qbtwnxr - Metamath Proof Explorer

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Theorem qbtwnxr 11905

Description: The rational numbers are dense in ℝ^*: any two extended real numbers have a rational between them. (Contributed by NM, 6-Feb-2007.) (Proof shortened by Mario Carneiro, 23-Aug-2015.)

**Assertion**
Ref	Expression
qbtwnxr	⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐴 < 𝐵) → ∃𝑥 ∈ ℚ (𝐴 < 𝑥 ∧ 𝑥 < 𝐵))

Distinct variable groups: 𝑥,𝐴 𝑥,𝐵

**Proof of Theorem qbtwnxr**
Step	Hyp	Ref	Expression
1		elxr 11826	. . 3 ⊢ (𝐴 ∈ ℝ^* ↔ (𝐴 ∈ ℝ ∨ 𝐴 = +∞ ∨ 𝐴 = -∞))
2		elxr 11826	. . . . 5 ⊢ (𝐵 ∈ ℝ^* ↔ (𝐵 ∈ ℝ ∨ 𝐵 = +∞ ∨ 𝐵 = -∞))
3		qbtwnre 11904	. . . . . . 7 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐴 < 𝐵) → ∃𝑥 ∈ ℚ (𝐴 < 𝑥 ∧ 𝑥 < 𝐵))
4	3	3expia 1259	. . . . . 6 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 < 𝐵 → ∃𝑥 ∈ ℚ (𝐴 < 𝑥 ∧ 𝑥 < 𝐵)))
5		simpl 472	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 = +∞) → 𝐴 ∈ ℝ)
6		peano2re 10088	. . . . . . . . . 10 ⊢ (𝐴 ∈ ℝ → (𝐴 + 1) ∈ ℝ)
7	6	adantr 480	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 = +∞) → (𝐴 + 1) ∈ ℝ)
8		ltp1 10740	. . . . . . . . . 10 ⊢ (𝐴 ∈ ℝ → 𝐴 < (𝐴 + 1))
9	8	adantr 480	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 = +∞) → 𝐴 < (𝐴 + 1))
10		qbtwnre 11904	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℝ ∧ (𝐴 + 1) ∈ ℝ ∧ 𝐴 < (𝐴 + 1)) → ∃𝑥 ∈ ℚ (𝐴 < 𝑥 ∧ 𝑥 < (𝐴 + 1)))
11	5, 7, 9, 10	syl3anc 1318	. . . . . . . 8 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 = +∞) → ∃𝑥 ∈ ℚ (𝐴 < 𝑥 ∧ 𝑥 < (𝐴 + 1)))
12		qre 11669	. . . . . . . . . . . . . 14 ⊢ (𝑥 ∈ ℚ → 𝑥 ∈ ℝ)
13		ltpnf 11830	. . . . . . . . . . . . . 14 ⊢ (𝑥 ∈ ℝ → 𝑥 < +∞)
14	12, 13	syl 17	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ ℚ → 𝑥 < +∞)
15	14	adantl 481	. . . . . . . . . . . 12 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 = +∞) ∧ 𝑥 ∈ ℚ) → 𝑥 < +∞)
16		simplr 788	. . . . . . . . . . . 12 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 = +∞) ∧ 𝑥 ∈ ℚ) → 𝐵 = +∞)
17	15, 16	breqtrrd 4611	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 = +∞) ∧ 𝑥 ∈ ℚ) → 𝑥 < 𝐵)
18	17	a1d 25	. . . . . . . . . 10 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 = +∞) ∧ 𝑥 ∈ ℚ) → (𝑥 < (𝐴 + 1) → 𝑥 < 𝐵))
19	18	anim2d 587	. . . . . . . . 9 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 = +∞) ∧ 𝑥 ∈ ℚ) → ((𝐴 < 𝑥 ∧ 𝑥 < (𝐴 + 1)) → (𝐴 < 𝑥 ∧ 𝑥 < 𝐵)))
20	19	reximdva 3000	. . . . . . . 8 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 = +∞) → (∃𝑥 ∈ ℚ (𝐴 < 𝑥 ∧ 𝑥 < (𝐴 + 1)) → ∃𝑥 ∈ ℚ (𝐴 < 𝑥 ∧ 𝑥 < 𝐵)))
21	11, 20	mpd 15	. . . . . . 7 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 = +∞) → ∃𝑥 ∈ ℚ (𝐴 < 𝑥 ∧ 𝑥 < 𝐵))
