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Theorem axpaschlem 24963
Description: Lemma for axpasch 24964. 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 9639 . . . . . . . 8  |-  1  e.  RR
2 0re 9640 . . . . . . . . . . 11  |-  0  e.  RR
32, 1elicc2i 11697 . . . . . . . . . 10  |-  ( T  e.  ( 0 [,] 1 )  <->  ( T  e.  RR  /\  0  <_  T  /\  T  <_  1
) )
43simp1bi 1022 . . . . . . . . 9  |-  ( T  e.  ( 0 [,] 1 )  ->  T  e.  RR )
54ad2antrl 733 . . . . . . . 8  |-  ( ( S  =  0  /\  ( T  e.  ( 0 [,] 1 )  /\  S  e.  ( 0 [,] 1 ) ) )  ->  T  e.  RR )
6 resubcl 9935 . . . . . . . 8  |-  ( ( 1  e.  RR  /\  T  e.  RR )  ->  ( 1  -  T
)  e.  RR )
71, 5, 6sylancr 668 . . . . . . 7  |-  ( ( S  =  0  /\  ( T  e.  ( 0 [,] 1 )  /\  S  e.  ( 0 [,] 1 ) ) )  ->  (
1  -  T )  e.  RR )
87recnd 9666 . . . . . 6  |-  ( ( S  =  0  /\  ( T  e.  ( 0 [,] 1 )  /\  S  e.  ( 0 [,] 1 ) ) )  ->  (
1  -  T )  e.  CC )
98mul02d 9828 . . . . 5  |-  ( ( S  =  0  /\  ( T  e.  ( 0 [,] 1 )  /\  S  e.  ( 0 [,] 1 ) ) )  ->  (
0  x.  ( 1  -  T ) )  =  0 )
109eqcomd 2456 . . . 4  |-  ( ( S  =  0  /\  ( T  e.  ( 0 [,] 1 )  /\  S  e.  ( 0 [,] 1 ) ) )  ->  0  =  ( 0  x.  ( 1  -  T
) ) )
112, 1elicc2i 11697 . . . . . . . . . 10  |-  ( S  e.  ( 0 [,] 1 )  <->  ( S  e.  RR  /\  0  <_  S  /\  S  <_  1
) )
1211simp1bi 1022 . . . . . . . . 9  |-  ( S  e.  ( 0 [,] 1 )  ->  S  e.  RR )
1312ad2antll 734 . . . . . . . 8  |-  ( ( S  =  0  /\  ( T  e.  ( 0 [,] 1 )  /\  S  e.  ( 0 [,] 1 ) ) )  ->  S  e.  RR )
14 resubcl 9935 . . . . . . . 8  |-  ( ( 1  e.  RR  /\  S  e.  RR )  ->  ( 1  -  S
)  e.  RR )
151, 13, 14sylancr 668 . . . . . . 7  |-  ( ( S  =  0  /\  ( T  e.  ( 0 [,] 1 )  /\  S  e.  ( 0 [,] 1 ) ) )  ->  (
1  -  S )  e.  RR )
1615recnd 9666 . . . . . 6  |-  ( ( S  =  0  /\  ( T  e.  ( 0 [,] 1 )  /\  S  e.  ( 0 [,] 1 ) ) )  ->  (
1  -  S )  e.  CC )
1716mulid2d 9658 . . . . 5  |-  ( ( S  =  0  /\  ( T  e.  ( 0 [,] 1 )  /\  S  e.  ( 0 [,] 1 ) ) )  ->  (
1  x.  ( 1  -  S ) )  =  ( 1  -  S ) )
18 oveq2 6296 . . . . . . 7  |-  ( S  =  0  ->  (
1  -  S )  =  ( 1  -  0 ) )
1918adantr 467 . . . . . 6  |-  ( ( S  =  0  /\  ( T  e.  ( 0 [,] 1 )  /\  S  e.  ( 0 [,] 1 ) ) )  ->  (
1  -  S )  =  ( 1  -  0 ) )
20 1m0e1 10717 . . . . . 6  |-  ( 1  -  0 )  =  1
2119, 20syl6eq 2500 . . . . 5  |-  ( ( S  =  0  /\  ( T  e.  ( 0 [,] 1 )  /\  S  e.  ( 0 [,] 1 ) ) )  ->  (
1  -  S )  =  1 )
2217, 21eqtr2d 2485 . . . 4  |-  ( ( S  =  0  /\  ( T  e.  ( 0 [,] 1 )  /\  S  e.  ( 0 [,] 1 ) ) )  ->  1  =  ( 1  x.  ( 1  -  S
) ) )
235recnd 9666 . . . . . 6  |-  ( ( S  =  0  /\  ( T  e.  ( 0 [,] 1 )  /\  S  e.  ( 0 [,] 1 ) ) )  ->  T  e.  CC )
2423mul02d 9828 . . . . 5  |-  ( ( S  =  0  /\  ( T  e.  ( 0 [,] 1 )  /\  S  e.  ( 0 [,] 1 ) ) )  ->  (
0  x.  T )  =  0 )
25 oveq2 6296 . . . . . . 7  |-  ( S  =  0  ->  (
1  x.  S )  =  ( 1  x.  0 ) )
