MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  ltanq Structured version   Unicode version

Theorem ltanq 9348
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 9325 . . 3  |-  +Q  :
( Q.  X.  Q. )
--> Q.
21fdmi 5735 . 2  |-  dom  +Q  =  ( Q.  X.  Q. )
3 ltrelnq 9303 . 2  |-  <Q  C_  ( Q.  X.  Q. )
4 0nnq 9301 . 2  |-  -.  (/)  e.  Q.
5 ordpinq 9320 . . . 4  |-  ( ( A  e.  Q.  /\  B  e.  Q. )  ->  ( A  <Q  B  <->  ( ( 1st `  A )  .N  ( 2nd `  B
) )  <N  (
( 1st `  B
)  .N  ( 2nd `  A ) ) ) )
653adant3 1016 . . 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 9302 . . . . . . 7  |-  ( C  e.  Q.  ->  C  e.  ( N.  X.  N. ) )
873ad2ant3 1019 . . . . . 6  |-  ( ( A  e.  Q.  /\  B  e.  Q.  /\  C  e.  Q. )  ->  C  e.  ( N.  X.  N. ) )
9 elpqn 9302 . . . . . . 7  |-  ( A  e.  Q.  ->  A  e.  ( N.  X.  N. ) )
1093ad2ant1 1017 . . . . . 6  |-  ( ( A  e.  Q.  /\  B  e.  Q.  /\  C  e.  Q. )  ->  A  e.  ( N.  X.  N. ) )
11 addpipq2 9313 . . . . . 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 661 . . . . 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 9302 . . . . . . 7  |-  ( B  e.  Q.  ->  B  e.  ( N.  X.  N. ) )
14133ad2ant2 1018 . . . . . 6  |-  ( ( A  e.  Q.  /\  B  e.  Q.  /\  C  e.  Q. )  ->  B  e.  ( N.  X.  N. ) )
15 addpipq2 9313 . . . . . 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 661 . . . . 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 4460 . . . 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 9315 . . . . . . . 8  |-  ( ( C  e.  Q.  /\  A  e.  Q. )  ->  ( C  +Q  A
)  =  ( /Q
`  ( C  +pQ  A ) ) )
1918ancoms 453 . . . . . . 7  |-  ( ( A  e.  Q.  /\  C  e.  Q. )  ->  ( C  +Q  A
)  =  ( /Q
`  ( C  +pQ  A ) ) )
20193adant2 1015 . . . . . 6  |-  ( ( A  e.  Q.  /\  B  e.  Q.  /\  C  e.  Q. )  ->  ( C  +Q  A )  =  ( /Q `  ( C  +pQ  A ) ) )
21 addpqnq 9315 . . . . . . . 8  |-  ( ( C  e.  Q.  /\  B  e.  Q. )  ->  ( C  +Q  B
)  =  ( /Q
`  ( C  +pQ  B ) ) )
2221ancoms 453 . . . . . . 7  |-  ( ( B  e.  Q.  /\  C  e.  Q. )  ->  ( C  +Q  B
)  =  ( /Q
`  ( C  +pQ  B ) ) )
23223adant1 1014 . . . . . 6  |-  ( ( A  e.  Q.  /\  B  e.  Q.  /\  C  e.  Q. )  ->  ( C  +Q  B )  =  ( /Q `  ( C  +pQ  B ) ) )
2420, 23breq12d 4460 . . . . 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 9347 . . . . 5  |-  ( ( C  +pQ  A ) 
<pQ  ( C  +pQ  B
)  <->  ( /Q `  ( C  +pQ  A ) )  <Q  ( /Q `  ( C  +pQ  B
) ) )
2624, 25syl6bbr 263 . . . 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 6815 . . . . . . . . . 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 9270 . . . . . . . . 9  |-  ( ( ( 2nd `  C
)  e.  N.  /\  ( 2nd `  C )  e.  N. )  -> 
( ( 2nd `  C
)  .N  ( 2nd `  C ) )  e. 
N. )
3028, 28, 29syl2anc 661 . . . . . . . 8  |-  ( ( A  e.  Q.  /\  B  e.  Q.  /\  C  e.  Q. )  ->  (
( 2nd `  C
)  .N  ( 2nd `  C ) )  e. 
N. )
31 ltmpi 9281 . . . . . . . 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 6815 . . . . . . . . . . 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 9270 . . . . . . . . . 10  |-  ( ( ( 2nd `  C
)  e.  N.  /\  ( 2nd `  B )  e.  N. )  -> 
( ( 2nd `  C
)  .N  ( 2nd `  B ) )  e. 
N. )
3628, 34, 35syl2anc 661 . . . . . . . . 9  |-  ( ( A  e.  Q.  /\  B  e.  Q.  /\  C  e.  Q. )  ->  (
( 2nd `  C
)  .N  ( 2nd `  B ) )  e. 
