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Theorem ruclem1 12822
Description: Lemma for ruc 12834 (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 6090 . . . . 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 4902 . . . . . . 7  |-  ( ( A  e.  RR  /\  B  e.  RR )  -> 
<. A ,  B >.  e.  ( RR  X.  RR ) )
63, 4, 5syl2anc 643 . . . . . 6  |-  ( ph  -> 
<. A ,  B >.  e.  ( RR  X.  RR ) )
7 ruclem1.5 . . . . . 6  |-  ( ph  ->  M  e.  RR )
8 simpr 448 . . . . . . . . . . 11  |-  ( ( x  =  <. A ,  B >.  /\  y  =  M )  ->  y  =  M )
98breq2d 4216 . . . . . . . . . 10  |-  ( ( x  =  <. A ,  B >.  /\  y  =  M )  ->  (
m  <  y  <->  m  <  M ) )
10 simpl 444 . . . . . . . . . . . 12  |-  ( ( x  =  <. A ,  B >.  /\  y  =  M )  ->  x  =  <. A ,  B >. )
1110fveq2d 5724 . . . . . . . . . . 11  |-  ( ( x  =  <. A ,  B >.  /\  y  =  M )  ->  ( 1st `  x )  =  ( 1st `  <. A ,  B >. )
)
1211opeq1d 3982 . . . . . . . . . 10  |-  ( ( x  =  <. A ,  B >.  /\  y  =  M )  ->  <. ( 1st `  x ) ,  m >.  =  <. ( 1st `  <. A ,  B >. ) ,  m >. )
1310fveq2d 5724 . . . . . . . . . . . . 13  |-  ( ( x  =  <. A ,  B >.  /\  y  =  M )  ->  ( 2nd `  x )  =  ( 2nd `  <. A ,  B >. )
)
1413oveq2d 6089 . . . . . . . . . . . 12  |-  ( ( x  =  <. A ,  B >.  /\  y  =  M )  ->  (
m  +  ( 2nd `  x ) )  =  ( m  +  ( 2nd `  <. A ,  B >. ) ) )
1514oveq1d 6088 . . . . . . . . . . 11  |-  ( ( x  =  <. A ,  B >.  /\  y  =  M )  ->  (
( m  +  ( 2nd `  x ) )  /  2 )  =  ( ( m  +  ( 2nd `  <. A ,  B >. )
)  /  2 ) )
1615, 13opeq12d 3984 . . . . . . . . . 10  |-  ( ( x  =  <. A ,  B >.  /\  y  =  M )  ->  <. (
( m  +  ( 2nd `  x ) )  /  2 ) ,  ( 2nd `  x
) >.  =  <. (
( m  +  ( 2nd `  <. A ,  B >. ) )  / 
2 ) ,  ( 2nd `  <. A ,  B >. ) >. )
179, 12, 16ifbieq12d 3753 . . . . . . . . 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 3268 . . . . . . . 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 6091 . . . . . . . . . 10  |-  ( ( x  =  <. A ,  B >.  /\  y  =  M )  ->  (
( 1st `  x
)  +  ( 2nd `  x ) )  =  ( ( 1st `  <. A ,  B >. )  +  ( 2nd `  <. A ,  B >. )
) )
2019oveq1d 6088 . . . . . . . . 9  |-  ( ( x  =  <. A ,  B >.  /\  y  =  M )  ->  (
( ( 1st `  x
)  +  ( 2nd `  x ) )  / 
2 )  =  ( ( ( 1st `  <. A ,  B >. )  +  ( 2nd `  <. A ,  B >. )
)  /  2 ) )
2120csbeq1d 3249 . . . . . . . 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 2467 . . . . . . 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 2435 . . . . . . 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 6098 . . . . . . . 8  |-  ( ( ( 1st `  <. A ,  B >. )  +  ( 2nd `  <. A ,  B >. )
)  /  2 )  e.  _V
25 opex 4419 . . . . . . . . 9  |-  <. ( 1st `  <. A ,  B >. ) ,  m >.  e. 
