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Theorem ruclem1 13821
Description: Lemma for ruc 13833 (the reals are uncountable). Substitutions for the function  D. (Contributed by Mario Carneiro, 28-May-2014.) (Revised by Fan Zheng, 6-Jun-2016.)
Hypotheses
Ref Expression
ruc.1  |-  ( ph  ->  F : NN --> RR )
ruc.2  |-  ( ph  ->  D  =  ( x  e.  ( RR  X.  RR ) ,  y  e.  RR  |->  [_ ( ( ( 1st `  x )  +  ( 2nd `  x
) )  /  2
)  /  m ]_ if ( m  <  y ,  <. ( 1st `  x
) ,  m >. , 
<. ( ( m  +  ( 2nd `  x ) )  /  2 ) ,  ( 2nd `  x
) >. ) ) )
ruclem1.3  |-  ( ph  ->  A  e.  RR )
ruclem1.4  |-  ( ph  ->  B  e.  RR )
ruclem1.5  |-  ( ph  ->  M  e.  RR )
ruclem1.6  |-  X  =  ( 1st `  ( <. A ,  B >. D M ) )
ruclem1.7  |-  Y  =  ( 2nd `  ( <. A ,  B >. D M ) )
Assertion
Ref Expression
ruclem1  |-  ( ph  ->  ( ( <. A ,  B >. D M )  e.  ( RR  X.  RR )  /\  X  =  if ( ( ( A  +  B )  /  2 )  < 
M ,  A , 
( ( ( ( A  +  B )  /  2 )  +  B )  /  2
) )  /\  Y  =  if ( ( ( A  +  B )  /  2 )  < 
M ,  ( ( A  +  B )  /  2 ) ,  B ) ) )
Distinct variable groups:    x, m, y, A    B, m, x, y    m, F, x, y    m, M, x, y
Allowed substitution hints:    ph( x, y, m)    D( x, y, m)    X( x, y, m)    Y( x, y, m)

Proof of Theorem ruclem1
StepHypRef Expression
1 ruc.2 . . . . . 6  |-  ( ph  ->  D  =  ( x  e.  ( RR  X.  RR ) ,  y  e.  RR  |->  [_ ( ( ( 1st `  x )  +  ( 2nd `  x
) )  /  2
)  /  m ]_ if ( m  <  y ,  <. ( 1st `  x
) ,  m >. , 
<. ( ( m  +  ( 2nd `  x ) )  /  2 ) ,  ( 2nd `  x
) >. ) ) )
21oveqd 6299 . . . . 5  |-  ( ph  ->  ( <. A ,  B >. D M )  =  ( <. A ,  B >. ( x  e.  ( RR  X.  RR ) ,  y  e.  RR  |->  [_ ( ( ( 1st `  x )  +  ( 2nd `  x ) )  /  2 )  /  m ]_ if ( m  <  y , 
<. ( 1st `  x
) ,  m >. , 
<. ( ( m  +  ( 2nd `  x ) )  /  2 ) ,  ( 2nd `  x
) >. ) ) M ) )
3 ruclem1.3 . . . . . . 7  |-  ( ph  ->  A  e.  RR )
4 ruclem1.4 . . . . . . 7  |-  ( ph  ->  B  e.  RR )
5 opelxpi 5030 . . . . . . 7  |-  ( ( A  e.  RR  /\  B  e.  RR )  -> 
<. A ,  B >.  e.  ( RR  X.  RR ) )
63, 4, 5syl2anc 661 . . . . . 6  |-  ( ph  -> 
<. A ,  B >.  e.  ( RR  X.  RR ) )
7 ruclem1.5 . . . . . 6  |-  ( ph  ->  M  e.  RR )
8 simpr 461 . . . . . . . . . . 11  |-  ( ( x  =  <. A ,  B >.  /\  y  =  M )  ->  y  =  M )
98breq2d 4459 . . . . . . . . . 10  |-  ( ( x  =  <. A ,  B >.  /\  y  =  M )  ->  (
m  <  y  <->  m  <  M ) )
10 simpl 457 . . . . . . . . . . . 12  |-  ( ( x  =  <. A ,  B >.  /\  y  =  M )  ->  x  =  <. A ,  B >. )
1110fveq2d 5868 . . . . . . . . . . 11  |-  ( ( x  =  <. A ,  B >.  /\  y  =  M )  ->  ( 1st `  x )  =  ( 1st `  <. A ,  B >. )
