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Theorem ruclem6 12529
Description: Lemma for ruc 12537. Domain and range of the interval sequence. (Contributed by Mario Carneiro, 28-May-2014.)
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
) >. ) ) )
ruc.4  |-  C  =  ( { <. 0 ,  <. 0 ,  1
>. >. }  u.  F
)
ruc.5  |-  G  =  seq  0 ( D ,  C )
Assertion
Ref Expression
ruclem6  |-  ( ph  ->  G : NN0 --> ( RR 
X.  RR ) )
Distinct variable groups:    x, m, y, F    m, G, x, y
Allowed substitution hints:    ph( x, y, m)    C( x, y, m)    D( x, y, m)

Proof of Theorem ruclem6
Dummy variables  w  z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ruc.5 . . . . . . 7  |-  G  =  seq  0 ( D ,  C )
21fveq1i 5542 . . . . . 6  |-  ( G `
 0 )  =  (  seq  0 ( D ,  C ) `
 0 )
3 0z 10051 . . . . . . 7  |-  0  e.  ZZ
4 seq1 11075 . . . . . . 7  |-  ( 0  e.  ZZ  ->  (  seq  0 ( D ,  C ) `  0
)  =  ( C `
 0 ) )
53, 4ax-mp 8 . . . . . 6  |-  (  seq  0 ( D ,  C ) `  0
)  =  ( C `
 0 )
62, 5eqtri 2316 . . . . 5  |-  ( G `
 0 )  =  ( C `  0
)
7 ruc.1 . . . . . 6  |-  ( ph  ->  F : NN --> RR )
8 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
) >. ) ) )
9 ruc.4 . . . . . 6  |-  C  =  ( { <. 0 ,  <. 0 ,  1
>. >. }  u.  F
)
107, 8, 9, 1ruclem4 12528 . . . . 5  |-  ( ph  ->  ( G `  0
)  =  <. 0 ,  1 >. )
116, 10syl5eqr 2342 . . . 4  |-  ( ph  ->  ( C `  0
)  =  <. 0 ,  1 >. )
12 0re 8854 . . . . 5  |-  0  e.  RR
13 1re 8853 . . . . 5  |-  1  e.  RR
14 opelxpi 4737 . . . . 5  |-  ( ( 0  e.  RR  /\  1  e.  RR )  -> 
<. 0 ,  1
>.  e.  ( RR  X.  RR ) )
1512, 13, 14mp2an 653 . . . 4  |-  <. 0 ,  1 >.  e.  ( RR  X.  RR )
1611, 15syl6eqel 2384 . . 3  |-  ( ph  ->  ( C `  0
)  e.  ( RR 
X.  RR ) )
17 1st2nd2 6175 . . . . . 6  |-  ( z  e.  ( RR  X.  RR )  ->  z  = 
<. ( 1st `  z
) ,  ( 2nd `  z ) >. )
1817ad2antrl 708 . . . . 5  |-  ( (
ph  /\  ( z  e.  ( RR  X.  RR )  /\  w  e.  RR ) )  ->  z  =  <. ( 1st `  z
) ,  ( 2nd `  z ) >. )
1918oveq1d 5889 . . . 4  |-  ( (
ph  /\  ( z  e.  ( RR  X.  RR )  /\  w  e.  RR ) )  ->  (
z D w )  =  ( <. ( 1st `  z ) ,  ( 2nd `  z
) >. D w ) )
207adantr 451 . . . . . 6  |-  ( (
ph  /\  ( z  e.  ( RR  X.  RR )  /\  w  e.  RR ) )  ->  F : NN --> RR )
218adantr 451 . . . . . 6  |-  ( (
ph  /\  ( z  e.  ( RR  X.  RR )  /\  w  e.  RR ) )  ->  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
) >. ) ) )
22 xp1st 6165 . . . . . . 7  |-  ( z  e.  ( RR  X.  RR )  ->  ( 1st `  z )  e.  RR )
2322ad2antrl 708 . . . . . 6  |-  ( (
ph  /\  ( z  e.  ( RR  X.  RR )  /\  w  e.  RR ) )  ->  ( 1st `  z )  e.  RR )
24 xp2nd 6166 . . . . . . 7  |-  ( z  e.  ( RR  X.  RR )  ->  ( 2nd `  z )  e.  RR )
2524ad2antrl 708 . . . . . 6  |-  ( (
ph  /\  ( z  e.  ( RR  X.  RR )  /\  w  e.  RR ) )  ->  ( 2nd `  z )  e.  RR )
26 simprr 733 . . . . . 6  |-  ( (
ph  /\  ( z  e.  ( RR  X.  RR )  /\  w  e.  RR ) )  ->  w  e.  RR )
27 eqid 2296 . . . . . 6  |-  ( 1st `  ( <. ( 1st `  z
) ,  ( 2nd `  z ) >. D w ) )  =  ( 1st `  ( <.
