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Theorem mulgt0sr 7752
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 7712 . . . . 5  |-  <R  C_  ( R.  X.  R. )
21brel 4672 . . . 4  |-  ( 0R 
<R  A  ->  ( 0R  e.  R.  /\  A  e.  R. ) )
32simprd 114 . . 3  |-  ( 0R 
<R  A  ->  A  e. 
R. )
41brel 4672 . . . 4  |-  ( 0R 
<R  B  ->  ( 0R  e.  R.  /\  B  e.  R. ) )
54simprd 114 . . 3  |-  ( 0R 
<R  B  ->  B  e. 
R. )
63, 5anim12i 338 . 2  |-  ( ( 0R  <R  A  /\  0R  <R  B )  -> 
( A  e.  R.  /\  B  e.  R. )
)
7 df-nr 7701 . . 3  |-  R.  =  ( ( P.  X.  P. ) /.  ~R  )
8 breq2 4002 . . . . 5  |-  ( [
<. x ,  y >. ]  ~R  =  A  -> 
( 0R  <R  [ <. x ,  y >. ]  ~R  <->  0R 
<R  A ) )
98anbi1d 465 . . . 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 5872 . . . . 5  |-  ( [
<. x ,  y >. ]  ~R  =  A  -> 
( [ <. x ,  y >. ]  ~R  .R 
[ <. z ,  w >. ]  ~R  )  =  ( A  .R  [ <. z ,  w >. ]  ~R  ) )
1110breq2d 4010 . . . 4  |-  ( [
<. x ,  y >. ]  ~R  =  A  -> 
( 0R  <R  ( [ <. x ,  y
>. ]  ~R  .R  [ <. z ,  w >. ]  ~R  )  <->  0R  <R  ( A  .R  [ <. z ,  w >. ]  ~R  ) ) )
129, 11imbi12d 234 . . 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 4002 . . . . 5  |-  ( [
<. z ,  w >. ]  ~R  =  B  -> 
( 0R  <R  [ <. z ,  w >. ]  ~R  <->  0R 
<R  B ) )
1413anbi2d 464 . . . 4  |-  ( [
<. z ,  w >. ]  ~R  =  B  -> 
( ( 0R  <R  A  /\  0R  <R  [ <. z ,  w >. ]  ~R  ) 
<->  ( 0R  <R  A  /\  0R  <R  B ) ) )
15 oveq2 5873 . . . . 5  |-  ( [
<. z ,  w >. ]  ~R  =  B  -> 
( A  .R  [ <. z ,  w >. ]  ~R  )  =  ( A  .R  B ) )
1615breq2d 4010 . . . 4  |-  ( [
<. z ,  w >. ]  ~R  =  B  -> 
( 0R  <R  ( A  .R  [ <. z ,  w >. ]  ~R  )  <->  0R 
<R  ( A  .R  B
) ) )
1714, 16imbi12d 234 . . 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 7722 . . . . 5  |-  ( 0R 
<R  [ <. x ,  y
>. ]  ~R  <->  y  <P  x )
19 gt0srpr 7722 . . . . 5  |-  ( 0R 
<R  [ <. z ,  w >. ]  ~R  <->  w  <P  z )
2018, 19anbi12i 460 . . . 4  |-  ( ( 0R  <R  [ <. x ,  y >. ]  ~R  /\  0R  <R  [ <. z ,  w >. ]  ~R  )  <->  ( y  <P  x  /\  w  <P  z ) )
21 ltexpri 7587 . . . . . . 7  |-  ( y 
<P  x  ->  E. v  e.  P.  ( y  +P.  v )  =  x )
22 ltexpri 7587 . . . . . . . . 9  |-  ( w 
<P  z  ->  E. u  e.  P.  ( w  +P.  u )  =  z )
23 addclpr 7511 . . . . . . . . . . . . . 14  |-  ( ( f  e.  P.  /\  g  e.  P. )  ->  ( f  +P.  g
)  e.  P. )
2423adantl 277 . . . . . . . . . . . . 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 536 . . . . . . . . . . . . . . 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 528 . . . . . . . . . . . . . . . . 17  |-  ( ( ( x  e.  P.  /\  y  e.  P. )  /\  ( z  e.  P.  /\  w  e.  P. )
)  ->  y  e.  P. )
2726ad2antrr 488 . . . . . . . . . . . . . . . 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 535 . . . . . . . . . . . . . . . 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 6018 . . . . . . . . . . . . . . 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 2253 . . . . . . . . . . . . . 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 536 . . . . . . . . . . . . . . 15  |-  ( ( ( ( x  e. 
P.  /\  y  e.  P. )  /\  (
z  e.  P.  /\  w  e.  P. )
)  /\  ( v  e.  P.  /\  ( y  +P.  v )  =  x ) )  ->  w  e.  P. )
3231adantr 276 . . . . . . . . . . . . . 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 7546 . . . . . . . . . . . . . 14  |-  ( ( x  e.  P.  /\  w  e.  P. )  ->  ( x  .P.  w
)  e.  P. )
3430, 32, 33syl2anc 411 . . . . . . . . . . . . 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 535 . . . . . . . . . . . . . . 15  |-  ( ( ( ( x  e. 
