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Theorem ltmnq 8564
Description: Ordering property of multiplication for positive fractions. Proposition 9-2.6(iii) of [Gleason] p. 120. (Contributed by NM, 6-Mar-1996.) (Revised by Mario Carneiro, 10-May-2013.) (New usage is discouraged.)
Assertion
Ref Expression
ltmnq  |-  ( C  e.  Q.  ->  ( A  <Q  B  <->  ( C  .Q  A )  <Q  ( C  .Q  B ) ) )

Proof of Theorem ltmnq
StepHypRef Expression
1 mulnqf 8541 . . 3  |-  .Q  :
( Q.  X.  Q. )
--> Q.
21fdmi 5332 . 2  |-  dom  .Q  =  ( Q.  X.  Q. )
3 ltrelnq 8518 . 2  |-  <Q  C_  ( Q.  X.  Q. )
4 0nnq 8516 . 2  |-  -.  (/)  e.  Q.
5 elpqn 8517 . . . . . . . . . 10  |-  ( C  e.  Q.  ->  C  e.  ( N.  X.  N. ) )
653ad2ant3 983 . . . . . . . . 9  |-  ( ( A  e.  Q.  /\  B  e.  Q.  /\  C  e.  Q. )  ->  C  e.  ( N.  X.  N. ) )
7 xp1st 6083 . . . . . . . . 9  |-  ( C  e.  ( N.  X.  N. )  ->  ( 1st `  C )  e.  N. )
86, 7syl 17 . . . . . . . 8  |-  ( ( A  e.  Q.  /\  B  e.  Q.  /\  C  e.  Q. )  ->  ( 1st `  C )  e. 
N. )
9 xp2nd 6084 . . . . . . . . 9  |-  ( C  e.  ( N.  X.  N. )  ->  ( 2nd `  C )  e.  N. )
106, 9syl 17 . . . . . . . 8  |-  ( ( A  e.  Q.  /\  B  e.  Q.  /\  C  e.  Q. )  ->  ( 2nd `  C )  e. 
N. )
11 mulclpi 8485 . . . . . . . 8  |-  ( ( ( 1st `  C
)  e.  N.  /\  ( 2nd `  C )  e.  N. )  -> 
( ( 1st `  C
)  .N  ( 2nd `  C ) )  e. 
N. )
128, 10, 11syl2anc 645 . . . . . . 7  |-  ( ( A  e.  Q.  /\  B  e.  Q.  /\  C  e.  Q. )  ->  (
( 1st `  C
)  .N  ( 2nd `  C ) )  e. 
N. )
13 ltmpi 8496 . . . . . . 7  |-  ( ( ( 1st `  C
)  .N  ( 2nd `  C ) )  e. 
N.  ->  ( ( ( 1st `  A )  .N  ( 2nd `  B
) )  <N  (
( 1st `  B
)  .N  ( 2nd `  A ) )  <->  ( (
( 1st `  C
)  .N  ( 2nd `  C ) )  .N  ( ( 1st `  A
)  .N  ( 2nd `  B ) ) ) 
<N  ( ( ( 1st `  C )  .N  ( 2nd `  C ) )  .N  ( ( 1st `  B )  .N  ( 2nd `  A ) ) ) ) )
1412, 13syl 17 . . . . . 6  |-  ( ( A  e.  Q.  /\  B  e.  Q.  /\  C  e.  Q. )  ->  (
( ( 1st `  A
)  .N  ( 2nd `  B ) )  <N 
( ( 1st `  B
)  .N  ( 2nd `  A ) )  <->  ( (
( 1st `  C
)  .N  ( 2nd `  C ) )  .N  ( ( 1st `  A
)  .N  ( 2nd `  B ) ) ) 
<N  ( ( ( 1st `  C )  .N  ( 2nd `  C ) )  .N  ( ( 1st `  B )  .N  ( 2nd `  A ) ) ) ) )
15 fvex 5472 . . . . . . . 8  |-  ( 1st `  C )  e.  _V
16 fvex 5472 . . . . . . . 8  |-  ( 2nd `  C )  e.  _V
17 fvex 5472 . . . . . . . 8  |-  ( 1st `  A )  e.  _V
18 mulcompi 8488 . . . . . . . 8  |-  ( x  .N  y )  =  ( y  .N  x
