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Theorem ruclem1 12509
Description: Lemma for ruc 12521 (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 5875 . . . . 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 4721 . . . . . . 7  |-  ( ( A  e.  RR  /\  B  e.  RR )  -> 
<. A ,  B >.  e.  ( RR  X.  RR ) )
63, 4, 5syl2anc 642 . . . . . 6  |-  ( ph  -> 
<. A ,  B >.  e.  ( RR  X.  RR ) )
7 ruclem1.5 . . . . . 6  |-  ( ph  ->  M  e.  RR )
8 simpr 447 . . . . . . . . . . 11  |-  ( ( x  =  <. A ,  B >.  /\  y  =  M )  ->  y  =  M )
98breq2d 4035 . . . . . . . . . 10  |-  ( ( x  =  <. A ,  B >.  /\  y  =  M )  ->  (
m  <  y  <->  m  <  M ) )
10 simpl 443 . . . . . . . . . . . 12  |-  ( ( x  =  <. A ,  B >.  /\  y  =  M )  ->  x  =  <. A ,  B >. )
1110fveq2d 5529 . . . . . . . . . . 11  |-  ( ( x  =  <. A ,  B >.  /\  y  =  M )  ->  ( 1st `  x )  =  ( 1st `  <. A ,  B >. )
)
1211opeq1d 3802 . . . . . . . . . 10  |-  ( ( x  =  <. A ,  B >.  /\  y  =  M )  ->  <. ( 1st `  x ) ,  m >.  =  <. ( 1st `  <. A ,  B >. ) ,  m >. )
1310fveq2d 5529 . . . . . . . . . . . . 13  |-  ( ( x  =  <. A ,  B >.  /\  y  =  M )  ->  ( 2nd `  x )  =  ( 2nd `  <. A ,  B >. )
)
1413oveq2d 5874 . . . . . . . . . . . 12  |-  ( ( x  =  <. A ,  B >.  /\  y  =  M )  ->  (
m  +  ( 2nd `  x ) )  =  ( m  +  ( 2nd `  <. A ,  B >. ) ) )
1514oveq1d 5873 . . . . . . . . . . 11  |-  ( ( x  =  <. A ,  B >.  /\  y  =  M )  ->  (
( m  +  ( 2nd `  x ) )  /  2 )  =  ( ( m  +  ( 2nd `  <. A ,  B >. )
)  /  2 ) )
1615, 13opeq12d 3804 . . . . . . . . . 10  |-  ( ( x  =  <. A ,  B >.  /\  y  =  M )  ->  <. (
( m  +  ( 2nd `  x ) )  /  2 ) ,  ( 2nd `  x
) >.  =  <. (
( m  +  ( 2nd `  <. A ,  B >. ) )  / 
2 ) ,  ( 2nd `  <. A ,  B >. ) >. )
179, 12, 16ifbieq12d 3587 . . . . . . . . 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 3106 . . . . . . . 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 5876 . . . . . . . . . 10  |-  ( ( x  =  <. A ,  B >.  /\  y  =  M )  ->  (
( 1st `  x
)  +  ( 2nd `  x ) )  =  ( ( 1st `  <. A ,  B >. )  +  ( 2nd `  <. A ,  B >. )
) )
2019oveq1d 5873 . . . . . . . . 9  |-  ( ( x  =  <. A ,  B >.  /\  y  =  M )  ->  (
( ( 1st `  x
)  +  ( 2nd `  x ) )  / 
2 )  =  ( ( ( 1st `  <. A ,  B >. )  +  ( 2nd `  <. A ,  B >. )
)  /  2 ) )
2120csbeq1d 3087 . . . . . . . 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 2315 . . . . . . 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 2283 . . . . . . 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 ovex 5883 . . . . . . . 8  |-  ( ( ( 1st `  <. A ,  B >. )  +  ( 2nd `  <. A ,  B >. )
)  /  2 )  e.  _V
25 opex 4237 . . . . . . . . 9  |-  <. ( 1st `  <. A ,  B >. ) ,  m >.  e. 
