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Theorem addnnnq0 7398
Description: Addition of nonnegative fractions in terms of natural numbers. (Contributed by Jim Kingdon, 22-Nov-2019.)
Assertion
Ref Expression
addnnnq0  |-  ( ( ( A  e.  om  /\  B  e.  N. )  /\  ( C  e.  om  /\  D  e.  N. )
)  ->  ( [ <. A ,  B >. ] ~Q0 +Q0  [ <. C ,  D >. ] ~Q0  )  =  [ <. (
( A  .o  D
)  +o  ( B  .o  C ) ) ,  ( B  .o  D ) >. ] ~Q0  )

Proof of Theorem addnnnq0
Dummy variables  x  y  z  w  v  u  t are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 opelxpi 4641 . . . 4  |-  ( ( A  e.  om  /\  B  e.  N. )  -> 
<. A ,  B >.  e.  ( om  X.  N. ) )
2 enq0ex 7388 . . . . 5  |- ~Q0  e.  _V
32ecelqsi 6563 . . . 4  |-  ( <. A ,  B >.  e.  ( om  X.  N. )  ->  [ <. A ,  B >. ] ~Q0  e.  ( ( om 
X.  N. ) /. ~Q0  ) )
41, 3syl 14 . . 3  |-  ( ( A  e.  om  /\  B  e.  N. )  ->  [ <. A ,  B >. ] ~Q0  e.  ( ( om  X.  N. ) /. ~Q0  ) )
5 opelxpi 4641 . . . 4  |-  ( ( C  e.  om  /\  D  e.  N. )  -> 
<. C ,  D >.  e.  ( om  X.  N. ) )
62ecelqsi 6563 . . . 4  |-  ( <. C ,  D >.  e.  ( om  X.  N. )  ->  [ <. C ,  D >. ] ~Q0  e.  ( ( om 
X.  N. ) /. ~Q0  ) )
75, 6syl 14 . . 3  |-  ( ( C  e.  om  /\  D  e.  N. )  ->  [ <. C ,  D >. ] ~Q0  e.  ( ( om  X.  N. ) /. ~Q0  ) )
84, 7anim12i 336 . 2  |-  ( ( ( A  e.  om  /\  B  e.  N. )  /\  ( C  e.  om  /\  D  e.  N. )
)  ->  ( [ <. A ,  B >. ] ~Q0  e.  ( ( om  X.  N. ) /. ~Q0  )  /\  [ <. C ,  D >. ] ~Q0  e.  ( ( om 
X.  N. ) /. ~Q0  ) ) )
9 eqid 2170 . . . 4  |-  [ <. A ,  B >. ] ~Q0  =  [ <. A ,  B >. ] ~Q0
10 eqid 2170 . . . 4  |-  [ <. C ,  D >. ] ~Q0  =  [ <. C ,  D >. ] ~Q0
119, 10pm3.2i 270 . . 3  |-  ( [
<. A ,  B >. ] ~Q0  =  [ <. A ,  B >. ] ~Q0  /\  [ <. C ,  D >. ] ~Q0  =  [ <. C ,  D >. ] ~Q0  )
12 eqid 2170 . . 3  |-  [ <. ( ( A  .o  D
)  +o  ( B  .o  C ) ) ,  ( B  .o  D ) >. ] ~Q0  =  [ <. (
( A  .o  D
)  +o  ( B  .o  C ) ) ,  ( B  .o  D ) >. ] ~Q0
13 opeq12 3765 . . . . . . . . 9  |-  ( ( w  =  A  /\  v  =  B )  -> 
<. w ,  v >.  =  <. A ,  B >. )
1413eceq1d 6545 . . . . . . . 8  |-  ( ( w  =  A  /\  v  =  B )  ->  [ <. w ,  v
>. ] ~Q0  =  [ <. A ,  B >. ] ~Q0  )
