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Theorem addnnnq0 7158
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 4509 . . . 4  |-  ( ( A  e.  om  /\  B  e.  N. )  -> 
<. A ,  B >.  e.  ( om  X.  N. ) )
2 enq0ex 7148 . . . . 5  |- ~Q0  e.  _V
32ecelqsi 6413 . . . 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 4509 . . . 4  |-  ( ( C  e.  om  /\  D  e.  N. )  -> 
<. C ,  D >.  e.  ( om  X.  N. ) )
62ecelqsi 6413 . . . 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 334 . 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 2100 . . . 4  |-  [ <. A ,  B >. ] ~Q0  =  [ <. A ,  B >. ] ~Q0
10 eqid 2100 . . . 4  |-  [ <. C ,  D >. ] ~Q0  =  [ <. C ,  D >. ] ~Q0
119, 10pm3.2i 268 . . 3  |-  ( [
<. A ,  B >. ] ~Q0  =  [ <. A ,  B >. ] ~Q0  /\  [ <. C ,  D >. ] ~Q0  =  [ <. C ,  D >. ] ~Q0  )
12 eqid 2100 . . 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 3654 . . . . . . . . 9  |-  ( ( w  =  A  /\  v  =  B )  -> 
<. w ,  v >.  =  <. A ,  B >. )
1413eceq1d 6395 . . . . . . . 8  |-  ( ( w  =  A  /\  v  =  B )  ->  [ <. w ,  v
>. ] ~Q0  =  [ <. A ,  B >. ] ~Q0  )
1514eqeq2d 2111 . . . . . . 7  |-  ( ( w  =  A  /\  v  =  B )  ->  ( [ <. A ,  B >. ] ~Q0  =  [ <. w ,  v >. ] ~Q0  <->  [ <. A ,  B >. ] ~Q0  =  [ <. A ,  B >. ] ~Q0  ) )
1615anbi1d 456 . . . . . 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 5721 . . . . . . . . . 10  |-  ( ( w  =  A  /\  v  =  B )  ->  ( w  .o  D
)  =  ( A  .o  D ) )
19 simpr 109 . . . . . . . . . . 11  |-  ( ( w  =  A  /\  v  =  B )  ->  v  =  B )
2019oveq1d 5721 . . . . . . . . . 10  |-  ( ( w  =  A  /\  v  =  B )  ->  ( v  .o  C
)  =  ( B  .o  C ) )
2118, 20oveq12d 5724 . . . . . . . . 9  |-  ( ( w  =  A  /\  v  =  B )  ->  ( ( w  .o  D )  +o  (
v  .o  C ) )  =  ( ( A  .o  D )  +o  ( B  .o  C ) ) )
2219oveq1d 5721 . . . . . . . . 9  |-  ( ( w  =  A  /\  v  =  B )  ->  ( v  .o  D
)  =  ( B  .o  D ) )
2321, 22opeq12d 3660 . . . . . . . 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 6395 . . . . . . 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 2111 . . . . . 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 460 . . . . 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 2730 . . . 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 3654 . . . . . . . . . 10  |-  ( ( u  =  C  /\  t  =  D )  -> 
<. u ,  t >.  =  <. C ,  D >. )
2928eceq1d 6395 . . . . . . . . 9  |-  ( ( u  =  C  /\  t  =  D )  ->  [ <. u ,  t
>. ] ~Q0  =  [ <. C ,  D >. ] ~Q0  )
3029eqeq2d 2111 . . . . . . . 8  |-  ( ( u  =  C  /\  t  =  D )  ->  ( [ <. C ,  D >. ] ~Q0  =  [ <. u ,  t >. ] ~Q0  <->  [ <. C ,  D >. ] ~Q0  =  [ <. C ,  D >. ] ~Q0  ) )
3130anbi2d 455 . . . . . . 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 5722 . . . . . . . . . . 11  |-  ( ( u  =  C  /\  t  =  D )  ->  ( w  .o  t
)  =  ( w  .o  D ) )
34 simpl 108 . . . . . . . . . . . 12  |-  ( ( u  =  C  /\  t  =  D )  ->  u  =  C )
3534oveq2d 5722 . . . . . . . . . . 11  |-  ( ( u  =  C  /\  t  =  D )  ->  ( v  .o  u
)  =  ( v  .o  C ) )
3633, 35oveq12d 5724 . . . . . . . . . 10  |-  ( ( u  =  C  /\  t  =  D )  ->  ( ( w  .o  t )  +o  (
v  .o  u ) )  =  ( ( w  .o  D )  +o  ( v  .o  C ) ) )
3732oveq2d 5722 . . . . . . . . . 10  |-  ( ( u  =  C  /\  t  =  D )  ->  ( v  .o  t
)  =  ( v  .o  D ) )
