Theorem qbtwnxr 10193

Description: The rational numbers are dense in RR*: 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 e. RR* /\ B e. RR* /\ A < B ) -> E. x e. QQ ( A < x /\ x < B ) )

Distinct variable groups:   x, A   x, B

Proof of Theorem qbtwnxr

Step	Hyp	Ref	Expression
1		elxr 9712	. . 3 |- ( A e. RR* <-> ( A e. RR \/ A = +oo \/ A = -oo ) )
2		elxr 9712	. . . . 5 |- ( B e. RR* <-> ( B e. RR \/ B = +oo \/ B = -oo ) )
3		qbtwnre 10192	. . . . . . 7 |- ( ( A e. RR /\ B e. RR /\ A < B ) -> E. x e. QQ ( A < x /\ x < B ) )
4	3	3expia 1195	. . . . . 6 |- ( ( A e. RR /\ B e. RR ) -> ( A < B -> E. x e. QQ ( A < x /\ x < B ) ) )
5		simpl 108	. . . . . . . . 9 |- ( ( A e. RR /\ B = +oo ) -> A e. RR )
6		peano2re 8034	. . . . . . . . . 10 |- ( A e. RR -> ( A + 1 ) e. RR )
7	6	adantr 274	. . . . . . . . 9 |- ( ( A e. RR /\ B = +oo ) -> ( A + 1 ) e. RR )
8		ltp1 8739	. . . . . . . . . 10 |- ( A e. RR -> A < ( A + 1 ) )
9	8	adantr 274	. . . . . . . . 9 |- ( ( A e. RR /\ B = +oo ) -> A < ( A + 1 ) )
10		qbtwnre 10192	. . . . . . . . 9 |- ( ( A e. RR /\ ( A + 1 ) e. RR /\ A < ( A + 1 ) ) -> E. x e. QQ ( A < x /\ x < ( A + 1 ) ) )
11	5, 7, 9, 10	syl3anc 1228	. . . . . . . 8 |- ( ( A e. RR /\ B = +oo ) -> E. x e. QQ ( A < x /\ x < ( A + 1 ) ) )
12		qre 9563	. . . . . . . . . . . . . 14 |- ( x e. QQ -> x e. RR )
13		ltpnf 9716	. . . . . . . . . . . . . 14 |- ( x e. RR -> x < +oo )
14	12, 13	syl 14	. . . . . . . . . . . . 13 |- ( x e. QQ -> x < +oo )
15	14	adantl 275	. . . . . . . . . . . 12 |- ( ( ( A e. RR /\ B = +oo ) /\ x e. QQ ) -> x < +oo )
16		simplr 520	. . . . . . . . . . . 12 |- ( ( ( A e. RR /\ B = +oo ) /\ x e. QQ ) -> B = +oo )
17	15, 16	breqtrrd 4010	. . . . . . . . . . 11 |- ( ( ( A e. RR /\ B = +oo ) /\ x e. QQ ) -> x < B )
18	17	a1d 22	. . . . . . . . . 10 |- ( ( ( A e. RR /\ B = +oo ) /\ x e. QQ ) -> ( x < ( A + 1 ) -> x < B ) )
19	18	anim2d 335	. . . . . . . . 9 |- ( ( ( A e. RR /\ B = +oo ) /\ x e. QQ ) -> ( ( A < x /\ x < ( A + 1 ) ) -> ( A < x /\ x < B ) ) )
20	19	reximdva 2568	. . . . . . . 8 |- ( ( A e. RR /\ B = +oo ) -> ( E. x e. QQ ( A < x /\ x < ( A + 1 ) ) -> E. x e. QQ ( A < x /\ x < B ) ) )
21	11, 20	mpd 13	. . . . . . 7 |- ( ( A e. RR /\ B = +oo ) -> E. x e. QQ ( A < x /\ x < B ) )
22	21	a1d 22	. . . . . 6 |- ( ( A e. RR /\ B = +oo ) -> ( A < B -> E. x e. QQ ( A < x /\ x < B ) ) )
23		rexr 7944	. . . . . . 7 |- ( A e. RR -> A e. RR* )
24		breq2 3986	. . . . . . . . 9 |- ( B = -oo -> ( A < B <-> A < -oo ) )
25	24	adantl 275	. . . . . . . 8 |- ( ( A e. RR* /\ B = -oo ) -> ( A < B <-> A < -oo ) )
26		nltmnf 9724	. . . . . . . . . 10 |- ( A e. RR* -> -. A < -oo )