)
1211opeq1d 4219 . . . . . . . . . 10  |-  ( ( x  =  <. A ,  B >.  /\  y  =  M )  ->  <. ( 1st `  x ) ,  m >.  =  <. ( 1st `  <. A ,  B >. ) ,  m >. )
1310fveq2d 5868 . . . . . . . . . . . . 13  |-  ( ( x  =  <. A ,  B >.  /\  y  =  M )  ->  ( 2nd `  x )  =  ( 2nd `  <. A ,  B >. )
)
1413oveq2d 6298 . . . . . . . . . . . 12  |-  ( ( x  =  <. A ,  B >.  /\  y  =  M )  ->  (
m  +  ( 2nd `  x ) )  =  ( m  +  ( 2nd `  <. A ,  B >. ) ) )
1514oveq1d 6297 . . . . . . . . . . 11  |-  ( ( x  =  <. A ,  B >.  /\  y  =  M )  ->  (
( m  +  ( 2nd `  x ) )  /  2 )  =  ( ( m  +  ( 2nd `  <. A ,  B >. )
)  /  2 ) )
1615, 13opeq12d 4221 . . . . . . . . . 10  |-  ( ( x  =  <. A ,  B >.  /\  y  =  M )  ->  <. (
( m  +  ( 2nd `  x ) )  /  2 ) ,  ( 2nd `  x
) >.  =  <. (
( m  +  ( 2nd `  <. A ,  B >. ) )  / 
2 ) ,  ( 2nd `  <. A ,  B >. ) >. )
179, 12, 16ifbieq12d 3966 . . . . . . . . 9  |-  ( ( x  =  <. A ,  B >.  /\  y  =  M )  ->  if ( m  <  y , 
<. ( 1st `  x
) ,  m >. , 
<. ( ( m  +  ( 2nd `  x ) )  /  2 ) ,  ( 2nd `  x
) >. )  =  if ( m  <  M ,  <. ( 1st `  <. A ,  B >. ) ,  m >. ,  <. (
( m  +  ( 2nd `  <. A ,  B >. ) )  / 
2 ) ,  ( 2nd `  <. A ,  B >. ) >. )
)
1817csbeq2dv 3835 . . . . . . . 8  |-  ( ( x  =  <. A ,  B >.  /\  y  =  M )  ->  [_ (
( ( 1st `  x
)  +  ( 2nd `  x ) )  / 
2 )  /  m ]_ if ( m  < 
y ,  <. ( 1st `  x ) ,  m >. ,  <. (
( m  +  ( 2nd `  x ) )  /  2 ) ,  ( 2nd `  x
) >. )  =  [_ ( ( ( 1st `  x )  +  ( 2nd `  x ) )  /  2 )  /  m ]_ if ( m  <  M ,  <. ( 1st `  <. A ,  B >. ) ,  m >. ,  <. (
( m  +  ( 2nd `  <. A ,  B >. ) )  / 
2 ) ,  ( 2nd `  <. A ,  B >. ) >. )
)
1911, 13oveq12d 6300 . . . . . . . . . 10  |-  ( ( x  =  <. A ,  B >.  /\  y  =  M )  ->  (
( 1st `  x
)  +  ( 2nd `  x ) )  =  ( ( 1st `  <. A ,  B >. )  +  ( 2nd `  <. A ,  B >. )
) )
2019oveq1d 6297 . . . . . . . . 9  |-  ( ( x  =  <. A ,  B >.  /\  y  =  M )  ->  (
( ( 1st `  x
)  +  ( 2nd `  x ) )  / 
2 )  =  ( ( ( 1st `  <. A ,  B >. )  +  ( 2nd `  <. A ,  B >. )
)  /  2 ) )
2120csbeq1d 3442 . . . . . . . 8  |-  ( ( x  =  <. A ,  B >.  /\  y  =  M )  ->  [_ (
( ( 1st `  x
)  +  ( 2nd `  x ) )  / 
2 )  /  m ]_ if ( m  < 
M ,  <. ( 1st `  <. A ,  B >. ) ,  m >. , 
<. ( ( m  +  ( 2nd `  <. A ,  B >. ) )  / 
2 ) ,  ( 2nd `  <. A ,  B >. ) >. )  =  [_ ( ( ( 1st `  <. A ,  B >. )  +  ( 2nd `  <. A ,  B >. ) )  / 
2 )  /  m ]_ if ( m  < 
M ,  <. ( 1st `  <. A ,  B >. ) ,  m >. , 
<. ( ( m  +  ( 2nd `  <. A ,  B >. ) )  / 
2 ) ,  ( 2nd `  <. A ,  B >. ) >. )
)
2218, 21eqtrd 2508 . . . . . . 7  |-  ( ( x  =  <. A ,  B >.  /\  y  =  M )  ->  [_ (
