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Theorem distrsrg 6901
Description: Multiplication of signed reals is distributive. (Contributed by Jim Kingdon, 4-Jan-2020.)
Assertion
Ref Expression
distrsrg  |-  ( ( A  e.  R.  /\  B  e.  R.  /\  C  e.  R. )  ->  ( A  .R  ( B  +R  C ) )  =  ( ( A  .R  B )  +R  ( A  .R  C ) ) )

Proof of Theorem distrsrg
Dummy variables  f  g  h  u  v  w  x  y  z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-nr 6869 . 2  |-  R.  =  ( ( P.  X.  P. ) /.  ~R  )
2 addsrpr 6887 . 2  |-  ( ( ( z  e.  P.  /\  w  e.  P. )  /\  ( v  e.  P.  /\  u  e.  P. )
)  ->  ( [ <. z ,  w >. ]  ~R  +R  [ <. v ,  u >. ]  ~R  )  =  [ <. (
z  +P.  v ) ,  ( w  +P.  u ) >. ]  ~R  )
3 mulsrpr 6888 . 2  |-  ( ( ( x  e.  P.  /\  y  e.  P. )  /\  ( ( z  +P.  v )  e.  P.  /\  ( w  +P.  u
)  e.  P. )
)  ->  ( [ <. x ,  y >. ]  ~R  .R  [ <. ( z  +P.  v ) ,  ( w  +P.  u ) >. ]  ~R  )  =  [ <. (
( x  .P.  (
z  +P.  v )
)  +P.  ( y  .P.  ( w  +P.  u
) ) ) ,  ( ( x  .P.  ( w  +P.  u ) )  +P.  ( y  .P.  ( z  +P.  v ) ) )
>. ]  ~R  )
4 mulsrpr 6888 . 2  |-  ( ( ( x  e.  P.  /\  y  e.  P. )  /\  ( z  e.  P.  /\  w  e.  P. )
)  ->  ( [ <. x ,  y >. ]  ~R  .R  [ <. z ,  w >. ]  ~R  )  =  [ <. (
( x  .P.  z
)  +P.  ( y  .P.  w ) ) ,  ( ( x  .P.  w )  +P.  (
y  .P.  z )
) >. ]  ~R  )
5 mulsrpr 6888 . 2  |-  ( ( ( x  e.  P.  /\  y  e.  P. )  /\  ( v  e.  P.  /\  u  e.  P. )
)  ->  ( [ <. x ,  y >. ]  ~R  .R  [ <. v ,  u >. ]  ~R  )  =  [ <. (
( x  .P.  v
)  +P.  ( y  .P.  u ) ) ,  ( ( x  .P.  u )  +P.  (
y  .P.  v )
) >. ]  ~R  )
6 addsrpr 6887 . 2  |-  ( ( ( ( ( x  .P.  z )  +P.  ( y  .P.  w
) )  e.  P.  /\  ( ( x  .P.  w )  +P.  (
y  .P.  z )
)  e.  P. )  /\  ( ( ( x  .P.  v )  +P.  ( y  .P.  u
) )  e.  P.  /\  ( ( x  .P.  u )  +P.  (
y  .P.  v )
)  e.  P. )
)  ->  ( [ <. ( ( x  .P.  z )  +P.  (
y  .P.  w )
) ,  ( ( x  .P.  w )  +P.  ( y  .P.  z ) ) >. ]  ~R  +R  [ <. ( ( x  .P.  v
)  +P.  ( y  .P.  u ) ) ,  ( ( x  .P.  u )  +P.  (
y  .P.  v )
) >. ]  ~R  )  =  [ <. ( ( ( x  .P.  z )  +P.  ( y  .P.  w ) )  +P.  ( ( x  .P.  v )  +P.  (
y  .P.  u )
) ) ,  ( ( ( x  .P.  w )  +P.  (
y  .P.  z )
)  +P.  ( (
x  .P.  u )  +P.  ( y  .P.  v
) ) ) >. ]  ~R  )
7 addclpr 6692 . . . 4  |-  ( ( z  e.  P.  /\  v  e.  P. )  ->  ( z  +P.  v
)  e.  P. )
87ad2ant2r 486 . . 3  |-  ( ( ( z  e.  P.  /\  w  e.  P. )  /\  ( v  e.  P.  /\  u  e.  P. )
)  ->  ( z  +P.  v )  e.  P. )
9 addclpr 6692 . . . 4  |-  ( ( w  e.  P.  /\  u  e.  P. )  ->  ( w  +P.  u
)  e.  P. )
109ad2ant2l 485 . . 3  |-  ( ( ( z  e.  P.  /\  w  e.  P. )  /\  ( v  e.  P.  /\  u  e.  P. )
)  ->  ( w  +P.  u )  e.  P. )
118, 10jca 294 . 2  |-  ( ( ( z  e.  P.  /\  w  e.  P. )  /\  ( v  e.  P.  /\  u  e.  P. )
)  ->  ( (
z  +P.  v )  e.  P.  /\  ( w  +P.  u )  e. 
