suplesup - Mathbox for Glauco Siliprandi

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Theorem suplesup 45378

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 11151	. . . . . 6 ⊢ ℝ ⊆ ℝ^*
3	1, 2	sstrdi 3942	. . . . 5 ⊢ (𝜑 → 𝐴 ⊆ ℝ^*)
4		supxrcl 13209	. . . . 5 ⊢ (𝐴 ⊆ ℝ^* → sup(𝐴, ℝ^, < ) ∈ ℝ^)
5	3, 4	syl 17	. . . 4 ⊢ (𝜑 → sup(𝐴, ℝ^, < ) ∈ ℝ^)
6	5	adantr 480	. . 3 ⊢ ((𝜑 ∧ sup(𝐴, ℝ^, < ) = +∞) → sup(𝐴, ℝ^, < ) ∈ ℝ^*)
7		eqidd 2732	. . . 4 ⊢ ((𝜑 ∧ sup(𝐴, ℝ^*, < ) = +∞) → +∞ = +∞)
8		simpr 484	. . . 4 ⊢ ((𝜑 ∧ sup(𝐴, ℝ^, < ) = +∞) → sup(𝐴, ℝ^, < ) = +∞)
9		peano2re 11281	. . . . . . . . . 10 ⊢ (𝑤 ∈ ℝ → (𝑤 + 1) ∈ ℝ)
10	9	adantl 481	. . . . . . . . 9 ⊢ (((𝜑 ∧ sup(𝐴, ℝ^*, < ) = +∞) ∧ 𝑤 ∈ ℝ) → (𝑤 + 1) ∈ ℝ)
11	3	adantr 480	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ sup(𝐴, ℝ^, < ) = +∞) → 𝐴 ⊆ ℝ^)
12		supxrunb2 13214	. . . . . . . . . . . 12 ⊢ (𝐴 ⊆ ℝ^* → (∀𝑟 ∈ ℝ ∃𝑥 ∈ 𝐴 𝑟 < 𝑥 ↔ sup(𝐴, ℝ^*, < ) = +∞))
13	11, 12	syl 17	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ sup(𝐴, ℝ^, < ) = +∞) → (∀𝑟 ∈ ℝ ∃𝑥 ∈ 𝐴 𝑟 < 𝑥 ↔ sup(𝐴, ℝ^, < ) = +∞))
14	8, 13	mpbird 257	. . . . . . . . . 10 ⊢ ((𝜑 ∧ sup(𝐴, ℝ^*, < ) = +∞) → ∀𝑟 ∈ ℝ ∃𝑥 ∈ 𝐴 𝑟 < 𝑥)
15	14	adantr 480	. . . . . . . . 9 ⊢ (((𝜑 ∧ sup(𝐴, ℝ^*, < ) = +∞) ∧ 𝑤 ∈ ℝ) → ∀𝑟 ∈ ℝ ∃𝑥 ∈ 𝐴 𝑟 < 𝑥)
16		breq1 5089	. . . . . . . . . . 11 ⊢ (𝑟 = (𝑤 + 1) → (𝑟 < 𝑥 ↔ (𝑤 + 1) < 𝑥))
17	16	rexbidv 3156	. . . . . . . . . 10 ⊢ (𝑟 = (𝑤 + 1) → (∃𝑥 ∈ 𝐴 𝑟 < 𝑥 ↔ ∃𝑥 ∈ 𝐴 (𝑤 + 1) < 𝑥))
18	17	rspcva 3570	. . . . . . . . 9 ⊢ (((𝑤 + 1) ∈ ℝ ∧ ∀𝑟 ∈ ℝ ∃𝑥 ∈ 𝐴 𝑟 < 𝑥) → ∃𝑥 ∈ 𝐴 (𝑤 + 1) < 𝑥)
19	10, 15, 18	syl2anc 584	. . . . . . . 8 ⊢ (((𝜑 ∧ sup(𝐴, ℝ^*, < ) = +∞) ∧ 𝑤 ∈ ℝ) → ∃𝑥 ∈ 𝐴 (𝑤 + 1) < 𝑥)
20		1rp 12889	. . . . . . . . . . . . . . . 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 3224	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ∀𝑦 ∈ ℝ⁺ ∃𝑧 ∈ 𝐵 (𝑥 − 𝑦) < 𝑧)
24		oveq2 7349	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑦 = 1 → (𝑥 − 𝑦) = (𝑥 − 1))
25	24	breq1d 5096	. . . . . . . . . . . . . . . . 17 ⊢ (𝑦 = 1 → ((𝑥 − 𝑦) < 𝑧 ↔ (𝑥 − 1) < 𝑧))
26	25	rexbidv 3156	. . . . . . . . . . . . . . . 16 ⊢ (𝑦 = 1 → (∃𝑧 ∈ 𝐵 (𝑥 − 𝑦) < 𝑧 ↔ ∃𝑧 ∈ 𝐵 (𝑥 − 1) < 𝑧))
27	26	rspcva 3570	. . . . . . . . . . . . . . 15 ⊢ ((1 ∈ ℝ⁺ ∧ ∀𝑦 ∈ ℝ⁺ ∃𝑧 ∈ 𝐵 (𝑥 − 𝑦) < 𝑧) → ∃𝑧 ∈ 𝐵 (𝑥 − 1) < 𝑧)
28	21, 23, 27	syl2anc 584	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ∃𝑧 ∈ 𝐵 (𝑥 − 1) < 𝑧)
29	28	adantlr 715	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑤 ∈ ℝ) ∧ 𝑥 ∈ 𝐴) → ∃𝑧 ∈ 𝐵 (𝑥 − 1) < 𝑧)
