qbtwnxr - Metamath Proof Explorer

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Theorem qbtwnxr 13242

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 13158	. . 3 ⊢ (𝐴 ∈ ℝ^* ↔ (𝐴 ∈ ℝ ∨ 𝐴 = +∞ ∨ 𝐴 = -∞))
2		elxr 13158	. . . . 5 ⊢ (𝐵 ∈ ℝ^* ↔ (𝐵 ∈ ℝ ∨ 𝐵 = +∞ ∨ 𝐵 = -∞))
3		qbtwnre 13241	. . . . . . 7 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐴 < 𝐵) → ∃𝑥 ∈ ℚ (𝐴 < 𝑥 ∧ 𝑥 < 𝐵))
4	3	3expia 1122	. . . . . 6 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 < 𝐵 → ∃𝑥 ∈ ℚ (𝐴 < 𝑥 ∧ 𝑥 < 𝐵)))
5		simpl 482	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 = +∞) → 𝐴 ∈ ℝ)
6		peano2re 11434	. . . . . . . . . 10 ⊢ (𝐴 ∈ ℝ → (𝐴 + 1) ∈ ℝ)
7	6	adantr 480	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 = +∞) → (𝐴 + 1) ∈ ℝ)
8		ltp1 12107	. . . . . . . . . 10 ⊢ (𝐴 ∈ ℝ → 𝐴 < (𝐴 + 1))
9	8	adantr 480	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 = +∞) → 𝐴 < (𝐴 + 1))
10		qbtwnre 13241	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℝ ∧ (𝐴 + 1) ∈ ℝ ∧ 𝐴 < (𝐴 + 1)) → ∃𝑥 ∈ ℚ (𝐴 < 𝑥 ∧ 𝑥 < (𝐴 + 1)))
11	5, 7, 9, 10	syl3anc 1373	. . . . . . . 8 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 = +∞) → ∃𝑥 ∈ ℚ (𝐴 < 𝑥 ∧ 𝑥 < (𝐴 + 1)))
12		qre 12995	. . . . . . . . . . . . . 14 ⊢ (𝑥 ∈ ℚ → 𝑥 ∈ ℝ)
13	12	ltpnfd 13163	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ ℚ → 𝑥 < +∞)
14	13	adantl 481	. . . . . . . . . . . 12 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 = +∞) ∧ 𝑥 ∈ ℚ) → 𝑥 < +∞)
15		simplr 769	. . . . . . . . . . . 12 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 = +∞) ∧ 𝑥 ∈ ℚ) → 𝐵 = +∞)
16	14, 15	breqtrrd 5171	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 = +∞) ∧ 𝑥 ∈ ℚ) → 𝑥 < 𝐵)
17	16	a1d 25	. . . . . . . . . 10 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 = +∞) ∧ 𝑥 ∈ ℚ) → (𝑥 < (𝐴 + 1) → 𝑥 < 𝐵))
18	17	anim2d 612	. . . . . . . . 9 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 = +∞) ∧ 𝑥 ∈ ℚ) → ((𝐴 < 𝑥 ∧ 𝑥 < (𝐴 + 1)) → (𝐴 < 𝑥 ∧ 𝑥 < 𝐵)))
19	18	reximdva 3168	. . . . . . . 8 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 = +∞) → (∃𝑥 ∈ ℚ (𝐴 < 𝑥 ∧ 𝑥 < (𝐴 + 1)) → ∃𝑥 ∈ ℚ (𝐴 < 𝑥 ∧ 𝑥 < 𝐵)))
