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Theorem mulgt0sr 8972
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 8938 . . . . 5  |-  <R  C_  ( R.  X.  R. )
21brel 4918 . . . 4  |-  ( 0R 
<R  A  ->  ( 0R  e.  R.  /\  A  e.  R. ) )
32simprd 450 . . 3  |-  ( 0R 
<R  A  ->  A  e. 
R. )
41brel 4918 . . . 4  |-  ( 0R 
<R  B  ->  ( 0R  e.  R.  /\  B  e.  R. ) )
54simprd 450 . . 3  |-  ( 0R 
<R  B  ->  B  e. 
R. )
63, 5anim12i 550 . 2  |-  ( ( 0R  <R  A  /\  0R  <R  B )  -> 
( A  e.  R.  /\  B  e.  R. )
)
7 df-nr 8927 . . 3  |-  R.  =  ( ( P.  X.  P. ) /.  ~R  )
8 breq2 4208 . . . . 5  |-  ( [
<. x ,  y >. ]  ~R  =  A  -> 
( 0R  <R  [ <. x ,  y >. ]  ~R  <->  0R 
<R  A ) )
98anbi1d 686 . . . 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 6080 . . . . 5  |-  ( [
<. x ,  y >. ]  ~R  =  A  -> 
( [ <. x ,  y >. ]  ~R  .R 
[ <. z ,  w >. ]  ~R  )  =  ( A  .R  [ <. z ,  w >. ]  ~R  ) )
1110breq2d 4216 . . . 4  |-  ( [
<. x ,  y >. ]  ~R  =  A  -> 
( 0R  <R  ( [ <. x ,  y
>. ]  ~R  .R  [ <. z ,  w >. ]  ~R  )  <->  0R  <R  ( A  .R  [ <. z ,  w >. ]  ~R  ) ) )
129, 11imbi12d 312 . . 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 4208 . . . . 5  |-  ( [
<. z ,  w >. ]  ~R  =  B  -> 
( 0R  <R  [ <. z ,  w >. ]  ~R  <->  0R 
<R  B ) )
1413anbi2d 685 . . . 4  |-  ( [
<. z ,  w >. ]  ~R  =  B  -> 
( ( 0R  <R  A  /\  0R  <R  [ <. z ,  w >. ]  ~R  ) 
<->  ( 0R  <R  A  /\  0R  <R  B ) ) )
15 oveq2 6081 . . . . 5  |-  ( [
<. z ,  w >. ]  ~R  =  B  -> 
( A  .R  [ <. z ,  w >. ]  ~R  )  =  ( A  .R  B ) )
1615breq2d 4216 . . . 4  |-  ( [
<. z ,  w >. ]  ~R  =  B  -> 
( 0R  <R  ( A  .R  [ <. z ,  w >. ]  ~R  )  <->  0R 
<R  ( A  .R  B
) ) )
1714, 16imbi12d 312 . . 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 8945 . . . . 5  |-  ( 0R 
<R  [ <. x ,  y
>. ]  ~R  <->  y  <P  x )
19 gt0srpr 8945 . . . . 5  |-  ( 0R 
<R  [ <. z ,  w >. ]  ~R  <->  w  <P  z )
2018, 19anbi12i 679 . . . 4  |-  ( ( 0R  <R  [ <. x ,  y >. ]  ~R  /\  0R  <R  [ <. z ,  w >. ]  ~R  )  <->  ( y  <P  x  /\  w  <P  z ) )
21 simprr 734 . . . . . 6  |-  ( ( ( x  e.  P.  /\  y  e.  P. )  /\  ( z  e.  P.  /\  w  e.  P. )
)  ->  w  e.  P. )
22 mulclpr 8889 . . . . . . . 8  |-  ( ( x  e.  P.  /\  z  e.  P. )  ->  ( x  .P.  z
)  e.  P. )
23 mulclpr 8889 . . . . . . . 8  |-  ( ( y  e.  P.  /\  w  e.  P. )  ->  ( y  .P.  w
)  e.  P. )
24 addclpr 8887 . . . . . . . 8  |-  ( ( ( x  .P.  z
)  e.  P.  /\  ( y  .P.  w
)  e.  P. )  ->  ( ( x  .P.  z )  +P.  (
y  .P.  w )
)  e.  P. )
2522, 23, 24syl2an 464 . . . . . . 7  |-  ( ( ( x  e.  P.  /\  z  e.  P. )  /\  ( y  e.  P.  /\  w  e.  P. )
)  ->  ( (
x  .P.  z )  +P.  ( y  .P.  w
) )  e.  P. )
2625an4s 800 . . . . . 6  |-  ( ( ( x  e.  P.  /\  y  e.  P. )  /\  ( z  e.  P.  /\  w  e.  P. )
)  ->  ( (
x  .P.  z )  +P.  ( y  .P.  w
) )  e.  P. )
27 ltexpri 8912 . . . . . . . . 9  |-  ( y 
<P  x  ->  E. v  e.  P.  ( y  +P.  v )  =  x )
28 ltexpri 8912 . . . . . . . . 9  |-  ( w 
<P  z  ->  E. u  e.  P.  ( w  +P.  u )  =  z )
