Theorem qbtwnre 9111

Description: The rational numbers are dense in RR: any two real numbers have a rational between them. Exercise 6 of [Apostol] p. 28. (Contributed by NM, 18-Nov-2004.)

Assertion

Ref	Expression
qbtwnre	|- ( ( 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 qbtwnre

Dummy variables m n are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		simp2 905	. . . 4 |- ( ( A e. RR /\ B e. RR /\ A < B ) -> B e. RR )
2		simp1 904	. . . 4 |- ( ( A e. RR /\ B e. RR /\ A < B ) -> A e. RR )
3	1, 2	resubcld 7379	. . 3 |- ( ( A e. RR /\ B e. RR /\ A < B ) -> ( B - A ) e. RR )
4		simp3 906	. . . 4 |- ( ( A e. RR /\ B e. RR /\ A < B ) -> A < B )
5	2, 1	posdifd 7523	. . . 4 |- ( ( A e. RR /\ B e. RR /\ A < B ) -> ( A < B <-> 0 < ( B - A ) ) )
6	4, 5	mpbid 135	. . 3 |- ( ( A e. RR /\ B e. RR /\ A < B ) -> 0 < ( B - A ) )
7		nnrecl 8179	. . 3 |- ( ( ( B - A ) e. RR /\ 0 < ( B - A ) ) -> E. n e. NN ( 1 / n ) < ( B - A ) )
8	3, 6, 7	syl2anc 391	. 2 |- ( ( A e. RR /\ B e. RR /\ A < B ) -> E. n e. NN ( 1 / n ) < ( B - A ) )
9	2	adantr 261	. . . . 5 |- ( ( ( A e. RR /\ B e. RR /\ A < B ) /\ ( n e. NN /\ ( 1 / n ) < ( B - A ) ) ) -> A e. RR )
10		2re 7985	. . . . . . 7 |- 2 e. RR
11	10	a1i 9	. . . . . 6 |- ( ( ( A e. RR /\ B e. RR /\ A < B ) /\ ( n e. NN /\ ( 1 / n ) < ( B - A ) ) ) -> 2 e. RR )
12		simprl 483	. . . . . . 7 |- ( ( ( A e. RR /\ B e. RR /\ A < B ) /\ ( n e. NN /\ ( 1 / n ) < ( B - A ) ) ) -> n e. NN )
13	12	nnred 7927	. . . . . 6 |- ( ( ( A e. RR /\ B e. RR /\ A < B ) /\ ( n e. NN /\ ( 1 / n ) < ( B - A ) ) ) -> n e. RR )
14	11, 13	remulcld 7056	. . . . 5 |- ( ( ( A e. RR /\ B e. RR /\ A < B ) /\ ( n e. NN /\ ( 1 / n ) < ( B - A ) ) ) -> ( 2 x. n ) e. RR )
15	9, 14	remulcld 7056	. . . 4 |- ( ( ( A e. RR /\ B e. RR /\ A < B ) /\ ( n e. NN /\ ( 1 / n ) < ( B - A ) ) ) -> ( A x. ( 2 x. n ) ) e. RR )
16		rebtwn2z 9109	. . . 4 |- ( ( A x. ( 2 x. n ) ) e. RR -> E. m e. ZZ ( m < ( A x. ( 2 x. n ) ) /\ ( A x. ( 2 x. n ) ) < ( m + 2 ) ) )
17	15, 16	syl 14	. . 3 |- ( ( ( A e. RR /\ B e. RR /\ A < B ) /\ ( n e. NN /\ ( 1 / n ) < ( B - A ) ) ) -> E. m e. ZZ ( m < ( A x. ( 2 x. n ) ) /\ ( A x. ( 2 x. n ) ) < ( m + 2 ) ) )
18		simprl 483	. . . . . 6 |- ( ( ( ( A e. RR /\ B e. RR /\ A < B ) /\ ( n e. NN /\ ( 1 / n ) < ( B - A ) ) ) /\ ( m e. ZZ /\ ( m < ( A x. ( 2 x. n ) ) /\ ( A x. ( 2 x. n ) ) < ( m + 2 ) ) ) ) -> m e. ZZ )
19		2z 8273	. . . . . . 7 |- 2 e. ZZ
