Users' Mathboxes Mathbox for Scott Fenton < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  axpaschlem Unicode version

Theorem axpaschlem 25783
Description: Lemma for axpasch 25784. Set up coefficents used in the proof. (Contributed by Scott Fenton, 5-Jun-2013.)
Assertion
Ref Expression
axpaschlem  |-  ( ( T  e.  ( 0 [,] 1 )  /\  S  e.  ( 0 [,] 1 ) )  ->  E. r  e.  ( 0 [,] 1 ) E. p  e.  ( 0 [,] 1 ) ( p  =  ( ( 1  -  r
)  x.  ( 1  -  T ) )  /\  r  =  ( ( 1  -  p
)  x.  ( 1  -  S ) )  /\  ( ( 1  -  r )  x.  T )  =  ( ( 1  -  p
)  x.  S ) ) )
Distinct variable groups:    T, p, r    S, p, r

Proof of Theorem axpaschlem
StepHypRef Expression
1 1re 9046 . . . . . . . 8  |-  1  e.  RR
2 0re 9047 . . . . . . . . . . 11  |-  0  e.  RR
32, 1elicc2i 10932 . . . . . . . . . 10  |-  ( T  e.  ( 0 [,] 1 )  <->  ( T  e.  RR  /\  0  <_  T  /\  T  <_  1
) )
43simp1bi 972 . . . . . . . . 9  |-  ( T  e.  ( 0 [,] 1 )  ->  T  e.  RR )
54ad2antrl 709 . . . . . . . 8  |-  ( ( S  =  0  /\  ( T  e.  ( 0 [,] 1 )  /\  S  e.  ( 0 [,] 1 ) ) )  ->  T  e.  RR )
6 resubcl 9321 . . . . . . . 8  |-  ( ( 1  e.  RR  /\  T  e.  RR )  ->  ( 1  -  T
)  e.  RR )
71, 5, 6sylancr 645 . . . . . . 7  |-  ( ( S  =  0  /\  ( T  e.  ( 0 [,] 1 )  /\  S  e.  ( 0 [,] 1 ) ) )  ->  (
1  -  T )  e.  RR )
87recnd 9070 . . . . . 6  |-  ( ( S  =  0  /\  ( T  e.  ( 0 [,] 1 )  /\  S  e.  ( 0 [,] 1 ) ) )  ->  (
1  -  T )  e.  CC )
98mul02d 9220 . . . . 5  |-  ( ( S  =  0  /\  ( T  e.  ( 0 [,] 1 )  /\  S  e.  ( 0 [,] 1 ) ) )  ->  (
0  x.  ( 1  -  T ) )  =  0 )
109eqcomd 2409 . . . 4  |-  ( ( S  =  0  /\  ( T  e.  ( 0 [,] 1 )  /\  S  e.  ( 0 [,] 1 ) ) )  ->  0  =  ( 0  x.  ( 1  -  T
) ) )
112, 1elicc2i 10932 . . . . . . . . . 10  |-  ( S  e.  ( 0 [,] 1 )  <->  ( S  e.  RR  /\  0  <_  S  /\  S  <_  1
) )
1211simp1bi 972 . . . . . . . . 9  |-  ( S  e.  ( 0 [,] 1 )  ->  S  e.  RR )
1312ad2antll 710 . . . . . . . 8  |-  ( ( S  =  0  /\  ( T  e.  ( 0 [,] 1 )  /\  S  e.  ( 0 [,] 1 ) ) )  ->  S  e.  RR )
14 resubcl 9321 . . . . . . . 8  |-  ( ( 1  e.  RR  /\  S  e.  RR )  ->  ( 1  -  S
)  e.  RR )
151, 13, 14sylancr 645 . . . . . . 7  |-  ( ( S  =  0  /\  ( T  e.  ( 0 [,] 1 )  /\  S  e.  ( 0 [,] 1 ) ) )  ->  (
1  -  S )  e.  RR )
1615recnd 9070 . . . . . 6  |-  ( ( S  =  0  /\  ( T  e.  ( 0 [,] 1 )  /\  S  e.  ( 0 [,] 1 ) ) )  ->  (
1  -  S )  e.  CC )
1716mulid2d 9062 . . . . 5  |-  ( ( S  =  0  /\  ( T  e.  ( 0 [,] 1 )  /\  S  e.  ( 0 [,] 1 ) ) )  ->  (
1  x.  ( 1  -  S ) )  =  ( 1  -  S ) )
18 oveq2 6048 . . . . . . 7  |-  ( S  =  0  ->  (
1  -  S )  =  ( 1  -  0 ) )
1918adantr 452 . . . . . 6  |-  ( ( S  =  0  /\  ( T  e.  ( 0 [,] 1 )  /\  S  e.  ( 0 [,] 1 ) ) )  ->  (
1  -  S )  =  ( 1  -  0 ) )
20 ax-1cn 9004 . . . . . . 7  |-  1  e.  CC
2120subid1i 9328 . . . . . 6  |-  ( 1  -  0 )  =  1
2219, 21syl6eq 2452 . . . . 5  |-  ( ( S  =  0  /\  ( T  e.  ( 0 [,] 1 )  /\  S  e.  ( 0 [,] 1 ) ) )  ->  (
1  -  S )  =  1 )
2317, 22eqtr2d 2437 . . . 4  |-  ( ( S  =  0  /\  ( T  e.  ( 0 [,] 1 )  /\  S  e.  ( 0 [,] 1 ) ) )  ->  1  =  ( 1  x.  ( 1  -  S
) ) )
245recnd 9070 . . . . . 6  |-  ( ( S  =  0  /\  ( T  e.  ( 0 [,] 1 )  /\  S  e.  ( 0 [,] 1 ) ) )  ->  T  e.  CC )
2524mul02d 9220 . . . . 5  |-  ( ( S  =  0  /\  ( T  e.  ( 0 [,] 1 )  /\  S  e.  ( 0 [,] 1 ) ) )  ->  (
0  x.  T )  =  0 )
26 oveq2 6048 . . . . . . 7  |-  ( S  =  0  ->  (
1  x.  S )  =  ( 1  x.  0 ) )
2726adantr 452 . . . . . 6  |-  ( ( S  =  0  /\  ( T  e.  ( 0 [,] 1 )  /\  S  e.  ( 0 [,] 1 ) ) )  ->  (
1  x.  S )  =  ( 1  x.  0 ) )
2820mul01i 9212 . . . . . 6  |-  ( 1  x.  0 )  =  0
2927, 28syl6eq 2452 . . . . 5  |-  ( ( S  =  0  /\  ( T  e.  ( 0 [,] 1 )  /\  S  e.  ( 0 [,] 1 ) ) )  ->  (
1  x.  S )  =  0 )
3025, 29eqtr4d 2439 . . . 4  |-  ( ( S  =  0  /\  ( T  e.  ( 0 [,] 1 )  /\  S  e.  ( 0 [,] 1 ) ) )  ->  (
0  x.  T )  =  ( 1  x.  S ) )
31 1elunit 10972 . . . . 5  |-  1  e.  ( 0 [,] 1
)
32 0elunit 10971 . . . . 5  |-  0  e.  ( 0 [,] 1
)
33 oveq2 6048 . . . . . . . . . 10  |-  ( r  =  1  ->  (
1  -  r )  =  ( 1  -  1 ) )
34 1m1e0 10024 . . . . . . . . . 10  |-  ( 1  -  1 )  =  0
3533, 34syl6eq 2452 . . . . . . . . 9  |-  ( r  =  1  ->  (
1  -  r )  =  0 )
3635oveq1d 6055 . . . . . . . 8  |-  ( r  =  1  ->  (
( 1  -  r
)  x.  ( 1  -  T ) )  =  ( 0  x.  ( 1  -  T
) ) )
3736eqeq2d 2415 . . . . . . 7  |-  ( r  =  1  ->  (
p  =  ( ( 1  -  r )  x.  ( 1  -  T ) )  <->  p  =  ( 0  x.  (
1  -  T ) ) ) )
38 eqeq1 2410 . . . . . . 7  |-  ( r  =  1  ->  (
r  =  ( ( 1  -  p )  x.  ( 1  -  S ) )  <->  1  =  ( ( 1  -  p )  x.  (
1  -  S ) ) ) )
3935oveq1d 6055 . . . . . . . 8  |-  ( r  =  1  ->  (
( 1  -  r
)  x.  T )  =  ( 0  x.  T ) )