30	29	3adant3 1132	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑤 ∈ ℝ) ∧ 𝑥 ∈ 𝐴 ∧ (𝑤 + 1) < 𝑥) → ∃𝑧 ∈ 𝐵 (𝑥 − 1) < 𝑧)
31		nfv 1915	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑧((𝜑 ∧ 𝑤 ∈ ℝ) ∧ 𝑥 ∈ 𝐴 ∧ (𝑤 + 1) < 𝑥)
32		simp11r 1286	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ 𝑤 ∈ ℝ) ∧ 𝑥 ∈ 𝐴 ∧ (𝑤 + 1) < 𝑥) ∧ 𝑧 ∈ 𝐵 ∧ (𝑥 − 1) < 𝑧) → 𝑤 ∈ ℝ)
33	2, 32	sselid 3927	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ 𝑤 ∈ ℝ) ∧ 𝑥 ∈ 𝐴 ∧ (𝑤 + 1) < 𝑥) ∧ 𝑧 ∈ 𝐵 ∧ (𝑥 − 1) < 𝑧) → 𝑤 ∈ ℝ^*)
34	1	sselda 3929	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝑥 ∈ ℝ)
35		1red 11108	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 1 ∈ ℝ)
36	34, 35	resubcld 11540	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝑥 − 1) ∈ ℝ)
37	36	adantlr 715	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑤 ∈ ℝ) ∧ 𝑥 ∈ 𝐴) → (𝑥 − 1) ∈ ℝ)
38	37	3adant3 1132	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑤 ∈ ℝ) ∧ 𝑥 ∈ 𝐴 ∧ (𝑤 + 1) < 𝑥) → (𝑥 − 1) ∈ ℝ)
39	38	3ad2ant1 1133	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ 𝑤 ∈ ℝ) ∧ 𝑥 ∈ 𝐴 ∧ (𝑤 + 1) < 𝑥) ∧ 𝑧 ∈ 𝐵 ∧ (𝑥 − 1) < 𝑧) → (𝑥 − 1) ∈ ℝ)
40	2, 39	sselid 3927	. . . . . . . . . . . . . . 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 3929	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐵) → 𝑧 ∈ ℝ^*)
43	42	adantlr 715	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑤 ∈ ℝ) ∧ 𝑧 ∈ 𝐵) → 𝑧 ∈ ℝ^*)
44	43	3ad2antl1 1186	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ 𝑤 ∈ ℝ) ∧ 𝑥 ∈ 𝐴 ∧ (𝑤 + 1) < 𝑥) ∧ 𝑧 ∈ 𝐵) → 𝑧 ∈ ℝ^*)
45	44	3adant3 1132	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ 𝑤 ∈ ℝ) ∧ 𝑥 ∈ 𝐴 ∧ (𝑤 + 1) < 𝑥) ∧ 𝑧 ∈ 𝐵 ∧ (𝑥 − 1) < 𝑧) → 𝑧 ∈ ℝ^*)
46		simp3 1138	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑤 ∈ ℝ) ∧ 𝑥 ∈ 𝐴 ∧ (𝑤 + 1) < 𝑥) → (𝑤 + 1) < 𝑥)
47		simp1r 1199	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑤 ∈ ℝ) ∧ 𝑥 ∈ 𝐴 ∧ (𝑤 + 1) < 𝑥) → 𝑤 ∈ ℝ)
48		1red 11108	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑤 ∈ ℝ) ∧ 𝑥 ∈ 𝐴 ∧ (𝑤 + 1) < 𝑥) → 1 ∈ ℝ)
49	34	adantlr 715	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ 𝑤 ∈ ℝ) ∧ 𝑥 ∈ 𝐴) → 𝑥 ∈ ℝ)
50	49	3adant3 1132	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑤 ∈ ℝ) ∧ 𝑥 ∈ 𝐴 ∧ (𝑤 + 1) < 𝑥) → 𝑥 ∈ ℝ)
51	47, 48, 50	ltaddsubd 11712	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑤 ∈ ℝ) ∧ 𝑥 ∈ 𝐴 ∧ (𝑤 + 1) < 𝑥) → ((𝑤 + 1) < 𝑥 ↔ 𝑤 < (𝑥 − 1)))
52	46, 51	mpbid 232	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑤 ∈ ℝ) ∧ 𝑥 ∈ 𝐴 ∧ (𝑤 + 1) < 𝑥) → 𝑤 < (𝑥 − 1))
