Theorem lt2mulnq 7600

Description: Ordering property of multiplication for positive fractions. (Contributed by Jim Kingdon, 18-Jul-2021.)

Assertion

Ref	Expression
lt2mulnq	|- ( ( ( A e. Q. /\ B e. Q. ) /\ ( C e. Q. /\ D e. Q. ) ) -> ( ( A <Q B /\ C <Q D ) -> ( A .Q C ) <Q ( B .Q D ) ) )

Proof of Theorem lt2mulnq

Step	Hyp	Ref	Expression
1		ltmnqg 7596	. . . . . 6 |- ( ( A e. Q. /\ B e. Q. /\ C e. Q. ) -> ( A <Q B <-> ( C .Q A ) <Q ( C .Q B ) ) )
2	1	3expa 1227	. . . . 5 |- ( ( ( A e. Q. /\ B e. Q. ) /\ C e. Q. ) -> ( A <Q B <-> ( C .Q A ) <Q ( C .Q B ) ) )
3	2	adantrr 479	. . . 4 |- ( ( ( A e. Q. /\ B e. Q. ) /\ ( C e. Q. /\ D e. Q. ) ) -> ( A <Q B <-> ( C .Q A ) <Q ( C .Q B ) ) )
4		mulcomnqg 7578	. . . . . . 7 |- ( ( C e. Q. /\ A e. Q. ) -> ( C .Q A ) = ( A .Q C ) )
5	4	ancoms 268	. . . . . 6 |- ( ( A e. Q. /\ C e. Q. ) -> ( C .Q A ) = ( A .Q C ) )
6	5	ad2ant2r 509	. . . . 5 |- ( ( ( A e. Q. /\ B e. Q. ) /\ ( C e. Q. /\ D e. Q. ) ) -> ( C .Q A ) = ( A .Q C ) )
7		mulcomnqg 7578	. . . . . . 7 |- ( ( C e. Q. /\ B e. Q. ) -> ( C .Q B ) = ( B .Q C ) )
8	7	ancoms 268	. . . . . 6 |- ( ( B e. Q. /\ C e. Q. ) -> ( C .Q B ) = ( B .Q C ) )
9	8	ad2ant2lr 510	. . . . 5 |- ( ( ( A e. Q. /\ B e. Q. ) /\ ( C e. Q. /\ D e. Q. ) ) -> ( C .Q B ) = ( B .Q C ) )
10	6, 9	breq12d 4096	. . . 4 |- ( ( ( A e. Q. /\ B e. Q. ) /\ ( C e. Q. /\ D e. Q. ) ) -> ( ( C .Q A ) <Q ( C .Q B ) <-> ( A .Q C ) <Q ( B .Q C ) ) )
11	3, 10	bitrd 188	. . 3 |- ( ( ( A e. Q. /\ B e. Q. ) /\ ( C e. Q. /\ D e. Q. ) ) -> ( A <Q B <-> ( A .Q C ) <Q ( B .Q C ) ) )
12		ltmnqg 7596	. . . . . 6 |- ( ( C e. Q. /\ D e. Q. /\ B e. Q. ) -> ( C <Q D <-> ( B .Q C ) <Q ( B .Q D ) ) )
13	12	3expa 1227	. . . . 5 |- ( ( ( C e. Q. /\ D e. Q. ) /\ B e. Q. ) -> ( C <Q D <-> ( B .Q C ) <Q ( B .Q D ) ) )
14	13	ancoms 268	. . . 4 |- ( ( B e. Q. /\ ( C e. Q. /\ D e. Q. ) ) -> ( C <Q D <-> ( B .Q C ) <Q ( B .Q D ) ) )
15	14	adantll 476	. . 3 |- ( ( ( A e. Q. /\ B e. Q. ) /\ ( C e. Q. /\ D e. Q. ) ) -> ( C <Q D <-> ( B .Q C ) <Q ( B .Q D ) ) )
16	11, 15	anbi12d 473	. 2 |- ( ( ( A e. Q. /\ B e. Q. ) /\ ( C e. Q. /\ D e. Q. ) ) -> ( ( A <Q B /\ C <Q D ) <-> ( ( A .Q C ) <Q ( B .Q C ) /\ ( B .Q C ) <Q ( B .Q D ) ) ) )
17		ltsonq 7593	. . 3 |- <Q Or Q.
18		ltrelnq 7560	. . 3 |- <Q C_ ( Q. X. Q. )
19	17, 18	sotri 5124	. 2 |- ( ( ( A .Q C ) <Q ( B .Q C ) /\ ( B .Q C ) <Q ( B .Q D ) ) -> ( A .Q C ) <Q ( B .Q D ) )
20	16, 19	biimtrdi 163	1 |- ( ( ( A e. Q. /\ B e. Q. ) /\ ( C e. Q. /\ D e. Q. ) ) -> ( ( A <Q B /\ C <Q D ) -> ( A .Q C ) <Q ( B .Q D ) ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1395   e. wcel 2200   class class class wbr 4083  (class class class)co 6007   Q.cnq 7475   .Q cmq 7478   <Q cltq 7480

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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-coll 4199  ax-sep 4202  ax-nul 4210  ax-pow 4258  ax-pr 4293  ax-un 4524  ax-setind 4629  ax-iinf 4680

This theorem depends on definitions:  df-bi 117  df-dc 840  df-3or 1003  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-ral 2513  df-rex 2514  df-reu 2515  df-rab 2517  df-v 2801  df-sbc 3029  df-csb 3125  df-dif 3199  df-un 3201  df-in 3203  df-ss 3210  df-nul 3492  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3889  df-int 3924  df-iun 3967  df-br 4084  df-opab 4146  df-mpt 4147  df-tr 4183  df-eprel 4380  df-id 4384  df-po 4387  df-iso 4388  df-iord 4457  df-on 4459  df-suc 4462  df-iom 4683  df-xp 4725  df-rel 4726  df-cnv 4727  df-co 4728  df-dm 4729  df-rn 4730  df-res 4731  df-ima 4732  df-iota 5278  df-fun 5320  df-fn 5321  df-f 5322  df-f1 5323  df-fo 5324  df-f1o 5325  df-fv 5326  df-ov 6010  df-oprab 6011  df-mpo 6012  df-1st 6292  df-2nd 6293  df-recs 6457  df-irdg 6522  df-oadd 6572  df-omul 6573  df-er 6688  df-ec 6690  df-qs 6694  df-ni 7499  df-mi 7501  df-lti 7502  df-mpq 7540  df-enq 7542  df-nqqs 7543  df-mqqs 7545  df-ltnqqs 7548

This theorem is referenced by:  mulnqprlemrl  7768  mulnqprlemru  7769