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Theorem mulcmpblnr 9355
Description: Lemma showing compatibility of multiplication. (Contributed by NM, 5-Sep-1995.) (New usage is discouraged.)
Assertion
Ref Expression
mulcmpblnr  |-  ( ( ( ( A  e. 
P.  /\  B  e.  P. )  /\  ( C  e.  P.  /\  D  e.  P. ) )  /\  ( ( F  e. 
P.  /\  G  e.  P. )  /\  ( R  e.  P.  /\  S  e.  P. ) ) )  ->  ( ( ( A  +P.  D )  =  ( B  +P.  C )  /\  ( F  +P.  S )  =  ( G  +P.  R
) )  ->  <. (
( A  .P.  F
)  +P.  ( B  .P.  G ) ) ,  ( ( A  .P.  G )  +P.  ( B  .P.  F ) )
>.  ~R  <. ( ( C  .P.  R )  +P.  ( D  .P.  S
) ) ,  ( ( C  .P.  S
)  +P.  ( D  .P.  R ) ) >.
) )

Proof of Theorem mulcmpblnr
StepHypRef Expression
1 mulclpr 9303 . . . . 5  |-  ( ( D  e.  P.  /\  F  e.  P. )  ->  ( D  .P.  F
)  e.  P. )
21ad2ant2lr 747 . . . 4  |-  ( ( ( C  e.  P.  /\  D  e.  P. )  /\  ( F  e.  P.  /\  G  e.  P. )
)  ->  ( D  .P.  F )  e.  P. )
3 rnlem 956 . . . . . . . . 9  |-  ( ( ( A  e.  P.  /\  B  e.  P. )  /\  ( F  e.  P.  /\  G  e.  P. )
)  <->  ( ( ( A  e.  P.  /\  F  e.  P. )  /\  ( B  e.  P.  /\  G  e.  P. )
)  /\  ( ( A  e.  P.  /\  G  e.  P. )  /\  ( B  e.  P.  /\  F  e.  P. ) ) ) )
4 mulclpr 9303 . . . . . . . . . . 11  |-  ( ( A  e.  P.  /\  F  e.  P. )  ->  ( A  .P.  F
)  e.  P. )
5 mulclpr 9303 . . . . . . . . . . 11  |-  ( ( B  e.  P.  /\  G  e.  P. )  ->  ( B  .P.  G
)  e.  P. )
6 addclpr 9301 . . . . . . . . . . 11  |-  ( ( ( A  .P.  F
)  e.  P.  /\  ( B  .P.  G )  e.  P. )  -> 
( ( A  .P.  F )  +P.  ( B  .P.  G ) )  e.  P. )
74, 5, 6syl2an 477 . . . . . . . . . 10  |-  ( ( ( A  e.  P.  /\  F  e.  P. )  /\  ( B  e.  P.  /\  G  e.  P. )
)  ->  ( ( A  .P.  F )  +P.  ( B  .P.  G
) )  e.  P. )
8 mulclpr 9303 . . . . . . . . . . 11  |-  ( ( A  e.  P.  /\  G  e.  P. )  ->  ( A  .P.  G
)  e.  P. )
9 mulclpr 9303 . . . . . . . . . . 11  |-  ( ( B  e.  P.  /\  F  e.  P. )  ->  ( B  .P.  F
)  e.  P. )
10 addclpr 9301 . . . . . . . . . . 11  |-  ( ( ( A  .P.  G
)  e.  P.  /\  ( B  .P.  F )  e.  P. )  -> 
( ( A  .P.  G )  +P.  ( B  .P.  F ) )  e.  P. )
118, 9, 10syl2an 477 . . . . . . . . . 10  |-  ( ( ( A  e.  P.  /\  G  e.  P. )  /\  ( B  e.  P.  /\  F  e.  P. )
)  ->  ( ( A  .P.  G )  +P.  ( B  .P.  F
) )  e.  P. )
127, 11anim12i 566 . . . . . . . . 9  |-  ( ( ( ( A  e. 