P. ) )
12 mulclpr 6727 . . . . 5  |-  ( ( x  e.  P.  /\  z  e.  P. )  ->  ( x  .P.  z
)  e.  P. )
1312ad2ant2r 486 . . . 4  |-  ( ( ( x  e.  P.  /\  y  e.  P. )  /\  ( z  e.  P.  /\  w  e.  P. )
)  ->  ( x  .P.  z )  e.  P. )
14 mulclpr 6727 . . . . 5  |-  ( ( y  e.  P.  /\  w  e.  P. )  ->  ( y  .P.  w
)  e.  P. )
1514ad2ant2l 485 . . . 4  |-  ( ( ( x  e.  P.  /\  y  e.  P. )  /\  ( z  e.  P.  /\  w  e.  P. )
)  ->  ( y  .P.  w )  e.  P. )
16 addclpr 6692 . . . 4  |-  ( ( ( x  .P.  z
)  e.  P.  /\  ( y  .P.  w
)  e.  P. )  ->  ( ( x  .P.  z )  +P.  (
y  .P.  w )
)  e.  P. )
1713, 15, 16syl2anc 397 . . 3  |-  ( ( ( x  e.  P.  /\  y  e.  P. )  /\  ( z  e.  P.  /\  w  e.  P. )
)  ->  ( (
x  .P.  z )  +P.  ( y  .P.  w
) )  e.  P. )
18 mulclpr 6727 . . . . 5  |-  ( ( x  e.  P.  /\  w  e.  P. )  ->  ( x  .P.  w
)  e.  P. )
1918ad2ant2rl 488 . . . 4  |-  ( ( ( x  e.  P.  /\  y  e.  P. )  /\  ( z  e.  P.  /\  w  e.  P. )
)  ->  ( x  .P.  w )  e.  P. )
20 mulclpr 6727 . . . . 5  |-  ( ( y  e.  P.  /\  z  e.  P. )  ->  ( y  .P.  z
)  e.  P. )
2120ad2ant2lr 487 . . . 4  |-  ( ( ( x  e.  P.  /\  y  e.  P. )  /\  ( z  e.  P.  /\  w  e.  P. )
)  ->  ( y  .P.  z )  e.  P. )
22 addclpr 6692 . . . 4  |-  ( ( ( x  .P.  w
)  e.  P.  /\  ( y  .P.  z
)  e.  P. )  ->  ( ( x  .P.  w )  +P.  (
y  .P.  z )
)  e.  P. )
2319, 21, 22syl2anc 397 . . 3  |-  ( ( ( x  e.  P.  /\  y  e.  P. )  /\  ( z  e.  P.  /\  w  e.  P. )
)  ->  ( (
x  .P.  w )  +P.  ( y  .P.  z
) )  e.  P. )
2417, 23jca 294 . 2  |-  ( ( ( x  e.  P.  /\  y  e.  P. )  /\  ( z  e.  P.  /\  w  e.  P. )
)  ->  ( (
( x  .P.  z
)  +P.  ( y  .P.  w ) )  e. 
