Theorem appdiv0nq 7526

Description: Approximate division for positive rationals. This can be thought of as a variation of appdivnq 7525 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 7374	. . 3 |- ( B e. Q. -> E. x e. Q. x <Q B )
2	1	adantr 274	. 2 |- ( ( B e. Q. /\ C e. Q. ) -> E. x e. Q. x <Q B )
3		appdivnq 7525	. . . . 5 |- ( ( x <Q B /\ C e. Q. ) -> E. m e. Q. ( x <Q ( m .Q C ) /\ ( m .Q C ) <Q B ) )
4		simpr 109	. . . . . 6 |- ( ( x <Q ( m .Q C ) /\ ( m .Q C ) <Q B ) -> ( m .Q C ) <Q B )
5	4	reximi 2567	. . . . 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 266	. . 3 |- ( ( C e. Q. /\ x <Q B ) -> E. m e. Q. ( m .Q C ) <Q B )
8	7	ad2ant2l 505	. 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 2592	1 |- ( ( B e. Q. /\ C e. Q. ) -> E. m e. Q. ( m .Q C ) <Q B )

Syntax hints:   -> wi 4   /\ wa 103   e. wcel 2141   E.wrex 2449   class class class wbr 3989  (class class class)co 5853   Q.cnq 7242   .Q cmq 7245   <Q cltq 7247

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

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-ral 2453  df-rex 2454  df-reu 2455  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-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-eprel 4274  df-id 4278  df-po 4281  df-iso 4282  df-iord 4351  df-on 4353  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-ov 5856  df-oprab 5857  df-mpo 5858  df-1st 6119  df-2nd 6120  df-recs 6284  df-irdg 6349  df-1o 6395  df-oadd 6399  df-omul 6400  df-er 6513  df-ec 6515  df-qs 6519  df-ni 7266  df-pli 7267  df-mi 7268  df-lti 7269  df-plpq 7306  df-mpq 7307  df-enq 7309  df-nqqs 7310  df-plqqs 7311  df-mqqs 7312  df-1nqqs 7313  df-rq 7314  df-ltnqqs 7315

This theorem is referenced by:  prmuloc  7528