)
19 mulasspi 8489 . . . . . . . 8  |-  ( ( x  .N  y )  .N  z )  =  ( x  .N  (
y  .N  z ) )
20 fvex 5472 . . . . . . . 8  |-  ( 2nd `  B )  e.  _V
2115, 16, 17, 18, 19, 20caov4 5985 . . . . . . 7  |-  ( ( ( 1st `  C
)  .N  ( 2nd `  C ) )  .N  ( ( 1st `  A
)  .N  ( 2nd `  B ) ) )  =  ( ( ( 1st `  C )  .N  ( 1st `  A
) )  .N  (
( 2nd `  C
)  .N  ( 2nd `  B ) ) )
22 fvex 5472 . . . . . . . 8  |-  ( 1st `  B )  e.  _V
23 fvex 5472 . . . . . . . 8  |-  ( 2nd `  A )  e.  _V
2415, 16, 22, 18, 19, 23caov4 5985 . . . . . . 7  |-  ( ( ( 1st `  C
)  .N  ( 2nd `  C ) )  .N  ( ( 1st `  B
)  .N  ( 2nd `  A ) ) )  =  ( ( ( 1st `  C )  .N  ( 1st `  B
) )  .N  (
( 2nd `  C
)  .N  ( 2nd `  A ) ) )
2521, 24breq12i 4006 . . . . . 6  |-  ( ( ( ( 1st `  C
)  .N  ( 2nd `  C ) )  .N  ( ( 1st `  A
)  .N  ( 2nd `  B ) ) ) 
<N  ( ( ( 1st `  C )  .N  ( 2nd `  C ) )  .N  ( ( 1st `  B )  .N  ( 2nd `  A ) ) )  <->  ( ( ( 1st `  C )  .N  ( 1st `  A
) )  .N  (
( 2nd `  C
)  .N  ( 2nd `  B ) ) ) 
<N  ( ( ( 1st `  C )  .N  ( 1st `  B ) )  .N  ( ( 2nd `  C )  .N  ( 2nd `  A ) ) ) )
2614, 25syl6bb 254 . . . . 5  |-  ( ( A  e.  Q.  /\  B  e.  Q.  /\  C  e.  Q. )  ->  (
( ( 1st `  A
)  .N  ( 2nd `  B ) )  <N 
( ( 1st `  B
)  .N  ( 2nd `  A ) )  <->  ( (
( 1st `  C
)  .N  ( 1st `  A ) )  .N  ( ( 2nd `  C
)  .N  ( 2nd `  B ) ) ) 
<N  ( ( ( 1st `  C )  .N  ( 1st `  B ) )  .N  ( ( 2nd `  C )  .N  ( 2nd `  A ) ) ) ) )
27 ordpipq 8534 . . . . 5  |-  ( <.
( ( 1st `  C
)  .N  ( 1st `  A ) ) ,  ( ( 2nd `  C
)  .N  ( 2nd `  A ) ) >.  <pQ 
<. ( ( 1st `  C
)  .N  ( 1st `  B ) ) ,  ( ( 2nd `  C
)  .N  ( 2nd `  B ) ) >.  <->  ( ( ( 1st `  C
)  .N  ( 1st `  A ) )  .N  ( ( 2nd `  C
)  .N  ( 2nd `  B ) ) ) 
<N  ( ( ( 1st `  C )  .N  ( 1st `  B ) )  .N  ( ( 2nd `  C )  .N  ( 2nd `  A ) ) ) )
2826, 27syl6bbr 256 . . . 4  |-  ( ( A  e.  Q.  /\  B  e.  Q.  /\  C  e.  Q. )  ->  (
( ( 1st `  A
)  .N  ( 2nd `  B ) )  <N 
( ( 1st `  B
)  .N  ( 2nd `  A ) )  <->  <. ( ( 1st `  C )  .N  ( 1st `  A
) ) ,  ( ( 2nd `  C
)  .N  ( 2nd `  A ) ) >.  <pQ 
<. ( ( 1st `  C
)  .N  ( 1st `  B ) ) ,  ( ( 2nd `  C
)  .N  ( 2nd `  B ) ) >.
) )
29 elpqn 8517 . . . . . . 7  |-  ( A  e.  Q.  ->  A  e.  ( N.  X.  N. ) )
30293ad2ant1 981 . . . . . 6  |-  ( ( A  e.  Q.  /\  B  e.  Q.  /\  C  e.  Q. )  ->  A  e.  ( N.  X.  N. ) )
31 mulpipq2 8531 . . . . . 6  |-  ( ( C  e.  ( N. 
