Theorem qbtwnxr 10329

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 9845	. . 3 |- ( A e. RR* <-> ( A e. RR \/ A = +oo \/ A = -oo ) )
2		elxr 9845	. . . . 5 |- ( B e. RR* <-> ( B e. RR \/ B = +oo \/ B = -oo ) )
3		qbtwnre 10328	. . . . . . 7 |- ( ( A e. RR /\ B e. RR /\ A < B ) -> E. x e. QQ ( A < x /\ x < B ) )
4	3	3expia 1207	. . . . . 6 |- ( ( A e. RR /\ B e. RR ) -> ( A < B -> E. x e. QQ ( A < x /\ x < B ) ) )
5		simpl 109	. . . . . . . . 9 |- ( ( A e. RR /\ B = +oo ) -> A e. RR )
6		peano2re 8157	. . . . . . . . . 10 |- ( A e. RR -> ( A + 1 ) e. RR )
7	6	adantr 276	. . . . . . . . 9 |- ( ( A e. RR /\ B = +oo ) -> ( A + 1 ) e. RR )
8		ltp1 8865	. . . . . . . . . 10 |- ( A e. RR -> A < ( A + 1 ) )
9	8	adantr 276	. . . . . . . . 9 |- ( ( A e. RR /\ B = +oo ) -> A < ( A + 1 ) )
10		qbtwnre 10328	. . . . . . . . 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 1249	. . . . . . . 8 |- ( ( A e. RR /\ B = +oo ) -> E. x e. QQ ( A < x /\ x < ( A + 1 ) ) )
12		qre 9693	. . . . . . . . . . . . . 14 |- ( x e. QQ -> x e. RR )
13		ltpnf 9849	. . . . . . . . . . . . . 14 |- ( x e. RR -> x < +oo )
14	12, 13	syl 14	. . . . . . . . . . . . 13 |- ( x e. QQ -> x < +oo )
15	14	adantl 277	. . . . . . . . . . . 12 |- ( ( ( A e. RR /\ B = +oo ) /\ x e. QQ ) -> x < +oo )
16		simplr 528	. . . . . . . . . . . 12 |- ( ( ( A e. RR /\ B = +oo ) /\ x e. QQ ) -> B = +oo )
17	15, 16	breqtrrd 4058	. . . . . . . . . . 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 337	. . . . . . . . 9 |- ( ( ( A e. RR /\ B = +oo ) /\ x e. QQ ) -> ( ( A < x /\ x < ( A + 1 ) ) -> ( A < x /\ x < B ) ) )
20	19	reximdva 2596	. . . . . . . 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 8067	. . . . . . 7 |- ( A e. RR -> A e. RR* )
24		breq2 4034	. . . . . . . . 9 |- ( B = -oo -> ( A < B <-> A < -oo ) )
25	24	adantl 277	. . . . . . . 8 |- ( ( A e. RR* /\ B = -oo ) -> ( A < B <-> A < -oo ) )
26		nltmnf 9857	. . . . . . . . . 10 |- ( A e. RR* -> -. A < -oo )
27	26	adantr 276	. . . . . . . . 9 |- ( ( A e. RR* /\ B = -oo ) -> -. A < -oo )
28	27	pm2.21d 620	. . . . . . . 8 |- ( ( A e. RR* /\ B = -oo ) -> ( A < -oo -> E. x e. QQ ( A < x /\ x < B ) ) )
29	25, 28	sylbid 150	. . . . . . 7 |- ( ( A e. RR* /\ B = -oo ) -> ( A < B -> E. x e. QQ ( A < x /\ x < B ) ) )
30	23, 29	sylan 283	. . . . . 6 |- ( ( A e. RR /\ B = -oo ) -> ( A < B -> E. x e. QQ ( A < x /\ x < B ) ) )
31	4, 22, 30	3jaodan 1317	. . . . 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 287	. . . 4 |- ( ( A e. RR /\ B e. RR* ) -> ( A < B -> E. x e. QQ ( A < x /\ x < B ) ) )
33		breq1 4033	. . . . . 6 |- ( A = +oo -> ( A < B <-> +oo < B ) )
34	33	adantr 276	. . . . 5 |- ( ( A = +oo /\ B e. RR* ) -> ( A < B <-> +oo < B ) )
