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Theorem ltanq 8563
Description: Ordering property of addition for positive fractions. Proposition 9-2.6(ii) of [Gleason] p. 120. (Contributed by NM, 6-Mar-1996.) (Revised by Mario Carneiro, 10-May-2013.) (New usage is discouraged.)
Assertion
Ref Expression
ltanq  |-  ( C  e.  Q.  ->  ( A  <Q  B  <->  ( C  +Q  A )  <Q  ( C  +Q  B ) ) )

Proof of Theorem ltanq
StepHypRef Expression
1 addnqf 8540 . . 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 ordpinq 8535 . . . 4  |-  ( ( A  e.  Q.  /\  B  e.  Q. )  ->  ( A  <Q  B  <->  ( ( 1st `  A )  .N  ( 2nd `  B
) )  <N  (
( 1st `  B
)  .N  ( 2nd `  A ) ) ) )
653adant3 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 ) ) ) )
7 elpqn 8517 . . . . . . 7  |-  ( C  e.  Q.  ->  C  e.  ( N.  X.  N. ) )
873ad2ant3 983 . . . . . 6  |-  ( ( A  e.  Q.  /\  B  e.  Q.  /\  C  e.  Q. )  ->  C  e.  ( N.  X.  N. ) )
9 elpqn 8517 . . . . . . 7  |-  ( A  e.  Q.  ->  A  e.  ( N.  X.  N. ) )
1093ad2ant1 981 . . . . . 6  |-  ( ( A  e.  Q.  /\  B  e.  Q.  /\  C  e.  Q. )  ->  A  e.  ( N.  X.  N. ) )
11 addpipq2 8528 . . . . . 6  |-  ( ( C  e.  ( N. 
X.  N. )  /\  A  e.  ( N.  X.  N. ) )  ->  ( C  +pQ  A )  = 
<. ( ( ( 1st `  C )  .N  ( 2nd `  A ) )  +N  ( ( 1st `  A )  .N  ( 2nd `  C ) ) ) ,  ( ( 2nd `  C )  .N  ( 2nd `  A
) ) >. )
128, 10, 11syl2anc 645 . . . . 5  |-  ( ( A  e.  Q.  /\  B  e.  Q.  /\  C  e.  Q. )  ->  ( C  +pQ  A )  = 
<. ( ( ( 1st `  C )  .N  ( 2nd `  A ) )  +N  ( ( 1st `  A )  .N  ( 2nd `  C ) ) ) ,  ( ( 2nd `  C )  .N  ( 2nd `  A
) ) >. )
13 elpqn 8517 . . . . . . 7  |-  ( B  e.  Q.  ->  B  e.  ( N.  X.  N. ) )
14133ad2ant2 982 . . . . . 6  |-  ( ( A  e.  Q.  /\  B  e.  Q.  /\  C  e.  Q. )  ->  B  e.  ( N.  X.  N. ) )
15 addpipq2 8528 . . . . . 6  |-  ( ( C  e.  ( N. 
X.  N. )  /\  B  e.  ( N.  X.  N. ) )  ->  ( C  +pQ  B )  = 
<. ( ( ( 1st `  C )  .N  ( 2nd `  B ) )  +N  ( ( 1st `  B )  .N  ( 2nd `  C ) ) ) ,  ( ( 2nd `  C )  .N  ( 2nd `  B
) ) >. )
168, 14, 15syl2anc 645 . . . . 5  |-  ( ( A  e.  Q.  /\  B  e.  Q.  /\  C  e.  Q. )  ->  ( C  +pQ  B )  = 
<. ( ( ( 1st `  C )  .N  ( 2nd `  B ) )  +N  ( ( 1st `  B )  .N  ( 2nd `  C ) ) ) ,  ( ( 2nd `  C )  .N  ( 2nd `  B
) ) >. )
1712, 16breq12d 4010 . . . 4  |-  ( ( A  e.  Q.  /\  B  e.  Q.  /\  C  e.  Q. )  ->  (
( C  +pQ  A
)  <pQ  ( C  +pQ  B )  <->  <. ( ( ( 1st `  C )  .N  ( 2nd `  A
) )  +N  (
( 1st `  A
)  .N  ( 2nd `  C ) ) ) ,  ( ( 2nd `  C )  .N  ( 2nd `  A ) )
>.  <pQ  <. ( ( ( 1st `  C )  .N  ( 2nd `  B
) )  +N  (
( 1st `  B
