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Theorem dfplpq2 6510
Description: Alternative definition of pre-addition on positive fractions. (Contributed by Jim Kingdon, 12-Sep-2019.)
Assertion
Ref Expression
dfplpq2  |-  +pQ  =  { <. <. x ,  y
>. ,  z >.  |  ( ( x  e.  ( N.  X.  N. )  /\  y  e.  ( N.  X.  N. )
)  /\  E. w E. v E. u E. f ( ( x  =  <. w ,  v
>.  /\  y  =  <. u ,  f >. )  /\  z  =  <. ( ( w  .N  f
)  +N  ( v  .N  u ) ) ,  ( v  .N  f ) >. )
) }
Distinct variable group:    x, y, z, w, v, u, f

Proof of Theorem dfplpq2
StepHypRef Expression
1 df-mpt2 5545 . 2  |-  ( x  e.  ( N.  X.  N. ) ,  y  e.  ( N.  X.  N. )  |->  <. ( ( ( 1st `  x )  .N  ( 2nd `  y
) )  +N  (
( 1st `  y
)  .N  ( 2nd `  x ) ) ) ,  ( ( 2nd `  x )  .N  ( 2nd `  y ) )
>. )  =  { <. <. x ,  y
>. ,  z >.  |  ( ( x  e.  ( N.  X.  N. )  /\  y  e.  ( N.  X.  N. )
)  /\  z  =  <. ( ( ( 1st `  x )  .N  ( 2nd `  y ) )  +N  ( ( 1st `  y )  .N  ( 2nd `  x ) ) ) ,  ( ( 2nd `  x )  .N  ( 2nd `  y
) ) >. ) }
2 df-plpq 6500 . 2  |-  +pQ  =  ( x  e.  ( N.  X.  N. ) ,  y  e.  ( N. 
X.  N. )  |->  <. (
( ( 1st `  x
)  .N  ( 2nd `  y ) )  +N  ( ( 1st `  y
)  .N  ( 2nd `  x ) ) ) ,  ( ( 2nd `  x )  .N  ( 2nd `  y ) )
>. )
3 1st2nd2 5829 . . . . . . . . . 10  |-  ( x  e.  ( N.  X.  N. )  ->  x  = 
<. ( 1st `  x
) ,  ( 2nd `  x ) >. )
43eqeq1d 2064 . . . . . . . . 9  |-  ( x  e.  ( N.  X.  N. )  ->  ( x  =  <. w ,  v
>. 
<-> 
<. ( 1st `  x
) ,  ( 2nd `  x ) >.  =  <. w ,  v >. )
)
5 1st2nd2 5829 . . . . . . . . . 10  |-  ( y  e.  ( N.  X.  N. )  ->  y  = 
<. ( 1st `  y
) ,  ( 2nd `  y ) >. )
65eqeq1d 2064 . . . . . . . . 9  |-  ( y  e.  ( N.  X.  N. )  ->  ( y  =  <. u ,  f
>. 
<-> 
<. ( 1st `  y
) ,  ( 2nd `  y ) >.  =  <. u ,  f >. )
)
74, 6bi2anan9 548 . . . . . . . 8  |-  ( ( x  e.  ( N. 
X.  N. )  /\  y  e.  ( N.  X.  N. ) )  ->  (
( x  =  <. w ,  v >.  /\  y  =  <. u ,  f
>. )  <->  ( <. ( 1st `  x ) ,  ( 2nd `  x
) >.  =  <. w ,  v >.  /\  <. ( 1st `  y ) ,  ( 2nd `  y
) >.  =  <. u ,  f >. )
) )
87anbi1d 446 . . . . . . 7  |-  ( ( x  e.  ( N. 
