ixxlb - Metamath Proof Explorer

	Metamath Proof Explorer		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > ixxlb		Structured version Visualization version GIF version

Theorem ixxlb 12763

Description: Extract the lower bound of an interval. (Contributed by Mario Carneiro, 17-Jun-2014.) (Revised by AV, 12-Sep-2020.)

**Hypotheses**
Ref	Expression
ixx.1	⊢ 𝑂 = (𝑥 ∈ ℝ^, 𝑦 ∈ ℝ^ ↦ {𝑧 ∈ ℝ^* ∣ (𝑥𝑅𝑧 ∧ 𝑧𝑆𝑦)})
ixxub.2	⊢ ((𝑤 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) → (𝑤 < 𝐵 → 𝑤𝑆𝐵))
ixxub.3	⊢ ((𝑤 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) → (𝑤𝑆𝐵 → 𝑤 ≤ 𝐵))
ixxub.4	⊢ ((𝐴 ∈ ℝ^* ∧ 𝑤 ∈ ℝ^*) → (𝐴 < 𝑤 → 𝐴𝑅𝑤))
ixxub.5	⊢ ((𝐴 ∈ ℝ^* ∧ 𝑤 ∈ ℝ^*) → (𝐴𝑅𝑤 → 𝐴 ≤ 𝑤))

**Assertion**
Ref	Expression
ixxlb	⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ (𝐴𝑂𝐵) ≠ ∅) → inf((𝐴𝑂𝐵), ℝ^*, < ) = 𝐴)

Distinct variable groups: 𝑥,𝑤,𝑦,𝑧,𝐴 𝑤,𝑂 𝑤,𝐵,𝑥,𝑦,𝑧 𝑥,𝑅,𝑦,𝑧 𝑥,𝑆,𝑦,𝑧 Allowed substitution hints: 𝑅(𝑤) 𝑆(𝑤) 𝑂(𝑥,𝑦,𝑧)

