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Theorem ltanq 8590
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 8567 . . 3  |-  +Q  :
( Q.  X.  Q. )
--> Q.
21fdmi 5359 . 2  |-  dom  +Q  =  ( Q.  X.  Q. )
3 ltrelnq 8545 . 2  |-  <Q  C_  ( Q.  X.  Q. )
4 0nnq 8543 . 2  |-  -.  (/)  e.  Q.
5 ordpinq 8562 . . . 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 8544 . . . . . . 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 8544 . . . . . . 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 8555 . . . . . 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 8544 . . . . . . 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 8555 . . . . . 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 4037 . . . 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 8557 . . . . . . . 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 8557 . . . . . . . 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 4037 . . . . 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 8589 . . . . 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 6111 . . . . . . . . . 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 8512 . . . . . . . . 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 8523 . . . . . . . 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 6111 . . . . . . . . . . 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 8512 . . . . . . . . . 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 6110 . . . . . . . . . . 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 6111 . . . . . . . . . . 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 8512 . . . . . . . . . 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 8512 . . . . . . . . 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 8522 . . . . . . . 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 8515 . . . . . . . . . 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 5499 . . . . . . . . . . 11  |-  ( 1st `  A )  e.  _V
50 fvex 5499 . . . . . . . . . . 11  |-  ( 2nd `  B )  e.  _V
51 fvex 5499 . . . . . . . . . . 11  |-  ( 2nd `  C )  e.  _V
52 mulcompi 8515 . . . . . . . . . . 11  |-  ( x  .N  y )  =  ( y  .N  x
)
53 mulasspi 8516 . . . . . . . . . . 11  |-  ( ( x  .N  y )  .N  z )  =  ( x  .N  (
y  .N  z ) )
5449, 50, 51, 52, 53, 51caov411 6013 . . . . . . . . . 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 2304 . . . . . . . . 9  |-  ( ( ( 2nd `  C
)  .N  ( 2nd `  C ) )  .N  ( ( 1st `  A
)  .N  ( 2nd `  B ) ) )  =  ( ( ( 2nd `  C )  .N  ( 2nd `  B
) )  .N  (
( 1st `  A
)  .N  ( 2nd `  C ) ) )
5655oveq2i 5830 . . . . . . . 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 8517 . . . . . . . 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 8515 . . . . . . . 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 2310 . . . . . . 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 8515 . . . . . . . . . 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 5499 . . . . . . . . . . 11  |-  ( 1st `  C )  e.  _V
62 fvex 5499 . . . . . . . . . . 11  |-  ( 2nd `  A )  e.  _V
6361, 62, 51, 52, 53, 50caov411 6013 . . . . . . . . . 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 2304 . . . . . . . . 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 8515 . . . . . . . . . 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 5499 . . . . . . . . . . 11  |-  ( 1st `  B )  e.  _V
6766, 62, 51, 52, 53, 51caov411 6013 . . . . . . . . . 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 2304 . . . . . . . . 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 5831 . . . . . . . 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 8517 . . . . . . . 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 8515 . . . . . . . 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 2310 . . . . . . 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 4033 . . . . . 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 8561 . . . . 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 5971 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 1628    e. wcel 1688   <.cop 3644   class class class wbr 4024    X. cxp 4686   ` cfv 5221  (class class class)co 5819   1stc1st 6081   2ndc2nd 6082   N.cnpi 8461    +N cpli 8462    .N cmi 8463    <N clti 8464    +pQ cplpq 8465    <pQ cltpq 8467   Q.cnq 8469   /Qcerq 8471    +Q cplq 8472    <Q cltq 8475
This theorem is referenced by:  ltaddnq  8593  ltbtwnnq  8597  addclpr  8637  distrlem4pr  8645  ltexprlem3  8657  ltexprlem4  8658  ltexprlem6  8660  prlem936  8666
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-gen 1538  ax-5 1549  ax-17 1608  ax-9 1641  ax-8 1648  ax-13 1690  ax-14 1692  ax-6 1707  ax-7 1712  ax-11 1719  ax-12 1869  ax-ext 2265  ax-sep 4142  ax-nul 4150  ax-pr 4213  ax-un 4511
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 1534  df-nf 1537  df-sb 1636  df-eu 2148  df-mo 2149  df-clab 2271  df-cleq 2277  df-clel 2280  df-nfc 2409  df-ne 2449  df-ral 2549  df-rex 2550  df-reu 2551  df-rmo 2552  df-rab 2553  df-v 2791  df-sbc 2993  df-csb 3083  df-dif 3156  df-un 3158  df-in 3160  df-ss 3167  df-pss 3169  df-nul 3457  df-if 3567  df-pw 3628  df-sn 3647  df-pr 3648  df-tp 3649  df-op 3650  df-uni 3829  df-iun 3908  df-br 4025  df-opab 4079  df-mpt 4080  df-tr 4115  df-eprel 4304  df-id 4308  df-po 4313  df-so 4314  df-fr 4351  df-we 4353  df-ord 4394  df-on 4395  df-lim 4396  df-suc 4397  df-om 4656  df-xp 4694  df-rel 4695  df-cnv 4696  df-co 4697  df-dm 4698  df-rn 4699  df-res 4700  df-ima 4701  df-fun 5223  df-fn 5224  df-f 5225  df-f1 5226  df-fo 5227  df-f1o 5228  df-fv 5229  df-ov 5822  df-oprab 5823  df-mpt2 5824  df-1st 6083  df-2nd 6084  df-recs 6383  df-rdg 6418  df-1o 6474  df-oadd 6478  df-omul 6479  df-er 6655  df-ni 8491  df-pli 8492  df-mi 8493  df-lti 8494  df-plpq 8527  df-ltpq 8529  df-enq 8530  df-nq 8531  df-erq 8532  df-plq 8533  df-1nq 8535  df-ltnq 8537
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