X.  N. )  /\  y  e.  ( N.  X.  N. ) )  ->  (
( ( x  = 
<. w ,  v >.  /\  y  =  <. u ,  f >. )  /\  z  =  <. ( ( w  .N  f
)  +N  ( u  .N  v ) ) ,  ( v  .N  f ) >. )  <->  ( ( <. ( 1st `  x
) ,  ( 2nd `  x ) >.  =  <. w ,  v >.  /\  <. ( 1st `  y ) ,  ( 2nd `  y
) >.  =  <. u ,  f >. )  /\  z  =  <. ( ( w  .N  f
)  +N  ( u  .N  v ) ) ,  ( v  .N  f ) >. )
) )
9 xp1st 5820 . . . . . . . . . . . . . 14  |-  ( y  e.  ( N.  X.  N. )  ->  ( 1st `  y )  e.  N. )
109ad2antlr 466 . . . . . . . . . . . . 13  |-  ( ( ( x  e.  ( N.  X.  N. )  /\  y  e.  ( N.  X.  N. ) )  /\  ( x  = 
<. w ,  v >.  /\  y  =  <. u ,  f >. )
)  ->  ( 1st `  y )  e.  N. )
117biimpa 284 . . . . . . . . . . . . . . 15  |-  ( ( ( x  e.  ( N.  X.  N. )  /\  y  e.  ( N.  X.  N. ) )  /\  ( x  = 
<. w ,  v >.  /\  y  =  <. u ,  f >. )
)  ->  ( <. ( 1st `  x ) ,  ( 2nd `  x
) >.  =  <. w ,  v >.  /\  <. ( 1st `  y ) ,  ( 2nd `  y
) >.  =  <. u ,  f >. )
)
1211simprd 111 . . . . . . . . . . . . . 14  |-  ( ( ( x  e.  ( N.  X.  N. )  /\  y  e.  ( N.  X.  N. ) )  /\  ( x  = 
<. w ,  v >.  /\  y  =  <. u ,  f >. )
)  ->  <. ( 1st `  y ) ,  ( 2nd `  y )
>.  =  <. u ,  f >. )
13 vex 2577 . . . . . . . . . . . . . . . . 17  |-  u  e. 
_V
14 vex 2577 . . . . . . . . . . . . . . . . 17  |-  f  e. 
_V
1513, 14opth2 4005 . . . . . . . . . . . . . . . 16  |-  ( <.
( 1st `  y
) ,  ( 2nd `  y ) >.  =  <. u ,  f >.  <->  ( ( 1st `  y )  =  u  /\  ( 2nd `  y )  =  f ) )
1615simplbi 263 . . . . . . . . . . . . . . 15  |-  ( <.
( 1st `  y
) ,  ( 2nd `  y ) >.  =  <. u ,  f >.  ->  ( 1st `  y )  =  u )
1716eleq1d 2122 . . . . . . . . . . . . . 14  |-  ( <.
( 1st `  y
) ,  ( 2nd `  y ) >.  =  <. u ,  f >.  ->  (
( 1st `  y
)  e.  N.  <->  u  e.  N. ) )
1812, 17syl 14 . . . . . . . . . . . . 13  |-  ( ( ( x  e.  ( N.  X.  N. )  /\  y  e.  ( N.  X.  N. ) )  /\  ( x  = 
<. w ,  v >.  /\  y  =  <. u ,  f >. )
)  ->  ( ( 1st `  y )  e. 
N. 
<->  u  e.  N. )
)
1910, 18mpbid 139 . . . . . . . . . . . 12  |-  ( ( ( x  e.  ( N.  X.  N. )  /\  y  e.  ( N.  X.  N. ) )  /\  ( x  = 
<. w ,  v >.  /\  y  =  <. u ,  f >. )
)  ->  u  e.  N. )
20 xp2nd 5821 . . . . . . . . . . . . . 14  |-  ( x  e.  ( N.  X.  N. )  ->  ( 2nd `  x )  e.  N. )
2120ad2antrr 465 . . . . . . . . . . . . 13  |-  ( ( ( x  e.  ( N.  X.  N. )  /\  y  e.  ( N.  X.  N. ) )  /\  ( x  = 
<. w ,  v >.  /\  y  =  <. u ,  f >. )
)  ->  ( 2nd `  x )  e.  N. )
2211simpld 109 . . . . . . . . . . . . . 14  |-  ( ( ( x  e.  ( N.  X.  N. )  /\  y  e.  ( N.  X.  N. ) )  /\  ( x  = 
<. w ,  v >.  /\  y  =  <. u ,  f >. )
)  ->  <. ( 1st `  x ) ,  ( 2nd `  x )
>.  =  <. w ,  v >. )
23 vex 2577 . . . . . . . . . . . . . . . . 17  |-  w  e. 
_V
24 vex 2577 . . . . . . . . . . . . . . . . 17  |-  v  e. 
_V
2523, 24opth2 4005 . . . . . . . . . . . . . . . 16  |-  ( <.