)  .N  ( 2nd `  C ) ) ) ,  ( ( 2nd `  C )  .N  ( 2nd `  B ) )
>. ) )
18 addpqnq 8530 . . . . . . . 8  |-  ( ( C  e.  Q.  /\  A  e.  Q. )  ->  ( C  +Q  A
)  =  ( /Q
`  ( C  +pQ  A ) ) )
1918ancoms 441 . . . . . . 7  |-  ( ( A  e.  Q.  /\  C  e.  Q. )  ->  ( C  +Q  A
)  =  ( /Q
`  ( C  +pQ  A ) ) )
20193adant2 979 . . . . . 6  |-  ( ( A  e.  Q.  /\  B  e.  Q.  /\  C  e.  Q. )  ->  ( C  +Q  A )  =  ( /Q `  ( C  +pQ  A ) ) )
21 addpqnq 8530 . . . . . . . 8  |-  ( ( C  e.  Q.  /\  B  e.  Q. )  ->  ( C  +Q  B
)  =  ( /Q
`  ( C  +pQ  B ) ) )
2221ancoms 441 . . . . . . 7  |-  ( ( B  e.  Q.  /\  C  e.  Q. )  ->  ( C  +Q  B
)  =  ( /Q
`  ( C  +pQ  B ) ) )
23223adant1 978 . . . . . 6  |-  ( ( A  e.  Q.  /\  B  e.  Q.  /\  C  e.  Q. )  ->  ( C  +Q  B )  =  ( /Q `  ( C  +pQ  B ) ) )
2420, 23breq12d 4010 . . . . 5  |-  ( ( 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 ) ) ) )
25 lterpq 8562 . . . . 5  |-  ( ( C  +pQ  A ) 
<pQ  ( C  +pQ  B
)  <->  ( /Q `  ( C  +pQ  A ) )  <Q  ( /Q `  ( C  +pQ  B
) ) )
2624, 25syl6bbr 256 . . . 4  |-  ( ( A  e.  Q.  /\  B  e.  Q.  /\  C  e.  Q. )  ->  (
( C  +Q  A
)  <Q  ( C  +Q  B )  <->  ( C  +pQ  A )  <pQ  ( C 
+pQ  B ) ) )
27 xp2nd 6084 . . . . . . . . . 10  |-  ( C  e.  ( N.  X.  N. )  ->  ( 2nd `  C )  e.  N. )
288, 27syl 17 . . . . . . . . 9  |-  ( ( A  e.  Q.  /\  B  e.  Q.  /\  C  e.  Q. )  ->  ( 2nd `  C )  e. 
N. )
29 mulclpi 8485 . . . . . . . . 9  |-  ( ( ( 2nd `  C
)  e.  N.  /\  ( 2nd `  C )  e.  N. )  -> 
( ( 2nd `  C
)  .N  ( 2nd `  C ) )  e. 
N. )
3028, 28, 29syl2anc 645 . . . . . . . 8  |-  ( ( A  e.  Q.  /\  B  e.  Q.  /\  C  e.  Q. )  ->  (
( 2nd `  C
)  .N  ( 2nd `  C ) )  e. 
N. )
31 ltmpi 8496 . . . . . . . 8  |-  ( ( ( 2nd `  C
)  .N  ( 2nd `  C ) )  e. 
N.  ->  ( ( ( 1st `  A )  .N  ( 2nd `  B
) )  <N  (
( 1st `  B
)  .N  ( 2nd `  A ) )  <->  ( (
( 2nd `  C
)  .N  ( 2nd `  C ) )  .N  ( ( 1st `  A
)  .N  ( 2nd `  B ) ) ) 
<N  ( ( ( 2nd `  C )  .N  ( 2nd `  C ) )  .N  ( ( 1st `  B )  .N  ( 2nd `  A ) ) ) ) )
3230, 31syl 17 . . . . . . 7  |-  ( ( A  e.  Q.  /\  B  e.  Q.  /\  C  e.  Q. )  ->  (
( ( 1st `  A
)  .N  ( 2nd `  B ) )  <N 
( ( 1st `  B
)  .N  ( 2nd `  A ) )  <->  ( (
( 2nd `  C
)  .N  ( 2nd `  C ) )  .N  ( ( 1st `  A
)  .N  ( 2nd `  B ) ) ) 
<N  ( ( ( 2nd `  C )  .N  ( 2nd `  C ) )  .N  ( ( 1st `  B )  .N  ( 2nd `  A ) ) ) ) )
33 xp2nd 6084 . . . . . . . . . . 11  |-  ( B  e.  ( N.  X.  N. )  ->  ( 2nd `  B )  e.  N. )
3414, 33syl 17 . . . . . . . . . 10  |-  ( ( A  e.  Q.  /\  B  e.  Q.  /\  C  e.  Q. )  ->  ( 2nd `  B )  e. 