22	21	a1d 25	. . . . . 6 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 = +∞) → (𝐴 < 𝐵 → ∃𝑥 ∈ ℚ (𝐴 < 𝑥 ∧ 𝑥 < 𝐵)))
23		rexr 9964	. . . . . . 7 ⊢ (𝐴 ∈ ℝ → 𝐴 ∈ ℝ^*)
24		breq2 4587	. . . . . . . . 9 ⊢ (𝐵 = -∞ → (𝐴 < 𝐵 ↔ 𝐴 < -∞))
25	24	adantl 481	. . . . . . . 8 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 = -∞) → (𝐴 < 𝐵 ↔ 𝐴 < -∞))
26		nltmnf 11839	. . . . . . . . . 10 ⊢ (𝐴 ∈ ℝ^* → ¬ 𝐴 < -∞)
27	26	adantr 480	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 = -∞) → ¬ 𝐴 < -∞)
28	27	pm2.21d 117	. . . . . . . 8 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 = -∞) → (𝐴 < -∞ → ∃𝑥 ∈ ℚ (𝐴 < 𝑥 ∧ 𝑥 < 𝐵)))
29	25, 28	sylbid 229	. . . . . . 7 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 = -∞) → (𝐴 < 𝐵 → ∃𝑥 ∈ ℚ (𝐴 < 𝑥 ∧ 𝑥 < 𝐵)))
30	23, 29	sylan 487	. . . . . 6 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 = -∞) → (𝐴 < 𝐵 → ∃𝑥 ∈ ℚ (𝐴 < 𝑥 ∧ 𝑥 < 𝐵)))
31	4, 22, 30	3jaodan 1386	. . . . 5 ⊢ ((𝐴 ∈ ℝ ∧ (𝐵 ∈ ℝ ∨ 𝐵 = +∞ ∨ 𝐵 = -∞)) → (𝐴 < 𝐵 → ∃𝑥 ∈ ℚ (𝐴 < 𝑥 ∧ 𝑥 < 𝐵)))
32	2, 31	sylan2b 491	. . . 4 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ^*) → (𝐴 < 𝐵 → ∃𝑥 ∈ ℚ (𝐴 < 𝑥 ∧ 𝑥 < 𝐵)))
33		breq1 4586	. . . . . 6 ⊢ (𝐴 = +∞ → (𝐴 < 𝐵 ↔ +∞ < 𝐵))
34	33	adantr 480	. . . . 5 ⊢ ((𝐴 = +∞ ∧ 𝐵 ∈ ℝ^*) → (𝐴 < 𝐵 ↔ +∞ < 𝐵))
35		pnfnlt 11838	. . . . . . 7 ⊢ (𝐵 ∈ ℝ^* → ¬ +∞ < 𝐵)
36	35	adantl 481	. . . . . 6 ⊢ ((𝐴 = +∞ ∧ 𝐵 ∈ ℝ^*) → ¬ +∞ < 𝐵)
37	36	pm2.21d 117	. . . . 5 ⊢ ((𝐴 = +∞ ∧ 𝐵 ∈ ℝ^*) → (+∞ < 𝐵 → ∃𝑥 ∈ ℚ (𝐴 < 𝑥 ∧ 𝑥 < 𝐵)))
38	34, 37	sylbid 229	. . . 4 ⊢ ((𝐴 = +∞ ∧ 𝐵 ∈ ℝ^*) → (𝐴 < 𝐵 → ∃𝑥 ∈ ℚ (𝐴 < 𝑥 ∧ 𝑥 < 𝐵)))
39		peano2rem 10227	. . . . . . . . . 10 ⊢ (𝐵 ∈ ℝ → (𝐵 − 1) ∈ ℝ)
40	39	adantl 481	. . . . . . . . 9 ⊢ ((𝐴 = -∞ ∧ 𝐵 ∈ ℝ) → (𝐵 − 1) ∈ ℝ)
41		simpr 476	. . . . . . . . 9 ⊢ ((𝐴 = -∞ ∧ 𝐵 ∈ ℝ) → 𝐵 ∈ ℝ)
42		ltm1 10742	. . . . . . . . . 10 ⊢ (𝐵 ∈ ℝ → (𝐵 − 1) < 𝐵)
43	42	adantl 481	. . . . . . . . 9 ⊢ ((𝐴 = -∞ ∧ 𝐵 ∈ ℝ) → (𝐵 − 1) < 𝐵)
44		qbtwnre 11904	. . . . . . . . 9 ⊢ (((𝐵 − 1) ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ (𝐵 − 1) < 𝐵) → ∃𝑥 ∈ ℚ ((𝐵 − 1) < 𝑥 ∧ 𝑥 < 𝐵))
45	40, 41, 43, 44	syl3anc 1318	. . . . . . . 8 ⊢ ((𝐴 = -∞ ∧ 𝐵 ∈ ℝ) → ∃𝑥 ∈ ℚ ((𝐵 − 1) < 𝑥 ∧ 𝑥 < 𝐵))
46		simpll 786	. . . . . . . . . . . 12 ⊢ (((𝐴 = -∞ ∧ 𝐵 ∈ ℝ) ∧ 𝑥 ∈ ℚ) → 𝐴 = -∞)
47	12	adantl 481	. . . . . . . . . . . . 13 ⊢ (((𝐴 = -∞ ∧ 𝐵 ∈ ℝ) ∧ 𝑥 ∈ ℚ) → 𝑥 ∈ ℝ)
48		mnflt 11833	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ ℝ → -∞ < 𝑥)
49	47, 48	syl 17	. . . . . . . . . . . 12 ⊢ (((𝐴 = -∞ ∧ 𝐵 ∈ ℝ) ∧ 𝑥 ∈ ℚ) → -∞ < 𝑥)
50	46, 49	eqbrtrd 4605	. . . . . . . . . . 11 ⊢ (((𝐴 = -∞ ∧ 𝐵 ∈ ℝ) ∧ 𝑥 ∈ ℚ) → 𝐴 < 𝑥)
51	50	a1d 25	. . . . . . . . . 10 ⊢ (((𝐴 = -∞ ∧ 𝐵 ∈ ℝ) ∧ 𝑥 ∈ ℚ) → ((𝐵 − 1) < 𝑥 → 𝐴 < 𝑥))
