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Theorem axpaschlem 23121
Description: Lemma for axpasch 23122. 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 9381 . . . . . . . 8  |-  1  e.  RR
2 0re 9382 . . . . . . . . . . 11  |-  0  e.  RR
32, 1elicc2i 11357 . . . . . . . . . 10  |-  ( T  e.  ( 0 [,] 1 )  <->  ( T  e.  RR  /\  0  <_  T  /\  T  <_  1
) )
43simp1bi 998 . . . . . . . . 9  |-  ( T  e.  ( 0 [,] 1 )  ->  T  e.  RR )
54ad2antrl 722 . . . . . . . 8  |-  ( ( S  =  0  /\  ( T  e.  ( 0 [,] 1 )  /\  S  e.  ( 0 [,] 1 ) ) )  ->  T  e.  RR )
6 resubcl 9669 . . . . . . . 8  |-  ( ( 1  e.  RR  /\  T  e.  RR )  ->  ( 1  -  T
)  e.  RR )
71, 5, 6sylancr 658 . . . . . . 7  |-  ( ( S  =  0  /\  ( T  e.  ( 0 [,] 1 )  /\  S  e.  ( 0 [,] 1 ) ) )  ->  (
1  -  T )  e.  RR )
87recnd 9408 . . . . . 6  |-  ( ( S  =  0  /\  ( T  e.  ( 0 [,] 1 )  /\  S  e.  ( 0 [,] 1 ) ) )  ->  (
1  -  T )  e.  CC )
98mul02d 9563 . . . . 5  |-  ( ( S  =  0  /\  ( T  e.  ( 0 [,] 1 )  /\  S  e.  ( 0 [,] 1 ) ) )  ->  (
0  x.  ( 1  -  T ) )  =  0 )
109eqcomd 2446 . . . 4  |-  ( ( S  =  0  /\  ( T  e.  ( 0 [,] 1 )  /\  S  e.  ( 0 [,] 1 ) ) )  ->  0  =  ( 0  x.  ( 1  -  T
) ) )
112, 1elicc2i 11357 . . . . . . . . . 10  |-  ( S  e.  ( 0 [,] 1 )  <->  ( S  e.  RR  /\  0  <_  S  /\  S  <_  1
) )
1211simp1bi 998 . . . . . . . . 9  |-  ( S  e.  ( 0 [,] 1 )  ->  S  e.  RR )
1312ad2antll 723 . . . . . . . 8  |-  ( ( S  =  0  /\  ( T  e.  ( 0 [,] 1 )  /\  S  e.  ( 0 [,] 1 ) ) )  ->  S  e.  RR )
14 resubcl 9669 . . . . . . . 8  |-  ( ( 1  e.  RR  /\  S  e.  RR )  ->  ( 1  -  S
)  e.  RR )
151, 13, 14sylancr 658 . . . . . . 7  |-  ( ( S  =  0  /\  ( T  e.  ( 0 [,] 1 )  /\  S  e.  ( 0 [,] 1 ) ) )  ->  (
1  -  S )  e.  RR )
1615recnd 9408 . . . . . 6  |-  ( ( S  =  0  /\  ( T  e.  ( 0 [,] 1 )  /\  S  e.  ( 0 [,] 1 ) ) )  ->  (
1  -  S )  e.  CC )
1716mulid2d 9400 . . . . 5  |-  ( ( S  =  0  /\  ( T  e.  ( 0 [,] 1 )  /\  S  e.  ( 0 [,] 1 ) ) )  ->  (
1  x.  ( 1  -  S ) )  =  ( 1  -  S ) )
18 oveq2 6098 . . . . . . 7  |-  ( S  =  0  ->  (
1  -  S )  =  ( 1  -  0 ) )
1918adantr 462 . . . . . 6  |-  ( ( S  =  0  /\  ( T  e.  ( 0 [,] 1 )  /\  S  e.  ( 0 [,] 1 ) ) )  ->  (
1  -  S )  =  ( 1  -  0 ) )
20 1m0e1 10428 . . . . . 6  |-  ( 1  -  0 )  =  1
2119, 20syl6eq 2489 . . . . 5  |-  ( ( S  =  0  /\  ( T  e.  ( 0 [,] 1 )  /\  S  e.  ( 0 [,] 1 ) ) )  ->  (
1  -  S )  =  1 )
2217, 21eqtr2d 2474 . . . 4  |-  ( ( S  =  0  /\  ( T  e.  ( 0 [,] 1 )  /\  S  e.  ( 0 [,] 1 ) ) )  ->  1  =  ( 1  x.  ( 1  -  S
) ) )
235recnd 9408 . . . . . 6  |-  ( ( S  =  0  /\  ( T  e.  ( 0 [,] 1 )  /\  S  e.  ( 0 [,] 1 ) ) )  ->  T  e.  CC )
2423mul02d 9563 . . . . 5  |-  ( ( S  =  0  /\  ( T  e.  ( 0 [,] 1 )  /\  S  e.  ( 0 [,] 1 ) ) )  ->  (
0  x.  T )  =  0 )
25 oveq2 6098 . . . . . . 7  |-  ( S  =  0  ->  (
1  x.  S )  =  ( 1  x.  0 ) )
