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Theorem mulgt0sr 7520
Description: The product of two positive signed reals is positive. (Contributed by NM, 13-May-1996.)
Assertion
Ref Expression
mulgt0sr  |-  ( ( 0R  <R  A  /\  0R  <R  B )  ->  0R  <R  ( A  .R  B ) )

Proof of Theorem mulgt0sr
Dummy variables  x  y  z  w  v  u  f  g  h are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ltrelsr 7481 . . . . 5  |-  <R  C_  ( R.  X.  R. )
21brel 4551 . . . 4  |-  ( 0R 
<R  A  ->  ( 0R  e.  R.  /\  A  e.  R. ) )
32simprd 113 . . 3  |-  ( 0R 
<R  A  ->  A  e. 
R. )
41brel 4551 . . . 4  |-  ( 0R 
<R  B  ->  ( 0R  e.  R.  /\  B  e.  R. ) )
54simprd 113 . . 3  |-  ( 0R 
<R  B  ->  B  e. 
R. )
63, 5anim12i 334 . 2  |-  ( ( 0R  <R  A  /\  0R  <R  B )  -> 
( A  e.  R.  /\  B  e.  R. )
)
7 df-nr 7470 . . 3  |-  R.  =  ( ( P.  X.  P. ) /.  ~R  )
8 breq2 3899 . . . . 5  |-  ( [
<. x ,  y >. ]  ~R  =  A  -> 
( 0R  <R  [ <. x ,  y >. ]  ~R  <->  0R 
<R  A ) )
98anbi1d 458 . . . 4  |-  ( [
<. x ,  y >. ]  ~R  =  A  -> 
( ( 0R  <R  [
<. x ,  y >. ]  ~R  /\  0R  <R  [
<. z ,  w >. ]  ~R  )  <->  ( 0R  <R  A  /\  0R  <R  [
<. z ,  w >. ]  ~R  ) ) )
10 oveq1 5735 . . . . 5  |-  ( [
<. x ,  y >. ]  ~R  =  A  -> 
( [ <. x ,  y >. ]  ~R  .R 
[ <. z ,  w >. ]  ~R  )  =  ( A  .R  [ <. z ,  w >. ]  ~R  ) )
1110breq2d 3907 . . . 4  |-  ( [
<. x ,  y >. ]  ~R  =  A  -> 
( 0R  <R  ( [ <. x ,  y
>. ]  ~R  .R  [ <. z ,  w >. ]  ~R  )  <->  0R  <R  ( A  .R  [ <. z ,  w >. ]  ~R  ) ) )
129, 11imbi12d 233 . . 3  |-  ( [
<. x ,  y >. ]  ~R  =  A  -> 
( ( ( 0R 
<R  [ <. x ,  y
>. ]  ~R  /\  0R  <R  [ <. z ,  w >. ]  ~R  )  ->  0R  <R  ( [ <. x ,  y >. ]  ~R  .R 
[ <. z ,  w >. ]  ~R  ) )  <-> 
( ( 0R  <R  A  /\  0R  <R  [ <. z ,  w >. ]  ~R  )  ->  0R  <R  ( A  .R  [ <. z ,  w >. ]  ~R  )
) ) )
13 breq2 3899 . . . . 5  |-  ( [
<. z ,  w >. ]  ~R  =  B  -> 
( 0R  <R  [ <. z ,  w >. ]  ~R  <->  0R 
<R  B ) )
1413anbi2d 457 . . . 4  |-  ( [
<. z ,  w >. ]  ~R  =  B  -> 
( ( 0R  <R  A  /\  0R  <R  [ <. z ,  w >. ]  ~R  ) 
<->  ( 0R  <R  A  /\  0R  <R  B ) ) )
15 oveq2 5736 . . . . 5  |-  ( [
<. z ,  w >. ]  ~R  =  B  -> 
( A  .R  [ <. z ,  w >. ]  ~R  )  =  ( A  .R  B ) )
1615breq2d 3907 . . . 4  |-  ( [
<. z ,  w >. ]  ~R  =  B  -> 
( 0R  <R  ( A  .R  [ <. z ,  w >. ]  ~R  )  <->  0R 
<R  ( A  .R  B
) ) )
1714, 16imbi12d 233 . . 3  |-  ( [
<. z ,  w >. ]  ~R  =  B  -> 
( ( ( 0R 
<R  A  /\  0R  <R  [
<. z ,  w >. ]  ~R  )  ->  0R  <R  ( A  .R  [ <. z ,  w >. ]  ~R  ) )  <->  ( ( 0R  <R  A  /\  0R  <R  B )  ->  0R  <R  ( A  .R  B
) ) ) )
18 gt0srpr 7491 . . . . 5  |-  ( 0R 
<R  [ <. x ,  y
>. ]  ~R  <->  y  <P  x )
19 gt0srpr 7491 . . . . 5  |-  ( 0R 
<R  [ <. z ,  w >. ]  ~R  <->  w  <P  z )
2018, 19anbi12i 453 . . . 4  |-  ( ( 0R  <R  [ <. x ,  y >. ]  ~R  /\  0R  <R  [ <. z ,  w >. ]  ~R  )  <->  ( y  <P  x  /\  w  <P  z ) )
21 ltexpri 7369 . . . . . . 7  |-  ( y 
<P  x  ->  E. v  e.  P.  ( y  +P.  v )  =  x )
22 ltexpri 7369 . . . . . . . . 9  |-  ( w 
<P  z  ->  E. u  e.  P.  ( w  +P.  u )  =  z )
23 addclpr 7293 . . . . . . . . . . . . . 14  |-  ( ( f  e.  P.  /\  g  e.  P. )  ->  ( f  +P.  g
)  e.  P. )
2423adantl 273 . . . . . . . . . . . . 13  |-  ( ( ( ( ( ( x  e.  P.  /\  y  e.  P. )  /\  ( z  e.  P.  /\  w  e.  P. )
)  /\  ( v  e.  P.  /\  ( y  +P.  v )  =  x ) )  /\  ( u  e.  P.  /\  ( w  +P.  u
)  =  z ) )  /\  ( f  e.  P.  /\  g  e.  P. ) )  -> 
( f  +P.  g
)  e.  P. )
25 simplrr 508 . . . . . . . . . . . . . . 15  |-  ( ( ( ( ( x  e.  P.  /\  y  e.  P. )  /\  (
z  e.  P.  /\  w  e.  P. )
)  /\  ( v  e.  P.  /\  ( y  +P.  v )  =  x ) )  /\  ( u  e.  P.  /\  ( w  +P.  u
)  =  z ) )  ->  ( y  +P.  v )  =  x )
26 simplr 502 . . . . . . . . . . . . . . . . 17  |-  ( ( ( x  e.  P.  /\  y  e.  P. )  /\  ( z  e.  P.  /\  w  e.  P. )
)  ->  y  e.  P. )
2726ad2antrr 477 . . . . . . . . . . . . . . . 16  |-  ( ( ( ( ( x  e.  P.  /\  y  e.  P. )  /\  (
z  e.  P.  /\  w  e.  P. )
)  /\  ( v  e.  P.  /\  ( y  +P.  v )  =  x ) )  /\  ( u  e.  P.  /\  ( w  +P.  u
)  =  z ) )  ->  y  e.  P. )
28 simplrl 507 . . . . . . . . . . . . . . . 16  |-  ( ( ( ( ( x  e.  P.  /\  y  e.  P. )  /\  (
z  e.  P.  /\  w  e.  P. )
)  /\  ( v  e.  P.  /\  ( y  +P.  v )  =  x ) )  /\  ( u  e.  P.  /\  ( w  +P.  u
)  =  z ) )  ->  v  e.  P. )
2924, 27, 28caovcld 5878 . . . . . . . . . . . . . . 15  |-  ( ( ( ( ( x  e.  P.  /\  y  e.  P. )  /\  (
z  e.  P.  /\  w  e.  P. )
)  /\  ( v  e.  P.  /\  ( y  +P.  v )  =  x ) )  /\  ( u  e.  P.  /\  ( w  +P.  u
)  =  z ) )  ->  ( y  +P.  v )  e.  P. )
3025, 29eqeltrrd 2192 . . . . . . . . . . . . . 14  |-  ( ( ( ( ( x  e.  P.  /\  y  e.  P. )  /\  (
z  e.  P.  /\  w  e.  P. )
)  /\  ( v  e.  P.  /\  ( y  +P.  v )  =  x ) )  /\  ( u  e.  P.  /\  ( w  +P.  u
)  =  z ) )  ->  x  e.  P. )
31 simplrr 508 . . . . . . . . . . . . . . 15  |-  ( ( ( ( x  e. 