27	26	adantr 274	. . . . . . . . 9 |- ( ( A e. RR* /\ B = -oo ) -> -. A < -oo )
28	27	pm2.21d 609	. . . . . . . 8 |- ( ( A e. RR* /\ B = -oo ) -> ( A < -oo -> E. x e. QQ ( A < x /\ x < B ) ) )
29	25, 28	sylbid 149	. . . . . . 7 |- ( ( A e. RR* /\ B = -oo ) -> ( A < B -> E. x e. QQ ( A < x /\ x < B ) ) )
30	23, 29	sylan 281	. . . . . 6 |- ( ( A e. RR /\ B = -oo ) -> ( A < B -> E. x e. QQ ( A < x /\ x < B ) ) )
31	4, 22, 30	3jaodan 1296	. . . . 5 |- ( ( A e. RR /\ ( B e. RR \/ B = +oo \/ B = -oo ) ) -> ( A < B -> E. x e. QQ ( A < x /\ x < B ) ) )
32	2, 31	sylan2b 285	. . . 4 |- ( ( A e. RR /\ B e. RR* ) -> ( A < B -> E. x e. QQ ( A < x /\ x < B ) ) )
33		breq1 3985	. . . . . 6 |- ( A = +oo -> ( A < B <-> +oo < B ) )
34	33	adantr 274	. . . . 5 |- ( ( A = +oo /\ B e. RR* ) -> ( A < B <-> +oo < B ) )
35		pnfnlt 9723	. . . . . . 7 |- ( B e. RR* -> -. +oo < B )
36	35	adantl 275	. . . . . 6 |- ( ( A = +oo /\ B e. RR* ) -> -. +oo < B )
37	36	pm2.21d 609	. . . . 5 |- ( ( A = +oo /\ B e. RR* ) -> ( +oo < B -> E. x e. QQ ( A < x /\ x < B ) ) )
38	34, 37	sylbid 149	. . . 4 |- ( ( A = +oo /\ B e. RR* ) -> ( A < B -> E. x e. QQ ( A < x /\ x < B ) ) )
39		peano2rem 8165	. . . . . . . . . 10 |- ( B e. RR -> ( B - 1 ) e. RR )
40	39	adantl 275	. . . . . . . . 9 |- ( ( A = -oo /\ B e. RR ) -> ( B - 1 ) e. RR )
41		simpr 109	. . . . . . . . 9 |- ( ( A = -oo /\ B e. RR ) -> B e. RR )
42		ltm1 8741	. . . . . . . . . 10 |- ( B e. RR -> ( B - 1 ) < B )
43	42	adantl 275	. . . . . . . . 9 |- ( ( A = -oo /\ B e. RR ) -> ( B - 1 ) < B )
44		qbtwnre 10192	. . . . . . . . 9 |- ( ( ( B - 1 ) e. RR /\ B e. RR /\ ( B - 1 ) < B ) -> E. x e. QQ ( ( B - 1 ) < x /\ x < B ) )
45	40, 41, 43, 44	syl3anc 1228	. . . . . . . 8 |- ( ( A = -oo /\ B e. RR ) -> E. x e. QQ ( ( B - 1 ) < x /\ x < B ) )
46		simpll 519	. . . . . . . . . . . 12 |- ( ( ( A = -oo /\ B e. RR ) /\ x e. QQ ) -> A = -oo )
47	12	adantl 275	. . . . . . . . . . . . 13 |- ( ( ( A = -oo /\ B e. RR ) /\ x e. QQ ) -> x e. RR )
48		mnflt 9719	. . . . . . . . . . . . 13 |- ( x e. RR -> -oo < x )
49	47, 48	syl 14	. . . . . . . . . . . 12 |- ( ( ( A = -oo /\ B e. RR ) /\ x e. QQ ) -> -oo < x )
50	46, 49	eqbrtrd 4004	. . . . . . . . . . 11 |- ( ( ( A = -oo /\ B e. RR ) /\ x e. QQ ) -> A < x )
51	50	a1d 22	. . . . . . . . . 10 |- ( ( ( A = -oo /\ B e. RR ) /\ x e. QQ ) -> ( ( B - 1 ) < x -> A < x ) )
52	51	anim1d 334	. . . . . . . . 9 |- ( ( ( A = -oo /\ B e. RR ) /\ x e. QQ ) -> ( ( ( B - 1 ) < x /\ x < B ) -> ( A < x /\ x < B ) ) )
53	52	reximdva 2568	. . . . . . . 8 |- ( ( A = -oo /\ B e. RR ) -> ( E. x e. QQ ( ( B - 1 ) < x /\ x < B ) -> E. x e. QQ ( A < x /\ x < B ) ) )