( 1st `  z
) ,  ( 2nd `  z ) >. D w ) )
28 eqid 2296 . . . . . 6  |-  ( 2nd `  ( <. ( 1st `  z
) ,  ( 2nd `  z ) >. D w ) )  =  ( 2nd `  ( <.
( 1st `  z
) ,  ( 2nd `  z ) >. D w ) )
2920, 21, 23, 25, 26, 27, 28ruclem1 12525 . . . . 5  |-  ( (
ph  /\  ( z  e.  ( RR  X.  RR )  /\  w  e.  RR ) )  ->  (
( <. ( 1st `  z
) ,  ( 2nd `  z ) >. D w )  e.  ( RR 
X.  RR )  /\  ( 1st `  ( <.
( 1st `  z
) ,  ( 2nd `  z ) >. D w ) )  =  if ( ( ( ( 1st `  z )  +  ( 2nd `  z
) )  /  2
)  <  w , 
( 1st `  z
) ,  ( ( ( ( ( 1st `  z )  +  ( 2nd `  z ) )  /  2 )  +  ( 2nd `  z
) )  /  2
) )  /\  ( 2nd `  ( <. ( 1st `  z ) ,  ( 2nd `  z
) >. D w ) )  =  if ( ( ( ( 1st `  z )  +  ( 2nd `  z ) )  /  2 )  <  w ,  ( ( ( 1st `  z
)  +  ( 2nd `  z ) )  / 
2 ) ,  ( 2nd `  z ) ) ) )
3029simp1d 967 . . . 4  |-  ( (
ph  /\  ( z  e.  ( RR  X.  RR )  /\  w  e.  RR ) )  ->  ( <. ( 1st `  z
) ,  ( 2nd `  z ) >. D w )  e.  ( RR 
X.  RR ) )
3119, 30eqeltrd 2370 . . 3  |-  ( (
ph  /\  ( z  e.  ( RR  X.  RR )  /\  w  e.  RR ) )  ->  (
z D w )  e.  ( RR  X.  RR ) )
32 nn0uz 10278 . . 3  |-  NN0  =  ( ZZ>= `  0 )
333a1i 10 . . 3  |-  ( ph  ->  0  e.  ZZ )
34 0p1e1 9855 . . . . . . 7  |-  ( 0  +  1 )  =  1
3534fveq2i 5544 . . . . . 6  |-  ( ZZ>= `  ( 0  +  1 ) )  =  (
ZZ>= `  1 )
36 nnuz 10279 . . . . . 6  |-  NN  =  ( ZZ>= `  1 )
3735, 36eqtr4i 2319 . . . . 5  |-  ( ZZ>= `  ( 0  +  1 ) )  =  NN
3837eleq2i 2360 . . . 4  |-  ( z  e.  ( ZZ>= `  (
0  +  1 ) )  <->  z  e.  NN )
399equncomi 3334 . . . . . . . 8  |-  C  =  ( F  u.  { <. 0 ,  <. 0 ,  1 >. >. } )
4039fveq1i 5542 . . . . . . 7  |-  ( C `
 z )  =  ( ( F  u.  {
<. 0 ,  <. 0 ,  1 >. >. } ) `  z
)
41 nnne0 9794 . . . . . . . . 9  |-  ( z  e.  NN  ->  z  =/=  0 )
4241necomd 2542 . . . . . . . 8  |-  ( z  e.  NN  ->  0  =/=  z )
43 fvunsn 5728 . . . . . . . 8  |-  ( 0  =/=  z  ->  (
( F  u.  { <. 0 ,  <. 0 ,  1 >. >. } ) `
 z )  =  ( F `  z
) )
4442, 43syl 15 . . . . . . 7  |-  ( z  e.  NN  ->  (
( F  u.  { <. 0 ,  <. 0 ,  1 >. >. } ) `
 z )  =  ( F `  z
) )
4540, 44syl5eq 2340 . . . . . 6  |-  ( z  e.  NN  ->  ( C `  z )  =  ( F `  z ) )
4645adantl 452 . . . . 5  |-  ( (
ph  /\  z  e.  NN )  ->  ( C `