20	11, 19	mpd 15	. . . . . . 7 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 = +∞) → ∃𝑥 ∈ ℚ (𝐴 < 𝑥 ∧ 𝑥 < 𝐵))
21	20	a1d 25	. . . . . 6 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 = +∞) → (𝐴 < 𝐵 → ∃𝑥 ∈ ℚ (𝐴 < 𝑥 ∧ 𝑥 < 𝐵)))
22		rexr 11307	. . . . . . 7 ⊢ (𝐴 ∈ ℝ → 𝐴 ∈ ℝ^*)
23		breq2 5147	. . . . . . . . 9 ⊢ (𝐵 = -∞ → (𝐴 < 𝐵 ↔ 𝐴 < -∞))
24	23	adantl 481	. . . . . . . 8 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 = -∞) → (𝐴 < 𝐵 ↔ 𝐴 < -∞))
25		nltmnf 13171	. . . . . . . . . 10 ⊢ (𝐴 ∈ ℝ^* → ¬ 𝐴 < -∞)
26	25	adantr 480	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 = -∞) → ¬ 𝐴 < -∞)
27	26	pm2.21d 121	. . . . . . . 8 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 = -∞) → (𝐴 < -∞ → ∃𝑥 ∈ ℚ (𝐴 < 𝑥 ∧ 𝑥 < 𝐵)))
28	24, 27	sylbid 240	. . . . . . 7 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 = -∞) → (𝐴 < 𝐵 → ∃𝑥 ∈ ℚ (𝐴 < 𝑥 ∧ 𝑥 < 𝐵)))
29	22, 28	sylan 580	. . . . . 6 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 = -∞) → (𝐴 < 𝐵 → ∃𝑥 ∈ ℚ (𝐴 < 𝑥 ∧ 𝑥 < 𝐵)))
30	4, 21, 29	3jaodan 1433	. . . . 5 ⊢ ((𝐴 ∈ ℝ ∧ (𝐵 ∈ ℝ ∨ 𝐵 = +∞ ∨ 𝐵 = -∞)) → (𝐴 < 𝐵 → ∃𝑥 ∈ ℚ (𝐴 < 𝑥 ∧ 𝑥 < 𝐵)))
31	2, 30	sylan2b 594	. . . 4 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ^*) → (𝐴 < 𝐵 → ∃𝑥 ∈ ℚ (𝐴 < 𝑥 ∧ 𝑥 < 𝐵)))
32		breq1 5146	. . . . . 6 ⊢ (𝐴 = +∞ → (𝐴 < 𝐵 ↔ +∞ < 𝐵))
33	32	adantr 480	. . . . 5 ⊢ ((𝐴 = +∞ ∧ 𝐵 ∈ ℝ^*) → (𝐴 < 𝐵 ↔ +∞ < 𝐵))
34		pnfnlt 13170	. . . . . . 7 ⊢ (𝐵 ∈ ℝ^* → ¬ +∞ < 𝐵)
35	34	adantl 481	. . . . . 6 ⊢ ((𝐴 = +∞ ∧ 𝐵 ∈ ℝ^*) → ¬ +∞ < 𝐵)
36	35	pm2.21d 121	. . . . 5 ⊢ ((𝐴 = +∞ ∧ 𝐵 ∈ ℝ^*) → (+∞ < 𝐵 → ∃𝑥 ∈ ℚ (𝐴 < 𝑥 ∧ 𝑥 < 𝐵)))
37	33, 36	sylbid 240	. . . 4 ⊢ ((𝐴 = +∞ ∧ 𝐵 ∈ ℝ^*) → (𝐴 < 𝐵 → ∃𝑥 ∈ ℚ (𝐴 < 𝑥 ∧ 𝑥 < 𝐵)))
38		peano2rem 11576	. . . . . . . . . 10 ⊢ (𝐵 ∈ ℝ → (𝐵 − 1) ∈ ℝ)
39	38	adantl 481	. . . . . . . . 9 ⊢ ((𝐴 = -∞ ∧ 𝐵 ∈ ℝ) → (𝐵 − 1) ∈ ℝ)
40		simpr 484	. . . . . . . . 9 ⊢ ((𝐴 = -∞ ∧ 𝐵 ∈ ℝ) → 𝐵 ∈ ℝ)
41		ltm1 12109	. . . . . . . . . 10 ⊢ (𝐵 ∈ ℝ → (𝐵 − 1) < 𝐵)
42	41	adantl 481	. . . . . . . . 9 ⊢ ((𝐴 = -∞ ∧ 𝐵 ∈ ℝ) → (𝐵 − 1) < 𝐵)
43		qbtwnre 13241	. . . . . . . . 9 ⊢ (((𝐵 − 1) ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ (𝐵 − 1) < 𝐵) → ∃𝑥 ∈ ℚ ((𝐵 − 1) < 𝑥 ∧ 𝑥 < 𝐵))