53	52	3ad2ant1 1133	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ 𝑤 ∈ ℝ) ∧ 𝑥 ∈ 𝐴 ∧ (𝑤 + 1) < 𝑥) ∧ 𝑧 ∈ 𝐵 ∧ (𝑥 − 1) < 𝑧) → 𝑤 < (𝑥 − 1))
54		simp3 1138	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ 𝑤 ∈ ℝ) ∧ 𝑥 ∈ 𝐴 ∧ (𝑤 + 1) < 𝑥) ∧ 𝑧 ∈ 𝐵 ∧ (𝑥 − 1) < 𝑧) → (𝑥 − 1) < 𝑧)
55	33, 40, 45, 53, 54	xrlttrd 13053	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑤 ∈ ℝ) ∧ 𝑥 ∈ 𝐴 ∧ (𝑤 + 1) < 𝑥) ∧ 𝑧 ∈ 𝐵 ∧ (𝑥 − 1) < 𝑧) → 𝑤 < 𝑧)
56	55	3exp 1119	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑤 ∈ ℝ) ∧ 𝑥 ∈ 𝐴 ∧ (𝑤 + 1) < 𝑥) → (𝑧 ∈ 𝐵 → ((𝑥 − 1) < 𝑧 → 𝑤 < 𝑧)))
57	31, 56	reximdai 3234	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑤 ∈ ℝ) ∧ 𝑥 ∈ 𝐴 ∧ (𝑤 + 1) < 𝑥) → (∃𝑧 ∈ 𝐵 (𝑥 − 1) < 𝑧 → ∃𝑧 ∈ 𝐵 𝑤 < 𝑧))
58	30, 57	mpd 15	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑤 ∈ ℝ) ∧ 𝑥 ∈ 𝐴 ∧ (𝑤 + 1) < 𝑥) → ∃𝑧 ∈ 𝐵 𝑤 < 𝑧)
59	58	3exp 1119	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑤 ∈ ℝ) → (𝑥 ∈ 𝐴 → ((𝑤 + 1) < 𝑥 → ∃𝑧 ∈ 𝐵 𝑤 < 𝑧)))
60	59	adantlr 715	. . . . . . . . 9 ⊢ (((𝜑 ∧ sup(𝐴, ℝ^*, < ) = +∞) ∧ 𝑤 ∈ ℝ) → (𝑥 ∈ 𝐴 → ((𝑤 + 1) < 𝑥 → ∃𝑧 ∈ 𝐵 𝑤 < 𝑧)))
61	60	rexlimdv 3131	. . . . . . . 8 ⊢ (((𝜑 ∧ sup(𝐴, ℝ^*, < ) = +∞) ∧ 𝑤 ∈ ℝ) → (∃𝑥 ∈ 𝐴 (𝑤 + 1) < 𝑥 → ∃𝑧 ∈ 𝐵 𝑤 < 𝑧))
62	19, 61	mpd 15	. . . . . . 7 ⊢ (((𝜑 ∧ sup(𝐴, ℝ^*, < ) = +∞) ∧ 𝑤 ∈ ℝ) → ∃𝑧 ∈ 𝐵 𝑤 < 𝑧)
63	2	a1i 11	. . . . . . . . . . . . 13 ⊢ (𝜑 → ℝ ⊆ ℝ^*)
64	63	sselda 3929	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑤 ∈ ℝ) → 𝑤 ∈ ℝ^*)
65	64	ad2antrr 726	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑤 ∈ ℝ) ∧ 𝑧 ∈ 𝐵) ∧ 𝑤 < 𝑧) → 𝑤 ∈ ℝ^*)
66	43	adantr 480	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑤 ∈ ℝ) ∧ 𝑧 ∈ 𝐵) ∧ 𝑤 < 𝑧) → 𝑧 ∈ ℝ^*)
67		simpr 484	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑤 ∈ ℝ) ∧ 𝑧 ∈ 𝐵) ∧ 𝑤 < 𝑧) → 𝑤 < 𝑧)
68	65, 66, 67	xrltled 13044	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑤 ∈ ℝ) ∧ 𝑧 ∈ 𝐵) ∧ 𝑤 < 𝑧) → 𝑤 ≤ 𝑧)
69	68	ex 412	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑤 ∈ ℝ) ∧ 𝑧 ∈ 𝐵) → (𝑤 < 𝑧 → 𝑤 ≤ 𝑧))
70	69	adantllr 719	. . . . . . . 8 ⊢ ((((𝜑 ∧ sup(𝐴, ℝ^*, < ) = +∞) ∧ 𝑤 ∈ ℝ) ∧ 𝑧 ∈ 𝐵) → (𝑤 < 𝑧 → 𝑤 ≤ 𝑧))
71	70	reximdva 3145	. . . . . . 7 ⊢ (((𝜑 ∧ sup(𝐴, ℝ^*, < ) = +∞) ∧ 𝑤 ∈ ℝ) → (∃𝑧 ∈ 𝐵 𝑤 < 𝑧 → ∃𝑧 ∈ 𝐵 𝑤 ≤ 𝑧))
72	62, 71	mpd 15	. . . . . 6 ⊢ (((𝜑 ∧ sup(𝐴, ℝ^*, < ) = +∞) ∧ 𝑤 ∈ ℝ) → ∃𝑧 ∈ 𝐵 𝑤 ≤ 𝑧)
73	72	ralrimiva 3124	. . . . 5 ⊢ ((𝜑 ∧ sup(𝐴, ℝ^*, < ) = +∞) → ∀𝑤 ∈ ℝ ∃𝑧 ∈ 𝐵 𝑤 ≤ 𝑧)
74		supxrunb1 13213	. . . . . . 7 ⊢ (𝐵 ⊆ ℝ^* → (∀𝑤 ∈ ℝ ∃𝑧 ∈ 𝐵 𝑤 ≤ 𝑧 ↔ sup(𝐵, ℝ^*, < ) = +∞))
75	41, 74	syl 17	. . . . . 6 ⊢ (𝜑 → (∀𝑤 ∈ ℝ ∃𝑧 ∈ 𝐵 𝑤 ≤ 𝑧 ↔ sup(𝐵, ℝ^*, < ) = +∞))
