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Theorem ltmnq 9415
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
Dummy variables  x  y  z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 mulnqf 9392 . . 3  |-  .Q  :
( Q.  X.  Q. )
--> Q.
21fdmi 5746 . 2  |-  dom  .Q  =  ( Q.  X.  Q. )
3 ltrelnq 9369 . 2  |-  <Q  C_  ( Q.  X.  Q. )
4 0nnq 9367 . 2  |-  -.  (/)  e.  Q.
5 elpqn 9368 . . . . . . . . . 10  |-  ( C  e.  Q.  ->  C  e.  ( N.  X.  N. ) )
653ad2ant3 1053 . . . . . . . . 9  |-  ( ( A  e.  Q.  /\  B  e.  Q.  /\  C  e.  Q. )  ->  C  e.  ( N.  X.  N. ) )
7 xp1st 6842 . . . . . . . . 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 6843 . . . . . . . . 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 9336 . . . . . . . 8  |-  ( ( ( 1st `  C
)  e.  N.  /\  ( 2nd `  C )  e.  N. )  -> 
( ( 1st `  C
)  .N  ( 2nd `  C ) )  e. 
N. )
128, 10, 11syl2anc 673 . . . . . . 7  |-  ( ( A  e.  Q.  /\  B  e.  Q.  /\  C  e.  Q. )  ->  (
( 1st `  C
)  .N  ( 2nd `  C ) )  e. 
N. )
13 ltmpi 9347 . . . . . . 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 5889 . . . . . . . 8  |-  ( 1st `  C )  e.  _V
16 fvex 5889 . . . . . . . 8  |-  ( 2nd `  C )  e.  _V
17 fvex 5889 . . . . . . . 8  |-  ( 1st `  A )  e.  _V
18 mulcompi 9339 . . . . . . . 8  |-  ( x  .N  y )  =  ( y  .N  x
)
19 mulasspi 9340 . . . . . . . 8  |-  ( ( x  .N  y )  .N  z )  =  ( x  .N  (
y  .N  z ) )
20 fvex 5889 . . . . . . . 8  |-  ( 2nd `  B )  e.  _V
2115, 16, 17, 18, 19, 20caov4 6519 . . . . . . 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 5889 . . . . . . . 8  |-  ( 1st `  B )  e.  _V
23 fvex 5889 . . . . . . . 8  |-  ( 2nd `  A )  e.  _V
2415, 16, 22, 18, 19, 23caov4 6519 . . . . . . 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 4404 . . . . . 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 269 . . . . 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 9385 . . . . 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 271 . . . 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 9368 . . . . . . 7  |-  ( A  e.  Q.  ->  A  e.  ( N.  X.  N. ) )
30293ad2ant1 1051 . . . . . 6  |-  ( ( A  e.  Q.  /\  B  e.  Q.  /\  C  e.  Q. )  ->  A  e.  ( N.  X.  N. ) )
31 mulpipq2 9382 . . . . . 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 673 . . . . 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 9368 . . . . . . 7  |-  ( B  e.  Q.  ->  B  e.  ( N.  X.  N. ) )
34333ad2ant2 1052 . . . . . 6  |-  ( ( A  e.  Q.  /\  B  e.  Q.  /\  C  e.  Q. )  ->  B  e.  ( N.  X.  N. ) )
35 mulpipq2 9382 . . . . . 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 673 . . . . 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 4408 . . . 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 264 . . 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 9386 . . . 4  |-  ( ( A  e.  Q.  /\  B  e.  Q. )  ->  ( A  <Q  B  <->  ( ( 1st `  A )  .N  ( 2nd `  B
) )  <N  (
( 1st `  B
)  .N  ( 2nd `  A ) ) ) )
40393adant3 1050 . . 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 9384 . . . . . . 7  |-  ( ( C  e.  Q.  /\  A  e.  Q. )  ->  ( C  .Q  A
