Theorem qdenre 11166

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 10213. (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 9617	. . . . 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 8300	. . 3 |- ( ( A e. RR /\ B e. RR+ ) -> ( A - B ) e. RR )
5	1, 3	readdcld 7949	. . 3 |- ( ( A e. RR /\ B e. RR+ ) -> ( A + B ) e. RR )
6		ltsubrp 9647	. . . 4 |- ( ( A e. RR /\ B e. RR+ ) -> ( A - B ) < A )
7		ltaddrp 9648	. . . 4 |- ( ( A e. RR /\ B e. RR+ ) -> A < ( A + B ) )
8	4, 1, 5, 6, 7	lttrd 8045	. . 3 |- ( ( A e. RR /\ B e. RR+ ) -> ( A - B ) < ( A + B ) )
9	4, 5, 8	3jca 1172	. 2 |- ( ( A e. RR /\ B e. RR+ ) -> ( ( A - B ) e. RR /\ ( A + B ) e. RR /\ ( A - B ) < ( A + B ) ) )
10		qbtwnre 10213	. . 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 9584	. . . . . 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 524	. . . . 5 |- ( ( ( A e. RR /\ B e. RR+ ) /\ x e. QQ ) -> A e. RR )
14	2	ad2antlr 486	. . . . 5 |- ( ( ( A e. RR /\ B e. RR+ ) /\ x e. QQ ) -> B e. RR )
15		absdiflt 11056	. . . . . 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 1233	. . . 4 |- ( ( ( A e. RR /\ B e. RR+ ) /\ x e. QQ ) -> ( ( ( A - B ) < x /\ x < ( A + B ) ) -> ( abs ` ( x - A ) ) < B ) )
18	17	reximdva 2572	. . 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 973   e. wcel 2141   E.wrex 2449   class class class wbr 3989   ` cfv 5198  (class class class)co 5853   RRcr 7773   + caddc 7777   < clt 7954   - cmin 8090   QQcq 9578   RR+crp 9610   abscabs 10961

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 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-coll 4104  ax-sep 4107  ax-nul 4115  ax-pow 4160  ax-pr 4194  ax-un 4418  ax-setind 4521  ax-iinf 4572  ax-cnex 7865  ax-resscn 7866  ax-1cn 7867  ax-1re 7868  ax-icn 7869  ax-addcl 7870  ax-addrcl 7871  ax-mulcl 7872  ax-mulrcl 7873  ax-addcom 7874  ax-mulcom 7875  ax-addass 7876  ax-mulass 7877  ax-distr 7878  ax-i2m1 7879  ax-0lt1 7880  ax-1rid 7881  ax-0id 7882  ax-rnegex 7883  ax-precex 7884  ax-cnre 7885  ax-pre-ltirr 7886  ax-pre-ltwlin 7887  ax-pre-lttrn 7888  ax-pre-apti 7889  ax-pre-ltadd 7890  ax-pre-mulgt0 7891  ax-pre-mulext 7892  ax-arch 7893  ax-caucvg 7894

This theorem depends on definitions:  df-bi 116  df-dc 830  df-3or 974  df-3an 975  df-tru 1351  df-fal 1354  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ne 2341  df-nel 2436  df-ral 2453  df-rex 2454  df-reu 2455  df-rmo 2456  df-rab 2457  df-v 2732  df-sbc 2956  df-csb 3050  df-dif 3123  df-un 3125  df-in 3127  df-ss 3134  df-nul 3415  df-if 3527  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-int 3832  df-iun 3875  df-br 3990  df-opab 4051  df-mpt 4052  df-tr 4088  df-id 4278  df-po 4281  df-iso 4282  df-iord 4351  df-on 4353  df-ilim 4354  df-suc 4356  df-iom 4575  df-xp 4617  df-rel 4618  df-cnv 4619  df-co 4620  df-dm 4621  df-rn 4622  df-res 4623  df-ima 4624  df-iota 5160  df-fun 5200  df-fn 5201  df-f 5202  df-f1 5203  df-fo 5204  df-f1o 5205  df-fv 5206  df-riota 5809  df-ov 5856  df-oprab 5857  df-mpo 5858  df-1st 6119  df-2nd 6120  df-recs 6284  df-frec 6370  df-pnf 7956  df-mnf 7957  df-xr 7958  df-ltxr 7959  df-le 7960  df-sub 8092  df-neg 8093  df-reap 8494  df-ap 8501  df-div 8590  df-inn 8879  df-2 8937  df-3 8938  df-4 8939  df-n0 9136  df-z 9213  df-uz 9488  df-q 9579  df-rp 9611  df-seqfrec 10402  df-exp 10476  df-cj 10806  df-re 10807  df-im 10808  df-rsqrt 10962  df-abs 10963

This theorem is referenced by:  qdencn  14059