P.  /\  F  e.  P. )  /\  ( B  e.  P.  /\  G  e.  P. ) )  /\  ( ( A  e. 
P.  /\  G  e.  P. )  /\  ( B  e.  P.  /\  F  e.  P. ) ) )  ->  ( ( ( A  .P.  F )  +P.  ( B  .P.  G ) )  e.  P.  /\  ( ( A  .P.  G )  +P.  ( B  .P.  F ) )  e.  P. ) )
133, 12sylbi 195 . . . . . . . 8  |-  ( ( ( A  e.  P.  /\  B  e.  P. )  /\  ( F  e.  P.  /\  G  e.  P. )
)  ->  ( (
( A  .P.  F
)  +P.  ( B  .P.  G ) )  e. 
P.  /\  ( ( A  .P.  G )  +P.  ( B  .P.  F
) )  e.  P. ) )
14 rnlem 956 . . . . . . . . 9  |-  ( ( ( C  e.  P.  /\  D  e.  P. )  /\  ( R  e.  P.  /\  S  e.  P. )
)  <->  ( ( ( C  e.  P.  /\  R  e.  P. )  /\  ( D  e.  P.  /\  S  e.  P. )
)  /\  ( ( C  e.  P.  /\  S  e.  P. )  /\  ( D  e.  P.  /\  R  e.  P. ) ) ) )
15 mulclpr 9303 . . . . . . . . . . 11  |-  ( ( C  e.  P.  /\  R  e.  P. )  ->  ( C  .P.  R
)  e.  P. )
16 mulclpr 9303 . . . . . . . . . . 11  |-  ( ( D  e.  P.  /\  S  e.  P. )  ->  ( D  .P.  S
)  e.  P. )
17 addclpr 9301 . . . . . . . . . . 11  |-  ( ( ( C  .P.  R
)  e.  P.  /\  ( D  .P.  S )  e.  P. )  -> 
( ( C  .P.  R )  +P.  ( D  .P.  S ) )  e.  P. )
1815, 16, 17syl2an 477 . . . . . . . . . 10  |-  ( ( ( C  e.  P.  /\  R  e.  P. )  /\  ( D  e.  P.  /\  S  e.  P. )
)  ->  ( ( C  .P.  R )  +P.  ( D  .P.  S
) )  e.  P. )
19 mulclpr 9303 . . . . . . . . . . 11  |-  ( ( C  e.  P.  /\  S  e.  P. )  ->  ( C  .P.  S
)  e.  P. )
20 mulclpr 9303 . . . . . . . . . . 11  |-  ( ( D  e.  P.  /\  R  e.  P. )  ->  ( D  .P.  R
)  e.  P. )
21 addclpr 9301 . . . . . . . . . . 11  |-  ( ( ( C  .P.  S
)  e.  P.  /\  ( D  .P.  R )  e.  P. )  -> 
( ( C  .P.  S )  +P.  ( D  .P.  R ) )  e.  P. )
2219, 20, 21syl2an 477 . . . . . . . . . 10  |-  ( ( ( C  e.  P.  /\  S  e.  P. )  /\  ( D  e.  P.  /\  R  e.  P. )
)  ->  ( ( C  .P.  S )  +P.  ( D  .P.  R
) )  e.  P. )
2318, 22anim12i 566 . . . . . . . . 9  |-  ( ( ( ( C  e. 
P.  /\  R  e.  P. )  /\  ( D  e.  P.  /\  S  e.  P. ) )  /\  ( ( C  e. 
P.  /\  S  e.  P. )  /\  ( D  e.  P.  /\  R  e.  P. ) ) )  ->  ( ( ( C  .P.  R )  +P.  ( D  .P.  S ) )  e.  P.  /\  ( ( C  .P.  S )  +P.  ( D  .P.  R ) )  e.  P. ) )
2414, 23sylbi 195 . . . . . . . 8  |-  ( ( ( C  e.  P.  /\  D  e.  P. )  /\  ( R  e.  P.  /\  S  e.  P. )
)  ->  ( (
( C  .P.  R
)  +P.  ( D  .P.  S ) )  e. 