P.  /\  y  e.  P. )  /\  (
z  e.  P.  /\  w  e.  P. )
)  /\  ( v  e.  P.  /\  ( y  +P.  v )  =  x ) )  ->  w  e.  P. )
3231adantr 272 . . . . . . . . . . . . . 14  |-  ( ( ( ( ( x  e.  P.  /\  y  e.  P. )  /\  (
z  e.  P.  /\  w  e.  P. )
)  /\  ( v  e.  P.  /\  ( y  +P.  v )  =  x ) )  /\  ( u  e.  P.  /\  ( w  +P.  u
)  =  z ) )  ->  w  e.  P. )
33 mulclpr 7328 . . . . . . . . . . . . . 14  |-  ( ( x  e.  P.  /\  w  e.  P. )  ->  ( x  .P.  w
)  e.  P. )
3430, 32, 33syl2anc 406 . . . . . . . . . . . . 13  |-  ( ( ( ( ( x  e.  P.  /\  y  e.  P. )  /\  (
z  e.  P.  /\  w  e.  P. )
)  /\  ( v  e.  P.  /\  ( y  +P.  v )  =  x ) )  /\  ( u  e.  P.  /\  ( w  +P.  u
)  =  z ) )  ->  ( x  .P.  w )  e.  P. )
35 simplrl 507 . . . . . . . . . . . . . . 15  |-  ( ( ( ( x  e. 
P.  /\  y  e.  P. )  /\  (
z  e.  P.  /\  w  e.  P. )
)  /\  ( v  e.  P.  /\  ( y  +P.  v )  =  x ) )  -> 
z  e.  P. )
3635adantr 272 . . . . . . . . . . . . . 14  |-  ( ( ( ( ( x  e.  P.  /\  y  e.  P. )  /\  (
z  e.  P.  /\  w  e.  P. )
)  /\  ( v  e.  P.  /\  ( y  +P.  v )  =  x ) )  /\  ( u  e.  P.  /\  ( w  +P.  u
)  =  z ) )  ->  z  e.  P. )
37 mulclpr 7328 . . . . . . . . . . . . . 14  |-  ( ( y  e.  P.  /\  z  e.  P. )  ->  ( y  .P.  z
)  e.  P. )
3827, 36, 37syl2anc 406 . . . . . . . . . . . . 13  |-  ( ( ( ( ( x  e.  P.  /\  y  e.  P. )  /\  (
z  e.  P.  /\  w  e.  P. )
)  /\  ( v  e.  P.  /\  ( y  +P.  v )  =  x ) )  /\  ( u  e.  P.  /\  ( w  +P.  u
)  =  z ) )  ->  ( y  .P.  z )  e.  P. )
3924, 34, 38caovcld 5878 . . . . . . . . . . . 12  |-  ( ( ( ( ( x  e.  P.  /\  y  e.  P. )  /\  (
z  e.  P.  /\  w  e.  P. )
)  /\  ( v  e.  P.  /\  ( y  +P.  v )  =  x ) )  /\  ( u  e.  P.  /\  ( w  +P.  u
)  =  z ) )  ->  ( (
x  .P.  w )  +P.  ( y  .P.  z
) )  e.  P. )
40 simprl 503 . . . . . . . . . . . . 13  |-  ( ( ( ( ( x  e.  P.  /\  y  e.  P. )  /\  (
z  e.  P.  /\  w  e.  P. )
)  /\  ( v  e.  P.  /\  ( y  +P.  v )  =  x ) )  /\  ( u  e.  P.  /\  ( w  +P.  u
)  =  z ) )  ->  u  e.  P. )
41 mulclpr 7328 . . . . . . . . . . . . 13  |-  ( ( v  e.  P.  /\  u  e.  P. )  ->  ( v  .P.  u
)  e.  P. )
4228, 40, 41syl2anc 406 . . . . . . . . . . . 12  |-  ( ( ( ( ( x  e.  P.  /\  y  e.  P. )  /\  (
z  e.  P.  /\  w  e.  P. )
)  /\  ( v  e.  P.  /\  ( y  +P.  v )  =  x ) )  /\  ( u  e.  P.  /\  ( w  +P.  u
)  =  z ) )  ->  ( v  .P.  u )  e.  P. )
43 ltaddpr 7353 . . . . . . . . . . . 12  |-  ( ( ( ( x  .P.  w )  +P.  (
y  .P.  z )
)  e.  P.  /\  ( v  .P.  u
)  e.  P. )  ->  ( ( x  .P.  w )  +P.  (
y  .P.  z )
)  <P  ( ( ( x  .P.  w )  +P.  ( y  .P.  z ) )  +P.  ( v  .P.  u
) ) )
4439, 42, 43syl2anc 406 . . . . . . . . . . 11  |-  ( ( ( ( ( x  e.  P.  /\  y  e.  P. )  /\  (
z  e.  P.  /\  w  e.  P. )
)  /\  ( v  e.  P.  /\  ( y  +P.  v )  =  x ) )  /\  ( u  e.  P.  /\  ( w  +P.  u
)  =  z ) )  ->  ( (
x  .P.  w )  +P.  ( y  .P.  z
) )  <P  (
( ( x  .P.  w )  +P.  (
y  .P.  z )
)  +P.  ( v  .P.  u ) ) )
45 simprr 504 . . . . . . . . . . . . . . 15  |-  ( ( ( ( ( x  e.  P.  /\  y  e.  P. )  /\  (
z  e.  P.  /\  w  e.  P. )
)  /\  ( v  e.  P.  /\  ( y  +P.  v )  =  x ) )  /\  ( u  e.  P.  /\  ( w  +P.  u