P.  /\  ( (
x  .P.  w )  +P.  ( y  .P.  z
) )  e.  P. ) )
25 mulclpr 6727 . . . . 5  |-  ( ( x  e.  P.  /\  v  e.  P. )  ->  ( x  .P.  v
)  e.  P. )
2625ad2ant2r 486 . . . 4  |-  ( ( ( x  e.  P.  /\  y  e.  P. )  /\  ( v  e.  P.  /\  u  e.  P. )
)  ->  ( x  .P.  v )  e.  P. )
27 mulclpr 6727 . . . . 5  |-  ( ( y  e.  P.  /\  u  e.  P. )  ->  ( y  .P.  u
)  e.  P. )
2827ad2ant2l 485 . . . 4  |-  ( ( ( x  e.  P.  /\  y  e.  P. )  /\  ( v  e.  P.  /\  u  e.  P. )
)  ->  ( y  .P.  u )  e.  P. )
29 addclpr 6692 . . . 4  |-  ( ( ( x  .P.  v
)  e.  P.  /\  ( y  .P.  u
)  e.  P. )  ->  ( ( x  .P.  v )  +P.  (
y  .P.  u )
)  e.  P. )
3026, 28, 29syl2anc 397 . . 3  |-  ( ( ( x  e.  P.  /\  y  e.  P. )  /\  ( v  e.  P.  /\  u  e.  P. )
)  ->  ( (
x  .P.  v )  +P.  ( y  .P.  u
) )  e.  P. )
31 mulclpr 6727 . . . . 5  |-  ( ( x  e.  P.  /\  u  e.  P. )  ->  ( x  .P.  u
)  e.  P. )
3231ad2ant2rl 488 . . . 4  |-  ( ( ( x  e.  P.  /\  y  e.  P. )  /\  ( v  e.  P.  /\  u  e.  P. )
)  ->  ( x  .P.  u )  e.  P. )
33 mulclpr 6727 . . . . 5  |-  ( ( y  e.  P.  /\  v  e.  P. )  ->  ( y  .P.  v
)  e.  P. )
3433ad2ant2lr 487 . . . 4  |-  ( ( ( x  e.  P.  /\  y  e.  P. )  /\  ( v  e.  P.  /\  u  e.  P. )
)  ->  ( y  .P.  v )  e.  P. )
35 addclpr 6692 . . . 4  |-  ( ( ( x  .P.  u
)  e.  P.  /\  ( y  .P.  v
)  e.  P. )  ->  ( ( x  .P.  u )  +P.  (
y  .P.  v )
)  e.  P. )
3632, 34, 35syl2anc 397 . . 3  |-  ( ( ( x  e.  P.  /\  y  e.  P. )  /\  ( v  e.  P.  /\  u  e.  P. )
)  ->  ( (
x  .P.  u )  +P.  ( y  .P.  v
) )  e.  P. )
3730, 36jca 294 . 2  |-  ( ( ( x  e.  P.  /\  y  e.  P. )  /\  ( v  e.  P.  /\  u  e.  P. )
)  ->  ( (
( x  .P.  v
)  +P.  ( y  .P.  u ) )  e. 
P.  /\  ( (
x  .P.  u )  +P.  ( y  .P.  v
) )  e.  P. ) )
38 simp1l 939 . . . . 5  |-  ( ( ( x  e.  P.  /\  y  e.  P. )  /\  ( z  e.  P.  /\  w  e.  P. )  /\  ( v  e.  P.  /\  u  e.  P. )
)  ->  x  e.  P. )
39 simp2l 941 . . . . 5  |-  ( ( ( x  e.  P.  /\  y  e.  P. )  /\  ( z  e.  P.  /\  w  e.  P. )  /\  ( v  e.  P.  /\  u  e.  P. )
)  ->  z  e.  P. )
40 simp3l 943 . . . . 5  |-  ( ( ( x  e.  P.  /\  y  e.  P. )  /\  ( z  e.  P.  /\  w  e.  P. )  /\  ( v  e.  P.  /\  u  e.  P. )
)  ->  v  e.  P. )
41 distrprg 6743 . . . . 5  |-  ( ( x  e.  P.  /\  z  e.  P.  /\  v  e.  P. )  ->  (
x  .P.  ( z  +P.  v ) )  =  ( ( x  .P.  z )  +P.  (
x  .P.  v )
) )
4238, 39, 40, 41syl3anc 1146 . . . 4  |-  ( ( ( x  e.  P.  /\  y  e.  P. )  /\  ( z  e.  P.  /\  w  e.  P. )  /\  ( v  e.  P.  /\  u  e.  P. )
)  ->  ( x  .P.  ( z  +P.  v
) )  =  ( ( x  .P.  z
)  +P.  ( x  .P.  v ) ) )