1514eqeq2d 2182 . . . . . . 7  |-  ( ( w  =  A  /\  v  =  B )  ->  ( [ <. A ,  B >. ] ~Q0  =  [ <. w ,  v >. ] ~Q0  <->  [ <. A ,  B >. ] ~Q0  =  [ <. A ,  B >. ] ~Q0  ) )
1615anbi1d 462 . . . . . 6  |-  ( ( w  =  A  /\  v  =  B )  ->  ( ( [ <. A ,  B >. ] ~Q0  =  [ <. w ,  v >. ] ~Q0  /\  [ <. C ,  D >. ] ~Q0  =  [ <. C ,  D >. ] ~Q0  )  <->  ( [ <. A ,  B >. ] ~Q0  =  [ <. A ,  B >. ] ~Q0  /\  [ <. C ,  D >. ] ~Q0  =  [ <. C ,  D >. ] ~Q0  ) ) )
17 simpl 108 . . . . . . . . . . 11  |-  ( ( w  =  A  /\  v  =  B )  ->  w  =  A )
1817oveq1d 5865 . . . . . . . . . 10  |-  ( ( w  =  A  /\  v  =  B )  ->  ( w  .o  D
)  =  ( A  .o  D ) )
19 simpr 109 . . . . . . . . . . 11  |-  ( ( w  =  A  /\  v  =  B )  ->  v  =  B )
2019oveq1d 5865 . . . . . . . . . 10  |-  ( ( w  =  A  /\  v  =  B )  ->  ( v  .o  C
)  =  ( B  .o  C ) )
2118, 20oveq12d 5868 . . . . . . . . 9  |-  ( ( w  =  A  /\  v  =  B )  ->  ( ( w  .o  D )  +o  (
v  .o  C ) )  =  ( ( A  .o  D )  +o  ( B  .o  C ) ) )
2219oveq1d 5865 . . . . . . . . 9  |-  ( ( w  =  A  /\  v  =  B )  ->  ( v  .o  D
)  =  ( B  .o  D ) )
2321, 22opeq12d 3771 . . . . . . . 8  |-  ( ( w  =  A  /\  v  =  B )  -> 
<. ( ( w  .o  D )  +o  (
v  .o  C ) ) ,  ( v  .o  D ) >.  =  <. ( ( A  .o  D )  +o  ( B  .o  C
) ) ,  ( B  .o  D )
>. )
2423eceq1d 6545 . . . . . . 7  |-  ( ( w  =  A  /\  v  =  B )  ->  [ <. ( ( w  .o  D )  +o  ( v  .o  C
) ) ,  ( v  .o  D )
>. ] ~Q0  =  [ <. ( ( A  .o  D )  +o  ( B  .o  C
) ) ,  ( B  .o  D )
>. ] ~Q0  )
2524eqeq2d 2182 . . . . . 6  |-  ( ( w  =  A  /\  v  =  B )  ->  ( [ <. (
( A  .o  D
)  +o  ( B  .o  C ) ) ,  ( B  .o  D ) >. ] ~Q0  =  [ <. (
( w  .o  D
)  +o  ( v  .o  C ) ) ,  ( v  .o  D ) >. ] ~Q0  <->  [ <. ( ( A  .o  D )  +o  ( B  .o  C
) ) ,  ( B  .o  D )
>. ] ~Q0  =  [ <. ( ( A  .o  D )  +o  ( B  .o  C
) ) ,  ( B  .o  D )
>. ] ~Q0  ) )
2616, 25anbi12d 470 . . . . 5  |-  ( ( w  =  A  /\  v  =  B )  ->  ( ( ( [
<. A ,  B >. ] ~Q0  =  [ <. w ,  v
>. ] ~Q0  /\  [ <. C ,  D >. ] ~Q0  =  [ <. C ,  D >. ] ~Q0  )  /\  [ <. (
( A  .o  D
)  +o  ( B  .o  C ) ) ,  ( B  .o  D ) >. ] ~Q0  =  [ <. (
( w  .o  D