4039eqeq1d 2412 . . . . . . 7  |-  ( r  =  1  ->  (
( ( 1  -  r )  x.  T
)  =  ( ( 1  -  p )  x.  S )  <->  ( 0  x.  T )  =  ( ( 1  -  p )  x.  S
) ) )
4137, 38, 403anbi123d 1254 . . . . . 6  |-  ( r  =  1  ->  (
( p  =  ( ( 1  -  r
)  x.  ( 1  -  T ) )  /\  r  =  ( ( 1  -  p
)  x.  ( 1  -  S ) )  /\  ( ( 1  -  r )  x.  T )  =  ( ( 1  -  p
)  x.  S ) )  <->  ( p  =  ( 0  x.  (
1  -  T ) )  /\  1  =  ( ( 1  -  p )  x.  (
1  -  S ) )  /\  ( 0  x.  T )  =  ( ( 1  -  p )  x.  S
) ) ) )
42 eqeq1 2410 . . . . . . 7  |-  ( p  =  0  ->  (
p  =  ( 0  x.  ( 1  -  T ) )  <->  0  =  ( 0  x.  (
1  -  T ) ) ) )
43 oveq2 6048 . . . . . . . . . 10  |-  ( p  =  0  ->  (
1  -  p )  =  ( 1  -  0 ) )
4443, 21syl6eq 2452 . . . . . . . . 9  |-  ( p  =  0  ->  (
1  -  p )  =  1 )
4544oveq1d 6055 . . . . . . . 8  |-  ( p  =  0  ->  (
( 1  -  p
)  x.  ( 1  -  S ) )  =  ( 1  x.  ( 1  -  S
) ) )
4645eqeq2d 2415 . . . . . . 7  |-  ( p  =  0  ->  (
1  =  ( ( 1  -  p )  x.  ( 1  -  S ) )  <->  1  =  ( 1  x.  (
1  -  S ) ) ) )
4744oveq1d 6055 . . . . . . . 8  |-  ( p  =  0  ->  (
( 1  -  p
)  x.  S )  =  ( 1  x.  S ) )
4847eqeq2d 2415 . . . . . . 7  |-  ( p  =  0  ->  (
( 0  x.  T
)  =  ( ( 1  -  p )  x.  S )  <->  ( 0  x.  T )  =  ( 1  x.  S
) ) )
4942, 46, 483anbi123d 1254 . . . . . 6  |-  ( p  =  0  ->  (
( p  =  ( 0  x.  ( 1  -  T ) )  /\  1  =  ( ( 1  -  p
)  x.  ( 1  -  S ) )  /\  ( 0  x.  T )  =  ( ( 1  -  p
)  x.  S ) )  <->  ( 0  =  ( 0  x.  (
1  -  T ) )  /\  1  =  ( 1  x.  (
1  -  S ) )  /\  ( 0  x.  T )  =  ( 1  x.  S
) ) ) )
5041, 49rspc2ev 3020 . . . . 5  |-  ( ( 1  e.  ( 0 [,] 1 )  /\  0  e.  ( 0 [,] 1 )  /\  ( 0  =  ( 0  x.  ( 1  -  T ) )  /\  1  =  ( 1  x.  ( 1  -  S ) )  /\  ( 0  x.  T )  =  ( 1  x.  S ) ) )  ->  E. r  e.  ( 0 [,] 1
) E. p  e.  ( 0 [,] 1
) ( p  =  ( ( 1  -  r )  x.  (
1  -  T ) )  /\  r  =  ( ( 1  -  p )  x.  (
1  -  S ) )  /\  ( ( 1  -  r )  x.  T )  =  ( ( 1  -  p )  x.  S
) ) )
5131, 32, 50mp3an12 1269 . . . 4  |-  ( ( 0  =  ( 0  x.  ( 1  -  T ) )  /\  1  =  ( 1  x.  ( 1  -  S ) )  /\  ( 0  x.  T
)  =  ( 1  x.  S ) )  ->  E. r  e.  ( 0 [,] 1 ) E. p  e.  ( 0 [,] 1 ) ( p  =  ( ( 1  -  r
)  x.  ( 1  -  T ) )  /\  r  =  ( ( 1  -  p
)  x.  ( 1  -  S ) )  /\  ( ( 1  -  r )  x.  T )  =  ( ( 1  -  p
)  x.  S ) ) )
5210, 23, 30, 51syl3anc 1184 . . 3  |-  ( ( S  =  0  /\  ( T  e.  ( 0 [,] 1 )  /\  S  e.  ( 0 [,] 1 ) ) )  ->  E. r  e.  ( 0 [,] 1
) E. p  e.  ( 0 [,] 1
) ( p  =  ( ( 1  -  r )  x.  (
1  -  T ) )  /\  r  =  ( ( 1  -  p )  x.  (
1  -  S ) )  /\  ( ( 1  -  r )  x.  T )  =  ( ( 1  -  p )  x.  S
) ) )
5352ex 424 . 2  |-  ( S  =  0  ->  (
( T  e.  ( 0 [,] 1 )  /\  S  e.  ( 0 [,] 1 ) )  ->  E. r  e.  ( 0 [,] 1
) E. p  e.  ( 0 [,] 1
) ( p  =  ( ( 1  -  r )  x.  (
1  -  T ) )  /\  r  =  ( ( 1  -  p )  x.  (
1  -  S ) )  /\  ( ( 1  -  r )  x.  T )  =  ( ( 1  -  p )  x.  S
) ) ) )
544ad2antrl 709 . . . . . . 7  |-  ( ( S  =/=  0  /\  ( T  e.  ( 0 [,] 1 )  /\  S  e.  ( 0 [,] 1 ) ) )  ->  T  e.  RR )
5512ad2antll 710 . . . . . . . 8  |-  ( ( S  =/=  0  /\  ( T  e.  ( 0 [,] 1 )  /\  S  e.  ( 0 [,] 1 ) ) )  ->  S  e.  RR )
5655, 54remulcld 9072 . . . . . . 7  |-  ( ( S  =/=  0  /\  ( T  e.  ( 0 [,] 1 )  /\  S  e.  ( 0 [,] 1 ) ) )  ->  ( S  x.  T )  e.  RR )
5754, 56resubcld 9421 . . . . . 6  |-  ( ( S  =/=  0  /\  ( T  e.  ( 0 [,] 1 )  /\  S  e.  ( 0 [,] 1 ) ) )  ->  ( T  -  ( S  x.  T ) )  e.  RR )
5855, 54readdcld 9071 . . . . . . 7  |-  ( ( S  =/=  0  /\  ( T  e.  ( 0 [,] 1 )  /\  S  e.  ( 0 [,] 1 ) ) )  ->  ( S  +  T )  e.  RR )
5958, 56resubcld 9421 . . . . . 6  |-  ( ( S  =/=  0  /\  ( T  e.  ( 0 [,] 1 )  /\  S  e.  ( 0 [,] 1 ) ) )  ->  (
( S  +  T
)  -  ( S  x.  T ) )  e.  RR )
601a1i 11 . . . . . . . . . . 11  |-  ( ( S  =/=  0  /\  ( T  e.  ( 0 [,] 1 )  /\  S  e.  ( 0 [,] 1 ) ) )  ->  1  e.  RR )
613simp2bi 973 . . . . . . . . . . . 12  |-  ( T  e.  ( 0 [,] 1 )  ->  0  <_  T )
6261ad2antrl 709 . . . . . . . . . . 11  |-  ( ( S  =/=  0  /\  ( T  e.  ( 0 [,] 1 )  /\  S  e.  ( 0 [,] 1 ) ) )  ->  0  <_  T )
6311simp3bi 974 . . . . . . . . . . . 12  |-  ( S  e.  ( 0 [,] 1 )  ->  S  <_  1 )
6463ad2antll 710 . . . . . . . . . . 11  |-  ( ( S  =/=  0  /\  ( T  e.  ( 0 [,] 1 )  /\  S  e.  ( 0 [,] 1 ) ) )  ->  S  <_  1 )
6555, 60, 54, 62, 64lemul1ad 9906 . . . . . . . . . 10  |-  ( ( S  =/=  0  /\  ( T  e.  ( 0 [,] 1 )  /\  S  e.  ( 0 [,] 1 ) ) )  ->  ( S  x.  T )  <_  ( 1  x.  T
) )
6654recnd 9070 . . . . . . . . . . 11  |-  ( ( S  =/=  0  /\  ( T  e.  ( 0 [,] 1 )  /\  S  e.  ( 0 [,] 1 ) ) )  ->  T  e.  CC )
6766mulid2d 9062 . . . . . . . . . 10  |-  ( ( S  =/=  0  /\  ( T  e.  ( 0 [,] 1 )  /\  S  e.  ( 0 [,] 1 ) ) )  ->  (
1  x.  T )  =  T )