N. )
37 xp1st 6814 . . . . . . . . . . 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 6815 . . . . . . . . . . 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 9270 . . . . . . . . . 10  |-  ( ( ( 1st `  C
)  e.  N.  /\  ( 2nd `  A )  e.  N. )  -> 
( ( 1st `  C
)  .N  ( 2nd `  A ) )  e. 
N. )
4238, 40, 41syl2anc 661 . . . . . . . . 9  |-  ( ( A  e.  Q.  /\  B  e.  Q.  /\  C  e.  Q. )  ->  (
( 1st `  C
)  .N  ( 2nd `  A ) )  e. 
N. )
43 mulclpi 9270 . . . . . . . . 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 661 . . . . . . . 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 9280 . . . . . . . 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 253 . . . . . 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 9273 . . . . . . . . . 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 5875 . . . . . . . . . . 11  |-  ( 1st `  A )  e.  _V
50 fvex 5875 . . . . . . . . . . 11  |-  ( 2nd `  B )  e.  _V
51 fvex 5875 . . . . . . . . . . 11  |-  ( 2nd `  C )  e.  _V
52 mulcompi 9273 . . . . . . . . . . 11  |-  ( x  .N  y )  =  ( y  .N  x
)
53 mulasspi 9274 . . . . . . . . . . 11  |-  ( ( x  .N  y )  .N  z )  =  ( x  .N  (
y  .N  z ) )
5449, 50, 51, 52, 53, 51caov411 6490 . . . . . . . . . 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 2496 . . . . . . . . 9  |-  ( ( ( 2nd `  C
)  .N  ( 2nd `  C ) )  .N  ( ( 1st `  A
)  .N  ( 2nd `  B ) ) )  =  ( ( ( 2nd `  C )  .N  ( 2nd `  B
) )  .N  (
( 1st `  A
)  .N  ( 2nd `  C ) ) )
5655oveq2i 6294 . . . . . . . 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 9275 . . . . . . . 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 9273 . . . . . . . 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 2502 . . . . . . 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 9273 . . . . . . . . . 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 5875 . . . . . . . . . . 11  |-  ( 1st `  C )  e.  _V
62 fvex 5875 . . . . . . . . . . 11  |-  ( 2nd `  A )  e.  _V
6361, 62, 51, 52, 53, 50caov411 6490 . . . . . . . . . 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 2496 . . . . . . . . 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 9273 . . . . . . . . . 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 5875 . . . . . . . . . . 11  |-  ( 1st `  B )  e.  _V
6766, 62, 51, 52, 53, 51caov411 6490 . . . . . . . . . 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 2496 . . . . . . . . 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 6295 . . . . . . . 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 9275 . . . . . . . 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 9273 . . . . . . . 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 2502 . . . . . . 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 4456 . . . . . 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 261 . . . . 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 9319 . . . . 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 263 . . . 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 286 . . 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 253 . 2  |-  ( ( A  e.  Q.  /\  B  e.  Q.  /\  C  e.  Q. )  ->  ( A  <Q  B  <->  ( C  +Q  A )  <Q  ( C  +Q  B ) ) )
792, 3, 4, 78ndmovord 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 184    /\ w3a 973    = wceq 1379    e. wcel 1767   <.cop 4033   class class class wbr 4447    X. cxp 4997   ` cfv 5587  (class class class)co 6283   1stc1st 6782   2ndc2nd 6783   N.cnpi 9221    +N cpli 9222    .N cmi 9223    <N clti 9224    +pQ cplpq 9225    <pQ cltpq 9227   Q.cnq 9229   /Qcerq 9231    +Q cplq 9232    <Q cltq 9235
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6575
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-reu 2821  df-rmo 2822  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-we 4840  df-ord 4881  df-on 4882  df-lim 4883  df-suc 4884  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5550  df-fun 5589  df-fn 5590  df-f 5591  df-f1 5592  df-fo 5593  df-f1o 5594  df-fv 5595  df-ov 6286  df-oprab 6287  df-mpt2 6288  df-om 6680  df-1st 6784  df-2nd 6785  df-recs 7042  df-rdg 7076  df-1o 7130  df-oadd 7134  df-omul 7135  df-er 7311  df-ni 9249  df-pli 9250  df-mi 9251  df-lti 9252  df-plpq 9285  df-ltpq 9287  df-enq 9288  df-nq 9289  df-erq 9290  df-plq 9291  df-1nq 9293  df-ltnq 9295
This theorem is referenced by:  ltaddnq  9351  ltbtwnnq  9355  addclpr  9395  distrlem4pr  9403  ltexprlem3  9415  ltexprlem4  9416  ltexprlem6  9418  prlem936  9424
  Copyright terms: Public domain W3C validator