P.  /\  y  e.  P. )  /\  (
z  e.  P.  /\  w  e.  P. )
)  /\  ( v  e.  P.  /\  ( y  +P.  v )  =  x ) )  -> 
z  e.  P. )
3635adantr 276 . . . . . . . . . . . . . 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 7546 . . . . . . . . . . . . . 14  |-  ( ( y  e.  P.  /\  z  e.  P. )  ->  ( y  .P.  z
)  e.  P. )
3827, 36, 37syl2anc 411 . . . . . . . . . . . . 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 6018 . . . . . . . . . . . 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 529 . . . . . . . . . . . . 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 7546 . . . . . . . . . . . . 13  |-  ( ( v  e.  P.  /\  u  e.  P. )  ->  ( v  .P.  u
)  e.  P. )
4228, 40, 41syl2anc 411 . . . . . . . . . . . 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 7571 . . . . . . . . . . . 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 411 . . . . . . . . . . 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 531 . . . . . . . . . . . . . . 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 5874 . . . . . . . . . . . . . . . 16  |-  ( ( ( y  +P.  v
)  =  x  /\  ( w  +P.  u )  =  z )  -> 
( ( y  +P.  v )  .P.  (
w  +P.  u )
)  =  ( x  .P.  z ) )
4746oveq1d 5880 . . . . . . . . . . . . . . 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 411 . . . . . . . . . . . . . 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 7562 . . . . . . . . . . . . . . . . . . 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 1238 . . . . . . . . . . . . . . . . . 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 5873 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( w  +P.  u )  =  z  ->  (
y  .P.  ( w  +P.  u ) )  =  ( y  .P.  z
) )
5251adantl 277 . . . . . . . . . . . . . . . . . . 19  |-  ( ( u  e.  P.  /\  ( w  +P.  u )  =  z )  -> 
( y  .P.  (
w  +P.  u )
)  =  ( y  .P.  z ) )
5352adantl 277 . . . . . . . . . . . . . . . . . 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 2210 . . . . . . . . . . . . . . . . 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 5880 . . . . . . . . . . . . . . . 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 7562 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( f  e.  P.  /\  g  e.  P.  /\  h  e.  P. )  ->  (
f  .P.  ( g  +P.  h ) )  =  ( ( f  .P.  g )  +P.  (
f  .P.  h )
) )
5756adantl 277 . . . . . . . . . . . . . . . . . . 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 7554 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( f  e.  P.  /\  g  e.  P. )  ->  ( f  .P.  g
)  =  ( g  .P.  f ) )
5958adantl 277 . . . . . . . . . . . . . . . . . . 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 6041 . . . . . . . . . . . . . . . . . 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 6041 . . . . . . . . . . . . . . . . . 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 5883 . . . . . . . . . . . . . . . . 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 7562 . . . . . . . . . . . . . . . . . 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 1238 . . . . . . . . . . . . . . . . 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 7546 . . . . . . . . . . . . . . . . . . 19  |-  ( ( y  e.  P.  /\  w  e.  P. )  ->  ( y  .P.  w
)  e.  P. )
6627, 32, 65syl2anc 411 . . . . . . . . . . . . . . . . . 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 7546 . . . . . . . . . . . . . . . . . . 19  |-  ( ( y  e.  P.  /\  u  e.  P. )  ->  ( y  .P.  u
)  e.  P. )
6827, 40, 67syl2anc 411 . . . . . . . . . . . . . . . . . 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 7546 . . . . . . . . . . . . . . . . . . 19  |-  ( ( v  e.  P.  /\  w  e.  P. )  ->  ( v  .P.  w
)  e.  P. )
7028, 32, 69syl2anc 411 . . . . . . . . . . . . . . . . . 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 7552 . . . . . . . . . . . . . . . . . . 19  |-  ( ( f  e.  P.  /\  g  e.  P. )  ->  ( f  +P.  g
)  =  ( g  +P.  f ) )
7271adantl 277 . . . . . . . . . . . . . . . . . 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 7553 . . . . . . . . . . . . . . . . . . 19  |-  ( ( f  e.  P.  /\  g  e.  P.  /\  h  e.  P. )  ->  (
( f  +P.  g
)  +P.  h )  =  ( f  +P.  ( g  +P.  h
) ) )
7473adantl 277 . . . . . . . . . . . . . . . . . 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 6049 . . . . . . . . . . . . . . . . 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 2218 . . . . . . . . . . . . . . . 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 6046 . . . . . . . . . . . . . . . 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 2218 . . . . . . . . . . . . . . 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 5872 . . . . . . . . . . . . . . . . . 18  |-  ( ( y  +P.  v )  =  x  ->  (
( y  +P.  v
)  .P.  w )  =  ( x  .P.  w ) )
8079adantl 277 . . . . . . . . . . . . . . . . 17  |-  ( ( v  e.  P.  /\  ( y  +P.  v
)  =  x )  ->  ( ( y  +P.  v )  .P.  w )  =  ( x  .P.  w ) )