44	39, 40, 42, 43	syl3anc 1373	. . . . . . . 8 ⊢ ((𝐴 = -∞ ∧ 𝐵 ∈ ℝ) → ∃𝑥 ∈ ℚ ((𝐵 − 1) < 𝑥 ∧ 𝑥 < 𝐵))
45		simpll 767	. . . . . . . . . . . 12 ⊢ (((𝐴 = -∞ ∧ 𝐵 ∈ ℝ) ∧ 𝑥 ∈ ℚ) → 𝐴 = -∞)
46	12	adantl 481	. . . . . . . . . . . . 13 ⊢ (((𝐴 = -∞ ∧ 𝐵 ∈ ℝ) ∧ 𝑥 ∈ ℚ) → 𝑥 ∈ ℝ)
47	46	mnfltd 13166	. . . . . . . . . . . 12 ⊢ (((𝐴 = -∞ ∧ 𝐵 ∈ ℝ) ∧ 𝑥 ∈ ℚ) → -∞ < 𝑥)
48	45, 47	eqbrtrd 5165	. . . . . . . . . . 11 ⊢ (((𝐴 = -∞ ∧ 𝐵 ∈ ℝ) ∧ 𝑥 ∈ ℚ) → 𝐴 < 𝑥)
49	48	a1d 25	. . . . . . . . . 10 ⊢ (((𝐴 = -∞ ∧ 𝐵 ∈ ℝ) ∧ 𝑥 ∈ ℚ) → ((𝐵 − 1) < 𝑥 → 𝐴 < 𝑥))
50	49	anim1d 611	. . . . . . . . 9 ⊢ (((𝐴 = -∞ ∧ 𝐵 ∈ ℝ) ∧ 𝑥 ∈ ℚ) → (((𝐵 − 1) < 𝑥 ∧ 𝑥 < 𝐵) → (𝐴 < 𝑥 ∧ 𝑥 < 𝐵)))
51	50	reximdva 3168	. . . . . . . 8 ⊢ ((𝐴 = -∞ ∧ 𝐵 ∈ ℝ) → (∃𝑥 ∈ ℚ ((𝐵 − 1) < 𝑥 ∧ 𝑥 < 𝐵) → ∃𝑥 ∈ ℚ (𝐴 < 𝑥 ∧ 𝑥 < 𝐵)))
52	44, 51	mpd 15	. . . . . . 7 ⊢ ((𝐴 = -∞ ∧ 𝐵 ∈ ℝ) → ∃𝑥 ∈ ℚ (𝐴 < 𝑥 ∧ 𝑥 < 𝐵))
53	52	a1d 25	. . . . . 6 ⊢ ((𝐴 = -∞ ∧ 𝐵 ∈ ℝ) → (𝐴 < 𝐵 → ∃𝑥 ∈ ℚ (𝐴 < 𝑥 ∧ 𝑥 < 𝐵)))
54		1re 11261	. . . . . . . . . 10 ⊢ 1 ∈ ℝ
55		mnflt 13165	. . . . . . . . . 10 ⊢ (1 ∈ ℝ → -∞ < 1)
56	54, 55	ax-mp 5	. . . . . . . . 9 ⊢ -∞ < 1
57		breq1 5146	. . . . . . . . 9 ⊢ (𝐴 = -∞ → (𝐴 < 1 ↔ -∞ < 1))
58	56, 57	mpbiri 258	. . . . . . . 8 ⊢ (𝐴 = -∞ → 𝐴 < 1)
59		ltpnf 13162	. . . . . . . . . 10 ⊢ (1 ∈ ℝ → 1 < +∞)
60	54, 59	ax-mp 5	. . . . . . . . 9 ⊢ 1 < +∞
61		breq2 5147	. . . . . . . . 9 ⊢ (𝐵 = +∞ → (1 < 𝐵 ↔ 1 < +∞))
62	60, 61	mpbiri 258	. . . . . . . 8 ⊢ (𝐵 = +∞ → 1 < 𝐵)
63		1z 12647	. . . . . . . . . 10 ⊢ 1 ∈ ℤ
64		zq 12996	. . . . . . . . . 10 ⊢ (1 ∈ ℤ → 1 ∈ ℚ)
65	63, 64	ax-mp 5	. . . . . . . . 9 ⊢ 1 ∈ ℚ
66		breq2 5147	. . . . . . . . . . 11 ⊢ (𝑥 = 1 → (𝐴 < 𝑥 ↔ 𝐴 < 1))
67		breq1 5146	. . . . . . . . . . 11 ⊢ (𝑥 = 1 → (𝑥 < 𝐵 ↔ 1 < 𝐵))
68	66, 67	anbi12d 632	. . . . . . . . . 10 ⊢ (𝑥 = 1 → ((𝐴 < 𝑥 ∧ 𝑥 < 𝐵) ↔ (𝐴 < 1 ∧ 1 < 𝐵)))
69	68	rspcev 3622	. . . . . . . . 9 ⊢ ((1 ∈ ℚ ∧ (𝐴 < 1 ∧ 1 < 𝐵)) → ∃𝑥 ∈ ℚ (𝐴 < 𝑥 ∧ 𝑥 < 𝐵))
70	65, 69	mpan 690	. . . . . . . 8 ⊢ ((𝐴 < 1 ∧ 1 < 𝐵) → ∃𝑥 ∈ ℚ (𝐴 < 𝑥 ∧ 𝑥 < 𝐵))
71	58, 62, 70	syl2an 596	. . . . . . 7 ⊢ ((𝐴 = -∞ ∧ 𝐵 = +∞) → ∃𝑥 ∈ ℚ (𝐴 < 𝑥 ∧ 𝑥 < 𝐵))
72	71	a1d 25	. . . . . 6 ⊢ ((𝐴 = -∞ ∧ 𝐵 = +∞) → (𝐴 < 𝐵 → ∃𝑥 ∈ ℚ (𝐴 < 𝑥 ∧ 𝑥 < 𝐵)))
73		3mix3 1333	. . . . . . . 8 ⊢ (𝐴 = -∞ → (𝐴 ∈ ℝ ∨ 𝐴 = +∞ ∨ 𝐴 = -∞))
