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Theorem mulcmpblnr 8905
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 8853 . . . . 5  |-  ( ( D  e.  P.  /\  F  e.  P. )  ->  ( D  .P.  F
)  e.  P. )
21ad2ant2lr 729 . . . 4  |-  ( ( ( C  e.  P.  /\  D  e.  P. )  /\  ( F  e.  P.  /\  G  e.  P. )
)  ->  ( D  .P.  F )  e.  P. )
3 rnlem 932 . . . . . . . . 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 8853 . . . . . . . . . . 11  |-  ( ( A  e.  P.  /\  F  e.  P. )  ->  ( A  .P.  F
)  e.  P. )
5 mulclpr 8853 . . . . . . . . . . 11  |-  ( ( B  e.  P.  /\  G  e.  P. )  ->  ( B  .P.  G
)  e.  P. )
6 addclpr 8851 . . . . . . . . . . 11  |-  ( ( ( A  .P.  F
)  e.  P.  /\  ( B  .P.  G )  e.  P. )  -> 
( ( A  .P.  F )  +P.  ( B  .P.  G ) )  e.  P. )
74, 5, 6syl2an 464 . . . . . . . . . 10  |-  ( ( ( A  e.  P.  /\  F  e.  P. )  /\  ( B  e.  P.  /\  G  e.  P. )
)  ->  ( ( A  .P.  F )  +P.  ( B  .P.  G
) )  e.  P. )
8 mulclpr 8853 . . . . . . . . . . 11  |-  ( ( A  e.  P.  /\  G  e.  P. )  ->  ( A  .P.  G
)  e.  P. )
9 mulclpr 8853 . . . . . . . . . . 11  |-  ( ( B  e.  P.  /\  F  e.  P. )  ->  ( B  .P.  F
)  e.  P. )
10 addclpr 8851 . . . . . . . . . . 11  |-  ( ( ( A  .P.  G
)  e.  P.  /\  ( B  .P.  F )  e.  P. )  -> 
( ( A  .P.  G )  +P.  ( B  .P.  F ) )  e.  P. )
118, 9, 10syl2an 464 . . . . . . . . . 10  |-  ( ( ( A  e.  P.  /\  G  e.  P. )  /\  ( B  e.  P.  /\  F  e.  P. )
)  ->  ( ( A  .P.  G )  +P.  ( B  .P.  F
) )  e.  P. )
127, 11anim12i 550 . . . . . . . . 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 188 . . . . . . . 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 932 . . . . . . . . 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 8853 . . . . . . . . . . 11  |-  ( ( C  e.  P.  /\  R  e.  P. )  ->  ( C  .P.  R
)  e.  P. )
16 mulclpr 8853 . . . . . . . . . . 11  |-  ( ( D  e.  P.  /\  S  e.  P. )  ->  ( D  .P.  S
)  e.  P. )
17 addclpr 8851 . . . . . . . . . . 11  |-  ( ( ( C  .P.  R
)  e.  P.  /\  ( D  .P.  S )  e.  P. )  -> 
( ( C  .P.  R )  +P.  ( D  .P.  S ) )  e.  P. )
1815, 16, 17syl2an 464 . . . . . . . . . 10  |-  ( ( ( C  e.  P.  /\  R  e.  P. )  /\  ( D  e.  P.  /\  S  e.  P. )
)  ->  ( ( C  .P.  R )  +P.  ( D  .P.  S
) )  e.  P. )
19 mulclpr 8853 . . . . . . . . . . 11  |-  ( ( C  e.  P.  /\  S  e.  P. )  ->  ( C  .P.  S
)  e.  P. )
20 mulclpr 8853 . . . . . . . . . . 11  |-  ( ( D  e.  P.  /\  R  e.  P. )  ->  ( D  .P.  R
)  e.  P. )
21 addclpr 8851 . . . . . . . . . . 11  |-  ( ( ( C  .P.  S
)  e.  P.  /\  ( D  .P.  R )  e.  P. )  -> 
( ( C  .P.  S )  +P.  ( D  .P.  R ) )  e.  P. )
2219, 20, 21syl2an 464 . . . . . . . . . 10  |-  ( ( ( C  e.  P.  /\  S  e.  P. )  /\  ( D  e.  P.  /\  R  e.  P. )
)  ->  ( ( C  .P.  S )  +P.  ( D  .P.  R
) )  e.  P. )
2318, 22anim12i 550 . . . . . . . . 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 188 . . . . . . . 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 8851 . . . . . . . . . . 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 8904 . . . . . . . . . . . . . 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 8879 . . . . . . . . . . . . . 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 30 . . . . . . . . . . . . 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 425 . . . . . . . . . . . 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 421 . . . . . . . . . . 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 730 . . . . . . . . 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 8898 . . . . . . . . 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 226 . . . . . . . 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 464 . . . . . . 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 801 . . . . . 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 800 . . . . 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 591 . . . 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 40 . . 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 419 . 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 801 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 359    = wceq 1649    e. wcel 1721   <.cop 3777   class class class wbr 4172  (class class class)co 6040   P.cnp 8690    +P. cpp 8692    .P. cmp 8693    ~R cer 8697
This theorem is referenced by:  mulsrpr  8907
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660  ax-inf2 7552
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-ral 2671  df-rex 2672  df-reu 2673  df-rmo 2674  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-pss 3296  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-tp 3782  df-op 3783  df-uni 3976  df-int 4011  df-iun 4055  df-br 4173  df-opab 4227  df-mpt 4228  df-tr 4263  df-eprel 4454  df-id 4458  df-po 4463  df-so 4464  df-fr 4501  df-we 4503  df-ord 4544  df-on 4545  df-lim 4546  df-suc 4547  df-om 4805  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-ov 6043  df-oprab 6044  df-mpt2 6045  df-1st 6308  df-2nd 6309  df-recs 6592  df-rdg 6627  df-1o 6683  df-oadd 6687  df-omul 6688  df-er 6864  df-ni 8705  df-pli 8706  df-mi 8707  df-lti 8708  df-plpq 8741  df-mpq 8742  df-ltpq 8743  df-enq 8744  df-nq 8745  df-erq 8746  df-plq 8747  df-mq 8748  df-1nq 8749  df-rq 8750  df-ltnq 8751  df-np 8814  df-plp 8816  df-mp 8817  df-ltp 8818  df-enr 8890
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