𝑤 ∈ ℝ⁺) ∧ 𝑥 ∈ 𝐴) ∧ (sup(𝐴, ℝ^, < ) − (𝑤 / 2)) < 𝑥) ∧ 𝑧 ∈ 𝐵) ∧ (𝑥 − (𝑤 / 2)) < 𝑧) → (sup(𝐴, ℝ^*, < ) − (𝑤 / 2)) ∈ ℝ)
171		simp-4r 796	. . . . . . . . . . . . . 14 ⊢ (((((((𝜑 ∧ sup(𝐴, ℝ^, < ) ∈ ℝ) ∧ 𝑤 ∈ ℝ⁺) ∧ 𝑥 ∈ 𝐴) ∧ (sup(𝐴, ℝ^, < ) − (𝑤 / 2)) < 𝑥) ∧ 𝑧 ∈ 𝐵) ∧ (𝑥 − (𝑤 / 2)) < 𝑧) → 𝑥 ∈ 𝐴)
172	164, 171, 34	syl2anc 596	. . . . . . . . . . . . 13 ⊢ (((((((𝜑 ∧ sup(𝐴, ℝ^, < ) ∈ ℝ) ∧ 𝑤 ∈ ℝ⁺) ∧ 𝑥 ∈ 𝐴) ∧ (sup(𝐴, ℝ^, < ) − (𝑤 / 2)) < 𝑥) ∧ 𝑧 ∈ 𝐵) ∧ (𝑥 − (𝑤 / 2)) < 𝑧) → 𝑥 ∈ ℝ)
173		simpllr 788	. . . . . . . . . . . . 13 ⊢ (((((((𝜑 ∧ sup(𝐴, ℝ^, < ) ∈ ℝ) ∧ 𝑤 ∈ ℝ⁺) ∧ 𝑥 ∈ 𝐴) ∧ (sup(𝐴, ℝ^, < ) − (𝑤 / 2)) < 𝑥) ∧ 𝑧 ∈ 𝐵) ∧ (𝑥 − (𝑤 / 2)) < 𝑧) → (sup(𝐴, ℝ^*, < ) − (𝑤 / 2)) < 𝑥)
174	170, 172, 169, 173	ltsub1dd 11935	. . . . . . . . . . . 12 ⊢ (((((((𝜑 ∧ sup(𝐴, ℝ^, < ) ∈ ℝ) ∧ 𝑤 ∈ ℝ⁺) ∧ 𝑥 ∈ 𝐴) ∧ (sup(𝐴, ℝ^, < ) − (𝑤 / 2)) < 𝑥) ∧ 𝑧 ∈ 𝐵) ∧ (𝑥 − (𝑤 / 2)) < 𝑧) → ((sup(𝐴, ℝ^*, < ) − (𝑤 / 2)) − (𝑤 / 2)) < (𝑥 − (𝑤 / 2)))
175		simpr 490	. . . . . . . . . . . 12 ⊢ (((((((𝜑 ∧ sup(𝐴, ℝ^, < ) ∈ ℝ) ∧ 𝑤 ∈ ℝ⁺) ∧ 𝑥 ∈ 𝐴) ∧ (sup(𝐴, ℝ^, < ) − (𝑤 / 2)) < 𝑥) ∧ 𝑧 ∈ 𝐵) ∧ (𝑥 − (𝑤 / 2)) < 𝑧) → (𝑥 − (𝑤 / 2)) < 𝑧)
176	157, 163, 166, 174, 175	xrlttrd 13295	. . . . . . . . . . 11 ⊢ (((((((𝜑 ∧ sup(𝐴, ℝ^, < ) ∈ ℝ) ∧ 𝑤 ∈ ℝ⁺) ∧ 𝑥 ∈ 𝐴) ∧ (sup(𝐴, ℝ^, < ) − (𝑤 / 2)) < 𝑥) ∧ 𝑧 ∈ 𝐵) ∧ (𝑥 − (𝑤 / 2)) < 𝑧) → ((sup(𝐴, ℝ^*, < ) − (𝑤 / 2)) − (𝑤 / 2)) < 𝑧)
