Theorem qbtwnxr 10670

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 10157	. . 3 |- ( A e. RR* <-> ( A e. RR \/ A = +oo \/ A = -oo ) )
2		elxr 10157	. . . . 5 |- ( B e. RR* <-> ( B e. RR \/ B = +oo \/ B = -oo ) )
3		qbtwnre 10669	. . . . . . 7 |- ( ( A e. RR /\ B e. RR /\ A < B ) -> E. x e. QQ ( A < x /\ x < B ) )
4	3	3expia 1236	. . . . . 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 8452	. . . . . . . . . 10 |- ( A e. RR -> ( A + 1 ) e. RR )
7	6	adantr 276	. . . . . . . . 9 |- ( ( A e. RR /\ B = +oo ) -> ( A + 1 ) e. RR )
8		ltp1 9164	. . . . . . . . . 10 |- ( A e. RR -> A < ( A + 1 ) )
9	8	adantr 276	. . . . . . . . 9 |- ( ( A e. RR /\ B = +oo ) -> A < ( A + 1 ) )
10		qbtwnre 10669	. . . . . . . . 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 1278	. . . . . . . 8 |- ( ( A e. RR /\ B = +oo ) -> E. x e. QQ ( A < x /\ x < ( A + 1 ) ) )
12		qre 10004	. . . . . . . . . . . . . 14 |- ( x e. QQ -> x e. RR )
13		ltpnf 10161	. . . . . . . . . . . . . 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 533	. . . . . . . . . . . 12 |- ( ( ( A e. RR /\ B = +oo ) /\ x e. QQ ) -> B = +oo )
17	15, 16	breqtrrd 4153	. . . . . . . . . . 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 2652	. . . . . . . 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 8361	. . . . . . 7 |- ( A e. RR -> A e. RR* )
24		breq2 4129	. . . . . . . . 9 |- ( B = -oo -> ( A < B <-> A < -oo ) )
25	24	adantl 277	. . . . . . . 8 |- ( ( A e. RR* /\ B = -oo ) -> ( A < B <-> A < -oo ) )
26		nltmnf 10169	. . . . . . . . . 10 |- ( A e. RR* -> -. A < -oo )
27	26	adantr 276	. . . . . . . . 9 |- ( ( A e. RR* /\ B = -oo ) -> -. A < -oo )
28	27	pm2.21d 628	. . . . . . . 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 1347	. . . . 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 4128	. . . . . 6 |- ( A = +oo -> ( A < B <-> +oo < B ) )
34	33	adantr 276	. . . . 5 |- ( ( A = +oo /\ B e. RR* ) -> ( A < B <-> +oo < B ) )
35		pnfnlt 10168	. . . . . . 7 |- ( B e. RR* -> -. +oo < B )
36	35	adantl 277	. . . . . 6 |- ( ( A = +oo /\ B e. RR* ) -> -. +oo < B )
37	36	pm2.21d 628	. . . . 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 8583	. . . . . . . . . 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 9166	. . . . . . . . . 10 |- ( B e. RR -> ( B - 1 ) < B )
43	42	adantl 277	. . . . . . . . 9 |- ( ( A = -oo /\ B e. RR ) -> ( B - 1 ) < B )
44		qbtwnre 10669	. . . . . . . . 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 1278	. . . . . . . 8 |- ( ( A = -oo /\ B e. RR ) -> E. x e. QQ ( ( B - 1 ) < x /\ x < B ) )
46		simpll 531	. . . . . . . . . . . 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 10164	. . . . . . . . . . . . 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 4147	. . . . . . . . . . 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 2652	. . . . . . . 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 8315	. . . . . . . . . 10 |- 1 e. RR
57		mnflt 10164	. . . . . . . . . 10 |- ( 1 e. RR -> -oo < 1 )
58	56, 57	ax-mp 5	. . . . . . . . 9 |- -oo < 1
59		breq1 4128	. . . . . . . . 9 |- ( A = -oo -> ( A < 1 <-> -oo < 1 ) )
60	58, 59	mpbiri 168	. . . . . . . 8 |- ( A = -oo -> A < 1 )
61		ltpnf 10161	. . . . . . . . . 10 |- ( 1 e. RR -> 1 < +oo )
62	56, 61	ax-mp 5	. . . . . . . . 9 |- 1 < +oo
63		breq2 4129	. . . . . . . . 9 |- ( B = +oo -> ( 1 < B <-> 1 < +oo ) )
64	62, 63	mpbiri 168	. . . . . . . 8 |- ( B = +oo -> 1 < B )
65		1z 9649	. . . . . . . . . 10 |- 1 e. ZZ
66		zq 10005	. . . . . . . . . 10 |- ( 1 e. ZZ -> 1 e. QQ )
67	65, 66	ax-mp 5	. . . . . . . . 9 |- 1 e. QQ
68		breq2 4129	. . . . . . . . . . 11 |- ( x = 1 -> ( A < x <-> A < 1 ) )
69		breq1 4128	. . . . . . . . . . 11 |- ( x = 1 -> ( x < B <-> 1 < B ) )
70	68, 69	anbi12d 477	. . . . . . . . . 10 |- ( x = 1 -> ( ( A < x /\ x < B ) <-> ( A < 1 /\ 1 < B ) ) )
71	70	rspcev 2929	. . . . . . . . 9 |- ( ( 1 e. QQ /\ ( A < 1 /\ 1 < B ) ) -> E. x e. QQ ( A < x /\ x < B ) )
72	67, 71	mpan 428	. . . . . . . 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 1199	. . . . . . . 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 1347	. . . . 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 1346	. . 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 1231	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 1008   /\ w3a 1009   = wceq 1402   e. wcel 2209   E.wrex 2529   class class class wbr 4125  (class class class)co 6075   RRcr 8168   1c1 8170   + caddc 8172   +oocpnf 8347   -oocmnf 8348   RR*cxr 8349   < clt 8350   - cmin 8487   ZZcz 9623   QQcq 9998

