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Theorem mulgt0sr 9481
Description: The product of two positive signed reals is positive. (Contributed by NM, 13-May-1996.) (New usage is discouraged.)
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 9444 . . . . 5  |-  <R  C_  ( R.  X.  R. )
21brel 5047 . . . 4  |-  ( 0R 
<R  A  ->  ( 0R  e.  R.  /\  A  e.  R. ) )
32simprd 463 . . 3  |-  ( 0R 
<R  A  ->  A  e. 
R. )
41brel 5047 . . . 4  |-  ( 0R 
<R  B  ->  ( 0R  e.  R.  /\  B  e.  R. ) )
54simprd 463 . . 3  |-  ( 0R 
<R  B  ->  B  e. 
R. )
63, 5anim12i 566 . 2  |-  ( ( 0R  <R  A  /\  0R  <R  B )  -> 
( A  e.  R.  /\  B  e.  R. )
)
7 df-nr 9433 . . 3  |-  R.  =  ( ( P.  X.  P. ) /.  ~R  )
8 breq2 4451 . . . . 5  |-  ( [
<. x ,  y >. ]  ~R  =  A  -> 
( 0R  <R  [ <. x ,  y >. ]  ~R  <->  0R 
<R  A ) )
98anbi1d 704 . . . 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 6290 . . . . 5  |-  ( [
<. x ,  y >. ]  ~R  =  A  -> 
( [ <. x ,  y >. ]  ~R  .R 
[ <. z ,  w >. ]  ~R  )  =  ( A  .R  [ <. z ,  w >. ]  ~R  ) )
1110breq2d 4459 . . . 4  |-  ( [
<. x ,  y >. ]  ~R  =  A  -> 
( 0R  <R  ( [ <. x ,  y
>. ]  ~R  .R  [ <. z ,  w >. ]  ~R  )  <->  0R  <R  ( A  .R  [ <. z ,  w >. ]  ~R  ) ) )
129, 11imbi12d 320 . . 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 4451 . . . . 5  |-  ( [
<. z ,  w >. ]  ~R  =  B  -> 
( 0R  <R  [ <. z ,  w >. ]  ~R  <->  0R 
<R  B ) )
1413anbi2d 703 . . . 4  |-  ( [
<. z ,  w >. ]  ~R  =  B  -> 
( ( 0R  <R  A  /\  0R  <R  [ <. z ,  w >. ]  ~R  ) 
<->  ( 0R  <R  A  /\  0R  <R  B ) ) )
15 oveq2 6291 . . . . 5  |-  ( [
<. z ,  w >. ]  ~R  =  B  -> 
( A  .R  [ <. z ,  w >. ]  ~R  )  =  ( A  .R  B ) )
1615breq2d 4459 . . . 4  |-  ( [
<. z ,  w >. ]  ~R  =  B  -> 
( 0R  <R  ( A  .R  [ <. z ,  w >. ]  ~R  )  <->  0R 
<R  ( A  .R  B
) ) )
1714, 16imbi12d 320 . . 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 9454 . . . . 5  |-  ( 0R 
<R  [ <. x ,  y
>. ]  ~R  <->  y  <P  x )
19 gt0srpr 9454 . . . . 5  |-  ( 0R 
<R  [ <. z ,  w >. ]  ~R  <->  w  <P  z )
2018, 19anbi12i 697 . . . 4  |-  ( ( 0R  <R  [ <. x ,  y >. ]  ~R  /\  0R  <R  [ <. z ,  w >. ]  ~R  )  <->  ( y  <P  x  /\  w  <P  z ) )
21 simprr 756 . . . . . 6  |-  ( ( ( x  e.  P.  /\  y  e.  P. )  /\  ( z  e.  P.  /\  w  e.  P. )
)  ->  w  e.  P. )
22 mulclpr 9397 . . . . . . . 8  |-  ( ( x  e.  P.  /\  z  e.  P. )  ->  ( x  .P.  z
)  e.  P. )
23 mulclpr 9397 . . . . . . . 8  |-  ( ( y  e.  P.  /\  w  e.  P. )  ->  ( y  .P.  w