2625adantr 467 . . . . . 6  |-  ( ( S  =  0  /\  ( T  e.  ( 0 [,] 1 )  /\  S  e.  ( 0 [,] 1 ) ) )  ->  (
1  x.  S )  =  ( 1  x.  0 ) )
27 ax-1cn 9594 . . . . . . 7  |-  1  e.  CC
2827mul01i 9820 . . . . . 6  |-  ( 1  x.  0 )  =  0
2926, 28syl6eq 2500 . . . . 5  |-  ( ( S  =  0  /\  ( T  e.  ( 0 [,] 1 )  /\  S  e.  ( 0 [,] 1 ) ) )  ->  (
1  x.  S )  =  0 )
3024, 29eqtr4d 2487 . . . 4  |-  ( ( S  =  0  /\  ( T  e.  ( 0 [,] 1 )  /\  S  e.  ( 0 [,] 1 ) ) )  ->  (
0  x.  T )  =  ( 1  x.  S ) )
31 1elunit 11748 . . . . 5  |-  1  e.  ( 0 [,] 1
)
32 0elunit 11747 . . . . 5  |-  0  e.  ( 0 [,] 1
)
33 oveq2 6296 . . . . . . . . . 10  |-  ( r  =  1  ->  (
1  -  r )  =  ( 1  -  1 ) )
34 1m1e0 10675 . . . . . . . . . 10  |-  ( 1  -  1 )  =  0
3533, 34syl6eq 2500 . . . . . . . . 9  |-  ( r  =  1  ->  (
1  -  r )  =  0 )
3635oveq1d 6303 . . . . . . . 8  |-  ( r  =  1  ->  (
( 1  -  r
)  x.  ( 1  -  T ) )  =  ( 0  x.  ( 1  -  T
) ) )
3736eqeq2d 2460 . . . . . . 7  |-  ( r  =  1  ->  (
p  =  ( ( 1  -  r )  x.  ( 1  -  T ) )  <->  p  =  ( 0  x.  (
1  -  T ) ) ) )
38 eqeq1 2454 . . . . . . 7  |-  ( r  =  1  ->  (
r  =  ( ( 1  -  p )  x.  ( 1  -  S ) )  <->  1  =  ( ( 1  -  p )  x.  (
1  -  S ) ) ) )
3935oveq1d 6303 . . . . . . . 8  |-  ( r  =  1  ->  (
( 1  -  r
)  x.  T )  =  ( 0  x.  T ) )
4039eqeq1d 2452 . . . . . . 7  |-  ( r  =  1  ->  (
( ( 1  -  r )  x.  T
)  =  ( ( 1  -  p )  x.  S )  <->  ( 0  x.  T )  =  ( ( 1  -  p )  x.  S
) ) )
4137, 38, 403anbi123d 1338 . . . . . 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 2454 . . . . . . 7  |-  ( p  =  0  ->  (
p  =  ( 0  x.  ( 1  -  T ) )  <->  0  =  ( 0  x.  (
1  -  T ) ) ) )
43 oveq2 6296 . . . . . . . . . 10  |-  ( p  =  0  ->  (
1  -  p )  =  ( 1  -  0 ) )
4443, 20syl6eq 2500 . . . . . . . . 9  |-  ( p  =  0  ->  (
1  -  p )  =  1 )
4544oveq1d 6303 . . . . . . . 8  |-  ( p  =  0  ->  (
( 1  -  p
)  x.  ( 1  -  S ) )  =  ( 1  x.  ( 1  -  S
) ) )
4645eqeq2d 2460 . . . . . . 7  |-  ( p  =  0  ->  (
1  =  ( ( 1  -  p )  x.  ( 1  -  S ) )  <->  1  =  ( 1  x.  (
1  -  S ) ) ) )
4744oveq1d 6303 . . . . . . . 8  |-  ( p  =  0  ->  (
( 1  -  p
)  x.  S )  =  ( 1  x.  S ) )
4847eqeq2d 2460 . . . . . . 7  |-  ( p  =  0  ->  (
( 0  x.  T
)  =  ( ( 1  -  p )  x.  S )  <->  ( 0  x.  T )  =  ( 1  x.  S
) ) )
4942, 46, 483anbi123d 1338 . . . . . 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 3160 . . . . 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 1353 . . . 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, 22, 30, 51syl3anc 1267 . . 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 436 . 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 733 . . . . . . 7  |-  ( ( S  =/=  0  /\  ( T  e.  ( 0 [,] 1 )  /\  S  e.  ( 0 [,] 1 ) ) )  ->  T  e.  RR )
5512ad2antll 734 . . . . . . . 8  |-  ( ( S  =/=  0  /\  ( T  e.  ( 0 [,] 1 )  /\  S  e.  ( 0 [,] 1 ) ) )  ->  S  e.  RR )
