Theorem qbtwnre 10691

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 1029	. . . 4 |- ( ( A e. RR /\ B e. RR /\ A < B ) -> B e. RR )
2		simp1 1028	. . . 4 |- ( ( A e. RR /\ B e. RR /\ A < B ) -> A e. RR )
3	1, 2	resubcld 8708	. . 3 |- ( ( A e. RR /\ B e. RR /\ A < B ) -> ( B - A ) e. RR )
4		simp3 1030	. . . 4 |- ( ( A e. RR /\ B e. RR /\ A < B ) -> A < B )
5	2, 1	posdifd 8860	. . . 4 |- ( ( A e. RR /\ B e. RR /\ A < B ) -> ( A < B <-> 0 < ( B - A ) ) )
6	4, 5	mpbid 147	. . 3 |- ( ( A e. RR /\ B e. RR /\ A < B ) -> 0 < ( B - A ) )
7		nnrecl 9561	. . 3 |- ( ( ( B - A ) e. RR /\ 0 < ( B - A ) ) -> E. n e. NN ( 1 / n ) < ( B - A ) )
8	3, 6, 7	syl2anc 415	. 2 |- ( ( A e. RR /\ B e. RR /\ A < B ) -> E. n e. NN ( 1 / n ) < ( B - A ) )
9	2	adantr 276	. . . . 5 |- ( ( ( A e. RR /\ B e. RR /\ A < B ) /\ ( n e. NN /\ ( 1 / n ) < ( B - A ) ) ) -> A e. RR )
10		2re 9374	. . . . . . 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 535	. . . . . . 7 |- ( ( ( A e. RR /\ B e. RR /\ A < B ) /\ ( n e. NN /\ ( 1 / n ) < ( B - A ) ) ) -> n e. NN )
13	12	nnred 9317	. . . . . 6 |- ( ( ( A e. RR /\ B e. RR /\ A < B ) /\ ( n e. NN /\ ( 1 / n ) < ( B - A ) ) ) -> n e. RR )
14	11, 13	remulcld 8356	. . . . 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 8356	. . . 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 10689	. . . 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 535	. . . . . 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 9672	. . . . . . 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 9772	. . . . 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 9466	. . . . . . 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 276	. . . . . 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 9353	. . . . 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 10024	. . . . 5 |- ( ( ( m + 2 ) e. ZZ /\ ( 2 x. n ) e. NN ) -> ( ( m + 2 ) / ( 2 x. n ) ) e. QQ )
27	21, 25, 26	syl2anc 415	. . . 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 546	. . . . 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 276	. . . . . 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 9768	. . . . . 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 10095	. . . . . 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 10145	. . . . 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 147	. . . 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 1068	. . . . 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 545	. . . . 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 542	. . . . 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 10690	. . . 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 4134	. . . . . 6 |- ( x = ( ( m + 2 ) / ( 2 x. n ) ) -> ( A < x <-> A < ( ( m + 2 ) / ( 2 x. n ) ) ) )
39		breq1 4133	. . . . . 6 |- ( x = ( ( m + 2 ) / ( 2 x. n ) ) -> ( x < B <-> ( ( m + 2 ) / ( 2 x. n ) ) < B ) )
40	38, 39	anbi12d 477	. . . . 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 2929	. . . 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 1276	. . 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 2673	. 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 2673	1 |- ( ( A e. RR /\ B e. RR /\ A < B ) -> E. x e. QQ ( A < x /\ x < B ) )

This proof depends on syntax axioms:   -> wi 4   /\ wa 104   /\ w3a 1009   = wceq 1402   e. wcel 2209   E.wrex 2529   class class class wbr 4130  (class class class)co 6085   RRcr 8178   0cc0 8179   1c1 8180   + caddc 8182   x. cmul 8184   < clt 8360   - cmin 8497   / cdiv 9002   NNcn 9304   2c2 9355   ZZcz 9644   QQcq 10019

This proof depends on 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 4249  ax-pow 4311  ax-pr 4346  ax-un 4578  ax-setind 4684  ax-cnex 8270  ax-resscn 8271  ax-1cn 8272  ax-1re 8273  ax-icn 8274  ax-addcl 8275  ax-addrcl 8276  ax-mulcl 8277  ax-mulrcl 8278  ax-addcom 8279  ax-mulcom 8280  ax-addass 8281  ax-mulass 8282  ax-distr 8283  ax-i2m1 8284  ax-0lt1 8285  ax-1rid 8286  ax-0id 8287  ax-rnegex 8288  ax-precex 8289  ax-cnre 8290  ax-pre-ltirr 8291  ax-pre-ltwlin 8292  ax-pre-lttrn 8293  ax-pre-apti 8294  ax-pre-ltadd 8295  ax-pre-mulgt0 8296  ax-pre-mulext 8297  ax-arch 8298

This proof 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 3690  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-int 3971  df-iun 4014  df-br 4131  df-opab 4193  df-mpt 4194  df-id 4438  df-po 4441  df-iso 4442  df-xp 4780  df-rel 4781  df-cnv 4782  df-co 4783  df-dm 4784  df-rn 4785  df-res 4786  df-ima 4787  df-iota 5337  df-fun 5379  df-fn 5380  df-f 5381  df-fv 5385  df-riota 6038  df-ov 6088  df-oprab 6089  df-mpo 6090  df-1st 6374  df-2nd 6375  df-pnf 8362  df-mnf 8363  df-xr 8364  df-ltxr 8365  df-le 8366  df-sub 8499  df-neg 8500  df-reap 8903  df-ap 8910  df-div 9003  df-inn 9305  df-2 9363  df-n0 9564  df-z 9645  df-uz 9922  df-q 10020  df-rp 10055

This theorem is used by:  qbtwnxr  10692  qdenre  11968  expcnvre  12270