Theorem lt2mulnq 7668

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 7664	. . . . . 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 7646	. . . . . . 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 7646	. . . . . . 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 4106	. . . 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 7664	. . . . . 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 7661	. . 3 |- <Q Or Q.
18		ltrelnq 7628	. . 3 |- <Q C_ ( Q. X. Q. )
19	17, 18	sotri 5139	. 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 2202   class class class wbr 4093  (class class class)co 6028   Q.cnq 7543   .Q cmq 7546   <Q cltq 7548

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 2204  ax-14 2205  ax-ext 2213  ax-coll 4209  ax-sep 4212  ax-nul 4220  ax-pow 4270  ax-pr 4305  ax-un 4536  ax-setind 4641  ax-iinf 4692

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 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ne 2404  df-ral 2516  df-rex 2517  df-reu 2518  df-rab 2520  df-v 2805  df-sbc 3033  df-csb 3129  df-dif 3203  df-un 3205  df-in 3207  df-ss 3214  df-nul 3497  df-pw 3658  df-sn 3679  df-pr 3680  df-op 3682  df-uni 3899  df-int 3934  df-iun 3977  df-br 4094  df-opab 4156  df-mpt 4157  df-tr 4193  df-eprel 4392  df-id 4396  df-po 4399  df-iso 4400  df-iord 4469  df-on 4471  df-suc 4474  df-iom 4695  df-xp 4737  df-rel 4738  df-cnv 4739  df-co 4740  df-dm 4741  df-rn 4742  df-res 4743  df-ima 4744  df-iota 5293  df-fun 5335  df-fn 5336  df-f 5337  df-f1 5338  df-fo 5339  df-f1o 5340  df-fv 5341  df-ov 6031  df-oprab 6032  df-mpo 6033  df-1st 6312  df-2nd 6313  df-recs 6514  df-irdg 6579  df-oadd 6629  df-omul 6630  df-er 6745  df-ec 6747  df-qs 6751  df-ni 7567  df-mi 7569  df-lti 7570  df-mpq 7608  df-enq 7610  df-nqqs 7611  df-mqqs 7613  df-ltnqqs 7616

This theorem is referenced by:  mulnqprlemrl  7836  mulnqprlemru  7837