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Theorem mulgt0sr 6920
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 6881 . . . . 5  |-  <R  C_  ( R.  X.  R. )
21brel 4420 . . . 4  |-  ( 0R 
<R  A  ->  ( 0R  e.  R.  /\  A  e.  R. ) )
32simprd 111 . . 3  |-  ( 0R 
<R  A  ->  A  e. 
R. )
41brel 4420 . . . 4  |-  ( 0R 
<R  B  ->  ( 0R  e.  R.  /\  B  e.  R. ) )
54simprd 111 . . 3  |-  ( 0R 
<R  B  ->  B  e. 
R. )
63, 5anim12i 325 . 2  |-  ( ( 0R  <R  A  /\  0R  <R  B )  -> 
( A  e.  R.  /\  B  e.  R. )
)
7 df-nr 6870 . . 3  |-  R.  =  ( ( P.  X.  P. ) /.  ~R  )
8 breq2 3796 . . . . 5  |-  ( [
<. x ,  y >. ]  ~R  =  A  -> 
( 0R  <R  [ <. x ,  y >. ]  ~R  <->  0R 
<R  A ) )
98anbi1d 446 . . . 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 5547 . . . . 5  |-  ( [
<. x ,  y >. ]  ~R  =  A  -> 
( [ <. x ,  y >. ]  ~R  .R 
[ <. z ,  w >. ]  ~R  )  =  ( A  .R  [ <. z ,  w >. ]  ~R  ) )
1110breq2d 3804 . . . 4  |-  ( [
<. x ,  y >. ]  ~R  =  A  -> 
( 0R  <R  ( [ <. x ,  y
>. ]  ~R  .R  [ <. z ,  w >. ]  ~R  )  <->  0R  <R  ( A  .R  [ <. z ,  w >. ]  ~R  ) ) )
129, 11imbi12d 227 . . 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 3796 . . . . 5  |-  ( [
<. z ,  w >. ]  ~R  =  B  -> 
( 0R  <R  [ <. z ,  w >. ]  ~R  <->  0R 
<R  B ) )
1413anbi2d 445 . . . 4  |-  ( [
<. z ,  w >. ]  ~R  =  B  -> 
( ( 0R  <R  A  /\  0R  <R  [ <. z ,  w >. ]  ~R  ) 
<->  ( 0R  <R  A  /\  0R  <R  B ) ) )
15 oveq2 5548 . . . . 5  |-  ( [
<. z ,  w >. ]  ~R  =  B  -> 
( A  .R  [ <. z ,  w >. ]  ~R  )  =  ( A  .R  B ) )
1615breq2d 3804 . . . 4  |-  ( [
<. z ,  w >. ]  ~R  =  B  -> 
( 0R  <R  ( A  .R  [ <. z ,  w >. ]  ~R  )  <->  0R 
<R  ( A  .R  B
) ) )
1714, 16imbi12d 227 . . 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 6891 . . . . 5  |-  ( 0R 
<R  [ <. x ,  y
>. ]  ~R  <->  y  <P  x )
19 gt0srpr 6891 . . . . 5  |-  ( 0R 
<R  [ <. z ,  w >. ]  ~R  <->  w  <P  z )
2018, 19anbi12i 441 . . . 4  |-  ( ( 0R  <R  [ <. x ,  y >. ]  ~R  /\  0R  <R  [ <. z ,  w >. ]  ~R  )  <->  ( y  <P  x  /\  w  <P  z ) )
21 ltexpri 6769 . . . . . . 7  |-  ( y 
<P  x  ->  E. v  e.  P.  ( y  +P.  v )  =  x )
22 ltexpri 6769 . . . . . . . . 9  |-  ( w 
<P  z  ->  E. u  e.  P.  ( w  +P.  u )  =  z )
23 addclpr 6693 . . . . . . . . . . . . . 14  |-  ( ( f  e.  P.  /\  g  e.  P. )  ->  ( f  +P.  g
)  e.  P. )
2423adantl 266 . . . . . . . . . . . . 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 496 . . . . . . . . . . . . . . 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 490 . . . . . . . . . . . . . . . . 17  |-  ( ( ( x  e.  P.  /\  y  e.  P. )  /\  ( z  e.  P.  /\  w  e.  P. )
)  ->  y  e.  P. )
2726ad2antrr 465 . . . . . . . . . . . . . . . 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 495 . . . . . . . . . . . . . . . 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 5682 . . . . . . . . . . . . . . 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 2131 . . . . . . . . . . . . . 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 496 . . . . . . . . . . . . . . 15  |-  ( ( ( ( x  e. 