5655, 54remulcld 9668 . . . . . . 7  |-  ( ( S  =/=  0  /\  ( T  e.  ( 0 [,] 1 )  /\  S  e.  ( 0 [,] 1 ) ) )  ->  ( S  x.  T )  e.  RR )
5754, 56resubcld 10044 . . . . . 6  |-  ( ( S  =/=  0  /\  ( T  e.  ( 0 [,] 1 )  /\  S  e.  ( 0 [,] 1 ) ) )  ->  ( T  -  ( S  x.  T ) )  e.  RR )
5855, 54readdcld 9667 . . . . . . 7  |-  ( ( S  =/=  0  /\  ( T  e.  ( 0 [,] 1 )  /\  S  e.  ( 0 [,] 1 ) ) )  ->  ( S  +  T )  e.  RR )
5958, 56resubcld 10044 . . . . . 6  |-  ( ( S  =/=  0  /\  ( T  e.  ( 0 [,] 1 )  /\  S  e.  ( 0 [,] 1 ) ) )  ->  (
( S  +  T
)  -  ( S  x.  T ) )  e.  RR )
60 1red 9655 . . . . . . . . . . 11  |-  ( ( S  =/=  0  /\  ( T  e.  ( 0 [,] 1 )  /\  S  e.  ( 0 [,] 1 ) ) )  ->  1  e.  RR )
613simp2bi 1023 . . . . . . . . . . . 12  |-  ( T  e.  ( 0 [,] 1 )  ->  0  <_  T )
6261ad2antrl 733 . . . . . . . . . . 11  |-  ( ( S  =/=  0  /\  ( T  e.  ( 0 [,] 1 )  /\  S  e.  ( 0 [,] 1 ) ) )  ->  0  <_  T )
6311simp3bi 1024 . . . . . . . . . . . 12  |-  ( S  e.  ( 0 [,] 1 )  ->  S  <_  1 )
6463ad2antll 734 . . . . . . . . . . 11  |-  ( ( S  =/=  0  /\  ( T  e.  ( 0 [,] 1 )  /\  S  e.  ( 0 [,] 1 ) ) )  ->  S  <_  1 )
6555, 60, 54, 62, 64lemul1ad 10543 . . . . . . . . . 10  |-  ( ( S  =/=  0  /\  ( T  e.  ( 0 [,] 1 )  /\  S  e.  ( 0 [,] 1 ) ) )  ->  ( S  x.  T )  <_  ( 1  x.  T
) )
6654recnd 9666 . . . . . . . . . . 11  |-  ( ( S  =/=  0  /\  ( T  e.  ( 0 [,] 1 )  /\  S  e.  ( 0 [,] 1 ) ) )  ->  T  e.  CC )
6766mulid2d 9658 . . . . . . . . . 10  |-  ( ( S  =/=  0  /\  ( T  e.  ( 0 [,] 1 )  /\  S  e.  ( 0 [,] 1 ) ) )  ->  (
1  x.  T )  =  T )
6865, 67breqtrd 4426 . . . . . . . . 9  |-  ( ( S  =/=  0  /\  ( T  e.  ( 0 [,] 1 )  /\  S  e.  ( 0 [,] 1 ) ) )  ->  ( S  x.  T )  <_  T )
6911simp2bi 1023 . . . . . . . . . . . 12  |-  ( S  e.  ( 0 [,] 1 )  ->  0  <_  S )
7069ad2antll 734 . . . . . . . . . . 11  |-  ( ( S  =/=  0  /\  ( T  e.  ( 0 [,] 1 )  /\  S  e.  ( 0 [,] 1 ) ) )  ->  0  <_  S )
71 simpl 459 . . . . . . . . . . 11  |-  ( ( S  =/=  0  /\  ( T  e.  ( 0 [,] 1 )  /\  S  e.  ( 0 [,] 1 ) ) )  ->  S  =/=  0 )
7255, 70, 71ne0gt0d 9769 . . . . . . . . . 10  |-  ( ( S  =/=  0  /\  ( T  e.  ( 0 [,] 1 )  /\  S  e.  ( 0 [,] 1 ) ) )  ->  0  <  S )
7355, 54ltaddpos2d 10195 . . . . . . . . . 10  |-  ( ( S  =/=  0  /\  ( T  e.  ( 0 [,] 1 )  /\  S  e.  ( 0 [,] 1 ) ) )  ->  (
0  <  S  <->  T  <  ( S  +  T ) ) )
7472, 73mpbid 214 . . . . . . . . 9  |-  ( ( S  =/=  0  /\  ( T  e.  ( 0 [,] 1 )  /\  S  e.  ( 0 [,] 1 ) ) )  ->  T  <  ( S  +  T
) )
7556, 54, 58, 68, 74lelttrd 9790 . . . . . . . 8  |-  ( ( S  =/=  0  /\  ( T  e.  ( 0 [,] 1 )  /\  S  e.  ( 0 [,] 1 ) ) )  ->  ( S  x.  T )  <  ( S  +  T
) )
7656, 58posdifd 10197 . . . . . . . 8  |-  ( ( S  =/=  0  /\  ( T  e.  ( 0 [,] 1 )  /\  S  e.  ( 0 [,] 1 ) ) )  ->  (
( S  x.  T
)  <  ( S  +  T )  <->  0  <  ( ( S  +  T
)  -  ( S  x.  T ) ) ) )
7775, 76mpbid 214 . . . . . . 7  |-  ( ( S  =/=  0  /\  ( T  e.  ( 0 [,] 1 )  /\  S  e.  ( 0 [,] 1 ) ) )  ->  0  <  ( ( S  +  T )  -  ( S  x.  T )