6865, 67breqtrd 4196 . . . . . . . . 9  |-  ( ( S  =/=  0  /\  ( T  e.  ( 0 [,] 1 )  /\  S  e.  ( 0 [,] 1 ) ) )  ->  ( S  x.  T )  <_  T )
6911simp2bi 973 . . . . . . . . . . . 12  |-  ( S  e.  ( 0 [,] 1 )  ->  0  <_  S )
7069ad2antll 710 . . . . . . . . . . 11  |-  ( ( S  =/=  0  /\  ( T  e.  ( 0 [,] 1 )  /\  S  e.  ( 0 [,] 1 ) ) )  ->  0  <_  S )
71 simpl 444 . . . . . . . . . . 11  |-  ( ( S  =/=  0  /\  ( T  e.  ( 0 [,] 1 )  /\  S  e.  ( 0 [,] 1 ) ) )  ->  S  =/=  0 )
7255, 70, 71ne0gt0d 9166 . . . . . . . . . 10  |-  ( ( S  =/=  0  /\  ( T  e.  ( 0 [,] 1 )  /\  S  e.  ( 0 [,] 1 ) ) )  ->  0  <  S )
7355, 54ltaddpos2d 9567 . . . . . . . . . 10  |-  ( ( S  =/=  0  /\  ( T  e.  ( 0 [,] 1 )  /\  S  e.  ( 0 [,] 1 ) ) )  ->  (
0  <  S  <->  T  <  ( S  +  T ) ) )
7472, 73mpbid 202 . . . . . . . . 9  |-  ( ( S  =/=  0  /\  ( T  e.  ( 0 [,] 1 )  /\  S  e.  ( 0 [,] 1 ) ) )  ->  T  <  ( S  +  T
) )
7556, 54, 58, 68, 74lelttrd 9184 . . . . . . . 8  |-  ( ( S  =/=  0  /\  ( T  e.  ( 0 [,] 1 )  /\  S  e.  ( 0 [,] 1 ) ) )  ->  ( S  x.  T )  <  ( S  +  T
) )
7656, 58posdifd 9569 . . . . . . . 8  |-  ( ( S  =/=  0  /\  ( T  e.  ( 0 [,] 1 )  /\  S  e.  ( 0 [,] 1 ) ) )  ->  (
( S  x.  T
)  <  ( S  +  T )  <->  0  <  ( ( S  +  T
)  -  ( S  x.  T ) ) ) )
7775, 76mpbid 202 . . . . . . 7  |-  ( ( S  =/=  0  /\  ( T  e.  ( 0 [,] 1 )  /\  S  e.  ( 0 [,] 1 ) ) )  ->  0  <  ( ( S  +  T )  -  ( S  x.  T )
) )
7877gt0ne0d 9547 . . . . . 6  |-  ( ( S  =/=  0  /\  ( T  e.  ( 0 [,] 1 )  /\  S  e.  ( 0 [,] 1 ) ) )  ->  (
( S  +  T
)  -  ( S  x.  T ) )  =/=  0 )
7957, 59, 78redivcld 9798 . . . . 5  |-  ( ( S  =/=  0  /\  ( T  e.  ( 0 [,] 1 )  /\  S  e.  ( 0 [,] 1 ) ) )  ->  (
( T  -  ( S  x.  T )
)  /  ( ( S  +  T )  -  ( S  x.  T ) ) )  e.  RR )
8054, 56subge0d 9572 . . . . . . 7  |-  ( ( S  =/=  0  /\  ( T  e.  ( 0 [,] 1 )  /\  S  e.  ( 0 [,] 1 ) ) )  ->  (
0  <_  ( T  -  ( S  x.  T ) )  <->  ( S  x.  T )  <_  T
) )
8168, 80mpbird 224 . . . . . 6  |-  ( ( S  =/=  0  /\  ( T  e.  ( 0 [,] 1 )  /\  S  e.  ( 0 [,] 1 ) ) )  ->  0  <_  ( T  -  ( S  x.  T )
) )
82 divge0 9835 . . . . . 6  |-  ( ( ( ( T  -  ( S  x.  T
) )  e.  RR  /\  0  <_  ( T  -  ( S  x.  T ) ) )  /\  ( ( ( S  +  T )  -  ( S  x.  T ) )  e.  RR  /\  0  < 
( ( S  +  T )  -  ( S  x.  T )
) ) )  -> 
0  <_  ( ( T  -  ( S  x.  T ) )  / 
( ( S  +  T )  -  ( S  x.  T )
) ) )
8357, 81, 59, 77, 82syl22anc 1185 . . . . 5  |-  ( ( S  =/=  0  /\  ( T  e.  ( 0 [,] 1 )  /\  S  e.  ( 0 [,] 1 ) ) )  ->  0  <_  ( ( T  -  ( S  x.  T
) )  /  (
( S  +  T
)  -  ( S  x.  T ) ) ) )
8454, 58, 74ltled 9177 . . . . . . . 8  |-  ( ( S  =/=  0  /\  ( T  e.  ( 0 [,] 1 )  /\  S  e.  ( 0 [,] 1 ) ) )  ->  T  <_  ( S  +  T
) )
8554, 58, 56, 84lesub1dd 9598 . . . . . . 7  |-  ( ( S  =/=  0  /\  ( T  e.  ( 0 [,] 1 )  /\  S  e.  ( 0 [,] 1 ) ) )  ->  ( T  -  ( S  x.  T ) )  <_ 
( ( S  +  T )  -  ( S  x.  T )
) )
8659recnd 9070 . . . . . . . 8  |-  ( ( S  =/=  0  /\  ( T  e.  ( 0 [,] 1 )  /\  S  e.  ( 0 [,] 1 ) ) )  ->  (
( S  +  T
)  -  ( S  x.  T ) )  e.  CC )
8786mulid2d 9062 . . . . . . 7  |-  ( ( S  =/=  0  /\  ( T  e.  ( 0 [,] 1 )  /\  S  e.  ( 0 [,] 1 ) ) )  ->  (
1  x.  ( ( S  +  T )  -  ( S  x.  T ) ) )  =  ( ( S  +  T )  -  ( S  x.  T
) ) )
8885, 87breqtrrd 4198 . . . . . 6  |-  ( ( S  =/=  0  /\  ( T  e.  ( 0 [,] 1 )  /\  S  e.  ( 0 [,] 1 ) ) )  ->  ( T  -  ( S  x.  T ) )  <_ 
( 1  x.  (
( S  +  T
)  -  ( S  x.  T ) ) ) )
89 ledivmul2 9843 . . . . . . 7  |-  ( ( ( T  -  ( S  x.  T )
)  e.  RR  /\  1  e.  RR  /\  (
( ( S  +  T )  -  ( S  x.  T )
)  e.  RR  /\  0  <  ( ( S  +  T )  -  ( S  x.  T
) ) ) )  ->  ( ( ( T  -  ( S  x.  T ) )  /  ( ( S  +  T )  -  ( S  x.  T
) ) )  <_ 
1  <->  ( T  -  ( S  x.  T
) )  <_  (
1  x.  ( ( S  +  T )  -  ( S  x.  T ) ) ) ) )
9057, 60, 59, 77, 89syl112anc 1188 . . . . . 6  |-  ( ( S  =/=  0  /\  ( T  e.  ( 0 [,] 1 )  /\  S  e.  ( 0 [,] 1 ) ) )  ->  (
( ( T  -  ( S  x.  T
) )  /  (
( S  +  T
)  -  ( S  x.  T ) ) )  <_  1  <->  ( T  -  ( S  x.  T ) )  <_ 
( 1  x.  (
( S  +  T
)  -  ( S  x.  T ) ) ) ) )
9188, 90mpbird 224 . . . . 5  |-  ( ( S  =/=  0  /\  ( T  e.  ( 0 [,] 1 )  /\  S  e.  ( 0 [,] 1 ) ) )  ->  (
( T  -  ( S  x.  T )
)  /  ( ( S  +  T )  -  ( S  x.  T ) ) )  <_  1 )
922, 1elicc2i 10932 . . . . 5  |-  ( ( ( T  -  ( S  x.  T )
)  /  ( ( S  +  T )  -  ( S  x.  T ) ) )  e.  ( 0 [,] 1 )  <->  ( (
( T  -  ( S  x.  T )
)  /  ( ( S  +  T )  -  ( S  x.  T ) ) )  e.  RR  /\  0  <_  ( ( T  -  ( S  x.  T
) )  /  (
( S  +  T
)  -  ( S  x.  T ) ) )  /\  ( ( T  -  ( S  x.  T ) )  /  ( ( S  +  T )  -  ( S  x.  T
) ) )  <_ 
1 ) )
9379, 83, 91, 92syl3anbrc 1138 . . . 4  |-  ( ( S  =/=  0  /\  ( T  e.  ( 0 [,] 1 )  /\  S  e.  ( 0 [,] 1 ) ) )  ->  (
