Theorem qdenre 11734

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 10493. (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 109	. . . 4 |- ( ( A e. RR /\ B e. RR+ ) -> A e. RR )
2		rpre 9873	. . . . 5 |- ( B e. RR+ -> B e. RR )
3	2	adantl 277	. . . 4 |- ( ( A e. RR /\ B e. RR+ ) -> B e. RR )
4	1, 3	resubcld 8543	. . 3 |- ( ( A e. RR /\ B e. RR+ ) -> ( A - B ) e. RR )
5	1, 3	readdcld 8192	. . 3 |- ( ( A e. RR /\ B e. RR+ ) -> ( A + B ) e. RR )
6		ltsubrp 9903	. . . 4 |- ( ( A e. RR /\ B e. RR+ ) -> ( A - B ) < A )
7		ltaddrp 9904	. . . 4 |- ( ( A e. RR /\ B e. RR+ ) -> A < ( A + B ) )
8	4, 1, 5, 6, 7	lttrd 8288	. . 3 |- ( ( A e. RR /\ B e. RR+ ) -> ( A - B ) < ( A + B ) )
9	4, 5, 8	3jca 1201	. 2 |- ( ( A e. RR /\ B e. RR+ ) -> ( ( A - B ) e. RR /\ ( A + B ) e. RR /\ ( A - B ) < ( A + B ) ) )
10		qbtwnre 10493	. . 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 9837	. . . . . 6 |- ( x e. QQ -> x e. RR )
12	11	adantl 277	. . . . 5 |- ( ( ( A e. RR /\ B e. RR+ ) /\ x e. QQ ) -> x e. RR )
13		simpll 527	. . . . 5 |- ( ( ( A e. RR /\ B e. RR+ ) /\ x e. QQ ) -> A e. RR )
14	2	ad2antlr 489	. . . . 5 |- ( ( ( A e. RR /\ B e. RR+ ) /\ x e. QQ ) -> B e. RR )
15		absdiflt 11624	. . . . . 6 |- ( ( x e. RR /\ A e. RR /\ B e. RR ) -> ( ( abs ` ( x - A ) ) < B <-> ( ( A - B ) < x /\ x < ( A + B ) ) ) )
16	15	biimprd 158	. . . . 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 1271	. . . 4 |- ( ( ( A e. RR /\ B e. RR+ ) /\ x e. QQ ) -> ( ( ( A - B ) < x /\ x < ( A + B ) ) -> ( abs ` ( x - A ) ) < B ) )
18	17	reximdva 2632	. . 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 104   /\ w3a 1002   e. wcel 2200   E.wrex 2509   class class class wbr 4083   ` cfv 5321  (class class class)co 6010   RRcr 8014   + caddc 8018   < clt 8197   - cmin 8333   QQcq 9831   RR+crp 9866   abscabs 11529

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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-coll 4199  ax-sep 4202  ax-nul 4210  ax-pow 4259  ax-pr 4294  ax-un 4525  ax-setind 4630  ax-iinf 4681  ax-cnex 8106  ax-resscn 8107  ax-1cn 8108  ax-1re 8109  ax-icn 8110  ax-addcl 8111  ax-addrcl 8112  ax-mulcl 8113  ax-mulrcl 8114  ax-addcom 8115  ax-mulcom 8116  ax-addass 8117  ax-mulass 8118  ax-distr 8119  ax-i2m1 8120  ax-0lt1 8121  ax-1rid 8122  ax-0id 8123  ax-rnegex 8124  ax-precex 8125  ax-cnre 8126  ax-pre-ltirr 8127  ax-pre-ltwlin 8128  ax-pre-lttrn 8129  ax-pre-apti 8130  ax-pre-ltadd 8131  ax-pre-mulgt0 8132  ax-pre-mulext 8133  ax-arch 8134  ax-caucvg 8135

This theorem depends on definitions:  df-bi 117  df-dc 840  df-3or 1003  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-nel 2496  df-ral 2513  df-rex 2514  df-reu 2515  df-rmo 2516  df-rab 2517  df-v 2801  df-sbc 3029  df-csb 3125  df-dif 3199  df-un 3201  df-in 3203  df-ss 3210  df-nul 3492  df-if 3603  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3889  df-int 3924  df-iun 3967  df-br 4084  df-opab 4146  df-mpt 4147  df-tr 4183  df-id 4385  df-po 4388  df-iso 4389  df-iord 4458  df-on 4460  df-ilim 4461  df-suc 4463  df-iom 4684  df-xp 4726  df-rel 4727  df-cnv 4728  df-co 4729  df-dm 4730  df-rn 4731  df-res 4732  df-ima 4733  df-iota 5281  df-fun 5323  df-fn 5324  df-f 5325  df-f1 5326  df-fo 5327  df-f1o 5328  df-fv 5329  df-riota 5963  df-ov 6013  df-oprab 6014  df-mpo 6015  df-1st 6295  df-2nd 6296  df-recs 6462  df-frec 6548  df-pnf 8199  df-mnf 8200  df-xr 8201  df-ltxr 8202  df-le 8203  df-sub 8335  df-neg 8336  df-reap 8738  df-ap 8745  df-div 8836  df-inn 9127  df-2 9185  df-3 9186  df-4 9187  df-n0 9386  df-z 9463  df-uz 9739  df-q 9832  df-rp 9867  df-seqfrec 10687  df-exp 10778  df-cj 11374  df-re 11375  df-im 11376  df-rsqrt 11530  df-abs 11531

This theorem is referenced by:  qdencn  16509