76	75	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ sup(𝐴, ℝ^, < ) = +∞) → (∀𝑤 ∈ ℝ ∃𝑧 ∈ 𝐵 𝑤 ≤ 𝑧 ↔ sup(𝐵, ℝ^, < ) = +∞))
77	73, 76	mpbid 232	. . . 4 ⊢ ((𝜑 ∧ sup(𝐴, ℝ^, < ) = +∞) → sup(𝐵, ℝ^, < ) = +∞)
78	7, 8, 77	3eqtr4d 2776	. . 3 ⊢ ((𝜑 ∧ sup(𝐴, ℝ^, < ) = +∞) → sup(𝐴, ℝ^, < ) = sup(𝐵, ℝ^*, < ))
79	6, 78	xreqled 45369	. 2 ⊢ ((𝜑 ∧ sup(𝐴, ℝ^, < ) = +∞) → sup(𝐴, ℝ^, < ) ≤ sup(𝐵, ℝ^*, < ))
80		supeq1 9324	. . . . . . 7 ⊢ (𝐴 = ∅ → sup(𝐴, ℝ^, < ) = sup(∅, ℝ^, < ))
81		xrsup0 13217	. . . . . . . 8 ⊢ sup(∅, ℝ^*, < ) = -∞
82	81	a1i 11	. . . . . . 7 ⊢ (𝐴 = ∅ → sup(∅, ℝ^*, < ) = -∞)
83	80, 82	eqtrd 2766	. . . . . 6 ⊢ (𝐴 = ∅ → sup(𝐴, ℝ^*, < ) = -∞)
84	83	adantl 481	. . . . 5 ⊢ ((𝜑 ∧ 𝐴 = ∅) → sup(𝐴, ℝ^*, < ) = -∞)
85		supxrcl 13209	. . . . . . . 8 ⊢ (𝐵 ⊆ ℝ^* → sup(𝐵, ℝ^, < ) ∈ ℝ^)
86	41, 85	syl 17	. . . . . . 7 ⊢ (𝜑 → sup(𝐵, ℝ^, < ) ∈ ℝ^)
87		mnfle 13029	. . . . . . 7 ⊢ (sup(𝐵, ℝ^, < ) ∈ ℝ^ → -∞ ≤ sup(𝐵, ℝ^*, < ))
88	86, 87	syl 17	. . . . . 6 ⊢ (𝜑 → -∞ ≤ sup(𝐵, ℝ^*, < ))
89	88	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ 𝐴 = ∅) → -∞ ≤ sup(𝐵, ℝ^*, < ))
90	84, 89	eqbrtrd 5108	. . . 4 ⊢ ((𝜑 ∧ 𝐴 = ∅) → sup(𝐴, ℝ^, < ) ≤ sup(𝐵, ℝ^, < ))
91	90	adantlr 715	. . 3 ⊢ (((𝜑 ∧ ¬ sup(𝐴, ℝ^, < ) = +∞) ∧ 𝐴 = ∅) → sup(𝐴, ℝ^, < ) ≤ sup(𝐵, ℝ^*, < ))
92		simpll 766	. . . 4 ⊢ (((𝜑 ∧ ¬ sup(𝐴, ℝ^*, < ) = +∞) ∧ ¬ 𝐴 = ∅) → 𝜑)
93	1	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ ¬ 𝐴 = ∅) → 𝐴 ⊆ ℝ)
94		neqne 2936	. . . . . . . . 9 ⊢ (¬ 𝐴 = ∅ → 𝐴 ≠ ∅)
95	94	adantl 481	. . . . . . . 8 ⊢ ((𝜑 ∧ ¬ 𝐴 = ∅) → 𝐴 ≠ ∅)
96		supxrgtmnf 13223	. . . . . . . 8 ⊢ ((𝐴 ⊆ ℝ ∧ 𝐴 ≠ ∅) → -∞ < sup(𝐴, ℝ^*, < ))
97	93, 95, 96	syl2anc 584	. . . . . . 7 ⊢ ((𝜑 ∧ ¬ 𝐴 = ∅) → -∞ < sup(𝐴, ℝ^*, < ))
98	97	adantlr 715	. . . . . 6 ⊢ (((𝜑 ∧ ¬ sup(𝐴, ℝ^, < ) = +∞) ∧ ¬ 𝐴 = ∅) → -∞ < sup(𝐴, ℝ^, < ))
99		simpr 484	. . . . . . . . 9 ⊢ ((𝜑 ∧ ¬ sup(𝐴, ℝ^, < ) = +∞) → ¬ sup(𝐴, ℝ^, < ) = +∞)
100		simpl 482	. . . . . . . . . 10 ⊢ ((𝜑 ∧ ¬ sup(𝐴, ℝ^*, < ) = +∞) → 𝜑)
101		nltpnft 13058	. . . . . . . . . 10 ⊢ (sup(𝐴, ℝ^, < ) ∈ ℝ^ → (sup(𝐴, ℝ^, < ) = +∞ ↔ ¬ sup(𝐴, ℝ^, < ) < +∞))
102	100, 5, 101	3syl 18	. . . . . . . . 9 ⊢ ((𝜑 ∧ ¬ sup(𝐴, ℝ^, < ) = +∞) → (sup(𝐴, ℝ^, < ) = +∞ ↔ ¬ sup(𝐴, ℝ^*, < ) < +∞))
103	99, 102	mtbid 324	. . . . . . . 8 ⊢ ((𝜑 ∧ ¬ sup(𝐴, ℝ^, < ) = +∞) → ¬ ¬ sup(𝐴, ℝ^, < ) < +∞)
104		notnotr 130	. . . . . . . 8 ⊢ (¬ ¬ sup(𝐴, ℝ^, < ) < +∞ → sup(𝐴, ℝ^, < ) < +∞)
105	103, 104	syl 17	. . . . . . 7 ⊢ ((𝜑 ∧ ¬ sup(𝐴, ℝ^, < ) = +∞) → sup(𝐴, ℝ^, < ) < +∞)
