Theorem qdenre 11144

Description: The rational numbers are dense in RR: any real number can be approximated with arbitrary precision by a rational number. For order theoretic density, see qbtwnre 10192. (Contributed by BJ, 15-Oct-2021.)

Assertion

Ref	Expression
qdenre	|- ( ( A e. RR /\ B e. RR+ ) -> E. x e. QQ ( abs ` ( x - A ) ) < B )

Distinct variable groups:   x, A   x, B

Proof of Theorem qdenre

Step	Hyp	Ref	Expression
1		simpl 108	. . . 4 |- ( ( A e. RR /\ B e. RR+ ) -> A e. RR )
2		rpre 9596	. . . . 5 |- ( B e. RR+ -> B e. RR )
3	2	adantl 275	. . . 4 |- ( ( A e. RR /\ B e. RR+ ) -> B e. RR )
4	1, 3	resubcld 8279	. . 3 |- ( ( A e. RR /\ B e. RR+ ) -> ( A - B ) e. RR )
5	1, 3	readdcld 7928	. . 3 |- ( ( A e. RR /\ B e. RR+ ) -> ( A + B ) e. RR )
6		ltsubrp 9626	. . . 4 |- ( ( A e. RR /\ B e. RR+ ) -> ( A - B ) < A )
7		ltaddrp 9627	. . . 4 |- ( ( A e. RR /\ B e. RR+ ) -> A < ( A + B ) )
8	4, 1, 5, 6, 7	lttrd 8024	. . 3 |- ( ( A e. RR /\ B e. RR+ ) -> ( A - B ) < ( A + B ) )
9	4, 5, 8	3jca 1167	. 2 |- ( ( A e. RR /\ B e. RR+ ) -> ( ( A - B ) e. RR /\ ( A + B ) e. RR /\ ( A - B ) < ( A + B ) ) )
10		qbtwnre 10192	. . 3 |- ( ( ( A - B ) e. RR /\ ( A + B ) e. RR /\ ( A - B ) < ( A + B ) ) -> E. x e. QQ ( ( A - B ) < x /\ x < ( A + B ) ) )
11		qre 9563	. . . . . 6 |- ( x e. QQ -> x e. RR )
12	11	adantl 275	. . . . 5 |- ( ( ( A e. RR /\ B e. RR+ ) /\ x e. QQ ) -> x e. RR )
13		simpll 519	. . . . 5 |- ( ( ( A e. RR /\ B e. RR+ ) /\ x e. QQ ) -> A e. RR )
14	2	ad2antlr 481	. . . . 5 |- ( ( ( A e. RR /\ B e. RR+ ) /\ x e. QQ ) -> B e. RR )
15		absdiflt 11034	. . . . . 6 |- ( ( x e. RR /\ A e. RR /\ B e. RR ) -> ( ( abs ` ( x - A ) ) < B <-> ( ( A - B ) < x /\ x < ( A + B ) ) ) )
16	15	biimprd 157	. . . . 5 |- ( ( x e. RR /\ A e. RR /\ B e. RR ) -> ( ( ( A - B ) < x /\ x < ( A + B ) ) -> ( abs ` ( x - A ) ) < B ) )
17	12, 13, 14, 16	syl3anc 1228	. . . 4 |- ( ( ( A e. RR /\ B e. RR+ ) /\ x e. QQ ) -> ( ( ( A - B ) < x /\ x < ( A + B ) ) -> ( abs ` ( x - A ) ) < B ) )
18	17	reximdva 2568	. . 3 |- ( ( A e. RR /\ B e. RR+ ) -> ( E. x e. QQ ( ( A - B ) < x /\ x < ( A + B ) ) -> E. x e. QQ ( abs ` ( x - A ) ) < B ) )
19	10, 18	syl5 32	. 2 |- ( ( A e. RR /\ B e. RR+ ) -> ( ( ( A - B ) e. RR /\ ( A + B ) e. RR /\ ( A - B ) < ( A + B ) ) -> E. x e. QQ ( abs ` ( x - A ) ) < B ) )
20	9, 19	mpd 13	1 |- ( ( A e. RR /\ B e. RR+ ) -> E. x e. QQ ( abs ` ( x - A ) ) < B )

Syntax hints:   -> wi 4   /\ wa 103   /\ w3a 968   e. wcel 2136   E.wrex 2445   class class class wbr 3982   ` cfv 5188  (class class class)co 5842   RRcr 7752   + caddc 7756   < clt 7933   - cmin 8069   QQcq 9557   RR+crp 9589   abscabs 10939

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-coll 4097  ax-sep 4100  ax-nul 4108  ax-pow 4153  ax-pr 4187  ax-un 4411  ax-setind 4514  ax-iinf 4565  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  ax-caucvg 7873

This theorem depends on definitions:  df-bi 116  df-dc 825  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-nul 3410  df-if 3521  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-tr 4081  df-id 4271  df-po 4274  df-iso 4275  df-iord 4344  df-on 4346  df-ilim 4347  df-suc 4349  df-iom 4568  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-f1 5193  df-fo 5194  df-f1o 5195  df-fv 5196  df-riota 5798  df-ov 5845  df-oprab 5846  df-mpo 5847  df-1st 6108  df-2nd 6109  df-recs 6273  df-frec 6359  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-3 8917  df-4 8918  df-n0 9115  df-z 9192  df-uz 9467  df-q 9558  df-rp 9590  df-seqfrec 10381  df-exp 10455  df-cj 10784  df-re 10785  df-im 10786  df-rsqrt 10940  df-abs 10941

This theorem is referenced by:  qdencn  13906