_V
26 opex 4237 . . . . . . . . 9  |-  <. (
( m  +  ( 2nd `  <. A ,  B >. ) )  / 
2 ) ,  ( 2nd `  <. A ,  B >. ) >.  e.  _V
2725, 26ifex 3623 . . . . . . . 8  |-  if ( m  <  M ,  <. ( 1st `  <. A ,  B >. ) ,  m >. ,  <. (
( m  +  ( 2nd `  <. A ,  B >. ) )  / 
2 ) ,  ( 2nd `  <. A ,  B >. ) >. )  e.  _V
2824, 27csbex 3092 . . . . . . 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
2922, 23, 28ovmpt2a 5978 . . . . . 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 >. ) >. )
)
306, 7, 29syl2anc 642 . . . . 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 >. ) >. )
)
312, 30eqtrd 2315 . . . 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 >. ) >. )
)
32 op1stg 6132 . . . . . . . . 9  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( 1st `  <. A ,  B >. )  =  A )
333, 4, 32syl2anc 642 . . . . . . . 8  |-  ( ph  ->  ( 1st `  <. A ,  B >. )  =  A )
34 op2ndg 6133 . . . . . . . . 9  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( 2nd `  <. A ,  B >. )  =  B )
353, 4, 34syl2anc 642 . . . . . . . 8  |-  ( ph  ->  ( 2nd `  <. A ,  B >. )  =  B )
3633, 35oveq12d 5876 . . . . . . 7  |-  ( ph  ->  ( ( 1st `  <. A ,  B >. )  +  ( 2nd `  <. A ,  B >. )
)  =  ( A  +  B ) )
3736oveq1d 5873 . . . . . 6  |-  ( ph  ->  ( ( ( 1st `  <. A ,  B >. )  +  ( 2nd `  <. A ,  B >. ) )  /  2
)  =  ( ( A  +  B )  /  2 ) )
3837csbeq1d 3087 . . . . 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 >. ) >. )
)
39 ovex 5883 . . . . . . 7  |-  ( ( A  +  B )  /  2 )  e. 
_V
40 nfcv 2419 . . . . . . 7  |-  F/_ m if ( ( ( A  +  B )  / 
2 )  <  M ,  <. ( 1st `  <. A ,  B >. ) ,  ( ( A  +  B )  / 
2 ) >. ,  <. ( ( ( ( A  +  B )  / 
2 )  +  ( 2nd `  <. A ,  B >. ) )  / 
2 ) ,  ( 2nd `  <. A ,  B >. ) >. )
41 breq1 4026 . . . . . . . 8  |-  ( m  =  ( ( A  +  B )  / 
2 )  ->  (
m  <  M  <->  ( ( A  +  B )  /  2 )  < 
M ) )
42 opeq2 3797 . . . . . . . 8  |-  ( m  =  ( ( A  +  B )  / 
2 )  ->  <. ( 1st `  <. A ,  B >. ) ,  m >.  = 
<. ( 1st `  <. A ,  B >. ) ,  ( ( A  +  B )  / 
2 ) >. )
43 oveq1 5865 . . . . . . . . . 10  |-  ( m  =  ( ( A  +  B )  / 
2 )  ->  (
m  +  ( 2nd `  <. A ,  B >. ) )  =  ( ( ( A  +  B )  /  2
)  +  ( 2nd `  <. A ,  B >. ) ) )
4443oveq1d 5873 . . . . . . . . 9  |-  ( m  =  ( ( A  +  B )  / 
2 )  ->  (
( m  +  ( 2nd `  <. A ,  B >. ) )  / 
2 )  =  ( ( ( ( A  +  B )  / 
2 )  +  ( 2nd `  <. A ,  B >. ) )  / 
2 ) )
4544opeq1d 3802 . . . . . . . 8  |-  ( m  =  ( ( A  +  B )  / 
2 )  ->  <. (
( m  +  ( 2nd `  <. A ,  B >. ) )  / 
2 ) ,  ( 2nd `  <. A ,  B >. ) >.  =  <. ( ( ( ( A  +  B )  / 
2 )  +  ( 2nd `  <. A ,  B >. ) )  / 
2 ) ,  ( 2nd `  <. A ,  B >. ) >. )
4641, 42, 45ifbieq12d 3587 . . . . . . 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 >. ) >. ) )
4739, 40, 46csbief 3122 . . . . . 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 >. ) >. )
4833opeq1d 3802 . . . . . . 7  |-  ( ph  -> 
<. ( 1st `  <. A ,  B >. ) ,  ( ( A  +  B )  / 