106	105	adantr 480	. . . . . 6 ⊢ (((𝜑 ∧ ¬ sup(𝐴, ℝ^, < ) = +∞) ∧ ¬ 𝐴 = ∅) → sup(𝐴, ℝ^, < ) < +∞)
107	98, 106	jca 511	. . . . 5 ⊢ (((𝜑 ∧ ¬ sup(𝐴, ℝ^, < ) = +∞) ∧ ¬ 𝐴 = ∅) → (-∞ < sup(𝐴, ℝ^, < ) ∧ sup(𝐴, ℝ^*, < ) < +∞))
108	92, 5	syl 17	. . . . . 6 ⊢ (((𝜑 ∧ ¬ sup(𝐴, ℝ^, < ) = +∞) ∧ ¬ 𝐴 = ∅) → sup(𝐴, ℝ^, < ) ∈ ℝ^*)
109		xrrebnd 13062	. . . . . 6 ⊢ (sup(𝐴, ℝ^, < ) ∈ ℝ^ → (sup(𝐴, ℝ^, < ) ∈ ℝ ↔ (-∞ < sup(𝐴, ℝ^, < ) ∧ sup(𝐴, ℝ^*, < ) < +∞)))
110	108, 109	syl 17	. . . . 5 ⊢ (((𝜑 ∧ ¬ sup(𝐴, ℝ^, < ) = +∞) ∧ ¬ 𝐴 = ∅) → (sup(𝐴, ℝ^, < ) ∈ ℝ ↔ (-∞ < sup(𝐴, ℝ^, < ) ∧ sup(𝐴, ℝ^, < ) < +∞)))
111	107, 110	mpbird 257	. . . 4 ⊢ (((𝜑 ∧ ¬ sup(𝐴, ℝ^, < ) = +∞) ∧ ¬ 𝐴 = ∅) → sup(𝐴, ℝ^, < ) ∈ ℝ)
112		nfv 1915	. . . . 5 ⊢ Ⅎ𝑤(𝜑 ∧ sup(𝐴, ℝ^*, < ) ∈ ℝ)
113	41	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ sup(𝐴, ℝ^, < ) ∈ ℝ) → 𝐵 ⊆ ℝ^)
114		simpr 484	. . . . 5 ⊢ ((𝜑 ∧ sup(𝐴, ℝ^, < ) ∈ ℝ) → sup(𝐴, ℝ^, < ) ∈ ℝ)
115	114	adantr 480	. . . . . . . 8 ⊢ (((𝜑 ∧ sup(𝐴, ℝ^, < ) ∈ ℝ) ∧ 𝑤 ∈ ℝ⁺) → sup(𝐴, ℝ^, < ) ∈ ℝ)
116		simpr 484	. . . . . . . . 9 ⊢ (((𝜑 ∧ sup(𝐴, ℝ^*, < ) ∈ ℝ) ∧ 𝑤 ∈ ℝ⁺) → 𝑤 ∈ ℝ⁺)
117	116	rphalfcld 12941	. . . . . . . 8 ⊢ (((𝜑 ∧ sup(𝐴, ℝ^*, < ) ∈ ℝ) ∧ 𝑤 ∈ ℝ⁺) → (𝑤 / 2) ∈ ℝ⁺)
118	115, 117	ltsubrpd 12961	. . . . . . 7 ⊢ (((𝜑 ∧ sup(𝐴, ℝ^, < ) ∈ ℝ) ∧ 𝑤 ∈ ℝ⁺) → (sup(𝐴, ℝ^, < ) − (𝑤 / 2)) < sup(𝐴, ℝ^*, < ))
119	3	ad2antrr 726	. . . . . . . 8 ⊢ (((𝜑 ∧ sup(𝐴, ℝ^, < ) ∈ ℝ) ∧ 𝑤 ∈ ℝ⁺) → 𝐴 ⊆ ℝ^)
120		rpre 12894	. . . . . . . . . . . 12 ⊢ (𝑤 ∈ ℝ⁺ → 𝑤 ∈ ℝ)
121		2re 12194	. . . . . . . . . . . . 13 ⊢ 2 ∈ ℝ
122	121	a1i 11	. . . . . . . . . . . 12 ⊢ (𝑤 ∈ ℝ⁺ → 2 ∈ ℝ)
123		2ne0 12224	. . . . . . . . . . . . 13 ⊢ 2 ≠ 0
124	123	a1i 11	. . . . . . . . . . . 12 ⊢ (𝑤 ∈ ℝ⁺ → 2 ≠ 0)
125	120, 122, 124	redivcld 11944	. . . . . . . . . . 11 ⊢ (𝑤 ∈ ℝ⁺ → (𝑤 / 2) ∈ ℝ)
126	125	adantl 481	. . . . . . . . . 10 ⊢ (((𝜑 ∧ sup(𝐴, ℝ^*, < ) ∈ ℝ) ∧ 𝑤 ∈ ℝ⁺) → (𝑤 / 2) ∈ ℝ)
127	115, 126	resubcld 11540	. . . . . . . . 9 ⊢ (((𝜑 ∧ sup(𝐴, ℝ^, < ) ∈ ℝ) ∧ 𝑤 ∈ ℝ⁺) → (sup(𝐴, ℝ^, < ) − (𝑤 / 2)) ∈ ℝ)
128	2, 127	sselid 3927	. . . . . . . 8 ⊢ (((𝜑 ∧ sup(𝐴, ℝ^, < ) ∈ ℝ) ∧ 𝑤 ∈ ℝ⁺) → (sup(𝐴, ℝ^, < ) − (𝑤 / 2)) ∈ ℝ^*)
129		supxrlub 13219	. . . . . . . 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 584	. . . . . . 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 232	. . . . . 6 ⊢ (((𝜑 ∧ sup(𝐴, ℝ^, < ) ∈ ℝ) ∧ 𝑤 ∈ ℝ⁺) → ∃𝑥 ∈ 𝐴 (sup(𝐴, ℝ^, < ) − (𝑤 / 2)) < 𝑥)
132		rphalfcl 12914	. . . . . . . . . . 11 ⊢ (𝑤 ∈ ℝ⁺ → (𝑤 / 2) ∈ ℝ⁺)