) )
7877gt0ne0d 10175 . . . . . 6  |-  ( ( S  =/=  0  /\  ( T  e.  ( 0 [,] 1 )  /\  S  e.  ( 0 [,] 1 ) ) )  ->  (
( S  +  T
)  -  ( S  x.  T ) )  =/=  0 )
7957, 59, 78redivcld 10432 . . . . 5  |-  ( ( S  =/=  0  /\  ( T  e.  ( 0 [,] 1 )  /\  S  e.  ( 0 [,] 1 ) ) )  ->  (
( T  -  ( S  x.  T )
)  /  ( ( S  +  T )  -  ( S  x.  T ) ) )  e.  RR )
8054, 56subge0d 10200 . . . . . . 7  |-  ( ( S  =/=  0  /\  ( T  e.  ( 0 [,] 1 )  /\  S  e.  ( 0 [,] 1 ) ) )  ->  (
0  <_  ( T  -  ( S  x.  T ) )  <->  ( S  x.  T )  <_  T
) )
8168, 80mpbird 236 . . . . . 6  |-  ( ( S  =/=  0  /\  ( T  e.  ( 0 [,] 1 )  /\  S  e.  ( 0 [,] 1 ) ) )  ->  0  <_  ( T  -  ( S  x.  T )
) )
82 divge0 10471 . . . . . 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 1268 . . . . 5  |-  ( ( S  =/=  0  /\  ( T  e.  ( 0 [,] 1 )  /\  S  e.  ( 0 [,] 1 ) ) )  ->  0  <_  ( ( T  -  ( S  x.  T
) )  /  (
( S  +  T
)  -  ( S  x.  T ) ) ) )
8454, 58, 74ltled 9780 . . . . . . . 8  |-  ( ( S  =/=  0  /\  ( T  e.  ( 0 [,] 1 )  /\  S  e.  ( 0 [,] 1 ) ) )  ->  T  <_  ( S  +  T
) )
8554, 58, 56, 84lesub1dd 10226 . . . . . . 7  |-  ( ( S  =/=  0  /\  ( T  e.  ( 0 [,] 1 )  /\  S  e.  ( 0 [,] 1 ) ) )  ->  ( T  -  ( S  x.  T ) )  <_ 
( ( S  +  T )  -  ( S  x.  T )
) )
8659recnd 9666 . . . . . . . 8  |-  ( ( S  =/=  0  /\  ( T  e.  ( 0 [,] 1 )  /\  S  e.  ( 0 [,] 1 ) ) )  ->  (
( S  +  T
)  -  ( S  x.  T ) )  e.  CC )
8786mulid2d 9658 . . . . . . 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 4428 . . . . . 6  |-  ( ( S  =/=  0  /\  ( T  e.  ( 0 [,] 1 )  /\  S  e.  ( 0 [,] 1 ) ) )  ->  ( T  -  ( S  x.  T ) )  <_ 
( 1  x.  (
( S  +  T
)  -  ( S  x.  T ) ) ) )
89 ledivmul2 10481 . . . . . . 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 1271 . . . . . 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 236 . . . . 5  |-  ( ( S  =/=  0  /\  ( T  e.  ( 0 [,] 1 )  /\  S  e.  ( 0 [,] 1 ) ) )  ->  (
( T  -  ( S  x.  T )
)  /  ( ( S  +  T )  -  ( S  x.  T ) ) )  <_  1 )
922, 1elicc2i 11697 . . . . 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 1191 . . . 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 10044 . . . . . 6  |-  ( ( S  =/=  0  /\  ( T  e.  ( 0 [,] 1 )  /\  S  e.  ( 0 [,] 1 ) ) )  ->  ( S  -  ( S  x.  T ) )  e.  RR )
9594, 59, 78redivcld 10432 . . . . 5  |-  ( ( S  =/=  0  /\  ( T  e.  ( 0 [,] 1 )  /\  S  e.  ( 0 [,] 1 ) ) )  ->  (
( S  -  ( S  x.  T )
)  /  ( ( S  +  T )  -  ( S  x.  T ) ) )  e.  RR )
963simp3bi 1024 . . . . . . . . . 10  |-  ( T  e.  ( 0 [,] 1 )  ->  T  <_  1 )
9796ad2antrl 733 . . . . . . . . 9  |-  ( ( S  =/=  0  /\  ( T  e.  ( 0 [,] 1 )  /\  S  e.  ( 0 [,] 1 ) ) )  ->  T  <_  1 )
9854, 60, 55, 70, 97lemul2ad 10544 . . . . . . . 8  |-  ( ( S  =/=  0  /\  ( T  e.  ( 0 [,] 1 )  /\  S  e.  ( 0 [,] 1 ) ) )  ->  ( S  x.  T )  <_  ( S  x.  1 ) )