)  +o  ( v  .o  C ) ) ,  ( v  .o  D ) >. ] ~Q0  )  <->  ( ( [
<. A ,  B >. ] ~Q0  =  [ <. A ,  B >. ] ~Q0  /\  [ <. C ,  D >. ] ~Q0  =  [ <. C ,  D >. ] ~Q0  )  /\  [ <. (
( A  .o  D
)  +o  ( B  .o  C ) ) ,  ( B  .o  D ) >. ] ~Q0  =  [ <. (
( A  .o  D
)  +o  ( B  .o  C ) ) ,  ( B  .o  D ) >. ] ~Q0  ) ) )
2726spc2egv 2820 . . . 4  |-  ( ( A  e.  om  /\  B  e.  N. )  ->  ( ( ( [
<. A ,  B >. ] ~Q0  =  [ <. A ,  B >. ] ~Q0  /\  [ <. C ,  D >. ] ~Q0  =  [ <. C ,  D >. ] ~Q0  )  /\  [ <. (
( A  .o  D
)  +o  ( B  .o  C ) ) ,  ( B  .o  D ) >. ] ~Q0  =  [ <. (
( A  .o  D
)  +o  ( B  .o  C ) ) ,  ( B  .o  D ) >. ] ~Q0  )  ->  E. w E. v ( ( [
<. A ,  B >. ] ~Q0  =  [ <. w ,  v
>. ] ~Q0  /\  [ <. C ,  D >. ] ~Q0  =  [ <. C ,  D >. ] ~Q0  )  /\  [ <. (
( A  .o  D
)  +o  ( B  .o  C ) ) ,  ( B  .o  D ) >. ] ~Q0  =  [ <. (
( w  .o  D
)  +o  ( v  .o  C ) ) ,  ( v  .o  D ) >. ] ~Q0  ) ) )
28 opeq12 3765 . . . . . . . . . 10  |-  ( ( u  =  C  /\  t  =  D )  -> 
<. u ,  t >.  =  <. C ,  D >. )
2928eceq1d 6545 . . . . . . . . 9  |-  ( ( u  =  C  /\  t  =  D )  ->  [ <. u ,  t
>. ] ~Q0  =  [ <. C ,  D >. ] ~Q0  )
3029eqeq2d 2182 . . . . . . . 8  |-  ( ( u  =  C  /\  t  =  D )  ->  ( [ <. C ,  D >. ] ~Q0  =  [ <. u ,  t >. ] ~Q0  <->  [ <. C ,  D >. ] ~Q0  =  [ <. C ,  D >. ] ~Q0  ) )
3130anbi2d 461 . . . . . . 7  |-  ( ( u  =  C  /\  t  =  D )  ->  ( ( [ <. A ,  B >. ] ~Q0  =  [ <. w ,  v >. ] ~Q0  /\  [ <. C ,  D >. ] ~Q0  =  [ <. u ,  t >. ] ~Q0  )  <->  ( [ <. A ,  B >. ] ~Q0  =  [ <. w ,  v >. ] ~Q0  /\  [ <. C ,  D >. ] ~Q0  =  [ <. C ,  D >. ] ~Q0  ) ) )
32 simpr 109 . . . . . . . . . . . 12  |-  ( ( u  =  C  /\  t  =  D )  ->  t  =  D )
3332oveq2d 5866 . . . . . . . . . . 11  |-  ( ( u  =  C  /\  t  =  D )  ->  ( w  .o  t
)  =  ( w  .o  D ) )
34 simpl 108 . . . . . . . . . . . 12  |-  ( ( u  =  C  /\  t  =  D )  ->  u  =  C )
3534oveq2d 5866 . . . . . . . . . . 11  |-  ( ( u  =  C  /\  t  =  D )  ->  ( v  .o  u
)  =  ( v  .o  C ) )
3633, 35oveq12d 5868 . . . . . . . . . 10  |-  ( ( u  =  C  /\  t  =  D )  ->  ( ( w  .o  t )  +o  (
v  .o  u ) )  =  ( ( w  .o  D )  +o  ( v  .o  C ) ) )