)  =  z ) )  ->  ( w  +P.  u )  =  z )
46 oveq12 5737 . . . . . . . . . . . . . . . 16  |-  ( ( ( y  +P.  v
)  =  x  /\  ( w  +P.  u )  =  z )  -> 
( ( y  +P.  v )  .P.  (
w  +P.  u )
)  =  ( x  .P.  z ) )
4746oveq1d 5743 . . . . . . . . . . . . . . 15  |-  ( ( ( y  +P.  v
)  =  x  /\  ( w  +P.  u )  =  z )  -> 
( ( ( y  +P.  v )  .P.  ( w  +P.  u
) )  +P.  (
( y  .P.  w
)  +P.  ( v  .P.  w ) ) )  =  ( ( x  .P.  z )  +P.  ( ( y  .P.  w )  +P.  (
v  .P.  w )
) ) )
4825, 45, 47syl2anc 406 . . . . . . . . . . . . . 14  |-  ( ( ( ( ( x  e.  P.  /\  y  e.  P. )  /\  (
z  e.  P.  /\  w  e.  P. )
)  /\  ( v  e.  P.  /\  ( y  +P.  v )  =  x ) )  /\  ( u  e.  P.  /\  ( w  +P.  u
)  =  z ) )  ->  ( (
( y  +P.  v
)  .P.  ( w  +P.  u ) )  +P.  ( ( y  .P.  w )  +P.  (
v  .P.  w )
) )  =  ( ( x  .P.  z
)  +P.  ( (
y  .P.  w )  +P.  ( v  .P.  w
) ) ) )
49 distrprg 7344 . . . . . . . . . . . . . . . . . . 19  |-  ( ( y  e.  P.  /\  w  e.  P.  /\  u  e.  P. )  ->  (
y  .P.  ( w  +P.  u ) )  =  ( ( y  .P.  w )  +P.  (
y  .P.  u )
) )
5027, 32, 40, 49syl3anc 1199 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( ( ( x  e.  P.  /\  y  e.  P. )  /\  (
z  e.  P.  /\  w  e.  P. )
)  /\  ( v  e.  P.  /\  ( y  +P.  v )  =  x ) )  /\  ( u  e.  P.  /\  ( w  +P.  u
)  =  z ) )  ->  ( y  .P.  ( w  +P.  u
) )  =  ( ( y  .P.  w
)  +P.  ( y  .P.  u ) ) )
51 oveq2 5736 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( w  +P.  u )  =  z  ->  (
y  .P.  ( w  +P.  u ) )  =  ( y  .P.  z
) )
5251adantl 273 . . . . . . . . . . . . . . . . . . 19  |-  ( ( u  e.  P.  /\  ( w  +P.  u )  =  z )  -> 
( y  .P.  (
w  +P.  u )
)  =  ( y  .P.  z ) )
5352adantl 273 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( ( ( x  e.  P.  /\  y  e.  P. )  /\  (
z  e.  P.  /\  w  e.  P. )
)  /\  ( v  e.  P.  /\  ( y  +P.  v )  =  x ) )  /\  ( u  e.  P.  /\  ( w  +P.  u
)  =  z ) )  ->  ( y  .P.  ( w  +P.  u
) )  =  ( y  .P.  z ) )
5450, 53eqtr3d 2149 . . . . . . . . . . . . . . . . 17  |-  ( ( ( ( ( x  e.  P.  /\  y  e.  P. )  /\  (
z  e.  P.  /\  w  e.  P. )
)  /\  ( v  e.  P.  /\  ( y  +P.  v )  =  x ) )  /\  ( u  e.  P.  /\  ( w  +P.  u
)  =  z ) )  ->  ( (
y  .P.  w )  +P.  ( y  .P.  u
) )  =  ( y  .P.  z ) )
5554oveq1d 5743 . . . . . . . . . . . . . . . 16  |-  ( ( ( ( ( x  e.  P.  /\  y  e.  P. )  /\  (
z  e.  P.  /\  w  e.  P. )
)  /\  ( v  e.  P.  /\  ( y  +P.  v )  =  x ) )  /\  ( u  e.  P.  /\  ( w  +P.  u
)  =  z ) )  ->  ( (
( y  .P.  w
)  +P.  ( y  .P.  u ) )  +P.  ( ( v  .P.  w )  +P.  (
v  .P.  u )
) )  =  ( ( y  .P.  z
)  +P.  ( (
v  .P.  w )  +P.  ( v  .P.  u
) ) ) )
56 distrprg 7344 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( f  e.  P.  /\  g  e.  P.  /\  h  e.  P. )  ->  (
f  .P.  ( g  +P.  h ) )  =  ( ( f  .P.  g )  +P.  (
f  .P.  h )
) )
5756adantl 273 . . . . . . . . . . . . . . . . . . 19  |-  ( ( ( ( ( ( x  e.  P.  /\  y  e.  P. )  /\  ( z  e.  P.  /\  w  e.  P. )
)  /\  ( v  e.  P.  /\  ( y  +P.  v )  =  x ) )  /\  ( u  e.  P.  /\  ( w  +P.  u
)  =  z ) )  /\  ( f  e.  P.  /\  g  e.  P.  /\  h  e. 
