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Theorem mulgt0sr 9284
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 9250 . . . . 5  |-  <R  C_  ( R.  X.  R. )
21brel 4899 . . . 4  |-  ( 0R 
<R  A  ->  ( 0R  e.  R.  /\  A  e.  R. ) )
32simprd 463 . . 3  |-  ( 0R 
<R  A  ->  A  e. 
R. )
41brel 4899 . . . 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 9239 . . 3  |-  R.  =  ( ( P.  X.  P. ) /.  ~R  )
8 breq2 4308 . . . . 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 6110 . . . . 5  |-  ( [
<. x ,  y >. ]  ~R  =  A  -> 
( [ <. x ,  y >. ]  ~R  .R 
[ <. z ,  w >. ]  ~R  )  =  ( A  .R  [ <. z ,  w >. ]  ~R  ) )
1110breq2d 4316 . . . 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 4308 . . . . 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 6111 . . . . 5  |-  ( [
<. z ,  w >. ]  ~R  =  B  -> 
( A  .R  [ <. z ,  w >. ]  ~R  )  =  ( A  .R  B ) )
1615breq2d 4316 . . . 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 9257 . . . . 5  |-  ( 0R 
<R  [ <. x ,  y
>. ]  ~R  <->  y  <P  x )
19 gt0srpr 9257 . . . . 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 9201 . . . . . . . 8  |-  ( ( x  e.  P.  /\  z  e.  P. )  ->  ( x  .P.  z
)  e.  P. )
23 mulclpr 9201 . . . . . . . 8  |-  ( ( y  e.  P.  /\  w  e.  P. )  ->  ( y  .P.  w
)  e.  P. )
24 addclpr 9199 . . . . . . . 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 822 . . . . . 6  |-  ( ( ( x  e.  P.  /\  y  e.  P. )  /\  ( z  e.  P.  /\  w  e.  P. )
)  ->  ( (
x  .P.  z )  +P.  ( y  .P.  w
) )  e.  P. )
27 ltexpri 9224 . . . . . . . . 9  |-  ( y 
<P  x  ->  E. v  e.  P.  ( y  +P.  v )  =  x )
28 ltexpri 9224 . . . . . . . . 9  |-  ( w 
<P  z  ->  E. u  e.  P.  ( w  +P.  u )  =  z )
29 mulclpr 9201 . . . . . . . . . . . . . . . . 17  |-  ( ( v  e.  P.  /\  w  e.  P. )  ->  ( v  .P.  w
)  e.  P. )
30 oveq12 6112 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( ( ( y  +P.  v
)  =  x  /\  ( w  +P.  u )  =  z )  -> 
( ( y  +P.  v )  .P.  (
w  +P.  u )
)  =  ( x  .P.  z ) )
3130oveq1d 6118 . . . . . . . . . . . . . . . . . . . . 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 9209 . . . . . . . . . . . . . . . . . . . . . . . . 25  |-  ( y  .P.  ( w  +P.  u ) )  =  ( ( y  .P.  w )  +P.  (
y  .P.  u )
)
33 oveq2 6111 . . . . . . . . . . . . . . . . . . . . . . . . 25  |-  ( ( w  +P.  u )  =  z  ->  (
y  .P.  ( w  +P.  u ) )  =  ( y  .P.  z
) )
3432, 33syl5eqr 2489 . . . . . . . . . . . . . . . . . . . . . . . 24  |-  ( ( w  +P.  u )  =  z  ->  (
( y  .P.  w
)  +P.  ( y  .P.  u ) )  =  ( y  .P.  z
) )
3534oveq1d 6118 . . . . . . . . . . . . . . . . . . . . . . 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 2987 . . . . . . . . . . . . . . . . . . . . . . . . . 26  |-  y  e. 
_V
37 vex 2987 . . . . . . . . . . . . . . . . . . . . . . . . . 26  |-  v  e. 
_V
38 vex 2987 . . . . . . . . . . . . . . . . . . . . . . . . . 26  |-  w  e. 
_V
39 mulcompr 9204 . . . . . . . . . . . . . . . . . . . . . . . . . 26  |-  ( f  .P.  g )  =  ( g  .P.  f
)
40 distrpr 9209 . . . . . . . . . . . . . . . . . . . . . . . . . 26  |-  ( f  .P.  ( g  +P.  h ) )  =  ( ( f  .P.  g )  +P.  (
f  .P.  h )
)
4136, 37, 38, 39, 40caovdir 6309 . . . . . . . . . . . . . . . . . . . . . . . . 25  |-  ( ( y  +P.  v )  .P.  w )  =  ( ( y  .P.  w )  +P.  (
v  .P.  w )
)
42 vex 2987 . . . . . . . . . . . . . . . . . . . . . . . . . 26  |-  u  e. 
_V
4336, 37, 42, 39, 40caovdir 6309 . . . . . . . . . . . . . . . . . . . . . . . . 25  |-  ( ( y  +P.  v )  .P.  u )  =  ( ( y  .P.  u )  +P.  (
v  .P.  u )
)
4441, 43oveq12i 6115 . . . . . . . . . . . . . . . . . . . . . . . 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 9209 . . . . . . . . . . . . . . . . . . . . . . . 24  |-  ( ( y  +P.  v )  .P.  ( w  +P.  u ) )  =  ( ( ( y  +P.  v )  .P.  w )  +P.  (
( y  +P.  v
)  .P.  u )
)
46 ovex 6128 . . . . . . . . . . . . . . . . . . . . . . . . 25  |-  ( y  .P.  w )  e. 