)  e.  P. )
24 addclpr 9395 . . . . . . . 8  |-  ( ( ( x  .P.  z
)  e.  P.  /\  ( y  .P.  w
)  e.  P. )  ->  ( ( x  .P.  z )  +P.  (
y  .P.  w )
)  e.  P. )
2522, 23, 24syl2an 477 . . . . . . 7  |-  ( ( ( x  e.  P.  /\  z  e.  P. )  /\  ( y  e.  P.  /\  w  e.  P. )
)  ->  ( (
x  .P.  z )  +P.  ( y  .P.  w
) )  e.  P. )
2625an4s 824 . . . . . 6  |-  ( ( ( x  e.  P.  /\  y  e.  P. )  /\  ( z  e.  P.  /\  w  e.  P. )
)  ->  ( (
x  .P.  z )  +P.  ( y  .P.  w
) )  e.  P. )
27 ltexpri 9420 . . . . . . . . 9  |-  ( y 
<P  x  ->  E. v  e.  P.  ( y  +P.  v )  =  x )
28 ltexpri 9420 . . . . . . . . 9  |-  ( w 
<P  z  ->  E. u  e.  P.  ( w  +P.  u )  =  z )
29 mulclpr 9397 . . . . . . . . . . . . . . . . 17  |-  ( ( v  e.  P.  /\  w  e.  P. )  ->  ( v  .P.  w
)  e.  P. )
30 oveq12 6292 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( ( ( y  +P.  v
)  =  x  /\  ( w  +P.  u )  =  z )  -> 
( ( y  +P.  v )  .P.  (
w  +P.  u )
)  =  ( x  .P.  z ) )
3130oveq1d 6298 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ( ( 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 )
) ) )
32 distrpr 9405 . . . . . . . . . . . . . . . . . . . . . . . . 25  |-  ( y  .P.  ( w  +P.  u ) )  =  ( ( y  .P.  w )  +P.  (
y  .P.  u )
)
33 oveq2 6291 . . . . . . . . . . . . . . . . . . . . . . . . 25  |-  ( ( w  +P.  u )  =  z  ->  (
y  .P.  ( w  +P.  u ) )  =  ( y  .P.  z
) )
3432, 33syl5eqr 2522 . . . . . . . . . . . . . . . . . . . . . . . 24  |-  ( ( w  +P.  u )  =  z  ->  (
( y  .P.  w
)  +P.  ( y  .P.  u ) )  =  ( y  .P.  z
) )
3534oveq1d 6298 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( ( 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 ) ) ) )
36 vex 3116 . . . . . . . . . . . . . . . . . . . . . . . . . 26  |-  y  e. 
_V
37 vex 3116 . . . . . . . . . . . . . . . . . . . . . . . . . 26  |-  v  e. 
_V
38 vex 3116 . . . . . . . . . . . . . . . . . . . . . . . . . 26  |-  w  e. 
_V
39 mulcompr 9400 . . . . . . . . . . . . . . . . . . . . . . . . . 26  |-  ( f  .P.  g )  =  ( g  .P.  f
)
40 distrpr 9405 . . . . . . . . . . . . . . . . . . . . . . . . . 26  |-  ( f  .P.  ( g  +P.  h ) )  =  ( ( f  .P.  g )  +P.  (
f  .P.  h )
)
4136, 37, 38, 39, 40caovdir 6492 . . . . . . . . . . . . . . . . . . . . . . . . 25  |-  ( ( y  +P.  v )  .P.  w )  =  ( ( y  .P.  w )  +P.  (
v  .P.  w )
)
42 vex 3116 . . . . . . . . . . . . . . . . . . . . . . . . . 26  |-  u  e. 