2625adantr 462 . . . . . 6  |-  ( ( S  =  0  /\  ( T  e.  ( 0 [,] 1 )  /\  S  e.  ( 0 [,] 1 ) ) )  ->  (
1  x.  S )  =  ( 1  x.  0 ) )
27 ax-1cn 9336 . . . . . . 7  |-  1  e.  CC
2827mul01i 9555 . . . . . 6  |-  ( 1  x.  0 )  =  0
2926, 28syl6eq 2489 . . . . 5  |-  ( ( S  =  0  /\  ( T  e.  ( 0 [,] 1 )  /\  S  e.  ( 0 [,] 1 ) ) )  ->  (
1  x.  S )  =  0 )
3024, 29eqtr4d 2476 . . . 4  |-  ( ( S  =  0  /\  ( T  e.  ( 0 [,] 1 )  /\  S  e.  ( 0 [,] 1 ) ) )  ->  (
0  x.  T )  =  ( 1  x.  S ) )
31 1elunit 11400 . . . . 5  |-  1  e.  ( 0 [,] 1
)
32 0elunit 11399 . . . . 5  |-  0  e.  ( 0 [,] 1
)
33 oveq2 6098 . . . . . . . . . 10  |-  ( r  =  1  ->  (
1  -  r )  =  ( 1  -  1 ) )
34 1m1e0 10386 . . . . . . . . . 10  |-  ( 1  -  1 )  =  0
3533, 34syl6eq 2489 . . . . . . . . 9  |-  ( r  =  1  ->  (
1  -  r )  =  0 )
3635oveq1d 6105 . . . . . . . 8  |-  ( r  =  1  ->  (
( 1  -  r
)  x.  ( 1  -  T ) )  =  ( 0  x.  ( 1  -  T
) ) )
3736eqeq2d 2452 . . . . . . 7  |-  ( r  =  1  ->  (
p  =  ( ( 1  -  r )  x.  ( 1  -  T ) )  <->  p  =  ( 0  x.  (
1  -  T ) ) ) )
38 eqeq1 2447 . . . . . . 7  |-  ( r  =  1  ->  (
r  =  ( ( 1  -  p )  x.  ( 1  -  S ) )  <->  1  =  ( ( 1  -  p )  x.  (
1  -  S ) ) ) )
3935oveq1d 6105 . . . . . . . 8  |-  ( r  =  1  ->  (
( 1  -  r
)  x.  T )  =  ( 0  x.  T ) )
4039eqeq1d 2449 . . . . . . 7  |-  ( r  =  1  ->  (
( ( 1  -  r )  x.  T
)  =  ( ( 1  -  p )  x.  S )  <->  ( 0  x.  T )  =  ( ( 1  -  p )  x.  S
) ) )
4137, 38, 403anbi123d 1284 . . . . . 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 2447 . . . . . . 7  |-  ( p  =  0  ->  (
p  =  ( 0  x.  ( 1  -  T ) )  <->  0  =  ( 0  x.  (
1  -  T ) ) ) )
43 oveq2 6098 . . . . . . . . . 10  |-  ( p  =  0  ->  (
1  -  p )  =  ( 1  -  0 ) )
4443, 20syl6eq 2489 . . . . . . . . 9  |-  ( p  =  0  ->  (
1  -  p )  =  1 )
4544oveq1d 6105 . . . . . . . 8  |-  ( p  =  0  ->  (
( 1  -  p
)  x.  ( 1  -  S ) )  =  ( 1  x.  ( 1  -  S
) ) )
4645eqeq2d 2452 . . . . . . 7  |-  ( p  =  0  ->  (
1  =  ( ( 1  -  p )  x.  ( 1  -  S ) )  <->  1  =  ( 1  x.  (
1  -  S ) ) ) )
4744oveq1d 6105 . . . . . . . 8  |-  ( p  =  0  ->  (
( 1  -  p
)  x.  S )  =  ( 1  x.  S ) )
4847eqeq2d 2452 . . . . . . 7  |-  ( p  =  0  ->  (
( 0  x.  T
)  =  ( ( 1  -  p )  x.  S )  <->  ( 0  x.  T )  =  ( 1  x.  S
) ) )
4942, 46, 483anbi123d 1284 . . . . . 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 3078 . . . . 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 1299 . . . 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 1213 . . 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 434 . 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 722 . . . . . . 7  |-  ( ( S  =/=  0  /\  ( T  e.  ( 0 [,] 1 )  /\  S  e.  ( 0 [,] 1 ) ) )  ->  T  e.  RR )
5512ad2antll 723 . . . . . . . 8  |-  ( ( S  =/=  0  /\  ( T  e.  ( 0 [,] 1 )  /\  S  e.  ( 0 [,] 1 ) ) )  ->  S  e.  RR )
