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Theorem mulcmpblnrlem 8711
Description: Lemma used in lemma showing compatibility of multiplication. (Contributed by NM, 4-Sep-1995.) (New usage is discouraged.)
Assertion
Ref Expression
mulcmpblnrlem  |-  ( ( ( A  +P.  D
)  =  ( B  +P.  C )  /\  ( F  +P.  S )  =  ( G  +P.  R ) )  ->  (
( D  .P.  F
)  +P.  ( (
( A  .P.  F
)  +P.  ( B  .P.  G ) )  +P.  ( ( C  .P.  S )  +P.  ( D  .P.  R ) ) ) )  =  ( ( D  .P.  F
)  +P.  ( (
( A  .P.  G
)  +P.  ( B  .P.  F ) )  +P.  ( ( C  .P.  R )  +P.  ( D  .P.  S ) ) ) ) )

Proof of Theorem mulcmpblnrlem
Dummy variables  x  y  z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 oveq1 5881 . . . . . . . . 9  |-  ( ( A  +P.  D )  =  ( B  +P.  C )  ->  ( ( A  +P.  D )  .P. 
F )  =  ( ( B  +P.  C
)  .P.  F )
)
2 distrpr 8668 . . . . . . . . . 10  |-  ( F  .P.  ( A  +P.  D ) )  =  ( ( F  .P.  A
)  +P.  ( F  .P.  D ) )
3 mulcompr 8663 . . . . . . . . . 10  |-  ( ( A  +P.  D )  .P.  F )  =  ( F  .P.  ( A  +P.  D ) )
4 mulcompr 8663 . . . . . . . . . . 11  |-  ( A  .P.  F )  =  ( F  .P.  A
)
5 mulcompr 8663 . . . . . . . . . . 11  |-  ( D  .P.  F )  =  ( F  .P.  D
)
64, 5oveq12i 5886 . . . . . . . . . 10  |-  ( ( A  .P.  F )  +P.  ( D  .P.  F ) )  =  ( ( F  .P.  A
)  +P.  ( F  .P.  D ) )
72, 3, 63eqtr4i 2326 . . . . . . . . 9  |-  ( ( A  +P.  D )  .P.  F )  =  ( ( A  .P.  F )  +P.  ( D  .P.  F ) )
8 distrpr 8668 . . . . . . . . . 10  |-  ( F  .P.  ( B  +P.  C ) )  =  ( ( F  .P.  B
)  +P.  ( F  .P.  C ) )
9 mulcompr 8663 . . . . . . . . . 10  |-  ( ( B  +P.  C )  .P.  F )  =  ( F  .P.  ( B  +P.  C ) )
10 mulcompr 8663 . . . . . . . . . . 11  |-  ( B  .P.  F )  =  ( F  .P.  B
)
11 mulcompr 8663 . . . . . . . . . . 11  |-  ( C  .P.  F )  =  ( F  .P.  C
)
1210, 11oveq12i 5886 . . . . . . . . . 10  |-  ( ( B  .P.  F )  +P.  ( C  .P.  F ) )  =  ( ( F  .P.  B
)  +P.  ( F  .P.  C ) )
138, 9, 123eqtr4i 2326 . . . . . . . . 9  |-  ( ( B  +P.  C )  .P.  F )  =  ( ( B  .P.  F )  +P.  ( C  .P.  F ) )
141, 7, 133eqtr3g 2351 . . . . . . . 8  |-  ( ( A  +P.  D )  =  ( B  +P.  C )  ->  ( ( A  .P.  F )  +P.  ( D  .P.  F
) )  =  ( ( B  .P.  F
)  +P.  ( C  .P.  F ) ) )