( T  -  ( S  x.  T )
)  /  ( ( S  +  T )  -  ( S  x.  T ) ) )  e.  ( 0 [,] 1 ) )
9455, 56resubcld 9421 . . . . . 6  |-  ( ( S  =/=  0  /\  ( T  e.  ( 0 [,] 1 )  /\  S  e.  ( 0 [,] 1 ) ) )  ->  ( S  -  ( S  x.  T ) )  e.  RR )
9594, 59, 78redivcld 9798 . . . . 5  |-  ( ( S  =/=  0  /\  ( T  e.  ( 0 [,] 1 )  /\  S  e.  ( 0 [,] 1 ) ) )  ->  (
( S  -  ( S  x.  T )
)  /  ( ( S  +  T )  -  ( S  x.  T ) ) )  e.  RR )
963simp3bi 974 . . . . . . . . . 10  |-  ( T  e.  ( 0 [,] 1 )  ->  T  <_  1 )
9796ad2antrl 709 . . . . . . . . 9  |-  ( ( S  =/=  0  /\  ( T  e.  ( 0 [,] 1 )  /\  S  e.  ( 0 [,] 1 ) ) )  ->  T  <_  1 )
9854, 60, 55, 70, 97lemul2ad 9907 . . . . . . . 8  |-  ( ( S  =/=  0  /\  ( T  e.  ( 0 [,] 1 )  /\  S  e.  ( 0 [,] 1 ) ) )  ->  ( S  x.  T )  <_  ( S  x.  1 ) )
9955recnd 9070 . . . . . . . . 9  |-  ( ( S  =/=  0  /\  ( T  e.  ( 0 [,] 1 )  /\  S  e.  ( 0 [,] 1 ) ) )  ->  S  e.  CC )
10099mulid1d 9061 . . . . . . . 8  |-  ( ( S  =/=  0  /\  ( T  e.  ( 0 [,] 1 )  /\  S  e.  ( 0 [,] 1 ) ) )  ->  ( S  x.  1 )  =  S )
10198, 100breqtrd 4196 . . . . . . 7  |-  ( ( S  =/=  0  /\  ( T  e.  ( 0 [,] 1 )  /\  S  e.  ( 0 [,] 1 ) ) )  ->  ( S  x.  T )  <_  S )
10255, 56subge0d 9572 . . . . . . 7  |-  ( ( S  =/=  0  /\  ( T  e.  ( 0 [,] 1 )  /\  S  e.  ( 0 [,] 1 ) ) )  ->  (
0  <_  ( S  -  ( S  x.  T ) )  <->  ( S  x.  T )  <_  S
) )
103101, 102mpbird 224 . . . . . 6  |-  ( ( S  =/=  0  /\  ( T  e.  ( 0 [,] 1 )  /\  S  e.  ( 0 [,] 1 ) ) )  ->  0  <_  ( S  -  ( S  x.  T )
) )
104 divge0 9835 . . . . . 6  |-  ( ( ( ( S  -  ( S  x.  T
) )  e.  RR  /\  0  <_  ( S  -  ( S  x.  T ) ) )  /\  ( ( ( S  +  T )  -  ( S  x.  T ) )  e.  RR  /\  0  < 
( ( S  +  T )  -  ( S  x.  T )
) ) )  -> 
0  <_  ( ( S  -  ( S  x.  T ) )  / 
( ( S  +  T )  -  ( S  x.  T )
) ) )
10594, 103, 59, 77, 104syl22anc 1185 . . . . 5  |-  ( ( S  =/=  0  /\  ( T  e.  ( 0 [,] 1 )  /\  S  e.  ( 0 [,] 1 ) ) )  ->  0  <_  ( ( S  -  ( S  x.  T
) )  /  (
( S  +  T
)  -  ( S  x.  T ) ) ) )
10655, 54addge01d 9570 . . . . . . . . 9  |-  ( ( S  =/=  0  /\  ( T  e.  ( 0 [,] 1 )  /\  S  e.  ( 0 [,] 1 ) ) )  ->  (
0  <_  T  <->  S  <_  ( S  +  T ) ) )
10762, 106mpbid 202 . . . . . . . 8  |-  ( ( S  =/=  0  /\  ( T  e.  ( 0 [,] 1 )  /\  S  e.  ( 0 [,] 1 ) ) )  ->  S  <_  ( S  +  T
) )
10855, 58, 56, 107lesub1dd 9598 . . . . . . 7  |-  ( ( S  =/=  0  /\  ( T  e.  ( 0 [,] 1 )  /\  S  e.  ( 0 [,] 1 ) ) )  ->  ( S  -  ( S  x.  T ) )  <_ 
( ( S  +  T )  -  ( S  x.  T )
) )
109108, 87breqtrrd 4198 . . . . . 6  |-  ( ( S  =/=  0  /\  ( T  e.  ( 0 [,] 1 )  /\  S  e.  ( 0 [,] 1 ) ) )  ->  ( S  -  ( S  x.  T ) )  <_ 
( 1  x.  (
( S  +  T
)  -  ( S  x.  T ) ) ) )
110 ledivmul2 9843 . . . . . . 7  |-  ( ( ( S  -  ( S  x.  T )
)  e.  RR  /\  1  e.  RR  /\  (
( ( S  +  T )  -  ( S  x.  T )
)  e.  RR  /\  0  <  ( ( S  +  T )  -  ( S  x.  T
) ) ) )  ->  ( ( ( S  -  ( S  x.  T ) )  /  ( ( S  +  T )  -  ( S  x.  T
) ) )  <_ 
1  <->  ( S  -  ( S  x.  T
) )  <_  (
1  x.  ( ( S  +  T )  -  ( S  x.  T ) ) ) ) )
11194, 60, 59, 77, 110syl112anc 1188 . . . . . 6  |-  ( ( S  =/=  0  /\  ( T  e.  ( 0 [,] 1 )  /\  S  e.  ( 0 [,] 1 ) ) )  ->  (
( ( S  -  ( S  x.  T
) )  /  (
( S  +  T
)  -  ( S  x.  T ) ) )  <_  1  <->  ( S  -  ( S  x.  T ) )  <_ 
( 1  x.  (
( S  +  T
)  -  ( S  x.  T ) ) ) ) )
112109, 111mpbird 224 . . . . 5  |-  ( ( S  =/=  0  /\  ( T  e.  ( 0 [,] 1 )  /\  S  e.  ( 0 [,] 1 ) ) )  ->  (
( S  -  ( S  x.  T )
)  /  ( ( S  +  T )  -  ( S  x.  T ) ) )  <_  1 )
1132, 1elicc2i 10932 . . . . 5  |-  ( ( ( S  -  ( S  x.  T )
)  /  ( ( S  +  T )  -  ( S  x.  T ) ) )  e.  ( 0 [,] 1 )  <->  ( (
( S  -  ( S  x.  T )
)  /  ( ( S  +  T )  -  ( S  x.  T ) ) )  e.  RR  /\  0  <_  ( ( S  -  ( S  x.  T
) )  /  (
( S  +  T
)  -  ( S  x.  T ) ) )  /\  ( ( S  -  ( S  x.  T ) )  /  ( ( S  +  T )  -  ( S  x.  T
) ) )  <_ 
1 ) )
11495, 105, 112, 113syl3anbrc 1138 . . . 4  |-  ( ( S  =/=  0  /\  ( T  e.  ( 0 [,] 1 )  /\  S  e.  ( 0 [,] 1 ) ) )  ->  (
( S  -  ( S  x.  T )
)  /  ( ( S  +  T )  -  ( S  x.  T ) ) )  e.  ( 0 [,] 1 ) )
1151, 54, 6sylancr 645 . . . . . . 7  |-  ( ( S  =/=  0  /\  ( T  e.  ( 0 [,] 1 )  /\  S  e.  ( 0 [,] 1 ) ) )  ->  (
1  -  T )  e.  RR )
116115recnd 9070 . . . . . 6  |-  ( ( S  =/=  0  /\  ( T  e.  ( 0 [,] 1 )  /\  S  e.  ( 0 [,] 1 ) ) )  ->  (
1  -  T )  e.  CC )
11799, 116, 86, 78div23d 9783 . . . . 5  |-  ( ( S  =/=  0  /\  ( T  e.  ( 0 [,] 1 )  /\  S  e.  ( 0 [,] 1 ) ) )  ->  (
( S  x.  (
1  -  T ) )  /  ( ( S  +  T )  -  ( S  x.  T ) ) )  =  ( ( S  /  ( ( S  +  T )  -  ( S  x.  T
) ) )  x.  ( 1  -  T
) ) )