43 simp1r 940 . . . . 5  |-  ( ( ( x  e.  P.  /\  y  e.  P. )  /\  ( z  e.  P.  /\  w  e.  P. )  /\  ( v  e.  P.  /\  u  e.  P. )
)  ->  y  e.  P. )
44 simp2r 942 . . . . 5  |-  ( ( ( x  e.  P.  /\  y  e.  P. )  /\  ( z  e.  P.  /\  w  e.  P. )  /\  ( v  e.  P.  /\  u  e.  P. )
)  ->  w  e.  P. )
45 simp3r 944 . . . . 5  |-  ( ( ( x  e.  P.  /\  y  e.  P. )  /\  ( z  e.  P.  /\  w  e.  P. )  /\  ( v  e.  P.  /\  u  e.  P. )
)  ->  u  e.  P. )
46 distrprg 6743 . . . . 5  |-  ( ( y  e.  P.  /\  w  e.  P.  /\  u  e.  P. )  ->  (
y  .P.  ( w  +P.  u ) )  =  ( ( y  .P.  w )  +P.  (
y  .P.  u )
) )
4743, 44, 45, 46syl3anc 1146 . . . 4  |-  ( ( ( x  e.  P.  /\  y  e.  P. )  /\  ( z  e.  P.  /\  w  e.  P. )  /\  ( v  e.  P.  /\  u  e.  P. )
)  ->  ( y  .P.  ( w  +P.  u
) )  =  ( ( y  .P.  w
)  +P.  ( y  .P.  u ) ) )
4842, 47oveq12d 5557 . . 3  |-  ( ( ( x  e.  P.  /\  y  e.  P. )  /\  ( z  e.  P.  /\  w  e.  P. )  /\  ( v  e.  P.  /\  u  e.  P. )
)  ->  ( (
x  .P.  ( z  +P.  v ) )  +P.  ( y  .P.  (
w  +P.  u )
) )  =  ( ( ( x  .P.  z )  +P.  (
x  .P.  v )
)  +P.  ( (
y  .P.  w )  +P.  ( y  .P.  u
) ) ) )
4938, 39, 12syl2anc 397 . . . 4  |-  ( ( ( x  e.  P.  /\  y  e.  P. )  /\  ( z  e.  P.  /\  w  e.  P. )  /\  ( v  e.  P.  /\  u  e.  P. )
)  ->  ( x  .P.  z )  e.  P. )
5038, 40, 25syl2anc 397 . . . 4  |-  ( ( ( x  e.  P.  /\  y  e.  P. )  /\  ( z  e.  P.  /\  w  e.  P. )  /\  ( v  e.  P.  /\  u  e.  P. )
)  ->  ( x  .P.  v )  e.  P. )
5143, 44, 14syl2anc 397 . . . 4  |-  ( ( ( x  e.  P.  /\  y  e.  P. )  /\  ( z  e.  P.  /\  w  e.  P. )  /\  ( v  e.  P.  /\  u  e.  P. )
)  ->  ( y  .P.  w )  e.  P. )
52 addcomprg 6733 . . . . 5  |-  ( ( f  e.  P.  /\  g  e.  P. )  ->  ( f  +P.  g
)  =  ( g  +P.  f ) )
5352adantl 266 . . . 4  |-  ( ( ( ( x  e. 
P.  /\  y  e.  P. )  /\  (
z  e.  P.  /\  w  e.  P. )  /\  ( v  e.  P.  /\  u  e.  P. )
)  /\  ( f  e.  P.  /\  g  e. 
P. ) )  -> 
( f  +P.  g
)  =  ( g  +P.  f ) )
54 addassprg 6734 . . . . 5  |-  ( ( f  e.  P.  /\  g  e.  P.  /\  h  e.  P. )  ->  (
( f  +P.  g
)  +P.  h )  =  ( f  +P.  ( g  +P.  h
) ) )
5554adantl 266 . . . 4  |-  ( ( ( ( x  e. 
P.  /\  y  e.  P. )  /\  (
z  e.  P.  /\  w  e.  P. )  /\  ( v  e.  P.  /\  u  e.  P. )
)  /\  ( f  e.  P.  /\  g  e. 
P.  /\  h  e.  P. ) )  ->  (
( f  +P.  g
)  +P.  h )  =  ( f  +P.  ( g  +P.  h
) ) )
5643, 45, 27syl2anc 397 . . . 4  |-  ( ( ( x  e.  P.  /\  y  e.  P. )  /\  ( z  e.  P.  /\  w  e.  P. )  /\  ( v  e.  P.  /\  u  e.  P. )
)  ->  ( y  .P.  u )  e.  P. )
57 addclpr 6692 . . . . 5  |-  ( ( f  e.  P.  /\  g  e.  P. )  ->  ( f  +P.  g
)  e.  P. )
5857adantl 266 . . . 4  |-  ( ( ( ( x  e. 
