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Theorem mulcmpblnrlem 8940
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 6080 . . . . . . . . 9  |-  ( ( A  +P.  D )  =  ( B  +P.  C )  ->  ( ( A  +P.  D )  .P. 
F )  =  ( ( B  +P.  C
)  .P.  F )
)
2 distrpr 8897 . . . . . . . . . 10  |-  ( F  .P.  ( A  +P.  D ) )  =  ( ( F  .P.  A
)  +P.  ( F  .P.  D ) )
3 mulcompr 8892 . . . . . . . . . 10  |-  ( ( A  +P.  D )  .P.  F )  =  ( F  .P.  ( A  +P.  D ) )
4 mulcompr 8892 . . . . . . . . . . 11  |-  ( A  .P.  F )  =  ( F  .P.  A
)
5 mulcompr 8892 . . . . . . . . . . 11  |-  ( D  .P.  F )  =  ( F  .P.  D
)
64, 5oveq12i 6085 . . . . . . . . . 10  |-  ( ( A  .P.  F )  +P.  ( D  .P.  F ) )  =  ( ( F  .P.  A
)  +P.  ( F  .P.  D ) )
72, 3, 63eqtr4i 2465 . . . . . . . . 9  |-  ( ( A  +P.  D )  .P.  F )  =  ( ( A  .P.  F )  +P.  ( D  .P.  F ) )
8 distrpr 8897 . . . . . . . . . 10  |-  ( F  .P.  ( B  +P.  C ) )  =  ( ( F  .P.  B
)  +P.  ( F  .P.  C ) )
9 mulcompr 8892 . . . . . . . . . 10  |-  ( ( B  +P.  C )  .P.  F )  =  ( F  .P.  ( B  +P.  C ) )
10 mulcompr 8892 . . . . . . . . . . 11  |-  ( B  .P.  F )  =  ( F  .P.  B
)
11 mulcompr 8892 . . . . . . . . . . 11  |-  ( C  .P.  F )  =  ( F  .P.  C
)
1210, 11oveq12i 6085 . . . . . . . . . 10  |-  ( ( B  .P.  F )  +P.  ( C  .P.  F ) )  =  ( ( F  .P.  B
)  +P.  ( F  .P.  C ) )
138, 9, 123eqtr4i 2465 . . . . . . . . 9  |-  ( ( B  +P.  C )  .P.  F )  =  ( ( B  .P.  F )  +P.  ( C  .P.  F ) )
141, 7, 133eqtr3g 2490 . . . . . . . 8  |-  ( ( A  +P.  D )  =  ( B  +P.  C )  ->  ( ( A  .P.  F )  +P.  ( D  .P.  F
) )  =  ( ( B  .P.  F
)  +P.  ( C  .P.  F ) ) )
1514oveq1d 6088 . . . . . . 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 8891 . . . . . . . 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 6081 . . . . . . . . . 10  |-  ( ( F  +P.  S )  =  ( G  +P.  R )  ->  ( C  .P.  ( F  +P.  S
) )  =  ( C  .P.  ( G  +P.  R ) ) )
18 distrpr 8897 . . . . . . . . . 10  |-  ( C  .P.  ( F  +P.  S ) )  =  ( ( C  .P.  F
)  +P.  ( C  .P.  S ) )
19 distrpr 8897 . . . . . . . . . 10  |-  ( C  .P.  ( G  +P.  R ) )  =  ( ( C  .P.  G
)  +P.  ( C  .P.  R ) )
2017, 18, 193eqtr3g 2490 . . . . . . . . 9  |-  ( ( F  +P.  S )  =  ( G  +P.  R )  ->  ( ( C  .P.  F )  +P.  ( C  .P.  S
) )  =  ( ( C  .P.  G
)  +P.  ( C  .P.  R ) ) )
2120oveq2d 6089 . . . . . . . 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 2479 . . . . . . 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 2487 . . . . . 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 6098 . . . . . . 7  |-  ( A  .P.  F )  e. 
_V
25 ovex 6098 . . . . . . 7  |-  ( D  .P.  F )  e. 
_V
26 ovex 6098 . . . . . . 7  |-  ( C  .P.  S )  e. 
_V
27 addcompr 8890 . . . . . . 7  |-  ( x  +P.  y )  =  ( y  +P.  x
)
28 addasspr 8891 . . . . . . 7  |-  ( ( x  +P.  y )  +P.  z )  =  ( x  +P.  (
y  +P.  z )
)
2924, 25, 26, 27, 28caov32 6266 . . . . . 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 6098 . . . . . . 7  |-  ( B  .P.  F )  e. 
_V
31 ovex 6098 . . . . . . 7  |-  ( C  .P.  G )  e. 
_V
32 ovex 6098 . . . . . . 7  |-  ( C  .P.  R )  e. 