3732oveq2d 5866 . . . . . . . . . 10  |-  ( ( u  =  C  /\  t  =  D )  ->  ( v  .o  t
)  =  ( v  .o  D ) )
3836, 37opeq12d 3771 . . . . . . . . 9  |-  ( ( u  =  C  /\  t  =  D )  -> 
<. ( ( w  .o  t )  +o  (
v  .o  u ) ) ,  ( v  .o  t ) >.  =  <. ( ( w  .o  D )  +o  ( v  .o  C
) ) ,  ( v  .o  D )
>. )
3938eceq1d 6545 . . . . . . . 8  |-  ( ( u  =  C  /\  t  =  D )  ->  [ <. ( ( w  .o  t )  +o  ( v  .o  u
) ) ,  ( v  .o  t )
>. ] ~Q0  =  [ <. ( ( w  .o  D )  +o  ( v  .o  C
) ) ,  ( v  .o  D )
>. ] ~Q0  )
4039eqeq2d 2182 . . . . . . 7  |-  ( ( u  =  C  /\  t  =  D )  ->  ( [ <. (
( A  .o  D
)  +o  ( B  .o  C ) ) ,  ( B  .o  D ) >. ] ~Q0  =  [ <. (
( w  .o  t
)  +o  ( v  .o  u ) ) ,  ( v  .o  t ) >. ] ~Q0  <->  [ <. ( ( A  .o  D )  +o  ( B  .o  C
) ) ,  ( B  .o  D )
>. ] ~Q0  =  [ <. ( ( w  .o  D )  +o  ( v  .o  C
) ) ,  ( v  .o  D )
>. ] ~Q0  ) )
4131, 40anbi12d 470 . . . . . 6  |-  ( ( u  =  C  /\  t  =  D )  ->  ( ( ( [
<. A ,  B >. ] ~Q0  =  [ <. w ,  v
>. ] ~Q0  /\  [ <. C ,  D >. ] ~Q0  =  [ <. u ,  t
>. ] ~Q0  )  /\  [ <. (
( A  .o  D
)  +o  ( B  .o  C ) ) ,  ( B  .o  D ) >. ] ~Q0  =  [ <. (
( w  .o  t
)  +o  ( v  .o  u ) ) ,  ( v  .o  t ) >. ] ~Q0  )  <->  ( ( [
<. A ,  B >. ] ~Q0  =  [ <. w ,  v
>. ] ~Q0  /\  [ <. C ,  D >. ] ~Q0  =  [ <. C ,  D >. ] ~Q0  )  /\  [ <. (
( A  .o  D
)  +o  ( B  .o  C ) ) ,  ( B  .o  D ) >. ] ~Q0  =  [ <. (
( w  .o  D
)  +o  ( v  .o  C ) ) ,  ( v  .o  D ) >. ] ~Q0  ) ) )
4241spc2egv 2820 . . . . 5  |-  ( ( C  e.  om  /\  D  e.  N. )  ->  ( ( ( [
<. A ,  B >. ] ~Q0  =  [ <. w ,  v
>. ] ~Q0  /\  [ <. C ,  D >. ] ~Q0  =  [ <. C ,  D >. ] ~Q0  )  /\  [ <. (
( A  .o  D
)  +o  ( B  .o  C ) ) ,  ( B  .o  D ) >. ] ~Q0  =  [ <. (
( w  .o  D
)  +o  ( v  .o  C ) ) ,  ( v  .o  D ) >. ] ~Q0  )  ->  E. u E. t ( ( [
<. A ,  B >. ] ~Q0  =  [ <. w ,  v
>. ] ~Q0  /\  [ <. C ,  D >. ] ~Q0  =  [ <. u ,  t
>. ] ~Q0  )  /\  [ <. (
( A  .o  D
)  +o  ( B  .o  C ) ) ,  ( B  .o  D ) >. ] ~Q0  =  [ <. (
( w  .o  t
)  +o  ( v  .o  u ) ) ,  ( v  .o  t ) >. ] ~Q0  ) ) )
43422eximdv 1875 . . . 4  |-  ( ( C  e.  om  /\  D  e.  N. )  ->  ( E. w E. v ( ( [
<. A ,  B >. ] ~Q0  =  [ <. w ,  v