)  =  ( /Q
`  ( C  .pQ  A ) ) )
4241ancoms 460 . . . . . 6  |-  ( ( A  e.  Q.  /\  C  e.  Q. )  ->  ( C  .Q  A
)  =  ( /Q
`  ( C  .pQ  A ) ) )
43423adant2 1049 . . . . 5  |-  ( ( A  e.  Q.  /\  B  e.  Q.  /\  C  e.  Q. )  ->  ( C  .Q  A )  =  ( /Q `  ( C  .pQ  A ) ) )
44 mulpqnq 9384 . . . . . . 7  |-  ( ( C  e.  Q.  /\  B  e.  Q. )  ->  ( C  .Q  B
)  =  ( /Q
`  ( C  .pQ  B ) ) )
4544ancoms 460 . . . . . 6  |-  ( ( B  e.  Q.  /\  C  e.  Q. )  ->  ( C  .Q  B
)  =  ( /Q
`  ( C  .pQ  B ) ) )
46453adant1 1048 . . . . 5  |-  ( ( A  e.  Q.  /\  B  e.  Q.  /\  C  e.  Q. )  ->  ( C  .Q  B )  =  ( /Q `  ( C  .pQ  B ) ) )
4743, 46breq12d 4408 . . . 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 9413 . . . 4  |-  ( ( C  .pQ  A ) 
<pQ  ( C  .pQ  B
)  <->  ( /Q `  ( C  .pQ  A ) )  <Q  ( /Q `  ( C  .pQ  B
) ) )
4947, 48syl6bbr 271 . . 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 293 . 2  |-  ( ( A  e.  Q.  /\  B  e.  Q.  /\  C  e.  Q. )  ->  ( A  <Q  B  <->  ( C  .Q  A )  <Q  ( C  .Q  B ) ) )
512, 3, 4, 50ndmovord 6478 1  |-  ( C  e.  Q.  ->  ( A  <Q  B  <->  ( C  .Q  A )  <Q  ( C  .Q  B ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 189    /\ w3a 1007    = wceq 1452    e. wcel 1904   <.cop 3965   class class class wbr 4395    X. cxp 4837   ` cfv 5589  (class class class)co 6308   1stc1st 6810   2ndc2nd 6811   N.cnpi 9287    .N cmi 9289    <N clti 9290    .pQ cmpq 9292    <pQ cltpq 9293   Q.cnq 9295   /Qcerq 9297    .Q cmq 9299    <Q cltq 9301
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-8 1906  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-sep 4518  ax-nul 4527  ax-pow 4579  ax-pr 4639  ax-un 6602
This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3or 1008  df-3an 1009  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-ral 2761  df-rex 2762  df-reu 2763  df-rmo 2764  df-rab 2765  df-v 3033  df-sbc 3256  df-csb 3350  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-pss 3406  df-nul 3723  df-if 3873  df-pw 3944  df-sn 3960  df-pr 3962  df-tp 3964  df-op 3966  df-uni 4191  df-iun 4271  df-br 4396  df-opab 4455  df-mpt 4456  df-tr 4491  df-eprel 4750  df-id 4754  df-po 4760  df-so 4761  df-fr 4798  df-we 4800  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-pred 5387  df-ord 5433  df-on 5434  df-lim 5435  df-suc 5436  df-iota 5553  df-fun 5591  df-fn 5592  df-f 5593  df-f1 5594  df-fo 5595  df-f1o 5596  df-fv 5597  df-ov 6311  df-oprab 6312  df-mpt2 6313  df-om 6712  df-1st 6812  df-2nd 6813  df-wrecs 7046  df-recs 7108  df-rdg 7146  df-1o 7200  df-oadd 7204  df-omul 7205  df-er 7381  df-ni 9315  df-mi 9317  df-lti 9318  df-mpq 9352  df-ltpq 9353  df-enq 9354  df-nq 9355  df-erq 9356  df-mq 9358  df-1nq 9359  df-ltnq 9361
This theorem is referenced by:  ltaddnq  9417  ltrnq  9422  addclprlem1  9459  mulclprlem  9462  mulclpr  9463  distrlem4pr  9469  1idpr  9472  prlem934  9476  prlem936  9490  reclem3pr  9492  reclem4pr  9493
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