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Theorem mulcmpblnrlem 9261
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 6119 . . . . . . . . 9  |-  ( ( A  +P.  D )  =  ( B  +P.  C )  ->  ( ( A  +P.  D )  .P. 
F )  =  ( ( B  +P.  C
)  .P.  F )
)
2 distrpr 9218 . . . . . . . . . 10  |-  ( F  .P.  ( A  +P.  D ) )  =  ( ( F  .P.  A
)  +P.  ( F  .P.  D ) )
3 mulcompr 9213 . . . . . . . . . 10  |-  ( ( A  +P.  D )  .P.  F )  =  ( F  .P.  ( A  +P.  D ) )
4 mulcompr 9213 . . . . . . . . . . 11  |-  ( A  .P.  F )  =  ( F  .P.  A
)
5 mulcompr 9213 . . . . . . . . . . 11  |-  ( D  .P.  F )  =  ( F  .P.  D
)
64, 5oveq12i 6124 . . . . . . . . . 10  |-  ( ( A  .P.  F )  +P.  ( D  .P.  F ) )  =  ( ( F  .P.  A
)  +P.  ( F  .P.  D ) )
72, 3, 63eqtr4i 2473 . . . . . . . . 9  |-  ( ( A  +P.  D )  .P.  F )  =  ( ( A  .P.  F )  +P.  ( D  .P.  F ) )
8 distrpr 9218 . . . . . . . . . 10  |-  ( F  .P.  ( B  +P.  C ) )  =  ( ( F  .P.  B
)  +P.  ( F  .P.  C ) )
9 mulcompr 9213 . . . . . . . . . 10  |-  ( ( B  +P.  C )  .P.  F )  =  ( F  .P.  ( B  +P.  C ) )
10 mulcompr 9213 . . . . . . . . . . 11  |-  ( B  .P.  F )  =  ( F  .P.  B
)
11 mulcompr 9213 . . . . . . . . . . 11  |-  ( C  .P.  F )  =  ( F  .P.  C
)
1210, 11oveq12i 6124 . . . . . . . . . 10  |-  ( ( B  .P.  F )  +P.  ( C  .P.  F ) )  =  ( ( F  .P.  B
)  +P.  ( F  .P.  C ) )
138, 9, 123eqtr4i 2473 . . . . . . . . 9  |-  ( ( B  +P.  C )  .P.  F )  =  ( ( B  .P.  F )  +P.  ( C  .P.  F ) )
141, 7, 133eqtr3g 2498 . . . . . . . 8  |-  ( ( A  +P.  D )  =  ( B  +P.  C )  ->  ( ( A  .P.  F )  +P.  ( D  .P.  F
) )  =  ( ( B  .P.  F
)  +P.  ( C  .P.  F ) ) )
1514oveq1d 6127 . . . . . . 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 9212 . . . . . . . 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 6120 . . . . . . . . . 10  |-  ( ( F  +P.  S )  =  ( G  +P.  R )  ->  ( C  .P.  ( F  +P.  S
) )  =  ( C  .P.  ( G  +P.  R ) ) )
18 distrpr 9218 . . . . . . . . . 10  |-  ( C  .P.  ( F  +P.  S ) )  =  ( ( C  .P.  F
)  +P.  ( C  .P.  S ) )
19 distrpr 9218 . . . . . . . . . 10  |-  ( C  .P.  ( G  +P.  R ) )  =  ( ( C  .P.  G
)  +P.  ( C  .P.  R ) )
2017, 18, 193eqtr3g 2498 . . . . . . . . 9  |-  ( ( F  +P.  S )  =  ( G  +P.  R )  ->  ( ( C  .P.  F )  +P.  ( C  .P.  S
) )  =  ( ( C  .P.  G
)  +P.  ( C  .P.  R ) ) )
2120oveq2d 6128 . . . . . . . 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 2487 . . . . . . 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 2495 . . . . . 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 6137 . . . . . . 7  |-  ( A  .P.  F )  e. 