5655, 54remulcld 9410 . . . . . . 7  |-  ( ( S  =/=  0  /\  ( T  e.  ( 0 [,] 1 )  /\  S  e.  ( 0 [,] 1 ) ) )  ->  ( S  x.  T )  e.  RR )
5754, 56resubcld 9772 . . . . . 6  |-  ( ( S  =/=  0  /\  ( T  e.  ( 0 [,] 1 )  /\  S  e.  ( 0 [,] 1 ) ) )  ->  ( T  -  ( S  x.  T ) )  e.  RR )
5855, 54readdcld 9409 . . . . . . 7  |-  ( ( S  =/=  0  /\  ( T  e.  ( 0 [,] 1 )  /\  S  e.  ( 0 [,] 1 ) ) )  ->  ( S  +  T )  e.  RR )
5958, 56resubcld 9772 . . . . . 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 999 . . . . . . . . . . . 12  |-  ( T  e.  ( 0 [,] 1 )  ->  0  <_  T )
6261ad2antrl 722 . . . . . . . . . . 11  |-  ( ( S  =/=  0  /\  ( T  e.  ( 0 [,] 1 )  /\  S  e.  ( 0 [,] 1 ) ) )  ->  0  <_  T )
6311simp3bi 1000 . . . . . . . . . . . 12  |-  ( S  e.  ( 0 [,] 1 )  ->  S  <_  1 )
6463ad2antll 723 . . . . . . . . . . 11  |-  ( ( S  =/=  0  /\  ( T  e.  ( 0 [,] 1 )  /\  S  e.  ( 0 [,] 1 ) ) )  ->  S  <_  1 )
6555, 60, 54, 62, 64lemul1ad 10268 . . . . . . . . . 10  |-  ( ( S  =/=  0  /\  ( T  e.  ( 0 [,] 1 )  /\  S  e.  ( 0 [,] 1 ) ) )  ->  ( S  x.  T )  <_  ( 1  x.  T
) )
6654recnd 9408 . . . . . . . . . . 11  |-  ( ( S  =/=  0  /\  ( T  e.  ( 0 [,] 1 )  /\  S  e.  ( 0 [,] 1 ) ) )  ->  T  e.  CC )
6766mulid2d 9400 . . . . . . . . . 10  |-  ( ( S  =/=  0  /\  ( T  e.  ( 0 [,] 1 )  /\  S  e.  ( 0 [,] 1 ) ) )  ->  (
1  x.  T )  =  T )
6865, 67breqtrd 4313 . . . . . . . . 9  |-  ( ( S  =/=  0  /\  ( T  e.  ( 0 [,] 1 )  /\  S  e.  ( 0 [,] 1 ) ) )  ->  ( S  x.  T )  <_  T )
6911simp2bi 999 . . . . . . . . . . . 12  |-  ( S  e.  ( 0 [,] 1 )  ->  0  <_  S )
7069ad2antll 723 . . . . . . . . . . 11  |-  ( ( S  =/=  0  /\  ( T  e.  ( 0 [,] 1 )  /\  S  e.  ( 0 [,] 1 ) ) )  ->  0  <_  S )
71 simpl 454 . . . . . . . . . . 11  |-  ( ( S  =/=  0  /\  ( T  e.  ( 0 [,] 1 )  /\  S  e.  ( 0 [,] 1 ) ) )  ->  S  =/=  0 )
7255, 70, 71ne0gt0d 9507 . . . . . . . . . 10  |-  ( ( S  =/=  0  /\  ( T  e.  ( 0 [,] 1 )  /\  S  e.  ( 0 [,] 1 ) ) )  ->  0  <  S )
7355, 54ltaddpos2d 9920 . . . . . . . . . 10  |-  ( ( S  =/=  0  /\  ( T  e.  ( 0 [,] 1 )  /\  S  e.  ( 0 [,] 1 ) ) )  ->  (
0  <  S  <->  T  <  ( S  +  T ) ) )
7472, 73mpbid 210 . . . . . . . . 9  |-  ( ( S  =/=  0  /\  ( T  e.  ( 0 [,] 1 )  /\  S  e.  ( 0 [,] 1 ) ) )  ->  T  <  ( S  +  T
) )
7556, 54, 58, 68, 74lelttrd 9525 . . . . . . . 8  |-  ( ( S  =/=  0  /\  ( T  e.  ( 0 [,] 1 )  /\  S  e.  ( 0 [,] 1 ) ) )  ->  ( S  x.  T )  <  ( S  +  T
) )
7656, 58posdifd 9922 . . . . . . . 8  |-  ( ( S  =/=  0  /\  ( T  e.  ( 0 [,] 1 )  /\  S  e.  ( 0 [,] 1 ) ) )  ->  (
( S  x.  T
)  <  ( S  +  T )  <->  0  <  ( ( S  +  T
)  -  ( S  x.  T ) ) ) )
7775, 76mpbid 210 . . . . . . 7  |-  ( ( S  =/=  0  /\  ( T  e.  ( 0 [,] 1 )  /\  S  e.  ( 0 [,] 1 ) ) )  ->  0  <  ( ( S  +  T )  -  ( S  x.  T )
) )