20	19	a1i 9	. . . . . 6 |- ( ( ( ( A e. RR /\ B e. RR /\ A < B ) /\ ( n e. NN /\ ( 1 / n ) < ( B - A ) ) ) /\ ( m e. ZZ /\ ( m < ( A x. ( 2 x. n ) ) /\ ( A x. ( 2 x. n ) ) < ( m + 2 ) ) ) ) -> 2 e. ZZ )
21	18, 20	zaddcld 8364	. . . . 5 |- ( ( ( ( A e. RR /\ B e. RR /\ A < B ) /\ ( n e. NN /\ ( 1 / n ) < ( B - A ) ) ) /\ ( m e. ZZ /\ ( m < ( A x. ( 2 x. n ) ) /\ ( A x. ( 2 x. n ) ) < ( m + 2 ) ) ) ) -> ( m + 2 ) e. ZZ )
22		2nn 8077	. . . . . . 7 |- 2 e. NN
23	22	a1i 9	. . . . . 6 |- ( ( ( ( A e. RR /\ B e. RR /\ A < B ) /\ ( n e. NN /\ ( 1 / n ) < ( B - A ) ) ) /\ ( m e. ZZ /\ ( m < ( A x. ( 2 x. n ) ) /\ ( A x. ( 2 x. n ) ) < ( m + 2 ) ) ) ) -> 2 e. NN )
24	12	adantr 261	. . . . . 6 |- ( ( ( ( A e. RR /\ B e. RR /\ A < B ) /\ ( n e. NN /\ ( 1 / n ) < ( B - A ) ) ) /\ ( m e. ZZ /\ ( m < ( A x. ( 2 x. n ) ) /\ ( A x. ( 2 x. n ) ) < ( m + 2 ) ) ) ) -> n e. NN )
25	23, 24	nnmulcld 7962	. . . . 5 |- ( ( ( ( A e. RR /\ B e. RR /\ A < B ) /\ ( n e. NN /\ ( 1 / n ) < ( B - A ) ) ) /\ ( m e. ZZ /\ ( m < ( A x. ( 2 x. n ) ) /\ ( A x. ( 2 x. n ) ) < ( m + 2 ) ) ) ) -> ( 2 x. n ) e. NN )
26		znq 8559	. . . . 5 |- ( ( ( m + 2 ) e. ZZ /\ ( 2 x. n ) e. NN ) -> ( ( m + 2 ) / ( 2 x. n ) ) e. QQ )
27	21, 25, 26	syl2anc 391	. . . 4 |- ( ( ( ( A e. RR /\ B e. RR /\ A < B ) /\ ( n e. NN /\ ( 1 / n ) < ( B - A ) ) ) /\ ( m e. ZZ /\ ( m < ( A x. ( 2 x. n ) ) /\ ( A x. ( 2 x. n ) ) < ( m + 2 ) ) ) ) -> ( ( m + 2 ) / ( 2 x. n ) ) e. QQ )
28		simprrr 492	. . . . 5 |- ( ( ( ( A e. RR /\ B e. RR /\ A < B ) /\ ( n e. NN /\ ( 1 / n ) < ( B - A ) ) ) /\ ( m e. ZZ /\ ( m < ( A x. ( 2 x. n ) ) /\ ( A x. ( 2 x. n ) ) < ( m + 2 ) ) ) ) -> ( A x. ( 2 x. n ) ) < ( m + 2 ) )
29	9	adantr 261	. . . . . 6 |- ( ( ( ( A e. RR /\ B e. RR /\ A < B ) /\ ( n e. NN /\ ( 1 / n ) < ( B - A ) ) ) /\ ( m e. ZZ /\ ( m < ( A x. ( 2 x. n ) ) /\ ( A x. ( 2 x. n ) ) < ( m + 2 ) ) ) ) -> A e. RR )
30	21	zred 8360	. . . . . 6 |- ( ( ( ( A e. RR /\ B e. RR /\ A < B ) /\ ( n e. NN /\ ( 1 / n ) < ( B - A ) ) ) /\ ( m e. ZZ /\ ( m < ( A x. ( 2 x. n ) ) /\ ( A x. ( 2 x. n ) ) < ( m + 2 ) ) ) ) -> ( m + 2 ) e. RR )
31	25	nnrpd 8621	. . . . . 6 |- ( ( ( ( A e. RR /\ B e. RR /\ A < B ) /\ ( n e. NN /\ ( 1 / n ) < ( B - A ) ) ) /\ ( m e. ZZ /\ ( m < ( A x. ( 2 x. n ) ) /\ ( A x. ( 2 x. n ) ) < ( m + 2 ) ) ) ) -> ( 2 x. n ) e. RR+ )