_V
25 ovex 6137 . . . . . . 7  |-  ( D  .P.  F )  e. 
_V
26 ovex 6137 . . . . . . 7  |-  ( C  .P.  S )  e. 
_V
27 addcompr 9211 . . . . . . 7  |-  ( x  +P.  y )  =  ( y  +P.  x
)
28 addasspr 9212 . . . . . . 7  |-  ( ( x  +P.  y )  +P.  z )  =  ( x  +P.  (
y  +P.  z )
)
2924, 25, 26, 27, 28caov32 6311 . . . . . 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 6137 . . . . . . 7  |-  ( B  .P.  F )  e. 
_V
31 ovex 6137 . . . . . . 7  |-  ( C  .P.  G )  e. 
_V
32 ovex 6137 . . . . . . 7  |-  ( C  .P.  R )  e. 
_V
3330, 31, 32, 27, 28caov12 6312 . . . . . 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 2498 . . . . 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 6128 . . . 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 6120 . . . . . . . . . . 11  |-  ( ( F  +P.  S )  =  ( G  +P.  R )  ->  ( D  .P.  ( F  +P.  S
) )  =  ( D  .P.  ( G  +P.  R ) ) )
37 distrpr 9218 . . . . . . . . . . 11  |-  ( D  .P.  ( F  +P.  S ) )  =  ( ( D  .P.  F
)  +P.  ( D  .P.  S ) )
38 distrpr 9218 . . . . . . . . . . 11  |-  ( D  .P.  ( G  +P.  R ) )  =  ( ( D  .P.  G
)  +P.  ( D  .P.  R ) )
3936, 37, 383eqtr3g 2498 . . . . . . . . . 10  |-  ( ( F  +P.  S )  =  ( G  +P.  R )  ->  ( ( D  .P.  F )  +P.  ( D  .P.  S
) )  =  ( ( D  .P.  G
)  +P.  ( D  .P.  R ) ) )
4039oveq2d 6128 . . . . . . . . 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 9212 . . . . . . . . 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 2493 . . . . . . . 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 6119 . . . . . . . . . 10  |-  ( ( A  +P.  D )  =  ( B  +P.  C )  ->  ( ( A  +P.  D )  .P. 
G )  =  ( ( B  +P.  C
)  .P.  G )
)
44 distrpr 9218 . . . . . . . . . . 11  |-  ( G  .P.  ( A  +P.  D ) )  =  ( ( G  .P.  A
)  +P.  ( G  .P.  D ) )
45 mulcompr 9213 . . . . . . . . . . 11  |-  ( ( A  +P.  D )  .P.  G )  =  ( G  .P.  ( A  +P.  D ) )
46 mulcompr 9213 . . . . . . . . . . . 12  |-  ( A  .P.  G )  =  ( G  .P.  A
)
47 mulcompr 9213 . . . . . . . . . . . 12  |-  ( D  .P.  G )  =  ( G  .P.  D
)
4846, 47oveq12i 6124 . . . . . . . . . . 11  |-  ( ( A  .P.  G )  +P.  ( D  .P.  G ) )  =  ( ( G  .P.  A
)  +P.  ( G  .P.  D ) )
4944, 45, 483eqtr4i 2473 . . . . . . . . . 10  |-  ( ( A  +P.  D )  .P.  G )  =  ( ( A  .P.  G )  +P.  ( D  .P.  G ) )
50 distrpr 9218 . . . . . . . . . . 11  |-  ( G  .P.  ( B  +P.  C ) )  =  ( ( G  .P.  B
)  +P.  ( G  .P.  C ) )
51 mulcompr 9213 . . . . . . . . . . 11  |-  ( ( B  +P.  C )  .P.  G )  =  ( G  .P.  ( B  +P.  C ) )
52 mulcompr 9213 . . . . . . . . . . . 12  |-  ( B  .P.  G )  =  ( G  .P.  B
)
53 mulcompr 9213 . . . . . . . . . . . 12  |-  ( C  .P.  G )  =  ( G  .P.  C
)
5452, 53oveq12i 6124 . . . . . . . . . . 11  |-  ( ( B  .P.  G )  +P.  ( C  .P.  G ) )  =  ( ( G  .P.  B
)  +P.  ( G  .P.  C ) )
5550, 51, 543eqtr4i 2473 . . . . . . . . . 10  |-  ( ( B  +P.  C )  .P.  G )  =  ( ( B  .P.  G )  +P.  ( C  .P.  G ) )
5643, 49, 553eqtr3g 2498 . . . . . . . . 9  |-  ( ( A  +P.  D )  =  ( B  +P.  C )  ->  ( ( A  .P.  G )  +P.  ( D  .P.  G
) )  =  ( ( B  .P.  G
)  +P.  ( C  .P.  G ) ) )
5756oveq1d 6127 . . . . . . . 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 2497 . . . . . . 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 6137 . . . . . . . 8  |-  ( A  .P.  G )  e. 
