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Theorem mulcmpblnr 8783
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 8731 . . . . 5  |-  ( ( D  e.  P.  /\  F  e.  P. )  ->  ( D  .P.  F
)  e.  P. )
21ad2ant2lr 728 . . . 4  |-  ( ( ( C  e.  P.  /\  D  e.  P. )  /\  ( F  e.  P.  /\  G  e.  P. )
)  ->  ( D  .P.  F )  e.  P. )
3 rnlem 931 . . . . . . . . 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 8731 . . . . . . . . . . 11  |-  ( ( A  e.  P.  /\  F  e.  P. )  ->  ( A  .P.  F
)  e.  P. )
5 mulclpr 8731 . . . . . . . . . . 11  |-  ( ( B  e.  P.  /\  G  e.  P. )  ->  ( B  .P.  G
)  e.  P. )
6 addclpr 8729 . . . . . . . . . . 11  |-  ( ( ( A  .P.  F
)  e.  P.  /\  ( B  .P.  G )  e.  P. )  -> 
( ( A  .P.  F )  +P.  ( B  .P.  G ) )  e.  P. )
74, 5, 6syl2an 463 . . . . . . . . . 10  |-  ( ( ( A  e.  P.  /\  F  e.  P. )  /\  ( B  e.  P.  /\  G  e.  P. )
)  ->  ( ( A  .P.  F )  +P.  ( B  .P.  G
) )  e.  P. )
8 mulclpr 8731 . . . . . . . . . . 11  |-  ( ( A  e.  P.  /\  G  e.  P. )  ->  ( A  .P.  G
)  e.  P. )
9 mulclpr 8731 . . . . . . . . . . 11  |-  ( ( B  e.  P.  /\  F  e.  P. )  ->  ( B  .P.  F
)  e.  P. )
10 addclpr 8729 . . . . . . . . . . 11  |-  ( ( ( A  .P.  G
)  e.  P.  /\  ( B  .P.  F )  e.  P. )  -> 
( ( A  .P.  G )  +P.  ( B  .P.  F ) )  e.  P. )
118, 9, 10syl2an 463 . . . . . . . . . 10  |-  ( ( ( A  e.  P.  /\  G  e.  P. )  /\  ( B  e.  P.  /\  F  e.  P. )
)  ->  ( ( A  .P.  G )  +P.  ( B  .P.  F
) )  e.  P. )
127, 11anim12i 549 . . . . . . . . 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 187 . . . . . . . 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 931 . . . . . . . . 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 8731 . . . . . . . . . . 11  |-  ( ( C  e.  P.  /\  R  e.  P. )  ->  ( C  .P.  R
)  e.  P. )
16 mulclpr 8731 . . . . . . . . . . 11  |-  ( ( D  e.  P.  /\  S  e.  P. )  ->  ( D  .P.  S
)  e.  P. )
17 addclpr 8729 . . . . . . . . . . 11  |-  ( ( ( C  .P.  R
)  e.  P.  /\  ( D  .P.  S )  e.  P. )  -> 
( ( C  .P.  R )  +P.  ( D  .P.  S ) )  e.  P. )
1815, 16, 17syl2an 463 . . . . . . . . . 10  |-  ( ( ( C  e.  P.  /\  R  e.  P. )  /\  ( D  e.  P.  /\  S  e.  P. )
)  ->  ( ( C  .P.  R )  +P.  ( D  .P.  S
) )  e.  P. )
19 mulclpr 8731 . . . . . . . . . . 11  |-  ( ( C  e.  P.  /\  S  e.  P. )  ->  ( C  .P.  S
)  e.  P. )
20 mulclpr 8731 . . . . . . . . . . 11  |-  ( ( D  e.  P.  /\  R  e.  P. )  ->  ( D  .P.  R
)  e.  P. )
21 addclpr 8729 . . . . . . . . . . 11  |-  ( ( ( C  .P.  S
)  e.  P.  /\  ( D  .P.  R )  e.  P. )  -> 
( ( C  .P.  S )  +P.  ( D  .P.  R ) )  e.  P. )
2219, 20, 21syl2an 463 . . . . . . . . . 10  |-  ( ( ( C  e.  P.  /\  S  e.  P. )  /\  ( D  e.  P.  /\  R  e.  P. )
)  ->  ( ( C  .P.  S )  +P.  ( D  .P.  R
) )  e.  P. )
2318, 22anim12i 549 . . . . . . . . 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 187 . . . . . . . 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 8729 . . . . . . . . . . 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 8782 . . . . . . . . . . . . . 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 8757 . . . . . . . . . . . . . 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 28 . . . . . . . . . . . . 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 424 . . . . . . . . . . . 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 ) ) ) ) ) )
3029imp3a 420 . . . . . . . . . . 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 15 . . . . . . . . . 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 729 . . . . . . . . 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 8776 . . . . . . . . 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 225 . . . . . . . 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 463 . . . . . . 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 800 . . . . . 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 799 . . . . 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 590 . . . 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 38 . . 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 418 . 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 800 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 set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1642    e. wcel 1710   <.cop 3719   class class class wbr 4102  (class class class)co 5942   P.cnp 8568    +P. cpp 8570    .P. cmp 8571    ~R cer 8575
This theorem is referenced by:  mulsrpr  8785
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-13 1712  ax-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1930  ax-ext 2339  ax-sep 4220  ax-nul 4228  ax-pow 4267  ax-pr 4293  ax-un 4591  ax-inf2 7429
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2213  df-mo 2214  df-clab 2345  df-cleq 2351  df-clel 2354  df-nfc 2483  df-ne 2523  df-ral 2624  df-rex 2625  df-reu 2626  df-rmo 2627  df-rab 2628  df-v 2866  df-sbc 3068  df-csb 3158  df-dif 3231  df-un 3233  df-in 3235  df-ss 3242  df-pss 3244  df-nul 3532  df-if 3642  df-pw 3703  df-sn 3722  df-pr 3723  df-tp 3724  df-op 3725  df-uni 3907  df-int 3942  df-iun 3986  df-br 4103  df-opab 4157  df-mpt 4158  df-tr 4193  df-eprel 4384  df-id 4388  df-po 4393  df-so 4394  df-fr 4431  df-we 4433  df-ord 4474  df-on 4475  df-lim 4476  df-suc 4477  df-om 4736  df-xp 4774  df-rel 4775  df-cnv 4776  df-co 4777  df-dm 4778  df-rn 4779  df-res 4780  df-ima 4781  df-iota 5298  df-fun 5336  df-fn 5337  df-f 5338  df-f1 5339  df-fo 5340  df-f1o 5341  df-fv 5342  df-ov 5945  df-oprab 5946  df-mpt2 5947  df-1st 6206  df-2nd 6207  df-recs 6472  df-rdg 6507  df-1o 6563  df-oadd 6567  df-omul 6568  df-er 6744  df-ni 8583  df-pli 8584  df-mi 8585  df-lti 8586  df-plpq 8619  df-mpq 8620  df-ltpq 8621  df-enq 8622  df-nq 8623  df-erq 8624  df-plq 8625  df-mq 8626  df-1nq 8627  df-rq 8628  df-ltnq 8629  df-np 8692  df-plp 8694  df-mp 8695  df-ltp 8696  df-enr 8768
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