177	153, 176	eqbrtrd 5126	. . . . . . . . . 10 ⊢ (((((((𝜑 ∧ sup(𝐴, ℝ^, < ) ∈ ℝ) ∧ 𝑤 ∈ ℝ⁺) ∧ 𝑥 ∈ 𝐴) ∧ (sup(𝐴, ℝ^, < ) − (𝑤 / 2)) < 𝑥) ∧ 𝑧 ∈ 𝐵) ∧ (𝑥 − (𝑤 / 2)) < 𝑧) → (sup(𝐴, ℝ^*, < ) − 𝑤) < 𝑧)
178	177	ex 418	. . . . . . . . 9 ⊢ ((((((𝜑 ∧ sup(𝐴, ℝ^, < ) ∈ ℝ) ∧ 𝑤 ∈ ℝ⁺) ∧ 𝑥 ∈ 𝐴) ∧ (sup(𝐴, ℝ^, < ) − (𝑤 / 2)) < 𝑥) ∧ 𝑧 ∈ 𝐵) → ((𝑥 − (𝑤 / 2)) < 𝑧 → (sup(𝐴, ℝ^*, < ) − 𝑤) < 𝑧))
179	178	reximdva 3175	. . . . . . . 8 ⊢ (((((𝜑 ∧ sup(𝐴, ℝ^, < ) ∈ ℝ) ∧ 𝑤 ∈ ℝ⁺) ∧ 𝑥 ∈ 𝐴) ∧ (sup(𝐴, ℝ^, < ) − (𝑤 / 2)) < 𝑥) → (∃𝑧 ∈ 𝐵 (𝑥 − (𝑤 / 2)) < 𝑧 → ∃𝑧 ∈ 𝐵 (sup(𝐴, ℝ^*, < ) − 𝑤) < 𝑧))
180	140, 179	mpd 16	. . . . . . 7 ⊢ (((((𝜑 ∧ sup(𝐴, ℝ^, < ) ∈ ℝ) ∧ 𝑤 ∈ ℝ⁺) ∧ 𝑥 ∈ 𝐴) ∧ (sup(𝐴, ℝ^, < ) − (𝑤 / 2)) < 𝑥) → ∃𝑧 ∈ 𝐵 (sup(𝐴, ℝ^*, < ) − 𝑤) < 𝑧)
181	180	rexlimdva2 3165	. . . . . 6 ⊢ (((𝜑 ∧ sup(𝐴, ℝ^, < ) ∈ ℝ) ∧ 𝑤 ∈ ℝ⁺) → (∃𝑥 ∈ 𝐴 (sup(𝐴, ℝ^, < ) − (𝑤 / 2)) < 𝑥 → ∃𝑧 ∈ 𝐵 (sup(𝐴, ℝ^*, < ) − 𝑤) < 𝑧))
182	131, 181	mpd 16	. . . . 5 ⊢ (((𝜑 ∧ sup(𝐴, ℝ^, < ) ∈ ℝ) ∧ 𝑤 ∈ ℝ⁺) → ∃𝑧 ∈ 𝐵 (sup(𝐴, ℝ^, < ) − 𝑤) < 𝑧)
183	112, 113, 114, 182	supxrgere 46453	. . . 4 ⊢ ((𝜑 ∧ sup(𝐴, ℝ^, < ) ∈ ℝ) → sup(𝐴, ℝ^, < ) ≤ sup(𝐵, ℝ^*, < ))
184	92, 111, 183	syl2anc 596	. . 3 ⊢ (((𝜑 ∧ ¬ sup(𝐴, ℝ^, < ) = +∞) ∧ ¬ 𝐴 = ∅) → sup(𝐴, ℝ^, < ) ≤ sup(𝐵, ℝ^*, < ))
185	91, 184	pm2.61dan 825	. 2 ⊢ ((𝜑 ∧ ¬ sup(𝐴, ℝ^, < ) = +∞) → sup(𝐴, ℝ^, < ) ≤ sup(𝐵, ℝ^*, < ))
186	79, 185	pm2.61dan 825	1 ⊢ (𝜑 → sup(𝐴, ℝ^, < ) ≤ sup(𝐵, ℝ^, < ))

