Theorem qbtwnre 10328

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 1000	. . . 4 |- ( ( A e. RR /\ B e. RR /\ A < B ) -> B e. RR )
2		simp1 999	. . . 4 |- ( ( A e. RR /\ B e. RR /\ A < B ) -> A e. RR )
3	1, 2	resubcld 8402	. . 3 |- ( ( A e. RR /\ B e. RR /\ A < B ) -> ( B - A ) e. RR )
4		simp3 1001	. . . 4 |- ( ( A e. RR /\ B e. RR /\ A < B ) -> A < B )
5	2, 1	posdifd 8553	. . . 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 9241	. . 3 |- ( ( ( B - A ) e. RR /\ 0 < ( B - A ) ) -> E. n e. NN ( 1 / n ) < ( B - A ) )
8	3, 6, 7	syl2anc 411	. 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 9054	. . . . . . 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 529	. . . . . . 7 |- ( ( ( A e. RR /\ B e. RR /\ A < B ) /\ ( n e. NN /\ ( 1 / n ) < ( B - A ) ) ) -> n e. NN )
13	12	nnred 8997	. . . . . 6 |- ( ( ( A e. RR /\ B e. RR /\ A < B ) /\ ( n e. NN /\ ( 1 / n ) < ( B - A ) ) ) -> n e. RR )
14	11, 13	remulcld 8052	. . . . 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 8052	. . . 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 10326	. . . 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 529	. . . . . 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 9348	. . . . . . 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 9446	. . . . 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 9146	. . . . . . 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 9033	. . . . 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 9692	. . . . 5 |- ( ( ( m + 2 ) e. ZZ /\ ( 2 x. n ) e. NN ) -> ( ( m + 2 ) / ( 2 x. n ) ) e. QQ )
27	21, 25, 26	syl2anc 411	. . . 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 540	. . . . 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 9442	. . . . . 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 9763	. . . . . 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 9813	. . . . 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 1039	. . . . 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 539	. . . . 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 536	. . . . 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 10327	. . . 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 4034	. . . . . 6 |- ( x = ( ( m + 2 ) / ( 2 x. n ) ) -> ( A < x <-> A < ( ( m + 2 ) / ( 2 x. n ) ) ) )
39		breq1 4033	. . . . . 6 |- ( x = ( ( m + 2 ) / ( 2 x. n ) ) -> ( x < B <-> ( ( m + 2 ) / ( 2 x. n ) ) < B ) )
40	38, 39	anbi12d 473	. . . . 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 2865	. . . 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 1247	. . 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 2616	. 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 2616	1 |- ( ( A e. RR /\ B e. RR /\ A < B ) -> E. x e. QQ ( A < x /\ x < B ) )

Syntax hints:   -> wi 4   /\ wa 104   /\ w3a 980   = wceq 1364   e. wcel 2164   E.wrex 2473   class class class wbr 4030  (class class class)co 5919   RRcr 7873   0cc0 7874   1c1 7875   + caddc 7877   x. cmul 7879   < clt 8056   - cmin 8192   / cdiv 8693   NNcn 8984   2c2 9035   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:  qbtwnxr  10329  qdenre  11349  expcnvre  11649