Theorem qdenre 11106

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 10160. (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 9568	. . . . 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 8257	. . 3 |- ( ( A e. RR /\ B e. RR+ ) -> ( A - B ) e. RR )
5	1, 3	readdcld 7908	. . 3 |- ( ( A e. RR /\ B e. RR+ ) -> ( A + B ) e. RR )
6		ltsubrp 9598	. . . 4 |- ( ( A e. RR /\ B e. RR+ ) -> ( A - B ) < A )
7		ltaddrp 9599	. . . 4 |- ( ( A e. RR /\ B e. RR+ ) -> A < ( A + B ) )
8	4, 1, 5, 6, 7	lttrd 8002	. . 3 |- ( ( A e. RR /\ B e. RR+ ) -> ( A - B ) < ( A + B ) )
9	4, 5, 8	3jca 1162	. 2 |- ( ( A e. RR /\ B e. RR+ ) -> ( ( A - B ) e. RR /\ ( A + B ) e. RR /\ ( A - B ) < ( A + B ) ) )
10		qbtwnre 10160	. . 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 9535	. . . . . 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 10996	. . . . . 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 1220	. . . 4 |- ( ( ( A e. RR /\ B e. RR+ ) /\ x e. QQ ) -> ( ( ( A - B ) < x /\ x < ( A + B ) ) -> ( abs ` ( x - A ) ) < B ) )
18	17	reximdva 2559	. . 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 963   e. wcel 2128   E.wrex 2436   class class class wbr 3966   ` cfv 5171  (class class class)co 5825   RRcr 7732   + caddc 7736   < clt 7913   - cmin 8047   QQcq 9529   RR+crp 9561   abscabs 10901

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 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-13 2130  ax-14 2131  ax-ext 2139  ax-coll 4080  ax-sep 4083  ax-nul 4091  ax-pow 4136  ax-pr 4170  ax-un 4394  ax-setind 4497  ax-iinf 4548  ax-cnex 7824  ax-resscn 7825  ax-1cn 7826  ax-1re 7827  ax-icn 7828  ax-addcl 7829  ax-addrcl 7830  ax-mulcl 7831  ax-mulrcl 7832  ax-addcom 7833  ax-mulcom 7834  ax-addass 7835  ax-mulass 7836  ax-distr 7837  ax-i2m1 7838  ax-0lt1 7839  ax-1rid 7840  ax-0id 7841  ax-rnegex 7842  ax-precex 7843  ax-cnre 7844  ax-pre-ltirr 7845  ax-pre-ltwlin 7846  ax-pre-lttrn 7847  ax-pre-apti 7848  ax-pre-ltadd 7849  ax-pre-mulgt0 7850  ax-pre-mulext 7851  ax-arch 7852  ax-caucvg 7853

This theorem depends on definitions:  df-bi 116  df-dc 821  df-3or 964  df-3an 965  df-tru 1338  df-fal 1341  df-nf 1441  df-sb 1743  df-eu 2009  df-mo 2010  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-ne 2328  df-nel 2423  df-ral 2440  df-rex 2441  df-reu 2442  df-rmo 2443  df-rab 2444  df-v 2714  df-sbc 2938  df-csb 3032  df-dif 3104  df-un 3106  df-in 3108  df-ss 3115  df-nul 3395  df-if 3506  df-pw 3545  df-sn 3566  df-pr 3567  df-op 3569  df-uni 3774  df-int 3809  df-iun 3852  df-br 3967  df-opab 4027  df-mpt 4028  df-tr 4064  df-id 4254  df-po 4257  df-iso 4258  df-iord 4327  df-on 4329  df-ilim 4330  df-suc 4332  df-iom 4551  df-xp 4593  df-rel 4594  df-cnv 4595  df-co 4596  df-dm 4597  df-rn 4598  df-res 4599  df-ima 4600  df-iota 5136  df-fun 5173  df-fn 5174  df-f 5175  df-f1 5176  df-fo 5177  df-f1o 5178  df-fv 5179  df-riota 5781  df-ov 5828  df-oprab 5829  df-mpo 5830  df-1st 6089  df-2nd 6090  df-recs 6253  df-frec 6339  df-pnf 7915  df-mnf 7916  df-xr 7917  df-ltxr 7918  df-le 7919  df-sub 8049  df-neg 8050  df-reap 8451  df-ap 8458  df-div 8547  df-inn 8835  df-2 8893  df-3 8894  df-4 8895  df-n0 9092  df-z 9169  df-uz 9441  df-q 9530  df-rp 9562  df-seqfrec 10349  df-exp 10423  df-cj 10746  df-re 10747  df-im 10748  df-rsqrt 10902  df-abs 10903

This theorem is referenced by:  qdencn  13640