3836, 37opeq12d 3660 . . . . . . . . 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 6395 . . . . . . . 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 2111 . . . . . . 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 460 . . . . . 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 2730 . . . . 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 1821 . . . 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 404 . . 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 426 . 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 6363 . . . 4  |-  ( ~Q0  e.  _V  ->  [ <. ( ( A  .o  D )  +o  ( B  .o  C
) ) ,  ( B  .o  D )
>. ] ~Q0  e.  _V )
472, 46ax-mp 7 . . 3  |-  [ <. ( ( A  .o  D
)  +o  ( B  .o  C ) ) ,  ( B  .o  D ) >. ] ~Q0  e.  _V
48 simp1 949 . . . . . . . 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 2108 . . . . . . 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 950 . . . . . . . 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 2108 . . . . . . 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 460 . . . . . 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 951 . . . . . . 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 2108 . . . . . 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 460 . . . . 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 1809 . . . 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 7156 . . . 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 7139 . . . 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 5824 . . 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 1272 . 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 930    = wceq 1299   E.wex 1436    e. wcel 1448   _Vcvv 2641   <.cop 3477   omcom 4442    X. cxp 4475  (class class class)co 5706    +o coa 6240    .o comu 6241   [cec 6357   /.cqs 6358   N.cnpi 6981   ~Q0 ceq0 6995   +Q0 cplq0 6998
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 584  ax-in2 585  ax-io 671  ax-5 1391  ax-7 1392  ax-gen 1393  ax-ie1 1437  ax-ie2 1438  ax-8 1450  ax-10 1451  ax-11 1452  ax-i12 1453  ax-bndl 1454  ax-4 1455  ax-13 1459  ax-14 1460  ax-17 1474  ax-i9 1478  ax-ial 1482  ax-i5r 1483  ax-ext 2082  ax-coll 3983  ax-sep 3986  ax-nul 3994  ax-pow 4038  ax-pr 4069  ax-un 4293  ax-setind 4390  ax-iinf 4440
This theorem depends on definitions:  df-bi 116  df-dc 787  df-3or 931  df-3an 932  df-tru 1302  df-fal 1305  df-nf 1405  df-sb 1704  df-eu 1963  df-mo 1964  df-clab 2087  df-cleq 2093  df-clel 2096  df-nfc 2229  df-ne 2268  df-ral 2380  df-rex 2381  df-reu 2382  df-rab 2384  df-v 2643  df-sbc 2863  df-csb 2956  df-dif 3023  df-un 3025  df-in 3027  df-ss 3034  df-nul 3311  df-pw 3459  df-sn 3480  df-pr 3481  df-op 3483  df-uni 3684  df-int 3719  df-iun 3762  df-br 3876  df-opab 3930  df-mpt 3931  df-tr 3967  df-id 4153  df-iord 4226  df-on 4228  df-suc 4231  df-iom 4443  df-xp 4483  df-rel 4484  df-cnv 4485  df-co 4486  df-dm 4487  df-rn 4488  df-res 4489  df-ima 4490  df-iota 5024  df-fun 5061  df-fn 5062  df-f 5063  df-f1 5064  df-fo 5065  df-f1o 5066  df-fv 5067  df-ov 5709  df-oprab 5710  df-mpo 5711  df-1st 5969  df-2nd 5970  df-recs 6132  df-irdg 6197  df-oadd 6247  df-omul 6248  df-er 6359  df-ec 6361  df-qs 6365  df-ni 7013  df-mi 7015  df-enq0 7133  df-nq0 7134  df-plq0 7136
This theorem is referenced by:  addclnq0  7160  nqpnq0nq  7162  nqnq0a  7163  nq0a0  7166  nnanq0  7167  distrnq0  7168  addassnq0  7171
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