>. ] ~Q0  /\  [ <. C ,  D >. ] ~Q0  =  [ <. C ,  D >. ] ~Q0  )  /\  [ <. (
( A  .o  D
)  +o  ( B  .o  C ) ) ,  ( B  .o  D ) >. ] ~Q0  =  [ <. (
( w  .o  D
)  +o  ( v  .o  C ) ) ,  ( v  .o  D ) >. ] ~Q0  )  ->  E. w E. v E. u E. t ( ( [
<. A ,  B >. ] ~Q0  =  [ <. w ,  v
>. ] ~Q0  /\  [ <. C ,  D >. ] ~Q0  =  [ <. u ,  t
>. ] ~Q0  )  /\  [ <. (
( A  .o  D
)  +o  ( B  .o  C ) ) ,  ( B  .o  D ) >. ] ~Q0  =  [ <. (
( w  .o  t
)  +o  ( v  .o  u ) ) ,  ( v  .o  t ) >. ] ~Q0  ) ) )
4427, 43sylan9 407 . . 3  |-  ( ( ( A  e.  om  /\  B  e.  N. )  /\  ( C  e.  om  /\  D  e.  N. )
)  ->  ( (
( [ <. A ,  B >. ] ~Q0  =  [ <. A ,  B >. ] ~Q0  /\  [ <. C ,  D >. ] ~Q0  =  [ <. C ,  D >. ] ~Q0  )  /\  [ <. ( ( A  .o  D
)  +o  ( B  .o  C ) ) ,  ( B  .o  D ) >. ] ~Q0  =  [ <. (
( A  .o  D
)  +o  ( B  .o  C ) ) ,  ( B  .o  D ) >. ] ~Q0  )  ->  E. w E. v E. u E. t ( ( [
<. A ,  B >. ] ~Q0  =  [ <. w ,  v
>. ] ~Q0  /\  [ <. C ,  D >. ] ~Q0  =  [ <. u ,  t
>. ] ~Q0  )  /\  [ <. (
( A  .o  D
)  +o  ( B  .o  C ) ) ,  ( B  .o  D ) >. ] ~Q0  =  [ <. (
( w  .o  t
)  +o  ( v  .o  u ) ) ,  ( v  .o  t ) >. ] ~Q0  ) ) )
4511, 12, 44mp2ani 430 . 2  |-  ( ( ( A  e.  om  /\  B  e.  N. )  /\  ( C  e.  om  /\  D  e.  N. )
)  ->  E. w E. v E. u E. t ( ( [
<. A ,  B >. ] ~Q0  =  [ <. w ,  v
>. ] ~Q0  /\  [ <. C ,  D >. ] ~Q0  =  [ <. u ,  t
>. ] ~Q0  )  /\  [ <. (
( A  .o  D
)  +o  ( B  .o  C ) ) ,  ( B  .o  D ) >. ] ~Q0  =  [ <. (
( w  .o  t
)  +o  ( v  .o  u ) ) ,  ( v  .o  t ) >. ] ~Q0  ) )
46 ecexg 6513 . . . 4  |-  ( ~Q0  e.  _V  ->  [ <. ( ( A  .o  D )  +o  ( B  .o  C
) ) ,  ( B  .o  D )
>. ] ~Q0  e.  _V )
472, 46ax-mp 5 . . 3  |-  [ <. ( ( A  .o  D
)  +o  ( B  .o  C ) ) ,  ( B  .o  D ) >. ] ~Q0  e.  _V
48 simp1 992 . . . . . . . 8  |-  ( ( x  =  [ <. A ,  B >. ] ~Q0  /\  y  =  [ <. C ,  D >. ] ~Q0  /\  z  =  [ <. (
( A  .o  D
)  +o  ( B  .o  C ) ) ,  ( B  .o  D ) >. ] ~Q0  )  ->  x  =  [ <. A ,  B >. ] ~Q0  )
4948eqeq1d 2179 . . . . . . 7  |-  ( ( x  =  [ <. A ,  B >. ] ~Q0  /\  y  =  [ <. C ,  D >. ] ~Q0  /\  z  =  [ <. (
( A  .o  D
)  +o  ( B  .o  C ) ) ,  ( B  .o  D ) >. ] ~Q0  )  ->  ( x  =  [ <. w ,  v
>. ] ~Q0  <->  [