29 mulclpr 8889 . . . . . . . . . . . . . . . . 17  |-  ( ( v  e.  P.  /\  w  e.  P. )  ->  ( v  .P.  w
)  e.  P. )
30 oveq12 6082 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( ( ( y  +P.  v
)  =  x  /\  ( w  +P.  u )  =  z )  -> 
( ( y  +P.  v )  .P.  (
w  +P.  u )
)  =  ( x  .P.  z ) )
3130oveq1d 6088 . . . . . . . . . . . . . . . . . . . . 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 8897 . . . . . . . . . . . . . . . . . . . . . . . . 25  |-  ( y  .P.  ( w  +P.  u ) )  =  ( ( y  .P.  w )  +P.  (
y  .P.  u )
)
33 oveq2 6081 . . . . . . . . . . . . . . . . . . . . . . . . 25  |-  ( ( w  +P.  u )  =  z  ->  (
y  .P.  ( w  +P.  u ) )  =  ( y  .P.  z
) )
3432, 33syl5eqr 2481 . . . . . . . . . . . . . . . . . . . . . . . 24  |-  ( ( w  +P.  u )  =  z  ->  (
( y  .P.  w
)  +P.  ( y  .P.  u ) )  =  ( y  .P.  z
) )
3534oveq1d 6088 . . . . . . . . . . . . . . . . . . . . . . 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 2951 . . . . . . . . . . . . . . . . . . . . . . . . . 26  |-  y  e. 
_V
37 vex 2951 . . . . . . . . . . . . . . . . . . . . . . . . . 26  |-  v  e. 
_V
38 vex 2951 . . . . . . . . . . . . . . . . . . . . . . . . . 26  |-  w  e. 
_V
39 mulcompr 8892 . . . . . . . . . . . . . . . . . . . . . . . . . 26  |-  ( f  .P.  g )  =  ( g  .P.  f
)
40 distrpr 8897 . . . . . . . . . . . . . . . . . . . . . . . . . 26  |-  ( f  .P.  ( g  +P.  h ) )  =  ( ( f  .P.  g )  +P.  (
f  .P.  h )
)
4136, 37, 38, 39, 40caovdir 6273 . . . . . . . . . . . . . . . . . . . . . . . . 25  |-  ( ( y  +P.  v )  .P.  w )  =  ( ( y  .P.  w )  +P.  (
v  .P.  w )
)
42 vex 2951 . . . . . . . . . . . . . . . . . . . . . . . . . 26  |-  u  e. 
_V
4336, 37, 42, 39, 40caovdir 6273 . . . . . . . . . . . . . . . . . . . . . . . . 25  |-  ( ( y  +P.  v )  .P.  u )  =  ( ( y  .P.  u )  +P.  (
v  .P.  u )
)
4441, 43oveq12i 6085 . . . . . . . . . . . . . . . . . . . . . . . 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 8897 . . . . . . . . . . . . . . . . . . . . . . . 24  |-  ( ( y  +P.  v )  .P.  ( w  +P.  u ) )  =  ( ( ( y  +P.  v )  .P.  w )  +P.  (
( y  +P.  v
)  .P.  u )
)
46 ovex 6098 . . . . . . . . . . . . . . . . . . . . . . . . 25  |-  ( y  .P.  w )  e. 
_V
47 ovex 6098 . . . . . . . . . . . . . . . . . . . . . . . . 25  |-  ( y  .P.  u )  e. 
_V
48 ovex 6098 . . . . . . . . . . . . . . . . . . . . . . . . 25  |-  ( v  .P.  w )  e. 
_V
49 addcompr 8890 . . . . . . . . . . . . . . . . . . . . . . . . 25  |-  ( f  +P.  g )  =  ( g  +P.  f
)
50 addasspr 8891 . . . . . . . . . . . . . . . . . . . . . . . . 25  |-  ( ( f  +P.  g )  +P.  h )  =  ( f  +P.  (
g  +P.  h )
)
51 ovex 6098 . . . . . . . . . . . . . . . . . . . . . . . . 25  |-  ( v  .P.  u )  e. 
_V
5246, 47, 48, 49, 50, 51caov4 6270 . . . . . . . . . . . . . . . . . . . . . . . 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 2465 . . . . . . . . . . . . . . . . . . . . . . 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 6098 . . . . . . . . . . . . . . . . . . . . . . . 24  |-  ( y  .P.  z )  e. 