35		pnfnlt 9856	. . . . . . 7 |- ( B e. RR* -> -. +oo < B )
36	35	adantl 277	. . . . . 6 |- ( ( A = +oo /\ B e. RR* ) -> -. +oo < B )
37	36	pm2.21d 620	. . . . 5 |- ( ( A = +oo /\ B e. RR* ) -> ( +oo < B -> E. x e. QQ ( A < x /\ x < B ) ) )
38	34, 37	sylbid 150	. . . 4 |- ( ( A = +oo /\ B e. RR* ) -> ( A < B -> E. x e. QQ ( A < x /\ x < B ) ) )
39		peano2rem 8288	. . . . . . . . . 10 |- ( B e. RR -> ( B - 1 ) e. RR )
40	39	adantl 277	. . . . . . . . 9 |- ( ( A = -oo /\ B e. RR ) -> ( B - 1 ) e. RR )
41		simpr 110	. . . . . . . . 9 |- ( ( A = -oo /\ B e. RR ) -> B e. RR )
42		ltm1 8867	. . . . . . . . . 10 |- ( B e. RR -> ( B - 1 ) < B )
43	42	adantl 277	. . . . . . . . 9 |- ( ( A = -oo /\ B e. RR ) -> ( B - 1 ) < B )
44		qbtwnre 10328	. . . . . . . . 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 1249	. . . . . . . 8 |- ( ( A = -oo /\ B e. RR ) -> E. x e. QQ ( ( B - 1 ) < x /\ x < B ) )
46		simpll 527	. . . . . . . . . . . 12 |- ( ( ( A = -oo /\ B e. RR ) /\ x e. QQ ) -> A = -oo )
47	12	adantl 277	. . . . . . . . . . . . 13 |- ( ( ( A = -oo /\ B e. RR ) /\ x e. QQ ) -> x e. RR )
48		mnflt 9852	. . . . . . . . . . . . 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 4052	. . . . . . . . . . 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 336	. . . . . . . . 9 |- ( ( ( A = -oo /\ B e. RR ) /\ x e. QQ ) -> ( ( ( B - 1 ) < x /\ x < B ) -> ( A < x /\ x < B ) ) )
53	52	reximdva 2596	. . . . . . . 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 8020	. . . . . . . . . 10 |- 1 e. RR
57		mnflt 9852	. . . . . . . . . 10 |- ( 1 e. RR -> -oo < 1 )
58	56, 57	ax-mp 5	. . . . . . . . 9 |- -oo < 1
59		breq1 4033	. . . . . . . . 9 |- ( A = -oo -> ( A < 1 <-> -oo < 1 ) )
60	58, 59	mpbiri 168	. . . . . . . 8 |- ( A = -oo -> A < 1 )
61		ltpnf 9849	. . . . . . . . . 10 |- ( 1 e. RR -> 1 < +oo )
62	56, 61	ax-mp 5	. . . . . . . . 9 |- 1 < +oo
63		breq2 4034	. . . . . . . . 9 |- ( B = +oo -> ( 1 < B <-> 1 < +oo ) )
64	62, 63	mpbiri 168	. . . . . . . 8 |- ( B = +oo -> 1 < B )
65		1z 9346	. . . . . . . . . 10 |- 1 e. ZZ
66		zq 9694	. . . . . . . . . 10 |- ( 1 e. ZZ -> 1 e. QQ )
67	65, 66	ax-mp 5	. . . . . . . . 9 |- 1 e. QQ
68		breq2 4034	. . . . . . . . . . 11 |- ( x = 1 -> ( A < x <-> A < 1 ) )
69		breq1 4033	. . . . . . . . . . 11 |- ( x = 1 -> ( x < B <-> 1 < B ) )
70	68, 69	anbi12d 473	. . . . . . . . . 10 |- ( x = 1 -> ( ( A < x /\ x < B ) <-> ( A < 1 /\ 1 < B ) ) )
71	70	rspcev 2865	. . . . . . . . 9 |- ( ( 1 e. QQ /\ ( A < 1 /\ 1 < B ) ) -> E. x e. QQ ( A < x /\ x < B ) )
72	67, 71	mpan 424	. . . . . . . 8 |- ( ( A < 1 /\ 1 < B ) -> E. x e. QQ ( A < x /\ x < B ) )
73	60, 64, 72	syl2an 289	. . . . . . 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 1170	. . . . . . . 8 |- ( A = -oo -> ( A e. RR \/ A = +oo \/ A = -oo ) )
