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Theorem ltmnq 9346
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 9323 . . 3  |-  .Q  :
( Q.  X.  Q. )
--> Q.
21fdmi 5734 . 2  |-  dom  .Q  =  ( Q.  X.  Q. )
3 ltrelnq 9300 . 2  |-  <Q  C_  ( Q.  X.  Q. )
4 0nnq 9298 . 2  |-  -.  (/)  e.  Q.
5 elpqn 9299 . . . . . . . . . 10  |-  ( C  e.  Q.  ->  C  e.  ( N.  X.  N. ) )
653ad2ant3 1019 . . . . . . . . 9  |-  ( ( A  e.  Q.  /\  B  e.  Q.  /\  C  e.  Q. )  ->  C  e.  ( N.  X.  N. ) )
7 xp1st 6811 . . . . . . . . 9  |-  ( C  e.  ( N.  X.  N. )  ->  ( 1st `  C )  e.  N. )
86, 7syl 16 . . . . . . . 8  |-  ( ( A  e.  Q.  /\  B  e.  Q.  /\  C  e.  Q. )  ->  ( 1st `  C )  e. 
N. )
9 xp2nd 6812 . . . . . . . . 9  |-  ( C  e.  ( N.  X.  N. )  ->  ( 2nd `  C )  e.  N. )
106, 9syl 16 . . . . . . . 8  |-  ( ( A  e.  Q.  /\  B  e.  Q.  /\  C  e.  Q. )  ->  ( 2nd `  C )  e. 
N. )
11 mulclpi 9267 . . . . . . . 8  |-  ( ( ( 1st `  C
)  e.  N.  /\  ( 2nd `  C )  e.  N. )  -> 
( ( 1st `  C
)  .N  ( 2nd `  C ) )  e. 
N. )
128, 10, 11syl2anc 661 . . . . . . 7  |-  ( ( A  e.  Q.  /\  B  e.  Q.  /\  C  e.  Q. )  ->  (
( 1st `  C
)  .N  ( 2nd `  C ) )  e. 
N. )
13 ltmpi 9278 . . . . . . 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 16 . . . . . 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 5874 . . . . . . . 8  |-  ( 1st `  C )  e.  _V
16 fvex 5874 . . . . . . . 8  |-  ( 2nd `  C )  e.  _V
17 fvex 5874 . . . . . . . 8  |-  ( 1st `  A )  e.  _V
18 mulcompi 9270 . . . . . . . 8  |-  ( x  .N  y )  =  ( y  .N  x
)
19 mulasspi 9271 . . . . . . . 8  |-  ( ( x  .N  y )  .N  z )  =  ( x  .N  (
y  .N  z ) )
20 fvex 5874 . . . . . . . 8  |-  ( 2nd `  B )  e.  _V
2115, 16, 17, 18, 19, 20caov4 6488 . . . . . . 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 5874 . . . . . . . 8  |-  ( 1st `  B )  e.  _V
23 fvex 5874 . . . . . . . 8  |-  ( 2nd `  A )  e.  _V
2415, 16, 22, 18, 19, 23caov4 6488 . . . . . . 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 4456 . . . . . 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 261 . . . . 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 9316 . . . . 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 263 . . . 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 9299 . . . . . . 7  |-  ( A  e.  Q.  ->  A  e.  ( N.  X.  N. ) )
30293ad2ant1 1017 . . . . . 6  |-  ( ( A  e.  Q.  /\  B  e.  Q.  /\  C  e.  Q. )  ->  A  e.  ( N.  X.  N. ) )
31 mulpipq2 9313 . . . . . 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 661 . . . . 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 9299 . . . . . . 7  |-  ( B  e.  Q.  ->  B  e.  ( N.  X.  N. ) )
34333ad2ant2 1018 . . . . . 6  |-  ( ( A  e.  Q.  /\  B  e.  Q.  /\  C  e.  Q. )  ->  B  e.  ( N.  X.  N. ) )
