Theorem qbtwnre 10034

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 982	. . . 4 |- ( ( A e. RR /\ B e. RR /\ A < B ) -> B e. RR )
2		simp1 981	. . . 4 |- ( ( A e. RR /\ B e. RR /\ A < B ) -> A e. RR )
3	1, 2	resubcld 8143	. . 3 |- ( ( A e. RR /\ B e. RR /\ A < B ) -> ( B - A ) e. RR )
4		simp3 983	. . . 4 |- ( ( A e. RR /\ B e. RR /\ A < B ) -> A < B )
5	2, 1	posdifd 8294	. . . 4 |- ( ( A e. RR /\ B e. RR /\ A < B ) -> ( A < B <-> 0 < ( B - A ) ) )
6	4, 5	mpbid 146	. . 3 |- ( ( A e. RR /\ B e. RR /\ A < B ) -> 0 < ( B - A ) )
7		nnrecl 8975	. . 3 |- ( ( ( B - A ) e. RR /\ 0 < ( B - A ) ) -> E. n e. NN ( 1 / n ) < ( B - A ) )
8	3, 6, 7	syl2anc 408	. 2 |- ( ( A e. RR /\ B e. RR /\ A < B ) -> E. n e. NN ( 1 / n ) < ( B - A ) )
9	2	adantr 274	. . . . 5 |- ( ( ( A e. RR /\ B e. RR /\ A < B ) /\ ( n e. NN /\ ( 1 / n ) < ( B - A ) ) ) -> A e. RR )
10		2re 8790	. . . . . . 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 520	. . . . . . 7 |- ( ( ( A e. RR /\ B e. RR /\ A < B ) /\ ( n e. NN /\ ( 1 / n ) < ( B - A ) ) ) -> n e. NN )
13	12	nnred 8733	. . . . . 6 |- ( ( ( A e. RR /\ B e. RR /\ A < B ) /\ ( n e. NN /\ ( 1 / n ) < ( B - A ) ) ) -> n e. RR )
14	11, 13	remulcld 7796	. . . . 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 7796	. . . 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 10032	. . . 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 520	. . . . . 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 9082	. . . . . . 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 9177	. . . . 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 8881	. . . . . . 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 274	. . . . . 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 8769	. . . . 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 9416	. . . . 5 |- ( ( ( m + 2 ) e. ZZ /\ ( 2 x. n ) e. NN ) -> ( ( m + 2 ) / ( 2 x. n ) ) e. QQ )
27	21, 25, 26	syl2anc 408	. . . 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 529	. . . . 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 274	. . . . . 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 9173	. . . . . 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 9482	. . . . . 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 9531	. . . . 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 146	. . . 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 1021	. . . . 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 528	. . . . 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 525	. . . . 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 10033	. . . 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 3933	. . . . . 6 |- ( x = ( ( m + 2 ) / ( 2 x. n ) ) -> ( A < x <-> A < ( ( m + 2 ) / ( 2 x. n ) ) ) )
39		breq1 3932	. . . . . 6 |- ( x = ( ( m + 2 ) / ( 2 x. n ) ) -> ( x < B <-> ( ( m + 2 ) / ( 2 x. n ) ) < B ) )
40	38, 39	anbi12d 464	. . . . 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 2789	. . . 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 1214	. . 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 2554	. 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 2554	1 |- ( ( A e. RR /\ B e. RR /\ A < B ) -> E. x e. QQ ( A < x /\ x < B ) )

Syntax hints:   -> wi 4   /\ wa 103   /\ w3a 962   = wceq 1331   e. wcel 1480   E.wrex 2417   class class class wbr 3929  (class class class)co 5774   RRcr 7619   0cc0 7620   1c1 7621   + caddc 7623   x. cmul 7625   < clt 7800   - cmin 7933   / cdiv 8432   NNcn 8720   2c2 8771   ZZcz 9054   QQcq 9411

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 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-sep 4046  ax-pow 4098  ax-pr 4131  ax-un 4355  ax-setind 4452  ax-cnex 7711  ax-resscn 7712  ax-1cn 7713  ax-1re 7714  ax-icn 7715  ax-addcl 7716  ax-addrcl 7717  ax-mulcl 7718  ax-mulrcl 7719  ax-addcom 7720  ax-mulcom 7721  ax-addass 7722  ax-mulass 7723  ax-distr 7724  ax-i2m1 7725  ax-0lt1 7726  ax-1rid 7727  ax-0id 7728  ax-rnegex 7729  ax-precex 7730  ax-cnre 7731  ax-pre-ltirr 7732  ax-pre-ltwlin 7733  ax-pre-lttrn 7734  ax-pre-apti 7735  ax-pre-ltadd 7736  ax-pre-mulgt0 7737  ax-pre-mulext 7738  ax-arch 7739

This theorem depends on definitions:  df-bi 116  df-3or 963  df-3an 964  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ne 2309  df-nel 2404  df-ral 2421  df-rex 2422  df-reu 2423  df-rmo 2424  df-rab 2425  df-v 2688  df-sbc 2910  df-csb 3004  df-dif 3073  df-un 3075  df-in 3077  df-ss 3084  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3737  df-int 3772  df-iun 3815  df-br 3930  df-opab 3990  df-mpt 3991  df-id 4215  df-po 4218  df-iso 4219  df-xp 4545  df-rel 4546  df-cnv 4547  df-co 4548  df-dm 4549  df-rn 4550  df-res 4551  df-ima 4552  df-iota 5088  df-fun 5125  df-fn 5126  df-f 5127  df-fv 5131  df-riota 5730  df-ov 5777  df-oprab 5778  df-mpo 5779  df-1st 6038  df-2nd 6039  df-pnf 7802  df-mnf 7803  df-xr 7804  df-ltxr 7805  df-le 7806  df-sub 7935  df-neg 7936  df-reap 8337  df-ap 8344  df-div 8433  df-inn 8721  df-2 8779  df-n0 8978  df-z 9055  df-uz 9327  df-q 9412  df-rp 9442

This theorem is referenced by:  qbtwnxr  10035  qdenre  10974  expcnvre  11272