52	51	anim1d 586	. . . . . . . . 9 ⊢ (((𝐴 = -∞ ∧ 𝐵 ∈ ℝ) ∧ 𝑥 ∈ ℚ) → (((𝐵 − 1) < 𝑥 ∧ 𝑥 < 𝐵) → (𝐴 < 𝑥 ∧ 𝑥 < 𝐵)))
53	52	reximdva 3000	. . . . . . . 8 ⊢ ((𝐴 = -∞ ∧ 𝐵 ∈ ℝ) → (∃𝑥 ∈ ℚ ((𝐵 − 1) < 𝑥 ∧ 𝑥 < 𝐵) → ∃𝑥 ∈ ℚ (𝐴 < 𝑥 ∧ 𝑥 < 𝐵)))
54	45, 53	mpd 15	. . . . . . 7 ⊢ ((𝐴 = -∞ ∧ 𝐵 ∈ ℝ) → ∃𝑥 ∈ ℚ (𝐴 < 𝑥 ∧ 𝑥 < 𝐵))
55	54	a1d 25	. . . . . 6 ⊢ ((𝐴 = -∞ ∧ 𝐵 ∈ ℝ) → (𝐴 < 𝐵 → ∃𝑥 ∈ ℚ (𝐴 < 𝑥 ∧ 𝑥 < 𝐵)))
56		1re 9918	. . . . . . . . . 10 ⊢ 1 ∈ ℝ
57		mnflt 11833	. . . . . . . . . 10 ⊢ (1 ∈ ℝ → -∞ < 1)
58	56, 57	ax-mp 5	. . . . . . . . 9 ⊢ -∞ < 1
59		breq1 4586	. . . . . . . . 9 ⊢ (𝐴 = -∞ → (𝐴 < 1 ↔ -∞ < 1))
60	58, 59	mpbiri 247	. . . . . . . 8 ⊢ (𝐴 = -∞ → 𝐴 < 1)
61		ltpnf 11830	. . . . . . . . . 10 ⊢ (1 ∈ ℝ → 1 < +∞)
62	56, 61	ax-mp 5	. . . . . . . . 9 ⊢ 1 < +∞
63		breq2 4587	. . . . . . . . 9 ⊢ (𝐵 = +∞ → (1 < 𝐵 ↔ 1 < +∞))
64	62, 63	mpbiri 247	. . . . . . . 8 ⊢ (𝐵 = +∞ → 1 < 𝐵)
65		1z 11284	. . . . . . . . . 10 ⊢ 1 ∈ ℤ
66		zq 11670	. . . . . . . . . 10 ⊢ (1 ∈ ℤ → 1 ∈ ℚ)
67	65, 66	ax-mp 5	. . . . . . . . 9 ⊢ 1 ∈ ℚ
68		breq2 4587	. . . . . . . . . . 11 ⊢ (𝑥 = 1 → (𝐴 < 𝑥 ↔ 𝐴 < 1))
69		breq1 4586	. . . . . . . . . . 11 ⊢ (𝑥 = 1 → (𝑥 < 𝐵 ↔ 1 < 𝐵))
70	68, 69	anbi12d 743	. . . . . . . . . 10 ⊢ (𝑥 = 1 → ((𝐴 < 𝑥 ∧ 𝑥 < 𝐵) ↔ (𝐴 < 1 ∧ 1 < 𝐵)))
71	70	rspcev 3282	. . . . . . . . 9 ⊢ ((1 ∈ ℚ ∧ (𝐴 < 1 ∧ 1 < 𝐵)) → ∃𝑥 ∈ ℚ (𝐴 < 𝑥 ∧ 𝑥 < 𝐵))
72	67, 71	mpan 702	. . . . . . . 8 ⊢ ((𝐴 < 1 ∧ 1 < 𝐵) → ∃𝑥 ∈ ℚ (𝐴 < 𝑥 ∧ 𝑥 < 𝐵))
73	60, 64, 72	syl2an 493	. . . . . . 7 ⊢ ((𝐴 = -∞ ∧ 𝐵 = +∞) → ∃𝑥 ∈ ℚ (𝐴 < 𝑥 ∧ 𝑥 < 𝐵))
74	73	a1d 25	. . . . . 6 ⊢ ((𝐴 = -∞ ∧ 𝐵 = +∞) → (𝐴 < 𝐵 → ∃𝑥 ∈ ℚ (𝐴 < 𝑥 ∧ 𝑥 < 𝐵)))
75		3mix3 1225	. . . . . . . 8 ⊢ (𝐴 = -∞ → (𝐴 ∈ ℝ ∨ 𝐴 = +∞ ∨ 𝐴 = -∞))
76	75, 1	sylibr 223	. . . . . . 7 ⊢ (𝐴 = -∞ → 𝐴 ∈ ℝ^*)
77	76, 29	sylan 487	. . . . . 6 ⊢ ((𝐴 = -∞ ∧ 𝐵 = -∞) → (𝐴 < 𝐵 → ∃𝑥 ∈ ℚ (𝐴 < 𝑥 ∧ 𝑥 < 𝐵)))
78	55, 74, 77	3jaodan 1386	. . . . 5 ⊢ ((𝐴 = -∞ ∧ (𝐵 ∈ ℝ ∨ 𝐵 = +∞ ∨ 𝐵 = -∞)) → (𝐴 < 𝐵 → ∃𝑥 ∈ ℚ (𝐴 < 𝑥 ∧ 𝑥 < 𝐵)))
79	2, 78	sylan2b 491	. . . 4 ⊢ ((𝐴 = -∞ ∧ 𝐵 ∈ ℝ^*) → (𝐴 < 𝐵 → ∃𝑥 ∈ ℚ (𝐴 < 𝑥 ∧ 𝑥 < 𝐵)))
80	32, 38, 79	3jaoian 1385	. . 3 ⊢ (((𝐴 ∈ ℝ ∨ 𝐴 = +∞ ∨ 𝐴 = -∞) ∧ 𝐵 ∈ ℝ^*) → (𝐴 < 𝐵 → ∃𝑥 ∈ ℚ (𝐴 < 𝑥 ∧ 𝑥 < 𝐵)))
81	1, 80	sylanb 488	. 2 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) → (𝐴 < 𝐵 → ∃𝑥 ∈ ℚ (𝐴 < 𝑥 ∧ 𝑥 < 𝐵)))
82	81	3impia 1253	1 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐴 < 𝐵) → ∃𝑥 ∈ ℚ (𝐴 < 𝑥 ∧ 𝑥 < 𝐵))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 195 ∧ wa 383 ∨ w3o 1030 ∧ w3a 1031 = wceq 1475 ∈ wcel 1977 ∃wrex 2897 class class class wbr 4583 (class class class)co 6549 ℝcr 9814 1c1 9816 + caddc 9818 +∞cpnf 9950 -∞cmnf 9951 ℝ^*cxr 9952 < clt 9953 − cmin 10145 ℤcz 11254 ℚcq 11664