<. A ,  B >. ] ~Q0  =  [ <. w ,  v
>. ] ~Q0  ) )
50 simp2 993 . . . . . . . 8  |-  ( ( x  =  [ <. A ,  B >. ] ~Q0  /\  y  =  [ <. C ,  D >. ] ~Q0  /\  z  =  [ <. (
( A  .o  D
)  +o  ( B  .o  C ) ) ,  ( B  .o  D ) >. ] ~Q0  )  ->  y  =  [ <. C ,  D >. ] ~Q0  )
5150eqeq1d 2179 . . . . . . 7  |-  ( ( x  =  [ <. A ,  B >. ] ~Q0  /\  y  =  [ <. C ,  D >. ] ~Q0  /\  z  =  [ <. (
( A  .o  D
)  +o  ( B  .o  C ) ) ,  ( B  .o  D ) >. ] ~Q0  )  ->  ( y  =  [ <. u ,  t
>. ] ~Q0  <->  [
<. C ,  D >. ] ~Q0  =  [ <. u ,  t
>. ] ~Q0  ) )
5249, 51anbi12d 470 . . . . . 6  |-  ( ( x  =  [ <. A ,  B >. ] ~Q0  /\  y  =  [ <. C ,  D >. ] ~Q0  /\  z  =  [ <. (
( A  .o  D
)  +o  ( B  .o  C ) ) ,  ( B  .o  D ) >. ] ~Q0  )  ->  ( (
x  =  [ <. w ,  v >. ] ~Q0  /\  y  =  [ <. u ,  t >. ] ~Q0  )  <-> 
( [ <. A ,  B >. ] ~Q0  =  [ <. w ,  v >. ] ~Q0  /\  [ <. C ,  D >. ] ~Q0  =  [ <. u ,  t >. ] ~Q0  ) ) )
53 simp3 994 . . . . . . 7  |-  ( ( x  =  [ <. A ,  B >. ] ~Q0  /\  y  =  [ <. C ,  D >. ] ~Q0  /\  z  =  [ <. (
( A  .o  D
)  +o  ( B  .o  C ) ) ,  ( B  .o  D ) >. ] ~Q0  )  ->  z  =  [ <. ( ( A  .o  D )  +o  ( B  .o  C
) ) ,  ( B  .o  D )
>. ] ~Q0  )
5453eqeq1d 2179 . . . . . 6  |-  ( ( x  =  [ <. A ,  B >. ] ~Q0  /\  y  =  [ <. C ,  D >. ] ~Q0  /\  z  =  [ <. (
( A  .o  D
)  +o  ( B  .o  C ) ) ,  ( B  .o  D ) >. ] ~Q0  )  ->  ( z  =  [ <. ( ( w  .o  t )  +o  ( v  .o  u
) ) ,  ( v  .o  t )
>. ] ~Q0  <->  [
<. ( ( A  .o  D )  +o  ( B  .o  C ) ) ,  ( B  .o  D ) >. ] ~Q0  =  [ <. (
( w  .o  t
)  +o  ( v  .o  u ) ) ,  ( v  .o  t ) >. ] ~Q0  ) )
5552, 54anbi12d 470 . . . . 5  |-  ( ( x  =  [ <. A ,  B >. ] ~Q0  /\  y  =  [ <. C ,  D >. ] ~Q0  /\  z  =  [ <. (
( A  .o  D
)  +o  ( B  .o  C ) ) ,  ( B  .o  D ) >. ] ~Q0  )  ->  ( (
( x  =  [ <. w ,  v >. ] ~Q0  /\  y  =  [ <. u ,  t >. ] ~Q0  )  /\  z  =  [ <. ( ( w  .o  t )  +o  ( v  .o  u
) ) ,  ( v  .o  t )
>. ] ~Q0  ) 
<->  ( ( [ <. A ,  B >. ] ~Q0  =  [ <. w ,  v >. ] ~Q0  /\  [ <. C ,  D >. ] ~Q0  =  [ <. u ,  t >. ] ~Q0  )  /\  [ <. ( ( A  .o  D
)  +o  ( B  .o  C ) ) ,  ( B  .o  D ) >. ] ~Q0  =  [ <. (
( w  .o  t
)  +o  ( v  .o  u ) ) ,  ( v  .o  t ) >. ] ~Q0  ) ) )