_V
26 opex 4419 . . . . . . . . 9  |-  <. (
( m  +  ( 2nd `  <. A ,  B >. ) )  / 
2 ) ,  ( 2nd `  <. A ,  B >. ) >.  e.  _V
2725, 26ifex 3789 . . . . . . . 8  |-  if ( m  <  M ,  <. ( 1st `  <. A ,  B >. ) ,  m >. ,  <. (
( m  +  ( 2nd `  <. A ,  B >. ) )  / 
2 ) ,  ( 2nd `  <. A ,  B >. ) >. )  e.  _V
2824, 27csbex 3254 . . . . . . 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 6196 . . . . . 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 643 . . . . 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 2467 . . . 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 6351 . . . . . . . . 9  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( 1st `  <. A ,  B >. )  =  A )
333, 4, 32syl2anc 643 . . . . . . . 8  |-  ( ph  ->  ( 1st `  <. A ,  B >. )  =  A )
34 op2ndg 6352 . . . . . . . . 9  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( 2nd `  <. A ,  B >. )  =  B )
353, 4, 34syl2anc 643 . . . . . . . 8  |-  ( ph  ->  ( 2nd `  <. A ,  B >. )  =  B )
3633, 35oveq12d 6091 . . . . . . 7  |-  ( ph  ->  ( ( 1st `  <. A ,  B >. )  +  ( 2nd `  <. A ,  B >. )
)  =  ( A  +  B ) )
3736oveq1d 6088 . . . . . 6  |-  ( ph  ->  ( ( ( 1st `  <. A ,  B >. )  +  ( 2nd `  <. A ,  B >. ) )  /  2
)  =  ( ( A  +  B )  /  2 ) )
3837csbeq1d 3249 . . . . 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 6098 . . . . . . 7  |-  ( ( A  +  B )  /  2 )  e. 
_V
40 nfcv 2571 . . . . . . 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 4207 . . . . . . . 8  |-  ( m  =  ( ( A  +  B )  / 
2 )  ->  (
m  <  M  <->  ( ( A  +  B )  /  2 )  < 
M ) )
42 opeq2 3977 . . . . . . . 8  |-  ( m  =  ( ( A  +  B )  / 
2 )  ->  <. ( 1st `  <. A ,  B >. ) ,  m >.  = 
<. ( 1st `  <. A ,  B >. ) ,  ( ( A  +  B )  / 
2 ) >. )
43 oveq1 6080 . . . . . . . . . 10  |-  ( m  =  ( ( A  +  B )  / 
2 )  ->  (
m  +  ( 2nd `  <. A ,  B >. ) )  =  ( ( ( A  +  B )  /  2
)  +  ( 2nd `  <. A ,  B >. ) ) )
4443oveq1d 6088 . . . . . . . . 9  |-  ( m  =  ( ( A  +  B )  / 
2 )  ->  (
( m  +  ( 2nd `  <. A ,  B >. ) )  / 
2 )  =  ( ( ( ( A  +  B )  / 
2 )  +  ( 2nd `  <. A ,  B >. ) )  / 
2 ) )
4544opeq1d 3982 . . . . . . . 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 3753 . . . . . . 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 3284 . . . . . 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 3982 . . . . . . 7  |-  ( ph  -> 
<. ( 1st `  <. A ,  B >. ) ,  ( ( A  +  B )  / 
2 ) >.  =  <. A ,  ( ( A  +  B )  / 
2 ) >. )
4935oveq2d 6089 . . . . . . . . 9  |-  ( ph  ->  ( ( ( A  +  B )  / 