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-cnex 9871 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-mulcom 9879 ax-addass 9880 ax-mulass 9881 ax-distr 9882 ax-i2m1 9883 ax-1ne0 9884 ax-1rid 9885 ax-rnegex 9886 ax-rrecex 9887 ax-cnre 9888 ax-pre-lttri 9889 ax-pre-lttrn 9890 ax-pre-ltadd 9891 ax-pre-mulgt0 9892 ax-pre-sup 9893

This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-reu 2903 df-rmo 2904 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-om 6958 df-1st 7059 df-2nd 7060 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-er 7629 df-en 7842 df-dom 7843 df-sdom 7844 df-sup 8231 df-inf 8232 df-pnf 9955 df-mnf 9956 df-xr 9957 df-ltxr 9958 df-le 9959 df-sub 10147 df-neg 10148 df-div 10564 df-nn 10898 df-n0 11170 df-z 11255 df-uz 11564 df-q 11665

This theorem is referenced by: qextltlem 11907 xralrple 11910 ixxub 12067 ixxlb 12068 ioo0 12071 ico0 12092 ioc0 12093 blssps 22039 blss 22040 blcld 22120 qdensere 22383 tgqioo 22411 dvlip2 23562 lhop2 23582 itgsubst 23616 itg2gt0cn 32635 qinioo 38609 qelioo 38620 qndenserrnbllem 39190

W3C validator