Theorem appdiv0nq 7566

Description: Approximate division for positive rationals. This can be thought of as a variation of appdivnq 7565 in which A is zero, although it can be stated and proved in terms of positive rationals alone, without zero as such. (Contributed by Jim Kingdon, 9-Dec-2019.)

Assertion

Ref	Expression
appdiv0nq	|- ( ( B e. Q. /\ C e. Q. ) -> E. m e. Q. ( m .Q C ) <Q B )

Distinct variable groups:   B, m   C, m

Proof of Theorem appdiv0nq

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		nsmallnqq 7414	. . 3 |- ( B e. Q. -> E. x e. Q. x <Q B )
2	1	adantr 276	. 2 |- ( ( B e. Q. /\ C e. Q. ) -> E. x e. Q. x <Q B )
3		appdivnq 7565	. . . . 5 |- ( ( x <Q B /\ C e. Q. ) -> E. m e. Q. ( x <Q ( m .Q C ) /\ ( m .Q C ) <Q B ) )
4		simpr 110	. . . . . 6 |- ( ( x <Q ( m .Q C ) /\ ( m .Q C ) <Q B ) -> ( m .Q C ) <Q B )
5	4	reximi 2574	. . . . 5 |- ( E. m e. Q. ( x <Q ( m .Q C ) /\ ( m .Q C ) <Q B ) -> E. m e. Q. ( m .Q C ) <Q B )
6	3, 5	syl 14	. . . 4 |- ( ( x <Q B /\ C e. Q. ) -> E. m e. Q. ( m .Q C ) <Q B )
7	6	ancoms 268	. . 3 |- ( ( C e. Q. /\ x <Q B ) -> E. m e. Q. ( m .Q C ) <Q B )
8	7	ad2ant2l 508	. 2 |- ( ( ( B e. Q. /\ C e. Q. ) /\ ( x e. Q. /\ x <Q B ) ) -> E. m e. Q. ( m .Q C ) <Q B )
9	2, 8	rexlimddv 2599	1 |- ( ( B e. Q. /\ C e. Q. ) -> E. m e. Q. ( m .Q C ) <Q B )

Syntax hints:   -> wi 4   /\ wa 104   e. wcel 2148   E.wrex 2456   class class class wbr 4005  (class class class)co 5878   Q.cnq 7282   .Q cmq 7285   <Q cltq 7287

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 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-coll 4120  ax-sep 4123  ax-nul 4131  ax-pow 4176  ax-pr 4211  ax-un 4435  ax-setind 4538  ax-iinf 4589

This theorem depends on definitions:  df-bi 117  df-dc 835  df-3or 979  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-ral 2460  df-rex 2461  df-reu 2462  df-rab 2464  df-v 2741  df-sbc 2965  df-csb 3060  df-dif 3133  df-un 3135  df-in 3137  df-ss 3144  df-nul 3425  df-pw 3579  df-sn 3600  df-pr 3601  df-op 3603  df-uni 3812  df-int 3847  df-iun 3890  df-br 4006  df-opab 4067  df-mpt 4068  df-tr 4104  df-eprel 4291  df-id 4295  df-po 4298  df-iso 4299  df-iord 4368  df-on 4370  df-suc 4373  df-iom 4592  df-xp 4634  df-rel 4635  df-cnv 4636  df-co 4637  df-dm 4638  df-rn 4639  df-res 4640  df-ima 4641  df-iota 5180  df-fun 5220  df-fn 5221  df-f 5222  df-f1 5223  df-fo 5224  df-f1o 5225  df-fv 5226  df-ov 5881  df-oprab 5882  df-mpo 5883  df-1st 6144  df-2nd 6145  df-recs 6309  df-irdg 6374  df-1o 6420  df-oadd 6424  df-omul 6425  df-er 6538  df-ec 6540  df-qs 6544  df-ni 7306  df-pli 7307  df-mi 7308  df-lti 7309  df-plpq 7346  df-mpq 7347  df-enq 7349  df-nqqs 7350  df-plqqs 7351  df-mqqs 7352  df-1nqqs 7353  df-rq 7354  df-ltnqqs 7355

This theorem is referenced by:  prmuloc  7568