9955recnd 9666 . . . . . . . . 9  |-  ( ( S  =/=  0  /\  ( T  e.  ( 0 [,] 1 )  /\  S  e.  ( 0 [,] 1 ) ) )  ->  S  e.  CC )
10099mulid1d 9657 . . . . . . . 8  |-  ( ( S  =/=  0  /\  ( T  e.  ( 0 [,] 1 )  /\  S  e.  ( 0 [,] 1 ) ) )  ->  ( S  x.  1 )  =  S )
10198, 100breqtrd 4426 . . . . . . 7  |-  ( ( S  =/=  0  /\  ( T  e.  ( 0 [,] 1 )  /\  S  e.  ( 0 [,] 1 ) ) )  ->  ( S  x.  T )  <_  S )
10255, 56subge0d 10200 . . . . . . 7  |-  ( ( S  =/=  0  /\  ( T  e.  ( 0 [,] 1 )  /\  S  e.  ( 0 [,] 1 ) ) )  ->  (
0  <_  ( S  -  ( S  x.  T ) )  <->  ( S  x.  T )  <_  S
) )
103101, 102mpbird 236 . . . . . 6  |-  ( ( S  =/=  0  /\  ( T  e.  ( 0 [,] 1 )  /\  S  e.  ( 0 [,] 1 ) ) )  ->  0  <_  ( S  -  ( S  x.  T )
) )
104 divge0 10471 . . . . . 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 1268 . . . . 5  |-  ( ( S  =/=  0  /\  ( T  e.  ( 0 [,] 1 )  /\  S  e.  ( 0 [,] 1 ) ) )  ->  0  <_  ( ( S  -  ( S  x.  T
) )  /  (
( S  +  T
)  -  ( S  x.  T ) ) ) )
10655, 54addge01d 10198 . . . . . . . . 9  |-  ( ( S  =/=  0  /\  ( T  e.  ( 0 [,] 1 )  /\  S  e.  ( 0 [,] 1 ) ) )  ->  (
0  <_  T  <->  S  <_  ( S  +  T ) ) )
10762, 106mpbid 214 . . . . . . . 8  |-  ( ( S  =/=  0  /\  ( T  e.  ( 0 [,] 1 )  /\  S  e.  ( 0 [,] 1 ) ) )  ->  S  <_  ( S  +  T
) )
10855, 58, 56, 107lesub1dd 10226 . . . . . . 7  |-  ( ( S  =/=  0  /\  ( T  e.  ( 0 [,] 1 )  /\  S  e.  ( 0 [,] 1 ) ) )  ->  ( S  -  ( S  x.  T ) )  <_ 
( ( S  +  T )  -  ( S  x.  T )
) )
109108, 87breqtrrd 4428 . . . . . 6  |-  ( ( S  =/=  0  /\  ( T  e.  ( 0 [,] 1 )  /\  S  e.  ( 0 [,] 1 ) ) )  ->  ( S  -  ( S  x.  T ) )  <_ 
( 1  x.  (
( S  +  T
)  -  ( S  x.  T ) ) ) )
110 ledivmul2 10481 . . . . . . 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 1271 . . . . . 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 236 . . . . 5  |-  ( ( S  =/=  0  /\  ( T  e.  ( 0 [,] 1 )  /\  S  e.  ( 0 [,] 1 ) ) )  ->  (
( S  -  ( S  x.  T )
)  /  ( ( S  +  T )  -  ( S  x.  T ) ) )  <_  1 )
1132, 1elicc2i 11697 . . . . 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 1191 . . . 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 668 . . . . . . 7  |-  ( ( S  =/=  0  /\  ( T  e.  ( 0 [,] 1 )  /\  S  e.  ( 0 [,] 1 ) ) )  ->  (
1  -  T )  e.  RR )
116115recnd 9666 . . . . . 6  |-  ( ( S  =/=  0  /\  ( T  e.  ( 0 [,] 1 )  /\  S  e.  ( 0 [,] 1 ) ) )  ->  (
1  -  T )  e.  CC )
11799, 116, 86, 78div23d 10417 . . . . 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 10049 . . . . . . . . 9  |-  ( ( S  e.  CC  /\  1  e.  CC  /\  T  e.  CC )  ->  ( S  x.  ( 1  -  T ) )  =  ( ( S  x.  1 )  -  ( S  x.  T
) ) )
11927, 118mp3an2 1351 . . . . . . . 8  |-  ( ( S  e.  CC  /\  T  e.  CC )  ->  ( S  x.  (
1  -  T ) )  =  ( ( S  x.  1 )  -  ( S  x.  T ) ) )
12099, 66, 119syl2anc 666 . . . . . . 7  |-  ( ( S  =/=  0  /\  ( T  e.  ( 0 [,] 1 )  /\  S  e.  ( 0 [,] 1 ) ) )  ->  ( S  x.  ( 1  -  T ) )  =  ( ( S  x.  1 )  -  ( S  x.  T