P.  /\  ( ( C  .P.  S )  +P.  ( D  .P.  R
) )  e.  P. ) )
25 addclpr 9301 . . . . . . . . . . 11  |-  ( ( ( ( A  .P.  F )  +P.  ( B  .P.  G ) )  e.  P.  /\  (
( C  .P.  S
)  +P.  ( D  .P.  R ) )  e. 
P. )  ->  (
( ( A  .P.  F )  +P.  ( B  .P.  G ) )  +P.  ( ( C  .P.  S )  +P.  ( D  .P.  R
) ) )  e. 
P. )
26 mulcmpblnrlem 9354 . . . . . . . . . . . . . 14  |-  ( ( ( 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 ) ) ) ) )
27 addcanpr 9329 . . . . . . . . . . . . . 14  |-  ( ( ( D  .P.  F
)  e.  P.  /\  ( ( ( A  .P.  F )  +P.  ( B  .P.  G
) )  +P.  (
( C  .P.  S
)  +P.  ( D  .P.  R ) ) )  e.  P. )  -> 
( ( ( 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 ) ) ) )  ->  ( (
( A  .P.  F
)  +P.  ( B  .P.  G ) )  +P.  ( ( C  .P.  S )  +P.  ( D  .P.  R ) ) )  =  ( ( ( A  .P.  G
)  +P.  ( B  .P.  F ) )  +P.  ( ( C  .P.  R )  +P.  ( D  .P.  S ) ) ) ) )
2826, 27syl5 32 . . . . . . . . . . . . 13  |-  ( ( ( D  .P.  F
)  e.  P.  /\  ( ( ( A  .P.  F )  +P.  ( B  .P.  G
) )  +P.  (
( C  .P.  S
)  +P.  ( D  .P.  R ) ) )  e.  P. )  -> 
( ( ( A  +P.  D )  =  ( B  +P.  C
)  /\  ( F  +P.  S )  =  ( G  +P.  R ) )  ->  ( (
( A  .P.  F
)  +P.  ( B  .P.  G ) )  +P.  ( ( C  .P.  S )  +P.  ( D  .P.  R ) ) )  =  ( ( ( A  .P.  G
)  +P.  ( B  .P.  F ) )  +P.  ( ( C  .P.  R )  +P.  ( D  .P.  S ) ) ) ) )
2928expcom 435 . . . . . . . . . . . 12  |-  ( ( ( ( A  .P.  F )  +P.  ( B  .P.  G ) )  +P.  ( ( C  .P.  S )  +P.  ( D  .P.  R
) ) )  e. 
P.  ->  ( ( D  .P.  F )  e. 
P.  ->  ( ( ( A  +P.  D )  =  ( B  +P.  C )  /\  ( F  +P.  S )  =  ( G  +P.  R
) )  ->  (
( ( A  .P.  F )  +P.  ( B  .P.  G ) )  +P.  ( ( C  .P.  S )  +P.  ( D  .P.  R
) ) )  =  ( ( ( A  .P.  G )  +P.  ( B  .P.  F
) )  +P.  (
( C  .P.  R
)  +P.  ( D  .P.  S ) ) ) ) ) )
3029impd 431 . . . . . . . . . . 11  |-  ( ( ( ( A  .P.  F )  +P.  ( B  .P.  G ) )  +P.  ( ( C  .P.  S )  +P.  ( D  .P.  R
) ) )  e. 
P.  ->  ( ( ( D  .P.  F )  e.  P.  /\  (
( A  +P.  D
)  =  ( B  +P.  C )  /\  ( F  +P.  S )  =  ( G  +P.  R ) ) )  -> 
( ( ( A  .P.  F )  +P.  ( B  .P.  G
) )  +P.  (
( C  .P.  S
)  +P.  ( D  .P.  R ) ) )  =  ( ( ( A  .P.  G )  +P.  ( B  .P.  F ) )  +P.  (
( C  .P.  R
)  +P.  ( D  .P.  S ) ) ) ) )
3125, 30syl 16 . . . . . . . . . 10  |-  ( ( ( ( A  .P.  F )  +P.  ( B  .P.  G ) )  e.  P.  /\  (
( C  .P.  S
)  +P.  ( D  .P.  R ) )  e. 