**Proof of Theorem ixxlb**
Step	Hyp	Ref	Expression
1		ixx.1	. . . . . . . . 9 ⊢ 𝑂 = (𝑥 ∈ ℝ^, 𝑦 ∈ ℝ^ ↦ {𝑧 ∈ ℝ^* ∣ (𝑥𝑅𝑧 ∧ 𝑧𝑆𝑦)})
2	1	elixx1 12750	. . . . . . . 8 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^) → (𝑤 ∈ (𝐴𝑂𝐵) ↔ (𝑤 ∈ ℝ^ ∧ 𝐴𝑅𝑤 ∧ 𝑤𝑆𝐵)))
3	2	3adant3 1128	. . . . . . 7 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ (𝐴𝑂𝐵) ≠ ∅) → (𝑤 ∈ (𝐴𝑂𝐵) ↔ (𝑤 ∈ ℝ^* ∧ 𝐴𝑅𝑤 ∧ 𝑤𝑆𝐵)))
4	3	biimpa 479	. . . . . 6 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ (𝐴𝑂𝐵) ≠ ∅) ∧ 𝑤 ∈ (𝐴𝑂𝐵)) → (𝑤 ∈ ℝ^* ∧ 𝐴𝑅𝑤 ∧ 𝑤𝑆𝐵))
5	4	simp1d 1138	. . . . 5 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ (𝐴𝑂𝐵) ≠ ∅) ∧ 𝑤 ∈ (𝐴𝑂𝐵)) → 𝑤 ∈ ℝ^*)
6	5	ex 415	. . . 4 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ (𝐴𝑂𝐵) ≠ ∅) → (𝑤 ∈ (𝐴𝑂𝐵) → 𝑤 ∈ ℝ^*))
7	6	ssrdv 3975	. . 3 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ (𝐴𝑂𝐵) ≠ ∅) → (𝐴𝑂𝐵) ⊆ ℝ^*)
8		infxrcl 12729	. . 3 ⊢ ((𝐴𝑂𝐵) ⊆ ℝ^* → inf((𝐴𝑂𝐵), ℝ^, < ) ∈ ℝ^)
9	7, 8	syl 17	. 2 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ (𝐴𝑂𝐵) ≠ ∅) → inf((𝐴𝑂𝐵), ℝ^, < ) ∈ ℝ^)
10		simp1 1132	. 2 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ (𝐴𝑂𝐵) ≠ ∅) → 𝐴 ∈ ℝ^*)
11		simprr 771	. . . . . 6 ⊢ ((((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ (𝐴𝑂𝐵) ≠ ∅) ∧ 𝑤 ∈ ℚ) ∧ (𝐴 < 𝑤 ∧ 𝑤 < inf((𝐴𝑂𝐵), ℝ^, < ))) → 𝑤 < inf((𝐴𝑂𝐵), ℝ^, < ))
12	7	ad2antrr 724	. . . . . . . 8 ⊢ ((((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ (𝐴𝑂𝐵) ≠ ∅) ∧ 𝑤 ∈ ℚ) ∧ (𝐴 < 𝑤 ∧ 𝑤 < inf((𝐴𝑂𝐵), ℝ^, < ))) → (𝐴𝑂𝐵) ⊆ ℝ^)
13		qre 12356	. . . . . . . . . . 11 ⊢ (𝑤 ∈ ℚ → 𝑤 ∈ ℝ)
14	13	rexrd 10693	. . . . . . . . . 10 ⊢ (𝑤 ∈ ℚ → 𝑤 ∈ ℝ^*)
15	14	ad2antlr 725	. . . . . . . . 9 ⊢ ((((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ (𝐴𝑂𝐵) ≠ ∅) ∧ 𝑤 ∈ ℚ) ∧ (𝐴 < 𝑤 ∧ 𝑤 < inf((𝐴𝑂𝐵), ℝ^, < ))) → 𝑤 ∈ ℝ^)
16		simprl 769	. . . . . . . . . 10 ⊢ ((((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ (𝐴𝑂𝐵) ≠ ∅) ∧ 𝑤 ∈ ℚ) ∧ (𝐴 < 𝑤 ∧ 𝑤 < inf((𝐴𝑂𝐵), ℝ^*, < ))) → 𝐴 < 𝑤)
17	10	ad2antrr 724	. . . . . . . . . . 11 ⊢ ((((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ (𝐴𝑂𝐵) ≠ ∅) ∧ 𝑤 ∈ ℚ) ∧ (𝐴 < 𝑤 ∧ 𝑤 < inf((𝐴𝑂𝐵), ℝ^, < ))) → 𝐴 ∈ ℝ^)
18		ixxub.4	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝑤 ∈ ℝ^*) → (𝐴 < 𝑤 → 𝐴𝑅𝑤))
19	17, 15, 18	syl2anc 586	. . . . . . . . . 10 ⊢ ((((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ (𝐴𝑂𝐵) ≠ ∅) ∧ 𝑤 ∈ ℚ) ∧ (𝐴 < 𝑤 ∧ 𝑤 < inf((𝐴𝑂𝐵), ℝ^*, < ))) → (𝐴 < 𝑤 → 𝐴𝑅𝑤))
20	16, 19	mpd 15	. . . . . . . . 9 ⊢ ((((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ (𝐴𝑂𝐵) ≠ ∅) ∧ 𝑤 ∈ ℚ) ∧ (𝐴 < 𝑤 ∧ 𝑤 < inf((𝐴𝑂𝐵), ℝ^*, < ))) → 𝐴𝑅𝑤)
21	9	ad2antrr 724	. . . . . . . . . . 11 ⊢ ((((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ (𝐴𝑂𝐵) ≠ ∅) ∧ 𝑤 ∈ ℚ) ∧ (𝐴 < 𝑤 ∧ 𝑤 < inf((𝐴𝑂𝐵), ℝ^, < ))) → inf((𝐴𝑂𝐵), ℝ^, < ) ∈ ℝ^*)
22		simpll2 1209	. . . . . . . . . . 11 ⊢ ((((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ (𝐴𝑂𝐵) ≠ ∅) ∧ 𝑤 ∈ ℚ) ∧ (𝐴 < 𝑤 ∧ 𝑤 < inf((𝐴𝑂𝐵), ℝ^, < ))) → 𝐵 ∈ ℝ^)