133	132	3ad2ant2 1134	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑤 ∈ ℝ⁺ ∧ 𝑥 ∈ 𝐴) → (𝑤 / 2) ∈ ℝ⁺)
134	23	3adant2 1131	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑤 ∈ ℝ⁺ ∧ 𝑥 ∈ 𝐴) → ∀𝑦 ∈ ℝ⁺ ∃𝑧 ∈ 𝐵 (𝑥 − 𝑦) < 𝑧)
135		oveq2 7349	. . . . . . . . . . . . 13 ⊢ (𝑦 = (𝑤 / 2) → (𝑥 − 𝑦) = (𝑥 − (𝑤 / 2)))
136	135	breq1d 5096	. . . . . . . . . . . 12 ⊢ (𝑦 = (𝑤 / 2) → ((𝑥 − 𝑦) < 𝑧 ↔ (𝑥 − (𝑤 / 2)) < 𝑧))
137	136	rexbidv 3156	. . . . . . . . . . 11 ⊢ (𝑦 = (𝑤 / 2) → (∃𝑧 ∈ 𝐵 (𝑥 − 𝑦) < 𝑧 ↔ ∃𝑧 ∈ 𝐵 (𝑥 − (𝑤 / 2)) < 𝑧))
138	137	rspcva 3570	. . . . . . . . . 10 ⊢ (((𝑤 / 2) ∈ ℝ⁺ ∧ ∀𝑦 ∈ ℝ⁺ ∃𝑧 ∈ 𝐵 (𝑥 − 𝑦) < 𝑧) → ∃𝑧 ∈ 𝐵 (𝑥 − (𝑤 / 2)) < 𝑧)
139	133, 134, 138	syl2anc 584	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑤 ∈ ℝ⁺ ∧ 𝑥 ∈ 𝐴) → ∃𝑧 ∈ 𝐵 (𝑥 − (𝑤 / 2)) < 𝑧)
140	139	ad5ant134 1369	. . . . . . . 8 ⊢ (((((𝜑 ∧ sup(𝐴, ℝ^, < ) ∈ ℝ) ∧ 𝑤 ∈ ℝ⁺) ∧ 𝑥 ∈ 𝐴) ∧ (sup(𝐴, ℝ^, < ) − (𝑤 / 2)) < 𝑥) → ∃𝑧 ∈ 𝐵 (𝑥 − (𝑤 / 2)) < 𝑧)
141		recn 11091	. . . . . . . . . . . . . . . . 17 ⊢ (sup(𝐴, ℝ^, < ) ∈ ℝ → sup(𝐴, ℝ^, < ) ∈ ℂ)
142	141	adantr 480	. . . . . . . . . . . . . . . 16 ⊢ ((sup(𝐴, ℝ^, < ) ∈ ℝ ∧ 𝑤 ∈ ℝ⁺) → sup(𝐴, ℝ^, < ) ∈ ℂ)
143	120	recnd 11135	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑤 ∈ ℝ⁺ → 𝑤 ∈ ℂ)
144	143	adantl 481	. . . . . . . . . . . . . . . . 17 ⊢ ((sup(𝐴, ℝ^*, < ) ∈ ℝ ∧ 𝑤 ∈ ℝ⁺) → 𝑤 ∈ ℂ)
145	144	halfcld 12361	. . . . . . . . . . . . . . . 16 ⊢ ((sup(𝐴, ℝ^*, < ) ∈ ℝ ∧ 𝑤 ∈ ℝ⁺) → (𝑤 / 2) ∈ ℂ)
146	142, 145, 145	subsub4d 11498	. . . . . . . . . . . . . . 15 ⊢ ((sup(𝐴, ℝ^, < ) ∈ ℝ ∧ 𝑤 ∈ ℝ⁺) → ((sup(𝐴, ℝ^, < ) − (𝑤 / 2)) − (𝑤 / 2)) = (sup(𝐴, ℝ^*, < ) − ((𝑤 / 2) + (𝑤 / 2))))
147	143	2halvesd 12362	. . . . . . . . . . . . . . . . 17 ⊢ (𝑤 ∈ ℝ⁺ → ((𝑤 / 2) + (𝑤 / 2)) = 𝑤)
148	147	oveq2d 7357	. . . . . . . . . . . . . . . 16 ⊢ (𝑤 ∈ ℝ⁺ → (sup(𝐴, ℝ^, < ) − ((𝑤 / 2) + (𝑤 / 2))) = (sup(𝐴, ℝ^, < ) − 𝑤))
149	148	adantl 481	. . . . . . . . . . . . . . 15 ⊢ ((sup(𝐴, ℝ^, < ) ∈ ℝ ∧ 𝑤 ∈ ℝ⁺) → (sup(𝐴, ℝ^, < ) − ((𝑤 / 2) + (𝑤 / 2))) = (sup(𝐴, ℝ^*, < ) − 𝑤))
150	146, 149	eqtr2d 2767	. . . . . . . . . . . . . 14 ⊢ ((sup(𝐴, ℝ^, < ) ∈ ℝ ∧ 𝑤 ∈ ℝ⁺) → (sup(𝐴, ℝ^, < ) − 𝑤) = ((sup(𝐴, ℝ^*, < ) − (𝑤 / 2)) − (𝑤 / 2)))
151	150	adantll 714	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ sup(𝐴, ℝ^, < ) ∈ ℝ) ∧ 𝑤 ∈ ℝ⁺) → (sup(𝐴, ℝ^, < ) − 𝑤) = ((sup(𝐴, ℝ^*, < ) − (𝑤 / 2)) − (𝑤 / 2)))
152	151	adantr 480	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ sup(𝐴, ℝ^, < ) ∈ ℝ) ∧ 𝑤 ∈ ℝ⁺) ∧ 𝑥 ∈ 𝐴) → (sup(𝐴, ℝ^, < ) − 𝑤) = ((sup(𝐴, ℝ^*, < ) − (𝑤 / 2)) − (𝑤 / 2)))