) ) )
121100oveq1d 6303 . . . . . . 7  |-  ( ( S  =/=  0  /\  ( T  e.  ( 0 [,] 1 )  /\  S  e.  ( 0 [,] 1 ) ) )  ->  (
( S  x.  1 )  -  ( S  x.  T ) )  =  ( S  -  ( S  x.  T
) ) )
122120, 121eqtrd 2484 . . . . . 6  |-  ( ( S  =/=  0  /\  ( T  e.  ( 0 [,] 1 )  /\  S  e.  ( 0 [,] 1 ) ) )  ->  ( S  x.  ( 1  -  T ) )  =  ( S  -  ( S  x.  T
) ) )
123122oveq1d 6303 . . . . 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 9666 . . . . . . . 8  |-  ( ( S  =/=  0  /\  ( T  e.  ( 0 [,] 1 )  /\  S  e.  ( 0 [,] 1 ) ) )  ->  ( T  -  ( S  x.  T ) )  e.  CC )
12586, 124, 86, 78divsubdird 10419 . . . . . . 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 9666 . . . . . . . . . 10  |-  ( ( S  =/=  0  /\  ( T  e.  ( 0 [,] 1 )  /\  S  e.  ( 0 [,] 1 ) ) )  ->  ( S  +  T )  e.  CC )
12756recnd 9666 . . . . . . . . . 10  |-  ( ( S  =/=  0  /\  ( T  e.  ( 0 [,] 1 )  /\  S  e.  ( 0 [,] 1 ) ) )  ->  ( S  x.  T )  e.  CC )
128126, 66, 127nnncan2d 10018 . . . . . . . . 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 9984 . . . . . . . . 9  |-  ( ( S  =/=  0  /\  ( T  e.  ( 0 [,] 1 )  /\  S  e.  ( 0 [,] 1 ) ) )  ->  (
( S  +  T
)  -  T )  =  S )
130128, 129eqtrd 2484 . . . . . . . 8  |-  ( ( S  =/=  0  /\  ( T  e.  ( 0 [,] 1 )  /\  S  e.  ( 0 [,] 1 ) ) )  ->  (
( ( S  +  T )  -  ( S  x.  T )
)  -  ( T  -  ( S  x.  T ) ) )  =  S )
131130oveq1d 6303 . . . . . . 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 10378 . . . . . . . 8  |-  ( ( S  =/=  0  /\  ( T  e.  ( 0 [,] 1 )  /\  S  e.  ( 0 [,] 1 ) ) )  ->  (
( ( S  +  T )  -  ( S  x.  T )
)  /  ( ( S  +  T )  -  ( S  x.  T ) ) )  =  1 )
133132oveq1d 6303 . . . . . . 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 2492 . . . . . 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 6303 . . . . 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 2492 . . . 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 668 . . . . . . 7  |-  ( ( S  =/=  0  /\  ( T  e.  ( 0 [,] 1 )  /\  S  e.  ( 0 [,] 1 ) ) )  ->  (
1  -  S )  e.  RR )
138137recnd 9666 . . . . . 6  |-  ( ( S  =/=  0  /\  ( T  e.  ( 0 [,] 1 )  /\  S  e.  ( 0 [,] 1 ) ) )  ->  (
1  -  S )  e.  CC )
13966, 138, 86, 78div23d 10417 . . . . 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 10049 . . . . . . . . 9  |-  ( ( T  e.  CC  /\  1  e.  CC  /\  S  e.  CC )  ->  ( T  x.  ( 1  -  S ) )  =  ( ( T  x.  1 )  -  ( T  x.  S
) ) )
14127, 140mp3an2 1351 . . . . . . . 8  |-  ( ( T  e.  CC  /\  S  e.  CC )  ->  ( T  x.  (
1  -  S ) )  =  ( ( T  x.  1 )  -  ( T  x.  S ) ) )
14266, 99, 141syl2anc 666 . . . . . . 7  |-  ( ( S  =/=  0  /\  ( T  e.  ( 0 [,] 1 )  /\  S  e.  ( 0 [,] 1 ) ) )  ->  ( T  x.  ( 1  -  S ) )  =  ( ( T  x.  1 )  -  ( T  x.  S
) ) )
14366mulid1d 9657 . . . . . . . 8  |-  ( ( S  =/=  0  /\  ( T  e.  ( 0 [,] 1 )  /\  S  e.  ( 0 [,] 1 ) ) )  ->  ( T  x.  1 )  =  T )
