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Theorem mulcmpblnrlem 8875
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 6021 . . . . . . . . 9  |-  ( ( A  +P.  D )  =  ( B  +P.  C )  ->  ( ( A  +P.  D )  .P. 
F )  =  ( ( B  +P.  C
)  .P.  F )
)
2 distrpr 8832 . . . . . . . . . 10  |-  ( F  .P.  ( A  +P.  D ) )  =  ( ( F  .P.  A
)  +P.  ( F  .P.  D ) )
3 mulcompr 8827 . . . . . . . . . 10  |-  ( ( A  +P.  D )  .P.  F )  =  ( F  .P.  ( A  +P.  D ) )
4 mulcompr 8827 . . . . . . . . . . 11  |-  ( A  .P.  F )  =  ( F  .P.  A
)
5 mulcompr 8827 . . . . . . . . . . 11  |-  ( D  .P.  F )  =  ( F  .P.  D
)
64, 5oveq12i 6026 . . . . . . . . . 10  |-  ( ( A  .P.  F )  +P.  ( D  .P.  F ) )  =  ( ( F  .P.  A
)  +P.  ( F  .P.  D ) )
72, 3, 63eqtr4i 2411 . . . . . . . . 9  |-  ( ( A  +P.  D )  .P.  F )  =  ( ( A  .P.  F )  +P.  ( D  .P.  F ) )
8 distrpr 8832 . . . . . . . . . 10  |-  ( F  .P.  ( B  +P.  C ) )  =  ( ( F  .P.  B
)  +P.  ( F  .P.  C ) )
9 mulcompr 8827 . . . . . . . . . 10  |-  ( ( B  +P.  C )  .P.  F )  =  ( F  .P.  ( B  +P.  C ) )
10 mulcompr 8827 . . . . . . . . . . 11  |-  ( B  .P.  F )  =  ( F  .P.  B
)
11 mulcompr 8827 . . . . . . . . . . 11  |-  ( C  .P.  F )  =  ( F  .P.  C
)
1210, 11oveq12i 6026 . . . . . . . . . 10  |-  ( ( B  .P.  F )  +P.  ( C  .P.  F ) )  =  ( ( F  .P.  B
)  +P.  ( F  .P.  C ) )
138, 9, 123eqtr4i 2411 . . . . . . . . 9  |-  ( ( B  +P.  C )  .P.  F )  =  ( ( B  .P.  F )  +P.  ( C  .P.  F ) )
141, 7, 133eqtr3g 2436 . . . . . . . 8  |-  ( ( A  +P.  D )  =  ( B  +P.  C )  ->  ( ( A  .P.  F )  +P.  ( D  .P.  F
) )  =  ( ( B  .P.  F
)  +P.  ( C  .P.  F ) ) )
1514oveq1d 6029 . . . . . . 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 8826 . . . . . . . 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 6022 . . . . . . . . . 10  |-  ( ( F  +P.  S )  =  ( G  +P.  R )  ->  ( C  .P.  ( F  +P.  S
) )  =  ( C  .P.  ( G  +P.  R ) ) )
18 distrpr 8832 . . . . . . . . . 10  |-  ( C  .P.  ( F  +P.  S ) )  =  ( ( C  .P.  F
)  +P.  ( C  .P.  S ) )
19 distrpr 8832 . . . . . . . . . 10  |-  ( C  .P.  ( G  +P.  R ) )  =  ( ( C  .P.  G
)  +P.  ( C  .P.  R ) )
2017, 18, 193eqtr3g 2436 . . . . . . . . 9  |-  ( ( F  +P.  S )  =  ( G  +P.  R )  ->  ( ( C  .P.  F )  +P.  ( C  .P.  S
) )  =  ( ( C  .P.  G
)  +P.  ( C  .P.  R ) ) )
2120oveq2d 6030 . . . . . . . 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 2425 . . . . . . 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 2433 . . . . . 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 6039 . . . . . . 7  |-  ( A  .P.  F )  e. 
