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Theorem mulcmpblnrlem 9436
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 6282 . . . . . . . . 9  |-  ( ( A  +P.  D )  =  ( B  +P.  C )  ->  ( ( A  +P.  D )  .P. 
F )  =  ( ( B  +P.  C
)  .P.  F )
)
2 distrpr 9395 . . . . . . . . . 10  |-  ( F  .P.  ( A  +P.  D ) )  =  ( ( F  .P.  A
)  +P.  ( F  .P.  D ) )
3 mulcompr 9390 . . . . . . . . . 10  |-  ( ( A  +P.  D )  .P.  F )  =  ( F  .P.  ( A  +P.  D ) )
4 mulcompr 9390 . . . . . . . . . . 11  |-  ( A  .P.  F )  =  ( F  .P.  A
)
5 mulcompr 9390 . . . . . . . . . . 11  |-  ( D  .P.  F )  =  ( F  .P.  D
)
64, 5oveq12i 6287 . . . . . . . . . 10  |-  ( ( A  .P.  F )  +P.  ( D  .P.  F ) )  =  ( ( F  .P.  A
)  +P.  ( F  .P.  D ) )
72, 3, 63eqtr4i 2499 . . . . . . . . 9  |-  ( ( A  +P.  D )  .P.  F )  =  ( ( A  .P.  F )  +P.  ( D  .P.  F ) )
8 distrpr 9395 . . . . . . . . . 10  |-  ( F  .P.  ( B  +P.  C ) )  =  ( ( F  .P.  B
)  +P.  ( F  .P.  C ) )
9 mulcompr 9390 . . . . . . . . . 10  |-  ( ( B  +P.  C )  .P.  F )  =  ( F  .P.  ( B  +P.  C ) )
10 mulcompr 9390 . . . . . . . . . . 11  |-  ( B  .P.  F )  =  ( F  .P.  B
)
11 mulcompr 9390 . . . . . . . . . . 11  |-  ( C  .P.  F )  =  ( F  .P.  C
)
1210, 11oveq12i 6287 . . . . . . . . . 10  |-  ( ( B  .P.  F )  +P.  ( C  .P.  F ) )  =  ( ( F  .P.  B
)  +P.  ( F  .P.  C ) )
138, 9, 123eqtr4i 2499 . . . . . . . . 9  |-  ( ( B  +P.  C )  .P.  F )  =  ( ( B  .P.  F )  +P.  ( C  .P.  F ) )
141, 7, 133eqtr3g 2524 . . . . . . . 8  |-  ( ( A  +P.  D )  =  ( B  +P.  C )  ->  ( ( A  .P.  F )  +P.  ( D  .P.  F
) )  =  ( ( B  .P.  F
)  +P.  ( C  .P.  F ) ) )
1514oveq1d 6290 . . . . . . 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 9389 . . . . . . . 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 6283 . . . . . . . . . 10  |-  ( ( F  +P.  S )  =  ( G  +P.  R )  ->  ( C  .P.  ( F  +P.  S
) )  =  ( C  .P.  ( G  +P.  R ) ) )
18 distrpr 9395 . . . . . . . . . 10  |-  ( C  .P.  ( F  +P.  S ) )  =  ( ( C  .P.  F
)  +P.  ( C  .P.  S ) )
19 distrpr 9395 . . . . . . . . . 10  |-  ( C  .P.  ( G  +P.  R ) )  =  ( ( C  .P.  G
)  +P.  ( C  .P.  R ) )
2017, 18, 193eqtr3g 2524 . . . . . . . . 9  |-  ( ( F  +P.  S )  =  ( G  +P.  R )  ->  ( ( C  .P.  F )  +P.  ( C  .P.  S
) )  =  ( ( C  .P.  G
)  +P.  ( C  .P.  R ) ) )
2120oveq2d 6291 . . . . . . . 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 2513 . . . . . . 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 2521 . . . . . 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 6300 . . . . . . 7  |-  ( A  .P.  F )  e. 
_V
25 ovex 6300 . . . . . . 7  |-  ( D  .P.  F )  e. 