14466, 99mulcomd 9661 . . . . . . . 8  |-  ( ( S  =/=  0  /\  ( T  e.  ( 0 [,] 1 )  /\  S  e.  ( 0 [,] 1 ) ) )  ->  ( T  x.  S )  =  ( S  x.  T ) )
145143, 144oveq12d 6306 . . . . . . 7  |-  ( ( S  =/=  0  /\  ( T  e.  ( 0 [,] 1 )  /\  S  e.  ( 0 [,] 1 ) ) )  ->  (
( T  x.  1 )  -  ( T  x.  S ) )  =  ( T  -  ( S  x.  T
) ) )
146142, 145eqtrd 2484 . . . . . 6  |-  ( ( S  =/=  0  /\  ( T  e.  ( 0 [,] 1 )  /\  S  e.  ( 0 [,] 1 ) ) )  ->  ( T  x.  ( 1  -  S ) )  =  ( T  -  ( S  x.  T
) ) )
147146oveq1d 6303 . . . . 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 9666 . . . . . . . 8  |-  ( ( S  =/=  0  /\  ( T  e.  ( 0 [,] 1 )  /\  S  e.  ( 0 [,] 1 ) ) )  ->  ( S  -  ( S  x.  T ) )  e.  CC )
14986, 148, 86, 78divsubdird 10419 . . . . . . 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 10018 . . . . . . . . 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 9985 . . . . . . . . 9  |-  ( ( S  =/=  0  /\  ( T  e.  ( 0 [,] 1 )  /\  S  e.  ( 0 [,] 1 ) ) )  ->  (
( S  +  T
)  -  S )  =  T )
152150, 151eqtrd 2484 . . . . . . . 8  |-  ( ( S  =/=  0  /\  ( T  e.  ( 0 [,] 1 )  /\  S  e.  ( 0 [,] 1 ) ) )  ->  (
( ( S  +  T )  -  ( S  x.  T )
)  -  ( S  -  ( S  x.  T ) ) )  =  T )
153152oveq1d 6303 . . . . . . 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 6303 . . . . . . 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 2492 . . . . . 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 6303 . . . . 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 2492 . . . 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 9661 . . . . . 6  |-  ( ( S  =/=  0  /\  ( T  e.  ( 0 [,] 1 )  /\  S  e.  ( 0 [,] 1 ) ) )  ->  ( S  x.  T )  =  ( T  x.  S ) )
159158oveq1d 6303 . . . . 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 10417 . . . . . 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 6303 . . . . . 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 2484 . . . . 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 10417 . . . . . 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 6303 . . . . . 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 2484 . . . . 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 2492 . . . 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 6296 . . . . . . . 8  |-  ( r  =  ( ( T  -  ( S  x.  T ) )  / 
( ( S  +  T )  -  ( S  x.  T )
) )  ->  (
1  -  r )  =  ( 1  -  ( ( T  -  ( S  x.  T
) )  /  (
( S  +  T
)  -  ( S  x.  T ) ) ) ) )
168167oveq1d 6303 . . . . . . 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 2460 . . . . . 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 2454 . . . . . 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 6303 . . . . . . 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 2452 . . . . . 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 1338 . . . . 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 2454 . . . . . 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 6296 . . . . . . . 8  |-  ( p  =  ( ( S  -  ( S  x.  T ) )  / 
