Theorem lt2mulnq 7737

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 7733	. . . . . 6 |- ( ( A e. Q. /\ B e. Q. /\ C e. Q. ) -> ( A <Q B <-> ( C .Q A ) <Q ( C .Q B ) ) )
2	1	3expa 1230	. . . . 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 7715	. . . . . . 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 7715	. . . . . . 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 4128	. . . 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 7733	. . . . . 6 |- ( ( C e. Q. /\ D e. Q. /\ B e. Q. ) -> ( C <Q D <-> ( B .Q C ) <Q ( B .Q D ) ) )
13	12	3expa 1230	. . . . 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 7730	. . 3 |- <Q Or Q.
18		ltrelnq 7697	. . 3 |- <Q C_ ( Q. X. Q. )
19	17, 18	sotri 5164	. 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 1398   e. wcel 2205   class class class wbr 4115  (class class class)co 6059   Q.cnq 7612   .Q cmq 7615   <Q cltq 7617

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 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2207  ax-14 2208  ax-ext 2216  ax-coll 4231  ax-sep 4234  ax-nul 4242  ax-pow 4293  ax-pr 4328  ax-un 4560  ax-setind 4665  ax-iinf 4716

This theorem depends on definitions:  df-bi 117  df-dc 843  df-3or 1006  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ne 2415  df-ral 2527  df-rex 2528  df-reu 2529  df-rab 2531  df-v 2817  df-sbc 3046  df-csb 3142  df-dif 3216  df-un 3218  df-in 3220  df-ss 3227  df-nul 3513  df-pw 3677  df-sn 3701  df-pr 3702  df-op 3704  df-uni 3921  df-int 3956  df-iun 3999  df-br 4116  df-opab 4178  df-mpt 4179  df-tr 4215  df-eprel 4416  df-id 4420  df-po 4423  df-iso 4424  df-iord 4493  df-on 4495  df-suc 4498  df-iom 4719  df-xp 4761  df-rel 4762  df-cnv 4763  df-co 4764  df-dm 4765  df-rn 4766  df-res 4767  df-ima 4768  df-iota 5318  df-fun 5360  df-fn 5361  df-f 5362  df-f1 5363  df-fo 5364  df-f1o 5365  df-fv 5366  df-ov 6062  df-oprab 6063  df-mpo 6064  df-1st 6348  df-2nd 6349  df-recs 6550  df-irdg 6615  df-oadd 6665  df-omul 6666  df-er 6781  df-ec 6783  df-qs 6787  df-ni 7636  df-mi 7638  df-lti 7639  df-mpq 7677  df-enq 7679  df-nqqs 7680  df-mqqs 7682  df-ltnqqs 7685

This theorem is referenced by:  mulnqprlemrl  7905  mulnqprlemru  7906