_V
26 ovex 6300 . . . . . . 7  |-  ( C  .P.  S )  e. 
_V
27 addcompr 9388 . . . . . . 7  |-  ( x  +P.  y )  =  ( y  +P.  x
)
28 addasspr 9389 . . . . . . 7  |-  ( ( x  +P.  y )  +P.  z )  =  ( x  +P.  (
y  +P.  z )
)
2924, 25, 26, 27, 28caov32 6477 . . . . . 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 6300 . . . . . . 7  |-  ( B  .P.  F )  e. 
_V
31 ovex 6300 . . . . . . 7  |-  ( C  .P.  G )  e. 
_V
32 ovex 6300 . . . . . . 7  |-  ( C  .P.  R )  e. 
_V
3330, 31, 32, 27, 28caov12 6478 . . . . . 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 2524 . . . . 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 6291 . . . 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 6283 . . . . . . . . . . 11  |-  ( ( F  +P.  S )  =  ( G  +P.  R )  ->  ( D  .P.  ( F  +P.  S
) )  =  ( D  .P.  ( G  +P.  R ) ) )
37 distrpr 9395 . . . . . . . . . . 11  |-  ( D  .P.  ( F  +P.  S ) )  =  ( ( D  .P.  F
)  +P.  ( D  .P.  S ) )
38 distrpr 9395 . . . . . . . . . . 11  |-  ( D  .P.  ( G  +P.  R ) )  =  ( ( D  .P.  G
)  +P.  ( D  .P.  R ) )
3936, 37, 383eqtr3g 2524 . . . . . . . . . 10  |-  ( ( F  +P.  S )  =  ( G  +P.  R )  ->  ( ( D  .P.  F )  +P.  ( D  .P.  S
) )  =  ( ( D  .P.  G
)  +P.  ( D  .P.  R ) ) )
4039oveq2d 6291 . . . . . . . . 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 9389 . . . . . . . . 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 2519 . . . . . . . 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 6282 . . . . . . . . . 10  |-  ( ( A  +P.  D )  =  ( B  +P.  C )  ->  ( ( A  +P.  D )  .P. 
G )  =  ( ( B  +P.  C
)  .P.  G )
)
44 distrpr 9395 . . . . . . . . . . 11  |-  ( G  .P.  ( A  +P.  D ) )  =  ( ( G  .P.  A
)  +P.  ( G  .P.  D ) )
45 mulcompr 9390 . . . . . . . . . . 11  |-  ( ( A  +P.  D )  .P.  G )  =  ( G  .P.  ( A  +P.  D ) )
46 mulcompr 9390 . . . . . . . . . . . 12  |-  ( A  .P.  G )  =  ( G  .P.  A
)
47 mulcompr 9390 . . . . . . . . . . . 12  |-  ( D  .P.  G )  =  ( G  .P.  D
)
4846, 47oveq12i 6287 . . . . . . . . . . 11  |-  ( ( A  .P.  G )  +P.  ( D  .P.  G ) )  =  ( ( G  .P.  A
)  +P.  ( G  .P.  D ) )
4944, 45, 483eqtr4i 2499 . . . . . . . . . 10  |-  ( ( A  +P.  D )  .P.  G )  =  ( ( A  .P.  G )  +P.  ( D  .P.  G ) )
50 distrpr 9395 . . . . . . . . . . 11  |-  ( G  .P.  ( B  +P.  C ) )  =  ( ( G  .P.  B
)  +P.  ( G  .P.  C ) )
51 mulcompr 9390 . . . . . . . . . . 11  |-  ( ( B  +P.  C )  .P.  G )  =  ( G  .P.  ( B  +P.  C ) )
52 mulcompr 9390 . . . . . . . . . . . 12  |-  ( B  .P.  G )  =  ( G  .P.  B
)
53 mulcompr 9390 . . . . . . . . . . . 12  |-  ( C  .P.  G )  =  ( G  .P.  C
)
5452, 53oveq12i 6287 . . . . . . . . . . 11  |-  ( ( B  .P.  G )  +P.  ( C  .P.  G ) )  =  ( ( G  .P.  B
)  +P.  ( G  .P.  C ) )
5550, 51, 543eqtr4i 2499 . . . . . . . . . 10  |-  ( ( B  +P.  C )  .P.  G )  =  ( ( B  .P.  G )  +P.  ( C  .P.  G ) )
5643, 49, 553eqtr3g 2524 . . . . . . . . 9  |-  ( ( A  +P.  D )  =  ( B  +P.  C )  ->  ( ( A  .P.  G )  +P.  ( D  .P.  G
) )  =  ( ( B  .P.  G
)  +P.  ( C  .P.  G ) ) )
5756oveq1d 6290 . . . . . . . 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 2523 . . . . . . 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 6300 . . . . . . . 8  |-  ( A  .P.  G )  e. 