( 1st `  x
) ,  ( 2nd `  x ) >.  =  <. w ,  v >.  <->  ( ( 1st `  x )  =  w  /\  ( 2nd `  x )  =  v ) )
2625simprbi 264 . . . . . . . . . . . . . . 15  |-  ( <.
( 1st `  x
) ,  ( 2nd `  x ) >.  =  <. w ,  v >.  ->  ( 2nd `  x )  =  v )
2726eleq1d 2122 . . . . . . . . . . . . . 14  |-  ( <.
( 1st `  x
) ,  ( 2nd `  x ) >.  =  <. w ,  v >.  ->  (
( 2nd `  x
)  e.  N.  <->  v  e.  N. ) )
2822, 27syl 14 . . . . . . . . . . . . 13  |-  ( ( ( x  e.  ( N.  X.  N. )  /\  y  e.  ( N.  X.  N. ) )  /\  ( x  = 
<. w ,  v >.  /\  y  =  <. u ,  f >. )
)  ->  ( ( 2nd `  x )  e. 
N. 
<->  v  e.  N. )
)
2921, 28mpbid 139 . . . . . . . . . . . 12  |-  ( ( ( x  e.  ( N.  X.  N. )  /\  y  e.  ( N.  X.  N. ) )  /\  ( x  = 
<. w ,  v >.  /\  y  =  <. u ,  f >. )
)  ->  v  e.  N. )
30 mulcompig 6487 . . . . . . . . . . . 12  |-  ( ( u  e.  N.  /\  v  e.  N. )  ->  ( u  .N  v
)  =  ( v  .N  u ) )
3119, 29, 30syl2anc 397 . . . . . . . . . . 11  |-  ( ( ( x  e.  ( N.  X.  N. )  /\  y  e.  ( N.  X.  N. ) )  /\  ( x  = 
<. w ,  v >.  /\  y  =  <. u ,  f >. )
)  ->  ( u  .N  v )  =  ( v  .N  u ) )
3231oveq2d 5556 . . . . . . . . . 10  |-  ( ( ( x  e.  ( N.  X.  N. )  /\  y  e.  ( N.  X.  N. ) )  /\  ( x  = 
<. w ,  v >.  /\  y  =  <. u ,  f >. )
)  ->  ( (
w  .N  f )  +N  ( u  .N  v ) )  =  ( ( w  .N  f )  +N  (
v  .N  u ) ) )
3332opeq1d 3583 . . . . . . . . 9  |-  ( ( ( x  e.  ( N.  X.  N. )  /\  y  e.  ( N.  X.  N. ) )  /\  ( x  = 
<. w ,  v >.  /\  y  =  <. u ,  f >. )
)  ->  <. ( ( w  .N  f )  +N  ( u  .N  v ) ) ,  ( v  .N  f
) >.  =  <. (
( w  .N  f
)  +N  ( v  .N  u ) ) ,  ( v  .N  f ) >. )
3433eqeq2d 2067 . . . . . . . 8  |-  ( ( ( x  e.  ( N.  X.  N. )  /\  y  e.  ( N.  X.  N. ) )  /\  ( x  = 
<. w ,  v >.  /\  y  =  <. u ,  f >. )
)  ->  ( z  =  <. ( ( w  .N  f )  +N  ( u  .N  v
) ) ,  ( v  .N  f )
>. 
<->  z  =  <. (
( w  .N  f
)  +N  ( v  .N  u ) ) ,  ( v  .N  f ) >. )
)
3534pm5.32da 433 . . . . . . 7  |-  ( ( x  e.  ( N. 
X.  N. )  /\  y  e.  ( N.  X.  N. ) )  ->  (
( ( x  = 
<. w ,  v >.  /\  y  =  <. u ,  f >. )  /\  z  =  <. ( ( w  .N  f
)  +N  ( u  .N  v ) ) ,  ( v  .N  f ) >. )  <->  ( ( x  =  <. w ,  v >.  /\  y  =  <. u ,  f
>. )  /\  z  =  <. ( ( w  .N  f )  +N  ( v  .N  u
) ) ,  ( v  .N  f )
>. ) ) )
368, 35bitr3d 183 . . . . . 6  |-  ( ( x  e.  ( N. 