7877gt0ne0d 9900 . . . . . 6  |-  ( ( S  =/=  0  /\  ( T  e.  ( 0 [,] 1 )  /\  S  e.  ( 0 [,] 1 ) ) )  ->  (
( S  +  T
)  -  ( S  x.  T ) )  =/=  0 )
7957, 59, 78redivcld 10155 . . . . 5  |-  ( ( S  =/=  0  /\  ( T  e.  ( 0 [,] 1 )  /\  S  e.  ( 0 [,] 1 ) ) )  ->  (
( T  -  ( S  x.  T )
)  /  ( ( S  +  T )  -  ( S  x.  T ) ) )  e.  RR )
8054, 56subge0d 9925 . . . . . . 7  |-  ( ( S  =/=  0  /\  ( T  e.  ( 0 [,] 1 )  /\  S  e.  ( 0 [,] 1 ) ) )  ->  (
0  <_  ( T  -  ( S  x.  T ) )  <->  ( S  x.  T )  <_  T
) )
8168, 80mpbird 232 . . . . . 6  |-  ( ( S  =/=  0  /\  ( T  e.  ( 0 [,] 1 )  /\  S  e.  ( 0 [,] 1 ) ) )  ->  0  <_  ( T  -  ( S  x.  T )
) )
82 divge0 10194 . . . . . 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 1214 . . . . 5  |-  ( ( S  =/=  0  /\  ( T  e.  ( 0 [,] 1 )  /\  S  e.  ( 0 [,] 1 ) ) )  ->  0  <_  ( ( T  -  ( S  x.  T
) )  /  (
( S  +  T
)  -  ( S  x.  T ) ) ) )
8454, 58, 74ltled 9518 . . . . . . . 8  |-  ( ( S  =/=  0  /\  ( T  e.  ( 0 [,] 1 )  /\  S  e.  ( 0 [,] 1 ) ) )  ->  T  <_  ( S  +  T
) )
8554, 58, 56, 84lesub1dd 9951 . . . . . . 7  |-  ( ( S  =/=  0  /\  ( T  e.  ( 0 [,] 1 )  /\  S  e.  ( 0 [,] 1 ) ) )  ->  ( T  -  ( S  x.  T ) )  <_ 
( ( S  +  T )  -  ( S  x.  T )
) )
8659recnd 9408 . . . . . . . 8  |-  ( ( S  =/=  0  /\  ( T  e.  ( 0 [,] 1 )  /\  S  e.  ( 0 [,] 1 ) ) )  ->  (
( S  +  T
)  -  ( S  x.  T ) )  e.  CC )
8786mulid2d 9400 . . . . . . 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 4315 . . . . . 6  |-  ( ( S  =/=  0  /\  ( T  e.  ( 0 [,] 1 )  /\  S  e.  ( 0 [,] 1 ) ) )  ->  ( T  -  ( S  x.  T ) )  <_ 
( 1  x.  (
( S  +  T
)  -  ( S  x.  T ) ) ) )
89 ledivmul2 10205 . . . . . . 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 1217 . . . . . 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 232 . . . . 5  |-  ( ( S  =/=  0  /\  ( T  e.  ( 0 [,] 1 )  /\  S  e.  ( 0 [,] 1 ) ) )  ->  (
( T  -  ( S  x.  T )
)  /  ( ( S  +  T )  -  ( S  x.  T ) ) )  <_  1 )
922, 1elicc2i 11357 . . . . 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 1167 . . . 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 9772 . . . . . 6  |-  ( ( S  =/=  0  /\  ( T  e.  ( 0 [,] 1 )  /\  S  e.  ( 0 [,] 1 ) ) )  ->  ( S  -  ( S  x.  T ) )  e.  RR )
9594, 59, 78redivcld 10155 . . . . 5  |-  ( ( S  =/=  0  /\  ( T  e.  ( 0 [,] 1 )  /\  S  e.  ( 0 [,] 1 ) ) )  ->  (
( S  -  ( S  x.  T )
)  /  ( ( S  +  T )  -  ( S  x.  T ) ) )  e.  RR )
963simp3bi 1000 . . . . . . . . . 10  |-  ( T  e.  ( 0 [,] 1 )  ->  T  <_  1 )
9796ad2antrl 722 . . . . . . . . 9  |-  ( ( S  =/=  0  /\  ( T  e.  ( 0 [,] 1 )  /\  S  e.  ( 0 [,] 1 ) ) )  ->  T  <_  1 )
9854, 60, 55, 70, 97lemul2ad 10269 . . . . . . . 8  |-  ( ( S  =/=  0  /\  ( T  e.  ( 0 [,] 1 )  /\  S  e.  ( 0 [,] 1 ) ) )  ->  ( S  x.  T )  <_  ( S  x.  1 ) )
9955recnd 9408 . . . . . . . . 9  |-  ( ( S  =/=  0  /\  ( T  e.  ( 0 [,] 1 )  /\  S  e.  ( 0 [,] 1 ) ) )  ->  S  e.  CC )
