Theorem appdiv0nq 7121

Description: Approximate division for positive rationals. This can be thought of as a variation of appdivnq 7120 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 6969	. . 3 |- ( B e. Q. -> E. x e. Q. x <Q B )
2	1	adantr 270	. 2 |- ( ( B e. Q. /\ C e. Q. ) -> E. x e. Q. x <Q B )
3		appdivnq 7120	. . . . 5 |- ( ( x <Q B /\ C e. Q. ) -> E. m e. Q. ( x <Q ( m .Q C ) /\ ( m .Q C ) <Q B ) )
4		simpr 108	. . . . . 6 |- ( ( x <Q ( m .Q C ) /\ ( m .Q C ) <Q B ) -> ( m .Q C ) <Q B )
5	4	reximi 2470	. . . . 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 264	. . 3 |- ( ( C e. Q. /\ x <Q B ) -> E. m e. Q. ( m .Q C ) <Q B )
8	7	ad2ant2l 492	. 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 2493	1 |- ( ( B e. Q. /\ C e. Q. ) -> E. m e. Q. ( m .Q C ) <Q B )

Syntax hints:   -> wi 4   /\ wa 102   e. wcel 1438   E.wrex 2360   class class class wbr 3845  (class class class)co 5652   Q.cnq 6837   .Q cmq 6840   <Q cltq 6842

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 579  ax-in2 580  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-13 1449  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-coll 3954  ax-sep 3957  ax-nul 3965  ax-pow 4009  ax-pr 4036  ax-un 4260  ax-setind 4353  ax-iinf 4403

This theorem depends on definitions:  df-bi 115  df-dc 781  df-3or 925  df-3an 926  df-tru 1292  df-fal 1295  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ne 2256  df-ral 2364  df-rex 2365  df-reu 2366  df-rab 2368  df-v 2621  df-sbc 2841  df-csb 2934  df-dif 3001  df-un 3003  df-in 3005  df-ss 3012  df-nul 3287  df-pw 3431  df-sn 3452  df-pr 3453  df-op 3455  df-uni 3654  df-int 3689  df-iun 3732  df-br 3846  df-opab 3900  df-mpt 3901  df-tr 3937  df-eprel 4116  df-id 4120  df-po 4123  df-iso 4124  df-iord 4193  df-on 4195  df-suc 4198  df-iom 4406  df-xp 4444  df-rel 4445  df-cnv 4446  df-co 4447  df-dm 4448  df-rn 4449  df-res 4450  df-ima 4451  df-iota 4980  df-fun 5017  df-fn 5018  df-f 5019  df-f1 5020  df-fo 5021  df-f1o 5022  df-fv 5023  df-ov 5655  df-oprab 5656  df-mpt2 5657  df-1st 5911  df-2nd 5912  df-recs 6070  df-irdg 6135  df-1o 6181  df-oadd 6185  df-omul 6186  df-er 6290  df-ec 6292  df-qs 6296  df-ni 6861  df-pli 6862  df-mi 6863  df-lti 6864  df-plpq 6901  df-mpq 6902  df-enq 6904  df-nqqs 6905  df-plqqs 6906  df-mqqs 6907  df-1nqqs 6908  df-rq 6909  df-ltnqqs 6910

This theorem is referenced by:  prmuloc  7123