X.  N. )  /\  A  e.  ( N.  X.  N. ) )  ->  ( C  .pQ  A )  = 
<. ( ( 1st `  C
)  .N  ( 1st `  A ) ) ,  ( ( 2nd `  C
)  .N  ( 2nd `  A ) ) >.
)
326, 30, 31syl2anc 645 . . . . 5  |-  ( ( A  e.  Q.  /\  B  e.  Q.  /\  C  e.  Q. )  ->  ( C  .pQ  A )  = 
<. ( ( 1st `  C
)  .N  ( 1st `  A ) ) ,  ( ( 2nd `  C
)  .N  ( 2nd `  A ) ) >.
)
33 elpqn 8517 . . . . . . 7  |-  ( B  e.  Q.  ->  B  e.  ( N.  X.  N. ) )
34333ad2ant2 982 . . . . . 6  |-  ( ( A  e.  Q.  /\  B  e.  Q.  /\  C  e.  Q. )  ->  B  e.  ( N.  X.  N. ) )
35 mulpipq2 8531 . . . . . 6  |-  ( ( C  e.  ( N. 
X.  N. )  /\  B  e.  ( N.  X.  N. ) )  ->  ( C  .pQ  B )  = 
<. ( ( 1st `  C
)  .N  ( 1st `  B ) ) ,  ( ( 2nd `  C
)  .N  ( 2nd `  B ) ) >.
)
366, 34, 35syl2anc 645 . . . . 5  |-  ( ( A  e.  Q.  /\  B  e.  Q.  /\  C  e.  Q. )  ->  ( C  .pQ  B )  = 
<. ( ( 1st `  C
)  .N  ( 1st `  B ) ) ,  ( ( 2nd `  C
)  .N  ( 2nd `  B ) ) >.
)
3732, 36breq12d 4010 . . . 4  |-  ( ( A  e.  Q.  /\  B  e.  Q.  /\  C  e.  Q. )  ->  (
( C  .pQ  A
)  <pQ  ( C  .pQ  B )  <->  <. ( ( 1st `  C )  .N  ( 1st `  A ) ) ,  ( ( 2nd `  C )  .N  ( 2nd `  A ) )
>.  <pQ  <. ( ( 1st `  C )  .N  ( 1st `  B ) ) ,  ( ( 2nd `  C )  .N  ( 2nd `  B ) )
>. ) )
3828, 37bitr4d 249 . . 3  |-  ( ( A  e.  Q.  /\  B  e.  Q.  /\  C  e.  Q. )  ->  (
( ( 1st `  A
)  .N  ( 2nd `  B ) )  <N 
( ( 1st `  B
)  .N  ( 2nd `  A ) )  <->  ( C  .pQ  A )  <pQ  ( C 
.pQ  B ) ) )
39 ordpinq 8535 . . . 4  |-  ( ( A  e.  Q.  /\  B  e.  Q. )  ->  ( A  <Q  B  <->  ( ( 1st `  A )  .N  ( 2nd `  B
) )  <N  (
( 1st `  B
)  .N  ( 2nd `  A ) ) ) )
40393adant3 980 . . 3  |-  ( ( A  e.  Q.  /\  B  e.  Q.  /\  C  e.  Q. )  ->  ( A  <Q  B  <->  ( ( 1st `  A )  .N  ( 2nd `  B
) )  <N  (
( 1st `  B
)  .N  ( 2nd `  A ) ) ) )
41 mulpqnq 8533 . . . . . . 7  |-  ( ( C  e.  Q.  /\  A  e.  Q. )  ->  ( C  .Q  A
)  =  ( /Q
`  ( C  .pQ  A ) ) )
4241ancoms 441 . . . . . 6  |-  ( ( A  e.  Q.  /\  C  e.  Q. )  ->  ( C  .Q  A
)  =  ( /Q
`  ( C  .pQ  A ) ) )
43423adant2 979 . . . . 5  |-  ( ( A  e.  Q.  /\  B  e.  Q.  /\  C  e.  Q. )  ->  ( C  .Q  A )  =  ( /Q `  ( C  .pQ  A ) ) )
44 mulpqnq 8533 . . . . . . 7  |-  ( ( C  e.  Q.  /\  B  e.  Q. )  ->  ( C  .Q  B
)  =  ( /Q
`  ( C  .pQ  B ) ) )
4544ancoms 441 . . . . . 6  |-  ( ( B  e.  Q.  /\  C  e.  Q. )  ->  ( C  .Q  B
)  =  ( /Q
`  ( C  .pQ  B ) ) )
46453adant1 978 . . . . 5  |-  ( ( A  e.  Q.  /\  B  e.  Q.  /\  C  e.  Q. )  ->  ( C  .Q  B )  =  ( /Q `  ( C  .pQ  B ) ) )
4743, 46breq12d 4010 . . . 4  |-  ( ( A  e.  Q.  /\  B  e.  Q.  /\  C  e.  Q. )  ->  (
( C  .Q  A
)  <Q  ( C  .Q  B )  <->  ( /Q `  ( C  .pQ  A