118 subdi 9423 . . . . . . . . 9  |-  ( ( S  e.  CC  /\  1  e.  CC  /\  T  e.  CC )  ->  ( S  x.  ( 1  -  T ) )  =  ( ( S  x.  1 )  -  ( S  x.  T
) ) )
11920, 118mp3an2 1267 . . . . . . . 8  |-  ( ( S  e.  CC  /\  T  e.  CC )  ->  ( S  x.  (
1  -  T ) )  =  ( ( S  x.  1 )  -  ( S  x.  T ) ) )
12099, 66, 119syl2anc 643 . . . . . . 7  |-  ( ( S  =/=  0  /\  ( T  e.  ( 0 [,] 1 )  /\  S  e.  ( 0 [,] 1 ) ) )  ->  ( S  x.  ( 1  -  T ) )  =  ( ( S  x.  1 )  -  ( S  x.  T
) ) )
121100oveq1d 6055 . . . . . . 7  |-  ( ( S  =/=  0  /\  ( T  e.  ( 0 [,] 1 )  /\  S  e.  ( 0 [,] 1 ) ) )  ->  (
( S  x.  1 )  -  ( S  x.  T ) )  =  ( S  -  ( S  x.  T
) ) )
122120, 121eqtrd 2436 . . . . . 6  |-  ( ( S  =/=  0  /\  ( T  e.  ( 0 [,] 1 )  /\  S  e.  ( 0 [,] 1 ) ) )  ->  ( S  x.  ( 1  -  T ) )  =  ( S  -  ( S  x.  T
) ) )
123122oveq1d 6055 . . . . 5  |-  ( ( S  =/=  0  /\  ( T  e.  ( 0 [,] 1 )  /\  S  e.  ( 0 [,] 1 ) ) )  ->  (
( S  x.  (
1  -  T ) )  /  ( ( S  +  T )  -  ( S  x.  T ) ) )  =  ( ( S  -  ( S  x.  T ) )  / 
( ( S  +  T )  -  ( S  x.  T )
) ) )
12457recnd 9070 . . . . . . . 8  |-  ( ( S  =/=  0  /\  ( T  e.  ( 0 [,] 1 )  /\  S  e.  ( 0 [,] 1 ) ) )  ->  ( T  -  ( S  x.  T ) )  e.  CC )
12586, 124, 86, 78divsubdird 9785 . . . . . . 7  |-  ( ( S  =/=  0  /\  ( T  e.  ( 0 [,] 1 )  /\  S  e.  ( 0 [,] 1 ) ) )  ->  (
( ( ( S  +  T )  -  ( S  x.  T
) )  -  ( T  -  ( S  x.  T ) ) )  /  ( ( S  +  T )  -  ( S  x.  T
) ) )  =  ( ( ( ( S  +  T )  -  ( S  x.  T ) )  / 
( ( S  +  T )  -  ( S  x.  T )
) )  -  (
( T  -  ( S  x.  T )
)  /  ( ( S  +  T )  -  ( S  x.  T ) ) ) ) )
12658recnd 9070 . . . . . . . . . 10  |-  ( ( S  =/=  0  /\  ( T  e.  ( 0 [,] 1 )  /\  S  e.  ( 0 [,] 1 ) ) )  ->  ( S  +  T )  e.  CC )
12756recnd 9070 . . . . . . . . . 10  |-  ( ( S  =/=  0  /\  ( T  e.  ( 0 [,] 1 )  /\  S  e.  ( 0 [,] 1 ) ) )  ->  ( S  x.  T )  e.  CC )
128126, 66, 127nnncan2d 9402 . . . . . . . . 9  |-  ( ( S  =/=  0  /\  ( T  e.  ( 0 [,] 1 )  /\  S  e.  ( 0 [,] 1 ) ) )  ->  (
( ( S  +  T )  -  ( S  x.  T )
)  -  ( T  -  ( S  x.  T ) ) )  =  ( ( S  +  T )  -  T ) )
12999, 66pncand 9368 . . . . . . . . 9  |-  ( ( S  =/=  0  /\  ( T  e.  ( 0 [,] 1 )  /\  S  e.  ( 0 [,] 1 ) ) )  ->  (
( S  +  T
)  -  T )  =  S )
130128, 129eqtrd 2436 . . . . . . . 8  |-  ( ( S  =/=  0  /\  ( T  e.  ( 0 [,] 1 )  /\  S  e.  ( 0 [,] 1 ) ) )  ->  (
( ( S  +  T )  -  ( S  x.  T )
)  -  ( T  -  ( S  x.  T ) ) )  =  S )
131130oveq1d 6055 . . . . . . 7  |-  ( ( S  =/=  0  /\  ( T  e.  ( 0 [,] 1 )  /\  S  e.  ( 0 [,] 1 ) ) )  ->  (
( ( ( S  +  T )  -  ( S  x.  T
) )  -  ( T  -  ( S  x.  T ) ) )  /  ( ( S  +  T )  -  ( S  x.  T
) ) )  =  ( S  /  (
( S  +  T
)  -  ( S  x.  T ) ) ) )
13286, 78dividd 9744 . . . . . . . 8  |-  ( ( S  =/=  0  /\  ( T  e.  ( 0 [,] 1 )  /\  S  e.  ( 0 [,] 1 ) ) )  ->  (
( ( S  +  T )  -  ( S  x.  T )
)  /  ( ( S  +  T )  -  ( S  x.  T ) ) )  =  1 )
133132oveq1d 6055 . . . . . . 7  |-  ( ( S  =/=  0  /\  ( T  e.  ( 0 [,] 1 )  /\  S  e.  ( 0 [,] 1 ) ) )  ->  (
( ( ( S  +  T )  -  ( S  x.  T
) )  /  (
( S  +  T
)  -  ( S  x.  T ) ) )  -  ( ( T  -  ( S  x.  T ) )  /  ( ( S  +  T )  -  ( S  x.  T
) ) ) )  =  ( 1  -  ( ( T  -  ( S  x.  T
) )  /  (
( S  +  T
)  -  ( S  x.  T ) ) ) ) )
134125, 131, 1333eqtr3d 2444 . . . . . 6  |-  ( ( S  =/=  0  /\  ( T  e.  ( 0 [,] 1 )  /\  S  e.  ( 0 [,] 1 ) ) )  ->  ( S  /  ( ( S  +  T )  -  ( S  x.  T
) ) )  =  ( 1  -  (
( T  -  ( S  x.  T )
)  /  ( ( S  +  T )  -  ( S  x.  T ) ) ) ) )
135134oveq1d 6055 . . . . 5  |-  ( ( S  =/=  0  /\  ( T  e.  ( 0 [,] 1 )  /\  S  e.  ( 0 [,] 1 ) ) )  ->  (
( S  /  (
( S  +  T
)  -  ( S  x.  T ) ) )  x.  ( 1  -  T ) )  =  ( ( 1  -  ( ( T  -  ( S  x.  T ) )  / 
( ( S  +  T )  -  ( S  x.  T )
) ) )  x.  ( 1  -  T
) ) )
136117, 123, 1353eqtr3d 2444 . . . 4  |-  ( ( S  =/=  0  /\  ( T  e.  ( 0 [,] 1 )  /\  S  e.  ( 0 [,] 1 ) ) )  ->  (
( S  -  ( S  x.  T )
)  /  ( ( S  +  T )  -  ( S  x.  T ) ) )  =  ( ( 1  -  ( ( T  -  ( S  x.  T ) )  / 
( ( S  +  T )  -  ( S  x.  T )
) ) )  x.  ( 1  -  T
) ) )
1371, 55, 14sylancr 645 . . . . . . 7  |-  ( ( S  =/=  0  /\  ( T  e.  ( 0 [,] 1 )  /\  S  e.  ( 0 [,] 1 ) ) )  ->  (
1  -  S )  e.  RR )
138137recnd 9070 . . . . . 6  |-  ( ( S  =/=  0  /\  ( T  e.  ( 0 [,] 1 )  /\  S  e.  ( 0 [,] 1 ) ) )  ->  (
1  -  S )  e.  CC )
13966, 138, 86, 78div23d 9783 . . . . 5  |-  ( ( S  =/=  0  /\  ( T  e.  ( 0 [,] 1 )  /\  S  e.  ( 0 [,] 1 ) ) )  ->  (
( T  x.  (
1  -  S ) )  /  ( ( S  +  T )  -  ( S  x.  T ) ) )  =  ( ( T  /  ( ( S  +  T )  -  ( S  x.  T
) ) )  x.  ( 1  -  S
) ) )
140 subdi 9423 . . . . . . . . 9  |-  ( ( T  e.  CC  /\  1  e.  CC  /\  S  e.  CC )  ->  ( T  x.  ( 1  -  S ) )  =  ( ( T  x.  1 )  -  ( T  x.  S
) ) )