Colors of variables: wff setvar class

This proof depends on syntax axioms: ¬ wn 3 → wi 4 ↔ wb 209 ∧ wa 401 ∧ w3a 1103 = wceq 1569 ∈ wcel 2144 ≠ wne 2955 ∀wral 3076 ∃wrex 3086 ⊆ wss 3896 ∅c0 4277 class class class wbr 5102 (class class class)co 7424 supcsup 9439 ℂcc 11205 ℝcr 11206 0cc0 11207 1c1 11208 + caddc 11210 +∞cpnf 11347 -∞cmnf 11348 ℝ^*cxr 11349 < clt 11350 ≤ cle 11351 − cmin 11548 / cdiv 11980 2c2 12404 ℝ⁺crp 13127

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1827 ax-4 1841 ax-5 1942 ax-6 1999 ax-7 2040 ax-8 2146 ax-9 2154 ax-10 2177 ax-11 2193 ax-12 2212 ax-ext 2732 ax-sep 5248 ax-nul 5259 ax-pow 5327 ax-pr 5391 ax-un 7755 ax-cnex 11263 ax-resscn 11264 ax-1cn 11265 ax-icn 11266 ax-addcl 11267 ax-addrcl 11268 ax-mulcl 11269 ax-mulrcl 11270 ax-mulcom 11271 ax-addass 11272 ax-mulass 11273 ax-distr 11274 ax-i2m1 11275 ax-1ne0 11276 ax-1rid 11277 ax-rnegex 11278 ax-rrecex 11279 ax-cnre 11280 ax-pre-lttri 11281 ax-pre-lttrn 11282 ax-pre-ltadd 11283 ax-pre-mulgt0 11284 ax-pre-sup 11285

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1572 df-fal 1582 df-ex 1812 df-nf 1816 df-sb 2099 df-mo 2564 df-eu 2594 df-clab 2739 df-cleq 2752 df-clel 2835 df-nfc 2909 df-ne 2956 df-nel 3062 df-ral 3077 df-rex 3087 df-rmo 3365 df-reu 3366 df-rab 3413 df-v 3452 df-sbc 3739 df-csb 3848 df-ss 3897 df-pss 4013 df-dif 4046 df-in 4090 df-un 4150 df-nul 4278 df-if 4482 df-pw 4558 df-sn 4585 df-pr 4587 df-op 4835 df-uni 4867 df-iun 4952 df-br 5103 df-opab 5167 df-mpt 5186 df-tr 5212 df-id 5546 df-eprel 5551 df-po 5559 df-so 5560 df-fr 5604 df-we 5606 df-xp 5657 df-rel 5658 df-cnv 5659 df-co 5660 df-dm 5661 df-rn 5662 df-res 5663 df-ima 5664 df-pred 6307 df-ord 6368 df-on 6369 df-lim 6370 df-suc 6371 df-iota 6497 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-riota 7381 df-ov 7427 df-oprab 7428 df-mpo 7429 df-om 7882 df-2nd 8006 df-frecs 8306 df-wrecs 8337 df-recs 8386 df-rdg 8425 df-er 8724 df-en 8981 df-dom 8982 df-sdom 8983 df-sup 9441 df-pnf 11352 df-mnf 11353 df-xr 11354 df-ltxr 11355 df-le 11356 df-sub 11550 df-neg 11551 df-div 11981 df-nn 12343 df-2 12412 df-rp 13128

This theorem is used by: (None)

W3C validator