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 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4244  ax-pow 4306  ax-pr 4341  ax-un 4573  ax-setind 4679  ax-cnex 8260  ax-resscn 8261  ax-1cn 8262  ax-1re 8263  ax-icn 8264  ax-addcl 8265  ax-addrcl 8266  ax-mulcl 8267  ax-mulrcl 8268  ax-addcom 8269  ax-mulcom 8270  ax-addass 8271  ax-mulass 8272  ax-distr 8273  ax-i2m1 8274  ax-0lt1 8275  ax-1rid 8276  ax-0id 8277  ax-rnegex 8278  ax-precex 8279  ax-cnre 8280  ax-pre-ltirr 8281  ax-pre-ltwlin 8282  ax-pre-lttrn 8283  ax-pre-apti 8284  ax-pre-ltadd 8285  ax-pre-mulgt0 8286  ax-pre-mulext 8287  ax-arch 8288

This theorem depends on definitions:  df-bi 117  df-3or 1010  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-nel 2516  df-ral 2533  df-rex 2534  df-reu 2535  df-rmo 2536  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-pw 3687  df-sn 3711  df-pr 3712  df-op 3714  df-uni 3931  df-int 3966  df-iun 4009  df-br 4126  df-opab 4188  df-mpt 4189  df-id 4433  df-po 4436  df-iso 4437  df-xp 4775  df-rel 4776  df-cnv 4777  df-co 4778  df-dm 4779  df-rn 4780  df-res 4781  df-ima 4782  df-iota 5332  df-fun 5374  df-fn 5375  df-f 5376  df-fv 5380  df-riota 6028  df-ov 6078  df-oprab 6079  df-mpo 6080  df-1st 6364  df-2nd 6365  df-pnf 8352  df-mnf 8353  df-xr 8354  df-ltxr 8355  df-le 8356  df-sub 8489  df-neg 8490  df-reap 8893  df-ap 8900  df-div 8993  df-inn 9284  df-2 9342  df-n0 9543  df-z 9624  df-uz 9901  df-q 9999  df-rp 10034

This theorem is referenced by:  ioo0  10672  ioom  10673  ico0  10674  ioc0  10675  blssps  15451  blss  15452  tgqioo  15579