Theorem qbtwnre 10192

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 988	. . . 4 |- ( ( A e. RR /\ B e. RR /\ A < B ) -> B e. RR )
2		simp1 987	. . . 4 |- ( ( A e. RR /\ B e. RR /\ A < B ) -> A e. RR )
3	1, 2	resubcld 8279	. . 3 |- ( ( A e. RR /\ B e. RR /\ A < B ) -> ( B - A ) e. RR )
4		simp3 989	. . . 4 |- ( ( A e. RR /\ B e. RR /\ A < B ) -> A < B )
5	2, 1	posdifd 8430	. . . 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 9112	. . 3 |- ( ( ( B - A ) e. RR /\ 0 < ( B - A ) ) -> E. n e. NN ( 1 / n ) < ( B - A ) )
8	3, 6, 7	syl2anc 409	. 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 8927	. . . . . . 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 521	. . . . . . 7 |- ( ( ( A e. RR /\ B e. RR /\ A < B ) /\ ( n e. NN /\ ( 1 / n ) < ( B - A ) ) ) -> n e. NN )
13	12	nnred 8870	. . . . . 6 |- ( ( ( A e. RR /\ B e. RR /\ A < B ) /\ ( n e. NN /\ ( 1 / n ) < ( B - A ) ) ) -> n e. RR )
14	11, 13	remulcld 7929	. . . . 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 7929	. . . 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 10190	. . . 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 521	. . . . . 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 9219	. . . . . . 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 9317	. . . . 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 9018	. . . . . . 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 8906	. . . . 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 9562	. . . . 5 |- ( ( ( m + 2 ) e. ZZ /\ ( 2 x. n ) e. NN ) -> ( ( m + 2 ) / ( 2 x. n ) ) e. QQ )
27	21, 25, 26	syl2anc 409	. . . 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 530	. . . . 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 9313	. . . . . 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 9630	. . . . . 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 9680	. . . . 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 1027	. . . . 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 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 ) ) ) ) -> m < ( A x. ( 2 x. n ) ) )
36		simplrr 526	. . . . 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 10191	. . . 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 3986	. . . . . 6 |- ( x = ( ( m + 2 ) / ( 2 x. n ) ) -> ( A < x <-> A < ( ( m + 2 ) / ( 2 x. n ) ) ) )
39		breq1 3985	. . . . . 6 |- ( x = ( ( m + 2 ) / ( 2 x. n ) ) -> ( x < B <-> ( ( m + 2 ) / ( 2 x. n ) ) < B ) )
40	38, 39	anbi12d 465	. . . . 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 2830	. . . 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 1226	. . 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 2588	. 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 2588	1 |- ( ( A e. RR /\ B e. RR /\ A < B ) -> E. x e. QQ ( A < x /\ x < B ) )

Syntax hints:   -> wi 4   /\ wa 103   /\ w3a 968   = wceq 1343   e. wcel 2136   E.wrex 2445   class class class wbr 3982  (class class class)co 5842   RRcr 7752   0cc0 7753   1c1 7754   + caddc 7756   x. cmul 7758   < clt 7933   - cmin 8069   / cdiv 8568   NNcn 8857   2c2 8908   ZZcz 9191   QQcq 9557

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 604  ax-in2 605  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-13 2138  ax-14 2139  ax-ext 2147  ax-sep 4100  ax-pow 4153  ax-pr 4187  ax-un 4411  ax-setind 4514  ax-cnex 7844  ax-resscn 7845  ax-1cn 7846  ax-1re 7847  ax-icn 7848  ax-addcl 7849  ax-addrcl 7850  ax-mulcl 7851  ax-mulrcl 7852  ax-addcom 7853  ax-mulcom 7854  ax-addass 7855  ax-mulass 7856  ax-distr 7857  ax-i2m1 7858  ax-0lt1 7859  ax-1rid 7860  ax-0id 7861  ax-rnegex 7862  ax-precex 7863  ax-cnre 7864  ax-pre-ltirr 7865  ax-pre-ltwlin 7866  ax-pre-lttrn 7867  ax-pre-apti 7868  ax-pre-ltadd 7869  ax-pre-mulgt0 7870  ax-pre-mulext 7871  ax-arch 7872

This theorem depends on definitions:  df-bi 116  df-3or 969  df-3an 970  df-tru 1346  df-fal 1349  df-nf 1449  df-sb 1751  df-eu 2017  df-mo 2018  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ne 2337  df-nel 2432  df-ral 2449  df-rex 2450  df-reu 2451  df-rmo 2452  df-rab 2453  df-v 2728  df-sbc 2952  df-csb 3046  df-dif 3118  df-un 3120  df-in 3122  df-ss 3129  df-pw 3561  df-sn 3582  df-pr 3583  df-op 3585  df-uni 3790  df-int 3825  df-iun 3868  df-br 3983  df-opab 4044  df-mpt 4045  df-id 4271  df-po 4274  df-iso 4275  df-xp 4610  df-rel 4611  df-cnv 4612  df-co 4613  df-dm 4614  df-rn 4615  df-res 4616  df-ima 4617  df-iota 5153  df-fun 5190  df-fn 5191  df-f 5192  df-fv 5196  df-riota 5798  df-ov 5845  df-oprab 5846  df-mpo 5847  df-1st 6108  df-2nd 6109  df-pnf 7935  df-mnf 7936  df-xr 7937  df-ltxr 7938  df-le 7939  df-sub 8071  df-neg 8072  df-reap 8473  df-ap 8480  df-div 8569  df-inn 8858  df-2 8916  df-n0 9115  df-z 9192  df-uz 9467  df-q 9558  df-rp 9590

This theorem is referenced by:  qbtwnxr  10193  qdenre  11144  expcnvre  11444