_V
4336, 37, 42, 39, 40caovdir 6492 . . . . . . . . . . . . . . . . . . . . . . . . 25  |-  ( ( y  +P.  v )  .P.  u )  =  ( ( y  .P.  u )  +P.  (
v  .P.  u )
)
4441, 43oveq12i 6295 . . . . . . . . . . . . . . . . . . . . . . . 24  |-  ( ( ( 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
) ) )
45 distrpr 9405 . . . . . . . . . . . . . . . . . . . . . . . 24  |-  ( ( y  +P.  v )  .P.  ( w  +P.  u ) )  =  ( ( ( y  +P.  v )  .P.  w )  +P.  (
( y  +P.  v
)  .P.  u )
)
46 ovex 6308 . . . . . . . . . . . . . . . . . . . . . . . . 25  |-  ( y  .P.  w )  e. 
_V
47 ovex 6308 . . . . . . . . . . . . . . . . . . . . . . . . 25  |-  ( y  .P.  u )  e. 
_V
48 ovex 6308 . . . . . . . . . . . . . . . . . . . . . . . . 25  |-  ( v  .P.  w )  e. 
_V
49 addcompr 9398 . . . . . . . . . . . . . . . . . . . . . . . . 25  |-  ( f  +P.  g )  =  ( g  +P.  f
)
50 addasspr 9399 . . . . . . . . . . . . . . . . . . . . . . . . 25  |-  ( ( f  +P.  g )  +P.  h )  =  ( f  +P.  (
g  +P.  h )
)
51 ovex 6308 . . . . . . . . . . . . . . . . . . . . . . . . 25  |-  ( v  .P.  u )  e. 
_V
5246, 47, 48, 49, 50, 51caov4 6489 . . . . . . . . . . . . . . . . . . . . . . . 24  |-  ( ( ( 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
) ) )
5344, 45, 523eqtr4i 2506 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( ( y  +P.  v )  .P.  ( w  +P.  u ) )  =  ( ( ( y  .P.  w )  +P.  ( y  .P.  u
) )  +P.  (
( v  .P.  w
)  +P.  ( v  .P.  u ) ) )
54 ovex 6308 . . . . . . . . . . . . . . . . . . . . . . . 24  |-  ( y  .P.  z )  e. 
_V
5548, 54, 51, 49, 50caov12 6486 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( ( v  .P.  w )  +P.  ( ( y  .P.  z )  +P.  ( v  .P.  u
) ) )  =  ( ( y  .P.  z )  +P.  (
( v  .P.  w
)  +P.  ( v  .P.  u ) ) )
5635, 53, 553eqtr4g 2533 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( ( w  +P.  u )  =  z  ->  (
( y  +P.  v
)  .P.  ( w  +P.  u ) )  =  ( ( v  .P.  w )  +P.  (
( y  .P.  z
)  +P.  ( v  .P.  u ) ) ) )
57 oveq1 6290 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( ( y  +P.  v )  =  x  ->  (
( y  +P.  v
)  .P.  w )  =  ( x  .P.  w ) )
5841, 57syl5eqr 2522 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( ( y  +P.  v )  =  x  ->  (
( y  .P.  w
)  +P.  ( v  .P.  w ) )  =  ( x  .P.  w
) )
5956, 58oveqan12rd 6303 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ( ( y  +P.  v
)  =  x  /\  ( 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
) ) )
6031, 59eqtr3d 2510 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( ( y  +P.  v
)  =  x  /\  ( 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
) ) )
61 addasspr 9399 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ( ( x  .P.  z
)  +P.  ( y  .P.  w ) )  +P.  ( v  .P.  w
) )  =  ( ( x  .P.  z
)  +P.  ( (
y  .P.  w )  +P.  ( v  .P.  w
) ) )
62 addcompr 9398 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ( ( x  .P.  z
)  +P.  ( y  .P.  w ) )  +P.  ( v  .P.  w
) )  =  ( ( v  .P.  w
)  +P.  ( (
x  .P.  z )  +P.  ( y  .P.  w
) ) )
6361, 62eqtr3i 2498 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( x  .P.  z )  +P.  ( ( y  .P.  w )  +P.  ( v  .P.  w
) ) )  =  ( ( v  .P.  w )  +P.  (
( x  .P.  z
)  +P.  ( y  .P.  w ) ) )
64 addasspr 9399 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ( ( 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 )
) ) )
65 ovex 6308 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( ( y  .P.  z )  +P.  ( v  .P.  u ) )  e. 