P.  /\  y  e.  P. )  /\  (
z  e.  P.  /\  w  e.  P. )
)  /\  ( v  e.  P.  /\  ( y  +P.  v )  =  x ) )  ->  w  e.  P. )
3231adantr 265 . . . . . . . . . . . . . 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 6728 . . . . . . . . . . . . . 14  |-  ( ( x  e.  P.  /\  w  e.  P. )  ->  ( x  .P.  w
)  e.  P. )
3430, 32, 33syl2anc 397 . . . . . . . . . . . . 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 495 . . . . . . . . . . . . . . 15  |-  ( ( ( ( x  e. 
P.  /\  y  e.  P. )  /\  (
z  e.  P.  /\  w  e.  P. )
)  /\  ( v  e.  P.  /\  ( y  +P.  v )  =  x ) )  -> 
z  e.  P. )
3635adantr 265 . . . . . . . . . . . . . 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 6728 . . . . . . . . . . . . . 14  |-  ( ( y  e.  P.  /\  z  e.  P. )  ->  ( y  .P.  z
)  e.  P. )
3827, 36, 37syl2anc 397 . . . . . . . . . . . . 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 5682 . . . . . . . . . . . 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 491 . . . . . . . . . . . . 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 6728 . . . . . . . . . . . . 13  |-  ( ( v  e.  P.  /\  u  e.  P. )  ->  ( v  .P.  u
)  e.  P. )
4228, 40, 41syl2anc 397 . . . . . . . . . . . 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 6753 . . . . . . . . . . . 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 397 . . . . . . . . . . 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 492 . . . . . . . . . . . . . . 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 5549 . . . . . . . . . . . . . . . 16  |-  ( ( ( y  +P.  v
)  =  x  /\  ( w  +P.  u )  =  z )  -> 
( ( y  +P.  v )  .P.  (
w  +P.  u )
)  =  ( x  .P.  z ) )
4746oveq1d 5555 . . . . . . . . . . . . . . 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 397 . . . . . . . . . . . . . 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 6744 . . . . . . . . . . . . . . . . . . 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 1146 . . . . . . . . . . . . . . . . . 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 5548 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( w  +P.  u )  =  z  ->  (
y  .P.  ( w  +P.  u ) )  =  ( y  .P.  z
) )
5251adantl 266 . . . . . . . . . . . . . . . . . . 19  |-  ( ( u  e.  P.  /\  ( w  +P.  u )  =  z )  -> 
( y  .P.  (
w  +P.  u )
)  =  ( y  .P.  z ) )
5352adantl 266 . . . . . . . . . . . . . . . . . 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 2090 . . . . . . . . . . . . . . . . 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 5555 . . . . . . . . . . . . . . . 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 6744 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( f  e.  P.  /\  g  e.  P.  /\  h  e.  P. )  ->  (
f  .P.  ( g  +P.  h ) )  =  ( ( f  .P.  g )  +P.  (
f  .P.  h )
) )
5756adantl 266 . . . . . . . . . . . . . . . . . . 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 6736 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( f  e.  P.  /\  g  e.  P. )  ->  ( f  .P.  g
)  =  ( g  .P.  f ) )
5958adantl 266 . . . . . . . . . . . . . . . . . . 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 5705 . . . . . . . . . . . . . . . . . 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 5705 . . . . . . . . . . . . . . . . . 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 5558 . . . . . . . . . . . . . . . . 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 6744 . . . . . . . . . . . . . . . . . 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 1146 . . . . . . . . . . . . . . . . 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 6728 . . . . . . . . . . . . . . . . . . 19  |-  ( ( y  e.  P.  /\  w  e.  P. )  ->  ( y  .P.  w
)  e.  P. )
6627, 32, 65syl2anc 397 . . . . . . . . . . . . . . . . . 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 6728 . . . . . . . . . . . . . . . . . . 19  |-  ( ( y  e.  P.  /\  u  e.  P. )  ->  ( y  .P.  u
)  e.  P. )
6827, 40, 67syl2anc 397 . . . . . . . . . . . . . . . . . 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 6728 . . . . . . . . . . . . . . . . . . 19  |-  ( ( v  e.  P.  /\  w  e.  P. )  ->  ( v  .P.  w
)  e.  P. )
7028, 32, 69syl2anc 397 . . . . . . . . . . . . . . . . . 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 6734 . . . . . . . . . . . . . . . . . . 19  |-  ( ( f  e.  P.  /\  g  e.  P. )  ->  ( f  +P.  g
)  =  ( g  +P.  f ) )
7271adantl 266 . . . . . . . . . . . . . . . . . 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 6735 . . . . . . . . . . . . . . . . . . 19  |-  ( ( f  e.  P.  /\  g  e.  P.  /\  h  e.  P. )  ->  (
( f  +P.  g
)  +P.  h )  =  ( f  +P.  ( g  +P.  h
) ) )