_V
47 ovex 6128 . . . . . . . . . . . . . . . . . . . . . . . . 25  |-  ( y  .P.  u )  e. 
_V
48 ovex 6128 . . . . . . . . . . . . . . . . . . . . . . . . 25  |-  ( v  .P.  w )  e. 
_V
49 addcompr 9202 . . . . . . . . . . . . . . . . . . . . . . . . 25  |-  ( f  +P.  g )  =  ( g  +P.  f
)
50 addasspr 9203 . . . . . . . . . . . . . . . . . . . . . . . . 25  |-  ( ( f  +P.  g )  +P.  h )  =  ( f  +P.  (
g  +P.  h )
)
51 ovex 6128 . . . . . . . . . . . . . . . . . . . . . . . . 25  |-  ( v  .P.  u )  e. 
_V
5246, 47, 48, 49, 50, 51caov4 6306 . . . . . . . . . . . . . . . . . . . . . . . 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 2473 . . . . . . . . . . . . . . . . . . . . . . 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 6128 . . . . . . . . . . . . . . . . . . . . . . . 24  |-  ( y  .P.  z )  e. 
_V
5548, 54, 51, 49, 50caov12 6303 . . . . . . . . . . . . . . . . . . . . . . 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 2500 . . . . . . . . . . . . . . . . . . . . . 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 6110 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( ( y  +P.  v )  =  x  ->  (
( y  +P.  v
)  .P.  w )  =  ( x  .P.  w ) )
5841, 57syl5eqr 2489 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( ( y  +P.  v )  =  x  ->  (
( y  .P.  w
)  +P.  ( v  .P.  w ) )  =  ( x  .P.  w
) )
5956, 58oveqan12rd 6123 . . . . . . . . . . . . . . . . . . . . 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 2477 . . . . . . . . . . . . . . . . . . . 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 9203 . . . . . . . . . . . . . . . . . . . . 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 9202 . . . . . . . . . . . . . . . . . . . . 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 2465 . . . . . . . . . . . . . . . . . . . 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 9203 . . . . . . . . . . . . . . . . . . . . 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 6128 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( ( y  .P.  z )  +P.  ( v  .P.  u ) )  e. 
_V
66 ovex 6128 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( x  .P.  w )  e. 
_V
6748, 65, 66, 49, 50caov32 6302 . . . . . . . . . . . . . . . . . . . . 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 9203 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( ( ( x  .P.  w
)  +P.  ( y  .P.  z ) )  +P.  ( v  .P.  u
) )  =  ( ( x  .P.  w
)  +P.  ( (
y  .P.  z )  +P.  ( v  .P.  u
) ) )
6968oveq2i 6114 . . . . . . . . . . . . . . . . . . . . 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 2473 . . . . . . . . . . . . . . . . . . . 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 2498 . . . . . . . . . . . . . . . . . . 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 9227 . . . . . . . . . . . . . . . . . . 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 2445 . . . . . . . . . . . . . . . . . . . 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 9216 . . . . . . . . . . . . . . . . . . . 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 2852 . . . . . . . . . . . 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 2847 . . . . . . . . 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 9255 . . . . . . 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 4316 . . . . . 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 9257 . . . . . 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 7203 . 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 1369    e. wcel 1756   E.wrex 2728   <.cop 3895   class class class wbr 4304  (class class class)co 6103   [cec 7111   P.cnp 9038    +P. cpp 9040    .P. cmp 9041    <P cltp 9042    ~R cer 9045   R.cnr 9046   0Rc0r 9047    .R cmr 9051    <R cltr 9052
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4425  ax-nul 4433  ax-pow 4482  ax-pr 4543  ax-un 6384  ax-inf2 7859
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2577  df-ne 2620  df-ral 2732  df-rex 2733  df-reu 2734  df-rmo 2735  df-rab 2736  df-v 2986  df-sbc 3199  df-csb 3301  df-dif 3343  df-un 3345  df-in 3347  df-ss 3354  df-pss 3356  df-nul 3650  df-if 3804  df-pw 3874  df-sn 3890  df-pr 3892  df-tp 3894  df-op 3896  df-uni 4104  df-int 4141  df-iun 4185  df-br 4305  df-opab 4363  df-mpt 4364  df-tr 4398  df-eprel 4644  df-id 4648  df-po 4653  df-so 4654  df-fr 4691  df-we 4693  df-ord 4734  df-on 4735  df-lim 4736  df-suc 4737  df-xp 4858  df-rel 4859  df-cnv 4860  df-co 4861  df-dm 4862  df-rn 4863  df-res 4864  df-ima 4865  df-iota 5393  df-fun 5432  df-fn 5433  df-f 5434  df-f1 5435  df-fo 5436  df-f1o 5437  df-fv 5438  df-ov 6106  df-oprab 6107  df-mpt2 6108  df-om 6489  df-1st 6589  df-2nd 6590  df-recs 6844  df-rdg 6878  df-1o 6932  df-oadd 6936  df-omul 6937  df-er 7113  df-ec 7115  df-qs 7119  df-ni 9053  df-pli 9054  df-mi 9055  df-lti 9056  df-plpq 9089  df-mpq 9090  df-ltpq 9091  df-enq 9092  df-nq 9093  df-erq 9094  df-plq 9095  df-mq 9096  df-1nq 9097  df-rq 9098  df-ltnq 9099  df-np 9162  df-1p 9163  df-plp 9164  df-mp 9165  df-ltp 9166  df-mpr 9237  df-enr 9238  df-nr 9239  df-mr 9241  df-ltr 9242  df-0r 9243
This theorem is referenced by:  sqgt0sr  9285  axpre-mulgt0  9347
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