2 )  +  ( 2nd `  <. A ,  B >. ) )  =  ( ( ( A  +  B )  / 
2 )  +  B
) )
5049oveq1d 6088 . . . . . . . 8  |-  ( ph  ->  ( ( ( ( A  +  B )  /  2 )  +  ( 2nd `  <. A ,  B >. )
)  /  2 )  =  ( ( ( ( A  +  B
)  /  2 )  +  B )  / 
2 ) )
5150, 35opeq12d 3984 . . . . . . 7  |-  ( ph  -> 
<. ( ( ( ( A  +  B )  /  2 )  +  ( 2nd `  <. A ,  B >. )
)  /  2 ) ,  ( 2nd `  <. A ,  B >. ) >.  =  <. ( ( ( ( A  +  B
)  /  2 )  +  B )  / 
2 ) ,  B >. )
5248, 51ifeq12d 3747 . . . . . 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 2479 . . . . 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 2467 . . . 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 2467 . . 3  |-  ( ph  ->  ( <. A ,  B >. D M )  =  if ( ( ( A  +  B )  /  2 )  < 
M ,  <. A , 
( ( A  +  B )  /  2
) >. ,  <. (
( ( ( A  +  B )  / 
2 )  +  B
)  /  2 ) ,  B >. )
)
563, 4readdcld 9107 . . . . . 6  |-  ( ph  ->  ( A  +  B
)  e.  RR )
5756rehalfcld 10206 . . . . 5  |-  ( ph  ->  ( ( A  +  B )  /  2
)  e.  RR )
58 opelxpi 4902 . . . . 5  |-  ( ( A  e.  RR  /\  ( ( A  +  B )  /  2
)  e.  RR )  ->  <. A ,  ( ( A  +  B
)  /  2 )
>.  e.  ( RR  X.  RR ) )
593, 57, 58syl2anc 643 . . . 4  |-  ( ph  -> 
<. A ,  ( ( A  +  B )  /  2 ) >.  e.  ( RR  X.  RR ) )
6057, 4readdcld 9107 . . . . . 6  |-  ( ph  ->  ( ( ( A  +  B )  / 
2 )  +  B
)  e.  RR )
6160rehalfcld 10206 . . . . 5  |-  ( ph  ->  ( ( ( ( A  +  B )  /  2 )  +  B )  /  2
)  e.  RR )
62 opelxpi 4902 . . . . 5  |-  ( ( ( ( ( ( A  +  B )  /  2 )  +  B )  /  2
)  e.  RR  /\  B  e.  RR )  -> 
<. ( ( ( ( A  +  B )  /  2 )  +  B )  /  2
) ,  B >.  e.  ( RR  X.  RR ) )
6361, 4, 62syl2anc 643 . . . 4  |-  ( ph  -> 
<. ( ( ( ( A  +  B )  /  2 )  +  B )  /  2
) ,  B >.  e.  ( RR  X.  RR ) )
64 ifcl 3767 . . . 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 643 . . 3  |-  ( ph  ->  if ( ( ( A  +  B )  /  2 )  < 
M ,  <. A , 
( ( A  +  B )  /  2
) >. ,  <. (
( ( ( A  +  B )  / 
2 )  +  B
)  /  2 ) ,  B >. )  e.  ( RR  X.  RR ) )
6655, 65eqeltrd 2509 . 2  |-  ( ph  ->  ( <. A ,  B >. D M )  e.  ( RR  X.  RR ) )
67 ruclem1.6 . . 3  |-  X  =  ( 1st `  ( <. A ,  B >. D M ) )
6855fveq2d 5724 . . . 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 5735 . . . . 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 6351 . . . . . . 7  |-  ( ( A  e.  RR  /\  ( ( A  +  B )  /  2
)  e.  _V )  ->  ( 1st `  <. A ,  ( ( A  +  B )  / 
2 ) >. )  =  A )
713, 39, 70sylancl 644 . . . . . 6  |-  ( ph  ->  ( 1st `  <. A ,  ( ( A  +  B )  / 
2 ) >. )  =  A )
72 ovex 6098 . . . . . . 7  |-  ( ( ( ( A  +  B )  /  2
)  +  B )  /  2 )  e. 