23		simp3 1134	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ (𝐴𝑂𝐵) ≠ ∅) → (𝐴𝑂𝐵) ≠ ∅)
24		n0 4312	. . . . . . . . . . . . . 14 ⊢ ((𝐴𝑂𝐵) ≠ ∅ ↔ ∃𝑤 𝑤 ∈ (𝐴𝑂𝐵))
25	23, 24	sylib 220	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ (𝐴𝑂𝐵) ≠ ∅) → ∃𝑤 𝑤 ∈ (𝐴𝑂𝐵))
26	9	adantr 483	. . . . . . . . . . . . . 14 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ (𝐴𝑂𝐵) ≠ ∅) ∧ 𝑤 ∈ (𝐴𝑂𝐵)) → inf((𝐴𝑂𝐵), ℝ^, < ) ∈ ℝ^)
27		simpl2 1188	. . . . . . . . . . . . . 14 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ (𝐴𝑂𝐵) ≠ ∅) ∧ 𝑤 ∈ (𝐴𝑂𝐵)) → 𝐵 ∈ ℝ^*)
28		infxrlb 12730	. . . . . . . . . . . . . . 15 ⊢ (((𝐴𝑂𝐵) ⊆ ℝ^* ∧ 𝑤 ∈ (𝐴𝑂𝐵)) → inf((𝐴𝑂𝐵), ℝ^*, < ) ≤ 𝑤)
29	7, 28	sylan 582	. . . . . . . . . . . . . 14 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ (𝐴𝑂𝐵) ≠ ∅) ∧ 𝑤 ∈ (𝐴𝑂𝐵)) → inf((𝐴𝑂𝐵), ℝ^*, < ) ≤ 𝑤)
30	4	simp3d 1140	. . . . . . . . . . . . . . 15 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ (𝐴𝑂𝐵) ≠ ∅) ∧ 𝑤 ∈ (𝐴𝑂𝐵)) → 𝑤𝑆𝐵)
31		ixxub.3	. . . . . . . . . . . . . . . 16 ⊢ ((𝑤 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) → (𝑤𝑆𝐵 → 𝑤 ≤ 𝐵))
32	5, 27, 31	syl2anc 586	. . . . . . . . . . . . . . 15 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ (𝐴𝑂𝐵) ≠ ∅) ∧ 𝑤 ∈ (𝐴𝑂𝐵)) → (𝑤𝑆𝐵 → 𝑤 ≤ 𝐵))
33	30, 32	mpd 15	. . . . . . . . . . . . . 14 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ (𝐴𝑂𝐵) ≠ ∅) ∧ 𝑤 ∈ (𝐴𝑂𝐵)) → 𝑤 ≤ 𝐵)
34	26, 5, 27, 29, 33	xrletrd 12558	. . . . . . . . . . . . 13 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ (𝐴𝑂𝐵) ≠ ∅) ∧ 𝑤 ∈ (𝐴𝑂𝐵)) → inf((𝐴𝑂𝐵), ℝ^*, < ) ≤ 𝐵)
35	25, 34	exlimddv 1936	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ (𝐴𝑂𝐵) ≠ ∅) → inf((𝐴𝑂𝐵), ℝ^*, < ) ≤ 𝐵)
36	35	ad2antrr 724	. . . . . . . . . . 11 ⊢ ((((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ (𝐴𝑂𝐵) ≠ ∅) ∧ 𝑤 ∈ ℚ) ∧ (𝐴 < 𝑤 ∧ 𝑤 < inf((𝐴𝑂𝐵), ℝ^, < ))) → inf((𝐴𝑂𝐵), ℝ^, < ) ≤ 𝐵)
37	15, 21, 22, 11, 36	xrltletrd 12557	. . . . . . . . . 10 ⊢ ((((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ (𝐴𝑂𝐵) ≠ ∅) ∧ 𝑤 ∈ ℚ) ∧ (𝐴 < 𝑤 ∧ 𝑤 < inf((𝐴𝑂𝐵), ℝ^*, < ))) → 𝑤 < 𝐵)
38		ixxub.2	. . . . . . . . . . 11 ⊢ ((𝑤 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) → (𝑤 < 𝐵 → 𝑤𝑆𝐵))
39	15, 22, 38	syl2anc 586	. . . . . . . . . 10 ⊢ ((((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ (𝐴𝑂𝐵) ≠ ∅) ∧ 𝑤 ∈ ℚ) ∧ (𝐴 < 𝑤 ∧ 𝑤 < inf((𝐴𝑂𝐵), ℝ^*, < ))) → (𝑤 < 𝐵 → 𝑤𝑆𝐵))
40	37, 39	mpd 15	. . . . . . . . 9 ⊢ ((((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ (𝐴𝑂𝐵) ≠ ∅) ∧ 𝑤 ∈ ℚ) ∧ (𝐴 < 𝑤 ∧ 𝑤 < inf((𝐴𝑂𝐵), ℝ^*, < ))) → 𝑤𝑆𝐵)
41	3	ad2antrr 724	. . . . . . . . 9 ⊢ ((((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ (𝐴𝑂𝐵) ≠ ∅) ∧ 𝑤 ∈ ℚ) ∧ (𝐴 < 𝑤 ∧ 𝑤 < inf((𝐴𝑂𝐵), ℝ^, < ))) → (𝑤 ∈ (𝐴𝑂𝐵) ↔ (𝑤 ∈ ℝ^ ∧ 𝐴𝑅𝑤 ∧ 𝑤𝑆𝐵)))
42	15, 20, 40, 41	mpbir3and 1338	. . . . . . . 8 ⊢ ((((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ (𝐴𝑂𝐵) ≠ ∅) ∧ 𝑤 ∈ ℚ) ∧ (𝐴 < 𝑤 ∧ 𝑤 < inf((𝐴𝑂𝐵), ℝ^*, < ))) → 𝑤 ∈ (𝐴𝑂𝐵))