_V
25 ovex 6039 . . . . . . 7  |-  ( D  .P.  F )  e. 
_V
26 ovex 6039 . . . . . . 7  |-  ( C  .P.  S )  e. 
_V
27 addcompr 8825 . . . . . . 7  |-  ( x  +P.  y )  =  ( y  +P.  x
)
28 addasspr 8826 . . . . . . 7  |-  ( ( x  +P.  y )  +P.  z )  =  ( x  +P.  (
y  +P.  z )
)
2924, 25, 26, 27, 28caov32 6207 . . . . . 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 6039 . . . . . . 7  |-  ( B  .P.  F )  e. 
_V
31 ovex 6039 . . . . . . 7  |-  ( C  .P.  G )  e. 
_V
32 ovex 6039 . . . . . . 7  |-  ( C  .P.  R )  e. 
_V
3330, 31, 32, 27, 28caov12 6208 . . . . . 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 2436 . . . . 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 6030 . . . 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 6022 . . . . . . . . . . 11  |-  ( ( F  +P.  S )  =  ( G  +P.  R )  ->  ( D  .P.  ( F  +P.  S
) )  =  ( D  .P.  ( G  +P.  R ) ) )
37 distrpr 8832 . . . . . . . . . . 11  |-  ( D  .P.  ( F  +P.  S ) )  =  ( ( D  .P.  F
)  +P.  ( D  .P.  S ) )
38 distrpr 8832 . . . . . . . . . . 11  |-  ( D  .P.  ( G  +P.  R ) )  =  ( ( D  .P.  G
)  +P.  ( D  .P.  R ) )
3936, 37, 383eqtr3g 2436 . . . . . . . . . 10  |-  ( ( F  +P.  S )  =  ( G  +P.  R )  ->  ( ( D  .P.  F )  +P.  ( D  .P.  S
) )  =  ( ( D  .P.  G
)  +P.  ( D  .P.  R ) ) )
4039oveq2d 6030 . . . . . . . . 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 8826 . . . . . . . . 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 2431 . . . . . . . 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 6021 . . . . . . . . . 10  |-  ( ( A  +P.  D )  =  ( B  +P.  C )  ->  ( ( A  +P.  D )  .P. 
G )  =  ( ( B  +P.  C
)  .P.  G )
)
44 distrpr 8832 . . . . . . . . . . 11  |-  ( G  .P.  ( A  +P.  D ) )  =  ( ( G  .P.  A
)  +P.  ( G  .P.  D ) )
45 mulcompr 8827 . . . . . . . . . . 11  |-  ( ( A  +P.  D )  .P.  G )  =  ( G  .P.  ( A  +P.  D ) )
46 mulcompr 8827 . . . . . . . . . . . 12  |-  ( A  .P.  G )  =  ( G  .P.  A
)
47 mulcompr 8827 . . . . . . . . . . . 12  |-  ( D  .P.  G )  =  ( G  .P.  D
)
4846, 47oveq12i 6026 . . . . . . . . . . 11  |-  ( ( A  .P.  G )  +P.  ( D  .P.  G ) )  =  ( ( G  .P.  A
)  +P.  ( G  .P.  D ) )
4944, 45, 483eqtr4i 2411 . . . . . . . . . 10  |-  ( ( A  +P.  D )  .P.  G )  =  ( ( A  .P.  G )  +P.  ( D  .P.  G ) )
50 distrpr 8832 . . . . . . . . . . 11  |-  ( G  .P.  ( B  +P.  C ) )  =  ( ( G  .P.  B
)  +P.  ( G  .P.  C ) )
51 mulcompr 8827 . . . . . . . . . . 11  |-  ( ( B  +P.  C )  .P.  G )  =  ( G  .P.  ( B  +P.  C ) )
52 mulcompr 8827 . . . . . . . . . . . 12  |-  ( B  .P.  G )  =  ( G  .P.  B
)
53 mulcompr 8827 . . . . . . . . . . . 12  |-  ( C  .P.  G )  =  ( G  .P.  C
)
5452, 53oveq12i 6026 . . . . . . . . . . 11  |-  ( ( B  .P.  G )  +P.  ( C  .P.  G ) )  =  ( ( G  .P.  B
)  +P.  ( G  .P.  C ) )
5550, 51, 543eqtr4i 2411 . . . . . . . . . 10  |-  ( ( B  +P.  C )  .P.  G )  =  ( ( B  .P.  G )  +P.  ( C  .P.  G ) )
5643, 49, 553eqtr3g 2436 . . . . . . . . 9  |-  ( ( A  +P.  D )  =  ( B  +P.  C )  ->  ( ( A  .P.  G )  +P.  ( D  .P.  G
) )  =  ( ( B  .P.  G
)  +P.  ( C  .P.  G ) ) )
5756oveq1d 6029 . . . . . . . 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 2435 . . . . . . 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 6039 . . . . . . . 8  |-  ( A  .P.  G )  e. 