10099mulid1d 9399 . . . . . . . 8  |-  ( ( S  =/=  0  /\  ( T  e.  ( 0 [,] 1 )  /\  S  e.  ( 0 [,] 1 ) ) )  ->  ( S  x.  1 )  =  S )
10198, 100breqtrd 4313 . . . . . . 7  |-  ( ( S  =/=  0  /\  ( T  e.  ( 0 [,] 1 )  /\  S  e.  ( 0 [,] 1 ) ) )  ->  ( S  x.  T )  <_  S )
10255, 56subge0d 9925 . . . . . . 7  |-  ( ( S  =/=  0  /\  ( T  e.  ( 0 [,] 1 )  /\  S  e.  ( 0 [,] 1 ) ) )  ->  (
0  <_  ( S  -  ( S  x.  T ) )  <->  ( S  x.  T )  <_  S
) )
103101, 102mpbird 232 . . . . . 6  |-  ( ( S  =/=  0  /\  ( T  e.  ( 0 [,] 1 )  /\  S  e.  ( 0 [,] 1 ) ) )  ->  0  <_  ( S  -  ( S  x.  T )
) )
104 divge0 10194 . . . . . 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 1214 . . . . 5  |-  ( ( S  =/=  0  /\  ( T  e.  ( 0 [,] 1 )  /\  S  e.  ( 0 [,] 1 ) ) )  ->  0  <_  ( ( S  -  ( S  x.  T
) )  /  (
( S  +  T
)  -  ( S  x.  T ) ) ) )
10655, 54addge01d 9923 . . . . . . . . 9  |-  ( ( S  =/=  0  /\  ( T  e.  ( 0 [,] 1 )  /\  S  e.  ( 0 [,] 1 ) ) )  ->  (
0  <_  T  <->  S  <_  ( S  +  T ) ) )
10762, 106mpbid 210 . . . . . . . 8  |-  ( ( S  =/=  0  /\  ( T  e.  ( 0 [,] 1 )  /\  S  e.  ( 0 [,] 1 ) ) )  ->  S  <_  ( S  +  T
) )
10855, 58, 56, 107lesub1dd 9951 . . . . . . 7  |-  ( ( S  =/=  0  /\  ( T  e.  ( 0 [,] 1 )  /\  S  e.  ( 0 [,] 1 ) ) )  ->  ( S  -  ( S  x.  T ) )  <_ 
( ( S  +  T )  -  ( S  x.  T )
) )
109108, 87breqtrrd 4315 . . . . . 6  |-  ( ( S  =/=  0  /\  ( T  e.  ( 0 [,] 1 )  /\  S  e.  ( 0 [,] 1 ) ) )  ->  ( S  -  ( S  x.  T ) )  <_ 
( 1  x.  (
( S  +  T
)  -  ( S  x.  T ) ) ) )
110 ledivmul2 10205 . . . . . . 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 1217 . . . . . 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 232 . . . . 5  |-  ( ( S  =/=  0  /\  ( T  e.  ( 0 [,] 1 )  /\  S  e.  ( 0 [,] 1 ) ) )  ->  (
( S  -  ( S  x.  T )
)  /  ( ( S  +  T )  -  ( S  x.  T ) ) )  <_  1 )
1132, 1elicc2i 11357 . . . . 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 1167 . . . 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 658 . . . . . . 7  |-  ( ( S  =/=  0  /\  ( T  e.  ( 0 [,] 1 )  /\  S  e.  ( 0 [,] 1 ) ) )  ->  (
1  -  T )  e.  RR )
116115recnd 9408 . . . . . 6  |-  ( ( S  =/=  0  /\  ( T  e.  ( 0 [,] 1 )  /\  S  e.  ( 0 [,] 1 ) ) )  ->  (
1  -  T )  e.  CC )
11799, 116, 86, 78div23d 10140 . . . . 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 9774 . . . . . . . . 9  |-  ( ( S  e.  CC  /\  1  e.  CC  /\  T  e.  CC )  ->  ( S  x.  ( 1  -  T ) )  =  ( ( S  x.  1 )  -  ( S  x.  T
) ) )
11927, 118mp3an2 1297 . . . . . . . 8  |-  ( ( S  e.  CC  /\  T  e.  CC )  ->  ( S  x.  (
1  -  T ) )  =  ( ( S  x.  1 )  -  ( S  x.  T ) ) )
12099, 66, 119syl2anc 656 . . . . . . 7  |-  ( ( S  =/=  0  /\  ( T  e.  ( 0 [,] 1 )  /\  S  e.  ( 0 [,] 1 ) ) )  ->  ( S  x.  ( 1  -  T ) )  =  ( ( S  x.  1 )  -  ( S  x.  T