N. )
35 mulclpi 8485 . . . . . . . . . 10  |-  ( ( ( 2nd `  C
)  e.  N.  /\  ( 2nd `  B )  e.  N. )  -> 
( ( 2nd `  C
)  .N  ( 2nd `  B ) )  e. 
N. )
3628, 34, 35syl2anc 645 . . . . . . . . 9  |-  ( ( A  e.  Q.  /\  B  e.  Q.  /\  C  e.  Q. )  ->  (
( 2nd `  C
)  .N  ( 2nd `  B ) )  e. 
N. )
37 xp1st 6083 . . . . . . . . . . 11  |-  ( C  e.  ( N.  X.  N. )  ->  ( 1st `  C )  e.  N. )
388, 37syl 17 . . . . . . . . . 10  |-  ( ( A  e.  Q.  /\  B  e.  Q.  /\  C  e.  Q. )  ->  ( 1st `  C )  e. 
N. )
39 xp2nd 6084 . . . . . . . . . . 11  |-  ( A  e.  ( N.  X.  N. )  ->  ( 2nd `  A )  e.  N. )
4010, 39syl 17 . . . . . . . . . 10  |-  ( ( A  e.  Q.  /\  B  e.  Q.  /\  C  e.  Q. )  ->  ( 2nd `  A )  e. 
N. )
41 mulclpi 8485 . . . . . . . . . 10  |-  ( ( ( 1st `  C
)  e.  N.  /\  ( 2nd `  A )  e.  N. )  -> 
( ( 1st `  C
)  .N  ( 2nd `  A ) )  e. 
N. )
4238, 40, 41syl2anc 645 . . . . . . . . 9  |-  ( ( A  e.  Q.  /\  B  e.  Q.  /\  C  e.  Q. )  ->  (
( 1st `  C
)  .N  ( 2nd `  A ) )  e. 
N. )
43 mulclpi 8485 . . . . . . . . 9  |-  ( ( ( ( 2nd `  C
)  .N  ( 2nd `  B ) )  e. 
N.  /\  ( ( 1st `  C )  .N  ( 2nd `  A
) )  e.  N. )  ->  ( ( ( 2nd `  C )  .N  ( 2nd `  B
) )  .N  (
( 1st `  C
)  .N  ( 2nd `  A ) ) )  e.  N. )
4436, 42, 43syl2anc 645 . . . . . . . 8  |-  ( ( A  e.  Q.  /\  B  e.  Q.  /\  C  e.  Q. )  ->  (
( ( 2nd `  C
)  .N  ( 2nd `  B ) )  .N  ( ( 1st `  C
)  .N  ( 2nd `  A ) ) )  e.  N. )
45 ltapi 8495 . . . . . . . 8  |-  ( ( ( ( 2nd `  C
)  .N  ( 2nd `  B ) )  .N  ( ( 1st `  C
)  .N  ( 2nd `  A ) ) )  e.  N.  ->  (
( ( ( 2nd `  C )  .N  ( 2nd `  C ) )  .N  ( ( 1st `  A )  .N  ( 2nd `  B ) ) )  <N  ( (
( 2nd `  C
)  .N  ( 2nd `  C ) )  .N  ( ( 1st `  B
)  .N  ( 2nd `  A ) ) )  <-> 
( ( ( ( 2nd `  C )  .N  ( 2nd `  B
) )  .N  (
( 1st `  C
)  .N  ( 2nd `  A ) ) )  +N  ( ( ( 2nd `  C )  .N  ( 2nd `  C
) )  .N  (
( 1st `  A
)  .N  ( 2nd `  B ) ) ) )  <N  ( (
( ( 2nd `  C
)  .N  ( 2nd `  B ) )  .N  ( ( 1st `  C
)  .N  ( 2nd `  A ) ) )  +N  ( ( ( 2nd `  C )  .N  ( 2nd `  C
) )  .N  (
( 1st `  B
)  .N  ( 2nd `  A ) ) ) ) ) )
4644, 45syl 17 . . . . . . 7  |-  ( ( A  e.  Q.  /\  B  e.  Q.  /\  C  e.  Q. )  ->  (
( ( ( 2nd `  C )  .N  ( 2nd `  C ) )  .N  ( ( 1st `  A )  .N  ( 2nd `  B ) ) )  <N  ( (