43	12, 42, 28	syl2anc 586	. . . . . . 7 ⊢ ((((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ (𝐴𝑂𝐵) ≠ ∅) ∧ 𝑤 ∈ ℚ) ∧ (𝐴 < 𝑤 ∧ 𝑤 < inf((𝐴𝑂𝐵), ℝ^, < ))) → inf((𝐴𝑂𝐵), ℝ^, < ) ≤ 𝑤)
44	21, 15	xrlenltd 10709	. . . . . . 7 ⊢ ((((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ (𝐴𝑂𝐵) ≠ ∅) ∧ 𝑤 ∈ ℚ) ∧ (𝐴 < 𝑤 ∧ 𝑤 < inf((𝐴𝑂𝐵), ℝ^, < ))) → (inf((𝐴𝑂𝐵), ℝ^, < ) ≤ 𝑤 ↔ ¬ 𝑤 < inf((𝐴𝑂𝐵), ℝ^*, < )))
45	43, 44	mpbid 234	. . . . . 6 ⊢ ((((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ (𝐴𝑂𝐵) ≠ ∅) ∧ 𝑤 ∈ ℚ) ∧ (𝐴 < 𝑤 ∧ 𝑤 < inf((𝐴𝑂𝐵), ℝ^, < ))) → ¬ 𝑤 < inf((𝐴𝑂𝐵), ℝ^, < ))
46	11, 45	pm2.65da 815	. . . . 5 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ (𝐴𝑂𝐵) ≠ ∅) ∧ 𝑤 ∈ ℚ) → ¬ (𝐴 < 𝑤 ∧ 𝑤 < inf((𝐴𝑂𝐵), ℝ^*, < )))
47	46	nrexdv 3272	. . . 4 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ (𝐴𝑂𝐵) ≠ ∅) → ¬ ∃𝑤 ∈ ℚ (𝐴 < 𝑤 ∧ 𝑤 < inf((𝐴𝑂𝐵), ℝ^*, < )))
48		qbtwnxr 12596	. . . . . 6 ⊢ ((𝐴 ∈ ℝ^* ∧ inf((𝐴𝑂𝐵), ℝ^, < ) ∈ ℝ^ ∧ 𝐴 < inf((𝐴𝑂𝐵), ℝ^, < )) → ∃𝑤 ∈ ℚ (𝐴 < 𝑤 ∧ 𝑤 < inf((𝐴𝑂𝐵), ℝ^, < )))
49	48	3expia 1117	. . . . 5 ⊢ ((𝐴 ∈ ℝ^* ∧ inf((𝐴𝑂𝐵), ℝ^, < ) ∈ ℝ^) → (𝐴 < inf((𝐴𝑂𝐵), ℝ^, < ) → ∃𝑤 ∈ ℚ (𝐴 < 𝑤 ∧ 𝑤 < inf((𝐴𝑂𝐵), ℝ^, < ))))
50	10, 9, 49	syl2anc 586	. . . 4 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ (𝐴𝑂𝐵) ≠ ∅) → (𝐴 < inf((𝐴𝑂𝐵), ℝ^, < ) → ∃𝑤 ∈ ℚ (𝐴 < 𝑤 ∧ 𝑤 < inf((𝐴𝑂𝐵), ℝ^, < ))))
51	47, 50	mtod 200	. . 3 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ (𝐴𝑂𝐵) ≠ ∅) → ¬ 𝐴 < inf((𝐴𝑂𝐵), ℝ^*, < ))
52	9, 10, 51	xrnltled 10711	. 2 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ (𝐴𝑂𝐵) ≠ ∅) → inf((𝐴𝑂𝐵), ℝ^*, < ) ≤ 𝐴)
53	4	simp2d 1139	. . . . 5 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ (𝐴𝑂𝐵) ≠ ∅) ∧ 𝑤 ∈ (𝐴𝑂𝐵)) → 𝐴𝑅𝑤)
54	10	adantr 483	. . . . . 6 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ (𝐴𝑂𝐵) ≠ ∅) ∧ 𝑤 ∈ (𝐴𝑂𝐵)) → 𝐴 ∈ ℝ^*)
55		ixxub.5	. . . . . 6 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝑤 ∈ ℝ^*) → (𝐴𝑅𝑤 → 𝐴 ≤ 𝑤))
56	54, 5, 55	syl2anc 586	. . . . 5 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ (𝐴𝑂𝐵) ≠ ∅) ∧ 𝑤 ∈ (𝐴𝑂𝐵)) → (𝐴𝑅𝑤 → 𝐴 ≤ 𝑤))
57	53, 56	mpd 15	. . . 4 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ (𝐴𝑂𝐵) ≠ ∅) ∧ 𝑤 ∈ (𝐴𝑂𝐵)) → 𝐴 ≤ 𝑤)
58	57	ralrimiva 3184	. . 3 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ (𝐴𝑂𝐵) ≠ ∅) → ∀𝑤 ∈ (𝐴𝑂𝐵)𝐴 ≤ 𝑤)
59		infxrgelb 12731	. . . 4 ⊢ (((𝐴𝑂𝐵) ⊆ ℝ^* ∧ 𝐴 ∈ ℝ^) → (𝐴 ≤ inf((𝐴𝑂𝐵), ℝ^, < ) ↔ ∀𝑤 ∈ (𝐴𝑂𝐵)𝐴 ≤ 𝑤))
60	7, 10, 59	syl2anc 586	. . 3 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ (𝐴𝑂𝐵) ≠ ∅) → (𝐴 ≤ inf((𝐴𝑂𝐵), ℝ^*, < ) ↔ ∀𝑤 ∈ (𝐴𝑂𝐵)𝐴 ≤ 𝑤))
61	58, 60	mpbird 259	. 2 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ (𝐴𝑂𝐵) ≠ ∅) → 𝐴 ≤ inf((𝐴𝑂𝐵), ℝ^*, < ))
62	9, 10, 52, 61	xrletrid 12551	1 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ (𝐴𝑂𝐵) ≠ ∅) → inf((𝐴𝑂𝐵), ℝ^*, < ) = 𝐴)