( ( 1st `  x
)  +  ( 2nd `  x ) )  / 
2 )  /  m ]_ if ( m  < 
y ,  <. ( 1st `  x ) ,  m >. ,  <. (
( m  +  ( 2nd `  x ) )  /  2 ) ,  ( 2nd `  x
) >. )  =  [_ ( ( ( 1st `  <. A ,  B >. )  +  ( 2nd `  <. A ,  B >. ) )  /  2
)  /  m ]_ if ( m  <  M ,  <. ( 1st `  <. A ,  B >. ) ,  m >. ,  <. (
( m  +  ( 2nd `  <. A ,  B >. ) )  / 
2 ) ,  ( 2nd `  <. A ,  B >. ) >. )
)
23 eqid 2467 . . . . . . 7  |-  ( x  e.  ( RR  X.  RR ) ,  y  e.  RR  |->  [_ ( ( ( 1st `  x )  +  ( 2nd `  x
) )  /  2
)  /  m ]_ if ( m  <  y ,  <. ( 1st `  x
) ,  m >. , 
<. ( ( m  +  ( 2nd `  x ) )  /  2 ) ,  ( 2nd `  x
) >. ) )  =  ( x  e.  ( RR  X.  RR ) ,  y  e.  RR  |->  [_ ( ( ( 1st `  x )  +  ( 2nd `  x ) )  /  2 )  /  m ]_ if ( m  <  y , 
<. ( 1st `  x
) ,  m >. , 
<. ( ( m  +  ( 2nd `  x ) )  /  2 ) ,  ( 2nd `  x
) >. ) )
24 opex 4711 . . . . . . . . 9  |-  <. ( 1st `  <. A ,  B >. ) ,  m >.  e. 
_V
25 opex 4711 . . . . . . . . 9  |-  <. (
( m  +  ( 2nd `  <. A ,  B >. ) )  / 
2 ) ,  ( 2nd `  <. A ,  B >. ) >.  e.  _V
2624, 25ifex 4008 . . . . . . . 8  |-  if ( m  <  M ,  <. ( 1st `  <. A ,  B >. ) ,  m >. ,  <. (
( m  +  ( 2nd `  <. A ,  B >. ) )  / 
2 ) ,  ( 2nd `  <. A ,  B >. ) >. )  e.  _V
2726csbex 4580 . . . . . . 7  |-  [_ (
( ( 1st `  <. A ,  B >. )  +  ( 2nd `  <. A ,  B >. )
)  /  2 )  /  m ]_ if ( m  <  M ,  <. ( 1st `  <. A ,  B >. ) ,  m >. ,  <. (
( m  +  ( 2nd `  <. A ,  B >. ) )  / 
2 ) ,  ( 2nd `  <. A ,  B >. ) >. )  e.  _V
2822, 23, 27ovmpt2a 6415 . . . . . 6  |-  ( (
<. A ,  B >.  e.  ( RR  X.  RR )  /\  M  e.  RR )  ->  ( <. A ,  B >. ( x  e.  ( RR  X.  RR ) ,  y  e.  RR  |->  [_ ( ( ( 1st `  x )  +  ( 2nd `  x
) )  /  2
)  /  m ]_ if ( m  <  y ,  <. ( 1st `  x
) ,  m >. , 
<. ( ( m  +  ( 2nd `  x ) )  /  2 ) ,  ( 2nd `  x
) >. ) ) M )  =  [_ (
( ( 1st `  <. A ,  B >. )  +  ( 2nd `  <. A ,  B >. )
)  /  2 )  /  m ]_ if ( m  <  M ,  <. ( 1st `  <. A ,  B >. ) ,  m >. ,  <. (
( m  +  ( 2nd `  <. A ,  B >. ) )  / 
2 ) ,  ( 2nd `  <. A ,  B >. ) >. )
)
296, 7, 28syl2anc 661 . . . . 5  |-  ( ph  ->  ( <. A ,  B >. ( x  e.  ( RR  X.  RR ) ,  y  e.  RR  |->  [_ ( ( ( 1st `  x )  +  ( 2nd `  x ) )  /  2 )  /  m ]_ if ( m  <  y , 
<. ( 1st `  x
) ,  m >. , 
<. ( ( m  +  ( 2nd `  x ) )  /  2 ) ,  ( 2nd `  x
) >. ) ) M )  =  [_ (
( ( 1st `  <. A ,  B >. )  +  ( 2nd `  <. A ,  B >. )
)  /  2 )  /  m ]_ if ( m  <  M ,  <. ( 1st `  <. A ,  B >. ) ,  m >. ,  <. (
( m  +  ( 2nd `  <. A ,  B >. ) )  / 
2 ) ,  ( 2nd `  <. A ,  B >. ) >. )
)
302, 29eqtrd 2508 . . . 4  |-  ( ph  ->  ( <. A ,  B >. D M )  = 