_V
60 ovex 6300 . . . . . . . 8  |-  ( D  .P.  S )  e. 
_V
6159, 25, 60, 27, 28caov12 6478 . . . . . . 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 6300 . . . . . . . 8  |-  ( B  .P.  G )  e. 
_V
63 ovex 6300 . . . . . . . 8  |-  ( D  .P.  R )  e. 
_V
6462, 31, 63, 27, 28caov32 6477 . . . . . . 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 2524 . . . . . 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 6290 . . . . 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 9389 . . . . 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 2517 . . . 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 2504 . . 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 6300 . . . 4  |-  ( ( B  .P.  G )  +P.  ( D  .P.  R ) )  e.  _V
71 ovex 6300 . . . 4  |-  ( ( A  .P.  F )  +P.  ( C  .P.  S ) )  e.  _V
7270, 71, 25, 27, 28caov13 6480 . . 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 9389 . . 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 2524 . 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 6481 . . 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 6286 . 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 6483 . . 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 6286 . 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 2524 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 1374  (class class class)co 6275    +P. cpp 9228    .P. cmp 9229
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-8 1764  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1961  ax-ext 2438  ax-sep 4561  ax-nul 4569  ax-pow 4618  ax-pr 4679  ax-un 6567  ax-inf2 8047
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 969  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2272  df-mo 2273  df-clab 2446  df-cleq 2452  df-clel 2455  df-nfc 2610  df-ne 2657  df-ral 2812  df-rex 2813  df-reu 2814  df-rmo 2815  df-rab 2816  df-v 3108  df-sbc 3325  df-csb 3429  df-dif 3472  df-un 3474  df-in 3476  df-ss 3483  df-pss 3485  df-nul 3779  df-if 3933  df-pw 4005  df-sn 4021  df-pr 4023  df-tp 4025  df-op 4027  df-uni 4239  df-iun 4320  df-br 4441  df-opab 4499  df-mpt 4500  df-tr 4534  df-eprel 4784  df-id 4788  df-po 4793  df-so 4794  df-fr 4831  df-we 4833  df-ord 4874  df-on 4875  df-lim 4876  df-suc 4877  df-xp 4998  df-rel 4999  df-cnv 5000  df-co 5001  df-dm 5002  df-rn 5003  df-res 5004  df-ima 5005  df-iota 5542  df-fun 5581  df-fn 5582  df-f 5583  df-f1 5584  df-fo 5585  df-f1o 5586  df-fv 5587  df-ov 6278  df-oprab 6279  df-mpt2 6280  df-om 6672  df-1st 6774  df-2nd 6775  df-recs 7032  df-rdg 7066  df-1o 7120  df-oadd 7124  df-omul 7125  df-er 7301  df-ni 9239  df-pli 9240  df-mi 9241  df-lti 9242  df-plpq 9275  df-mpq 9276  df-ltpq 9277  df-enq 9278  df-nq 9279  df-erq 9280  df-plq 9281  df-mq 9282  df-1nq 9283  df-rq 9284  df-ltnq 9285  df-np 9348  df-plp 9350  df-mp 9351
This theorem is referenced by:  mulcmpblnr  9437
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