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Theorem ltanq 8837
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 8814 . . 3  |-  +Q  :
( Q.  X.  Q. )
--> Q.
21fdmi 5587 . 2  |-  dom  +Q  =  ( Q.  X.  Q. )
3 ltrelnq 8792 . 2  |-  <Q  C_  ( Q.  X.  Q. )
4 0nnq 8790 . 2  |-  -.  (/)  e.  Q.
5 ordpinq 8809 . . . 4  |-  ( ( A  e.  Q.  /\  B  e.  Q. )  ->  ( A  <Q  B  <->  ( ( 1st `  A )  .N  ( 2nd `  B
) )  <N  (
( 1st `  B
)  .N  ( 2nd `  A ) ) ) )
653adant3 977 . . 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 8791 . . . . . . 7  |-  ( C  e.  Q.  ->  C  e.  ( N.  X.  N. ) )
873ad2ant3 980 . . . . . 6  |-  ( ( A  e.  Q.  /\  B  e.  Q.  /\  C  e.  Q. )  ->  C  e.  ( N.  X.  N. ) )
9 elpqn 8791 . . . . . . 7  |-  ( A  e.  Q.  ->  A  e.  ( N.  X.  N. ) )
1093ad2ant1 978 . . . . . 6  |-  ( ( A  e.  Q.  /\  B  e.  Q.  /\  C  e.  Q. )  ->  A  e.  ( N.  X.  N. ) )
11 addpipq2 8802 . . . . . 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 643 . . . . 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 8791 . . . . . . 7  |-  ( B  e.  Q.  ->  B  e.  ( N.  X.  N. ) )
14133ad2ant2 979 . . . . . 6  |-  ( ( A  e.  Q.  /\  B  e.  Q.  /\  C  e.  Q. )  ->  B  e.  ( N.  X.  N. ) )
15 addpipq2 8802 . . . . . 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 643 . . . . 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 4217 . . . 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 8804 . . . . . . . 8  |-  ( ( C  e.  Q.  /\  A  e.  Q. )  ->  ( C  +Q  A
)  =  ( /Q
`  ( C  +pQ  A ) ) )
1918ancoms 440 . . . . . . 7  |-  ( ( A  e.  Q.  /\  C  e.  Q. )  ->  ( C  +Q  A
)  =  ( /Q
`  ( C  +pQ  A ) ) )
20193adant2 976 . . . . . 6  |-  ( ( A  e.  Q.  /\  B  e.  Q.  /\  C  e.  Q. )  ->  ( C  +Q  A )  =  ( /Q `  ( C  +pQ  A ) ) )
21 addpqnq 8804 . . . . . . . 8  |-  ( ( C  e.  Q.  /\  B  e.  Q. )  ->  ( C  +Q  B
)  =  ( /Q
`  ( C  +pQ  B ) ) )
2221ancoms 440 . . . . . . 7  |-  ( ( B  e.  Q.  /\  C  e.  Q. )  ->  ( C  +Q  B
)  =  ( /Q
`  ( C  +pQ  B ) ) )
23223adant1 975 . . . . . 6  |-  ( ( A  e.  Q.  /\  B  e.  Q.  /\  C  e.  Q. )  ->  ( C  +Q  B )  =  ( /Q `  ( C  +pQ  B ) ) )
2420, 23breq12d 4217 . . . . 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 8836 . . . . 5  |-  ( ( C  +pQ  A ) 
<pQ  ( C  +pQ  B
)  <->  ( /Q `  ( C  +pQ  A ) )  <Q  ( /Q `  ( C  +pQ  B
) ) )
2624, 25syl6bbr 255 . . . 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 6368 . . . . . . . . . 10  |-  ( C  e.  ( N.  X.  N. )  ->  ( 2nd `  C )  e.  N. )
288, 27syl 16 . . . . . . . . 9  |-  ( ( A  e.  Q.  /\  B  e.  Q.  /\  C  e.  Q. )  ->  ( 2nd `  C )  e. 
N. )
29 mulclpi 8759 . . . . . . . . 9  |-  ( ( ( 2nd `  C
)  e.  N.  /\  ( 2nd `  C )  e.  N. )  -> 
( ( 2nd `  C
)  .N  ( 2nd `  C ) )  e. 
N. )
3028, 28, 29syl2anc 643 . . . . . . . 8  |-  ( ( A  e.  Q.  /\  B  e.  Q.  /\  C  e.  Q. )  ->  (
( 2nd `  C
)  .N  ( 2nd `  C ) )  e. 
N. )
31 ltmpi 8770 . . . . . . . 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 16 . . . . . . 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 6368 . . . . . . . . . . 11  |-  ( B  e.  ( N.  X.  N. )  ->  ( 2nd `  B )  e.  N. )
3414, 33syl 16 . . . . . . . . . 10  |-  ( ( A  e.  Q.  /\  B  e.  Q.  /\  C  e.  Q. )  ->  ( 2nd `  B )  e. 
N. )
35 mulclpi 8759 . . . . . . . . . 10  |-  ( ( ( 2nd `  C
)  e.  N.  /\  ( 2nd `  B )  e.  N. )  -> 
( ( 2nd `  C
)  .N  ( 2nd `  B ) )  e. 
N. )
3628, 34, 35syl2anc 643 . . . . . . . . 9  |-  ( ( A  e.  Q.  /\  B  e.  Q.  /\  C  e.  Q. )  ->  (
( 2nd `  C
)  .N  ( 2nd `  B ) )  e. 