2 ) >.  =  <. A ,  ( ( A  +  B )  / 
2 ) >. )
4935oveq2d 5874 . . . . . . . . 9  |-  ( ph  ->  ( ( ( A  +  B )  / 
2 )  +  ( 2nd `  <. A ,  B >. ) )  =  ( ( ( A  +  B )  / 
2 )  +  B
) )
5049oveq1d 5873 . . . . . . . 8  |-  ( ph  ->  ( ( ( ( A  +  B )  /  2 )  +  ( 2nd `  <. A ,  B >. )
)  /  2 )  =  ( ( ( ( A  +  B
)  /  2 )  +  B )  / 
2 ) )
5150, 35opeq12d 3804 . . . . . . 7  |-  ( ph  -> 
<. ( ( ( ( A  +  B )  /  2 )  +  ( 2nd `  <. A ,  B >. )
)  /  2 ) ,  ( 2nd `  <. A ,  B >. ) >.  =  <. ( ( ( ( A  +  B
)  /  2 )  +  B )  / 
2 ) ,  B >. )
5248, 51ifeq12d 3581 . . . . . 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 >. ) )
5347, 52syl5eq 2327 . . . . 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 >. )
)
5438, 53eqtrd 2315 . . . 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 >. )
)
5531, 54eqtrd 2315 . . 3  |-  ( ph  ->  ( <. A ,  B >. D M )  =  if ( ( ( A  +  B )  /  2 )  < 
M ,  <. A , 
( ( A  +  B )  /  2
) >. ,  <. (
( ( ( A  +  B )  / 
2 )  +  B
)  /  2 ) ,  B >. )
)
563, 4readdcld 8862 . . . . . 6  |-  ( ph  ->  ( A  +  B
)  e.  RR )
5756rehalfcld 9958 . . . . 5  |-  ( ph  ->  ( ( A  +  B )  /  2
)  e.  RR )
58 opelxpi 4721 . . . . 5  |-  ( ( A  e.  RR  /\  ( ( A  +  B )  /  2
)  e.  RR )  ->  <. A ,  ( ( A  +  B
)  /  2 )
>.  e.  ( RR  X.  RR ) )
593, 57, 58syl2anc 642 . . . 4  |-  ( ph  -> 
<. A ,  ( ( A  +  B )  /  2 ) >.  e.  ( RR  X.  RR ) )
6057, 4readdcld 8862 . . . . . 6  |-  ( ph  ->  ( ( ( A  +  B )  / 
2 )  +  B
)  e.  RR )
6160rehalfcld 9958 . . . . 5  |-  ( ph  ->  ( ( ( ( A  +  B )  /  2 )  +  B )  /  2
)  e.  RR )
62 opelxpi 4721 . . . . 5  |-  ( ( ( ( ( ( A  +  B )  /  2 )  +  B )  /  2
)  e.  RR  /\  B  e.  RR )  -> 
<. ( ( ( ( A  +  B )  /  2 )  +  B )  /  2
) ,  B >.  e.  ( RR  X.  RR ) )
6361, 4, 62syl2anc 642 . . . 4  |-  ( ph  -> 
<. ( ( ( ( A  +  B )  /  2 )  +  B )  /  2
) ,  B >.  e.  ( RR  X.  RR ) )
64 ifcl 3601 . . . 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 ) )
6559, 63, 64syl2anc 642 . . 3  |-  ( ph  ->  if ( ( ( A  +  B )  /  2 )  < 
M ,  <. A , 
( ( A  +  B )  /  2
) >. ,  <. (
( ( ( A  +  B )  / 
2 )  +  B
)  /  2 ) ,  B >. )  e.  ( RR  X.  RR ) )
6655, 65eqeltrd 2357 . 2  |-  ( ph  ->  ( <. A ,  B >. D M )  e.  ( RR  X.  RR ) )
67 ruclem1.6 . . 3  |-  X  =  ( 1st `  ( <. A ,  B >. D M ) )
6855fveq2d 5529 . . . 4  |-  ( ph  ->  ( 1st `  ( <. A ,  B >. D M ) )  =  ( 1st `  if ( ( ( A  +  B )  / 
2 )  <  M ,  <. A ,  ( ( A  +  B
)  /  2 )
>. ,  <. ( ( ( ( A  +  B )  /  2
)  +  B )  /  2 ) ,  B >. ) ) )
69 fvif 5540 . . . . 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 >. )
)
70 op1stg 6132 . . . . . . 7  |-  ( ( A  e.  RR  /\  ( ( A  +  B )  /  2
)  e.  _V )  ->  ( 1st `  <. A ,  ( ( A  +  B )  / 
2 ) >. )  =  A )
713, 39, 70sylancl 643 . . . . . 6  |-  ( ph  ->  ( 1st `  <. A ,  ( ( A  +  B )  / 
2 ) >. )  =  A )
72 ovex 5883 . . . . . . 7  |-  ( ( ( ( A  +  B )  /  2
)  +  B )  /  2 )  e. 