54	45, 53	mpd 13	. . . . . . 7 |- ( ( A = -oo /\ B e. RR ) -> E. x e. QQ ( A < x /\ x < B ) )
55	54	a1d 22	. . . . . 6 |- ( ( A = -oo /\ B e. RR ) -> ( A < B -> E. x e. QQ ( A < x /\ x < B ) ) )
56		1re 7898	. . . . . . . . . 10 |- 1 e. RR
57		mnflt 9719	. . . . . . . . . 10 |- ( 1 e. RR -> -oo < 1 )
58	56, 57	ax-mp 5	. . . . . . . . 9 |- -oo < 1
59		breq1 3985	. . . . . . . . 9 |- ( A = -oo -> ( A < 1 <-> -oo < 1 ) )
60	58, 59	mpbiri 167	. . . . . . . 8 |- ( A = -oo -> A < 1 )
61		ltpnf 9716	. . . . . . . . . 10 |- ( 1 e. RR -> 1 < +oo )
62	56, 61	ax-mp 5	. . . . . . . . 9 |- 1 < +oo
63		breq2 3986	. . . . . . . . 9 |- ( B = +oo -> ( 1 < B <-> 1 < +oo ) )
64	62, 63	mpbiri 167	. . . . . . . 8 |- ( B = +oo -> 1 < B )
65		1z 9217	. . . . . . . . . 10 |- 1 e. ZZ
66		zq 9564	. . . . . . . . . 10 |- ( 1 e. ZZ -> 1 e. QQ )
67	65, 66	ax-mp 5	. . . . . . . . 9 |- 1 e. QQ
68		breq2 3986	. . . . . . . . . . 11 |- ( x = 1 -> ( A < x <-> A < 1 ) )
69		breq1 3985	. . . . . . . . . . 11 |- ( x = 1 -> ( x < B <-> 1 < B ) )
70	68, 69	anbi12d 465	. . . . . . . . . 10 |- ( x = 1 -> ( ( A < x /\ x < B ) <-> ( A < 1 /\ 1 < B ) ) )
71	70	rspcev 2830	. . . . . . . . 9 |- ( ( 1 e. QQ /\ ( A < 1 /\ 1 < B ) ) -> E. x e. QQ ( A < x /\ x < B ) )
72	67, 71	mpan 421	. . . . . . . 8 |- ( ( A < 1 /\ 1 < B ) -> E. x e. QQ ( A < x /\ x < B ) )
73	60, 64, 72	syl2an 287	. . . . . . 7 |- ( ( A = -oo /\ B = +oo ) -> E. x e. QQ ( A < x /\ x < B ) )
74	73	a1d 22	. . . . . 6 |- ( ( A = -oo /\ B = +oo ) -> ( A < B -> E. x e. QQ ( A < x /\ x < B ) ) )
75		3mix3 1158	. . . . . . . 8 |- ( A = -oo -> ( A e. RR \/ A = +oo \/ A = -oo ) )
76	75, 1	sylibr 133	. . . . . . 7 |- ( A = -oo -> A e. RR* )
77	76, 29	sylan 281	. . . . . 6 |- ( ( A = -oo /\ B = -oo ) -> ( A < B -> E. x e. QQ ( A < x /\ x < B ) ) )
78	55, 74, 77	3jaodan 1296	. . . . 5 |- ( ( A = -oo /\ ( B e. RR \/ B = +oo \/ B = -oo ) ) -> ( A < B -> E. x e. QQ ( A < x /\ x < B ) ) )
79	2, 78	sylan2b 285	. . . 4 |- ( ( A = -oo /\ B e. RR* ) -> ( A < B -> E. x e. QQ ( A < x /\ x < B ) ) )
80	32, 38, 79	3jaoian 1295	. . 3 |- ( ( ( A e. RR \/ A = +oo \/ A = -oo ) /\ B e. RR* ) -> ( A < B -> E. x e. QQ ( A < x /\ x < B ) ) )
81	1, 80	sylanb 282	. 2 |- ( ( A e. RR* /\ B e. RR* ) -> ( A < B -> E. x e. QQ ( A < x /\ x < B ) ) )
82	81	3impia 1190	1 |- ( ( A e. RR* /\ B e. RR* /\ A < B ) -> E. x e. QQ ( A < x /\ x < B ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 103   <-> wb 104   \/ w3o 967   /\ w3a 968   = wceq 1343   e. wcel 2136   E.wrex 2445   class class class wbr 3982  (class class class)co 5842   RRcr 7752   1c1 7754   + caddc 7756   +oocpnf 7930   -oocmnf 7931   RR*cxr 7932   < clt 7933   - cmin 8069   ZZcz 9191   QQcq 9557