1514oveq1d 5889 . . . . . . 7  |-  ( ( A  +P.  D )  =  ( B  +P.  C )  ->  ( (
( A  .P.  F
)  +P.  ( D  .P.  F ) )  +P.  ( C  .P.  S
) )  =  ( ( ( B  .P.  F )  +P.  ( C  .P.  F ) )  +P.  ( C  .P.  S ) ) )
16 addasspr 8662 . . . . . . . 8  |-  ( ( ( B  .P.  F
)  +P.  ( C  .P.  F ) )  +P.  ( C  .P.  S
) )  =  ( ( B  .P.  F
)  +P.  ( ( C  .P.  F )  +P.  ( C  .P.  S
) ) )
17 oveq2 5882 . . . . . . . . . 10  |-  ( ( F  +P.  S )  =  ( G  +P.  R )  ->  ( C  .P.  ( F  +P.  S
) )  =  ( C  .P.  ( G  +P.  R ) ) )
18 distrpr 8668 . . . . . . . . . 10  |-  ( C  .P.  ( F  +P.  S ) )  =  ( ( C  .P.  F
)  +P.  ( C  .P.  S ) )
19 distrpr 8668 . . . . . . . . . 10  |-  ( C  .P.  ( G  +P.  R ) )  =  ( ( C  .P.  G
)  +P.  ( C  .P.  R ) )
2017, 18, 193eqtr3g 2351 . . . . . . . . 9  |-  ( ( F  +P.  S )  =  ( G  +P.  R )  ->  ( ( C  .P.  F )  +P.  ( C  .P.  S
) )  =  ( ( C  .P.  G
)  +P.  ( C  .P.  R ) ) )
2120oveq2d 5890 . . . . . . . 8  |-  ( ( F  +P.  S )  =  ( G  +P.  R )  ->  ( ( B  .P.  F )  +P.  ( ( C  .P.  F )  +P.  ( C  .P.  S ) ) )  =  ( ( B  .P.  F )  +P.  ( ( C  .P.  G )  +P.  ( C  .P.  R
) ) ) )
2216, 21syl5eq 2340 . . . . . . 7  |-  ( ( F  +P.  S )  =  ( G  +P.  R )  ->  ( (
( B  .P.  F
)  +P.  ( C  .P.  F ) )  +P.  ( C  .P.  S
) )  =  ( ( B  .P.  F
)  +P.  ( ( C  .P.  G )  +P.  ( C  .P.  R
) ) ) )
2315, 22sylan9eq 2348 . . . . . 6  |-  ( ( ( A  +P.  D
)  =  ( B  +P.  C )  /\  ( F  +P.  S )  =  ( G  +P.  R ) )  ->  (
( ( A  .P.  F )  +P.  ( D  .P.  F ) )  +P.  ( C  .P.  S ) )  =  ( ( B  .P.  F
)  +P.  ( ( C  .P.  G )  +P.  ( C  .P.  R
) ) ) )
24 ovex 5899 . . . . . . 7  |-  ( A  .P.  F )  e. 
_V
25 ovex 5899 . . . . . . 7  |-  ( D  .P.  F )  e. 
_V
26 ovex 5899 . . . . . . 7  |-  ( C  .P.  S )  e. 
_V
27 addcompr 8661 . . . . . . 7  |-  ( x  +P.  y )  =  ( y  +P.  x
)
28 addasspr 8662 . . . . . . 7  |-  ( ( x  +P.  y )  +P.  z )  =  ( x  +P.  (
y  +P.  z )
)
2924, 25, 26, 27, 28caov32 6063 . . . . . 6  |-  ( ( ( A  .P.  F
)  +P.  ( D  .P.  F ) )  +P.  ( C  .P.  S
) )  =  ( ( ( A  .P.  F )  +P.  ( C  .P.  S ) )  +P.  ( D  .P.  F ) )
30 ovex 5899 . . . . . . 7  |-  ( B  .P.  F )  e. 
_V
31 ovex 5899 . . . . . . 7  |-  ( C  .P.  G )  e. 
_V
32 ovex 5899 . . . . . . 7  |-  ( C  .P.  R )  e. 