[_ ( ( ( 1st `  <. A ,  B >. )  +  ( 2nd `  <. A ,  B >. ) )  / 
2 )  /  m ]_ if ( m  < 
M ,  <. ( 1st `  <. A ,  B >. ) ,  m >. , 
<. ( ( m  +  ( 2nd `  <. A ,  B >. ) )  / 
2 ) ,  ( 2nd `  <. A ,  B >. ) >. )
)
31 op1stg 6793 . . . . . . . . 9  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( 1st `  <. A ,  B >. )  =  A )
323, 4, 31syl2anc 661 . . . . . . . 8  |-  ( ph  ->  ( 1st `  <. A ,  B >. )  =  A )
33 op2ndg 6794 . . . . . . . . 9  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( 2nd `  <. A ,  B >. )  =  B )
343, 4, 33syl2anc 661 . . . . . . . 8  |-  ( ph  ->  ( 2nd `  <. A ,  B >. )  =  B )
3532, 34oveq12d 6300 . . . . . . 7  |-  ( ph  ->  ( ( 1st `  <. A ,  B >. )  +  ( 2nd `  <. A ,  B >. )
)  =  ( A  +  B ) )
3635oveq1d 6297 . . . . . 6  |-  ( ph  ->  ( ( ( 1st `  <. A ,  B >. )  +  ( 2nd `  <. A ,  B >. ) )  /  2
)  =  ( ( A  +  B )  /  2 ) )
3736csbeq1d 3442 . . . . 5  |-  ( ph  ->  [_ ( ( ( 1st `  <. A ,  B >. )  +  ( 2nd `  <. A ,  B >. ) )  / 
2 )  /  m ]_ if ( m  < 
M ,  <. ( 1st `  <. A ,  B >. ) ,  m >. , 
<. ( ( m  +  ( 2nd `  <. A ,  B >. ) )  / 
2 ) ,  ( 2nd `  <. A ,  B >. ) >. )  =  [_ ( ( A  +  B )  / 
2 )  /  m ]_ if ( m  < 
M ,  <. ( 1st `  <. A ,  B >. ) ,  m >. , 
<. ( ( m  +  ( 2nd `  <. A ,  B >. ) )  / 
2 ) ,  ( 2nd `  <. A ,  B >. ) >. )
)
38 ovex 6307 . . . . . . 7  |-  ( ( A  +  B )  /  2 )  e. 
_V
39 nfcv 2629 . . . . . . 7  |-  F/_ m if ( ( ( A  +  B )  / 
2 )  <  M ,  <. ( 1st `  <. A ,  B >. ) ,  ( ( A  +  B )  / 
2 ) >. ,  <. ( ( ( ( A  +  B )  / 
2 )  +  ( 2nd `  <. A ,  B >. ) )  / 
2 ) ,  ( 2nd `  <. A ,  B >. ) >. )
40 breq1 4450 . . . . . . . 8  |-  ( m  =  ( ( A  +  B )  / 
2 )  ->  (
m  <  M  <->  ( ( A  +  B )  /  2 )  < 
M ) )
41 opeq2 4214 . . . . . . . 8  |-  ( m  =  ( ( A  +  B )  / 
2 )  ->  <. ( 1st `  <. A ,  B >. ) ,  m >.  = 
<. ( 1st `  <. A ,  B >. ) ,  ( ( A  +  B )  / 
2 ) >. )
42 oveq1 6289 . . . . . . . . . 10  |-  ( m  =  ( ( A  +  B )  / 
2 )  ->  (
m  +  ( 2nd `  <. A ,  B >. ) )  =  ( ( ( A  +  B )  /  2
)  +  ( 2nd `  <. A ,  B >. ) ) )
4342oveq1d 6297 . . . . . . . . 9  |-  ( m  =  ( ( A  +  B )  / 
2 )  ->  (
( m  +  ( 2nd `  <. A ,  B >. ) )  / 
2 )  =  ( ( ( ( A  +  B )  / 
2 )  +  ( 2nd `  <. A ,  B >. ) )  / 
2 ) )
4443opeq1d 4219 . . . . . . . 8  |-  ( m  =  ( ( A  +  B )  / 
2 )  ->  <. (
( m  +  ( 2nd `  <. A ,  B >. ) )  / 
2 ) ,  ( 2nd `  <. A ,  B >. ) >.  =  <. ( ( ( ( A  +  B )  / 
2 )  +  ( 2nd `  <. A ,  B >. ) )  / 
2 ) ,  ( 2nd `  <. A ,  B >. ) >. )
4540, 41, 44ifbieq12d 3966 . . . . . . 7  |-  ( m  =  ( ( A  +  B )  / 
