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Theorem ltanq 8595
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
Dummy variables  x  y  z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 addnqf 8572 . . 3  |-  +Q  :
( Q.  X.  Q. )
--> Q.
21fdmi 5394 . 2  |-  dom  +Q  =  ( Q.  X.  Q. )
3 ltrelnq 8550 . 2  |-  <Q  C_  ( Q.  X.  Q. )
4 0nnq 8548 . 2  |-  -.  (/)  e.  Q.
5 ordpinq 8567 . . . 4  |-  ( ( A  e.  Q.  /\  B  e.  Q. )  ->  ( A  <Q  B  <->  ( ( 1st `  A )  .N  ( 2nd `  B
) )  <N  (
( 1st `  B
)  .N  ( 2nd `  A ) ) ) )
653adant3 975 . . 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 8549 . . . . . . 7  |-  ( C  e.  Q.  ->  C  e.  ( N.  X.  N. ) )
873ad2ant3 978 . . . . . 6  |-  ( ( A  e.  Q.  /\  B  e.  Q.  /\  C  e.  Q. )  ->  C  e.  ( N.  X.  N. ) )
9 elpqn 8549 . . . . . . 7  |-  ( A  e.  Q.  ->  A  e.  ( N.  X.  N. ) )
1093ad2ant1 976 . . . . . 6  |-  ( ( A  e.  Q.  /\  B  e.  Q.  /\  C  e.  Q. )  ->  A  e.  ( N.  X.  N. ) )
11 addpipq2 8560 . . . . . 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 642 . . . . 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 8549 . . . . . . 7  |-  ( B  e.  Q.  ->  B  e.  ( N.  X.  N. ) )
14133ad2ant2 977 . . . . . 6  |-  ( ( A  e.  Q.  /\  B  e.  Q.  /\  C  e.  Q. )  ->  B  e.  ( N.  X.  N. ) )
15 addpipq2 8560 . . . . . 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 642 . . . . 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 4036 . . . 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 8562 . . . . . . . 8  |-  ( ( C  e.  Q.  /\  A  e.  Q. )  ->  ( C  +Q  A
)  =  ( /Q
`  ( C  +pQ  A ) ) )
1918ancoms 439 . . . . . . 7  |-  ( ( A  e.  Q.  /\  C  e.  Q. )  ->  ( C  +Q  A
)  =  ( /Q
`  ( C  +pQ  A ) ) )
20193adant2 974 . . . . . 6  |-  ( ( A  e.  Q.  /\  B  e.  Q.  /\  C  e.  Q. )  ->  ( C  +Q  A )  =  ( /Q `  ( C  +pQ  A ) ) )
21 addpqnq 8562 . . . . . . . 8  |-  ( ( C  e.  Q.  /\  B  e.  Q. )  ->  ( C  +Q  B
)  =  ( /Q
`  ( C  +pQ  B ) ) )
2221ancoms 439 . . . . . . 7  |-  ( ( B  e.  Q.  /\  C  e.  Q. )  ->  ( C  +Q  B
)  =  ( /Q
`  ( C  +pQ  B ) ) )
23223adant1 973 . . . . . 6  |-  ( ( A  e.  Q.  /\  B  e.  Q.  /\  C  e.  Q. )  ->  ( C  +Q  B )  =  ( /Q `  ( C  +pQ  B ) ) )
2420, 23breq12d 4036 . . . . 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 8594 . . . . 5  |-  ( ( C  +pQ  A ) 
<pQ  ( C  +pQ  B
)  <->  ( /Q `  ( C  +pQ  A ) )  <Q  ( /Q `  ( C  +pQ  B
) ) )
2624, 25syl6bbr 254 . . . 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 6150 . . . . . . . . . 10  |-  ( C  e.  ( N.  X.  N. )  ->  ( 2nd `  C )  e.  N. )
288, 27syl 15 . . . . . . . . 9  |-  ( ( A  e.  Q.  /\  B  e.  Q.  /\  C  e.  Q. )  ->  ( 2nd `  C )  e. 
N. )
29 mulclpi 8517 . . . . . . . . 9  |-  ( ( ( 2nd `  C
)  e.  N.  /\  ( 2nd `  C )  e.  N. )  -> 
( ( 2nd `  C
)  .N  ( 2nd `  C ) )  e. 
N. )
3028, 28, 29syl2anc 642 . . . . . . . 8  |-  ( ( A  e.  Q.  /\  B  e.  Q.  /\  C  e.  Q. )  ->  (
( 2nd `  C
)  .N  ( 2nd `  C ) )  e. 
N. )
31 ltmpi 8528 . . . . . . . 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 15 . . . . . . 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 6150 . . . . . . . . . . 11  |-  ( B  e.  ( N.  X.  N. )  ->  ( 2nd `  B )  e.  N. )
3414, 33syl 15 . . . . . . . . . 10  |-  ( ( A  e.  Q.  /\  B  e.  Q.  /\  C  e.  Q. )  ->  ( 2nd `  B )  e. 
N. )
35 mulclpi 8517 . . . . . . . . . 10  |-  ( ( ( 2nd `  C
)  e.  N.  /\  ( 2nd `  B )  e.  N. )  -> 
( ( 2nd `  C
)  .N  ( 2nd `  B ) )  e. 