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-13 2138  ax-14 2139  ax-ext 2147  ax-sep 4100  ax-pow 4153  ax-pr 4187  ax-un 4411  ax-setind 4514  ax-cnex 7844  ax-resscn 7845  ax-1cn 7846  ax-1re 7847  ax-icn 7848  ax-addcl 7849  ax-addrcl 7850  ax-mulcl 7851  ax-mulrcl 7852  ax-addcom 7853  ax-mulcom 7854  ax-addass 7855  ax-mulass 7856  ax-distr 7857  ax-i2m1 7858  ax-0lt1 7859  ax-1rid 7860  ax-0id 7861  ax-rnegex 7862  ax-precex 7863  ax-cnre 7864  ax-pre-ltirr 7865  ax-pre-ltwlin 7866  ax-pre-lttrn 7867  ax-pre-apti 7868  ax-pre-ltadd 7869  ax-pre-mulgt0 7870  ax-pre-mulext 7871  ax-arch 7872

This theorem depends on definitions:  df-bi 116  df-3or 969  df-3an 970  df-tru 1346  df-fal 1349  df-nf 1449  df-sb 1751  df-eu 2017  df-mo 2018  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ne 2337  df-nel 2432  df-ral 2449  df-rex 2450  df-reu 2451  df-rmo 2452  df-rab 2453  df-v 2728  df-sbc 2952  df-csb 3046  df-dif 3118  df-un 3120  df-in 3122  df-ss 3129  df-pw 3561  df-sn 3582  df-pr 3583  df-op 3585  df-uni 3790  df-int 3825  df-iun 3868  df-br 3983  df-opab 4044  df-mpt 4045  df-id 4271  df-po 4274  df-iso 4275  df-xp 4610  df-rel 4611  df-cnv 4612  df-co 4613  df-dm 4614  df-rn 4615  df-res 4616  df-ima 4617  df-iota 5153  df-fun 5190  df-fn 5191  df-f 5192  df-fv 5196  df-riota 5798  df-ov 5845  df-oprab 5846  df-mpo 5847  df-1st 6108  df-2nd 6109  df-pnf 7935  df-mnf 7936  df-xr 7937  df-ltxr 7938  df-le 7939  df-sub 8071  df-neg 8072  df-reap 8473  df-ap 8480  df-div 8569  df-inn 8858  df-2 8916  df-n0 9115  df-z 9192  df-uz 9467  df-q 9558  df-rp 9590

This theorem is referenced by:  ioo0  10195  ioom  10196  ico0  10197  ioc0  10198  blssps  13067  blss  13068  tgqioo  13187