) )  <Q  ( /Q `  ( C  .pQ  B ) ) ) )
48 lterpq 8562 . . . 4  |-  ( ( C  .pQ  A ) 
<pQ  ( C  .pQ  B
)  <->  ( /Q `  ( C  .pQ  A ) )  <Q  ( /Q `  ( C  .pQ  B
) ) )
4947, 48syl6bbr 256 . . 3  |-  ( ( A  e.  Q.  /\  B  e.  Q.  /\  C  e.  Q. )  ->  (
( C  .Q  A
)  <Q  ( C  .Q  B )  <->  ( C  .pQ  A )  <pQ  ( C 
.pQ  B ) ) )
5038, 40, 493bitr4d 278 . 2  |-  ( ( A  e.  Q.  /\  B  e.  Q.  /\  C  e.  Q. )  ->  ( A  <Q  B  <->  ( C  .Q  A )  <Q  ( C  .Q  B ) ) )
512, 3, 4, 50ndmovord 5944 1  |-  ( C  e.  Q.  ->  ( A  <Q  B  <->  ( C  .Q  A )  <Q  ( C  .Q  B ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 6    <-> wb 178    /\ w3a 939    = wceq 1619    e. wcel 1621   <.cop 3617   class class class wbr 3997    X. cxp 4659   ` cfv 4673  (class class class)co 5792   1stc1st 6054   2ndc2nd 6055   N.cnpi 8434    .N cmi 8436    <N clti 8437    .pQ cmpq 8439    <pQ cltpq 8440   Q.cnq 8442   /Qcerq 8444    .Q cmq 8446    <Q cltq 8448
This theorem is referenced by:  ltaddnq  8566  ltrnq  8571  addclprlem1  8608  mulclprlem  8611  mulclpr  8612  distrlem4pr  8618  1idpr  8621  prlem934  8625  prlem936  8639  reclem3pr  8641  reclem4pr  8642
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-13 1625  ax-14 1626  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1927  ax-ext 2239  ax-sep 4115  ax-nul 4123  ax-pr 4186  ax-un 4484
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 940  df-3an 941  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1884  df-eu 2122  df-mo 2123  df-clab 2245  df-cleq 2251  df-clel 2254  df-nfc 2383  df-ne 2423  df-ral 2523  df-rex 2524  df-reu 2525  df-rmo 2526  df-rab 2527  df-v 2765  df-sbc 2967  df-csb 3057  df-dif 3130  df-un 3132  df-in 3134  df-ss 3141  df-pss 3143  df-nul 3431  df-if 3540  df-pw 3601  df-sn 3620  df-pr 3621  df-tp 3622  df-op 3623  df-uni 3802  df-iun 3881  df-br 3998  df-opab 4052  df-mpt 4053  df-tr 4088  df-eprel 4277  df-id 4281  df-po 4286  df-so 4287  df-fr 4324  df-we 4326  df-ord 4367  df-on 4368  df-lim 4369  df-suc 4370  df-om 4629  df-xp 4675  df-rel 4676  df-cnv 4677  df-co 4678  df-dm 4679  df-rn 4680  df-res 4681  df-ima 4682  df-fun 4683  df-fn 4684  df-f 4685  df-f1 4686  df-fo 4687  df-f1o 4688  df-fv 4689  df-ov 5795  df-oprab 5796  df-mpt2 5797  df-1st 6056  df-2nd 6057  df-recs 6356  df-rdg 6391  df-1o 6447  df-oadd 6451  df-omul 6452  df-er 6628  df-ni 8464  df-mi 8466  df-lti 8467  df-mpq 8501  df-ltpq 8502  df-enq 8503  df-nq 8504  df-erq 8505  df-mq 8507  df-1nq 8508  df-ltnq 8510
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