8180ad2antlr 489 . . . . . . . . . . . . . . . 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 2210 . . . . . . . . . . . . . . 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 5883 . . . . . . . . . . . . . 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 2210 . . . . . . . . . . . . 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 7546 . . . . . . . . . . . . . . . 16  |-  ( ( x  e.  P.  /\  z  e.  P. )  ->  ( x  .P.  z
)  e.  P. )
8630, 36, 85syl2anc 411 . . . . . . . . . . . . . . 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 7553 . . . . . . . . . . . . . . 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 1238 . . . . . . . . . . . . . 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 7511 . . . . . . . . . . . . . . . 16  |-  ( ( ( x  .P.  z
)  e.  P.  /\  ( y  .P.  w
)  e.  P. )  ->  ( ( x  .P.  z )  +P.  (
y  .P.  w )
)  e.  P. )
9086, 66, 89syl2anc 411 . . . . . . . . . . . . . . 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 7552 . . . . . . . . . . . . . . 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 411 . . . . . . . . . . . . . 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 2210 . . . . . . . . . . . . 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 6018 . . . . . . . . . . . . . . 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 7553 . . . . . . . . . . . . . . 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 1238 . . . . . . . . . . . . . 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 6045 . . . . . . . . . . . . . 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 7553 . . . . . . . . . . . . . . . 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 1238 . . . . . . . . . . . . . . 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 5881 . . . . . . . . . . . . . 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 2218 . . . . . . . . . . . . 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 2216 . . . . . . . . . . . 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 6018 . . . . . . . . . . . . 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 7590 . . . . . . . . . . . . 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 1238 . . . . . . . . . . . 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 4026 . . . . . . . . . 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 2593 . . . . . . . . 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 2593 . . . . . . 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 254 . . . . 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 7720 . . . . . . 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 4010 . . . . . 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 7722 . . . . . 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, 115bitrdi 196 . . . . 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 169 . . . 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, 117biimtrid 152 . . 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 6613 . 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 104    /\ w3a 978    = wceq 1353    e. wcel 2146   E.wrex 2454   <.cop 3592   class class class wbr 3998  (class class class)co 5865   [cec 6523   P.cnp 7265    +P. cpp 7267    .P. cmp 7268    <P cltp 7269    ~R cer 7270   R.cnr 7271   0Rc0r 7272    .R cmr 7276    <R cltr 7277
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 614  ax-in2 615  ax-io 709  ax-5 1445  ax-7 1446  ax-gen 1447  ax-ie1 1491  ax-ie2 1492  ax-8 1502  ax-10 1503  ax-11 1504  ax-i12 1505  ax-bndl 1507  ax-4 1508  ax-17 1524  ax-i9 1528  ax-ial 1532  ax-i5r 1533  ax-13 2148  ax-14 2149  ax-ext 2157  ax-coll 4113  ax-sep 4116  ax-nul 4124  ax-pow 4169  ax-pr 4203  ax-un 4427  ax-setind 4530  ax-iinf 4581
This theorem depends on definitions:  df-bi 117  df-dc 835  df-3or 979  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1459  df-sb 1761  df-eu 2027  df-mo 2028  df-clab 2162  df-cleq 2168  df-clel 2171  df-nfc 2306  df-ne 2346  df-ral 2458  df-rex 2459  df-reu 2460  df-rab 2462  df-v 2737  df-sbc 2961  df-csb 3056  df-dif 3129  df-un 3131  df-in 3133  df-ss 3140  df-nul 3421  df-pw 3574  df-sn 3595  df-pr 3596  df-op 3598  df-uni 3806  df-int 3841  df-iun 3884  df-br 3999  df-opab 4060  df-mpt 4061  df-tr 4097  df-eprel 4283  df-id 4287  df-po 4290  df-iso 4291  df-iord 4360  df-on 4362  df-suc 4365  df-iom 4584  df-xp 4626  df-rel 4627  df-cnv 4628  df-co 4629  df-dm 4630  df-rn 4631  df-res 4632  df-ima 4633  df-iota 5170  df-fun 5210  df-fn 5211  df-f 5212  df-f1 5213  df-fo 5214  df-f1o 5215  df-fv 5216  df-ov 5868  df-oprab 5869  df-mpo 5870  df-1st 6131  df-2nd 6132  df-recs 6296  df-irdg 6361  df-1o 6407  df-2o 6408  df-oadd 6411  df-omul 6412  df-er 6525  df-ec 6527  df-qs 6531  df-ni 7278  df-pli 7279  df-mi 7280  df-lti 7281  df-plpq 7318  df-mpq 7319  df-enq 7321  df-nqqs 7322  df-plqqs 7323  df-mqqs 7324  df-1nqqs 7325  df-rq 7326  df-ltnqqs 7327  df-enq0 7398  df-nq0 7399  df-0nq0 7400  df-plq0 7401  df-mq0 7402  df-inp 7440  df-i1p 7441  df-iplp 7442  df-imp 7443  df-iltp 7444  df-enr 7700  df-nr 7701  df-mr 7703  df-ltr 7704  df-0r 7705
This theorem is referenced by:  axpre-mulgt0  7861
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