2 )  ->  if ( m  <  M ,  <. ( 1st `  <. A ,  B >. ) ,  m >. ,  <. (
( m  +  ( 2nd `  <. A ,  B >. ) )  / 
2 ) ,  ( 2nd `  <. A ,  B >. ) >. )  =  if ( ( ( A  +  B )  /  2 )  < 
M ,  <. ( 1st `  <. A ,  B >. ) ,  ( ( A  +  B )  /  2 ) >. ,  <. ( ( ( ( A  +  B
)  /  2 )  +  ( 2nd `  <. A ,  B >. )
)  /  2 ) ,  ( 2nd `  <. A ,  B >. ) >. ) )
4638, 39, 45csbief 3460 . . . . . 6  |-  [_ (
( A  +  B
)  /  2 )  /  m ]_ if ( m  <  M ,  <. ( 1st `  <. A ,  B >. ) ,  m >. ,  <. (
( m  +  ( 2nd `  <. A ,  B >. ) )  / 
2 ) ,  ( 2nd `  <. A ,  B >. ) >. )  =  if ( ( ( A  +  B )  /  2 )  < 
M ,  <. ( 1st `  <. A ,  B >. ) ,  ( ( A  +  B )  /  2 ) >. ,  <. ( ( ( ( A  +  B
)  /  2 )  +  ( 2nd `  <. A ,  B >. )
)  /  2 ) ,  ( 2nd `  <. A ,  B >. ) >. )
4732opeq1d 4219 . . . . . . 7  |-  ( ph  -> 
<. ( 1st `  <. A ,  B >. ) ,  ( ( A  +  B )  / 
2 ) >.  =  <. A ,  ( ( A  +  B )  / 
2 ) >. )
4834oveq2d 6298 . . . . . . . . 9  |-  ( ph  ->  ( ( ( A  +  B )  / 
2 )  +  ( 2nd `  <. A ,  B >. ) )  =  ( ( ( A  +  B )  / 
2 )  +  B
) )
4948oveq1d 6297 . . . . . . . 8  |-  ( ph  ->  ( ( ( ( A  +  B )  /  2 )  +  ( 2nd `  <. A ,  B >. )
)  /  2 )  =  ( ( ( ( A  +  B
)  /  2 )  +  B )  / 
2 ) )
5049, 34opeq12d 4221 . . . . . . 7  |-  ( ph  -> 
<. ( ( ( ( A  +  B )  /  2 )  +  ( 2nd `  <. A ,  B >. )
)  /  2 ) ,  ( 2nd `  <. A ,  B >. ) >.  =  <. ( ( ( ( A  +  B
)  /  2 )  +  B )  / 
2 ) ,  B >. )
5147, 50ifeq12d 3959 . . . . . 6  |-  ( ph  ->  if ( ( ( A  +  B )  /  2 )  < 
M ,  <. ( 1st `  <. A ,  B >. ) ,  ( ( A  +  B )  /  2 ) >. ,  <. ( ( ( ( A  +  B
)  /  2 )  +  ( 2nd `  <. A ,  B >. )
)  /  2 ) ,  ( 2nd `  <. A ,  B >. ) >. )  =  if ( ( ( A  +  B )  /  2
)  <  M ,  <. A ,  ( ( A  +  B )  /  2 ) >. ,  <. ( ( ( ( A  +  B
)  /  2 )  +  B )  / 
2 ) ,  B >. ) )
5246, 51syl5eq 2520 . . . . 5  |-  ( ph  ->  [_ ( ( A  +  B )  / 
2 )  /  m ]_ if ( m  < 
M ,  <. ( 1st `  <. A ,  B >. ) ,  m >. , 
<. ( ( m  +  ( 2nd `  <. A ,  B >. ) )  / 
2 ) ,  ( 2nd `  <. A ,  B >. ) >. )  =  if ( ( ( A  +  B )  /  2 )  < 
M ,  <. A , 
( ( A  +  B )  /  2
) >. ,  <. (
( ( ( A  +  B )  / 
2 )  +  B
)  /  2 ) ,  B >. )
)
5337, 52eqtrd 2508 . . . 4  |-  ( ph  ->  [_ ( ( ( 1st `  <. A ,  B >. )  +  ( 2nd `  <. A ,  B >. ) )  / 
2 )  /  m ]_ if ( m  < 
M ,  <. ( 1st `  <. A ,  B >. ) ,  m >. , 
<. ( ( m  +  ( 2nd `  <. A ,  B >. ) )  / 
2 ) ,  ( 2nd `  <. A ,  B >. ) >. )  =  if ( ( ( A  +  B )  /  2 )  < 
M ,  <. A , 
( ( A  +  B )  /  2
) >. ,  <. (
( ( ( A  +  B )  / 
2 )  +  B
)  /  2 ) ,  B >. )
)