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 208 ∧ wa 398 ∧ w3a 1083 = wceq 1537 ∃wex 1780 ∈ wcel 2114 ≠ wne 3018 ∀wral 3140 ∃wrex 3141 {crab 3144 ⊆ wss 3938 ∅c0 4293 class class class wbr 5068 (class class class)co 7158 ∈ cmpo 7160 infcinf 8907 ℝ^*cxr 10676 < clt 10677 ≤ cle 10678 ℚcq 12351

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 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 ax-cnex 10595 ax-resscn 10596 ax-1cn 10597 ax-icn 10598 ax-addcl 10599 ax-addrcl 10600 ax-mulcl 10601 ax-mulrcl 10602 ax-mulcom 10603 ax-addass 10604 ax-mulass 10605 ax-distr 10606 ax-i2m1 10607 ax-1ne0 10608 ax-1rid 10609 ax-rnegex 10610 ax-rrecex 10611 ax-cnre 10612 ax-pre-lttri 10613 ax-pre-lttrn 10614 ax-pre-ltadd 10615 ax-pre-mulgt0 10616 ax-pre-sup 10617

This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-nel 3126 df-ral 3145 df-rex 3146 df-reu 3147 df-rmo 3148 df-rab 3149 df-v 3498 df-sbc 3775 df-csb 3886 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-pss 3956 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-tp 4574 df-op 4576 df-uni 4841 df-iun 4923 df-br 5069 df-opab 5131 df-mpt 5149 df-tr 5175 df-id 5462 df-eprel 5467 df-po 5476 df-so 5477 df-fr 5516 df-we 5518 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-pred 6150 df-ord 6196 df-on 6197 df-lim 6198 df-suc 6199 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-f1 6362 df-fo 6363 df-f1o 6364 df-fv 6365 df-riota 7116 df-ov 7161 df-oprab 7162 df-mpo 7163 df-om 7583 df-1st 7691 df-2nd 7692 df-wrecs 7949 df-recs 8010 df-rdg 8048 df-er 8291 df-en 8512 df-dom 8513 df-sdom 8514 df-sup 8908 df-inf 8909 df-pnf 10679 df-mnf 10680 df-xr 10681 df-ltxr 10682 df-le 10683 df-sub 10874 df-neg 10875 df-div 11300 df-nn 11641 df-n0 11901 df-z 11985 df-uz 12247 df-q 12352

This theorem is referenced by: ioorf 24176 ioorinv2 24178 ioossioobi 41800

W3C validator