_V
60 ovex 6137 . . . . . . . 8  |-  ( D  .P.  S )  e. 
_V
6159, 25, 60, 27, 28caov12 6312 . . . . . . 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 6137 . . . . . . . 8  |-  ( B  .P.  G )  e. 
_V
63 ovex 6137 . . . . . . . 8  |-  ( D  .P.  R )  e. 
_V
6462, 31, 63, 27, 28caov32 6311 . . . . . . 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 2498 . . . . . 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 6127 . . . . 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 9212 . . . . 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 2491 . . . 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 2478 . . 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 6137 . . . 4  |-  ( ( B  .P.  G )  +P.  ( D  .P.  R ) )  e.  _V
71 ovex 6137 . . . 4  |-  ( ( A  .P.  F )  +P.  ( C  .P.  S ) )  e.  _V
7270, 71, 25, 27, 28caov13 6314 . . 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 9212 . . 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 2498 . 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 6315 . . 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 6123 . 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 6317 . . 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 6123 . 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 2498 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 setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1369  (class class class)co 6112    +P. cpp 9049    .P. cmp 9050
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4434  ax-nul 4442  ax-pow 4491  ax-pr 4552  ax-un 6393  ax-inf2 7868
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2577  df-ne 2622  df-ral 2741  df-rex 2742  df-reu 2743  df-rmo 2744  df-rab 2745  df-v 2995  df-sbc 3208  df-csb 3310  df-dif 3352  df-un 3354  df-in 3356  df-ss 3363  df-pss 3365  df-nul 3659  df-if 3813  df-pw 3883  df-sn 3899  df-pr 3901  df-tp 3903  df-op 3905  df-uni 4113  df-iun 4194  df-br 4314  df-opab 4372  df-mpt 4373  df-tr 4407  df-eprel 4653  df-id 4657  df-po 4662  df-so 4663  df-fr 4700  df-we 4702  df-ord 4743  df-on 4744  df-lim 4745  df-suc 4746  df-xp 4867  df-rel 4868  df-cnv 4869  df-co 4870  df-dm 4871  df-rn 4872  df-res 4873  df-ima 4874  df-iota 5402  df-fun 5441  df-fn 5442  df-f 5443  df-f1 5444  df-fo 5445  df-f1o 5446  df-fv 5447  df-ov 6115  df-oprab 6116  df-mpt2 6117  df-om 6498  df-1st 6598  df-2nd 6599  df-recs 6853  df-rdg 6887  df-1o 6941  df-oadd 6945  df-omul 6946  df-er 7122  df-ni 9062  df-pli 9063  df-mi 9064  df-lti 9065  df-plpq 9098  df-mpq 9099  df-ltpq 9100  df-enq 9101  df-nq 9102  df-erq 9103  df-plq 9104  df-mq 9105  df-1nq 9106  df-rq 9107  df-ltnq 9108  df-np 9171  df-plp 9173  df-mp 9174
This theorem is referenced by:  mulcmpblnr  9262
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