56554exbidv 1863 . . . 4  |-  ( ( x  =  [ <. A ,  B >. ] ~Q0  /\  y  =  [ <. C ,  D >. ] ~Q0  /\  z  =  [ <. (
( A  .o  D
)  +o  ( B  .o  C ) ) ,  ( B  .o  D ) >. ] ~Q0  )  ->  ( E. w E. v E. u E. t ( ( x  =  [ <. w ,  v >. ] ~Q0  /\  y  =  [ <. u ,  t >. ] ~Q0  )  /\  z  =  [ <. ( ( w  .o  t )  +o  (
v  .o  u ) ) ,  ( v  .o  t ) >. ] ~Q0  )  <->  E. w E. v E. u E. t ( ( [ <. A ,  B >. ] ~Q0  =  [ <. w ,  v >. ] ~Q0  /\  [ <. C ,  D >. ] ~Q0  =  [ <. u ,  t >. ] ~Q0  )  /\  [ <. ( ( A  .o  D
)  +o  ( B  .o  C ) ) ,  ( B  .o  D ) >. ] ~Q0  =  [ <. (
( w  .o  t
)  +o  ( v  .o  u ) ) ,  ( v  .o  t ) >. ] ~Q0  ) ) )
57 addnq0mo 7396 . . . 4  |-  ( ( x  e.  ( ( om  X.  N. ) /. ~Q0  )  /\  y  e.  ( ( om  X.  N. ) /. ~Q0  ) )  ->  E* z E. w E. v E. u E. t ( ( x  =  [ <. w ,  v >. ] ~Q0  /\  y  =  [ <. u ,  t >. ] ~Q0  )  /\  z  =  [ <. ( ( w  .o  t )  +o  ( v  .o  u
) ) ,  ( v  .o  t )
>. ] ~Q0  ) )
58 dfplq0qs 7379 . . . 4  |- +Q0  =  { <. <. x ,  y >. ,  z
>.  |  ( (
x  e.  ( ( om  X.  N. ) /. ~Q0  )  /\  y  e.  ( ( om  X.  N. ) /. ~Q0  ) )  /\  E. w E. v E. u E. t ( ( x  =  [ <. w ,  v >. ] ~Q0  /\  y  =  [ <. u ,  t >. ] ~Q0  )  /\  z  =  [ <. ( ( w  .o  t )  +o  (
v  .o  u ) ) ,  ( v  .o  t ) >. ] ~Q0  ) ) }
5956, 57, 58ovig 5971 . . 3  |-  ( ( [ <. A ,  B >. ] ~Q0  e.  ( ( om  X.  N. ) /. ~Q0  )  /\  [ <. C ,  D >. ] ~Q0  e.  ( ( om 
X.  N. ) /. ~Q0  )  /\  [ <. ( ( A  .o  D
)  +o  ( B  .o  C ) ) ,  ( B  .o  D ) >. ] ~Q0  e.  _V )  -> 
( E. w E. v E. u E. t
( ( [ <. A ,  B >. ] ~Q0  =  [ <. w ,  v >. ] ~Q0  /\  [ <. C ,  D >. ] ~Q0  =  [ <. u ,  t >. ] ~Q0  )  /\  [ <. ( ( A  .o  D
)  +o  ( B  .o  C ) ) ,  ( B  .o  D ) >. ] ~Q0  =  [ <. (
( w  .o  t
)  +o  ( v  .o  u ) ) ,  ( v  .o  t ) >. ] ~Q0  )  ->  ( [ <. A ,  B >. ] ~Q0 +Q0  [ <. C ,  D >. ] ~Q0  )  =  [ <. (
( A  .o  D
)  +o  ( B  .o  C ) ) ,  ( B  .o  D ) >. ] ~Q0  ) )
6047, 59mp3an3 1321 . 2  |-  ( ( [ <. A ,  B >. ] ~Q0  e.  ( ( om  X.  N. ) /. ~Q0  )  /\  [ <. C ,  D >. ] ~Q0  e.  ( ( om 
X.  N. ) /. ~Q0  ) )  ->  ( E. w E. v E. u E. t ( ( [ <. A ,  B >. ] ~Q0  =  [ <. w ,  v >. ] ~Q0  /\  [ <. C ,  D >. ] ~Q0  =  [ <. u ,  t >. ] ~Q0  )  /\  [ <. ( ( A  .o  D