5430, 53eqtrd 2508 . . 3  |-  ( ph  ->  ( <. A ,  B >. D M )  =  if ( ( ( A  +  B )  /  2 )  < 
M ,  <. A , 
( ( A  +  B )  /  2
) >. ,  <. (
( ( ( A  +  B )  / 
2 )  +  B
)  /  2 ) ,  B >. )
)
553, 4readdcld 9619 . . . . . 6  |-  ( ph  ->  ( A  +  B
)  e.  RR )
5655rehalfcld 10781 . . . . 5  |-  ( ph  ->  ( ( A  +  B )  /  2
)  e.  RR )
57 opelxpi 5030 . . . . 5  |-  ( ( A  e.  RR  /\  ( ( A  +  B )  /  2
)  e.  RR )  ->  <. A ,  ( ( A  +  B
)  /  2 )
>.  e.  ( RR  X.  RR ) )
583, 56, 57syl2anc 661 . . . 4  |-  ( ph  -> 
<. A ,  ( ( A  +  B )  /  2 ) >.  e.  ( RR  X.  RR ) )
5956, 4readdcld 9619 . . . . . 6  |-  ( ph  ->  ( ( ( A  +  B )  / 
2 )  +  B
)  e.  RR )
6059rehalfcld 10781 . . . . 5  |-  ( ph  ->  ( ( ( ( A  +  B )  /  2 )  +  B )  /  2
)  e.  RR )
61 opelxpi 5030 . . . . 5  |-  ( ( ( ( ( ( A  +  B )  /  2 )  +  B )  /  2
)  e.  RR  /\  B  e.  RR )  -> 
<. ( ( ( ( A  +  B )  /  2 )  +  B )  /  2
) ,  B >.  e.  ( RR  X.  RR ) )
6260, 4, 61syl2anc 661 . . . 4  |-  ( ph  -> 
<. ( ( ( ( A  +  B )  /  2 )  +  B )  /  2
) ,  B >.  e.  ( RR  X.  RR ) )
63 ifcl 3981 . . . 4  |-  ( (
<. A ,  ( ( A  +  B )  /  2 ) >.  e.  ( RR  X.  RR )  /\  <. ( ( ( ( A  +  B
)  /  2 )  +  B )  / 
2 ) ,  B >.  e.  ( RR  X.  RR ) )  ->  if ( ( ( A  +  B )  / 
2 )  <  M ,  <. A ,  ( ( A  +  B
)  /  2 )
>. ,  <. ( ( ( ( A  +  B )  /  2
)  +  B )  /  2 ) ,  B >. )  e.  ( RR  X.  RR ) )
6458, 62, 63syl2anc 661 . . 3  |-  ( ph  ->  if ( ( ( A  +  B )  /  2 )  < 
M ,  <. A , 
( ( A  +  B )  /  2
) >. ,  <. (
( ( ( A  +  B )  / 
2 )  +  B
)  /  2 ) ,  B >. )  e.  ( RR  X.  RR ) )
6554, 64eqeltrd 2555 . 2  |-  ( ph  ->  ( <. A ,  B >. D M )  e.  ( RR  X.  RR ) )
66 ruclem1.6 . . 3  |-  X  =  ( 1st `  ( <. A ,  B >. D M ) )
6754fveq2d 5868 . . . 4  |-  ( ph  ->  ( 1st `  ( <. A ,  B >. D M ) )  =  ( 1st `  if ( ( ( A  +  B )  / 
2 )  <  M ,  <. A ,  ( ( A  +  B
)  /  2 )
>. ,  <. ( ( ( ( A  +  B )  /  2
)  +  B )  /  2 ) ,  B >. ) ) )
68 fvif 5875 . . . . 5  |-  ( 1st `  if ( ( ( A  +  B )  /  2 )  < 
M ,  <. A , 
( ( A  +  B )  /  2
) >. ,  <. (
( ( ( A  +  B )  / 
2 )  +  B
)  /  2 ) ,  B >. )
)  =  if ( ( ( A  +  B )  /  2
)  <  M , 
( 1st `  <. A ,  ( ( A  +  B )  / 
2 ) >. ) ,  ( 1st `  <. ( ( ( ( A  +  B )  / 
2 )  +  B
)  /  2 ) ,  B >. )
)
69 op1stg 6793 . . . . . . 7  |-  ( ( A  e.  RR  /\  ( ( A  +  B )  /  2
)  e.  _V )  ->  ( 1st `  <. A ,  ( ( A  +  B )  / 
2 ) >. )  =  A )
703, 38, 69sylancl 662 . . . . . 6  |-  ( ph  ->  ( 1st `  <. A ,  ( ( A  +  B )  / 
2 ) >. )  =  A )
71 ovex 6307 . . . . . . 7  |-  ( ( ( ( A  +  B )  /  2
)  +  B )  /  2 )  e. 