153	152	ad3antrrr 730	. . . . . . . . . . 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 11540	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ sup(𝐴, ℝ^, < ) ∈ ℝ) ∧ 𝑤 ∈ ℝ⁺) → ((sup(𝐴, ℝ^, < ) − (𝑤 / 2)) − (𝑤 / 2)) ∈ ℝ)
155	154	adantr 480	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ sup(𝐴, ℝ^, < ) ∈ ℝ) ∧ 𝑤 ∈ ℝ⁺) ∧ 𝑥 ∈ 𝐴) → ((sup(𝐴, ℝ^, < ) − (𝑤 / 2)) − (𝑤 / 2)) ∈ ℝ)
156	155	ad3antrrr 730	. . . . . . . . . . . . 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 3927	. . . . . . . . . . . 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 681	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑤 ∈ ℝ⁺) ∧ 𝑥 ∈ 𝐴) → 𝑥 ∈ ℝ)
159	125	ad2antlr 727	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑤 ∈ ℝ⁺) ∧ 𝑥 ∈ 𝐴) → (𝑤 / 2) ∈ ℝ)
160	158, 159	resubcld 11540	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑤 ∈ ℝ⁺) ∧ 𝑥 ∈ 𝐴) → (𝑥 − (𝑤 / 2)) ∈ ℝ)
161	160	adantllr 719	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ sup(𝐴, ℝ^*, < ) ∈ ℝ) ∧ 𝑤 ∈ ℝ⁺) ∧ 𝑥 ∈ 𝐴) → (𝑥 − (𝑤 / 2)) ∈ ℝ)
162	161	ad3antrrr 730	. . . . . . . . . . . . 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 3927	. . . . . . . . . . . 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 786	. . . . . . . . . . . . 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 768	. . . . . . . . . . . . 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 584	. . . . . . . . . . . 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 787	. . . . . . . . . . . . . 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 735	. . . . . . . . . . . . . . 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 12363	. . . . . . . . . . . . . 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 11540	. . . . . . . . . . . . 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 783	. . . . . . . . . . . . . 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 584	. . . . . . . . . . . . 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 775	. . . . . . . . . . . . 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 11724	. . . . . . . . . . . 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 484	. . . . . . . . . . . 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 13053	. . . . . . . . . . 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 5108	. . . . . . . . . 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 412	. . . . . . . . 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 3145	. . . . . . . 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 15	. . . . . . 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 3135	. . . . . 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 15	. . . . 5 ⊢ (((𝜑 ∧ sup(𝐴, ℝ^, < ) ∈ ℝ) ∧ 𝑤 ∈ ℝ⁺) → ∃𝑧 ∈ 𝐵 (sup(𝐴, ℝ^, < ) − 𝑤) < 𝑧)
