Theorem qdenre 11367

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 10346. (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 9735	. . . . 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 8407	. . 3 |- ( ( A e. RR /\ B e. RR+ ) -> ( A - B ) e. RR )
5	1, 3	readdcld 8056	. . 3 |- ( ( A e. RR /\ B e. RR+ ) -> ( A + B ) e. RR )
6		ltsubrp 9765	. . . 4 |- ( ( A e. RR /\ B e. RR+ ) -> ( A - B ) < A )
7		ltaddrp 9766	. . . 4 |- ( ( A e. RR /\ B e. RR+ ) -> A < ( A + B ) )
8	4, 1, 5, 6, 7	lttrd 8152	. . 3 |- ( ( A e. RR /\ B e. RR+ ) -> ( A - B ) < ( A + B ) )
9	4, 5, 8	3jca 1179	. 2 |- ( ( A e. RR /\ B e. RR+ ) -> ( ( A - B ) e. RR /\ ( A + B ) e. RR /\ ( A - B ) < ( A + B ) ) )
10		qbtwnre 10346	. . 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 9699	. . . . . 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 11257	. . . . . 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 1249	. . . 4 |- ( ( ( A e. RR /\ B e. RR+ ) /\ x e. QQ ) -> ( ( ( A - B ) < x /\ x < ( A + B ) ) -> ( abs ` ( x - A ) ) < B ) )
18	17	reximdva 2599	. . 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 980   e. wcel 2167   E.wrex 2476   class class class wbr 4033   ` cfv 5258  (class class class)co 5922   RRcr 7878   + caddc 7882   < clt 8061   - cmin 8197   QQcq 9693   RR+crp 9728   abscabs 11162

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 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-13 2169  ax-14 2170  ax-ext 2178  ax-coll 4148  ax-sep 4151  ax-nul 4159  ax-pow 4207  ax-pr 4242  ax-un 4468  ax-setind 4573  ax-iinf 4624  ax-cnex 7970  ax-resscn 7971  ax-1cn 7972  ax-1re 7973  ax-icn 7974  ax-addcl 7975  ax-addrcl 7976  ax-mulcl 7977  ax-mulrcl 7978  ax-addcom 7979  ax-mulcom 7980  ax-addass 7981  ax-mulass 7982  ax-distr 7983  ax-i2m1 7984  ax-0lt1 7985  ax-1rid 7986  ax-0id 7987  ax-rnegex 7988  ax-precex 7989  ax-cnre 7990  ax-pre-ltirr 7991  ax-pre-ltwlin 7992  ax-pre-lttrn 7993  ax-pre-apti 7994  ax-pre-ltadd 7995  ax-pre-mulgt0 7996  ax-pre-mulext 7997  ax-arch 7998  ax-caucvg 7999

This theorem depends on definitions:  df-bi 117  df-dc 836  df-3or 981  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ne 2368  df-nel 2463  df-ral 2480  df-rex 2481  df-reu 2482  df-rmo 2483  df-rab 2484  df-v 2765  df-sbc 2990  df-csb 3085  df-dif 3159  df-un 3161  df-in 3163  df-ss 3170  df-nul 3451  df-if 3562  df-pw 3607  df-sn 3628  df-pr 3629  df-op 3631  df-uni 3840  df-int 3875  df-iun 3918  df-br 4034  df-opab 4095  df-mpt 4096  df-tr 4132  df-id 4328  df-po 4331  df-iso 4332  df-iord 4401  df-on 4403  df-ilim 4404  df-suc 4406  df-iom 4627  df-xp 4669  df-rel 4670  df-cnv 4671  df-co 4672  df-dm 4673  df-rn 4674  df-res 4675  df-ima 4676  df-iota 5219  df-fun 5260  df-fn 5261  df-f 5262  df-f1 5263  df-fo 5264  df-f1o 5265  df-fv 5266  df-riota 5877  df-ov 5925  df-oprab 5926  df-mpo 5927  df-1st 6198  df-2nd 6199  df-recs 6363  df-frec 6449  df-pnf 8063  df-mnf 8064  df-xr 8065  df-ltxr 8066  df-le 8067  df-sub 8199  df-neg 8200  df-reap 8602  df-ap 8609  df-div 8700  df-inn 8991  df-2 9049  df-3 9050  df-4 9051  df-n0 9250  df-z 9327  df-uz 9602  df-q 9694  df-rp 9729  df-seqfrec 10540  df-exp 10631  df-cj 11007  df-re 11008  df-im 11009  df-rsqrt 11163  df-abs 11164

This theorem is referenced by:  qdencn  15671