_V
73 op1stg 6132 . . . . . . 7  |-  ( ( ( ( ( ( A  +  B )  /  2 )  +  B )  /  2
)  e.  _V  /\  B  e.  RR )  ->  ( 1st `  <. ( ( ( ( A  +  B )  / 
2 )  +  B
)  /  2 ) ,  B >. )  =  ( ( ( ( A  +  B
)  /  2 )  +  B )  / 
2 ) )
7472, 4, 73sylancr 644 . . . . . 6  |-  ( ph  ->  ( 1st `  <. ( ( ( ( A  +  B )  / 
2 )  +  B
)  /  2 ) ,  B >. )  =  ( ( ( ( A  +  B
)  /  2 )  +  B )  / 
2 ) )
7571, 74ifeq12d 3581 . . . . 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 ) ) )
7669, 75syl5eq 2327 . . . 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
) ) )
7768, 76eqtrd 2315 . . 3  |-  ( ph  ->  ( 1st `  ( <. A ,  B >. D M ) )  =  if ( ( ( A  +  B )  /  2 )  < 
M ,  A , 
( ( ( ( A  +  B )  /  2 )  +  B )  /  2
) ) )
7867, 77syl5eq 2327 . 2  |-  ( ph  ->  X  =  if ( ( ( A  +  B )  /  2
)  <  M ,  A ,  ( (
( ( A  +  B )  /  2
)  +  B )  /  2 ) ) )
79 ruclem1.7 . . 3  |-  Y  =  ( 2nd `  ( <. A ,  B >. D M ) )
8055fveq2d 5529 . . . 4  |-  ( ph  ->  ( 2nd `  ( <. A ,  B >. D M ) )  =  ( 2nd `  if ( ( ( A  +  B )  / 
2 )  <  M ,  <. A ,  ( ( A  +  B
)  /  2 )
>. ,  <. ( ( ( ( A  +  B )  /  2
)  +  B )  /  2 ) ,  B >. ) ) )
81 fvif 5540 . . . . 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 >. )
)
82 op2ndg 6133 . . . . . . 7  |-  ( ( A  e.  RR  /\  ( ( A  +  B )  /  2
)  e.  _V )  ->  ( 2nd `  <. A ,  ( ( A  +  B )  / 
2 ) >. )  =  ( ( A  +  B )  / 
2 ) )
833, 39, 82sylancl 643 . . . . . 6  |-  ( ph  ->  ( 2nd `  <. A ,  ( ( A  +  B )  / 
2 ) >. )  =  ( ( A  +  B )  / 
2 ) )
84 op2ndg 6133 . . . . . . 7  |-  ( ( ( ( ( ( A  +  B )  /  2 )  +  B )  /  2
)  e.  _V  /\  B  e.  RR )  ->  ( 2nd `  <. ( ( ( ( A  +  B )  / 
2 )  +  B
)  /  2 ) ,  B >. )  =  B )
8572, 4, 84sylancr 644 . . . . . 6  |-  ( ph  ->  ( 2nd `  <. ( ( ( ( A  +  B )  / 
2 )  +  B
)  /  2 ) ,  B >. )  =  B )
8683, 85ifeq12d 3581 . . . . 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
) )
8781, 86syl5eq 2327 . . . 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 ) )
8880, 87eqtrd 2315 . . 3  |-  ( ph  ->  ( 2nd `  ( <. A ,  B >. D M ) )  =  if ( ( ( A  +  B )  /  2 )  < 
M ,  ( ( A  +  B )  /  2 ) ,  B ) )
8979, 88syl5eq 2327 . 2  |-  ( ph  ->  Y  =  if ( ( ( A  +  B )  /  2
)  <  M , 
( ( A  +  B )  /  2
) ,  B ) )
9066, 78, 893jca 1132 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 set class
Syntax hints:    -> wi 4    /\ wa 358    /\ w3a 934    = wceq 1623    e. wcel 1684   _Vcvv 2788   [_csb 3081   ifcif 3565   <.cop 3643   class class class wbr 4023    X. cxp 4687   -->wf 5251   ` cfv 5255  (class class class)co 5858    e. cmpt2 5860   1stc1st 6120   2ndc2nd 6121   RRcr 8736    + caddc 8740    < clt 8867    / cdiv 9423   NNcn 9746   2c2 9795
This theorem is referenced by:  ruclem2  12510  ruclem3  12511  ruclem6  12513
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-po 4314  df-so 4315  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-riota 6304  df-er 6660  df-en 6864  df-dom 6865  df-sdom 6866  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-div 9424  df-2 9804
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