( 2nd `  C
)  .N  ( 2nd `  C ) )  .N  ( ( 1st `  B
)  .N  ( 2nd `  A ) ) )  <-> 
( ( ( ( 2nd `  C )  .N  ( 2nd `  B
) )  .N  (
( 1st `  C
)  .N  ( 2nd `  A ) ) )  +N  ( ( ( 2nd `  C )  .N  ( 2nd `  C
) )  .N  (
( 1st `  A
)  .N  ( 2nd `  B ) ) ) )  <N  ( (
( ( 2nd `  C
)  .N  ( 2nd `  B ) )  .N  ( ( 1st `  C
)  .N  ( 2nd `  A ) ) )  +N  ( ( ( 2nd `  C )  .N  ( 2nd `  C
) )  .N  (
( 1st `  B
)  .N  ( 2nd `  A ) ) ) ) ) )
4732, 46bitrd 246 . . . . . 6  |-  ( ( A  e.  Q.  /\  B  e.  Q.  /\  C  e.  Q. )  ->  (
( ( 1st `  A
)  .N  ( 2nd `  B ) )  <N 
( ( 1st `  B
)  .N  ( 2nd `  A ) )  <->  ( (
( ( 2nd `  C
)  .N  ( 2nd `  B ) )  .N  ( ( 1st `  C
)  .N  ( 2nd `  A ) ) )  +N  ( ( ( 2nd `  C )  .N  ( 2nd `  C
) )  .N  (
( 1st `  A
)  .N  ( 2nd `  B ) ) ) )  <N  ( (
( ( 2nd `  C
)  .N  ( 2nd `  B ) )  .N  ( ( 1st `  C
)  .N  ( 2nd `  A ) ) )  +N  ( ( ( 2nd `  C )  .N  ( 2nd `  C
) )  .N  (
( 1st `  B
)  .N  ( 2nd `  A ) ) ) ) ) )
48 mulcompi 8488 . . . . . . . . . 10  |-  ( ( ( 2nd `  C
)  .N  ( 2nd `  C ) )  .N  ( ( 1st `  A
)  .N  ( 2nd `  B ) ) )  =  ( ( ( 1st `  A )  .N  ( 2nd `  B
) )  .N  (
( 2nd `  C
)  .N  ( 2nd `  C ) ) )
49 fvex 5472 . . . . . . . . . . 11  |-  ( 1st `  A )  e.  _V
50 fvex 5472 . . . . . . . . . . 11  |-  ( 2nd `  B )  e.  _V
51 fvex 5472 . . . . . . . . . . 11  |-  ( 2nd `  C )  e.  _V
52 mulcompi 8488 . . . . . . . . . . 11  |-  ( x  .N  y )  =  ( y  .N  x
)
53 mulasspi 8489 . . . . . . . . . . 11  |-  ( ( x  .N  y )  .N  z )  =  ( x  .N  (
y  .N  z ) )
5449, 50, 51, 52, 53, 51caov411 5986 . . . . . . . . . 10  |-  ( ( ( 1st `  A
)  .N  ( 2nd `  B ) )  .N  ( ( 2nd `  C
)  .N  ( 2nd `  C ) ) )  =  ( ( ( 2nd `  C )  .N  ( 2nd `  B
) )  .N  (
( 1st `  A
)  .N  ( 2nd `  C ) ) )
5548, 54eqtri 2278 . . . . . . . . 9  |-  ( ( ( 2nd `  C
)  .N  ( 2nd `  C ) )  .N  ( ( 1st `  A
)  .N  ( 2nd `  B ) ) )  =  ( ( ( 2nd `  C )  .N  ( 2nd `  B
) )  .N  (
( 1st `  A
)  .N  ( 2nd `  C ) ) )
5655oveq2i 5803 . . . . . . . 8  |-  ( ( ( ( 2nd `  C
)  .N  ( 2nd `  B ) )  .N  ( ( 1st `  C
)  .N  ( 2nd `  A ) ) )  +N  ( ( ( 2nd `  C )  .N  ( 2nd `  C
) )  .N  (
( 1st `  A
)  .N  ( 2nd `  B ) ) ) )  =  ( ( ( ( 2nd `  C
)  .N  ( 2nd `  B ) )  .N  ( ( 1st `  C