P. ) )  -> 
( f  .P.  (
g  +P.  h )
)  =  ( ( f  .P.  g )  +P.  ( f  .P.  h ) ) )
58 mulcomprg 7336 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( f  e.  P.  /\  g  e.  P. )  ->  ( f  .P.  g
)  =  ( g  .P.  f ) )
5958adantl 273 . . . . . . . . . . . . . . . . . . 19  |-  ( ( ( ( ( ( x  e.  P.  /\  y  e.  P. )  /\  ( z  e.  P.  /\  w  e.  P. )
)  /\  ( v  e.  P.  /\  ( y  +P.  v )  =  x ) )  /\  ( u  e.  P.  /\  ( w  +P.  u
)  =  z ) )  /\  ( f  e.  P.  /\  g  e.  P. ) )  -> 
( f  .P.  g
)  =  ( g  .P.  f ) )
6057, 27, 28, 32, 24, 59caovdir2d 5901 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( ( ( x  e.  P.  /\  y  e.  P. )  /\  (
z  e.  P.  /\  w  e.  P. )
)  /\  ( v  e.  P.  /\  ( y  +P.  v )  =  x ) )  /\  ( u  e.  P.  /\  ( w  +P.  u
)  =  z ) )  ->  ( (
y  +P.  v )  .P.  w )  =  ( ( y  .P.  w
)  +P.  ( v  .P.  w ) ) )
6157, 27, 28, 40, 24, 59caovdir2d 5901 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( ( ( x  e.  P.  /\  y  e.  P. )  /\  (
z  e.  P.  /\  w  e.  P. )
)  /\  ( v  e.  P.  /\  ( y  +P.  v )  =  x ) )  /\  ( u  e.  P.  /\  ( w  +P.  u
)  =  z ) )  ->  ( (
y  +P.  v )  .P.  u )  =  ( ( y  .P.  u
)  +P.  ( v  .P.  u ) ) )
6260, 61oveq12d 5746 . . . . . . . . . . . . . . . . 17  |-  ( ( ( ( ( x  e.  P.  /\  y  e.  P. )  /\  (
z  e.  P.  /\  w  e.  P. )
)  /\  ( v  e.  P.  /\  ( y  +P.  v )  =  x ) )  /\  ( u  e.  P.  /\  ( w  +P.  u
)  =  z ) )  ->  ( (
( y  +P.  v
)  .P.  w )  +P.  ( ( y  +P.  v )  .P.  u
) )  =  ( ( ( y  .P.  w )  +P.  (
v  .P.  w )
)  +P.  ( (
y  .P.  u )  +P.  ( v  .P.  u
) ) ) )
63 distrprg 7344 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( y  +P.  v
)  e.  P.  /\  w  e.  P.  /\  u  e.  P. )  ->  (
( y  +P.  v
)  .P.  ( w  +P.  u ) )  =  ( ( ( y  +P.  v )  .P.  w )  +P.  (
( y  +P.  v
)  .P.  u )
) )
6429, 32, 40, 63syl3anc 1199 . . . . . . . . . . . . . . . . 17  |-  ( ( ( ( ( x  e.  P.  /\  y  e.  P. )  /\  (
z  e.  P.  /\  w  e.  P. )
)  /\  ( v  e.  P.  /\  ( y  +P.  v )  =  x ) )  /\  ( u  e.  P.  /\  ( w  +P.  u
)  =  z ) )  ->  ( (
y  +P.  v )  .P.  ( w  +P.  u
) )  =  ( ( ( y  +P.  v )  .P.  w
)  +P.  ( (
y  +P.  v )  .P.  u ) ) )
65 mulclpr 7328 . . . . . . . . . . . . . . . . . . 19  |-  ( ( y  e.  P.  /\  w  e.  P. )  ->  ( y  .P.  w
)  e.  P. )
6627, 32, 65syl2anc 406 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( ( ( x  e.  P.  /\  y  e.  P. )  /\  (
z  e.  P.  /\  w  e.  P. )
)  /\  ( v  e.  P.  /\  ( y  +P.  v )  =  x ) )  /\  ( u  e.  P.  /\  ( w  +P.  u
)  =  z ) )  ->  ( y  .P.  w )  e.  P. )
67 mulclpr 7328 . . . . . . . . . . . . . . . . . . 19  |-  ( ( y  e.  P.  /\  u  e.  P. )  ->  ( y  .P.  u
)  e.  P. )
6827, 40, 67syl2anc 406 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( ( ( x  e.  P.  /\  y  e.  P. )  /\  (
z  e.  P.  /\  w  e.  P. )
)  /\  ( v  e.  P.  /\  ( y  +P.  v )  =  x ) )  /\  ( u  e.  P.  /\  ( w  +P.  u
)  =  z ) )  ->  ( y  .P.  u )  e.  P. )
69 mulclpr 7328 . . . . . . . . . . . . . . . . . . 19  |-  ( ( v  e.  P.  /\  w  e.  P. )  ->  ( v  .P.  w
)  e.  P. )
7028, 32, 69syl2anc 406 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( ( ( x  e.  P.  /\  y  e.  P. )  /\  (
z  e.  P.  /\  w  e.  P. )
)  /\  ( v  e.  P.  /\  ( y  +P.  v )  =  x ) )  /\  ( u  e.  P.  /\  ( w  +P.  u
)  =  z ) )  ->  ( v  .P.  w )  e.  P. )
71 addcomprg 7334 . . . . . . . . . . . . . . . . . . 19  |-  ( ( f  e.  P.  /\  g  e.  P. )  ->  ( f  +P.  g
)  =  ( g  +P.  f ) )
7271adantl 273 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( ( ( ( x  e.  P.  /\  y  e.  P. )  /\  ( z  e.  P.  /\  w  e.  P. )
)  /\  ( v  e.  P.  /\  ( y  +P.  v )  =  x ) )  /\  ( u  e.  P.  /\  ( w  +P.  u
)  =  z ) )  /\  ( f  e.  P.  /\  g  e.  P. ) )  -> 
( f  +P.  g
)  =  ( g  +P.  f ) )
73 addassprg 7335 . . . . . . . . . . . . . . . . . . 19  |-  ( ( f  e.  P.  /\  g  e.  P.  /\  h  e.  P. )  ->  (
( f  +P.  g
)  +P.  h )  =  ( f  +P.  ( g  +P.  h
) ) )
7473adantl 273 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( ( ( ( x  e.  P.  /\  y  e.  P. )  /\  ( z  e.  P.  /\  w  e.  P. )
)  /\  ( v  e.  P.  /\  ( y  +P.  v )  =  x ) )  /\  ( u  e.  P.  /\  ( w  +P.  u
)  =  z ) )  /\  ( f  e.  P.  /\  g  e.  P.  /\  h  e. 