) ) )
121100oveq1d 6105 . . . . . . 7  |-  ( ( S  =/=  0  /\  ( T  e.  ( 0 [,] 1 )  /\  S  e.  ( 0 [,] 1 ) ) )  ->  (
( S  x.  1 )  -  ( S  x.  T ) )  =  ( S  -  ( S  x.  T
) ) )
122120, 121eqtrd 2473 . . . . . 6  |-  ( ( S  =/=  0  /\  ( T  e.  ( 0 [,] 1 )  /\  S  e.  ( 0 [,] 1 ) ) )  ->  ( S  x.  ( 1  -  T ) )  =  ( S  -  ( S  x.  T
) ) )
123122oveq1d 6105 . . . . 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 9408 . . . . . . . 8  |-  ( ( S  =/=  0  /\  ( T  e.  ( 0 [,] 1 )  /\  S  e.  ( 0 [,] 1 ) ) )  ->  ( T  -  ( S  x.  T ) )  e.  CC )
12586, 124, 86, 78divsubdird 10142 . . . . . . 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 9408 . . . . . . . . . 10  |-  ( ( S  =/=  0  /\  ( T  e.  ( 0 [,] 1 )  /\  S  e.  ( 0 [,] 1 ) ) )  ->  ( S  +  T )  e.  CC )
12756recnd 9408 . . . . . . . . . 10  |-  ( ( S  =/=  0  /\  ( T  e.  ( 0 [,] 1 )  /\  S  e.  ( 0 [,] 1 ) ) )  ->  ( S  x.  T )  e.  CC )
128126, 66, 127nnncan2d 9750 . . . . . . . . 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 9716 . . . . . . . . 9  |-  ( ( S  =/=  0  /\  ( T  e.  ( 0 [,] 1 )  /\  S  e.  ( 0 [,] 1 ) ) )  ->  (
( S  +  T
)  -  T )  =  S )
130128, 129eqtrd 2473 . . . . . . . 8  |-  ( ( S  =/=  0  /\  ( T  e.  ( 0 [,] 1 )  /\  S  e.  ( 0 [,] 1 ) ) )  ->  (
( ( S  +  T )  -  ( S  x.  T )
)  -  ( T  -  ( S  x.  T ) ) )  =  S )
131130oveq1d 6105 . . . . . . 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 10101 . . . . . . . 8  |-  ( ( S  =/=  0  /\  ( T  e.  ( 0 [,] 1 )  /\  S  e.  ( 0 [,] 1 ) ) )  ->  (
( ( S  +  T )  -  ( S  x.  T )
)  /  ( ( S  +  T )  -  ( S  x.  T ) ) )  =  1 )
133132oveq1d 6105 . . . . . . 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 2481 . . . . . 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 6105 . . . . 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 2481 . . . 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 658 . . . . . . 7  |-  ( ( S  =/=  0  /\  ( T  e.  ( 0 [,] 1 )  /\  S  e.  ( 0 [,] 1 ) ) )  ->  (
1  -  S )  e.  RR )
138137recnd 9408 . . . . . 6  |-  ( ( S  =/=  0  /\  ( T  e.  ( 0 [,] 1 )  /\  S  e.  ( 0 [,] 1 ) ) )  ->  (
1  -  S )  e.  CC )
13966, 138, 86, 78div23d 10140 . . . . 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 9774 . . . . . . . . 9  |-  ( ( T  e.  CC  /\  1  e.  CC  /\  S  e.  CC )  ->  ( T  x.  ( 1  -  S ) )  =  ( ( T  x.  1 )  -  ( T  x.  S
) ) )
14127, 140mp3an2 1297 . . . . . . . 8  |-  ( ( T  e.  CC  /\  S  e.  CC )  ->  ( T  x.  (
1  -  S ) )  =  ( ( T  x.  1 )  -  ( T  x.  S ) ) )
14266, 99, 141syl2anc 656 . . . . . . 7  |-  ( ( S  =/=  0  /\  ( T  e.  ( 0 [,] 1 )  /\  S  e.  ( 0 [,] 1 ) ) )  ->  ( T  x.  ( 1  -  S ) )  =  ( ( T  x.  1 )  -  ( T  x.  S
) ) )
14366mulid1d 9399 . . . . . . . 8  |-  ( ( S  =/=  0  /\  ( T  e.  ( 0 [,] 1 )  /\  S  e.  ( 0 [,] 1 ) ) )  ->  ( T  x.  1 )  =  T )