14120, 140mp3an2 1267 . . . . . . . 8  |-  ( ( T  e.  CC  /\  S  e.  CC )  ->  ( T  x.  (
1  -  S ) )  =  ( ( T  x.  1 )  -  ( T  x.  S ) ) )
14266, 99, 141syl2anc 643 . . . . . . 7  |-  ( ( S  =/=  0  /\  ( T  e.  ( 0 [,] 1 )  /\  S  e.  ( 0 [,] 1 ) ) )  ->  ( T  x.  ( 1  -  S ) )  =  ( ( T  x.  1 )  -  ( T  x.  S
) ) )
14366mulid1d 9061 . . . . . . . 8  |-  ( ( S  =/=  0  /\  ( T  e.  ( 0 [,] 1 )  /\  S  e.  ( 0 [,] 1 ) ) )  ->  ( T  x.  1 )  =  T )
14466, 99mulcomd 9065 . . . . . . . 8  |-  ( ( S  =/=  0  /\  ( T  e.  ( 0 [,] 1 )  /\  S  e.  ( 0 [,] 1 ) ) )  ->  ( T  x.  S )  =  ( S  x.  T ) )
145143, 144oveq12d 6058 . . . . . . 7  |-  ( ( S  =/=  0  /\  ( T  e.  ( 0 [,] 1 )  /\  S  e.  ( 0 [,] 1 ) ) )  ->  (
( T  x.  1 )  -  ( T  x.  S ) )  =  ( T  -  ( S  x.  T
) ) )
146142, 145eqtrd 2436 . . . . . 6  |-  ( ( S  =/=  0  /\  ( T  e.  ( 0 [,] 1 )  /\  S  e.  ( 0 [,] 1 ) ) )  ->  ( T  x.  ( 1  -  S ) )  =  ( T  -  ( S  x.  T
) ) )
147146oveq1d 6055 . . . . 5  |-  ( ( S  =/=  0  /\  ( T  e.  ( 0 [,] 1 )  /\  S  e.  ( 0 [,] 1 ) ) )  ->  (
( T  x.  (
1  -  S ) )  /  ( ( S  +  T )  -  ( S  x.  T ) ) )  =  ( ( T  -  ( S  x.  T ) )  / 
( ( S  +  T )  -  ( S  x.  T )
) ) )
14894recnd 9070 . . . . . . . 8  |-  ( ( S  =/=  0  /\  ( T  e.  ( 0 [,] 1 )  /\  S  e.  ( 0 [,] 1 ) ) )  ->  ( S  -  ( S  x.  T ) )  e.  CC )
14986, 148, 86, 78divsubdird 9785 . . . . . . 7  |-  ( ( S  =/=  0  /\  ( T  e.  ( 0 [,] 1 )  /\  S  e.  ( 0 [,] 1 ) ) )  ->  (
( ( ( S  +  T )  -  ( S  x.  T
) )  -  ( S  -  ( S  x.  T ) ) )  /  ( ( S  +  T )  -  ( S  x.  T
) ) )  =  ( ( ( ( S  +  T )  -  ( S  x.  T ) )  / 
( ( S  +  T )  -  ( S  x.  T )
) )  -  (
( S  -  ( S  x.  T )
)  /  ( ( S  +  T )  -  ( S  x.  T ) ) ) ) )
150126, 99, 127nnncan2d 9402 . . . . . . . . 9  |-  ( ( S  =/=  0  /\  ( T  e.  ( 0 [,] 1 )  /\  S  e.  ( 0 [,] 1 ) ) )  ->  (
( ( S  +  T )  -  ( S  x.  T )
)  -  ( S  -  ( S  x.  T ) ) )  =  ( ( S  +  T )  -  S ) )
15199, 66pncan2d 9369 . . . . . . . . 9  |-  ( ( S  =/=  0  /\  ( T  e.  ( 0 [,] 1 )  /\  S  e.  ( 0 [,] 1 ) ) )  ->  (
( S  +  T
)  -  S )  =  T )
152150, 151eqtrd 2436 . . . . . . . 8  |-  ( ( S  =/=  0  /\  ( T  e.  ( 0 [,] 1 )  /\  S  e.  ( 0 [,] 1 ) ) )  ->  (
( ( S  +  T )  -  ( S  x.  T )
)  -  ( S  -  ( S  x.  T ) ) )  =  T )
153152oveq1d 6055 . . . . . . 7  |-  ( ( S  =/=  0  /\  ( T  e.  ( 0 [,] 1 )  /\  S  e.  ( 0 [,] 1 ) ) )  ->  (
( ( ( S  +  T )  -  ( S  x.  T
) )  -  ( S  -  ( S  x.  T ) ) )  /  ( ( S  +  T )  -  ( S  x.  T
) ) )  =  ( T  /  (
( S  +  T
)  -  ( S  x.  T ) ) ) )
154132oveq1d 6055 . . . . . . 7  |-  ( ( S  =/=  0  /\  ( T  e.  ( 0 [,] 1 )  /\  S  e.  ( 0 [,] 1 ) ) )  ->  (
( ( ( S  +  T )  -  ( S  x.  T
) )  /  (
( S  +  T
)  -  ( S  x.  T ) ) )  -  ( ( S  -  ( S  x.  T ) )  /  ( ( S  +  T )  -  ( S  x.  T
) ) ) )  =  ( 1  -  ( ( S  -  ( S  x.  T
) )  /  (
( S  +  T
)  -  ( S  x.  T ) ) ) ) )
155149, 153, 1543eqtr3d 2444 . . . . . 6  |-  ( ( S  =/=  0  /\  ( T  e.  ( 0 [,] 1 )  /\  S  e.  ( 0 [,] 1 ) ) )  ->  ( T  /  ( ( S  +  T )  -  ( S  x.  T
) ) )  =  ( 1  -  (
( S  -  ( S  x.  T )
)  /  ( ( S  +  T )  -  ( S  x.  T ) ) ) ) )
156155oveq1d 6055 . . . . 5  |-  ( ( S  =/=  0  /\  ( T  e.  ( 0 [,] 1 )  /\  S  e.  ( 0 [,] 1 ) ) )  ->  (
( T  /  (
( S  +  T
)  -  ( S  x.  T ) ) )  x.  ( 1  -  S ) )  =  ( ( 1  -  ( ( S  -  ( S  x.  T ) )  / 
( ( S  +  T )  -  ( S  x.  T )
) ) )  x.  ( 1  -  S
) ) )
157139, 147, 1563eqtr3d 2444 . . . 4  |-  ( ( S  =/=  0  /\  ( T  e.  ( 0 [,] 1 )  /\  S  e.  ( 0 [,] 1 ) ) )  ->  (
( T  -  ( S  x.  T )
)  /  ( ( S  +  T )  -  ( S  x.  T ) ) )  =  ( ( 1  -  ( ( S  -  ( S  x.  T ) )  / 
( ( S  +  T )  -  ( S  x.  T )
) ) )  x.  ( 1  -  S
) ) )
15899, 66mulcomd 9065 . . . . . 6  |-  ( ( S  =/=  0  /\  ( T  e.  ( 0 [,] 1 )  /\  S  e.  ( 0 [,] 1 ) ) )  ->  ( S  x.  T )  =  ( T  x.  S ) )
159158oveq1d 6055 . . . . 5  |-  ( ( S  =/=  0  /\  ( T  e.  ( 0 [,] 1 )  /\  S  e.  ( 0 [,] 1 ) ) )  ->  (
( S  x.  T
)  /  ( ( S  +  T )  -  ( S  x.  T ) ) )  =  ( ( T  x.  S )  / 
( ( S  +  T )  -  ( S  x.  T )
) ) )
16099, 66, 86, 78div23d 9783 . . . . . 6  |-  ( ( S  =/=  0  /\  ( T  e.  ( 0 [,] 1 )  /\  S  e.  ( 0 [,] 1 ) ) )  ->  (
( S  x.  T
)  /  ( ( S  +  T )  -  ( S  x.  T ) ) )  =  ( ( S  /  ( ( S  +  T )  -  ( S  x.  T
) ) )  x.  T ) )
161134oveq1d 6055 . . . . . 6  |-  ( ( S  =/=  0  /\  ( T  e.  ( 0 [,] 1 )  /\  S  e.  ( 0 [,] 1 ) ) )  ->  (
( S  /  (
( S  +  T
)  -  ( S  x.  T ) ) )  x.  T )  =  ( ( 1  -  ( ( T  -  ( S  x.  T ) )  / 
( ( S  +  T )  -  ( S  x.  T )
) ) )  x.  T ) )
162160, 161eqtrd 2436 . . . . 5  |-  ( ( S  =/=  0  /\  ( T  e.  ( 0 [,] 1 )  /\  S  e.  ( 0 [,] 1 ) ) )  ->  (
( S  x.  T