)  +o  ( B  .o  C ) ) ,  ( B  .o  D ) >. ] ~Q0  =  [ <. (
( w  .o  t
)  +o  ( v  .o  u ) ) ,  ( v  .o  t ) >. ] ~Q0  )  ->  ( [ <. A ,  B >. ] ~Q0 +Q0  [ <. C ,  D >. ] ~Q0  )  =  [ <. (
( A  .o  D
)  +o  ( B  .o  C ) ) ,  ( B  .o  D ) >. ] ~Q0  ) )
618, 45, 60sylc 62 1  |-  ( ( ( A  e.  om  /\  B  e.  N. )  /\  ( C  e.  om  /\  D  e.  N. )
)  ->  ( [ <. A ,  B >. ] ~Q0 +Q0  [ <. C ,  D >. ] ~Q0  )  =  [ <. (
( A  .o  D
)  +o  ( B  .o  C ) ) ,  ( B  .o  D ) >. ] ~Q0  )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    /\ w3a 973    = wceq 1348   E.wex 1485    e. wcel 2141   _Vcvv 2730   <.cop 3584   omcom 4572    X. cxp 4607  (class class class)co 5850    +o coa 6389    .o comu 6390   [cec 6507   /.cqs 6508   N.cnpi 7221   ~Q0 ceq0 7235   +Q0 cplq0 7238
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-coll 4102  ax-sep 4105  ax-nul 4113  ax-pow 4158  ax-pr 4192  ax-un 4416  ax-setind 4519  ax-iinf 4570
This theorem depends on definitions:  df-bi 116  df-dc 830  df-3or 974  df-3an 975  df-tru 1351  df-fal 1354  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ne 2341  df-ral 2453  df-rex 2454  df-reu 2455  df-rab 2457  df-v 2732  df-sbc 2956  df-csb 3050  df-dif 3123  df-un 3125  df-in 3127  df-ss 3134  df-nul 3415  df-pw 3566  df-sn 3587  df-pr 3588  df-op 3590  df-uni 3795  df-int 3830  df-iun 3873  df-br 3988  df-opab 4049  df-mpt 4050  df-tr 4086  df-id 4276  df-iord 4349  df-on 4351  df-suc 4354  df-iom 4573  df-xp 4615  df-rel 4616  df-cnv 4617  df-co 4618  df-dm 4619  df-rn 4620  df-res 4621  df-ima 4622  df-iota 5158  df-fun 5198  df-fn 5199  df-f 5200  df-f1 5201  df-fo 5202  df-f1o 5203  df-fv 5204  df-ov 5853  df-oprab 5854  df-mpo 5855  df-1st 6116  df-2nd 6117  df-recs 6281  df-irdg 6346  df-oadd 6396  df-omul 6397  df-er 6509  df-ec 6511  df-qs 6515  df-ni 7253  df-mi 7255  df-enq0 7373  df-nq0 7374  df-plq0 7376
This theorem is referenced by:  addclnq0  7400  nqpnq0nq  7402  nqnq0a  7403  nq0a0  7406  nnanq0  7407  distrnq0  7408  addassnq0  7411
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