_V
3330, 31, 32, 27, 28caov12 6064 . . . . . 6  |-  ( ( B  .P.  F )  +P.  ( ( C  .P.  G )  +P.  ( C  .P.  R
) ) )  =  ( ( C  .P.  G )  +P.  ( ( B  .P.  F )  +P.  ( C  .P.  R ) ) )
3423, 29, 333eqtr3g 2351 . . . . 5  |-  ( ( ( A  +P.  D
)  =  ( B  +P.  C )  /\  ( F  +P.  S )  =  ( G  +P.  R ) )  ->  (
( ( A  .P.  F )  +P.  ( C  .P.  S ) )  +P.  ( D  .P.  F ) )  =  ( ( C  .P.  G
)  +P.  ( ( B  .P.  F )  +P.  ( C  .P.  R
) ) ) )
3534oveq2d 5890 . . . 4  |-  ( ( ( A  +P.  D
)  =  ( B  +P.  C )  /\  ( F  +P.  S )  =  ( G  +P.  R ) )  ->  (
( ( B  .P.  G )  +P.  ( D  .P.  R ) )  +P.  ( ( ( A  .P.  F )  +P.  ( C  .P.  S ) )  +P.  ( D  .P.  F ) ) )  =  ( ( ( B  .P.  G
)  +P.  ( D  .P.  R ) )  +P.  ( ( C  .P.  G )  +P.  ( ( B  .P.  F )  +P.  ( C  .P.  R ) ) ) ) )
36 oveq2 5882 . . . . . . . . . . 11  |-  ( ( F  +P.  S )  =  ( G  +P.  R )  ->  ( D  .P.  ( F  +P.  S
) )  =  ( D  .P.  ( G  +P.  R ) ) )
37 distrpr 8668 . . . . . . . . . . 11  |-  ( D  .P.  ( F  +P.  S ) )  =  ( ( D  .P.  F
)  +P.  ( D  .P.  S ) )
38 distrpr 8668 . . . . . . . . . . 11  |-  ( D  .P.  ( G  +P.  R ) )  =  ( ( D  .P.  G
)  +P.  ( D  .P.  R ) )
3936, 37, 383eqtr3g 2351 . . . . . . . . . 10  |-  ( ( F  +P.  S )  =  ( G  +P.  R )  ->  ( ( D  .P.  F )  +P.  ( D  .P.  S
) )  =  ( ( D  .P.  G
)  +P.  ( D  .P.  R ) ) )
4039oveq2d 5890 . . . . . . . . 9  |-  ( ( F  +P.  S )  =  ( G  +P.  R )  ->  ( ( A  .P.  G )  +P.  ( ( D  .P.  F )  +P.  ( D  .P.  S ) ) )  =  ( ( A  .P.  G )  +P.  ( ( D  .P.  G )  +P.  ( D  .P.  R
) ) ) )
41 addasspr 8662 . . . . . . . . 9  |-  ( ( ( A  .P.  G
)  +P.  ( D  .P.  G ) )  +P.  ( D  .P.  R
) )  =  ( ( A  .P.  G
)  +P.  ( ( D  .P.  G )  +P.  ( D  .P.  R
) ) )
4240, 41syl6eqr 2346 . . . . . . . 8  |-  ( ( F  +P.  S )  =  ( G  +P.  R )  ->  ( ( A  .P.  G )  +P.  ( ( D  .P.  F )  +P.  ( D  .P.  S ) ) )  =  ( ( ( A  .P.  G
)  +P.  ( D  .P.  G ) )  +P.  ( D  .P.  R
) ) )
43 oveq1 5881 . . . . . . . . . 10  |-  ( ( A  +P.  D )  =  ( B  +P.  C )  ->  ( ( A  +P.  D )  .P. 