)  /  ( ( S  +  T )  -  ( S  x.  T ) ) )  =  ( ( 1  -  ( ( T  -  ( S  x.  T ) )  / 
( ( S  +  T )  -  ( S  x.  T )
) ) )  x.  T ) )
16366, 99, 86, 78div23d 9783 . . . . . 6  |-  ( ( S  =/=  0  /\  ( T  e.  ( 0 [,] 1 )  /\  S  e.  ( 0 [,] 1 ) ) )  ->  (
( T  x.  S
)  /  ( ( S  +  T )  -  ( S  x.  T ) ) )  =  ( ( T  /  ( ( S  +  T )  -  ( S  x.  T
) ) )  x.  S ) )
164155oveq1d 6055 . . . . . 6  |-  ( ( S  =/=  0  /\  ( T  e.  ( 0 [,] 1 )  /\  S  e.  ( 0 [,] 1 ) ) )  ->  (
( T  /  (
( S  +  T
)  -  ( S  x.  T ) ) )  x.  S )  =  ( ( 1  -  ( ( S  -  ( S  x.  T ) )  / 
( ( S  +  T )  -  ( S  x.  T )
) ) )  x.  S ) )
165163, 164eqtrd 2436 . . . . 5  |-  ( ( S  =/=  0  /\  ( T  e.  ( 0 [,] 1 )  /\  S  e.  ( 0 [,] 1 ) ) )  ->  (
( T  x.  S
)  /  ( ( S  +  T )  -  ( S  x.  T ) ) )  =  ( ( 1  -  ( ( S  -  ( S  x.  T ) )  / 
( ( S  +  T )  -  ( S  x.  T )
) ) )  x.  S ) )
166159, 162, 1653eqtr3d 2444 . . . 4  |-  ( ( S  =/=  0  /\  ( T  e.  ( 0 [,] 1 )  /\  S  e.  ( 0 [,] 1 ) ) )  ->  (
( 1  -  (
( T  -  ( S  x.  T )
)  /  ( ( S  +  T )  -  ( S  x.  T ) ) ) )  x.  T )  =  ( ( 1  -  ( ( S  -  ( S  x.  T ) )  / 
( ( S  +  T )  -  ( S  x.  T )
) ) )  x.  S ) )
167 oveq2 6048 . . . . . . . 8  |-  ( r  =  ( ( T  -  ( S  x.  T ) )  / 
( ( S  +  T )  -  ( S  x.  T )
) )  ->  (
1  -  r )  =  ( 1  -  ( ( T  -  ( S  x.  T
) )  /  (
( S  +  T
)  -  ( S  x.  T ) ) ) ) )
168167oveq1d 6055 . . . . . . 7  |-  ( r  =  ( ( T  -  ( S  x.  T ) )  / 
( ( S  +  T )  -  ( S  x.  T )
) )  ->  (
( 1  -  r
)  x.  ( 1  -  T ) )  =  ( ( 1  -  ( ( T  -  ( S  x.  T ) )  / 
( ( S  +  T )  -  ( S  x.  T )
) ) )  x.  ( 1  -  T
) ) )
169168eqeq2d 2415 . . . . . 6  |-  ( r  =  ( ( T  -  ( S  x.  T ) )  / 
( ( S  +  T )  -  ( S  x.  T )
) )  ->  (
p  =  ( ( 1  -  r )  x.  ( 1  -  T ) )  <->  p  =  ( ( 1  -  ( ( T  -  ( S  x.  T
) )  /  (
( S  +  T
)  -  ( S  x.  T ) ) ) )  x.  (
1  -  T ) ) ) )
170 eqeq1 2410 . . . . . 6  |-  ( r  =  ( ( T  -  ( S  x.  T ) )  / 
( ( S  +  T )  -  ( S  x.  T )
) )  ->  (
r  =  ( ( 1  -  p )  x.  ( 1  -  S ) )  <->  ( ( T  -  ( S  x.  T ) )  / 
( ( S  +  T )  -  ( S  x.  T )
) )  =  ( ( 1  -  p
)  x.  ( 1  -  S ) ) ) )
171167oveq1d 6055 . . . . . . 7  |-  ( r  =  ( ( T  -  ( S  x.  T ) )  / 
( ( S  +  T )  -  ( S  x.  T )
) )  ->  (
( 1  -  r
)  x.  T )  =  ( ( 1  -  ( ( T  -  ( S  x.  T ) )  / 
( ( S  +  T )  -  ( S  x.  T )
) ) )  x.  T ) )
172171eqeq1d 2412 . . . . . 6  |-  ( r  =  ( ( T  -  ( S  x.  T ) )  / 
( ( S  +  T )  -  ( S  x.  T )
) )  ->  (
( ( 1  -  r )  x.  T
)  =  ( ( 1  -  p )  x.  S )  <->  ( (
1  -  ( ( T  -  ( S  x.  T ) )  /  ( ( S  +  T )  -  ( S  x.  T
) ) ) )  x.  T )  =  ( ( 1  -  p )  x.  S
) ) )
173169, 170, 1723anbi123d 1254 . . . . 5  |-  ( r  =  ( ( T  -  ( S  x.  T ) )  / 
( ( S  +  T )  -  ( S  x.  T )
) )  ->  (
( p  =  ( ( 1  -  r
)  x.  ( 1  -  T ) )  /\  r  =  ( ( 1  -  p
)  x.  ( 1  -  S ) )  /\  ( ( 1  -  r )  x.  T )  =  ( ( 1  -  p
)  x.  S ) )  <->  ( p  =  ( ( 1  -  ( ( T  -  ( S  x.  T
) )  /  (
( S  +  T
)  -  ( S  x.  T ) ) ) )  x.  (
1  -  T ) )  /\  ( ( T  -  ( S  x.  T ) )  /  ( ( S  +  T )  -  ( S  x.  T
) ) )  =  ( ( 1  -  p )  x.  (
1  -  S ) )  /\  ( ( 1  -  ( ( T  -  ( S  x.  T ) )  /  ( ( S  +  T )  -  ( S  x.  T
) ) ) )  x.  T )  =  ( ( 1  -  p )  x.  S
) ) ) )
174 eqeq1 2410 . . . . . 6  |-  ( p  =  ( ( S  -  ( S  x.  T ) )  / 
( ( S  +  T )  -  ( S  x.  T )
) )  ->  (
p  =  ( ( 1  -  ( ( T  -  ( S  x.  T ) )  /  ( ( S  +  T )  -  ( S  x.  T
) ) ) )  x.  ( 1  -  T ) )  <->  ( ( S  -  ( S  x.  T ) )  / 
( ( S  +  T )  -  ( S  x.  T )
) )  =  ( ( 1  -  (
( T  -  ( S  x.  T )
)  /  ( ( S  +  T )  -  ( S  x.  T ) ) ) )  x.  ( 1  -  T ) ) ) )
175 oveq2 6048 . . . . . . . 8  |-  ( p  =  ( ( S  -  ( S  x.  T ) )  / 
( ( S  +  T )  -  ( S  x.  T )
) )  ->  (
1  -  p )  =  ( 1  -  ( ( S  -  ( S  x.  T
) )  /  (
( S  +  T
)  -  ( S  x.  T ) ) ) ) )
176175oveq1d 6055 . . . . . . 7  |-  ( p  =  ( ( S  -  ( S  x.  T ) )  / 
( ( S  +  T )  -  ( S  x.  T )
) )  ->  (
( 1  -  p
)  x.  ( 1  -  S ) )  =  ( ( 1  -  ( ( S  -  ( S  x.  T ) )  / 
( ( S  +  T )  -  ( S  x.  T )
) ) )  x.  ( 1  -  S
) ) )
177176eqeq2d 2415 . . . . . 6  |-  ( p  =  ( ( S  -  ( S  x.  T ) )  / 
( ( S  +  T )  -  ( S  x.  T )
) )  ->  (
( ( T  -  ( S  x.  T
) )  /  (
( S  +  T
)  -  ( S  x.  T ) ) )  =  ( ( 1  -  p )  x.  ( 1  -  S ) )  <->  ( ( T  -  ( S  x.  T ) )  / 
( ( S  +  T )  -  ( S  x.  T )
) )  =  ( ( 1  -  (
( S  -  ( S  x.  T )
)  /  ( ( S  +  T )  -  ( S  x.  T ) ) ) )  x.  ( 1  -  S ) ) ) )
178175oveq1d 6055 . . . . . . 7  |-  ( p  =  ( ( S  -  ( S  x.  T ) )  / 
( ( S  +  T )  -  ( S  x.  T )
) )  ->  (
( 1  -  p
)  x.  S )  =  ( ( 1  -  ( ( S  -  ( S  x.  T ) )  / 