)  .N  ( 2nd `  A ) ) )  +N  ( ( ( 2nd `  C )  .N  ( 2nd `  B
) )  .N  (
( 1st `  A
)  .N  ( 2nd `  C ) ) ) )
57 distrpi 8490 . . . . . . . 8  |-  ( ( ( 2nd `  C
)  .N  ( 2nd `  B ) )  .N  ( ( ( 1st `  C )  .N  ( 2nd `  A ) )  +N  ( ( 1st `  A )  .N  ( 2nd `  C ) ) ) )  =  ( ( ( ( 2nd `  C )  .N  ( 2nd `  B ) )  .N  ( ( 1st `  C )  .N  ( 2nd `  A ) ) )  +N  ( ( ( 2nd `  C
)  .N  ( 2nd `  B ) )  .N  ( ( 1st `  A
)  .N  ( 2nd `  C ) ) ) )
58 mulcompi 8488 . . . . . . . 8  |-  ( ( ( 2nd `  C
)  .N  ( 2nd `  B ) )  .N  ( ( ( 1st `  C )  .N  ( 2nd `  A ) )  +N  ( ( 1st `  A )  .N  ( 2nd `  C ) ) ) )  =  ( ( ( ( 1st `  C )  .N  ( 2nd `  A ) )  +N  ( ( 1st `  A )  .N  ( 2nd `  C ) ) )  .N  ( ( 2nd `  C )  .N  ( 2nd `  B
) ) )
5956, 57, 583eqtr2i 2284 . . . . . . 7  |-  ( ( ( ( 2nd `  C
)  .N  ( 2nd `  B ) )  .N  ( ( 1st `  C
)  .N  ( 2nd `  A ) ) )  +N  ( ( ( 2nd `  C )  .N  ( 2nd `  C
) )  .N  (
( 1st `  A
)  .N  ( 2nd `  B ) ) ) )  =  ( ( ( ( 1st `  C
)  .N  ( 2nd `  A ) )  +N  ( ( 1st `  A
)  .N  ( 2nd `  C ) ) )  .N  ( ( 2nd `  C )  .N  ( 2nd `  B ) ) )
60 mulcompi 8488 . . . . . . . . . 10  |-  ( ( ( 2nd `  C
)  .N  ( 2nd `  B ) )  .N  ( ( 1st `  C
)  .N  ( 2nd `  A ) ) )  =  ( ( ( 1st `  C )  .N  ( 2nd `  A
) )  .N  (
( 2nd `  C
)  .N  ( 2nd `  B ) ) )
61 fvex 5472 . . . . . . . . . . 11  |-  ( 1st `  C )  e.  _V
62 fvex 5472 . . . . . . . . . . 11  |-  ( 2nd `  A )  e.  _V
6361, 62, 51, 52, 53, 50caov411 5986 . . . . . . . . . 10  |-  ( ( ( 1st `  C
)  .N  ( 2nd `  A ) )  .N  ( ( 2nd `  C
)  .N  ( 2nd `  B ) ) )  =  ( ( ( 2nd `  C )  .N  ( 2nd `  A
) )  .N  (
( 1st `  C
)  .N  ( 2nd `  B ) ) )
6460, 63eqtri 2278 . . . . . . . . 9  |-  ( ( ( 2nd `  C
)  .N  ( 2nd `  B ) )  .N  ( ( 1st `  C
)  .N  ( 2nd `  A ) ) )  =  ( ( ( 2nd `  C )  .N  ( 2nd `  A
) )  .N  (
( 1st `  C
)  .N  ( 2nd `  B ) ) )
65 mulcompi 8488 . . . . . . . . . 10  |-  ( ( ( 2nd `  C
)  .N  ( 2nd `  C ) )  .N  ( ( 1st `  B
)  .N  ( 2nd `  A ) ) )  =  ( ( ( 1st `  B )  .N  ( 2nd `  A
) )  .N  (
( 2nd `  C
)  .N  ( 2nd `  C ) ) )
66 fvex 5472 . . . . . . . . . . 11  |-  ( 1st `  B )  e.  _V
6766, 62, 51, 52, 53, 51caov411 5986 . . . . . . . . . 10  |-  ( ( ( 1st `  B
)  .N  ( 2nd `  A ) )  .N  ( ( 2nd `  C