32	29, 30, 31	ltmuldivd 8670	. . . . 5 |- ( ( ( ( A e. RR /\ B e. RR /\ A < B ) /\ ( n e. NN /\ ( 1 / n ) < ( B - A ) ) ) /\ ( m e. ZZ /\ ( m < ( A x. ( 2 x. n ) ) /\ ( A x. ( 2 x. n ) ) < ( m + 2 ) ) ) ) -> ( ( A x. ( 2 x. n ) ) < ( m + 2 ) <-> A < ( ( m + 2 ) / ( 2 x. n ) ) ) )
33	28, 32	mpbid 135	. . . 4 |- ( ( ( ( A e. RR /\ B e. RR /\ A < B ) /\ ( n e. NN /\ ( 1 / n ) < ( B - A ) ) ) /\ ( m e. ZZ /\ ( m < ( A x. ( 2 x. n ) ) /\ ( A x. ( 2 x. n ) ) < ( m + 2 ) ) ) ) -> A < ( ( m + 2 ) / ( 2 x. n ) ) )
34		simpll2 944	. . . . 5 |- ( ( ( ( A e. RR /\ B e. RR /\ A < B ) /\ ( n e. NN /\ ( 1 / n ) < ( B - A ) ) ) /\ ( m e. ZZ /\ ( m < ( A x. ( 2 x. n ) ) /\ ( A x. ( 2 x. n ) ) < ( m + 2 ) ) ) ) -> B e. RR )
35		simprrl 491	. . . . 5 |- ( ( ( ( A e. RR /\ B e. RR /\ A < B ) /\ ( n e. NN /\ ( 1 / n ) < ( B - A ) ) ) /\ ( m e. ZZ /\ ( m < ( A x. ( 2 x. n ) ) /\ ( A x. ( 2 x. n ) ) < ( m + 2 ) ) ) ) -> m < ( A x. ( 2 x. n ) ) )
36		simplrr 488	. . . . 5 |- ( ( ( ( A e. RR /\ B e. RR /\ A < B ) /\ ( n e. NN /\ ( 1 / n ) < ( B - A ) ) ) /\ ( m e. ZZ /\ ( m < ( A x. ( 2 x. n ) ) /\ ( A x. ( 2 x. n ) ) < ( m + 2 ) ) ) ) -> ( 1 / n ) < ( B - A ) )
37	18, 24, 29, 34, 35, 36	qbtwnrelemcalc 9110	. . . 4 |- ( ( ( ( A e. RR /\ B e. RR /\ A < B ) /\ ( n e. NN /\ ( 1 / n ) < ( B - A ) ) ) /\ ( m e. ZZ /\ ( m < ( A x. ( 2 x. n ) ) /\ ( A x. ( 2 x. n ) ) < ( m + 2 ) ) ) ) -> ( ( m + 2 ) / ( 2 x. n ) ) < B )
38		breq2 3768	. . . . . 6 |- ( x = ( ( m + 2 ) / ( 2 x. n ) ) -> ( A < x <-> A < ( ( m + 2 ) / ( 2 x. n ) ) ) )
39		breq1 3767	. . . . . 6 |- ( x = ( ( m + 2 ) / ( 2 x. n ) ) -> ( x < B <-> ( ( m + 2 ) / ( 2 x. n ) ) < B ) )
40	38, 39	anbi12d 442	. . . . 5 |- ( x = ( ( m + 2 ) / ( 2 x. n ) ) -> ( ( A < x /\ x < B ) <-> ( A < ( ( m + 2 ) / ( 2 x. n ) ) /\ ( ( m + 2 ) / ( 2 x. n ) ) < B ) ) )
41	40	rspcev 2656	. . . 4 |- ( ( ( ( m + 2 ) / ( 2 x. n ) ) e. QQ /\ ( A < ( ( m + 2 ) / ( 2 x. n ) ) /\ ( ( m + 2 ) / ( 2 x. n ) ) < B ) ) -> E. x e. QQ ( A < x /\ x < B ) )
42	27, 33, 37, 41	syl12anc 1133	. . 3 |- ( ( ( ( A e. RR /\ B e. RR /\ A < B ) /\ ( n e. NN /\ ( 1 / n ) < ( B - A ) ) ) /\ ( m e. ZZ /\ ( m < ( A x. ( 2 x. n ) ) /\ ( A x. ( 2 x. n ) ) < ( m + 2 ) ) ) ) -> E. x e. QQ ( A < x /\ x < B ) )
43	17, 42	rexlimddv 2437	. 2 |- ( ( ( A e. RR /\ B e. RR /\ A < B ) /\ ( n e. NN /\ ( 1 / n ) < ( B - A ) ) ) -> E. x e. QQ ( A < x /\ x < B ) )
44	8, 43	rexlimddv 2437	1 |- ( ( A e. RR /\ B e. RR /\ A < B ) -> E. x e. QQ ( A < x /\ x < B ) )