76	75, 1	sylibr 134	. . . . . . 7 |- ( A = -oo -> A e. RR* )
77	76, 29	sylan 283	. . . . . 6 |- ( ( A = -oo /\ B = -oo ) -> ( A < B -> E. x e. QQ ( A < x /\ x < B ) ) )
78	55, 74, 77	3jaodan 1317	. . . . 5 |- ( ( A = -oo /\ ( B e. RR \/ B = +oo \/ B = -oo ) ) -> ( A < B -> E. x e. QQ ( A < x /\ x < B ) ) )
79	2, 78	sylan2b 287	. . . 4 |- ( ( A = -oo /\ B e. RR* ) -> ( A < B -> E. x e. QQ ( A < x /\ x < B ) ) )
80	32, 38, 79	3jaoian 1316	. . 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 284	. 2 |- ( ( A e. RR* /\ B e. RR* ) -> ( A < B -> E. x e. QQ ( A < x /\ x < B ) ) )
82	81	3impia 1202	1 |- ( ( A e. RR* /\ B e. RR* /\ A < B ) -> E. x e. QQ ( A < x /\ x < B ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 104   <-> wb 105   \/ w3o 979   /\ w3a 980   = wceq 1364   e. wcel 2164   E.wrex 2473   class class class wbr 4030  (class class class)co 5919   RRcr 7873   1c1 7875   + caddc 7877   +oocpnf 8053   -oocmnf 8054   RR*cxr 8055   < clt 8056   - cmin 8192   ZZcz 9320   QQcq 9687

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2166  ax-14 2167  ax-ext 2175  ax-sep 4148  ax-pow 4204  ax-pr 4239  ax-un 4465  ax-setind 4570  ax-cnex 7965  ax-resscn 7966  ax-1cn 7967  ax-1re 7968  ax-icn 7969  ax-addcl 7970  ax-addrcl 7971  ax-mulcl 7972  ax-mulrcl 7973  ax-addcom 7974  ax-mulcom 7975  ax-addass 7976  ax-mulass 7977  ax-distr 7978  ax-i2m1 7979  ax-0lt1 7980  ax-1rid 7981  ax-0id 7982  ax-rnegex 7983  ax-precex 7984  ax-cnre 7985  ax-pre-ltirr 7986  ax-pre-ltwlin 7987  ax-pre-lttrn 7988  ax-pre-apti 7989  ax-pre-ltadd 7990  ax-pre-mulgt0 7991  ax-pre-mulext 7992  ax-arch 7993

This theorem depends on definitions:  df-bi 117  df-3or 981  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ne 2365  df-nel 2460  df-ral 2477  df-rex 2478  df-reu 2479  df-rmo 2480  df-rab 2481  df-v 2762  df-sbc 2987  df-csb 3082  df-dif 3156  df-un 3158  df-in 3160  df-ss 3167  df-pw 3604  df-sn 3625  df-pr 3626  df-op 3628  df-uni 3837  df-int 3872  df-iun 3915  df-br 4031  df-opab 4092  df-mpt 4093  df-id 4325  df-po 4328  df-iso 4329  df-xp 4666  df-rel 4667  df-cnv 4668  df-co 4669  df-dm 4670  df-rn 4671  df-res 4672  df-ima 4673  df-iota 5216  df-fun 5257  df-fn 5258  df-f 5259  df-fv 5263  df-riota 5874  df-ov 5922  df-oprab 5923  df-mpo 5924  df-1st 6195  df-2nd 6196  df-pnf 8058  df-mnf 8059  df-xr 8060  df-ltxr 8061  df-le 8062  df-sub 8194  df-neg 8195  df-reap 8596  df-ap 8603  df-div 8694  df-inn 8985  df-2 9043  df-n0 9244  df-z 9321  df-uz 9596  df-q 9688  df-rp 9723

This theorem is referenced by:  ioo0  10331  ioom  10332  ico0  10333  ioc0  10334  blssps  14606  blss  14607  tgqioo  14734