P. )  ->  (
( ( D  .P.  F )  e.  P.  /\  ( ( A  +P.  D )  =  ( B  +P.  C )  /\  ( F  +P.  S )  =  ( G  +P.  R ) ) )  -> 
( ( ( A  .P.  F )  +P.  ( B  .P.  G
) )  +P.  (
( C  .P.  S
)  +P.  ( D  .P.  R ) ) )  =  ( ( ( A  .P.  G )  +P.  ( B  .P.  F ) )  +P.  (
( C  .P.  R
)  +P.  ( D  .P.  S ) ) ) ) )
3231ad2ant2rl 748 . . . . . . . . 9  |-  ( ( ( ( ( A  .P.  F )  +P.  ( B  .P.  G
) )  e.  P.  /\  ( ( A  .P.  G )  +P.  ( B  .P.  F ) )  e.  P. )  /\  ( ( ( C  .P.  R )  +P.  ( D  .P.  S
) )  e.  P.  /\  ( ( C  .P.  S )  +P.  ( D  .P.  R ) )  e.  P. ) )  ->  ( ( ( D  .P.  F )  e.  P.  /\  (
( A  +P.  D
)  =  ( B  +P.  C )  /\  ( F  +P.  S )  =  ( G  +P.  R ) ) )  -> 
( ( ( A  .P.  F )  +P.  ( B  .P.  G
) )  +P.  (
( C  .P.  S
)  +P.  ( D  .P.  R ) ) )  =  ( ( ( A  .P.  G )  +P.  ( B  .P.  F ) )  +P.  (
( C  .P.  R
)  +P.  ( D  .P.  S ) ) ) ) )
33 enrbreq 9348 . . . . . . . . 9  |-  ( ( ( ( ( A  .P.  F )  +P.  ( B  .P.  G
) )  e.  P.  /\  ( ( A  .P.  G )  +P.  ( B  .P.  F ) )  e.  P. )  /\  ( ( ( C  .P.  R )  +P.  ( D  .P.  S
) )  e.  P.  /\  ( ( C  .P.  S )  +P.  ( D  .P.  R ) )  e.  P. ) )  ->  ( <. (
( A  .P.  F
)  +P.  ( B  .P.  G ) ) ,  ( ( A  .P.  G )  +P.  ( B  .P.  F ) )
>.  ~R  <. ( ( C  .P.  R )  +P.  ( D  .P.  S
) ) ,  ( ( C  .P.  S
)  +P.  ( D  .P.  R ) ) >.  <->  ( ( ( A  .P.  F )  +P.  ( B  .P.  G ) )  +P.  ( ( C  .P.  S )  +P.  ( D  .P.  R
) ) )  =  ( ( ( A  .P.  G )  +P.  ( B  .P.  F
) )  +P.  (
( C  .P.  R
)  +P.  ( D  .P.  S ) ) ) ) )
3432, 33sylibrd 234 . . . . . . . 8  |-  ( ( ( ( ( A  .P.  F )  +P.  ( B  .P.  G
) )  e.  P.  /\  ( ( A  .P.  G )  +P.  ( B  .P.  F ) )  e.  P. )  /\  ( ( ( C  .P.  R )  +P.  ( D  .P.  S
) )  e.  P.  /\  ( ( C  .P.  S )  +P.  ( D  .P.  R ) )  e.  P. ) )  ->  ( ( ( D  .P.  F )  e.  P.  /\  (
( A  +P.  D
)  =  ( B  +P.  C )  /\  ( F  +P.  S )  =  ( G  +P.  R ) ) )  ->  <. ( ( A  .P.  F )  +P.  ( B  .P.  G ) ) ,  ( ( A  .P.  G )  +P.  ( B  .P.  F
) ) >.  ~R  <. ( ( C  .P.  R
)  +P.  ( D  .P.  S ) ) ,  ( ( C  .P.  S )  +P.  ( D  .P.  R ) )
>. ) )
3513, 24, 34syl2an 477 . . . . . . 7  |-  ( ( ( ( A  e. 