_V
73 op1stg 6351 . . . . . . 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 645 . . . . . 6  |-  ( ph  ->  ( 1st `  <. ( ( ( ( A  +  B )  / 
2 )  +  B
)  /  2 ) ,  B >. )  =  ( ( ( ( A  +  B
)  /  2 )  +  B )  / 
2 ) )
7571, 74ifeq12d 3747 . . . . 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 2479 . . . 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 2467 . . 3  |-  ( ph  ->  ( 1st `  ( <. A ,  B >. D M ) )  =  if ( ( ( A  +  B )  /  2 )  < 
M ,  A , 
( ( ( ( A  +  B )  /  2 )  +  B )  /  2
) ) )
7867, 77syl5eq 2479 . 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 5724 . . . 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 5735 . . . . 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 6352 . . . . . . 7  |-  ( ( A  e.  RR  /\  ( ( A  +  B )  /  2
)  e.  _V )  ->  ( 2nd `  <. A ,  ( ( A  +  B )  / 
2 ) >. )  =  ( ( A  +  B )  / 
2 ) )
833, 39, 82sylancl 644 . . . . . 6  |-  ( ph  ->  ( 2nd `  <. A ,  ( ( A  +  B )  / 
2 ) >. )  =  ( ( A  +  B )  / 
2 ) )
84 op2ndg 6352 . . . . . . 7  |-  ( ( ( ( ( ( A  +  B )  /  2 )  +  B )  /  2
)  e.  _V  /\  B  e.  RR )  ->  ( 2nd `  <. ( ( ( ( A  +  B )  / 
2 )  +  B
)  /  2 ) ,  B >. )  =  B )
8572, 4, 84sylancr 645 . . . . . 6  |-  ( ph  ->  ( 2nd `  <. ( ( ( ( A  +  B )  / 
2 )  +  B
)  /  2 ) ,  B >. )  =  B )
8683, 85ifeq12d 3747 . . . . 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 2479 . . . 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 2467 . . 3  |-  ( ph  ->  ( 2nd `  ( <. A ,  B >. D M ) )  =  if ( ( ( A  +  B )  /  2 )  < 
M ,  ( ( A  +  B )  /  2 ) ,  B ) )
8979, 88syl5eq 2479 . 2  |-  ( ph  ->  Y  =  if ( ( ( A  +  B )  /  2
)  <  M , 
( ( A  +  B )  /  2
) ,  B ) )
9066, 78, 893jca 1134 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 359    /\ w3a 936    = wceq 1652    e. wcel 1725   _Vcvv 2948   [_csb 3243   ifcif 3731   <.cop 3809   class class class wbr 4204    X. cxp 4868   -->wf 5442   ` cfv 5446  (class class class)co 6073    e. cmpt2 6075   1stc1st 6339   2ndc2nd 6340   RRcr 8981    + caddc 8985    < clt 9112    / cdiv 9669   NNcn 9992   2c2 10041
This theorem is referenced by:  ruclem2  12823  ruclem3  12824  ruclem6  12826
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693  ax-resscn 9039  ax-1cn 9040  ax-icn 9041  ax-addcl 9042  ax-addrcl 9043  ax-mulcl 9044  ax-mulrcl 9045  ax-mulcom 9046  ax-addass 9047  ax-mulass 9048  ax-distr 9049  ax-i2m1 9050  ax-1ne0 9051  ax-1rid 9052  ax-rnegex 9053  ax-rrecex 9054  ax-cnre 9055  ax-pre-lttri 9056  ax-pre-lttrn 9057  ax-pre-ltadd 9058  ax-pre-mulgt0 9059
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-nel 2601  df-ral 2702  df-rex 2703  df-reu 2704  df-rmo 2705  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-br 4205  df-opab 4259  df-mpt 4260  df-id 4490  df-po 4495  df-so 4496  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-1st 6341  df-2nd 6342  df-riota 6541  df-er 6897  df-en 7102  df-dom 7103  df-sdom 7104  df-pnf 9114  df-mnf 9115  df-xr 9116  df-ltxr 9117  df-le 9118  df-sub 9285  df-neg 9286  df-div 9670  df-2 10050
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