183	112, 113, 114, 182	supxrgere 45372	. . . 4 ⊢ ((𝜑 ∧ sup(𝐴, ℝ^, < ) ∈ ℝ) → sup(𝐴, ℝ^, < ) ≤ sup(𝐵, ℝ^*, < ))
184	92, 111, 183	syl2anc 584	. . 3 ⊢ (((𝜑 ∧ ¬ sup(𝐴, ℝ^, < ) = +∞) ∧ ¬ 𝐴 = ∅) → sup(𝐴, ℝ^, < ) ≤ sup(𝐵, ℝ^*, < ))
185	91, 184	pm2.61dan 812	. 2 ⊢ ((𝜑 ∧ ¬ sup(𝐴, ℝ^, < ) = +∞) → sup(𝐴, ℝ^, < ) ≤ sup(𝐵, ℝ^*, < ))
186	79, 185	pm2.61dan 812	1 ⊢ (𝜑 → sup(𝐴, ℝ^, < ) ≤ sup(𝐵, ℝ^, < ))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 206 ∧ wa 395 ∧ w3a 1086 = wceq 1541 ∈ wcel 2111 ≠ wne 2928 ∀wral 3047 ∃wrex 3056 ⊆ wss 3897 ∅c0 4278 class class class wbr 5086 (class class class)co 7341 supcsup 9319 ℂcc 10999 ℝcr 11000 0cc0 11001 1c1 11002 + caddc 11004 +∞cpnf 11138 -∞cmnf 11139 ℝ^*cxr 11140 < clt 11141 ≤ cle 11142 − cmin 11339 / cdiv 11769 2c2 12175 ℝ⁺crp 12885

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 5229 ax-nul 5239 ax-pow 5298 ax-pr 5365 ax-un 7663 ax-cnex 11057 ax-resscn 11058 ax-1cn 11059 ax-icn 11060 ax-addcl 11061 ax-addrcl 11062 ax-mulcl 11063 ax-mulrcl 11064 ax-mulcom 11065 ax-addass 11066 ax-mulass 11067 ax-distr 11068 ax-i2m1 11069 ax-1ne0 11070 ax-1rid 11071 ax-rnegex 11072 ax-rrecex 11073 ax-cnre 11074 ax-pre-lttri 11075 ax-pre-lttrn 11076 ax-pre-ltadd 11077 ax-pre-mulgt0 11078 ax-pre-sup 11079

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 3737 df-csb 3846 df-dif 3900 df-un 3902 df-in 3904 df-ss 3914 df-pss 3917 df-nul 4279 df-if 4471 df-pw 4547 df-sn 4572 df-pr 4574 df-op 4578 df-uni 4855 df-iun 4938 df-br 5087 df-opab 5149 df-mpt 5168 df-tr 5194 df-id 5506 df-eprel 5511 df-po 5519 df-so 5520 df-fr 5564 df-we 5566 df-xp 5617 df-rel 5618 df-cnv 5619 df-co 5620 df-dm 5621 df-rn 5622 df-res 5623 df-ima 5624 df-pred 6243 df-ord 6304 df-on 6305 df-lim 6306 df-suc 6307 df-iota 6432 df-fun 6478 df-fn 6479 df-f 6480 df-f1 6481 df-fo 6482 df-f1o 6483 df-fv 6484 df-riota 7298 df-ov 7344 df-oprab 7345 df-mpo 7346 df-om 7792 df-2nd 7917 df-frecs 8206 df-wrecs 8237 df-recs 8286 df-rdg 8324 df-er 8617 df-en 8865 df-dom 8866 df-sdom 8867 df-sup 9321 df-pnf 11143 df-mnf 11144 df-xr 11145 df-ltxr 11146 df-le 11147 df-sub 11341 df-neg 11342 df-div 11770 df-nn 12121 df-2 12183 df-rp 12886

This theorem is referenced by: (None)

W3C validator