P.  /\  y  e.  P. )  /\  (
z  e.  P.  /\  w  e.  P. )  /\  ( v  e.  P.  /\  u  e.  P. )
)  /\  ( f  e.  P.  /\  g  e. 
P. ) )  -> 
( f  +P.  g
)  e.  P. )
5949, 50, 51, 53, 55, 56, 58caov4d 5712 . . 3  |-  ( ( ( x  e.  P.  /\  y  e.  P. )  /\  ( z  e.  P.  /\  w  e.  P. )  /\  ( v  e.  P.  /\  u  e.  P. )
)  ->  ( (
( x  .P.  z
)  +P.  ( x  .P.  v ) )  +P.  ( ( y  .P.  w )  +P.  (
y  .P.  u )
) )  =  ( ( ( x  .P.  z )  +P.  (
y  .P.  w )
)  +P.  ( (
x  .P.  v )  +P.  ( y  .P.  u
) ) ) )
6048, 59eqtrd 2088 . 2  |-  ( ( ( x  e.  P.  /\  y  e.  P. )  /\  ( z  e.  P.  /\  w  e.  P. )  /\  ( v  e.  P.  /\  u  e.  P. )
)  ->  ( (
x  .P.  ( z  +P.  v ) )  +P.  ( y  .P.  (
w  +P.  u )
) )  =  ( ( ( x  .P.  z )  +P.  (
y  .P.  w )
)  +P.  ( (
x  .P.  v )  +P.  ( y  .P.  u
) ) ) )
61 distrprg 6743 . . . . 5  |-  ( ( x  e.  P.  /\  w  e.  P.  /\  u  e.  P. )  ->  (
x  .P.  ( w  +P.  u ) )  =  ( ( x  .P.  w )  +P.  (
x  .P.  u )
) )
6238, 44, 45, 61syl3anc 1146 . . . 4  |-  ( ( ( x  e.  P.  /\  y  e.  P. )  /\  ( z  e.  P.  /\  w  e.  P. )  /\  ( v  e.  P.  /\  u  e.  P. )
)  ->  ( x  .P.  ( w  +P.  u
) )  =  ( ( x  .P.  w
)  +P.  ( x  .P.  u ) ) )
63 distrprg 6743 . . . . 5  |-  ( ( y  e.  P.  /\  z  e.  P.  /\  v  e.  P. )  ->  (
y  .P.  ( z  +P.  v ) )  =  ( ( y  .P.  z )  +P.  (
y  .P.  v )
) )
6443, 39, 40, 63syl3anc 1146 . . . 4  |-  ( ( ( x  e.  P.  /\  y  e.  P. )  /\  ( z  e.  P.  /\  w  e.  P. )  /\  ( v  e.  P.  /\  u  e.  P. )
)  ->  ( y  .P.  ( z  +P.  v
) )  =  ( ( y  .P.  z
)  +P.  ( y  .P.  v ) ) )
6562, 64oveq12d 5557 . . 3  |-  ( ( ( x  e.  P.  /\  y  e.  P. )  /\  ( z  e.  P.  /\  w  e.  P. )  /\  ( v  e.  P.  /\  u  e.  P. )
)  ->  ( (
x  .P.  ( w  +P.  u ) )  +P.  ( y  .P.  (
z  +P.  v )
) )  =  ( ( ( x  .P.  w )  +P.  (
x  .P.  u )
)  +P.  ( (
y  .P.  z )  +P.  ( y  .P.  v
) ) ) )
6638, 44, 18syl2anc 397 . . . 4  |-  ( ( ( x  e.  P.  /\  y  e.  P. )  /\  ( z  e.  P.  /\  w  e.  P. )  /\  ( v  e.  P.  /\  u  e.  P. )
)  ->  ( x  .P.  w )  e.  P. )
6738, 45, 31syl2anc 397 . . . 4  |-  ( ( ( x  e.  P.  /\  y  e.  P. )  /\  ( z  e.  P.  /\  w  e.  P. )  /\  ( v  e.  P.  /\  u  e.  P. )
)  ->  ( x  .P.  u )  e.  P. )
6843, 39, 20syl2anc 397 . . . 4  |-  ( ( ( x  e.  P.  /\  y  e.  P. )  /\  ( z  e.  P.  /\  w  e.  P. )  /\  ( v  e.  P.  /\  u  e.  P. )
)  ->  ( y  .P.  z )  e.  P. )
6943, 40, 33syl2anc 397 . . . 4  |-  ( ( ( x  e.  P.  /\  y  e.  P. )  /\  ( z  e.  P.  /\  w  e.  P. )  /\  ( v  e.  P.  /\  u  e.  P. )
)  ->  ( y  .P.  v )  e.  P. )