( ( S  +  T )  -  ( S  x.  T )
) )  ->  (
1  -  p )  =  ( 1  -  ( ( S  -  ( S  x.  T
) )  /  (
( S  +  T
)  -  ( S  x.  T ) ) ) ) )
176175oveq1d 6303 . . . . . . 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 2460 . . . . . 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 6303 . . . . . . 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 2460 . . . . . 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 1338 . . . . 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 3160 . . . 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 1279 . . 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 436 . 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 2706 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 setvar class
Syntax hints:    -> wi 4    <-> wb 188    /\ wa 371    /\ w3a 984    = wceq 1443    e. wcel 1886    =/= wne 2621   E.wrex 2737   class class class wbr 4401  (class class class)co 6288   CCcc 9534   RRcr 9535   0cc0 9536   1c1 9537    + caddc 9539    x. cmul 9541    < clt 9672    <_ cle 9673    - cmin 9857    / cdiv 10266   [,]cicc 11635
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1668  ax-4 1681  ax-5 1757  ax-6 1804  ax-7 1850  ax-8 1888  ax-9 1895  ax-10 1914  ax-11 1919  ax-12 1932  ax-13 2090  ax-ext 2430  ax-sep 4524  ax-nul 4533  ax-pow 4580  ax-pr 4638  ax-un 6580  ax-cnex 9592  ax-resscn 9593  ax-1cn 9594  ax-icn 9595  ax-addcl 9596  ax-addrcl 9597  ax-mulcl 9598  ax-mulrcl 9599  ax-mulcom 9600  ax-addass 9601  ax-mulass 9602  ax-distr 9603  ax-i2m1 9604  ax-1ne0 9605  ax-1rid 9606  ax-rnegex 9607  ax-rrecex 9608  ax-cnre 9609  ax-pre-lttri 9610  ax-pre-lttrn 9611  ax-pre-ltadd 9612  ax-pre-mulgt0 9613
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3or 985  df-3an 986  df-tru 1446  df-ex 1663  df-nf 1667  df-sb 1797  df-eu 2302  df-mo 2303  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2580  df-ne 2623  df-nel 2624  df-ral 2741  df-rex 2742  df-reu 2743  df-rmo 2744  df-rab 2745  df-v 3046  df-sbc 3267  df-csb 3363  df-dif 3406  df-un 3408  df-in 3410  df-ss 3417  df-nul 3731  df-if 3881  df-pw 3952  df-sn 3968  df-pr 3970  df-op 3974  df-uni 4198  df-br 4402  df-opab 4461  df-mpt 4462  df-id 4748  df-po 4754  df-so 4755  df-xp 4839  df-rel 4840  df-cnv 4841  df-co 4842  df-dm 4843  df-rn 4844  df-res 4845  df-ima 4846  df-iota 5545  df-fun 5583  df-fn 5584  df-f 5585  df-f1 5586  df-fo 5587  df-f1o 5588  df-fv 5589  df-riota 6250  df-ov 6291  df-oprab 6292  df-mpt2 6293  df-er 7360  df-en 7567  df-dom 7568  df-sdom 7569  df-pnf 9674  df-mnf 9675  df-xr 9676  df-ltxr 9677  df-le 9678  df-sub 9859  df-neg 9860  df-div 10267  df-icc 11639
This theorem is referenced by:  axpasch  24964
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