X.  N. )  /\  y  e.  ( N.  X.  N. ) )  ->  (
( ( <. ( 1st `  x ) ,  ( 2nd `  x
) >.  =  <. w ,  v >.  /\  <. ( 1st `  y ) ,  ( 2nd `  y
) >.  =  <. u ,  f >. )  /\  z  =  <. ( ( w  .N  f
)  +N  ( u  .N  v ) ) ,  ( v  .N  f ) >. )  <->  ( ( x  =  <. w ,  v >.  /\  y  =  <. u ,  f
>. )  /\  z  =  <. ( ( w  .N  f )  +N  ( v  .N  u
) ) ,  ( v  .N  f )
>. ) ) )
37364exbidv 1766 . . . . 5  |-  ( ( x  e.  ( N. 
X.  N. )  /\  y  e.  ( N.  X.  N. ) )  ->  ( E. w E. v E. u E. f ( ( <. ( 1st `  x
) ,  ( 2nd `  x ) >.  =  <. w ,  v >.  /\  <. ( 1st `  y ) ,  ( 2nd `  y
) >.  =  <. u ,  f >. )  /\  z  =  <. ( ( w  .N  f
)  +N  ( u  .N  v ) ) ,  ( v  .N  f ) >. )  <->  E. w E. v E. u E. f ( ( x  =  <. w ,  v >.  /\  y  =  <. u ,  f
>. )  /\  z  =  <. ( ( w  .N  f )  +N  ( v  .N  u
) ) ,  ( v  .N  f )
>. ) ) )
38 xp1st 5820 . . . . . . 7  |-  ( x  e.  ( N.  X.  N. )  ->  ( 1st `  x )  e.  N. )
3938, 20jca 294 . . . . . 6  |-  ( x  e.  ( N.  X.  N. )  ->  ( ( 1st `  x )  e.  N.  /\  ( 2nd `  x )  e. 
N. ) )
40 xp2nd 5821 . . . . . . 7  |-  ( y  e.  ( N.  X.  N. )  ->  ( 2nd `  y )  e.  N. )
419, 40jca 294 . . . . . 6  |-  ( y  e.  ( N.  X.  N. )  ->  ( ( 1st `  y )  e.  N.  /\  ( 2nd `  y )  e. 
N. ) )
42 simpll 489 . . . . . . . . . . 11  |-  ( ( ( w  =  ( 1st `  x )  /\  v  =  ( 2nd `  x ) )  /\  ( u  =  ( 1st `  y
)  /\  f  =  ( 2nd `  y ) ) )  ->  w  =  ( 1st `  x
) )
43 simprr 492 . . . . . . . . . . 11  |-  ( ( ( w  =  ( 1st `  x )  /\  v  =  ( 2nd `  x ) )  /\  ( u  =  ( 1st `  y
)  /\  f  =  ( 2nd `  y ) ) )  ->  f  =  ( 2nd `  y
) )
4442, 43oveq12d 5558 . . . . . . . . . 10  |-  ( ( ( w  =  ( 1st `  x )  /\  v  =  ( 2nd `  x ) )  /\  ( u  =  ( 1st `  y
)  /\  f  =  ( 2nd `  y ) ) )  ->  (
w  .N  f )  =  ( ( 1st `  x )  .N  ( 2nd `  y ) ) )
45 simprl 491 . . . . . . . . . . 11  |-  ( ( ( w  =  ( 1st `  x )  /\  v  =  ( 2nd `  x ) )  /\  ( u  =  ( 1st `  y
)  /\  f  =  ( 2nd `  y ) ) )  ->  u  =  ( 1st `  y
) )
46 simplr 490 . . . . . . . . . . 11  |-  ( ( ( w  =  ( 1st `  x )  /\  v  =  ( 2nd `  x ) )  /\  ( u  =  ( 1st `  y
)  /\  f  =  ( 2nd `  y ) ) )  ->  v  =  ( 2nd `  x
) )
4745, 46oveq12d 5558 . . . . . . . . . 10  |-  ( ( ( w  =  ( 1st `  x )  /\  v  =  ( 2nd `  x ) )  /\  ( u  =  ( 1st `  y