N. )
3628, 34, 35syl2anc 642 . . . . . . . . 9  |-  ( ( A  e.  Q.  /\  B  e.  Q.  /\  C  e.  Q. )  ->  (
( 2nd `  C
)  .N  ( 2nd `  B ) )  e. 
N. )
37 xp1st 6149 . . . . . . . . . . 11  |-  ( C  e.  ( N.  X.  N. )  ->  ( 1st `  C )  e.  N. )
388, 37syl 15 . . . . . . . . . 10  |-  ( ( A  e.  Q.  /\  B  e.  Q.  /\  C  e.  Q. )  ->  ( 1st `  C )  e. 
N. )
39 xp2nd 6150 . . . . . . . . . . 11  |-  ( A  e.  ( N.  X.  N. )  ->  ( 2nd `  A )  e.  N. )
4010, 39syl 15 . . . . . . . . . 10  |-  ( ( A  e.  Q.  /\  B  e.  Q.  /\  C  e.  Q. )  ->  ( 2nd `  A )  e. 
N. )
41 mulclpi 8517 . . . . . . . . . 10  |-  ( ( ( 1st `  C
)  e.  N.  /\  ( 2nd `  A )  e.  N. )  -> 
( ( 1st `  C
)  .N  ( 2nd `  A ) )  e. 
N. )
4238, 40, 41syl2anc 642 . . . . . . . . 9  |-  ( ( A  e.  Q.  /\  B  e.  Q.  /\  C  e.  Q. )  ->  (
( 1st `  C
)  .N  ( 2nd `  A ) )  e. 
N. )
43 mulclpi 8517 . . . . . . . . 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 642 . . . . . . . 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 8527 . . . . . . . 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 15 . . . . . . 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 244 . . . . . 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 8520 . . . . . . . . . 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 5539 . . . . . . . . . . 11  |-  ( 1st `  A )  e.  _V
50 fvex 5539 . . . . . . . . . . 11  |-  ( 2nd `  B )  e.  _V
51 fvex 5539 . . . . . . . . . . 11  |-  ( 2nd `  C )  e.  _V
52 mulcompi 8520 . . . . . . . . . . 11  |-  ( x  .N  y )  =  ( y  .N  x
)
53 mulasspi 8521 . . . . . . . . . . 11  |-  ( ( x  .N  y )  .N  z )  =  ( x  .N  (
y  .N  z ) )
5449, 50, 51, 52, 53, 51caov411 6052 . . . . . . . . . 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 2303 . . . . . . . . 9  |-  ( ( ( 2nd `  C
)  .N  ( 2nd `  C ) )  .N  ( ( 1st `  A
)  .N  ( 2nd `  B ) ) )  =  ( ( ( 2nd `  C )  .N  ( 2nd `  B
) )  .N  (
( 1st `  A
)  .N  ( 2nd `  C ) ) )
5655oveq2i 5869 . . . . . . . 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 8522 . . . . . . . 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 8520 . . . . . . . 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 2309 . . . . . . 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 8520 . . . . . . . . . 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 5539 . . . . . . . . . . 11  |-  ( 1st `  C )  e.  _V
62 fvex 5539 . . . . . . . . . . 11  |-  ( 2nd `  A )  e.  _V
6361, 62, 51, 52, 53, 50caov411 6052 . . . . . . . . . 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 2303 . . . . . . . . 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 8520 . . . . . . . . . 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 5539 . . . . . . . . . . 11  |-  ( 1st `  B )  e.  _V
6766, 62, 51, 52, 53, 51caov411 6052 . . . . . . . . . 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 2303 . . . . . . . . 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 5870 . . . . . . . 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 8522 . . . . . . . 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 8520 . . . . . . . 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 2309 . . . . . . 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 4032 . . . . . 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 252 . . . . 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 8566 . . . . 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 254 . . . 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 277 . . 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 244 . 2  |-  ( ( A  e.  Q.  /\  B  e.  Q.  /\  C  e.  Q. )  ->  ( A  <Q  B  <->  ( C  +Q  A )  <Q  ( C  +Q  B ) ) )
792, 3, 4, 78ndmovord 6010 1  |-  ( C  e.  Q.  ->  ( A  <Q  B  <->  ( C  +Q  A )  <Q  ( C  +Q  B ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ w3a 934    = wceq 1623    e. wcel 1684   <.cop 3643   class class class wbr 4023    X. cxp 4687   ` cfv 5255  (class class class)co 5858   1stc1st 6120   2ndc2nd 6121   N.cnpi 8466    +N cpli 8467    .N cmi 8468    <N clti 8469    +pQ cplpq 8470    <pQ cltpq 8472   Q.cnq 8474   /Qcerq 8476    +Q cplq 8477    <Q cltq 8480
This theorem is referenced by:  ltaddnq  8598  ltbtwnnq  8602  addclpr  8642  distrlem4pr  8650  ltexprlem3  8662  ltexprlem4  8663  ltexprlem6  8665  prlem936  8671
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-recs 6388  df-rdg 6423  df-1o 6479  df-oadd 6483  df-omul 6484  df-er 6660  df-ni 8496  df-pli 8497  df-mi 8498  df-lti 8499  df-plpq 8532  df-ltpq 8534  df-enq 8535  df-nq 8536  df-erq 8537  df-plq 8538  df-1nq 8540  df-ltnq 8542
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