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Theorem ltmnq 9382
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 9359 . . 3  |-  .Q  :
( Q.  X.  Q. )
--> Q.
21fdmi 5721 . 2  |-  dom  .Q  =  ( Q.  X.  Q. )
3 ltrelnq 9336 . 2  |-  <Q  C_  ( Q.  X.  Q. )
4 0nnq 9334 . 2  |-  -.  (/)  e.  Q.
5 elpqn 9335 . . . . . . . . . 10  |-  ( C  e.  Q.  ->  C  e.  ( N.  X.  N. ) )
653ad2ant3 1022 . . . . . . . . 9  |-  ( ( A  e.  Q.  /\  B  e.  Q.  /\  C  e.  Q. )  ->  C  e.  ( N.  X.  N. ) )
7 xp1st 6816 . . . . . . . . 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 6817 . . . . . . . . 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 9303 . . . . . . . 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 9314 . . . . . . 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 5861 . . . . . . . 8  |-  ( 1st `  C )  e.  _V
16 fvex 5861 . . . . . . . 8  |-  ( 2nd `  C )  e.  _V
17 fvex 5861 . . . . . . . 8  |-  ( 1st `  A )  e.  _V
18 mulcompi 9306 . . . . . . . 8  |-  ( x  .N  y )  =  ( y  .N  x
)
19 mulasspi 9307 . . . . . . . 8  |-  ( ( x  .N  y )  .N  z )  =  ( x  .N  (
y  .N  z ) )
20 fvex 5861 . . . . . . . 8  |-  ( 2nd `  B )  e.  _V
2115, 16, 17, 18, 19, 20caov4 6489 . . . . . . 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 5861 . . . . . . . 8  |-  ( 1st `  B )  e.  _V
23 fvex 5861 . . . . . . . 8  |-  ( 2nd `  A )  e.  _V
2415, 16, 22, 18, 19, 23caov4 6489 . . . . . . 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 4406 . . . . . 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 263 . . . . 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 9352 . . . . 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 265 . . . 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 9335 . . . . . . 7  |-  ( A  e.  Q.  ->  A  e.  ( N.  X.  N. ) )
30293ad2ant1 1020 . . . . . 6  |-  ( ( A  e.  Q.  /\  B  e.  Q.  /\  C  e.  Q. )  ->  A  e.  ( N.  X.  N. ) )
31 mulpipq2 9349 . . . . . 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 9335 . . . . . . 7  |-  ( B  e.  Q.  ->  B  e.  ( N.  X.  N. ) )
34333ad2ant2 1021 . . . . . 6  |-  ( ( A  e.  Q.  /\  B  e.  Q.  /\  C  e.  Q. )  ->  B  e.  ( N.  X.  N. ) )
35 mulpipq2 9349 . . . . . 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 4410 . . . 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 258 . . 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 9353 . . . 4  |-  ( ( A  e.  Q.  /\  B  e.  Q. )  ->  ( A  <Q  B  <->  ( ( 1st `  A )  .N  ( 2nd `  B
) )  <N  (
( 1st `  B
)  .N  ( 2nd `  A ) ) ) )
40393adant3 1019 . . 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 9351 . . . . . . 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 1018 . . . . 5  |-  ( ( A  e.  Q.  /\  B  e.  Q.  /\  C  e.  Q. )  ->  ( C  .Q  A )  =  ( /Q `  ( C  .pQ  A ) ) )
44 mulpqnq 9351 . . . . . . 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 1017 . . . . 5  |-  ( ( A  e.  Q.  /\  B  e.  Q.  /\  C  e.  Q. )  ->  ( C  .Q  B )  =  ( /Q `  ( C  .pQ  B ) ) )
4743, 46breq12d 4410 . . . 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 9380 . . . 4  |-  ( ( C  .pQ  A ) 
<pQ  ( C  .pQ  B
)  <->  ( /Q `  ( C  .pQ  A ) )  <Q  ( /Q `  ( C  .pQ  B
) ) )
4947, 48syl6bbr 265 . . 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 287 . 2  |-  ( ( A  e.  Q.  /\  B  e.  Q.  /\  C  e.  Q. )  ->  ( A  <Q  B  <->  ( C  .Q  A )  <Q  ( C  .Q  B ) ) )
512, 3, 4, 50ndmovord 6448 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 186    /\ w3a 976    = wceq 1407    e. wcel 1844   <.cop 3980   class class class wbr 4397    X. cxp 4823   ` cfv 5571  (class class class)co 6280   1stc1st 6784   2ndc2nd 6785   N.cnpi 9254    .N cmi 9256    <N clti 9257    .pQ cmpq 9259    <pQ cltpq 9260   Q.cnq 9262   /Qcerq 9264    .Q cmq 9266    <Q cltq 9268
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1641  ax-4 1654  ax-5 1727  ax-6 1773  ax-7 1816  ax-8 1846  ax-9 1848  ax-10 1863  ax-11 1868  ax-12 1880  ax-13 2028  ax-ext 2382  ax-sep 4519  ax-nul 4527  ax-pow 4574  ax-pr 4632  ax-un 6576
This theorem depends on definitions:  df-bi 187  df-or 370  df-an 371  df-3or 977  df-3an 978  df-tru 1410  df-ex 1636  df-nf 1640  df-sb 1766  df-eu 2244  df-mo 2245  df-clab 2390  df-cleq 2396  df-clel 2399  df-nfc 2554  df-ne 2602  df-ral 2761  df-rex 2762  df-reu 2763  df-rmo 2764  df-rab 2765  df-v 3063  df-sbc 3280  df-csb 3376  df-dif 3419  df-un 3421  df-in 3423  df-ss 3430  df-pss 3432  df-nul 3741  df-if 3888  df-pw 3959  df-sn 3975  df-pr 3977  df-tp 3979  df-op 3981  df-uni 4194  df-iun 4275  df-br 4398  df-opab 4456  df-mpt 4457  df-tr 4492  df-eprel 4736  df-id 4740  df-po 4746  df-so 4747  df-fr 4784  df-we 4786  df-xp 4831  df-rel 4832  df-cnv 4833  df-co 4834  df-dm 4835  df-rn 4836  df-res 4837  df-ima 4838  df-pred 5369  df-ord 5415  df-on 5416  df-lim 5417  df-suc 5418  df-iota 5535  df-fun 5573  df-fn 5574  df-f 5575  df-f1 5576  df-fo 5577  df-f1o 5578  df-fv 5579  df-ov 6283  df-oprab 6284  df-mpt2 6285  df-om 6686  df-1st 6786  df-2nd 6787  df-wrecs 7015  df-recs 7077  df-rdg 7115  df-1o 7169  df-oadd 7173  df-omul 7174  df-er 7350  df-ni 9282  df-mi 9284  df-lti 9285  df-mpq 9319  df-ltpq 9320  df-enq 9321  df-nq 9322  df-erq 9323  df-mq 9325  df-1nq 9326  df-ltnq 9328
This theorem is referenced by:  ltaddnq  9384  ltrnq  9389  addclprlem1  9426  mulclprlem  9429  mulclpr  9430  distrlem4pr  9436  1idpr  9439  prlem934  9443  prlem936  9457  reclem3pr  9459  reclem4pr  9460
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