)  /\  f  =  ( 2nd `  y ) ) )  ->  (
u  .N  v )  =  ( ( 1st `  y )  .N  ( 2nd `  x ) ) )
4844, 47oveq12d 5558 . . . . . . . . 9  |-  ( ( ( w  =  ( 1st `  x )  /\  v  =  ( 2nd `  x ) )  /\  ( u  =  ( 1st `  y
)  /\  f  =  ( 2nd `  y ) ) )  ->  (
( w  .N  f
)  +N  ( u  .N  v ) )  =  ( ( ( 1st `  x )  .N  ( 2nd `  y
) )  +N  (
( 1st `  y
)  .N  ( 2nd `  x ) ) ) )
4946, 43oveq12d 5558 . . . . . . . . 9  |-  ( ( ( w  =  ( 1st `  x )  /\  v  =  ( 2nd `  x ) )  /\  ( u  =  ( 1st `  y
)  /\  f  =  ( 2nd `  y ) ) )  ->  (
v  .N  f )  =  ( ( 2nd `  x )  .N  ( 2nd `  y ) ) )
5048, 49opeq12d 3585 . . . . . . . 8  |-  ( ( ( w  =  ( 1st `  x )  /\  v  =  ( 2nd `  x ) )  /\  ( u  =  ( 1st `  y
)  /\  f  =  ( 2nd `  y ) ) )  ->  <. (
( w  .N  f
)  +N  ( u  .N  v ) ) ,  ( v  .N  f ) >.  =  <. ( ( ( 1st `  x
)  .N  ( 2nd `  y ) )  +N  ( ( 1st `  y
)  .N  ( 2nd `  x ) ) ) ,  ( ( 2nd `  x )  .N  ( 2nd `  y ) )
>. )
5150eqeq2d 2067 . . . . . . 7  |-  ( ( ( w  =  ( 1st `  x )  /\  v  =  ( 2nd `  x ) )  /\  ( u  =  ( 1st `  y
)  /\  f  =  ( 2nd `  y ) ) )  ->  (
z  =  <. (
( w  .N  f
)  +N  ( u  .N  v ) ) ,  ( v  .N  f ) >.  <->  z  =  <. ( ( ( 1st `  x )  .N  ( 2nd `  y ) )  +N  ( ( 1st `  y )  .N  ( 2nd `  x ) ) ) ,  ( ( 2nd `  x )  .N  ( 2nd `  y
) ) >. )
)
5251copsex4g 4012 . . . . . 6  |-  ( ( ( ( 1st `  x
)  e.  N.  /\  ( 2nd `  x )  e.  N. )  /\  ( ( 1st `  y
)  e.  N.  /\  ( 2nd `  y )  e.  N. ) )  ->  ( E. w E. v E. u E. f ( ( <.
( 1st `  x
) ,  ( 2nd `  x ) >.  =  <. w ,  v >.  /\  <. ( 1st `  y ) ,  ( 2nd `  y
) >.  =  <. u ,  f >. )  /\  z  =  <. ( ( w  .N  f
)  +N  ( u  .N  v ) ) ,  ( v  .N  f ) >. )  <->  z  =  <. ( ( ( 1st `  x )  .N  ( 2nd `  y
) )  +N  (
( 1st `  y
)  .N  ( 2nd `  x ) ) ) ,  ( ( 2nd `  x )  .N  ( 2nd `  y ) )
>. ) )
5339, 41, 52syl2an 277 . . . . 5  |-  ( ( x  e.  ( N. 
X.  N. )  /\  y  e.  ( N.  X.  N. ) )  ->  ( E. w E. v E. u E. f ( ( <. ( 1st `  x
) ,  ( 2nd `  x ) >.  =  <. w ,  v >.  /\  <. ( 1st `  y ) ,  ( 2nd `  y
) >.  =  <. u ,  f >. )  /\  z  =  <. ( ( w  .N  f
)  +N  ( u  .N  v ) ) ,  ( v  .N  f ) >. )  <->  z  =  <. ( ( ( 1st `  x )  .N  ( 2nd `  y
) )  +N  (
( 1st `  y
)  .N  ( 2nd `  x ) ) ) ,  ( ( 2nd `  x )  .N  ( 2nd `  y ) )
>. ) )
5437, 53bitr3d 183 . . . 4  |-  ( ( x  e.  ( N. 