( ( S  +  T )  -  ( S  x.  T )
) ) )  x.  S ) )
179178eqeq2d 2415 . . . . . 6  |-  ( p  =  ( ( S  -  ( S  x.  T ) )  / 
( ( S  +  T )  -  ( S  x.  T )
) )  ->  (
( ( 1  -  ( ( T  -  ( S  x.  T
) )  /  (
( S  +  T
)  -  ( S  x.  T ) ) ) )  x.  T
)  =  ( ( 1  -  p )  x.  S )  <->  ( (
1  -  ( ( T  -  ( S  x.  T ) )  /  ( ( S  +  T )  -  ( S  x.  T
) ) ) )  x.  T )  =  ( ( 1  -  ( ( S  -  ( S  x.  T
) )  /  (
( S  +  T
)  -  ( S  x.  T ) ) ) )  x.  S
) ) )
180174, 177, 1793anbi123d 1254 . . . . 5  |-  ( p  =  ( ( S  -  ( S  x.  T ) )  / 
( ( S  +  T )  -  ( S  x.  T )
) )  ->  (
( p  =  ( ( 1  -  (
( T  -  ( S  x.  T )
)  /  ( ( S  +  T )  -  ( S  x.  T ) ) ) )  x.  ( 1  -  T ) )  /\  ( ( T  -  ( S  x.  T ) )  / 
( ( S  +  T )  -  ( S  x.  T )
) )  =  ( ( 1  -  p
)  x.  ( 1  -  S ) )  /\  ( ( 1  -  ( ( T  -  ( S  x.  T ) )  / 
( ( S  +  T )  -  ( S  x.  T )
) ) )  x.  T )  =  ( ( 1  -  p
)  x.  S ) )  <->  ( ( ( S  -  ( S  x.  T ) )  /  ( ( S  +  T )  -  ( S  x.  T
) ) )  =  ( ( 1  -  ( ( T  -  ( S  x.  T
) )  /  (
( S  +  T
)  -  ( S  x.  T ) ) ) )  x.  (
1  -  T ) )  /\  ( ( T  -  ( S  x.  T ) )  /  ( ( S  +  T )  -  ( S  x.  T
) ) )  =  ( ( 1  -  ( ( S  -  ( S  x.  T
) )  /  (
( S  +  T
)  -  ( S  x.  T ) ) ) )  x.  (
1  -  S ) )  /\  ( ( 1  -  ( ( T  -  ( S  x.  T ) )  /  ( ( S  +  T )  -  ( S  x.  T
) ) ) )  x.  T )  =  ( ( 1  -  ( ( S  -  ( S  x.  T
) )  /  (
( S  +  T
)  -  ( S  x.  T ) ) ) )  x.  S
) ) ) )
181173, 180rspc2ev 3020 . . . 4  |-  ( ( ( ( T  -  ( S  x.  T
) )  /  (
( S  +  T
)  -  ( S  x.  T ) ) )  e.  ( 0 [,] 1 )  /\  ( ( S  -  ( S  x.  T
) )  /  (
( S  +  T
)  -  ( S  x.  T ) ) )  e.  ( 0 [,] 1 )  /\  ( ( ( S  -  ( S  x.  T ) )  / 
( ( S  +  T )  -  ( S  x.  T )
) )  =  ( ( 1  -  (
( T  -  ( S  x.  T )
)  /  ( ( S  +  T )  -  ( S  x.  T ) ) ) )  x.  ( 1  -  T ) )  /\  ( ( T  -  ( S  x.  T ) )  / 
( ( S  +  T )  -  ( S  x.  T )
) )  =  ( ( 1  -  (
( S  -  ( S  x.  T )
)  /  ( ( S  +  T )  -  ( S  x.  T ) ) ) )  x.  ( 1  -  S ) )  /\  ( ( 1  -  ( ( T  -  ( S  x.  T ) )  / 
( ( S  +  T )  -  ( S  x.  T )
) ) )  x.  T )  =  ( ( 1  -  (
( S  -  ( S  x.  T )
)  /  ( ( S  +  T )  -  ( S  x.  T ) ) ) )  x.  S ) ) )  ->  E. r  e.  ( 0 [,] 1
) E. p  e.  ( 0 [,] 1
) ( p  =  ( ( 1  -  r )  x.  (
1  -  T ) )  /\  r  =  ( ( 1  -  p )  x.  (
1  -  S ) )  /\  ( ( 1  -  r )  x.  T )  =  ( ( 1  -  p )  x.  S
) ) )
18293, 114, 136, 157, 166, 181syl113anc 1196 . . 3  |-  ( ( S  =/=  0  /\  ( T  e.  ( 0 [,] 1 )  /\  S  e.  ( 0 [,] 1 ) ) )  ->  E. r  e.  ( 0 [,] 1
) E. p  e.  ( 0 [,] 1
) ( p  =  ( ( 1  -  r )  x.  (
1  -  T ) )  /\  r  =  ( ( 1  -  p )  x.  (
1  -  S ) )  /\  ( ( 1  -  r )  x.  T )  =  ( ( 1  -  p )  x.  S
) ) )
183182ex 424 . 2  |-  ( S  =/=  0  ->  (
( T  e.  ( 0 [,] 1 )  /\  S  e.  ( 0 [,] 1 ) )  ->  E. r  e.  ( 0 [,] 1
) E. p  e.  ( 0 [,] 1
) ( p  =  ( ( 1  -  r )  x.  (
1  -  T ) )  /\  r  =  ( ( 1  -  p )  x.  (
1  -  S ) )  /\  ( ( 1  -  r )  x.  T )  =  ( ( 1  -  p )  x.  S
) ) ) )
18453, 183pm2.61ine 2643 1  |-  ( ( T  e.  ( 0 [,] 1 )  /\  S  e.  ( 0 [,] 1 ) )  ->  E. r  e.  ( 0 [,] 1 ) E. p  e.  ( 0 [,] 1 ) ( p  =  ( ( 1  -  r
)  x.  ( 1  -  T ) )  /\  r  =  ( ( 1  -  p
)  x.  ( 1  -  S ) )  /\  ( ( 1  -  r )  x.  T )  =  ( ( 1  -  p
)  x.  S ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    /\ w3a 936    = wceq 1649    e. wcel 1721    =/= wne 2567   E.wrex 2667   class class class wbr 4172  (class class class)co 6040   CCcc 8944   RRcr 8945   0cc0 8946   1c1 8947    + caddc 8949    x. cmul 8951    < clt 9076    <_ cle 9077    - cmin 9247    / cdiv 9633   [,]cicc 10875
This theorem is referenced by:  axpasch  25784
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660  ax-cnex 9002  ax-resscn 9003  ax-1cn 9004  ax-icn 9005  ax-addcl 9006  ax-addrcl 9007  ax-mulcl 9008  ax-mulrcl 9009  ax-mulcom 9010  ax-addass 9011  ax-mulass 9012  ax-distr 9013  ax-i2m1 9014  ax-1ne0 9015  ax-1rid 9016  ax-rnegex 9017  ax-rrecex 9018  ax-cnre 9019  ax-pre-lttri 9020  ax-pre-lttrn 9021  ax-pre-ltadd 9022  ax-pre-mulgt0 9023
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-nel 2570  df-ral 2671  df-rex 2672  df-reu 2673  df-rmo 2674  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-op 3783  df-uni 3976  df-br 4173  df-opab 4227  df-mpt 4228  df-id 4458  df-po 4463  df-so 4464  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-ov 6043  df-oprab 6044  df-mpt2 6045  df-riota 6508  df-er 6864  df-en 7069  df-dom 7070  df-sdom 7071  df-pnf 9078  df-mnf 9079  df-xr 9080  df-ltxr 9081  df-le 9082  df-sub 9249  df-neg 9250  df-div 9634  df-icc 10879
  Copyright terms: Public domain W3C validator