_V
72 op1stg 6793 . . . . . . 7  |-  ( ( ( ( ( ( A  +  B )  /  2 )  +  B )  /  2
)  e.  _V  /\  B  e.  RR )  ->  ( 1st `  <. ( ( ( ( A  +  B )  / 
2 )  +  B
)  /  2 ) ,  B >. )  =  ( ( ( ( A  +  B
)  /  2 )  +  B )  / 
2 ) )
7371, 4, 72sylancr 663 . . . . . 6  |-  ( ph  ->  ( 1st `  <. ( ( ( ( A  +  B )  / 
2 )  +  B
)  /  2 ) ,  B >. )  =  ( ( ( ( A  +  B
)  /  2 )  +  B )  / 
2 ) )
7470, 73ifeq12d 3959 . . . . 5  |-  ( ph  ->  if ( ( ( A  +  B )  /  2 )  < 
M ,  ( 1st `  <. A ,  ( ( A  +  B
)  /  2 )
>. ) ,  ( 1st `  <. ( ( ( ( A  +  B
)  /  2 )  +  B )  / 
2 ) ,  B >. ) )  =  if ( ( ( A  +  B )  / 
2 )  <  M ,  A ,  ( ( ( ( A  +  B )  /  2
)  +  B )  /  2 ) ) )
7568, 74syl5eq 2520 . . . 4  |-  ( ph  ->  ( 1st `  if ( ( ( A  +  B )  / 
2 )  <  M ,  <. A ,  ( ( A  +  B
)  /  2 )
>. ,  <. ( ( ( ( A  +  B )  /  2
)  +  B )  /  2 ) ,  B >. ) )  =  if ( ( ( A  +  B )  /  2 )  < 
M ,  A , 
( ( ( ( A  +  B )  /  2 )  +  B )  /  2
) ) )
7667, 75eqtrd 2508 . . 3  |-  ( ph  ->  ( 1st `  ( <. A ,  B >. D M ) )  =  if ( ( ( A  +  B )  /  2 )  < 
M ,  A , 
( ( ( ( A  +  B )  /  2 )  +  B )  /  2
) ) )
7766, 76syl5eq 2520 . 2  |-  ( ph  ->  X  =  if ( ( ( A  +  B )  /  2
)  <  M ,  A ,  ( (
( ( A  +  B )  /  2
)  +  B )  /  2 ) ) )
78 ruclem1.7 . . 3  |-  Y  =  ( 2nd `  ( <. A ,  B >. D M ) )
7954fveq2d 5868 . . . 4  |-  ( ph  ->  ( 2nd `  ( <. A ,  B >. D M ) )  =  ( 2nd `  if ( ( ( A  +  B )  / 
2 )  <  M ,  <. A ,  ( ( A  +  B
)  /  2 )
>. ,  <. ( ( ( ( A  +  B )  /  2
)  +  B )  /  2 ) ,  B >. ) ) )
80 fvif 5875 . . . . 5  |-  ( 2nd `  if ( ( ( A  +  B )  /  2 )  < 
M ,  <. A , 
( ( A  +  B )  /  2
) >. ,  <. (
( ( ( A  +  B )  / 
2 )  +  B
)  /  2 ) ,  B >. )
)  =  if ( ( ( A  +  B )  /  2
)  <  M , 
( 2nd `  <. A ,  ( ( A  +  B )  / 
2 ) >. ) ,  ( 2nd `  <. ( ( ( ( A  +  B )  / 
2 )  +  B
)  /  2 ) ,  B >. )
)
81 op2ndg 6794 . . . . . . 7  |-  ( ( A  e.  RR  /\  ( ( A  +  B )  /  2
)  e.  _V )  ->  ( 2nd `  <. A ,  ( ( A  +  B )  / 
2 ) >. )  =  ( ( A  +  B )  / 
2 ) )
823, 38, 81sylancl 662 . . . . . 6  |-  ( ph  ->  ( 2nd `  <. A ,  ( ( A  +  B )  / 
2 ) >. )  =  ( ( A  +  B )  / 
2 ) )
83 op2ndg 6794 . . . . . . 7  |-  ( ( ( ( ( ( A  +  B )  /  2 )  +  B )  /  2
)  e.  _V  /\  B  e.  RR )  ->  ( 2nd `  <. ( ( ( ( A  +  B )  / 
2 )  +  B
)  /  2 ) ,  B >. )  =  B )
8471, 4, 83sylancr 663 . . . . . 6  |-  ( ph  ->  ( 2nd `  <. ( ( ( ( A  +  B )  / 
2 )  +  B
)  /  2 ) ,  B >. )  =  B )
8582, 84ifeq12d 3959 . . . . 5  |-  ( ph  ->  if ( ( ( A  +  B )  /  2 )  < 
