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Theorem lt2mulnq 7403
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
StepHypRef Expression
1 ltmnqg 7399 . . . . . 6  |-  ( ( A  e.  Q.  /\  B  e.  Q.  /\  C  e.  Q. )  ->  ( A  <Q  B  <->  ( C  .Q  A )  <Q  ( C  .Q  B ) ) )
213expa 1203 . . . . 5  |-  ( ( ( A  e.  Q.  /\  B  e.  Q. )  /\  C  e.  Q. )  ->  ( A  <Q  B  <-> 
( C  .Q  A
)  <Q  ( C  .Q  B ) ) )
32adantrr 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 7381 . . . . . . 7  |-  ( ( C  e.  Q.  /\  A  e.  Q. )  ->  ( C  .Q  A
)  =  ( A  .Q  C ) )
54ancoms 268 . . . . . 6  |-  ( ( A  e.  Q.  /\  C  e.  Q. )  ->  ( C  .Q  A
)  =  ( A  .Q  C ) )
65ad2ant2r 509 . . . . 5  |-  ( ( ( A  e.  Q.  /\  B  e.  Q. )  /\  ( C  e.  Q.  /\  D  e.  Q. )
)  ->  ( C  .Q  A )  =  ( A  .Q  C ) )
7 mulcomnqg 7381 . . . . . . 7  |-  ( ( C  e.  Q.  /\  B  e.  Q. )  ->  ( C  .Q  B
)  =  ( B  .Q  C ) )
87ancoms 268 . . . . . 6  |-  ( ( B  e.  Q.  /\  C  e.  Q. )  ->  ( C  .Q  B
)  =  ( B  .Q  C ) )
98ad2ant2lr 510 . . . . 5  |-  ( ( ( A  e.  Q.  /\  B  e.  Q. )  /\  ( C  e.  Q.  /\  D  e.  Q. )
)  ->  ( C  .Q  B )  =  ( B  .Q  C ) )
106, 9breq12d 4016 . . . 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 ) ) )
113, 10bitrd 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 7399 . . . . . 6  |-  ( ( C  e.  Q.  /\  D  e.  Q.  /\  B  e.  Q. )  ->  ( C  <Q  D  <->  ( B  .Q  C )  <Q  ( B  .Q  D ) ) )
13123expa 1203 . . . . 5  |-  ( ( ( C  e.  Q.  /\  D  e.  Q. )  /\  B  e.  Q. )  ->  ( C  <Q  D  <-> 
( B  .Q  C
)  <Q  ( B  .Q  D ) ) )
1413ancoms 268 . . . 4  |-  ( ( B  e.  Q.  /\  ( C  e.  Q.  /\  D  e.  Q. )
)  ->  ( C  <Q  D  <->  ( B  .Q  C )  <Q  ( B  .Q  D ) ) )
1514adantll 476 . . 3  |-  ( ( ( A  e.  Q.  /\  B  e.  Q. )  /\  ( C  e.  Q.  /\  D  e.  Q. )
)  ->  ( C  <Q  D  <->  ( B  .Q  C )  <Q  ( B  .Q  D ) ) )
1611, 15anbi12d 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 7396 . . 3  |-  <Q  Or  Q.
18 ltrelnq 7363 . . 3  |-  <Q  C_  ( Q.  X.  Q. )
1917, 18sotri 5024 . 2  |-  ( ( ( A  .Q  C
)  <Q  ( B  .Q  C )  /\  ( B  .Q  C )  <Q 
( B  .Q  D
) )  ->  ( A  .Q  C )  <Q 
( B  .Q  D
) )
2016, 19syl6bi 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
) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 104    <-> wb 105    = wceq 1353    e. wcel 2148   class class class wbr 4003  (class class class)co 5874   Q.cnq 7278    .Q cmq 7281    <Q cltq 7283
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 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-coll 4118  ax-sep 4121  ax-nul 4129  ax-pow 4174  ax-pr 4209  ax-un 4433  ax-setind 4536  ax-iinf 4587
This theorem depends on definitions:  df-bi 117  df-dc 835  df-3or 979  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-ral 2460  df-rex 2461  df-reu 2462  df-rab 2464  df-v 2739  df-sbc 2963  df-csb 3058  df-dif 3131  df-un 3133  df-in 3135  df-ss 3142  df-nul 3423  df-pw 3577  df-sn 3598  df-pr 3599  df-op 3601  df-uni 3810  df-int 3845  df-iun 3888  df-br 4004  df-opab 4065  df-mpt 4066  df-tr 4102  df-eprel 4289  df-id 4293  df-po 4296  df-iso 4297  df-iord 4366  df-on 4368  df-suc 4371  df-iom 4590  df-xp 4632  df-rel 4633  df-cnv 4634  df-co 4635  df-dm 4636  df-rn 4637  df-res 4638  df-ima 4639  df-iota 5178  df-fun 5218  df-fn 5219  df-f 5220  df-f1 5221  df-fo 5222  df-f1o 5223  df-fv 5224  df-ov 5877  df-oprab 5878  df-mpo 5879  df-1st 6140  df-2nd 6141  df-recs 6305  df-irdg 6370  df-oadd 6420  df-omul 6421  df-er 6534  df-ec 6536  df-qs 6540  df-ni 7302  df-mi 7304  df-lti 7305  df-mpq 7343  df-enq 7345  df-nqqs 7346  df-mqqs 7348  df-ltnqqs 7351
This theorem is referenced by:  mulnqprlemrl  7571  mulnqprlemru  7572
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