Syntax hints:   -> wi 4   /\ wa 97   /\ w3a 885   = wceq 1243   e. wcel 1393   E.wrex 2307   class class class wbr 3764  (class class class)co 5512   RRcr 6888   0cc0 6889   1c1 6890   + caddc 6892   x. cmul 6894   < clt 7060   - cmin 7182   / cdiv 7651   NNcn 7914   2c2 7964   ZZcz 8245   QQcq 8554

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-13 1404  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-coll 3872  ax-sep 3875  ax-nul 3883  ax-pow 3927  ax-pr 3944  ax-un 4170  ax-setind 4262  ax-iinf 4311  ax-cnex 6975  ax-resscn 6976  ax-1cn 6977  ax-1re 6978  ax-icn 6979  ax-addcl 6980  ax-addrcl 6981  ax-mulcl 6982  ax-mulrcl 6983  ax-addcom 6984  ax-mulcom 6985  ax-addass 6986  ax-mulass 6987  ax-distr 6988  ax-i2m1 6989  ax-1rid 6991  ax-0id 6992  ax-rnegex 6993  ax-precex 6994  ax-cnre 6995  ax-pre-ltirr 6996  ax-pre-ltwlin 6997  ax-pre-lttrn 6998  ax-pre-apti 6999  ax-pre-ltadd 7000  ax-pre-mulgt0 7001  ax-pre-mulext 7002  ax-arch 7003

This theorem depends on definitions:  df-bi 110  df-dc 743  df-3or 886  df-3an 887  df-tru 1246  df-fal 1249  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ne 2206  df-nel 2207  df-ral 2311  df-rex 2312  df-reu 2313  df-rmo 2314  df-rab 2315  df-v 2559  df-sbc 2765  df-csb 2853  df-dif 2920  df-un 2922  df-in 2924  df-ss 2931  df-nul 3225  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384  df-uni 3581  df-int 3616  df-iun 3659  df-br 3765  df-opab 3819  df-mpt 3820  df-tr 3855  df-eprel 4026  df-id 4030  df-po 4033  df-iso 4034  df-iord 4103  df-on 4105  df-suc 4108  df-iom 4314  df-xp 4351  df-rel 4352  df-cnv 4353  df-co 4354  df-dm 4355  df-rn 4356  df-res 4357  df-ima 4358  df-iota 4867  df-fun 4904  df-fn 4905  df-f 4906  df-f1 4907  df-fo 4908  df-f1o 4909  df-fv 4910  df-riota 5468  df-ov 5515  df-oprab 5516  df-mpt2 5517  df-1st 5767  df-2nd 5768  df-recs 5920  df-irdg 5957  df-1o 6001  df-2o 6002  df-oadd 6005  df-omul 6006  df-er 6106  df-ec 6108  df-qs 6112  df-ni 6402  df-pli 6403  df-mi 6404  df-lti 6405  df-plpq 6442  df-mpq 6443  df-enq 6445  df-nqqs 6446  df-plqqs 6447  df-mqqs 6448  df-1nqqs 6449  df-rq 6450  df-ltnqqs 6451  df-enq0 6522  df-nq0 6523  df-0nq0 6524  df-plq0 6525  df-mq0 6526  df-inp 6564  df-i1p 6565  df-iplp 6566  df-iltp 6568  df-enr 6811  df-nr 6812  df-ltr 6815  df-0r 6816  df-1r 6817  df-0 6896  df-1 6897  df-r 6899  df-lt 6902  df-pnf 7062  df-mnf 7063  df-xr 7064  df-ltxr 7065  df-le 7066  df-sub 7184  df-neg 7185  df-reap 7566  df-ap 7573  df-div 7652  df-inn 7915  df-2 7973  df-n0 8182  df-z 8246  df-uz 8474  df-q 8555  df-rp 8584

This theorem is referenced by:  qdenre  9798