74	73, 1	sylibr 234	. . . . . . 7 ⊢ (𝐴 = -∞ → 𝐴 ∈ ℝ^*)
75	74, 28	sylan 580	. . . . . 6 ⊢ ((𝐴 = -∞ ∧ 𝐵 = -∞) → (𝐴 < 𝐵 → ∃𝑥 ∈ ℚ (𝐴 < 𝑥 ∧ 𝑥 < 𝐵)))
76	53, 72, 75	3jaodan 1433	. . . . 5 ⊢ ((𝐴 = -∞ ∧ (𝐵 ∈ ℝ ∨ 𝐵 = +∞ ∨ 𝐵 = -∞)) → (𝐴 < 𝐵 → ∃𝑥 ∈ ℚ (𝐴 < 𝑥 ∧ 𝑥 < 𝐵)))
77	2, 76	sylan2b 594	. . . 4 ⊢ ((𝐴 = -∞ ∧ 𝐵 ∈ ℝ^*) → (𝐴 < 𝐵 → ∃𝑥 ∈ ℚ (𝐴 < 𝑥 ∧ 𝑥 < 𝐵)))
78	31, 37, 77	3jaoian 1432	. . 3 ⊢ (((𝐴 ∈ ℝ ∨ 𝐴 = +∞ ∨ 𝐴 = -∞) ∧ 𝐵 ∈ ℝ^*) → (𝐴 < 𝐵 → ∃𝑥 ∈ ℚ (𝐴 < 𝑥 ∧ 𝑥 < 𝐵)))
79	1, 78	sylanb 581	. 2 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) → (𝐴 < 𝐵 → ∃𝑥 ∈ ℚ (𝐴 < 𝑥 ∧ 𝑥 < 𝐵)))
80	79	3impia 1118	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 206 ∧ wa 395 ∨ w3o 1086 ∧ w3a 1087 = wceq 1540 ∈ wcel 2108 ∃wrex 3070 class class class wbr 5143 (class class class)co 7431 ℝcr 11154 1c1 11156 + caddc 11158 +∞cpnf 11292 -∞cmnf 11293 ℝ^*cxr 11294 < clt 11295 − cmin 11492 ℤcz 12613 ℚcq 12990

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-cnex 11211 ax-resscn 11212 ax-1cn 11213 ax-icn 11214 ax-addcl 11215 ax-addrcl 11216 ax-mulcl 11217 ax-mulrcl 11218 ax-mulcom 11219 ax-addass 11220 ax-mulass 11221 ax-distr 11222 ax-i2m1 11223 ax-1ne0 11224 ax-1rid 11225 ax-rnegex 11226 ax-rrecex 11227 ax-cnre 11228 ax-pre-lttri 11229 ax-pre-lttrn 11230 ax-pre-ltadd 11231 ax-pre-mulgt0 11232 ax-pre-sup 11233

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3380 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-pred 6321 df-ord 6387 df-on 6388 df-lim 6389 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-1st 8014 df-2nd 8015 df-frecs 8306 df-wrecs 8337 df-recs 8411 df-rdg 8450 df-er 8745 df-en 8986 df-dom 8987 df-sdom 8988 df-sup 9482 df-inf 9483 df-pnf 11297 df-mnf 11298 df-xr 11299 df-ltxr 11300 df-le 11301 df-sub 11494 df-neg 11495 df-div 11921 df-nn 12267 df-n0 12527 df-z 12614 df-uz 12879 df-q 12991

This theorem is referenced by: qextltlem 13244 xralrple 13247 ixxub 13408 ixxlb 13409 ioo0 13412 ico0 13433 ioc0 13434 blssps 24434 blss 24435 blcld 24518 qdensere 24790 tgqioo 24821 dvlip2 26034 lhop2 26054 itgsubst 26090 itg2gt0cn 37682 qinioo 45548 qelioo 45559 qndenserrnbllem 46309

W3C validator