N. )
37 xp1st 6367 . . . . . . . . . . 11  |-  ( C  e.  ( N.  X.  N. )  ->  ( 1st `  C )  e.  N. )
388, 37syl 16 . . . . . . . . . 10  |-  ( ( A  e.  Q.  /\  B  e.  Q.  /\  C  e.  Q. )  ->  ( 1st `  C )  e. 
N. )
39 xp2nd 6368 . . . . . . . . . . 11  |-  ( A  e.  ( N.  X.  N. )  ->  ( 2nd `  A )  e.  N. )
4010, 39syl 16 . . . . . . . . . 10  |-  ( ( A  e.  Q.  /\  B  e.  Q.  /\  C  e.  Q. )  ->  ( 2nd `  A )  e. 
N. )
41 mulclpi 8759 . . . . . . . . . 10  |-  ( ( ( 1st `  C
)  e.  N.  /\  ( 2nd `  A )  e.  N. )  -> 
( ( 1st `  C
)  .N  ( 2nd `  A ) )  e. 
N. )
4238, 40, 41syl2anc 643 . . . . . . . . 9  |-  ( ( A  e.  Q.  /\  B  e.  Q.  /\  C  e.  Q. )  ->  (
( 1st `  C
)  .N  ( 2nd `  A ) )  e. 
N. )
43 mulclpi 8759 . . . . . . . . 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 643 . . . . . . . 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 8769 . . . . . . . 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 16 . . . . . . 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 245 . . . . . 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 8762 . . . . . . . . . 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 5733 . . . . . . . . . . 11  |-  ( 1st `  A )  e.  _V
50 fvex 5733 . . . . . . . . . . 11  |-  ( 2nd `  B )  e.  _V
51 fvex 5733 . . . . . . . . . . 11  |-  ( 2nd `  C )  e.  _V
52 mulcompi 8762 . . . . . . . . . . 11  |-  ( x  .N  y )  =  ( y  .N  x
)
53 mulasspi 8763 . . . . . . . . . . 11  |-  ( ( x  .N  y )  .N  z )  =  ( x  .N  (
y  .N  z ) )
5449, 50, 51, 52, 53, 51caov411 6270 . . . . . . . . . 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 2455 . . . . . . . . 9  |-  ( ( ( 2nd `  C
)  .N  ( 2nd `  C ) )  .N  ( ( 1st `  A
)  .N  ( 2nd `  B ) ) )  =  ( ( ( 2nd `  C )  .N  ( 2nd `  B
) )  .N  (
( 1st `  A
)  .N  ( 2nd `  C ) ) )
5655oveq2i 6083 . . . . . . . 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 8764 . . . . . . . 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 8762 . . . . . . . 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 2461 . . . . . . 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 8762 . . . . . . . . . 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 5733 . . . . . . . . . . 11  |-  ( 1st `  C )  e.  _V
62 fvex 5733 . . . . . . . . . . 11  |-  ( 2nd `  A )  e.  _V
6361, 62, 51, 52, 53, 50caov411 6270 . . . . . . . . . 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 2455 . . . . . . . . 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 8762 . . . . . . . . . 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 5733 . . . . . . . . . . 11  |-  ( 1st `  B )  e.  _V
6766, 62, 51, 52, 53, 51caov411 6270 . . . . . . . . . 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 2455 . . . . . . . . 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 6084 . . . . . . . 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 8764 . . . . . . . 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 8762 . . . . . . . 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 2461 . . . . . . 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 4213 . . . . . 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 253 . . . . 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 8808 . . . . 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 255 . . . 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 278 . . 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 245 . 2  |-  ( ( A  e.  Q.  /\  B  e.  Q.  /\  C  e.  Q. )  ->  ( A  <Q  B  <->  ( C  +Q  A )  <Q  ( C  +Q  B ) ) )
792, 3, 4, 78ndmovord 6228 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 177    /\ w3a 936    = wceq 1652    e. wcel 1725   <.cop 3809   class class class wbr 4204    X. cxp 4867   ` cfv 5445  (class class class)co 6072   1stc1st 6338   2ndc2nd 6339   N.cnpi 8708    +N cpli 8709    .N cmi 8710    <N clti 8711    +pQ cplpq 8712    <pQ cltpq 8714   Q.cnq 8716   /Qcerq 8718    +Q cplq 8719    <Q cltq 8722
This theorem is referenced by:  ltaddnq  8840  ltbtwnnq  8844  addclpr  8884  distrlem4pr  8892  ltexprlem3  8904  ltexprlem4  8905  ltexprlem6  8907  prlem936  8913
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4692
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-reu 2704  df-rmo 2705  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-pss 3328  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-tp 3814  df-op 3815  df-uni 4008  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-tr 4295  df-eprel 4486  df-id 4490  df-po 4495  df-so 4496  df-fr 4533  df-we 4535  df-ord 4576  df-on 4577  df-lim 4578  df-suc 4579  df-om 4837  df-xp 4875  df-rel 4876  df-cnv 4877  df-co 4878  df-dm 4879  df-rn 4880  df-res 4881  df-ima 4882  df-iota 5409  df-fun 5447  df-fn 5448  df-f 5449  df-f1 5450  df-fo 5451  df-f1o 5452  df-fv 5453  df-ov 6075  df-oprab 6076  df-mpt2 6077  df-1st 6340  df-2nd 6341  df-recs 6624  df-rdg 6659  df-1o 6715  df-oadd 6719  df-omul 6720  df-er 6896  df-ni 8738  df-pli 8739  df-mi 8740  df-lti 8741  df-plpq 8774  df-ltpq 8776  df-enq 8777  df-nq 8778  df-erq 8779  df-plq 8780  df-1nq 8782  df-ltnq 8784
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