 z )  =  ( F `  z
) )
47 ffvelrn 5679 . . . . . 6  |-  ( ( F : NN --> RR  /\  z  e.  NN )  ->  ( F `  z
)  e.  RR )
487, 47sylan 457 . . . . 5  |-  ( (
ph  /\  z  e.  NN )  ->  ( F `
 z )  e.  RR )
4946, 48eqeltrd 2370 . . . 4  |-  ( (
ph  /\  z  e.  NN )  ->  ( C `
 z )  e.  RR )
5038, 49sylan2b 461 . . 3  |-  ( (
ph  /\  z  e.  ( ZZ>= `  ( 0  +  1 ) ) )  ->  ( C `  z )  e.  RR )
5116, 31, 32, 33, 50seqf2 11081 . 2  |-  ( ph  ->  seq  0 ( D ,  C ) : NN0 --> ( RR  X.  RR ) )
521feq1i 5399 . 2  |-  ( G : NN0 --> ( RR 
X.  RR )  <->  seq  0
( D ,  C
) : NN0 --> ( RR 
X.  RR ) )
5351, 52sylibr 203 1  |-  ( ph  ->  G : NN0 --> ( RR 
X.  RR ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1632    e. wcel 1696    =/= wne 2459   [_csb 3094    u. cun 3163   ifcif 3578   {csn 3653   <.cop 3656   class class class wbr 4039    X. cxp 4703   -->wf 5267   ` cfv 5271  (class class class)co 5874    e. cmpt2 5876   1stc1st 6136   2ndc2nd 6137   RRcr 8752   0cc0 8753   1c1 8754    + caddc 8756    < clt 8883    / cdiv 9439   NNcn 9762   2c2 9811   NN0cn0 9981   ZZcz 10040   ZZ>=cuz 10246    seq cseq 11062
This theorem is referenced by:  ruclem8  12531  ruclem9  12532  ruclem10  12533  ruclem11  12534  ruclem12  12535
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-cnex 8809  ax-resscn 8810  ax-1cn 8811  ax-icn 8812  ax-addcl 8813  ax-addrcl 8814  ax-mulcl 8815  ax-mulrcl 8816  ax-mulcom 8817  ax-addass 8818  ax-mulass 8819  ax-distr 8820  ax-i2m1 8821  ax-1ne0 8822  ax-1rid 8823  ax-rnegex 8824  ax-rrecex 8825  ax-cnre 8826  ax-pre-lttri 8827  ax-pre-lttrn 8828  ax-pre-ltadd 8829  ax-pre-mulgt0 8830
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 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rmo 2564  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-1st 6138  df-2nd 6139  df-riota 6320  df-recs 6404  df-rdg 6439  df-er 6676  df-en 6880  df-dom 6881  df-sdom 6882  df-pnf 8885  df-mnf 8886  df-xr 8887  df-ltxr 8888  df-le 8889  df-sub 9055  df-neg 9056  df-div 9440  df-nn 9763  df-2 9820  df-n0 9982  df-z 10041  df-uz 10247  df-fz 10799  df-seq 11063
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