_V
5548, 54, 51, 49, 50caov12 6267 . . . . . . . . . . . . . . . . . . . . . . 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 2492 . . . . . . . . . . . . . . . . . . . . . 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 6080 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( ( y  +P.  v )  =  x  ->  (
( y  +P.  v
)  .P.  w )  =  ( x  .P.  w ) )
5841, 57syl5eqr 2481 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( ( y  +P.  v )  =  x  ->  (
( y  .P.  w
)  +P.  ( v  .P.  w ) )  =  ( x  .P.  w
) )
5956, 58oveqan12rd 6093 . . . . . . . . . . . . . . . . . . . . 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 2469 . . . . . . . . . . . . . . . . . . . 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 8891 . . . . . . . . . . . . . . . . . . . . 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 8890 . . . . . . . . . . . . . . . . . . . . 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 2457 . . . . . . . . . . . . . . . . . . . 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 8891 . . . . . . . . . . . . . . . . . . . . 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 6098 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( ( y  .P.  z )  +P.  ( v  .P.  u ) )  e. 
_V
66 ovex 6098 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( x  .P.  w )  e. 
_V
6748, 65, 66, 49, 50caov32 6266 . . . . . . . . . . . . . . . . . . . . 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 8891 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( ( ( x  .P.  w
)  +P.  ( y  .P.  z ) )  +P.  ( v  .P.  u
) )  =  ( ( x  .P.  w
)  +P.  ( (
y  .P.  z )  +P.  ( v  .P.  u
) ) )
6968oveq2i 6084 . . . . . . . . . . . . . . . . . . . . 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 2465 . . . . . . . . . . . . . . . . . . . 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 2490 . . . . . . . . . . . . . . . . . . 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 8915 . . . . . . . . . . . . . . . . . . 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 30 . . . . . . . . . . . . . . . . . 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 2437 . . . . . . . . . . . . . . . . . . . 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 8904 . . . . . . . . . . . . . . . . . . . 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 209 . . . . . . . . . . . . . . . . . . 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 453 . . . . . . . . . . . . . . . . . 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 42 . . . . . . . . . . . . . . . . 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 458 . . . . . . . . . . . . . . . 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 23 . . . . . . . . . . . . . . 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 590 . . . . . . . . . . . . . 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 79 . . . . . . . . . . . . 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 2821 . . . . . . . . . . . 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 602 . . . . . . . . . . 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 83 . . . . . . . . . 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 2816 . . . . . . . . 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 36 . . . . . . . 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 419 . . . . . . 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 29 . . . . . 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 643 . . . . 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 8943 . . . . . . 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 4216 . . . . . 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 8945 . . . . . 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 253 . . . . 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 226 . . . 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 209 . . 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 6987 . 2  |-  ( ( A  e.  R.  /\  B  e.  R. )  ->  ( ( 0R  <R  A  /\  0R  <R  B )  ->  0R  <R  ( A  .R  B ) ) )
986, 97mpcom 34 1  |-  ( ( 0R  <R  A  /\  0R  <R  B )  ->  0R  <R  ( A  .R  B ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1652    e. wcel 1725   E.wrex 2698   <.cop 3809   class class class wbr 4204  (class class class)co 6073   [cec 6895   P.cnp 8726    +P. cpp 8728    .P. cmp 8729    <P cltp 8730    ~R cer 8733   R.cnr 8734   0Rc0r 8735    .R cmr 8739    <R cltr 8740
This theorem is referenced by:  sqgt0sr  8973  axpre-mulgt0  9035
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693  ax-inf2 7588
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-reu 2704  df-rmo 2705  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-pss 3328  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-tp 3814  df-op 3815  df-uni 4008  df-int 4043  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-tr 4295  df-eprel 4486  df-id 4490  df-po 4495  df-so 4496  df-fr 4533  df-we 4535  df-ord 4576  df-on 4577  df-lim 4578  df-suc 4579  df-om 4838  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-1st 6341  df-2nd 6342  df-recs 6625  df-rdg 6660  df-1o 6716  df-oadd 6720  df-omul 6721  df-er 6897  df-ec 6899  df-qs 6903  df-ni 8741  df-pli 8742  df-mi 8743  df-lti 8744  df-plpq 8777  df-mpq 8778  df-ltpq 8779  df-enq 8780  df-nq 8781  df-erq 8782  df-plq 8783  df-mq 8784  df-1nq 8785  df-rq 8786  df-ltnq 8787  df-np 8850  df-1p 8851  df-plp 8852  df-mp 8853  df-ltp 8854  df-mpr 8925  df-enr 8926  df-nr 8927  df-mr 8929  df-ltr 8930  df-0r 8931
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