Theorem appdiv0nq 7697

Description: Approximate division for positive rationals. This can be thought of as a variation of appdivnq 7696 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 7545	. . 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 7696	. . . . 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 2604	. . . . 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 2629	1 |- ( ( B e. Q. /\ C e. Q. ) -> E. m e. Q. ( m .Q C ) <Q B )

Syntax hints:   -> wi 4   /\ wa 104   e. wcel 2177   E.wrex 2486   class class class wbr 4051  (class class class)co 5957   Q.cnq 7413   .Q cmq 7416   <Q cltq 7418

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 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-13 2179  ax-14 2180  ax-ext 2188  ax-coll 4167  ax-sep 4170  ax-nul 4178  ax-pow 4226  ax-pr 4261  ax-un 4488  ax-setind 4593  ax-iinf 4644

This theorem depends on definitions:  df-bi 117  df-dc 837  df-3or 982  df-3an 983  df-tru 1376  df-fal 1379  df-nf 1485  df-sb 1787  df-eu 2058  df-mo 2059  df-clab 2193  df-cleq 2199  df-clel 2202  df-nfc 2338  df-ne 2378  df-ral 2490  df-rex 2491  df-reu 2492  df-rab 2494  df-v 2775  df-sbc 3003  df-csb 3098  df-dif 3172  df-un 3174  df-in 3176  df-ss 3183  df-nul 3465  df-pw 3623  df-sn 3644  df-pr 3645  df-op 3647  df-uni 3857  df-int 3892  df-iun 3935  df-br 4052  df-opab 4114  df-mpt 4115  df-tr 4151  df-eprel 4344  df-id 4348  df-po 4351  df-iso 4352  df-iord 4421  df-on 4423  df-suc 4426  df-iom 4647  df-xp 4689  df-rel 4690  df-cnv 4691  df-co 4692  df-dm 4693  df-rn 4694  df-res 4695  df-ima 4696  df-iota 5241  df-fun 5282  df-fn 5283  df-f 5284  df-f1 5285  df-fo 5286  df-f1o 5287  df-fv 5288  df-ov 5960  df-oprab 5961  df-mpo 5962  df-1st 6239  df-2nd 6240  df-recs 6404  df-irdg 6469  df-1o 6515  df-oadd 6519  df-omul 6520  df-er 6633  df-ec 6635  df-qs 6639  df-ni 7437  df-pli 7438  df-mi 7439  df-lti 7440  df-plpq 7477  df-mpq 7478  df-enq 7480  df-nqqs 7481  df-plqqs 7482  df-mqqs 7483  df-1nqqs 7484  df-rq 7485  df-ltnqqs 7486

This theorem is referenced by:  prmuloc  7699