)  .N  ( 2nd `  C ) ) )  =  ( ( ( 2nd `  C )  .N  ( 2nd `  A
) )  .N  (
( 1st `  B
)  .N  ( 2nd `  C ) ) )
6865, 67eqtri 2278 . . . . . . . . 9  |-  ( ( ( 2nd `  C
)  .N  ( 2nd `  C ) )  .N  ( ( 1st `  B
)  .N  ( 2nd `  A ) ) )  =  ( ( ( 2nd `  C )  .N  ( 2nd `  A
) )  .N  (
( 1st `  B
)  .N  ( 2nd `  C ) ) )
6964, 68oveq12i 5804 . . . . . . . 8  |-  ( ( ( ( 2nd `  C
)  .N  ( 2nd `  B ) )  .N  ( ( 1st `  C
)  .N  ( 2nd `  A ) ) )  +N  ( ( ( 2nd `  C )  .N  ( 2nd `  C
) )  .N  (
( 1st `  B
)  .N  ( 2nd `  A ) ) ) )  =  ( ( ( ( 2nd `  C
)  .N  ( 2nd `  A ) )  .N  ( ( 1st `  C
)  .N  ( 2nd `  B ) ) )  +N  ( ( ( 2nd `  C )  .N  ( 2nd `  A
) )  .N  (
( 1st `  B
)  .N  ( 2nd `  C ) ) ) )
70 distrpi 8490 . . . . . . . 8  |-  ( ( ( 2nd `  C
)  .N  ( 2nd `  A ) )  .N  ( ( ( 1st `  C )  .N  ( 2nd `  B ) )  +N  ( ( 1st `  B )  .N  ( 2nd `  C ) ) ) )  =  ( ( ( ( 2nd `  C )  .N  ( 2nd `  A ) )  .N  ( ( 1st `  C )  .N  ( 2nd `  B ) ) )  +N  ( ( ( 2nd `  C
)  .N  ( 2nd `  A ) )  .N  ( ( 1st `  B
)  .N  ( 2nd `  C ) ) ) )
71 mulcompi 8488 . . . . . . . 8  |-  ( ( ( 2nd `  C
)  .N  ( 2nd `  A ) )  .N  ( ( ( 1st `  C )  .N  ( 2nd `  B ) )  +N  ( ( 1st `  B )  .N  ( 2nd `  C ) ) ) )  =  ( ( ( ( 1st `  C )  .N  ( 2nd `  B ) )  +N  ( ( 1st `  B )  .N  ( 2nd `  C ) ) )  .N  ( ( 2nd `  C )  .N  ( 2nd `  A
) ) )
7269, 70, 713eqtr2i 2284 . . . . . . 7  |-  ( ( ( ( 2nd `  C
)  .N  ( 2nd `  B ) )  .N  ( ( 1st `  C
)  .N  ( 2nd `  A ) ) )  +N  ( ( ( 2nd `  C )  .N  ( 2nd `  C
) )  .N  (
( 1st `  B
)  .N  ( 2nd `  A ) ) ) )  =  ( ( ( ( 1st `  C
)  .N  ( 2nd `  B ) )  +N  ( ( 1st `  B
)  .N  ( 2nd `  C ) ) )  .N  ( ( 2nd `  C )  .N  ( 2nd `  A ) ) )
7359, 72breq12i 4006 . . . . . 6  |-  ( ( ( ( ( 2nd `  C )  .N  ( 2nd `  B ) )  .N  ( ( 1st `  C )  .N  ( 2nd `  A ) ) )  +N  ( ( ( 2nd `  C
)  .N  ( 2nd `  C ) )  .N  ( ( 1st `  A
)  .N  ( 2nd `  B ) ) ) )  <N  ( (
( ( 2nd `  C
)  .N  ( 2nd `  B ) )  .N  ( ( 1st `  C
)  .N  ( 2nd `  A ) ) )  +N  ( ( ( 2nd `  C )  .N  ( 2nd `  C
) )  .N  (
( 1st `  B
)  .N  ( 2nd `  A ) ) ) )  <->  ( ( ( ( 1st `  C
)  .N  ( 2nd `  A ) )  +N  ( ( 1st `  A
)  .N  ( 2nd `  C ) ) )  .N  ( ( 2nd `  C )  .N  ( 2nd `  B ) ) )  <N  ( (
( ( 1st `  C