_V
60 ovex 6039 . . . . . . . 8  |-  ( D  .P.  S )  e. 
_V
6159, 25, 60, 27, 28caov12 6208 . . . . . . 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 6039 . . . . . . . 8  |-  ( B  .P.  G )  e. 
_V
63 ovex 6039 . . . . . . . 8  |-  ( D  .P.  R )  e. 
_V
6462, 31, 63, 27, 28caov32 6207 . . . . . . 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 2436 . . . . . 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 6029 . . . . 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 8826 . . . . 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 2429 . . . 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 2416 . . 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 6039 . . . 4  |-  ( ( B  .P.  G )  +P.  ( D  .P.  R ) )  e.  _V
71 ovex 6039 . . . 4  |-  ( ( A  .P.  F )  +P.  ( C  .P.  S ) )  e.  _V
7270, 71, 25, 27, 28caov13 6210 . . 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 8826 . . 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 2436 . 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 6211 . . 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 6025 . 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 6213 . . 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 6025 . 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 2436 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 359    = wceq 1649  (class class class)co 6014    +P. cpp 8663    .P. cmp 8664
This theorem is referenced by:  mulcmpblnr  8876
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2362  ax-sep 4265  ax-nul 4273  ax-pow 4312  ax-pr 4338  ax-un 4635  ax-inf2 7523
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2236  df-mo 2237  df-clab 2368  df-cleq 2374  df-clel 2377  df-nfc 2506  df-ne 2546  df-ral 2648  df-rex 2649  df-reu 2650  df-rmo 2651  df-rab 2652  df-v 2895  df-sbc 3099  df-csb 3189  df-dif 3260  df-un 3262  df-in 3264  df-ss 3271  df-pss 3273  df-nul 3566  df-if 3677  df-pw 3738  df-sn 3757  df-pr 3758  df-tp 3759  df-op 3760  df-uni 3952  df-iun 4031  df-br 4148  df-opab 4202  df-mpt 4203  df-tr 4238  df-eprel 4429  df-id 4433  df-po 4438  df-so 4439  df-fr 4476  df-we 4478  df-ord 4519  df-on 4520  df-lim 4521  df-suc 4522  df-om 4780  df-xp 4818  df-rel 4819  df-cnv 4820  df-co 4821  df-dm 4822  df-rn 4823  df-res 4824  df-ima 4825  df-iota 5352  df-fun 5390  df-fn 5391  df-f 5392  df-f1 5393  df-fo 5394  df-f1o 5395  df-fv 5396  df-ov 6017  df-oprab 6018  df-mpt2 6019  df-1st 6282  df-2nd 6283  df-recs 6563  df-rdg 6598  df-1o 6654  df-oadd 6658  df-omul 6659  df-er 6835  df-ni 8676  df-pli 8677  df-mi 8678  df-lti 8679  df-plpq 8712  df-mpq 8713  df-ltpq 8714  df-enq 8715  df-nq 8716  df-erq 8717  df-plq 8718  df-mq 8719  df-1nq 8720  df-rq 8721  df-ltnq 8722  df-np 8785  df-plp 8787  df-mp 8788
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