7473adantl 266 . . . . . . . . . . . . . . . . . 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 5713 . . . . . . . . . . . . . . . . 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 2098 . . . . . . . . . . . . . . . 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 5710 . . . . . . . . . . . . . . . 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 2098 . . . . . . . . . . . . . . 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 5547 . . . . . . . . . . . . . . . . . 18  |-  ( ( y  +P.  v )  =  x  ->  (
( y  +P.  v
)  .P.  w )  =  ( x  .P.  w ) )
8079adantl 266 . . . . . . . . . . . . . . . . 17  |-  ( ( v  e.  P.  /\  ( y  +P.  v
)  =  x )  ->  ( ( y  +P.  v )  .P.  w )  =  ( x  .P.  w ) )
8180ad2antlr 466 . . . . . . . . . . . . . . . 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 2090 . . . . . . . . . . . . . . 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 5558 . . . . . . . . . . . . . 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 2090 . . . . . . . . . . . . 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 6728 . . . . . . . . . . . . . . . 16  |-  ( ( x  e.  P.  /\  z  e.  P. )  ->  ( x  .P.  z
)  e.  P. )
8630, 36, 85syl2anc 397 . . . . . . . . . . . . . . 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 6735 . . . . . . . . . . . . . . 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 1146 . . . . . . . . . . . . . 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 6693 . . . . . . . . . . . . . . . 16  |-  ( ( ( x  .P.  z
)  e.  P.  /\  ( y  .P.  w
)  e.  P. )  ->  ( ( x  .P.  z )  +P.  (
y  .P.  w )
)  e.  P. )
9086, 66, 89syl2anc 397 . . . . . . . . . . . . . . 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 6734 . . . . . . . . . . . . . . 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 397 . . . . . . . . . . . . . 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 2090 . . . . . . . . . . . . 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 5682 . . . . . . . . . . . . . . 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 6735 . . . . . . . . . . . . . . 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 1146 . . . . . . . . . . . . . 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 5709 . . . . . . . . . . . . . 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 6735 . . . . . . . . . . . . . . . 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 1146 . . . . . . . . . . . . . . 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 5556 . . . . . . . . . . . . . 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 2098 . . . . . . . . . . . . 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 2096 . . . . . . . . . . . 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 5682 . . . . . . . . . . . . 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 6772 . . . . . . . . . . . . 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 1146 . . . . . . . . . . . 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 3818 . . . . . . . . . 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 2451 . . . . . . . . 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 2451 . . . . . . 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 246 . . . . 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 6889 . . . . . . 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 3804 . . . . . 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 6891 . . . . . 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 189 . . . . 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 162 . . . 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 145 . . 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 6225 . 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 101    /\ w3a 896    = wceq 1259    e. wcel 1409   E.wrex 2324   <.cop 3406   class class class wbr 3792  (class class class)co 5540   [cec 6135   P.cnp 6447    +P. cpp 6449    .P. cmp 6450    <P cltp 6451    ~R cer 6452   R.cnr 6453   0Rc0r 6454    .R cmr 6458    <R cltr 6459
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-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 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-eprel 4054  df-id 4058  df-po 4061  df-iso 4062  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-1o 6032  df-2o 6033  df-oadd 6036  df-omul 6037  df-er 6137  df-ec 6139  df-qs 6143  df-ni 6460  df-pli 6461  df-mi 6462  df-lti 6463  df-plpq 6500  df-mpq 6501  df-enq 6503  df-nqqs 6504  df-plqqs 6505  df-mqqs 6506  df-1nqqs 6507  df-rq 6508  df-ltnqqs 6509  df-enq0 6580  df-nq0 6581  df-0nq0 6582  df-plq0 6583  df-mq0 6584  df-inp 6622  df-i1p 6623  df-iplp 6624  df-imp 6625  df-iltp 6626  df-enr 6869  df-nr 6870  df-mr 6872  df-ltr 6873  df-0r 6874
This theorem is referenced by:  axpre-mulgt0  7019
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