_V
66 ovex 6308 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( x  .P.  w )  e. 
_V
6748, 65, 66, 49, 50caov32 6485 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ( ( 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
) ) )
68 addasspr 9399 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( ( ( x  .P.  w
)  +P.  ( y  .P.  z ) )  +P.  ( v  .P.  u
) )  =  ( ( x  .P.  w
)  +P.  ( (
y  .P.  z )  +P.  ( v  .P.  u
) ) )
6968oveq2i 6294 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ( 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
) ) ) )
7064, 67, 693eqtr4i 2506 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( ( 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
) ) )
7160, 63, 703eqtr3g 2531 . . . . . . . . . . . . . . . . . . 19  |-  ( ( ( y  +P.  v
)  =  x  /\  ( 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 )
) ) )
72 addcanpr 9423 . . . . . . . . . . . . . . . . . . 19  |-  ( ( ( v  .P.  w
)  e.  P.  /\  ( ( x  .P.  z )  +P.  (
y  .P.  w )
)  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
) ) ) )
7371, 72syl5 32 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( v  .P.  w
)  e.  P.  /\  ( ( x  .P.  z )  +P.  (
y  .P.  w )
)  e.  P. )  ->  ( ( ( y  +P.  v )  =  x  /\  ( w  +P.  u )  =  z )  ->  (
( x  .P.  z
)  +P.  ( y  .P.  w ) )  =  ( ( ( x  .P.  w )  +P.  ( y  .P.  z
) )  +P.  (
v  .P.  u )
) ) )
74 eqcom 2476 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( ( x  .P.  z
)  +P.  ( y  .P.  w ) )  =  ( ( ( x  .P.  w )  +P.  ( y  .P.  z
) )  +P.  (
v  .P.  u )
)  <->  ( ( ( x  .P.  w )  +P.  ( y  .P.  z ) )  +P.  ( v  .P.  u
) )  =  ( ( x  .P.  z
)  +P.  ( y  .P.  w ) ) )
75 ltaddpr2 9412 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( ( x  .P.  z
)  +P.  ( y  .P.  w ) )  e. 
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  ( ( x  .P.  z )  +P.  ( y  .P.  w
) ) ) )
7674, 75syl5bi 217 . . . . . . . . . . . . . . . . . . 19  |-  ( ( ( x  .P.  z
)  +P.  ( y  .P.  w ) )  e. 
P.  ->  ( ( ( x  .P.  z )  +P.  ( y  .P.  w ) )  =  ( ( ( x  .P.  w )  +P.  ( y  .P.  z
) )  +P.  (
v  .P.  u )
)  ->  ( (
x  .P.  w )  +P.  ( y  .P.  z
) )  <P  (
( x  .P.  z
)  +P.  ( y  .P.  w ) ) ) )
7776adantl 466 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( v  .P.  w
)  e.  P.  /\  ( ( x  .P.  z )  +P.  (
y  .P.  w )
)  e.  P. )  ->  ( ( ( x  .P.  z )  +P.  ( y  .P.  w