_V
3330, 31, 32, 27, 28caov12 6267 . . . . . 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 2490 . . . . 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 6089 . . . 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 6081 . . . . . . . . . . 11  |-  ( ( F  +P.  S )  =  ( G  +P.  R )  ->  ( D  .P.  ( F  +P.  S
) )  =  ( D  .P.  ( G  +P.  R ) ) )
37 distrpr 8897 . . . . . . . . . . 11  |-  ( D  .P.  ( F  +P.  S ) )  =  ( ( D  .P.  F
)  +P.  ( D  .P.  S ) )
38 distrpr 8897 . . . . . . . . . . 11  |-  ( D  .P.  ( G  +P.  R ) )  =  ( ( D  .P.  G
)  +P.  ( D  .P.  R ) )
3936, 37, 383eqtr3g 2490 . . . . . . . . . 10  |-  ( ( F  +P.  S )  =  ( G  +P.  R )  ->  ( ( D  .P.  F )  +P.  ( D  .P.  S
) )  =  ( ( D  .P.  G
)  +P.  ( D  .P.  R ) ) )
4039oveq2d 6089 . . . . . . . . 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 8891 . . . . . . . . 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 2485 . . . . . . . 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 6080 . . . . . . . . . 10  |-  ( ( A  +P.  D )  =  ( B  +P.  C )  ->  ( ( A  +P.  D )  .P. 
G )  =  ( ( B  +P.  C
)  .P.  G )
)
44 distrpr 8897 . . . . . . . . . . 11  |-  ( G  .P.  ( A  +P.  D ) )  =  ( ( G  .P.  A
)  +P.  ( G  .P.  D ) )
45 mulcompr 8892 . . . . . . . . . . 11  |-  ( ( A  +P.  D )  .P.  G )  =  ( G  .P.  ( A  +P.  D ) )
46 mulcompr 8892 . . . . . . . . . . . 12  |-  ( A  .P.  G )  =  ( G  .P.  A
)
47 mulcompr 8892 . . . . . . . . . . . 12  |-  ( D  .P.  G )  =  ( G  .P.  D
)
4846, 47oveq12i 6085 . . . . . . . . . . 11  |-  ( ( A  .P.  G )  +P.  ( D  .P.  G ) )  =  ( ( G  .P.  A
)  +P.  ( G  .P.  D ) )
4944, 45, 483eqtr4i 2465 . . . . . . . . . 10  |-  ( ( A  +P.  D )  .P.  G )  =  ( ( A  .P.  G )  +P.  ( D  .P.  G ) )
50 distrpr 8897 . . . . . . . . . . 11  |-  ( G  .P.  ( B  +P.  C ) )  =  ( ( G  .P.  B
)  +P.  ( G  .P.  C ) )
51 mulcompr 8892 . . . . . . . . . . 11  |-  ( ( B  +P.  C )  .P.  G )  =  ( G  .P.  ( B  +P.  C ) )
52 mulcompr 8892 . . . . . . . . . . . 12  |-  ( B  .P.  G )  =  ( G  .P.  B
)
53 mulcompr 8892 . . . . . . . . . . . 12  |-  ( C  .P.  G )  =  ( G  .P.  C
)
5452, 53oveq12i 6085 . . . . . . . . . . 11  |-  ( ( B  .P.  G )  +P.  ( C  .P.  G ) )  =  ( ( G  .P.  B
)  +P.  ( G  .P.  C ) )
5550, 51, 543eqtr4i 2465 . . . . . . . . . 10  |-  ( ( B  +P.  C )  .P.  G )  =  ( ( B  .P.  G )  +P.  ( C  .P.  G ) )
5643, 49, 553eqtr3g 2490 . . . . . . . . 9  |-  ( ( A  +P.  D )  =  ( B  +P.  C )  ->  ( ( A  .P.  G )  +P.  ( D  .P.  G
) )  =  ( ( B  .P.  G
)  +P.  ( C  .P.  G ) ) )
5756oveq1d 6088 . . . . . . . 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 2489 . . . . . . 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 6098 . . . . . . . 8  |-  ( A  .P.  G )  e. 
_V
60 ovex 6098 . . . . . . . 8  |-  ( D  .P.  S )  e. 
_V
6159, 25, 60, 27, 28caov12 6267 . . . . . . 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 6098 . . . . . . . 8  |-  ( B  .P.  G )  e. 
_V
63 ovex 6098 . . . . . . . 8  |-  ( D  .P.  R )  e. 
_V
6462, 31, 63, 27, 28caov32 6266 . . . . . . 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 2490 . . . . . 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 6088 . . . . 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 8891 . . . . 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 2483 . . . 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 2470 . . 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 6098 . . . 4  |-  ( ( B  .P.  G )  +P.  ( D  .P.  R ) )  e.  _V
71 ovex 6098 . . . 4  |-  ( ( A  .P.  F )  +P.  ( C  .P.  S ) )  e.  _V
7270, 71, 25, 27, 28caov13 6269 . . 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 8891 . . 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 2490 . 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 6270 . . 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 6084 . 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 6272 . . 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 6084 . 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 2490 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 1652  (class class class)co 6073    +P. cpp 8728    .P. cmp 8729
This theorem is referenced by:  mulcmpblnr  8941
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 4693  ax-inf2 7588
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 4838  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-1st 6341  df-2nd 6342  df-recs 6625  df-rdg 6660  df-1o 6716  df-oadd 6720  df-omul 6721  df-er 6897  df-ni 8741  df-pli 8742  df-mi 8743  df-lti 8744  df-plpq 8777  df-mpq 8778  df-ltpq 8779  df-enq 8780  df-nq 8781  df-erq 8782  df-plq 8783  df-mq 8784  df-1nq 8785  df-rq 8786  df-ltnq 8787  df-np 8850  df-plp 8852  df-mp 8853
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