35 mulpipq2 9313 . . . . . 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 661 . . . . 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 4460 . . . 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 256 . . 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 9317 . . . 4  |-  ( ( A  e.  Q.  /\  B  e.  Q. )  ->  ( A  <Q  B  <->  ( ( 1st `  A )  .N  ( 2nd `  B
) )  <N  (
( 1st `  B
)  .N  ( 2nd `  A ) ) ) )
40393adant3 1016 . . 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 9315 . . . . . . 7  |-  ( ( C  e.  Q.  /\  A  e.  Q. )  ->  ( C  .Q  A
)  =  ( /Q
`  ( C  .pQ  A ) ) )
4241ancoms 453 . . . . . 6  |-  ( ( A  e.  Q.  /\  C  e.  Q. )  ->  ( C  .Q  A
)  =  ( /Q
`  ( C  .pQ  A ) ) )
43423adant2 1015 . . . . 5  |-  ( ( A  e.  Q.  /\  B  e.  Q.  /\  C  e.  Q. )  ->  ( C  .Q  A )  =  ( /Q `  ( C  .pQ  A ) ) )
44 mulpqnq 9315 . . . . . . 7  |-  ( ( C  e.  Q.  /\  B  e.  Q. )  ->  ( C  .Q  B
)  =  ( /Q
`  ( C  .pQ  B ) ) )
4544ancoms 453 . . . . . 6  |-  ( ( B  e.  Q.  /\  C  e.  Q. )  ->  ( C  .Q  B
)  =  ( /Q
`  ( C  .pQ  B ) ) )
46453adant1 1014 . . . . 5  |-  ( ( A  e.  Q.  /\  B  e.  Q.  /\  C  e.  Q. )  ->  ( C  .Q  B )  =  ( /Q `  ( C  .pQ  B ) ) )
4743, 46breq12d 4460 . . . 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 9344 . . . 4  |-  ( ( C  .pQ  A ) 
<pQ  ( C  .pQ  B
)  <->  ( /Q `  ( C  .pQ  A ) )  <Q  ( /Q `  ( C  .pQ  B
) ) )
4947, 48syl6bbr 263 . . 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 285 . 2  |-  ( ( A  e.  Q.  /\  B  e.  Q.  /\  C  e.  Q. )  ->  ( A  <Q  B  <->  ( C  .Q  A )  <Q  ( C  .Q  B ) ) )
512, 3, 4, 50ndmovord 6447 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 184    /\ w3a 973    = wceq 1379    e. wcel 1767   <.cop 4033   class class class wbr 4447    X. cxp 4997   ` cfv 5586  (class class class)co 6282   1stc1st 6779   2ndc2nd 6780   N.cnpi 9218    .N cmi 9220    <N clti 9221    .pQ cmpq 9223    <pQ cltpq 9224   Q.cnq 9226   /Qcerq 9228    .Q cmq 9230    <Q cltq 9232
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6574
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-reu 2821  df-rmo 2822  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-we 4840  df-ord 4881  df-on 4882  df-lim 4883  df-suc 4884  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5549  df-fun 5588  df-fn 5589  df-f 5590  df-f1 5591  df-fo 5592  df-f1o 5593  df-fv 5594  df-ov 6285  df-oprab 6286  df-mpt2 6287  df-om 6679  df-1st 6781  df-2nd 6782  df-recs 7039  df-rdg 7073  df-1o 7127  df-oadd 7131  df-omul 7132  df-er 7308  df-ni 9246  df-mi 9248  df-lti 9249  df-mpq 9283  df-ltpq 9284  df-enq 9285  df-nq 9286  df-erq 9287  df-mq 9289  df-1nq 9290  df-ltnq 9292
This theorem is referenced by:  ltaddnq  9348  ltrnq  9353  addclprlem1  9390  mulclprlem  9393  mulclpr  9394  distrlem4pr  9400  1idpr  9403  prlem934  9407  prlem936  9421  reclem3pr  9423  reclem4pr  9424
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