) )  =  ( ( ( x  .P.  w )  +P.  (
y  .P.  z )
)  +P.  ( v  .P.  u ) )  -> 
( ( x  .P.  w )  +P.  (
y  .P.  z )
)  <P  ( ( x  .P.  z )  +P.  ( y  .P.  w
) ) ) )
7873, 77syld 44 . . . . . . . . . . . . . . . . 17  |-  ( ( ( v  .P.  w
)  e.  P.  /\  ( ( x  .P.  z )  +P.  (
y  .P.  w )
)  e.  P. )  ->  ( ( ( y  +P.  v )  =  x  /\  ( w  +P.  u )  =  z )  ->  (
( x  .P.  w
)  +P.  ( y  .P.  z ) )  <P 
( ( x  .P.  z )  +P.  (
y  .P.  w )
) ) )
7929, 78sylan 471 . . . . . . . . . . . . . . . 16  |-  ( ( ( v  e.  P.  /\  w  e.  P. )  /\  ( ( x  .P.  z )  +P.  (
y  .P.  w )
)  e.  P. )  ->  ( ( ( y  +P.  v )  =  x  /\  ( w  +P.  u )  =  z )  ->  (
( x  .P.  w
)  +P.  ( y  .P.  z ) )  <P 
( ( x  .P.  z )  +P.  (
y  .P.  w )
) ) )
8079a1d 25 . . . . . . . . . . . . . . 15  |-  ( ( ( v  e.  P.  /\  w  e.  P. )  /\  ( ( x  .P.  z )  +P.  (
y  .P.  w )
)  e.  P. )  ->  ( u  e.  P.  ->  ( ( ( y  +P.  v )  =  x  /\  ( w  +P.  u )  =  z )  ->  (
( x  .P.  w
)  +P.  ( y  .P.  z ) )  <P 
( ( x  .P.  z )  +P.  (
y  .P.  w )
) ) ) )
8180exp4a 606 . . . . . . . . . . . . . 14  |-  ( ( ( v  e.  P.  /\  w  e.  P. )  /\  ( ( x  .P.  z )  +P.  (
y  .P.  w )
)  e.  P. )  ->  ( u  e.  P.  ->  ( ( y  +P.  v )  =  x  ->  ( ( w  +P.  u )  =  z  ->  ( (
x  .P.  w )  +P.  ( y  .P.  z
) )  <P  (
( x  .P.  z
)  +P.  ( y  .P.  w ) ) ) ) ) )
8281com34 83 . . . . . . . . . . . . 13  |-  ( ( ( v  e.  P.  /\  w  e.  P. )  /\  ( ( x  .P.  z )  +P.  (
y  .P.  w )
)  e.  P. )  ->  ( u  e.  P.  ->  ( ( w  +P.  u )  =  z  ->  ( ( y  +P.  v )  =  x  ->  ( (
x  .P.  w )  +P.  ( y  .P.  z
) )  <P  (
( x  .P.  z
)  +P.  ( y  .P.  w ) ) ) ) ) )
8382rexlimdv 2953 . . . . . . . . . . . 12  |-  ( ( ( v  e.  P.  /\  w  e.  P. )  /\  ( ( x  .P.  z )  +P.  (
y  .P.  w )
)  e.  P. )  ->  ( E. u  e. 
P.  ( w  +P.  u )  =  z  ->  ( ( y  +P.  v )  =  x  ->  ( (
x  .P.  w )  +P.  ( y  .P.  z
) )  <P  (
( x  .P.  z
)  +P.  ( y  .P.  w ) ) ) ) )
8483expl 618 . . . . . . . . . . 11  |-  ( v  e.  P.  ->  (
( w  e.  P.  /\  ( ( x  .P.  z )  +P.  (
y  .P.  w )
)  e.  P. )  ->  ( E. u  e. 
P.  ( w  +P.  u )  =  z  ->  ( ( y  +P.  v )  =  x  ->  ( (
x  .P.  w )  +P.  ( y  .P.  z
) )  <P  (
( x  .P.  z
)  +P.  ( y  .P.  w ) ) ) ) ) )
8584com24 87 . . . . . . . . . 10  |-  ( v  e.  P.  ->  (
( y  +P.  v
)  =  x  -> 
( E. u  e. 
P.  ( w  +P.  u )  =  z  ->  ( ( w  e.  P.  /\  (
( x  .P.  z
)  +P.  ( y  .P.  w ) )  e. 