P. ) )  -> 
( ( f  +P.  g )  +P.  h
)  =  ( f  +P.  ( g  +P.  h ) ) )
7566, 68, 70, 72, 74, 42, 24caov4d 5909 . . . . . . . . . . . . . . . . 17  |-  ( ( ( ( ( x  e.  P.  /\  y  e.  P. )  /\  (
z  e.  P.  /\  w  e.  P. )
)  /\  ( v  e.  P.  /\  ( y  +P.  v )  =  x ) )  /\  ( u  e.  P.  /\  ( w  +P.  u
)  =  z ) )  ->  ( (
( y  .P.  w
)  +P.  ( y  .P.  u ) )  +P.  ( ( v  .P.  w )  +P.  (
v  .P.  u )
) )  =  ( ( ( y  .P.  w )  +P.  (
v  .P.  w )
)  +P.  ( (
y  .P.  u )  +P.  ( v  .P.  u
) ) ) )
7662, 64, 753eqtr4d 2157 . . . . . . . . . . . . . . . 16  |-  ( ( ( ( ( x  e.  P.  /\  y  e.  P. )  /\  (
z  e.  P.  /\  w  e.  P. )
)  /\  ( v  e.  P.  /\  ( y  +P.  v )  =  x ) )  /\  ( u  e.  P.  /\  ( w  +P.  u
)  =  z ) )  ->  ( (
y  +P.  v )  .P.  ( w  +P.  u
) )  =  ( ( ( y  .P.  w )  +P.  (
y  .P.  u )
)  +P.  ( (
v  .P.  w )  +P.  ( v  .P.  u
) ) ) )
7770, 38, 42, 72, 74caov12d 5906 . . . . . . . . . . . . . . . 16  |-  ( ( ( ( ( x  e.  P.  /\  y  e.  P. )  /\  (
z  e.  P.  /\  w  e.  P. )
)  /\  ( v  e.  P.  /\  ( y  +P.  v )  =  x ) )  /\  ( u  e.  P.  /\  ( w  +P.  u
)  =  z ) )  ->  ( (
v  .P.  w )  +P.  ( ( y  .P.  z )  +P.  (
v  .P.  u )
) )  =  ( ( y  .P.  z
)  +P.  ( (
v  .P.  w )  +P.  ( v  .P.  u
) ) ) )
7855, 76, 773eqtr4d 2157 . . . . . . . . . . . . . . 15  |-  ( ( ( ( ( x  e.  P.  /\  y  e.  P. )  /\  (
z  e.  P.  /\  w  e.  P. )
)  /\  ( v  e.  P.  /\  ( y  +P.  v )  =  x ) )  /\  ( u  e.  P.  /\  ( w  +P.  u
)  =  z ) )  ->  ( (
y  +P.  v )  .P.  ( w  +P.  u
) )  =  ( ( v  .P.  w
)  +P.  ( (
y  .P.  z )  +P.  ( v  .P.  u
) ) ) )
79 oveq1 5735 . . . . . . . . . . . . . . . . . 18  |-  ( ( y  +P.  v )  =  x  ->  (
( y  +P.  v
)  .P.  w )  =  ( x  .P.  w ) )
8079adantl 273 . . . . . . . . . . . . . . . . 17  |-  ( ( v  e.  P.  /\  ( y  +P.  v
)  =  x )  ->  ( ( y  +P.  v )  .P.  w )  =  ( x  .P.  w ) )
8180ad2antlr 478 . . . . . . . . . . . . . . . 16  |-  ( ( ( ( ( x  e.  P.  /\  y  e.  P. )  /\  (
z  e.  P.  /\  w  e.  P. )
)  /\  ( v  e.  P.  /\  ( y  +P.  v )  =  x ) )  /\  ( u  e.  P.  /\  ( w  +P.  u
)  =  z ) )  ->  ( (
y  +P.  v )  .P.  w )  =  ( x  .P.  w ) )
8260, 81eqtr3d 2149 . . . . . . . . . . . . . . 15  |-  ( ( ( ( ( x  e.  P.  /\  y  e.  P. )  /\  (
z  e.  P.  /\  w  e.  P. )
)  /\  ( v  e.  P.  /\  ( y  +P.  v )  =  x ) )  /\  ( u  e.  P.  /\  ( w  +P.  u
)  =  z ) )  ->  ( (
y  .P.  w )  +P.  ( v  .P.  w
) )  =  ( x  .P.  w ) )
8378, 82oveq12d 5746 . . . . . . . . . . . . . 14  |-  ( ( ( ( ( x  e.  P.  /\  y  e.  P. )  /\  (
z  e.  P.  /\  w  e.  P. )
)  /\  ( v  e.  P.  /\  ( y  +P.  v )  =  x ) )  /\  ( u  e.  P.  /\  ( w  +P.  u
)  =  z ) )  ->  ( (
( y  +P.  v
)  .P.  ( w  +P.  u ) )  +P.  ( ( y  .P.  w )  +P.  (
v  .P.  w )
) )  =  ( ( ( v  .P.  w )  +P.  (
( y  .P.  z
)  +P.  ( v  .P.  u ) ) )  +P.  ( x  .P.  w ) ) )
8448, 83eqtr3d 2149 . . . . . . . . . . . . 13  |-  ( ( ( ( ( x  e.  P.  /\  y  e.  P. )  /\  (
z  e.  P.  /\  w  e.  P. )
)  /\  ( v  e.  P.  /\  ( y  +P.  v )  =  x ) )  /\  ( u  e.  P.  /\  ( w  +P.  u
)  =  z ) )  ->  ( (
x  .P.  z )  +P.  ( ( y  .P.  w )  +P.  (
v  .P.  w )
) )  =  ( ( ( v  .P.  w )  +P.  (
( y  .P.  z
)  +P.  ( v  .P.  u ) ) )  +P.  ( x  .P.  w ) ) )
85 mulclpr 7328 . . . . . . . . . . . . . . . 16  |-  ( ( x  e.  P.  /\  z  e.  P. )  ->  ( x  .P.  z
)  e.  P. )
8630, 36, 85syl2anc 406 . . . . . . . . . . . . . . 15  |-  ( ( ( ( ( x  e.  P.  /\  y  e.  P. )  /\  (
z  e.  P.  /\  w  e.  P. )
)  /\  ( v  e.  P.  /\  ( y  +P.  v )  =  x ) )  /\  ( u  e.  P.  /\  ( w  +P.  u
)  =  z ) )  ->  ( x  .P.  z )  e.  P. )
87 addassprg 7335 . . . . . . . . . . . . . . 15  |-  ( ( ( x  .P.  z
)  e.  P.  /\  ( y  .P.  w
)  e.  P.  /\  ( v  .P.  w
)  e.  P. )  ->  ( ( ( x  .P.  z )  +P.  ( y  .P.  w