X.  N. )  /\  y  e.  ( N.  X.  N. ) )  ->  ( E. w E. v E. u E. f ( ( x  =  <. w ,  v >.  /\  y  =  <. u ,  f
>. )  /\  z  =  <. ( ( w  .N  f )  +N  ( v  .N  u
) ) ,  ( v  .N  f )
>. )  <->  z  =  <. ( ( ( 1st `  x
)  .N  ( 2nd `  y ) )  +N  ( ( 1st `  y
)  .N  ( 2nd `  x ) ) ) ,  ( ( 2nd `  x )  .N  ( 2nd `  y ) )
>. ) )
5554pm5.32i 435 . . 3  |-  ( ( ( x  e.  ( N.  X.  N. )  /\  y  e.  ( N.  X.  N. ) )  /\  E. w E. v E. u E. f
( ( x  = 
<. w ,  v >.  /\  y  =  <. u ,  f >. )  /\  z  =  <. ( ( w  .N  f
)  +N  ( v  .N  u ) ) ,  ( v  .N  f ) >. )
)  <->  ( ( x  e.  ( N.  X.  N. )  /\  y  e.  ( N.  X.  N. ) )  /\  z  =  <. ( ( ( 1st `  x )  .N  ( 2nd `  y
) )  +N  (
( 1st `  y
)  .N  ( 2nd `  x ) ) ) ,  ( ( 2nd `  x )  .N  ( 2nd `  y ) )
>. ) )
5655oprabbii 5588 . 2  |-  { <. <.
x ,  y >. ,  z >.  |  ( ( x  e.  ( N.  X.  N. )  /\  y  e.  ( N.  X.  N. ) )  /\  E. w E. v E. u E. f
( ( x  = 
<. w ,  v >.  /\  y  =  <. u ,  f >. )  /\  z  =  <. ( ( w  .N  f
)  +N  ( v  .N  u ) ) ,  ( v  .N  f ) >. )
) }  =  { <. <. x ,  y
>. ,  z >.  |  ( ( x  e.  ( N.  X.  N. )  /\  y  e.  ( N.  X.  N. )
)  /\  z  =  <. ( ( ( 1st `  x )  .N  ( 2nd `  y ) )  +N  ( ( 1st `  y )  .N  ( 2nd `  x ) ) ) ,  ( ( 2nd `  x )  .N  ( 2nd `  y
) ) >. ) }
571, 2, 563eqtr4i 2086 1  |-  +pQ  =  { <. <. x ,  y
>. ,  z >.  |  ( ( x  e.  ( N.  X.  N. )  /\  y  e.  ( N.  X.  N. )
)  /\  E. w E. v E. u E. f ( ( x  =  <. w ,  v
>.  /\  y  =  <. u ,  f >. )  /\  z  =  <. ( ( w  .N  f
)  +N  ( v  .N  u ) ) ,  ( v  .N  f ) >. )
) }
Colors of variables: wff set class
Syntax hints:    /\ wa 101    <-> wb 102    = wceq 1259   E.wex 1397    e. wcel 1409   <.cop 3406    X. cxp 4371   ` cfv 4930  (class class class)co 5540   {coprab 5541    |-> cmpt2 5542   1stc1st 5793   2ndc2nd 5794   N.cnpi 6428    +N cpli 6429    .N cmi 6430    +pQ cplpq 6432
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-in1 554  ax-in2 555  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-13 1420  ax-14 1421  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038  ax-coll 3900  ax-sep 3903  ax-nul 3911  ax-pow 3955  ax-pr 3972  ax-un 4198  ax-setind 4290  ax-iinf 4339
This theorem depends on definitions:  df-bi 114  df-3an 898  df-tru 1262  df-fal 1265  df-nf 1366  df-sb 1662  df-eu 1919  df-mo 1920  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-ne 2221  df-ral 2328  df-rex 2329  df-reu 2330  df-rab 2332  df-v 2576  df-sbc 2788  df-csb 2881  df-dif 2948  df-un 2950  df-in 2952  df-ss 2959  df-nul 3253  df-pw 3389  df-sn 3409  df-pr 3410  df-op 3412  df-uni 3609  df-int 3644  df-iun 3687  df-br 3793  df-opab 3847  df-mpt 3848  df-tr 3883  df-id 4058  df-iord 4131  df-on 4133  df-suc 4136  df-iom 4342  df-xp 4379  df-rel 4380  df-cnv 4381  df-co 4382  df-dm 4383  df-rn 4384  df-res 4385  df-ima 4386  df-iota 4895  df-fun 4932  df-fn 4933  df-f 4934  df-f1 4935  df-fo 4936  df-f1o 4937  df-fv 4938  df-ov 5543  df-oprab 5544  df-mpt2 5545  df-1st 5795  df-2nd 5796  df-recs 5951  df-irdg 5988  df-oadd 6036  df-omul 6037  df-ni 6460  df-mi 6462  df-plpq 6500
This theorem is referenced by:  addpipqqs  6526
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