M ,  ( 2nd `  <. A ,  ( ( A  +  B
)  /  2 )
>. ) ,  ( 2nd `  <. ( ( ( ( A  +  B
)  /  2 )  +  B )  / 
2 ) ,  B >. ) )  =  if ( ( ( A  +  B )  / 
2 )  <  M ,  ( ( A  +  B )  / 
2 ) ,  B
) )
8680, 85syl5eq 2520 . . . 4  |-  ( ph  ->  ( 2nd `  if ( ( ( A  +  B )  / 
2 )  <  M ,  <. A ,  ( ( A  +  B
)  /  2 )
>. ,  <. ( ( ( ( A  +  B )  /  2
)  +  B )  /  2 ) ,  B >. ) )  =  if ( ( ( A  +  B )  /  2 )  < 
M ,  ( ( A  +  B )  /  2 ) ,  B ) )
8779, 86eqtrd 2508 . . 3  |-  ( ph  ->  ( 2nd `  ( <. A ,  B >. D M ) )  =  if ( ( ( A  +  B )  /  2 )  < 
M ,  ( ( A  +  B )  /  2 ) ,  B ) )
8878, 87syl5eq 2520 . 2  |-  ( ph  ->  Y  =  if ( ( ( A  +  B )  /  2
)  <  M , 
( ( A  +  B )  /  2
) ,  B ) )
8965, 77, 883jca 1176 1  |-  ( ph  ->  ( ( <. A ,  B >. D M )  e.  ( RR  X.  RR )  /\  X  =  if ( ( ( A  +  B )  /  2 )  < 
M ,  A , 
( ( ( ( A  +  B )  /  2 )  +  B )  /  2
) )  /\  Y  =  if ( ( ( A  +  B )  /  2 )  < 
M ,  ( ( A  +  B )  /  2 ) ,  B ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    /\ w3a 973    = wceq 1379    e. wcel 1767   _Vcvv 3113   [_csb 3435   ifcif 3939   <.cop 4033   class class class wbr 4447    X. cxp 4997   -->wf 5582   ` cfv 5586  (class class class)co 6282    |-> cmpt2 6284   1stc1st 6779   2ndc2nd 6780   RRcr 9487    + caddc 9491    < clt 9624    / cdiv 10202   NNcn 10532   2c2 10581
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6574  ax-resscn 9545  ax-1cn 9546  ax-icn 9547  ax-addcl 9548  ax-addrcl 9549  ax-mulcl 9550  ax-mulrcl 9551  ax-mulcom 9552  ax-addass 9553  ax-mulass 9554  ax-distr 9555  ax-i2m1 9556  ax-1ne0 9557  ax-1rid 9558  ax-rnegex 9559  ax-rrecex 9560  ax-cnre 9561  ax-pre-lttri 9562  ax-pre-lttrn 9563  ax-pre-ltadd 9564  ax-pre-mulgt0 9565
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-fal 1385  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2819  df-rex 2820  df-reu 2821  df-rmo 2822  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-br 4448  df-opab 4506  df-mpt 4507  df-id 4795  df-po 4800  df-so 4801  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5549  df-fun 5588  df-fn 5589  df-f 5590  df-f1 5591  df-fo 5592  df-f1o 5593  df-fv 5594  df-riota 6243  df-ov 6285  df-oprab 6286  df-mpt2 6287  df-1st 6781  df-2nd 6782  df-er 7308  df-en 7514  df-dom 7515  df-sdom 7516  df-pnf 9626  df-mnf 9627  df-xr 9628  df-ltxr 9629  df-le 9630  df-sub 9803  df-neg 9804  df-div 10203  df-2 10590
This theorem is referenced by:  ruclem2  13822  ruclem3  13823  ruclem6  13825
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