7066, 67, 68, 53, 55, 69, 58caov4d 5712 . . 3  |-  ( ( ( x  e.  P.  /\  y  e.  P. )  /\  ( z  e.  P.  /\  w  e.  P. )  /\  ( v  e.  P.  /\  u  e.  P. )
)  ->  ( (
( x  .P.  w
)  +P.  ( x  .P.  u ) )  +P.  ( ( y  .P.  z )  +P.  (
y  .P.  v )
) )  =  ( ( ( x  .P.  w )  +P.  (
y  .P.  z )
)  +P.  ( (
x  .P.  u )  +P.  ( y  .P.  v
) ) ) )
7165, 70eqtrd 2088 . 2  |-  ( ( ( x  e.  P.  /\  y  e.  P. )  /\  ( z  e.  P.  /\  w  e.  P. )  /\  ( v  e.  P.  /\  u  e.  P. )
)  ->  ( (
x  .P.  ( w  +P.  u ) )  +P.  ( y  .P.  (
z  +P.  v )
) )  =  ( ( ( x  .P.  w )  +P.  (
y  .P.  z )
)  +P.  ( (
x  .P.  u )  +P.  ( y  .P.  v
) ) ) )
721, 2, 3, 4, 5, 6, 11, 24, 37, 60, 71ecovidi 6248 1  |-  ( ( A  e.  R.  /\  B  e.  R.  /\  C  e.  R. )  ->  ( A  .R  ( B  +R  C ) )  =  ( ( A  .R  B )  +R  ( A  .R  C ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 101    /\ w3a 896    = wceq 1259    e. wcel 1409  (class class class)co 5539   P.cnp 6446    +P. cpp 6448    .P. cmp 6449    ~R cer 6451   R.cnr 6452    +R cplr 6456    .R cmr 6457
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 3899  ax-sep 3902  ax-nul 3910  ax-pow 3954  ax-pr 3971  ax-un 4197  ax-setind 4289  ax-iinf 4338
This theorem depends on definitions:  df-bi 114  df-dc 754  df-3or 897  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 2787  df-csb 2880  df-dif 2947  df-un 2949  df-in 2951  df-ss 2958  df-nul 3252  df-pw 3388  df-sn 3408  df-pr 3409  df-op 3411  df-uni 3608  df-int 3643  df-iun 3686  df-br 3792  df-opab 3846  df-mpt 3847  df-tr 3882  df-eprel 4053  df-id 4057  df-po 4060  df-iso 4061  df-iord 4130  df-on 4132  df-suc 4135  df-iom 4341  df-xp 4378  df-rel 4379  df-cnv 4380  df-co 4381  df-dm 4382  df-rn 4383  df-res 4384  df-ima 4385  df-iota 4894  df-fun 4931  df-fn 4932  df-f 4933  df-f1 4934  df-fo 4935  df-f1o 4936  df-fv 4937  df-ov 5542  df-oprab 5543  df-mpt2 5544  df-1st 5794  df-2nd 5795  df-recs 5950  df-irdg 5987  df-1o 6031  df-2o 6032  df-oadd 6035  df-omul 6036  df-er 6136  df-ec 6138  df-qs 6142  df-ni 6459  df-pli 6460  df-mi 6461  df-lti 6462  df-plpq 6499  df-mpq 6500  df-enq 6502  df-nqqs 6503  df-plqqs 6504  df-mqqs 6505  df-1nqqs 6506  df-rq 6507  df-ltnqqs 6508  df-enq0 6579  df-nq0 6580  df-0nq0 6581  df-plq0 6582  df-mq0 6583  df-inp 6621  df-iplp 6623  df-imp 6624  df-enr 6868  df-nr 6869  df-plr 6870  df-mr 6871
This theorem is referenced by:  pn0sr  6913  axmulass  7004  axdistr  7005
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