)  .N  ( 2nd `  B ) )  +N  ( ( 1st `  B
)  .N  ( 2nd `  C ) ) )  .N  ( ( 2nd `  C )  .N  ( 2nd `  A ) ) ) )
7447, 73syl6bb 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  ( 2nd `  A ) )  +N  ( ( 1st `  A
)  .N  ( 2nd `  C ) ) )  .N  ( ( 2nd `  C )  .N  ( 2nd `  B ) ) )  <N  ( (
( ( 1st `  C
)  .N  ( 2nd `  B ) )  +N  ( ( 1st `  B
)  .N  ( 2nd `  C ) ) )  .N  ( ( 2nd `  C )  .N  ( 2nd `  A ) ) ) ) )
75 ordpipq 8534 . . . . 5  |-  ( <.
( ( ( 1st `  C )  .N  ( 2nd `  A ) )  +N  ( ( 1st `  A )  .N  ( 2nd `  C ) ) ) ,  ( ( 2nd `  C )  .N  ( 2nd `  A
) ) >.  <pQ  <. (
( ( 1st `  C
)  .N  ( 2nd `  B ) )  +N  ( ( 1st `  B
)  .N  ( 2nd `  C ) ) ) ,  ( ( 2nd `  C )  .N  ( 2nd `  B ) )
>. 
<->  ( ( ( ( 1st `  C )  .N  ( 2nd `  A
) )  +N  (
( 1st `  A
)  .N  ( 2nd `  C ) ) )  .N  ( ( 2nd `  C )  .N  ( 2nd `  B ) ) )  <N  ( (
( ( 1st `  C
)  .N  ( 2nd `  B ) )  +N  ( ( 1st `  B
)  .N  ( 2nd `  C ) ) )  .N  ( ( 2nd `  C )  .N  ( 2nd `  A ) ) ) )
7674, 75syl6bbr 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  ( 2nd `  A ) )  +N  ( ( 1st `  A
)  .N  ( 2nd `  C ) ) ) ,  ( ( 2nd `  C )  .N  ( 2nd `  A ) )
>.  <pQ  <. ( ( ( 1st `  C )  .N  ( 2nd `  B
) )  +N  (
( 1st `  B
)  .N  ( 2nd `  C ) ) ) ,  ( ( 2nd `  C )  .N  ( 2nd `  B ) )
>. ) )
7717, 26, 763bitr4rd 279 . . 3  |-  ( ( A  e.  Q.  /\  B  e.  Q.  /\  C  e.  Q. )  ->  (
( ( 1st `  A
)  .N  ( 2nd `  B ) )  <N 
( ( 1st `  B
)  .N  ( 2nd `  A ) )  <->  ( C  +Q  A )  <Q  ( C  +Q  B ) ) )
786, 77bitrd 246 . 2  |-  ( ( A  e.  Q.  /\  B  e.  Q.  /\  C  e.  Q. )  ->  ( A  <Q  B  <->  ( C  +Q  A )  <Q  ( C  +Q  B ) ) )
792, 3, 4, 78ndmovord 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 cpli 8435    .N cmi 8436    <N clti 8437    +pQ cplpq 8438    <pQ cltpq 8440   Q.cnq 8442   /Qcerq 8444    +Q cplq 8445    <Q cltq 8448
This theorem is referenced by:  ltaddnq  8566  ltbtwnnq  8570  addclpr  8610  distrlem4pr  8618  ltexprlem3  8630  ltexprlem4  8631  ltexprlem6  8633  prlem936  8639
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-pli 8465  df-mi 8466  df-lti 8467  df-plpq 8500  df-ltpq 8502  df-enq 8503  df-nq 8504  df-erq 8505  df-plq 8506  df-1nq 8508  df-ltnq 8510
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