G )  =  ( ( B  +P.  C
)  .P.  G )
)
44 distrpr 8668 . . . . . . . . . . 11  |-  ( G  .P.  ( A  +P.  D ) )  =  ( ( G  .P.  A
)  +P.  ( G  .P.  D ) )
45 mulcompr 8663 . . . . . . . . . . 11  |-  ( ( A  +P.  D )  .P.  G )  =  ( G  .P.  ( A  +P.  D ) )
46 mulcompr 8663 . . . . . . . . . . . 12  |-  ( A  .P.  G )  =  ( G  .P.  A
)
47 mulcompr 8663 . . . . . . . . . . . 12  |-  ( D  .P.  G )  =  ( G  .P.  D
)
4846, 47oveq12i 5886 . . . . . . . . . . 11  |-  ( ( A  .P.  G )  +P.  ( D  .P.  G ) )  =  ( ( G  .P.  A
)  +P.  ( G  .P.  D ) )
4944, 45, 483eqtr4i 2326 . . . . . . . . . 10  |-  ( ( A  +P.  D )  .P.  G )  =  ( ( A  .P.  G )  +P.  ( D  .P.  G ) )
50 distrpr 8668 . . . . . . . . . . 11  |-  ( G  .P.  ( B  +P.  C ) )  =  ( ( G  .P.  B
)  +P.  ( G  .P.  C ) )
51 mulcompr 8663 . . . . . . . . . . 11  |-  ( ( B  +P.  C )  .P.  G )  =  ( G  .P.  ( B  +P.  C ) )
52 mulcompr 8663 . . . . . . . . . . . 12  |-  ( B  .P.  G )  =  ( G  .P.  B
)
53 mulcompr 8663 . . . . . . . . . . . 12  |-  ( C  .P.  G )  =  ( G  .P.  C
)
5452, 53oveq12i 5886 . . . . . . . . . . 11  |-  ( ( B  .P.  G )  +P.  ( C  .P.  G ) )  =  ( ( G  .P.  B
)  +P.  ( G  .P.  C ) )
5550, 51, 543eqtr4i 2326 . . . . . . . . . 10  |-  ( ( B  +P.  C )  .P.  G )  =  ( ( B  .P.  G )  +P.  ( C  .P.  G ) )
5643, 49, 553eqtr3g 2351 . . . . . . . . 9  |-  ( ( A  +P.  D )  =  ( B  +P.  C )  ->  ( ( A  .P.  G )  +P.  ( D  .P.  G
) )  =  ( ( B  .P.  G
)  +P.  ( C  .P.  G ) ) )
5756oveq1d 5889 . . . . . . . 8  |-  ( ( A  +P.  D )  =  ( B  +P.  C )  ->  ( (
( A  .P.  G
)  +P.  ( D  .P.  G ) )  +P.  ( D  .P.  R
) )  =  ( ( ( B  .P.  G )  +P.  ( C  .P.  G ) )  +P.  ( D  .P.  R ) ) )
5842, 57sylan9eqr 2350 . . . . . . 7  |-  ( ( ( A  +P.  D
)  =  ( B  +P.  C )  /\  ( F  +P.  S )  =  ( G  +P.  R ) )  ->  (
( A  .P.  G
)  +P.  ( ( D  .P.  F )  +P.  ( D  .P.  S
) ) )  =  ( ( ( B  .P.  G )  +P.  ( C  .P.  G
) )  +P.  ( D  .P.  R ) ) )
59 ovex 5899 . . . . . . . 8  |-  ( A  .P.  G )  e. 
_V
60 ovex 5899 . . . . . . . 8  |-  ( D  .P.  S )  e. 
_V
6159, 25, 60, 27, 28caov12 6064 . . . . . . 7  |-  ( ( A  .P.  G )  +P.  ( ( D  .P.  F )  +P.  ( D  .P.  S
) ) )  =  ( ( D  .P.  F )  +P.  ( ( A  .P.  G )  +P.  ( D  .P.  S ) ) )
62 ovex 5899 . . . . . . . 8  |-  ( B  .P.  G )  e. 
_V
63 ovex 5899 . . . . . . . 8  |-  ( D  .P.  R )  e. 