) )  +P.  (
v  .P.  w )
)  =  ( ( x  .P.  z )  +P.  ( ( y  .P.  w )  +P.  ( v  .P.  w
) ) ) )
8886, 66, 70, 87syl3anc 1199 . . . . . . . . . . . . . 14  |-  ( ( ( ( ( x  e.  P.  /\  y  e.  P. )  /\  (
z  e.  P.  /\  w  e.  P. )
)  /\  ( v  e.  P.  /\  ( y  +P.  v )  =  x ) )  /\  ( u  e.  P.  /\  ( w  +P.  u
)  =  z ) )  ->  ( (
( x  .P.  z
)  +P.  ( y  .P.  w ) )  +P.  ( v  .P.  w
) )  =  ( ( x  .P.  z
)  +P.  ( (
y  .P.  w )  +P.  ( v  .P.  w
) ) ) )
89 addclpr 7293 . . . . . . . . . . . . . . . 16  |-  ( ( ( x  .P.  z
)  e.  P.  /\  ( y  .P.  w
)  e.  P. )  ->  ( ( x  .P.  z )  +P.  (
y  .P.  w )
)  e.  P. )
9086, 66, 89syl2anc 406 . . . . . . . . . . . . . . 15  |-  ( ( ( ( ( x  e.  P.  /\  y  e.  P. )  /\  (
z  e.  P.  /\  w  e.  P. )
)  /\  ( v  e.  P.  /\  ( y  +P.  v )  =  x ) )  /\  ( u  e.  P.  /\  ( w  +P.  u
)  =  z ) )  ->  ( (
x  .P.  z )  +P.  ( y  .P.  w
) )  e.  P. )
91 addcomprg 7334 . . . . . . . . . . . . . . 15  |-  ( ( ( ( x  .P.  z )  +P.  (
y  .P.  w )
)  e.  P.  /\  ( v  .P.  w
)  e.  P. )  ->  ( ( ( x  .P.  z )  +P.  ( y  .P.  w
) )  +P.  (
v  .P.  w )
)  =  ( ( v  .P.  w )  +P.  ( ( x  .P.  z )  +P.  ( y  .P.  w
) ) ) )
9290, 70, 91syl2anc 406 . . . . . . . . . . . . . 14  |-  ( ( ( ( ( x  e.  P.  /\  y  e.  P. )  /\  (
z  e.  P.  /\  w  e.  P. )
)  /\  ( v  e.  P.  /\  ( y  +P.  v )  =  x ) )  /\  ( u  e.  P.  /\  ( w  +P.  u
)  =  z ) )  ->  ( (
( x  .P.  z
)  +P.  ( y  .P.  w ) )  +P.  ( v  .P.  w
) )  =  ( ( v  .P.  w
)  +P.  ( (
x  .P.  z )  +P.  ( y  .P.  w
) ) ) )
9388, 92eqtr3d 2149 . . . . . . . . . . . . 13  |-  ( ( ( ( ( x  e.  P.  /\  y  e.  P. )  /\  (
z  e.  P.  /\  w  e.  P. )
)  /\  ( v  e.  P.  /\  ( y  +P.  v )  =  x ) )  /\  ( u  e.  P.  /\  ( w  +P.  u
)  =  z ) )  ->  ( (
x  .P.  z )  +P.  ( ( y  .P.  w )  +P.  (
v  .P.  w )
) )  =  ( ( v  .P.  w
)  +P.  ( (
x  .P.  z )  +P.  ( y  .P.  w
) ) ) )
9424, 38, 42caovcld 5878 . . . . . . . . . . . . . . 15  |-  ( ( ( ( ( x  e.  P.  /\  y  e.  P. )  /\  (
z  e.  P.  /\  w  e.  P. )
)  /\  ( v  e.  P.  /\  ( y  +P.  v )  =  x ) )  /\  ( u  e.  P.  /\  ( w  +P.  u
)  =  z ) )  ->  ( (
y  .P.  z )  +P.  ( v  .P.  u
) )  e.  P. )
95 addassprg 7335 . . . . . . . . . . . . . . 15  |-  ( ( ( v  .P.  w
)  e.  P.  /\  ( x  .P.  w )  e.  P.  /\  (
( y  .P.  z
)  +P.  ( v  .P.  u ) )  e. 
P. )  ->  (
( ( v  .P.  w )  +P.  (
x  .P.  w )
)  +P.  ( (
y  .P.  z )  +P.  ( v  .P.  u
) ) )  =  ( ( v  .P.  w )  +P.  (
( x  .P.  w
)  +P.  ( (
y  .P.  z )  +P.  ( v  .P.  u
) ) ) ) )
9670, 34, 94, 95syl3anc 1199 . . . . . . . . . . . . . 14  |-  ( ( ( ( ( x  e.  P.  /\  y  e.  P. )  /\  (
z  e.  P.  /\  w  e.  P. )
)  /\  ( v  e.  P.  /\  ( y  +P.  v )  =  x ) )  /\  ( u  e.  P.  /\  ( w  +P.  u
)  =  z ) )  ->  ( (
( v  .P.  w
)  +P.  ( x  .P.  w ) )  +P.  ( ( y  .P.  z )  +P.  (
v  .P.  u )
) )  =  ( ( v  .P.  w
)  +P.  ( (
x  .P.  w )  +P.  ( ( y  .P.  z )  +P.  (
v  .P.  u )
) ) ) )
9770, 94, 34, 72, 74caov32d 5905 . . . . . . . . . . . . . 14  |-  ( ( ( ( ( x  e.  P.  /\  y  e.  P. )  /\  (
z  e.  P.  /\  w  e.  P. )
)  /\  ( v  e.  P.  /\  ( y  +P.  v )  =  x ) )  /\  ( u  e.  P.  /\  ( w  +P.  u
)  =  z ) )  ->  ( (
( v  .P.  w
)  +P.  ( (
y  .P.  z )  +P.  ( v  .P.  u
) ) )  +P.  ( x  .P.  w
) )  =  ( ( ( v  .P.  w )  +P.  (
x  .P.  w )
)  +P.  ( (
y  .P.  z )  +P.  ( v  .P.  u
) ) ) )
98 addassprg 7335 . . . . . . . . . . . . . . . 16  |-  ( ( ( x  .P.  w
)  e.  P.  /\  ( y  .P.  z
)  e.  P.  /\  ( v  .P.  u
)  e.  P. )  ->  ( ( ( x  .P.  w )  +P.  ( y  .P.  z
) )  +P.  (
v  .P.  u )
)  =  ( ( x  .P.  w )  +P.  ( ( y  .P.  z )  +P.  ( v  .P.  u
) ) ) )
9934, 38, 42, 98syl3anc 1199 . . . . . . . . . . . . . . 15  |-  ( ( ( ( ( x  e.  P.  /\  y  e.  P. )  /\  (