P. )  ->  (
( x  .P.  w
)  +P.  ( y  .P.  z ) )  <P 
( ( x  .P.  z )  +P.  (
y  .P.  w )
) ) ) ) )
8685rexlimiv 2949 . . . . . . . . 9  |-  ( E. v  e.  P.  (
y  +P.  v )  =  x  ->  ( E. u  e.  P.  (
w  +P.  u )  =  z  ->  ( ( w  e.  P.  /\  ( ( x  .P.  z )  +P.  (
y  .P.  w )
)  e.  P. )  ->  ( ( x  .P.  w )  +P.  (
y  .P.  z )
)  <P  ( ( x  .P.  z )  +P.  ( y  .P.  w
) ) ) ) )
8727, 28, 86syl2im 38 . . . . . . . 8  |-  ( y 
<P  x  ->  ( w 
<P  z  ->  ( ( w  e.  P.  /\  ( ( x  .P.  z )  +P.  (
y  .P.  w )
)  e.  P. )  ->  ( ( x  .P.  w )  +P.  (
y  .P.  z )
)  <P  ( ( x  .P.  z )  +P.  ( y  .P.  w
) ) ) ) )
8887imp 429 . . . . . . 7  |-  ( ( y  <P  x  /\  w  <P  z )  -> 
( ( w  e. 
P.  /\  ( (
x  .P.  z )  +P.  ( y  .P.  w
) )  e.  P. )  ->  ( ( x  .P.  w )  +P.  ( y  .P.  z
) )  <P  (
( x  .P.  z
)  +P.  ( y  .P.  w ) ) ) )
8988com12 31 . . . . . 6  |-  ( ( w  e.  P.  /\  ( ( x  .P.  z )  +P.  (
y  .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 ) ) ) )
9021, 26, 89syl2anc 661 . . . . 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 )
) ) )
91 mulsrpr 9452 . . . . . . 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  )
9291breq2d 4459 . . . . . 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  ) )
93 gt0srpr 9454 . . . . . 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 ) ) )
9492, 93syl6bb 261 . . . . 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 ) ) ) )
9590, 94sylibrd 234 . . . 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  ) ) )
9620, 95syl5bi 217 . . 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  ) ) )
977, 12, 17, 962ecoptocl 7402 . 2  |-  ( ( A  e.  R.  /\  B  e.  R. )  ->  ( ( 0R  <R  A  /\  0R  <R  B )  ->  0R  <R  ( A  .R  B ) ) )
986, 97mpcom 36 1  |-  ( ( 0R  <R  A  /\  0R  <R  B )  ->  0R  <R  ( A  .R  B ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1379    e. wcel 1767   E.wrex 2815   <.cop 4033   class class class wbr 4447  (class class class)co 6283   [cec 7309   P.cnp 9236    +P. cpp 9238    .P. cmp 9239    <P cltp 9240    ~R cer 9241   R.cnr 9242   0Rc0r 9243    .R cmr 9247    <R cltr 9248
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6575  ax-inf2 8057
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-reu 2821  df-rmo 2822  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-int 4283  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-we 4840  df-ord 4881  df-on 4882  df-lim 4883  df-suc 4884  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5550  df-fun 5589  df-fn 5590  df-f 5591  df-f1 5592  df-fo 5593  df-f1o 5594  df-fv 5595  df-ov 6286  df-oprab 6287  df-mpt2 6288  df-om 6680  df-1st 6784  df-2nd 6785  df-recs 7042  df-rdg 7076  df-1o 7130  df-oadd 7134  df-omul 7135  df-er 7311  df-ec 7313  df-qs 7317  df-ni 9249  df-pli 9250  df-mi 9251  df-lti 9252  df-plpq 9285  df-mpq 9286  df-ltpq 9287  df-enq 9288  df-nq 9289  df-erq 9290  df-plq 9291  df-mq 9292  df-1nq 9293  df-rq 9294  df-ltnq 9295  df-np 9358  df-1p 9359  df-plp 9360  df-mp 9361  df-ltp 9362  df-enr 9432  df-nr 9433  df-mr 9435  df-ltr 9436  df-0r 9437
This theorem is referenced by:  sqgt0sr  9482  axpre-mulgt0  9544
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