_V
6462, 31, 63, 27, 28caov32 6063 . . . . . . 7  |-  ( ( ( B  .P.  G
)  +P.  ( C  .P.  G ) )  +P.  ( D  .P.  R
) )  =  ( ( ( B  .P.  G )  +P.  ( D  .P.  R ) )  +P.  ( C  .P.  G ) )
6558, 61, 643eqtr3g 2351 . . . . . 6  |-  ( ( ( A  +P.  D
)  =  ( B  +P.  C )  /\  ( F  +P.  S )  =  ( G  +P.  R ) )  ->  (
( D  .P.  F
)  +P.  ( ( A  .P.  G )  +P.  ( D  .P.  S
) ) )  =  ( ( ( B  .P.  G )  +P.  ( D  .P.  R
) )  +P.  ( C  .P.  G ) ) )
6665oveq1d 5889 . . . . 5  |-  ( ( ( A  +P.  D
)  =  ( B  +P.  C )  /\  ( F  +P.  S )  =  ( G  +P.  R ) )  ->  (
( ( D  .P.  F )  +P.  ( ( A  .P.  G )  +P.  ( D  .P.  S ) ) )  +P.  ( ( B  .P.  F )  +P.  ( C  .P.  R ) ) )  =  ( ( ( ( B  .P.  G )  +P.  ( D  .P.  R ) )  +P.  ( C  .P.  G ) )  +P.  (
( B  .P.  F
)  +P.  ( C  .P.  R ) ) ) )
67 addasspr 8662 . . . . 5  |-  ( ( ( ( B  .P.  G )  +P.  ( D  .P.  R ) )  +P.  ( C  .P.  G ) )  +P.  (
( B  .P.  F
)  +P.  ( C  .P.  R ) ) )  =  ( ( ( B  .P.  G )  +P.  ( D  .P.  R ) )  +P.  (
( C  .P.  G
)  +P.  ( ( B  .P.  F )  +P.  ( C  .P.  R
) ) ) )
6866, 67syl6eq 2344 . . . 4  |-  ( ( ( A  +P.  D
)  =  ( B  +P.  C )  /\  ( F  +P.  S )  =  ( G  +P.  R ) )  ->  (
( ( D  .P.  F )  +P.  ( ( A  .P.  G )  +P.  ( D  .P.  S ) ) )  +P.  ( ( B  .P.  F )  +P.  ( C  .P.  R ) ) )  =  ( ( ( B  .P.  G
)  +P.  ( D  .P.  R ) )  +P.  ( ( C  .P.  G )  +P.  ( ( B  .P.  F )  +P.  ( C  .P.  R ) ) ) ) )
6935, 68eqtr4d 2331 . . 3  |-  ( ( ( A  +P.  D
)  =  ( B  +P.  C )  /\  ( F  +P.  S )  =  ( G  +P.  R ) )  ->  (
( ( B  .P.  G )  +P.  ( D  .P.  R ) )  +P.  ( ( ( A  .P.  F )  +P.  ( C  .P.  S ) )  +P.  ( D  .P.  F ) ) )  =  ( ( ( D  .P.  F
)  +P.  ( ( A  .P.  G )  +P.  ( D  .P.  S
) ) )  +P.  ( ( B  .P.  F )  +P.  ( C  .P.  R ) ) ) )
70 ovex 5899 . . . 4  |-  ( ( B  .P.  G )  +P.  ( D  .P.  R ) )  e.  _V
71 ovex 5899 . . . 4  |-  ( ( A  .P.  F )  +P.  ( C  .P.  S ) )  e.  _V
7270, 71, 25, 27, 28caov13 6066 . . 3  |-  ( ( ( B  .P.  G
)  +P.  ( D  .P.  R ) )  +P.  ( ( ( A  .P.  F )  +P.  ( C  .P.  S
) )  +P.  ( D  .P.  F ) ) )  =  ( ( D  .P.  F )  +P.  ( ( ( A  .P.  F )  +P.  ( C  .P.  S ) )  +P.  (
( B  .P.  G
)  +P.  ( D  .P.  R ) ) ) )
73 addasspr 8662 . . 3  |-  ( ( ( D  .P.  F
)  +P.  ( ( A  .P.  G )  +P.  ( D  .P.  S
) ) )  +P.  ( ( B  .P.  F )  +P.  ( C  .P.  R ) ) )  =  ( ( D  .P.  F )  +P.  ( ( ( A  .P.  G )  +P.  ( D  .P.  S ) )  +P.  (
( B  .P.  F
)  +P.  ( C  .P.  R ) ) ) )