z  e.  P.  /\  w  e.  P. )
)  /\  ( v  e.  P.  /\  ( y  +P.  v )  =  x ) )  /\  ( u  e.  P.  /\  ( w  +P.  u
)  =  z ) )  ->  ( (
( x  .P.  w
)  +P.  ( y  .P.  z ) )  +P.  ( v  .P.  u
) )  =  ( ( x  .P.  w
)  +P.  ( (
y  .P.  z )  +P.  ( v  .P.  u
) ) ) )
10099oveq2d 5744 . . . . . . . . . . . . . 14  |-  ( ( ( ( ( x  e.  P.  /\  y  e.  P. )  /\  (
z  e.  P.  /\  w  e.  P. )
)  /\  ( v  e.  P.  /\  ( y  +P.  v )  =  x ) )  /\  ( u  e.  P.  /\  ( w  +P.  u
)  =  z ) )  ->  ( (
v  .P.  w )  +P.  ( ( ( x  .P.  w )  +P.  ( y  .P.  z
) )  +P.  (
v  .P.  u )
) )  =  ( ( v  .P.  w
)  +P.  ( (
x  .P.  w )  +P.  ( ( y  .P.  z )  +P.  (
v  .P.  u )
) ) ) )
10196, 97, 1003eqtr4d 2157 . . . . . . . . . . . . 13  |-  ( ( ( ( ( x  e.  P.  /\  y  e.  P. )  /\  (
z  e.  P.  /\  w  e.  P. )
)  /\  ( v  e.  P.  /\  ( y  +P.  v )  =  x ) )  /\  ( u  e.  P.  /\  ( w  +P.  u
)  =  z ) )  ->  ( (
( v  .P.  w
)  +P.  ( (
y  .P.  z )  +P.  ( v  .P.  u
) ) )  +P.  ( x  .P.  w
) )  =  ( ( v  .P.  w
)  +P.  ( (
( x  .P.  w
)  +P.  ( y  .P.  z ) )  +P.  ( v  .P.  u
) ) ) )
10284, 93, 1013eqtr3d 2155 . . . . . . . . . . . 12  |-  ( ( ( ( ( x  e.  P.  /\  y  e.  P. )  /\  (
z  e.  P.  /\  w  e.  P. )
)  /\  ( v  e.  P.  /\  ( y  +P.  v )  =  x ) )  /\  ( u  e.  P.  /\  ( w  +P.  u
)  =  z ) )  ->  ( (
v  .P.  w )  +P.  ( ( x  .P.  z )  +P.  (
y  .P.  w )
) )  =  ( ( v  .P.  w
)  +P.  ( (
( x  .P.  w
)  +P.  ( y  .P.  z ) )  +P.  ( v  .P.  u
) ) ) )
10324, 39, 42caovcld 5878 . . . . . . . . . . . . 13  |-  ( ( ( ( ( x  e.  P.  /\  y  e.  P. )  /\  (
z  e.  P.  /\  w  e.  P. )
)  /\  ( v  e.  P.  /\  ( y  +P.  v )  =  x ) )  /\  ( u  e.  P.  /\  ( w  +P.  u
)  =  z ) )  ->  ( (
( x  .P.  w
)  +P.  ( y  .P.  z ) )  +P.  ( v  .P.  u
) )  e.  P. )
104 addcanprg 7372 . . . . . . . . . . . . 13  |-  ( ( ( v  .P.  w
)  e.  P.  /\  ( ( x  .P.  z )  +P.  (
y  .P.  w )
)  e.  P.  /\  ( ( ( x  .P.  w )  +P.  ( y  .P.  z
) )  +P.  (
v  .P.  u )
)  e.  P. )  ->  ( ( ( v  .P.  w )  +P.  ( ( x  .P.  z )  +P.  (
y  .P.  w )
) )  =  ( ( v  .P.  w
)  +P.  ( (
( x  .P.  w
)  +P.  ( y  .P.  z ) )  +P.  ( v  .P.  u
) ) )  -> 
( ( x  .P.  z )  +P.  (
y  .P.  w )
)  =  ( ( ( x  .P.  w
)  +P.  ( y  .P.  z ) )  +P.  ( v  .P.  u
) ) ) )
10570, 90, 103, 104syl3anc 1199 . . . . . . . . . . . 12  |-  ( ( ( ( ( x  e.  P.  /\  y  e.  P. )  /\  (
z  e.  P.  /\  w  e.  P. )
)  /\  ( v  e.  P.  /\  ( y  +P.  v )  =  x ) )  /\  ( u  e.  P.  /\  ( w  +P.  u
)  =  z ) )  ->  ( (
( v  .P.  w
)  +P.  ( (
x  .P.  z )  +P.  ( y  .P.  w
) ) )  =  ( ( v  .P.  w )  +P.  (
( ( x  .P.  w )  +P.  (
y  .P.  z )
)  +P.  ( v  .P.  u ) ) )  ->  ( ( x  .P.  z )  +P.  ( y  .P.  w
) )  =  ( ( ( x  .P.  w )  +P.  (
y  .P.  z )
)  +P.  ( v  .P.  u ) ) ) )
106102, 105mpd 13 . . . . . . . . . . 11  |-  ( ( ( ( ( x  e.  P.  /\  y  e.  P. )  /\  (
z  e.  P.  /\  w  e.  P. )
)  /\  ( v  e.  P.  /\  ( y  +P.  v )  =  x ) )  /\  ( u  e.  P.  /\  ( w  +P.  u
)  =  z ) )  ->  ( (
x  .P.  z )  +P.  ( y  .P.  w
) )  =  ( ( ( x  .P.  w )  +P.  (
y  .P.  z )
)  +P.  ( v  .P.  u ) ) )
10744, 106breqtrrd 3921 . . . . . . . . . 10  |-  ( ( ( ( ( x  e.  P.  /\  y  e.  P. )  /\  (
z  e.  P.  /\  w  e.  P. )
)  /\  ( v  e.  P.  /\  ( y  +P.  v )  =  x ) )  /\  ( u  e.  P.  /\  ( w  +P.  u
)  =  z ) )  ->  ( (
x  .P.  w )  +P.  ( y  .P.  z
) )  <P  (
( x  .P.  z
)  +P.  ( y  .P.  w ) ) )
108107rexlimdvaa 2524 . . . . . . . . 9  |-  ( ( ( ( x  e. 
P.  /\  y  e.  P. )  /\  (
z  e.  P.  /\  w  e.  P. )
)  /\  ( v  e.  P.  /\  ( y  +P.  v )  =  x ) )  -> 
( E. u  e. 
P.  ( w  +P.  u )  =  z  ->  ( ( x  .P.  w )  +P.  ( y  .P.  z
) )  <P  (
( x  .P.  z
)  +P.  ( y  .P.  w ) ) ) )
10922, 108syl5 32 . . . . . . . 8  |-  ( ( ( ( x  e. 
