Theorem lt2mulnq 7560

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 7556	. . . . . 6 |- ( ( A e. Q. /\ B e. Q. /\ C e. Q. ) -> ( A <Q B <-> ( C .Q A ) <Q ( C .Q B ) ) )
2	1	3expa 1208	. . . . 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 7538	. . . . . . 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 7538	. . . . . . 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 4075	. . . 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 7556	. . . . . 6 |- ( ( C e. Q. /\ D e. Q. /\ B e. Q. ) -> ( C <Q D <-> ( B .Q C ) <Q ( B .Q D ) ) )
13	12	3expa 1208	. . . . 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 7553	. . 3 |- <Q Or Q.
18		ltrelnq 7520	. . 3 |- <Q C_ ( Q. X. Q. )
19	17, 18	sotri 5100	. 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 1375   e. wcel 2180   class class class wbr 4062  (class class class)co 5974   Q.cnq 7435   .Q cmq 7438   <Q cltq 7440

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 713  ax-5 1473  ax-7 1474  ax-gen 1475  ax-ie1 1519  ax-ie2 1520  ax-8 1530  ax-10 1531  ax-11 1532  ax-i12 1533  ax-bndl 1535  ax-4 1536  ax-17 1552  ax-i9 1556  ax-ial 1560  ax-i5r 1561  ax-13 2182  ax-14 2183  ax-ext 2191  ax-coll 4178  ax-sep 4181  ax-nul 4189  ax-pow 4237  ax-pr 4272  ax-un 4501  ax-setind 4606  ax-iinf 4657

This theorem depends on definitions:  df-bi 117  df-dc 839  df-3or 984  df-3an 985  df-tru 1378  df-fal 1381  df-nf 1487  df-sb 1789  df-eu 2060  df-mo 2061  df-clab 2196  df-cleq 2202  df-clel 2205  df-nfc 2341  df-ne 2381  df-ral 2493  df-rex 2494  df-reu 2495  df-rab 2497  df-v 2781  df-sbc 3009  df-csb 3105  df-dif 3179  df-un 3181  df-in 3183  df-ss 3190  df-nul 3472  df-pw 3631  df-sn 3652  df-pr 3653  df-op 3655  df-uni 3868  df-int 3903  df-iun 3946  df-br 4063  df-opab 4125  df-mpt 4126  df-tr 4162  df-eprel 4357  df-id 4361  df-po 4364  df-iso 4365  df-iord 4434  df-on 4436  df-suc 4439  df-iom 4660  df-xp 4702  df-rel 4703  df-cnv 4704  df-co 4705  df-dm 4706  df-rn 4707  df-res 4708  df-ima 4709  df-iota 5254  df-fun 5296  df-fn 5297  df-f 5298  df-f1 5299  df-fo 5300  df-f1o 5301  df-fv 5302  df-ov 5977  df-oprab 5978  df-mpo 5979  df-1st 6256  df-2nd 6257  df-recs 6421  df-irdg 6486  df-oadd 6536  df-omul 6537  df-er 6650  df-ec 6652  df-qs 6656  df-ni 7459  df-mi 7461  df-lti 7462  df-mpq 7500  df-enq 7502  df-nqqs 7503  df-mqqs 7505  df-ltnqqs 7508

This theorem is referenced by:  mulnqprlemrl  7728  mulnqprlemru  7729