7469, 72, 733eqtr3g 2351 . 2  |-  ( ( ( A  +P.  D
)  =  ( B  +P.  C )  /\  ( F  +P.  S )  =  ( G  +P.  R ) )  ->  (
( D  .P.  F
)  +P.  ( (
( A  .P.  F
)  +P.  ( C  .P.  S ) )  +P.  ( ( B  .P.  G )  +P.  ( D  .P.  R ) ) ) )  =  ( ( D  .P.  F
)  +P.  ( (
( A  .P.  G
)  +P.  ( D  .P.  S ) )  +P.  ( ( B  .P.  F )  +P.  ( C  .P.  R ) ) ) ) )
7524, 26, 62, 27, 28, 63caov4 6067 . . 3  |-  ( ( ( A  .P.  F
)  +P.  ( C  .P.  S ) )  +P.  ( ( B  .P.  G )  +P.  ( D  .P.  R ) ) )  =  ( ( ( A  .P.  F
)  +P.  ( B  .P.  G ) )  +P.  ( ( C  .P.  S )  +P.  ( D  .P.  R ) ) )
7675oveq2i 5885 . 2  |-  ( ( D  .P.  F )  +P.  ( ( ( A  .P.  F )  +P.  ( C  .P.  S ) )  +P.  (
( B  .P.  G
)  +P.  ( D  .P.  R ) ) ) )  =  ( ( D  .P.  F )  +P.  ( ( ( A  .P.  F )  +P.  ( B  .P.  G ) )  +P.  (
( C  .P.  S
)  +P.  ( D  .P.  R ) ) ) )
7759, 60, 30, 27, 28, 32caov42 6069 . . 3  |-  ( ( ( A  .P.  G
)  +P.  ( D  .P.  S ) )  +P.  ( ( B  .P.  F )  +P.  ( C  .P.  R ) ) )  =  ( ( ( A  .P.  G
)  +P.  ( B  .P.  F ) )  +P.  ( ( C  .P.  R )  +P.  ( D  .P.  S ) ) )
7877oveq2i 5885 . 2  |-  ( ( D  .P.  F )  +P.  ( ( ( A  .P.  G )  +P.  ( D  .P.  S ) )  +P.  (
( B  .P.  F
)  +P.  ( C  .P.  R ) ) ) )  =  ( ( D  .P.  F )  +P.  ( ( ( A  .P.  G )  +P.  ( B  .P.  F ) )  +P.  (
( C  .P.  R
)  +P.  ( D  .P.  S ) ) ) )
7974, 76, 783eqtr3g 2351 1  |-  ( ( ( A  +P.  D
)  =  ( B  +P.  C )  /\  ( F  +P.  S )  =  ( G  +P.  R ) )  ->  (
( D  .P.  F
)  +P.  ( (
( A  .P.  F
)  +P.  ( B  .P.  G ) )  +P.  ( ( C  .P.  S )  +P.  ( D  .P.  R ) ) ) )  =  ( ( D  .P.  F
)  +P.  ( (
( A  .P.  G
)  +P.  ( B  .P.  F ) )  +P.  ( ( C  .P.  R )  +P.  ( D  .P.  S ) ) ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1632  (class class class)co 5874    +P. cpp 8499    .P. cmp 8500
This theorem is referenced by:  mulcmpblnr  8712
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-inf2 7358
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 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-reu 2563  df-rmo 2564  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-1st 6138  df-2nd 6139  df-recs 6404  df-rdg 6439  df-1o 6495  df-oadd 6499  df-omul 6500  df-er 6676  df-ni 8512  df-pli 8513  df-mi 8514  df-lti 8515  df-plpq 8548  df-mpq 8549  df-ltpq 8550  df-enq 8551  df-nq 8552  df-erq 8553  df-plq 8554  df-mq 8555  df-1nq 8556  df-rq 8557  df-ltnq 8558  df-np 8621  df-plp 8623  df-mp 8624
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