P.  /\  B  e.  P. )  /\  ( F  e.  P.  /\  G  e.  P. ) )  /\  ( ( C  e. 
P.  /\  D  e.  P. )  /\  ( R  e.  P.  /\  S  e.  P. ) ) )  ->  ( ( ( D  .P.  F )  e.  P.  /\  (
( A  +P.  D
)  =  ( B  +P.  C )  /\  ( F  +P.  S )  =  ( G  +P.  R ) ) )  ->  <. ( ( A  .P.  F )  +P.  ( B  .P.  G ) ) ,  ( ( A  .P.  G )  +P.  ( B  .P.  F
) ) >.  ~R  <. ( ( C  .P.  R
)  +P.  ( D  .P.  S ) ) ,  ( ( C  .P.  S )  +P.  ( D  .P.  R ) )
>. ) )
3635an42s 823 . . . . . 6  |-  ( ( ( ( A  e. 
P.  /\  B  e.  P. )  /\  ( C  e.  P.  /\  D  e.  P. ) )  /\  ( ( R  e. 
P.  /\  S  e.  P. )  /\  ( F  e.  P.  /\  G  e.  P. ) ) )  ->  ( ( ( D  .P.  F )  e.  P.  /\  (
( A  +P.  D
)  =  ( B  +P.  C )  /\  ( F  +P.  S )  =  ( G  +P.  R ) ) )  ->  <. ( ( A  .P.  F )  +P.  ( B  .P.  G ) ) ,  ( ( A  .P.  G )  +P.  ( B  .P.  F
) ) >.  ~R  <. ( ( C  .P.  R
)  +P.  ( D  .P.  S ) ) ,  ( ( C  .P.  S )  +P.  ( D  .P.  R ) )
>. ) )
3736an4s 822 . . . . 5  |-  ( ( ( ( A  e. 
P.  /\  B  e.  P. )  /\  ( R  e.  P.  /\  S  e.  P. ) )  /\  ( ( C  e. 
P.  /\  D  e.  P. )  /\  ( F  e.  P.  /\  G  e.  P. ) ) )  ->  ( ( ( D  .P.  F )  e.  P.  /\  (
( A  +P.  D
)  =  ( B  +P.  C )  /\  ( F  +P.  S )  =  ( G  +P.  R ) ) )  ->  <. ( ( A  .P.  F )  +P.  ( B  .P.  G ) ) ,  ( ( A  .P.  G )  +P.  ( B  .P.  F
) ) >.  ~R  <. ( ( C  .P.  R
)  +P.  ( D  .P.  S ) ) ,  ( ( C  .P.  S )  +P.  ( D  .P.  R ) )
>. ) )
3837exp4b 607 . . . 4  |-  ( ( ( A  e.  P.  /\  B  e.  P. )  /\  ( R  e.  P.  /\  S  e.  P. )
)  ->  ( (
( C  e.  P.  /\  D  e.  P. )  /\  ( F  e.  P.  /\  G  e.  P. )
)  ->  ( ( D  .P.  F )  e. 
P.  ->  ( ( ( A  +P.  D )  =  ( B  +P.  C )  /\  ( F  +P.  S )  =  ( G  +P.  R
) )  ->  <. (
( A  .P.  F
)  +P.  ( B  .P.  G ) ) ,  ( ( A  .P.  G )  +P.  ( B  .P.  F ) )
>.  ~R  <. ( ( C  .P.  R )  +P.  ( D  .P.  S
) ) ,  ( ( C  .P.  S
)  +P.  ( D  .P.  R ) ) >.
) ) ) )
392, 38mpdi 42 . . 3  |-  ( ( ( A  e.  P.  /\  B  e.  P. )  /\  ( R  e.  P.  /\  S  e.  P. )
)  ->  ( (
( C  e.  P.  /\  D  e.  P. )  /\  ( F  e.  P.  /\  G  e.  P. )
)  ->  ( (
( A  +P.  D
)  =  ( B  +P.  C )  /\  ( F  +P.  S )  =  ( G  +P.  R ) )  ->  <. (
( A  .P.  F
)  +P.  ( B  .P.  G ) ) ,  ( ( A  .P.  G )  +P.  ( B  .P.  F ) )
>.  ~R  <. ( ( C  .P.  R )  +P.  ( D  .P.  S
) ) ,  ( ( C  .P.  S
)  +P.  ( D  .P.  R ) ) >.