14466, 99mulcomd 9403 . . . . . . . 8  |-  ( ( S  =/=  0  /\  ( T  e.  ( 0 [,] 1 )  /\  S  e.  ( 0 [,] 1 ) ) )  ->  ( T  x.  S )  =  ( S  x.  T ) )
145143, 144oveq12d 6108 . . . . . . 7  |-  ( ( S  =/=  0  /\  ( T  e.  ( 0 [,] 1 )  /\  S  e.  ( 0 [,] 1 ) ) )  ->  (
( T  x.  1 )  -  ( T  x.  S ) )  =  ( T  -  ( S  x.  T
) ) )
146142, 145eqtrd 2473 . . . . . 6  |-  ( ( S  =/=  0  /\  ( T  e.  ( 0 [,] 1 )  /\  S  e.  ( 0 [,] 1 ) ) )  ->  ( T  x.  ( 1  -  S ) )  =  ( T  -  ( S  x.  T
) ) )
147146oveq1d 6105 . . . . 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 9408 . . . . . . . 8  |-  ( ( S  =/=  0  /\  ( T  e.  ( 0 [,] 1 )  /\  S  e.  ( 0 [,] 1 ) ) )  ->  ( S  -  ( S  x.  T ) )  e.  CC )
14986, 148, 86, 78divsubdird 10142 . . . . . . 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 9750 . . . . . . . . 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 9717 . . . . . . . . 9  |-  ( ( S  =/=  0  /\  ( T  e.  ( 0 [,] 1 )  /\  S  e.  ( 0 [,] 1 ) ) )  ->  (
( S  +  T
)  -  S )  =  T )
152150, 151eqtrd 2473 . . . . . . . 8  |-  ( ( S  =/=  0  /\  ( T  e.  ( 0 [,] 1 )  /\  S  e.  ( 0 [,] 1 ) ) )  ->  (
( ( S  +  T )  -  ( S  x.  T )
)  -  ( S  -  ( S  x.  T ) ) )  =  T )
153152oveq1d 6105 . . . . . . 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 6105 . . . . . . 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 2481 . . . . . 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 6105 . . . . 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 2481 . . . 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 9403 . . . . . 6  |-  ( ( S  =/=  0  /\  ( T  e.  ( 0 [,] 1 )  /\  S  e.  ( 0 [,] 1 ) ) )  ->  ( S  x.  T )  =  ( T  x.  S ) )
159158oveq1d 6105 . . . . 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 10140 . . . . . 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 6105 . . . . . 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 2473 . . . . 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 10140 . . . . . 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 6105 . . . . . 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 2473 . . . . 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 2481 . . . 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 6098 . . . . . . . 8  |-  ( r  =  ( ( T  -  ( S  x.  T ) )  / 
( ( S  +  T )  -  ( S  x.  T )
) )  ->  (
1  -  r )  =  ( 1  -  ( ( T  -  ( S  x.  T
) )  /  (
( S  +  T
)  -  ( S  x.  T ) ) ) ) )
168167oveq1d 6105 . . . . . . 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 2452 . . . . . 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 2447 . . . . . 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 6105 . . . . . . 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 2449 . . . . . 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 1284 . . . . 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 2447 . . . . . 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 6098 . . . . . . . 8  |-  ( p  =  ( ( S  -  ( S  x.  T ) )  / 