P.  /\  y  e.  P. )  /\  (
z  e.  P.  /\  w  e.  P. )
)  /\  ( v  e.  P.  /\  ( y  +P.  v )  =  x ) )  -> 
( w  <P  z  ->  ( ( x  .P.  w )  +P.  (
y  .P.  z )
)  <P  ( ( x  .P.  z )  +P.  ( y  .P.  w
) ) ) )
110109rexlimdvaa 2524 . . . . . . 7  |-  ( ( ( x  e.  P.  /\  y  e.  P. )  /\  ( z  e.  P.  /\  w  e.  P. )
)  ->  ( E. v  e.  P.  (
y  +P.  v )  =  x  ->  ( w 
<P  z  ->  ( ( x  .P.  w )  +P.  ( y  .P.  z ) )  <P 
( ( x  .P.  z )  +P.  (
y  .P.  w )
) ) ) )
11121, 110syl5 32 . . . . . 6  |-  ( ( ( x  e.  P.  /\  y  e.  P. )  /\  ( z  e.  P.  /\  w  e.  P. )
)  ->  ( y  <P  x  ->  ( w  <P  z  ->  ( (
x  .P.  w )  +P.  ( y  .P.  z
) )  <P  (
( x  .P.  z
)  +P.  ( y  .P.  w ) ) ) ) )
112111impd 252 . . . . 5  |-  ( ( ( x  e.  P.  /\  y  e.  P. )  /\  ( z  e.  P.  /\  w  e.  P. )
)  ->  ( (
y  <P  x  /\  w  <P  z )  ->  (
( x  .P.  w
)  +P.  ( y  .P.  z ) )  <P 
( ( x  .P.  z )  +P.  (
y  .P.  w )
) ) )
113 mulsrpr 7489 . . . . . . 7  |-  ( ( ( 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  )
114113breq2d 3907 . . . . . 6  |-  ( ( ( x  e.  P.  /\  y  e.  P. )  /\  ( z  e.  P.  /\  w  e.  P. )
)  ->  ( 0R  <R  ( [ <. x ,  y >. ]  ~R  .R 
[ <. z ,  w >. ]  ~R  )  <->  0R  <R  [
<. ( ( x  .P.  z )  +P.  (
y  .P.  w )
) ,  ( ( x  .P.  w )  +P.  ( y  .P.  z ) ) >. ]  ~R  ) )
115 gt0srpr 7491 . . . . . 6  |-  ( 0R 
<R  [ <. ( ( x  .P.  z )  +P.  ( y  .P.  w
) ) ,  ( ( x  .P.  w
)  +P.  ( y  .P.  z ) ) >. ]  ~R  <->  ( ( x  .P.  w )  +P.  ( y  .P.  z
) )  <P  (
( x  .P.  z
)  +P.  ( y  .P.  w ) ) )
116114, 115syl6bb 195 . . . . 5  |-  ( ( ( x  e.  P.  /\  y  e.  P. )  /\  ( z  e.  P.  /\  w  e.  P. )
)  ->  ( 0R  <R  ( [ <. x ,  y >. ]  ~R  .R 
[ <. z ,  w >. ]  ~R  )  <->  ( (
x  .P.  w )  +P.  ( y  .P.  z
) )  <P  (
( x  .P.  z
)  +P.  ( y  .P.  w ) ) ) )
117112, 116sylibrd 168 . . . 4  |-  ( ( ( x  e.  P.  /\  y  e.  P. )  /\  ( z  e.  P.  /\  w  e.  P. )
)  ->  ( (
y  <P  x  /\  w  <P  z )  ->  0R  <R  ( [ <. x ,  y >. ]  ~R  .R 
[ <. z ,  w >. ]  ~R  ) ) )
11820, 117syl5bi 151 . . 3  |-  ( ( ( x  e.  P.  /\  y  e.  P. )  /\  ( z  e.  P.  /\  w  e.  P. )
)  ->  ( ( 0R  <R  [ <. x ,  y >. ]  ~R  /\  0R  <R  [ <. z ,  w >. ]  ~R  )  ->  0R  <R  ( [ <. x ,  y >. ]  ~R  .R  [ <. z ,  w >. ]  ~R  ) ) )
1197, 12, 17, 1182ecoptocl 6471 . 2  |-  ( ( A  e.  R.  /\  B  e.  R. )  ->  ( ( 0R  <R  A  /\  0R  <R  B )  ->  0R  <R  ( A  .R  B ) ) )
1206, 119mpcom 36 1  |-  ( ( 0R  <R  A  /\  0R  <R  B )  ->  0R  <R  ( A  .R  B ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    /\ w3a 945    = wceq 1314    e. wcel 1463   E.wrex 2391   <.cop 3496   class class class wbr 3895  (class class class)co 5728   [cec 6381   P.cnp 7047    +P. cpp 7049    .P. cmp 7050    <P cltp 7051    ~R cer 7052   R.cnr 7053   0Rc0r 7054    .R cmr 7058    <R cltr 7059
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 586  ax-in2 587  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-13 1474  ax-14 1475  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097  ax-coll 4003  ax-sep 4006  ax-nul 4014  ax-pow 4058  ax-pr 4091  ax-un 4315  ax-setind 4412  ax-iinf 4462
This theorem depends on definitions:  df-bi 116  df-dc 803  df-3or 946  df-3an 947  df-tru 1317  df-fal 1320  df-nf 1420  df-sb 1719  df-eu 1978  df-mo 1979  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2244  df-ne 2283  df-ral 2395  df-rex 2396  df-reu 2397  df-rab 2399  df-v 2659  df-sbc 2879  df-csb 2972  df-dif 3039  df-un 3041  df-in 3043  df-ss 3050  df-nul 3330  df-pw 3478  df-sn 3499  df-pr 3500  df-op 3502  df-uni 3703  df-int 3738  df-iun 3781  df-br 3896  df-opab 3950  df-mpt 3951  df-tr 3987  df-eprel 4171  df-id 4175  df-po 4178  df-iso 4179  df-iord 4248  df-on 4250  df-suc 4253  df-iom 4465  df-xp 4505  df-rel 4506  df-cnv 4507  df-co 4508  df-dm 4509  df-rn 4510  df-res 4511  df-ima 4512  df-iota 5046  df-fun 5083  df-fn 5084  df-f 5085  df-f1 5086  df-fo 5087  df-f1o 5088  df-fv 5089  df-ov 5731  df-oprab 5732  df-mpo 5733  df-1st 5992  df-2nd 5993  df-recs 6156  df-irdg 6221  df-1o 6267  df-2o 6268  df-oadd 6271  df-omul 6272  df-er 6383  df-ec 6385  df-qs 6389  df-ni 7060  df-pli 7061  df-mi 7062  df-lti 7063  df-plpq 7100  df-mpq 7101  df-enq 7103  df-nqqs 7104  df-plqqs 7105  df-mqqs 7106  df-1nqqs 7107  df-rq 7108  df-ltnqqs 7109  df-enq0 7180  df-nq0 7181  df-0nq0 7182  df-plq0 7183  df-mq0 7184  df-inp 7222  df-i1p 7223  df-iplp 7224  df-imp 7225  df-iltp 7226  df-enr 7469  df-nr 7470  df-mr 7472  df-ltr 7473  df-0r 7474
This theorem is referenced by:  axpre-mulgt0  7622
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