) ) )
4039imp 429 . 2  |-  ( ( ( ( A  e. 
P.  /\  B  e.  P. )  /\  ( R  e.  P.  /\  S  e.  P. ) )  /\  ( ( C  e. 
P.  /\  D  e.  P. )  /\  ( F  e.  P.  /\  G  e.  P. ) ) )  ->  ( ( ( A  +P.  D )  =  ( B  +P.  C )  /\  ( F  +P.  S )  =  ( G  +P.  R
) )  ->  <. (
( A  .P.  F
)  +P.  ( B  .P.  G ) ) ,  ( ( A  .P.  G )  +P.  ( B  .P.  F ) )
>.  ~R  <. ( ( C  .P.  R )  +P.  ( D  .P.  S
) ) ,  ( ( C  .P.  S
)  +P.  ( D  .P.  R ) ) >.
) )
4140an42s 823 1  |-  ( ( ( ( A  e. 
P.  /\  B  e.  P. )  /\  ( C  e.  P.  /\  D  e.  P. ) )  /\  ( ( F  e. 
P.  /\  G  e.  P. )  /\  ( R  e.  P.  /\  S  e.  P. ) ) )  ->  ( ( ( A  +P.  D )  =  ( B  +P.  C )  /\  ( F  +P.  S )  =  ( G  +P.  R
) )  ->  <. (
( A  .P.  F
)  +P.  ( B  .P.  G ) ) ,  ( ( A  .P.  G )  +P.  ( B  .P.  F ) )
>.  ~R  <. ( ( C  .P.  R )  +P.  ( D  .P.  S
) ) ,  ( ( C  .P.  S
)  +P.  ( D  .P.  R ) ) >.
) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1370    e. wcel 1758   <.cop 3994   class class class wbr 4403  (class class class)co 6203   P.cnp 9140    +P. cpp 9142    .P. cmp 9143    ~R cer 9147
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-sep 4524  ax-nul 4532  ax-pow 4581  ax-pr 4642  ax-un 6485  ax-inf2 7961
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-ral 2804  df-rex 2805  df-reu 2806  df-rmo 2807  df-rab 2808  df-v 3080  df-sbc 3295  df-csb 3399  df-dif 3442  df-un 3444  df-in 3446  df-ss 3453  df-pss 3455  df-nul 3749  df-if 3903  df-pw 3973  df-sn 3989  df-pr 3991  df-tp 3993  df-op 3995  df-uni 4203  df-int 4240  df-iun 4284  df-br 4404  df-opab 4462  df-mpt 4463  df-tr 4497  df-eprel 4743  df-id 4747  df-po 4752  df-so 4753  df-fr 4790  df-we 4792  df-ord 4833  df-on 4834  df-lim 4835  df-suc 4836  df-xp 4957  df-rel 4958  df-cnv 4959  df-co 4960  df-dm 4961  df-rn 4962  df-res 4963  df-ima 4964  df-iota 5492  df-fun 5531  df-fn 5532  df-f 5533  df-f1 5534  df-fo 5535  df-f1o 5536  df-fv 5537  df-ov 6206  df-oprab 6207  df-mpt2 6208  df-om 6590  df-1st 6690  df-2nd 6691  df-recs 6945  df-rdg 6979  df-1o 7033  df-oadd 7037  df-omul 7038  df-er 7214  df-ni 9155  df-pli 9156  df-mi 9157  df-lti 9158  df-plpq 9191  df-mpq 9192  df-ltpq 9193  df-enq 9194  df-nq 9195  df-erq 9196  df-plq 9197  df-mq 9198  df-1nq 9199  df-rq 9200  df-ltnq 9201  df-np 9264  df-plp 9266  df-mp 9267  df-ltp 9268  df-enr 9340
This theorem is referenced by:  mulsrpr  9357
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