( ( S  +  T )  -  ( S  x.  T )
) )  ->  (
1  -  p )  =  ( 1  -  ( ( S  -  ( S  x.  T
) )  /  (
( S  +  T
)  -  ( S  x.  T ) ) ) ) )
176175oveq1d 6105 . . . . . . 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 2452 . . . . . 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 6105 . . . . . . 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 2452 . . . . . 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 1284 . . . . 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 3078 . . . 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 1225 . . 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 434 . 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 2685 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 184    /\ wa 369    /\ w3a 960    = wceq 1364    e. wcel 1761    =/= wne 2604   E.wrex 2714   class class class wbr 4289  (class class class)co 6090   CCcc 9276   RRcr 9277   0cc0 9278   1c1 9279    + caddc 9281    x. cmul 9283    < clt 9414    <_ cle 9415    - cmin 9591    / cdiv 9989   [,]cicc 11299
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1713  ax-7 1733  ax-8 1763  ax-9 1765  ax-10 1780  ax-11 1785  ax-12 1797  ax-13 1948  ax-ext 2422  ax-sep 4410  ax-nul 4418  ax-pow 4467  ax-pr 4528  ax-un 6371  ax-cnex 9334  ax-resscn 9335  ax-1cn 9336  ax-icn 9337  ax-addcl 9338  ax-addrcl 9339  ax-mulcl 9340  ax-mulrcl 9341  ax-mulcom 9342  ax-addass 9343  ax-mulass 9344  ax-distr 9345  ax-i2m1 9346  ax-1ne0 9347  ax-1rid 9348  ax-rnegex 9349  ax-rrecex 9350  ax-cnre 9351  ax-pre-lttri 9352  ax-pre-lttrn 9353  ax-pre-ltadd 9354  ax-pre-mulgt0 9355
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 961  df-3an 962  df-tru 1367  df-ex 1592  df-nf 1595  df-sb 1706  df-eu 2261  df-mo 2262  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-nel 2607  df-ral 2718  df-rex 2719  df-reu 2720  df-rmo 2721  df-rab 2722  df-v 2972  df-sbc 3184  df-csb 3286  df-dif 3328  df-un 3330  df-in 3332  df-ss 3339  df-nul 3635  df-if 3789  df-pw 3859  df-sn 3875  df-pr 3877  df-op 3881  df-uni 4089  df-br 4290  df-opab 4348  df-mpt 4349  df-id 4632  df-po 4637  df-so 4638  df-xp 4842  df-rel 4843  df-cnv 4844  df-co 4845  df-dm 4846  df-rn 4847  df-res 4848  df-ima 4849  df-iota 5378  df-fun 5417  df-fn 5418  df-f 5419  df-f1 5420  df-fo 5421  df-f1o 5422  df-fv 5423  df-riota 6049  df-ov 6093  df-oprab 6094  df-mpt2 6095  df-er 7097  df-en 7307  df-dom 7308  df-sdom 7309  df-pnf 9416  df-mnf 9417  df-xr 9418  df-ltxr 9419  df-le 9420  df-sub 9593  df-neg 9594  df-div 9990  df-icc 11303
This theorem is referenced by:  axpasch  23122
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